Richard A. Gerwin

Los Alamos National Laboratory, Los Alamos, New Mexico

Integrity of the Plasma Magnetic Nozzle

NASA/TP—2009-213439

December 2009
NASA STI Program . . . in Profi le

Since its founding, NASA has been dedicated to the 

advancement of aeronautics and space science. The 

NASA Scientifi c and Technical Information (STI) 

program plays a key part in helping NASA maintain 

this important role.

The NASA STI Program operates under the auspices 

of the Agency Chief Information Offi cer. It collects, 

organizes, provides for archiving, and disseminates 

NASA’s STI. The NASA STI program provides access 

to the NASA Aeronautics and Space Database and 

its public interface, the NASA Technical Reports 

Server, thus providing one of the largest collections 

of aeronautical and space science STI in the world. 

Results are published in both non-NASA channels 

and by NASA in the NASA STI Report Series, which 

includes the following report types:

• TECHNICAL PUBLICATION. Reports of 

completed research or a major signifi cant phase 

of research that present the results of NASA 

programs and include extensive data or theoretical 

analysis. Includes compilations of signifi cant 

scientifi c and technical data and information 

deemed to be of continuing reference value. 

NASA counterpart of peer-reviewed formal 

professional papers but has less stringent 

limitations on manuscript length and extent of 

graphic presentations.

• TECHNICAL MEMORANDUM. Scientifi c 

and technical fi ndings that are preliminary or 

of specialized interest, e.g., quick release 

reports, working papers, and bibliographies that 

contain minimal annotation. Does not contain 

extensive analysis.

• CONTRACTOR REPORT. Scientifi c and 

technical fi ndings by NASA-sponsored 

contractors and grantees.

• CONFERENCE PUBLICATION. Collected 

papers from scientifi c and technical 

conferences, symposia, seminars, or other 

meetings sponsored or cosponsored by NASA.

• SPECIAL PUBLICATION. Scientifi c, 

technical, or historical information from 

NASA programs, projects, and missions, often 

concerned with subjects having substantial 

public interest.

• TECHNICAL TRANSLATION. Englishlanguage translations of foreign scientifi c and 

technical material pertinent to NASA’s mission.

Specialized services also include creating custom 

thesauri, building customized databases, organizing 

and publishing research results.

For more information about the NASA STI 

program, see the following:

• Access the NASA STI program home page at 

http://www.sti.nasa.gov

• E-mail your question via the Internet to help@

sti.nasa.gov

• Fax your question to the NASA STI Help Desk 

at 443–757–5803

• Telephone the NASA STI Help Desk at

 443–757–5802

• Write to:

 NASA Center for AeroSpace Information (CASI)

 7115 Standard Drive

 Hanover, MD 21076–1320
Richard A. Gerwin

Los Alamos National Laboratory, Los Alamos, New Mexico

Integrity of the Plasma Magnetic Nozzle

NASA/TP—2009-213439

December 2009

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared under C–32057–G
Acknowledgments

The author expresses gratitude to Dr. Peter J. Turchi and Dr. Dan Winske for their very helpful discussions. The author also 

wishes to thank the team of reviewers Dr. Ronald C. Davidson and Dr. Michael R. LaPointe for the time and effort they 

spent in reviewing and improving the manuscript. In addition, the author gratefully thanks Craig H. Williams and 

Laura R. Becker for signifi cantly improving the clarity of presentation of this multifaceted subject.

Available from

NASA Center for Aerospace Information

7115 Standard Drive

Hanover, MD 21076–1320

National Technical Information Service

5285 Port Royal Road

Springfi eld, VA 22161

Available electronically at http://gltrs.grc.nasa.gov

Level of Review: This material has been technically reviewed by three expert reviewers. 
NASA/TP—2009-213439 iii 

Contents 

Summary ....................................................................................................................................................................... 1

1.0 Introduction ............................................................................................................................................................. 1

1.1 Outline of Report .............................................................................................................................................. 5

1.2 Basic Notation and Basic Approach ................................................................................................................. 6

2.0 General Physics Description of Magnetic Nozzle................................................................................................... 6

2.1 Bernoulli Equation for Nozzle Flow ................................................................................................................ 7

2.2 Diffusive Intermixing in Plasma-Field Interface .............................................................................................. 9

2.2.1 Diffusion of Wall-Confined Plasma........................................................................................................ 9

2.2.2 Diffusion of Magnetically Confined Plasma ......................................................................................... 10

2.2.3 Diffusion of Confining Magnetic Field................................................................................................. 13

2.3 Hall Effect, Plasma Rotation, and Electron Pressure in Ohm’s Law ............................................................. 14

2.3.1 No Plasma Rotation, No Electron Pressure ........................................................................................... 15

2.3.1.1 No Hall current allowed .............................................................................................. 15

2.3.1.2 Hall current allowed (voltage shorted out) .................................................................. 16

2.3.2 Plasma Rotation and Electron Pressure................................................................................................. 18

2.3.2.1 No Hall current allowed .............................................................................................. 19

2.3.2.2 Hall current allowed (voltage shorted out) .................................................................. 19

2.3.3 Diffusion of Field Into Plasma With Hall Effect .................................................................................. 22

2.4 Summary......................................................................................................................................................... 27

3.0 Resistivity From Gradient-Driven Microinstabilities ........................................................................................... 32

3.1 Interface Width in Breech of Nozzle .............................................................................................................. 33

3.2 Linear Theory of Lower Hybrid Drift (LHD) Instability ............................................................................... 43

3.3 Quasi-Linear Evolution of LHD Instability and Anomalous Resistivity ....................................................... 46

3.4 Alternative Models for Saturation of LHD Microturbulence ......................................................................... 53

3.5 Comparison of Anomalous and Classical Resistivities .................................................................................. 58

4.0 Rayleigh-Taylor- (RT-) Type Instabilities ............................................................................................................ 60

4.1 RT-Type Instabilities at Initial Impact With Magnetic Field ......................................................................... 60

4.2 RT-Type Instabilities Downstream From Injection Region ........................................................................... 62

4.2.1 Surface Instability of Static Plasma in Region of Adverse Curvature .................................................. 63

4.2.2 Surface Instability of Flowing Propellant in Region of Adverse Curvature ......................................... 66

4.3 Finite Larmor Radius (FLR) Stabilization of RT-Type Flute Modes ............................................................ 69

4.3.1 Example: Estimate of Hall Term in Coaxial MHD Thruster ................................................................ 72

4.3.2 Concluding Thoughts ............................................................................................................................ 73

4.4 Short-Wavelength Gravity-Driven Instabilities .............................................................................................. 74

5.0 Summary and Conclusions .................................................................................................................................... 75

Appendix A.—Symbols and Expressions ................................................................................................................... 79

Appendix B.—Influence of Dwell Time on Plasma-Field Mixing Layer Width ........................................................ 83

Appendix C.—Planar Similarity Solution for Resistive Diffusion of Field Into Plasma ............................................ 85

Appendix D.—Azimuthal Magnetic Field When Hall Voltage Is Shorted Out .......................................................... 87

Appendix E.—Inward Radial Drift of Propellant ....................................................................................................... 89

Appendix F.—Derivation of Some Results Noted in Presentation of Initial Boundary-Layer Width ........................ 91

F.1 Radial Mobility Drift Distance of Electron Versus c/ωp i .............................................................................. 91

F.2 Radial Diffusion Distance of Electron Versus c/ωp i ...................................................................................... 91

F.3 Fluid Deceleration Neglected in Ion Trajectory Equation ............................................................................. 91

F.4 Azimuthal Current Carried by the Ions During One-Fourth of a Gyroperiod ............................................... 92

Appendix G.—Effect of Electron Pressure on the Initial Plasma-Field Mixing Layer Width.................................... 93

Appendix H.—Relation Between Global Beta and Radial Deceleration .................................................................... 97

Appendix I.—Initial Ratio of Ion Gyroradius to Plasma-Field Interface Width ....................................................... 101
NASA/TP—2009-213439 iv 

Appendix J.—Generalized Plasma-Field Interface Thickness at the Throat Calculation and Example .................. 103

J.1 Calculation of Plasma-Field Interface Thickness and Fraction of Attached Plasma Discussion and 

Example ....................................................................................................................................................... 103

J.2 Calculation of the Fraction of Attached Plasma—An Example ................................................................. 104

Appendix K.—Raleigh-Taylor Instability: Further Considerations .......................................................................... 107

References ................................................................................................................................................................. 111
NASA/TP—2009-213439 1 

Integrity of the Plasma Magnetic Nozzle

Richard A. Gerwin 

Los Alamos National Laboratory 

Los Alamos, New Mexico 87545 

Summary 

The injection of hot (100s of eV) plasma propellant into a 

nozzle composed of shaped magnetic flux to convert the 

plasma thermal energy into directed thrust is fundamental to 

enabling high-specific-impulse (10 000s of seconds) and 

high-specific-power (10s of kW/kg) piloted interplanetary 

propulsion. This report pertains to the theoretical physics 

governing certain aspects of the flow of plasma propellant 

through a magnetic nozzle, primarily the integrity of the 

interface between the plasma and the nozzle’s magnetic 

field, for these operational parameters. An expression for the 

initial thickness of the interface is derived and found to be 

significant (on the order of 10–2 m). A comparison is made 

between classical resistivity and gradient-driven Lower 

Hybrid Drift microturbulent (anomalous) resistivity, from 

which an algorithm is derived that obtains interface 

thickening as a time integral, that is then related to the 

nozzle-shaped geometry of the interface. An algorithm characterizing the plasma temperature, density, and velocity 

dependencies is derived and found to be comparable to 

classical resistivity at local plasma temperatures on the order 

of 200 eV. Macroscopic flute-mode instabilities within the 

interface in regions of adverse magnetic curvature are discussed and a practical growth rate formula for magnetic 

nozzle design is derived. It is calculated that only one to two 

e-foldings of the most unstable Rayleigh-Taylor (RT) mode 

would occur. For a more complete treatment of the RT effect 

it will be necessary to include the Hall effect as well as ion 

magnetoviscosity. The necessity of incorporating the Hall 

effect into Ohm’s law is discussed, where the full Hall current is able to flow and concomitant plasma rotation allowed. 

In that case, a critical nozzle length expression is derived 

below which the interface thickness is limited to about 1 ion 

gyroradius. 

1.0 Introduction 

This report pertains to the flow of plasma propellant 

through a nozzle composed of shaped magnetic flux, with 

application to the propulsion of space vehicles. The integrity 

of the interface between plasma and nozzle magnetic field 

is a particular concern. We consider at the outset a set of 

coaxial, circular, highly conducting coils in vacuum carrying 

azimuthal currents. Such a coil set produces a longitudinal 

magnetic field B having cylindrical vector components Br

and BZ. In terms of a “long-thin” geometric approximation, 

the latter component would be dominant. Plasma thrusters, 

in their simplest form, involve the nozzle-based acceleration 

of hot plasma propellant along the nozzle-shaped longitudinal magnetic field. 

Figure 1 illustrates a simplified few-coil magnetic nozzle 

cross section without and with the propellant flowing, 

respectively. An engineering design would incorporate a 

large number of incremental contiguous coils. The magnetic 

nozzle configuration is nominally axially symmetric. It 

consists of a figure-of-revolution around the dash-dot axis of 

symmetry. The current-carrying magnetic-field coils are 

represented in cross section as rectangles. They carry adjustable azimuthal currents in the θ- (azimuthal angle) direction. 

The magnetic field lines produced by these coils lie in the 

meridional r,Z-plane. Some of them are represented here as 

solid lines closed around the coils (div B = 0). The chamber 

wall of the nozzle is not shown here, but would lie just inside 

of the magnetic field coils, at a slightly smaller radius. 

Figure 1(a) illustrates magnetic field lines in vacuum, 

without injection of hot plasma propellant. The incipient 

nozzle breech, nozzle throat, and nozzle exit regions are 

indicated. An engineering design would incorporate a large 

number of incremental contiguous magnetic coils to smooth 

out and control the flow of plasma propellant. 

Figure 1(b) illustrates magnetic field lines, before and 

after being distorted by conducting plasma propellant 

injected into the breech. Magnetic flux is squeezed between 

the highly conducting plasma propellant and the yet more 

highly conducting metallic magnetic field coils. In the idealized magnetohydrodynamics (MHD) model, the initial 

magnetic flux in the chamber volume never penetrates either 

the coils or the plasma. The dotted arrows indicate magnitude and direction of the plasma flow velocity. The highly 

conducting edge plasma flows along the distorted nozzle 

magnetic field lines. (The pressure of the conducting plasma 

distorts the initial magnetic field lines.) Concomitantly, the 

exterior magnetic field acts like a flexible containment wall, 

radially confining the interior plasma. 

The directed plasma velocity is small in the breech and 

large at the exit. Stagnation enthalpy is converted into flow 

energy by the converging-diverging magnetic nozzle geometry. Magnetic trim coils may be needed beyond the exit 

region to straighten out the flow downstream. 

Adjustments of details of the configuration geometry, as 

well as adjustments of the propellant’s injection geometry, and 

the initial time dependence of the setting-up injection rate may 

be necessary to achieve a final desired configuration of steady 

flow. 
NASA/TP—2009-213439 2 

Resistive diffusion of the squeezed magnetic flux into the 

slightly resistive external magnetic coils can be mitigated, in 

principle, by time-dependent programming of the coil’s 

circuit currents or by the use of superconducting coils. Of 

course, neither of these options is available for the prevention of resistive interdiffusion of magnetic flux and plasma 

propellant. Certain instability mechanisms similarly cause 

mixing of plasma and magnetic field regions. These resistive 

and dynamical processes are both important because they 

degrade the performance of the magnetic nozzle, by allowing 

plasma attachment to the closed external magnetic field 

lines. The potential harm due to these processes thus constitutes the motivation for this report, which deals with the 

interface where the nozzle magnetic field meets the plasma. 

When a highly conductive volume of plasma (centered on 

the axis) is axially driven into such a longitudinal magnetic 

field, azimuthal diamagnetic currents are induced to flow in 

the plasma. These induced plasma currents may be regarded 

as images of the external currents. The plasma currents are 

directed to reduce the internal magnetic field within the 

plasma and increase the field on the outside of the plasma. 

The net result is as if magnetic flux had been transferred 

from the plasma volume to the external volume. From Faraday’s law, the total magnetic flux is conserved in the overall 

transverse cross section because the external coils are highly 

conducting. Pictorially, the intruding plasma pushes magnetic flux out of the way. 

In an extreme idealization of this process (derived from 

very highly conducting plasma), all of the internal magnetic 

flux will have been expelled from the plasma to augment the 

external magnetic flux that confines the plasma laterally. 

Such confinement is affected by external magnetic pressure 

of the field (ultimately supported by the external coils). This 

external magnetic pressure acts inward across the plasma 

boundary to balance the outward lateral thermal and inertial 

pressure of the plasma. 

The magnetic nozzle concept discussed in this report is 

based upon the above-described idealization (illustrated in 
NASA/TP—2009-213439 3 

Fig. 1), notably incorporating a “sharp-boundary” model of 

the field-free plasma. Moreover, the chosen configuration is 

the best that could be hoped for from the viewpoint of the 

plasma-detachment problem, since no plasma resides on 

magnetic flux. The report focuses upon the degradation of 

this ideal zero-order configuration due to the occurrence of 

fundamental microphysical processes. Concomitantly, the 

detachment problem again becomes a concern. 

A more realistic illustration that anticipates the degradation from an ideal configuration is shown in Figure 2. Starting at the top, distributed coils carry the electric current that 

generates the longitudinal magnetic field (B). An insulating 

liner is the physical surface that prevents the flow of radial 

current and thus provides the Hall voltage. (Without radial 

current, the edge plasma will not spinup.) The confining 

magnetic field occupies a region largely plasma-free near the 

coils and liner, where it is generally parallel to the axis of 

symmetry. The field then becomes more diffuse in the direction towards the axis of symmetry. This is the resistively 

expanding plasma-magnetic field (hereafter plasma-field) 

mixing layer δ, possibly due to anomalous resistivity, which 

contains both plasma and magnetic field. The core plasma is 

envisioned to be field free. 

Starting from the left in Figure 2, the initial penetration 

depth of ions into the confining magnetic field ∆ is upstream 

of the throat. The resistively expanding plasma-field mixing 

layer δ increases from the initial penetration depth of the 

ions. A region of possibly Rayleigh-Taylor (RT) unstable 

plasma due to adverse longitudinal curvature may exist just 

upstream on the throat. A smooth throat is formed by gradually 
NASA/TP—2009-213439 4 

increasing the azimuthal current in the coils about halfway 

down the complete nozzle, then gradually decreasing the coil 

currents further downstream towards the exit. Beginning at the 

throat, there is a precipitous drop in plasma temperature, 

which continues downstream (so the plasma resistivity is 

likely to be classical downstream of the throat). Beyond the 

“Exit” there will be supplementary magnetic coils to straighten out the flow of the core plasma. This process will involve 

resistive energy losses. 

The injection of hot hydrogen-plasma propellant into the 

breech of a magnetic nozzle, with subsequent conversion of 

its thermal energy into directed energy (and momentum), 

constitutes one of the important approaches to high-specificimpulse propulsion of space vehicles, as envisioned for 

certain interplanetary missions (Refs. 1 to 5). A magnetic 

field configuration comprising axially symmetric (Br, B0) 

field lines (in terms of cylindrical coordinates), can be 

shaped into a converging-diverging nozzle by properly 

placed coaxial solenoidal coils. It offers advantages over 

wall-confined flow for the controlled nozzle flow application 

of a high-temperature fully ionized propellant gas. 

One such advantage is derived from the nozzle magnetic 

field imposing an inward radial pressure balancing the outward plasma pressure, presumably isolating the plasma from 

the vessel wall (the coil shields). Such radial magnetic confinement of high-electrical-conductivity propellant can in 

principle reduce wall erosion by flowing plasma and reduce 

contamination of the plasma by heavy, high-atomic-number 

wall impurities. Radial magnetic confinement, if successful, 

thus mitigates unwanted mass entrainment and consequent 

reductions in specific impulse as well as mitigates the loss of 

thruster efficiency from inadvertent diversion of enthalpy 

into impurity radiation. 

Another advantage is the opportunity for expeditious 

experimental development since propellant flow control along 

shaped magnetic field lines can be controlled and optimized in 

a convenient, iterative manner during operation. Such optimization can be realized by adjusting the size and placement of 

coils and their currents. Still another possible advantage may 

be magnetic control of the thrust vector if slight deviations 

from axial symmetry are tolerated near the nozzle exit. 

Viability of the magnetic nozzle concept in its simplest 

form rests upon the integrity of a well-defined axially symmetric plasma-magnetic field interface. However, some 

fundamental microphysical processes, in the form of 

electron-ion collisions and microinstabilities, give cause for 

concern about the maintainability of a sharp interface. The 

sharp plasma edge adjacent to the surrounding magnetic flux 

constitutes an ideal axisymmetric equilibrium configuration 

in the sense of a pencil perfectly balanced on its point; it is 

nevertheless nonequilibrium in that there are closely accessible asymmetric configurations of lower energy. Such 

processes macroscopically broaden the interface by either 

classical or anomalous resistive diffusion, the former due to 

coulomb scattering of electrons on ions, and the latter due to 

time-dependent three-dimensional electric and magnetic 

fluctuations. 

If the desired interface were to become diffuse because of 

the action of these processes, the consequent enhanced 

intermixing of the periphery of the propellant core with the 

inner edge of the magnetic nozzle field (propellant “attachment”) would create a “detachment problem.” Inadvertent 

attachment of propellant to magnetic flux becomes ultimately manifest as resistive drag acting on the attached 

portion of the exhaust plume. That is, the attached part of the 

exhaust plume axially stretches and bends the returning 

magnetic flux lines. Because axial stretching and bending of 

returning magnetic field lines requires energy, the energy 

drain is manifested as a resistive drag on the propellant’s 

egress. Exhaust velocity and thrust thus become degraded. 

This situation motivates attention to the attachment process. 

In this report, a process of “instant attachment” of newly 

injected propellant to the magnetic nozzle flux is described. It 

is not accessible to standard MHD simulations, being characterized by Hall effects and individual particle (kinetic) effects. 

Examination of interface physics during the first quarter ion 

gyroperiod after injection is performed. It shows an initial 

boundary-layer thickness in the breech of the nozzle, which 

proves to be on the order of the ion inertia length, p i c ω (the 

speed of light divided by the ion plasma frequency). This layer 

thickness in the breech of the nozzle constitutes an initial 

condition subject to further resistive broadening (Fig. 2). 

One of the inferences of this report is as follows: Despite 

the substantial body of excellent theory and some limited 

experimental information that have been accrued, there is not 

yet a sufficiently comprehensive and accurate subgrid model 

readily available for magnetic nozzle design. The effects of 

nonzero beta (beta, β, is defined as the local ratio of total 

plasma pressure to magnetic pressure, Ptot/PB, where PB = 

B2

/2μ0; thus, β = 2μ0 Ptot/B2) in gradient-driven microturbulence are not yet well understood, and β ranges from practically zero in the external magnetic field to a large value 

within the internal plasma. Moreover, there is not yet agreement on the numerical factor in anomalous resistivity in the 

plasma edge layer, although there is agreement on the parameter dependencies. Such a model ideally would be applied 

in tandem with available two-dimensional axially symmetric 

MHD computer simulations (such as MACH2, see Ref. 3), 

with the intent to optimize the magnetic nozzle configuration 

while minimizing the detachment problem. 

On the other hand, within the magnetic-fusion-energy 

community and within the space-plasma physics community, 

there exists the knowledge base and the expertise for development of the needed theoretical and computational tools. 

The development of such a synergistic design capability to 

supplement the MHD codes would require a highly detailed 

and computationally intensive effort by a team of scientists 

collectively familiar with both the numerical issues and the 

physics issues. 
NASA/TP—2009-213439 5 

The body of this report is intended to provide substance to 

these general remarks. Some of the key journal articles are 

reviewed without dwelling overly on their mathematical 

details. (It proved necessary to collect and study many 

papers to assimilate a certain perspective from which to 

identify key contributions.) 

1.1 Outline of Report 

Section 2.0, “General Physics Description of Magnetic 

Nozzle,” presents general physics descriptions of magnetic 

nozzle flow, and the flow of a bounded, resistive fluid along 

magnetic nozzle field lines is discussed therein, along with 

associated processes that may be potentially deleterious to 

the plasma-field interface in the magnetic nozzle. The significance of local propellant resistivity and of local radial 

pressure gradients, in connection with diffusive broadening 

of the desired axially symmetric interface at the free-surface 

boundary of the propellant region, will be indicated. The 

eventual connection due to microturbulence between effective resistivity and the steepness of edge gradients will also 

be pointed out. 

We emphasize in Section 2.0 that a local treatment of diffusive intermixing of plasma and field needs to be made well 

defined by specifying certain global electrophysical boundary conditions. The need arises because of the Hall effect, 

which introduces additional dominant physics considerations 

when the electron gyrofrequency in the plasma-field interface is much larger than the electron collision frequency. The 

global electrophysical boundary conditions directly influence 

the nature of the local Ohm’s law, which, in turn, is a key 

player in calculating the evolution of the plasma-field 

interface. 

The conventional application of the simple Ohm’s law 

requires that the radial Hall voltage not be shorted out and 

concomitantly, that the Hall current not be allowed to flow. 

An important demonstration of this assumption lies in the 

opposite case, in which there ensues a severe reduction of 

effective plasma electrical conductivity due to Hall current, 

which is termed “magnetoresistance.” Magnetoresistance 

would be a feature of any simulation or experiment that 

allows Hall current but disallows plasma rotation. A practical consequence of magnetoresistance germane to 

magnetic nozzle operation would be a severe enhancement in 

the rate of diffusive plasma-field intermixing. Such considerations would apply no matter whether the fundamental 

resistivity is classical or anomalous. In the magnetic nozzle 

geometry, however, having a flow of Hall current without 

rotation of propellant would be unrealistic. Therefore, rotation also has been included here, with attention to satisfying 

boundary conditions in the propellant injection region. 

Allowing unhindered plasma rotation self-consistently 

with the presence of Hall current and with no Hall voltage 

drastically changes the character of the plasma-field 

transition layer from the artificial nonrotating case. We show 

that the layer thickness is limited to about the size of the ion 

gyroradius, provided that the nozzle length is shorter than a 

certain critical length. That length is λe mi me , namely 

the product of a representative electron mean free path (possibly anomalously short) with the square root of the ion-toelectron mass ratio. 

The quasi-radial Hall current produces an azimuthal 

(JHall × B) force that spins up the edge plasma. In a steadyflow model that satisfies the up-stream boundary conditions, 

the Hall-driven rotational velocity increases going downstream. The originally shorted Hall voltage is eventually 

reestablished downstream in the reference frame of the 

rotating edge plasma after sufficient spinup has been 

achieved. This explains why the interface layer broadening, 

downstream of the above-mentioned critical length, reverts 

to the resistive layer that it would have been had the Hall 

current not been allowed to flow. In this regard, it should be 

noted that viscous interaction of the plasma’s edge layer with 

the core plasma would slow down the spatial rate of Hall 

spinup of the edge layer, and would therefore increase the 

above-mentioned critical length. 

In Section 2.0 we suggest the possibility of backup 

approaches to magnetic nozzle utilization. In the event that a 

sharply defined plasma-field interface cannot be achieved 

and maintained throughout the nozzle, nozzle-based acceleration of attached plasma along with core plasma is to be 

expected because of the converging-diverging property of 

the attached external annular regions of magnetic flux. The 

problem of maximizing thrust then occurs at the nozzle exit, 

where accelerated attached plasma must be freed from 

returning magnetic flux. One approach to that detachment 

problem, which supports nozzle efficiency, has been documented, and the reference is listed. 

Section 2.0 also contains a summary of nominal working 

parameters (density, temperature, and magnetic field 

strength) that may be relevant to some interplanetary missions and laboratory experiments. These are referred to 

occasionally throughout the report in connection with certain 

estimates of plasma properties. 

In Section 3.0, “Resistivity From Gradient-Driven Microinstabilities,” we first address the initial interface width (in 

the breech) of the plasma-field mixing layer. This result is 

relevant to an important microinstability, the Lower Hybrid 

Drift (LHD) instability, regarding its linear and nonlinear 

evolution. Then we review the linear theory for that microinstability, which is suspected to cause the broadening of 

magnetic, shear-free plasma-field interfaces. This is a relatively robust, gradient-driven small-scale instability. Some 

stabilizing features are pointed out. 

Also within Section 3.0, the most difficult aspect of the 

microturbulence issue is addressed. It involves the connection of LHD microturbulence with anomalous resistivity, as 

arising from the nonlinear evolution and saturation of 

gradient-driven microinstabilities. The subject is reviewed 

and includes quasi-linear and alternative models. The review 

also discusses some attempts at numerical simulation. A 
NASA/TP—2009-213439 6 

practical formula for resistive interface broadening for general resistivity is identified, which can be applied to postprocessing of MHD simulations of magnetic nozzle flow. A 

comparison of classical and anomalous resistivity from one 

of the simulations is presented, as a function of plasma 

density and temperature in hydrogen propellant. 

Section 4.0, “Rayleigh-Taylor- (RT-) Type Instabilities,” 

deals with RT-type instabilities driven by adverse curvature. 

The presence of these RT modes leads to a lack of nozzlebased control of directed propellant flow. It is important to 

be aware that, in addition to diffusive broadening, the boundary surface of the propellant is free to deform, especially 

along the direction perpendicular to the field lines, namely 

the azimuthal direction. Deformations of the plasma surface 

in the azimuthal direction do not involve the expenditure of 

energy on local bending or stretching of field lines. The 

propellant’s surface therefore is vulnerable to “flute modes,” 

in spite of plasma pressure being nominally balanced by 

magnetic pressure at the interface. Specifically, regions of 

adverse streamline curvature and adverse magnetic curvature 

(center-of-curvature lies within the plasma) are susceptible 

to RT deformations (Fig. 2). The distinction between the 

propellant streamlines and the magnetic field lines is important in the injection region (see Sec. 4.0). 

The RT modes will be discussed within the context of 

magnetic nozzle operation, including the effect of finite 

Larmor radius (FLR) stabilization. A physical derivation of 

the growth rate will be presented that incorporates the simultaneous influences of plasma pressure and plasma flow with 

curved streamlines. A practical formula for postprocessing 

use with axisymmetric MHD simulations of magnetic nozzle 

flow is obtained. 

It is recognized that a more rigorous treatment of adverse 

curvature instabilities lies within the subject of “ballooning” 

modes, which takes into account that the adverse curvature 

regions have limited length. This is a complicated subject 

from which practical formulae applicable to magnetic nozzle 

flow are not readily available. It deserves a special treatment 

that is beyond the purview of this report. A reference to work 

on the destabilization of ballooning modes in mirror confined plasmas is given. 

The FLR stabilization of the long-wavelength RT branch 

is reviewed and is utilized to construct a practical formula 

for nozzle design. Enhancement of the RT growth rate by 

Hall effects is mentioned, with references, but these are not 

included in this report. It is observed that adverse curvature 

of propellant streamlines in the injection region is practically 

unavoidable. The short-wavelength branch of the RT modes 

is also reviewed, including the absence of their FLR stabilization, as well as their close connection with the previously 

treated LHD instability. For the short-wavelength RT 

branch, the relative importance of curvature relative to the 

plasma pressure gradient is examined and found to be important in the plasma injection region. 

Section 5.0 contains a final summary and discussion. 

Appendix A provides a list of the symbols used in this 

report, and Appendixes B through K expand on various 

concepts presented in this report (see “Contents”). 

1.2 Basic Notation and Basic Approach 

We use uppercase V for macroscopic fluid velocities as in 

the MHD model and Vth i for the ion thermal velocity. 

Lowercase v is used for particle velocities, as in connection 

with the Boltzman-Vlasov equation. 

The evolution of the diffusive plasma layer at the edge of 

the plasma is described with a local Cartesian coordinate 

system. This approach is valid whenever the thickness of the 

diffusive layer is small compared with the radius of curvature the short way around. As an example of the validity of 

this approach, see Section 2.3.3, “Diffusion of Field Into 

Plasma With Hall Effect,” which describes the penetration of 

an external magnetic field into the plasma in full cylindrical 

coordinates. Whenever the diffusive magnetic layer is thin 

compared to the radius of curvature, then the radial variable 

r varies negligibly over the region of interest. It thus cancels 

out as a constant, and the cylindrical diffusion equations 

reduce to a Cartesian representation of the diffusion process. 

Appendix B suggests a general and simple computational 

approach for obtaining the evolution of the plasma diffusive 

edge layer. One uses the exact resistive diffusivity D inside 

of the time integral. The time integral is replaced by an axial 

one-dimensional spatial integral involving the axial velocity. 

The axial velocity is expressed in terms of the density variation and the area variation along the axial direction using 

global mass flow conservation. The density, temperature, 

and velocity variations along the flow can be obtained from 

the quasi-one-dimensional model. If D is based upon classical resistivity, one can take its temperature dependence from 

the quasi-one-dimensional model. If it is based upon anomalous resistivity, for example, from the LHD instability, 

one can also use the quasi-one-dimensional model for the 

other scaling variables. 

2.0 General Physics Description of 

Magnetic Nozzle 

In this section, we consider the physics associated with 

the momentum equations of the electrons and ions. The 

notation utilized is as follows. The magnetic nozzle field is 

composed of the vector field B = (Br, 0, BZ), referring to 

cylindrical coordinates (r, θ, Z). The magnitude of the vector 

B is B. The Z-axis constitutes the symmetry axis of the 

nozzle geometry. The unit vector along B is bˆ . The plasma 

velocity vector (ion-fluid velocity vector) component along 

bˆ is V// (see Fig. 2). 

Let θ

ˆ be a unit vector in the azimuthal direction. Then 

the velocity vector component across the flux surfaces in the 
NASA/TP—2009-213439 7 

quasi-radial direction xˆ = θ

ˆ × bˆ is V⊥. The unit vector xˆ

is in the direction normal to the axially symmetric flux surfaces (Fig. 2). Propellant rotation is presently precluded (but 

included later), so that the total velocity vector of the propellant now is V b xˆ ˆ =V// +V⊥ . 

The steady-state momentum equation for the plasma (suppressing viscosity) is obtained by adding the momentum 

equations of electrons and ions, while neglecting electron 

inertia and recognizing that electron-ion collisions cannot 

alter the total momentum: 

ρV ⋅ ∇V + ∇Ptot = J × B (1) 

Here, ρ is the plasma mass density, Ptot is the plasma pressure (electron pressure plus ion pressure), and J is the current density vector to be obtained from Ohm’s law. When V⊥

<< V//, as is generally desired for magnetic-field guided flow 

of plasma, then the dominant components of Equation (1) along 

and across bˆ in the interface region are as follows: 

2

//

tot

1 0

2 l l

V P

    ∂ +∂ =         ρ  

 (2) 

2

//

tot ˆ V P

R ⊥

  ρ − +∇ = ×      

x JB (3) 

In Equation (2), l is the distance along the local field line (or 

streamline in the field-free region of core plasma). In Equation (3), R is the local longitudinal radius of curvature of the 

field line, and it is assumed for definiteness that the local 

curvature is adverse, that is, convex outwards. (If otherwise, 

replace R by –R.) The symbol ∇⊥Ptot represents the component of pressure gradient normal to the flux surfaces, along 

xˆ , where ∇⊥ represents the component of the gradient 

vector along the quasi-radial direction. 

It is assumed below that the plasma pressure is comparable to the magnetic pressure (β ≈ 1) in the plasma-field 

mixing layer; that is, for a nominal plasma-pressure versus 

magnetic-pressure confinement condition. This is a balance 

of pressures. Within the plasma core the local β is therefore 

very large, and within the external magnetic field region the 

local β is very small. The local β is of order 1 within the 

mixing layer, but also globally the internal plasma pressure 

must balance the external magnetic pressure across the interface. The latter viewpoint corresponds to the concept of a 

“global beta.” 

Strictly speaking, Equation (2) as it stands really is only 

valid along the actual flow streamlines, which can differ 

from the magnetic field lines because of resistive diffusion. 

Because the flow vector in the plasma-field mixing layer is 

referred to the magnetic field lines, however, Equation (2), 

which leads to the Bernoulli equation, is only an approximation within the plasma-field mixing layer. Its validity 

depends upon // V V ⊥ << 1. The flow streamlines of the 

plasma that have become attached to magnetic flux in the 

plasma-field mixing layer can be expected to differ only 

slightly from the magnetic field lines because of resistive 

diffusion. 

In particular, one finds V V DV ⊥ // // ≈ δ 0.5( ) . Here, D is 

a representative resistive diffusivity (D = η/μ0 in mks units), 

and δ is the characteristic diffusive width of the plasma-field 

mixing layer. For the parameters of interest and assuming 

classical resistivity due to coulomb scattering, the ratio 

V V ⊥ // proves to be on the order of 0.001. Thus, the flow 

lines would approximately follow the field lines even if there 

were substantial anomalous enhancement of resistivity (in 

D). In this connection, it turns out that δ scales as D , so an 

anomaly factor of 100 in D only increases V V ⊥ // by a 

factor of 10. 

2.1 Bernoulli Equation for Nozzle Flow 

Use of the Bernoulli equation for nozzle flow illustrates 

the conversion of thermal energy into directed kinetic energy—

the fundamental operation of a convergent-divergent rocket 

nozzle. A simple example of this is provided when a person 

blows up a balloon and then lets the compressed gas stream 

out of the blow hole. The total momentum is zero if there are 

no external forces acting on the balloon; so when gas streams 

out the back, the balloon has to dart forward to preserve the 

total momentum at zero. The total energy is initially entirely 

thermal. The thermal energy has been converted into directed 

kinetic energy, so the gas temperature must drop. Temperature 

is a measure of thermal energy. If the gas were a plasma, 

classical resistivity would increase. 

Integration of Equation (2), assuming that pressure is 

related to density adiabatically along a streamline, yields the 

Bernoulli equation. Thus, Ptotρ

–Ɣ is constant along a streamline. By assuming that the adiabatic index Ɣ = 1 + 2/Q = 5/3 

(where Q is the degrees of freedom; i.e., 3), there are at least 

occasional collisions. The energy-balance equation in its 

entropy-production form yields this adiabatic relation when 

there are neither sources nor sinks of mass and heat along a 

streamline. The choice Ɣ = 5/3 assumes that charged particles undergo at least several collisions during their transit 

of the nozzle, so that each charged particle samples all three 

degrees of freedom. 

The result of integrating Equation (2) along a streamline 

or flux line is 

( )

2 2

// constant

2 1

V S

+ =

γ − (4) 
NASA/TP—2009-213439 8 

Here, S is the local speed of sound, which is related to pressure and density by 

2 tot tot d

d

P P S = = ρ ργ (5) 

Equation (4) can also be related directly to the equation of 

total energy balance. With help of Equation (5), Equation (4) 

becomes a statement of conservation of total energy per unit 

mass along the stationary flow configuration. That is, the 

sum of the kinetic energy per unit mass, the thermal energy 

per unit mass, and the work done by unit mass against the 

ambient pressure constitute a sum that is conserved along a 

streamline. 

The Bernoulli equation displays the acceleration properties of a converging-diverging long-thin nozzle when one 

recalls the “choked flow” condition at the nozzle throat, 

namely that the flow velocity equals the sound velocity 

there. Denoting values in the throat by subscript “t,” 

V VS // ttt ≡ = (6) 

For simplicity, we shall assume infinite contraction and 

expansion ratios for the converging-diverging nozzle. It can 

be demonstrated, however, that a realistic moderate contraction ratio of 2, together with a moderate expansion ratio of 3, 

yields results very close to those obtained from assuming 

these ratios to be infinite. A long-thin coaxial tube of fluid 

(either central or annular) of variable cross section A has the 

property that mass flow rate m is conserved along the tube. 

Therefore, V// ≈ 0 is set in the breech (beginning) of the 

nozzle in Equation (4), because A is assumed very large 

(infinite) in the breech. The “constant” in Equation (4) is 

then determined to be 

( )

2

constant

1

bS = γ − (7) 

Here, Sb is the speed of sound in the breech of the nozzle. 

Use of Equations (6) and (7) in Equation (4) yields the flowspeed of propellant in the throat, 

2 2 2

4

3

t t b V = S = S (8) 

where Ɣ = 5/3 has been used. 

The exit velocity or specific impulse can be obtained from 

Equation (4) by recalling that propellant cools down as it 

expands through the nozzle. For a fully expanded flow, 

Equations (4) with (7) then provide the exit velocity as 

follows: 

ex b V = 3S (9a) 

With Equation (8) this also implies that 

Vex = 2Vt (9b) 

Use of the adiabatic relation in the sound speeds in Equation (8) then yields the ratio of ion density in the throat to ion 

density in the breech, namely as 

3

2 0.6495

4

3 2

3

 = ≈ 

 



 =

bi

t i

n

n

 (10) 

Since temperature, T, scales adiabatically as n

2/3 along a 

streamline, Equation (10) implies that the ratio of temperature in the throat to temperature in the breech is given 

exactly by 

4

3 = Tb

T

 (11) 

Before applying the above results, a perspective regarding 

acceleration by nozzle flow is presented. The above principles of nozzle flow and the realization of high exhaust 

velocities govern not only the field-free core plasma, but also 

govern plasma that already has interdiffused with the field 

and become attached. 

Three conditions, however, must be met for nozzle-based 

acceleration to be useful on attached plasma. First, electrical 

conductivity must be sufficiently large within the nozzle so 

that the plasma velocity is approximately confined along 

magnetic field lines, notwithstanding the presence of crossfield diffusion. Second, annular flux tubes carrying attached 

plasma must themselves exhibit convergence-divergence similar to the plasma core, so that nozzle-based acceleration occurs 

within each annular incremental flux tube. Third, the plasmadetachment problem must be addressed and solved at the exit. 

Reference 6 and references therein discuss nozzle-based 

acceleration of highly conducting attached plasma inserted in 

the breech within coaxial magnetic flux tubes (no field-free 

plasma) and includes George Marklin’s numerical calculation illustrating temperature reduction downstream of the 

throat of the nozzle. When longitudinal and azimuthal 

magnetic fields and longitudinal and rotating flows are all 

included, the nozzle-based acceleration process still can be 

reduced to a generalized Bernoulli equation. Reference 6 

recognizes the importance of solving the detachment problem, 

but does not consider special shaping of diverging magnetic 

field lines for that purpose. This has been carried out in Reference 7, however, which does describe a means of successful 

detachment with minimal loss of efficiency, while assuming 

that downstream plasma has only classical resistivity. 

Classical resistivity ηcl is the smallest possible. It therefore leads to the most difficult detachment problem. Because 

ηcl increases downstream as T–3/2 due to cooling of the 

expanding flow, the detachment process with ηcl can be 

successfully effected (Ref. 7). It proves necessary to outfit 

the vehicle with coaxial solenoidal trim-coils. They must be 

placed downstream in the diverging region of the magnetic 

nozzle, in a manner that ensures that the magnetic nozzle 

field lines are only weakly divergent, thereby minimizing 

 
NASA/TP—2009-213439 9 

TABLE I.—SAMPLE PARAMETERS RELEVANT TO SPACE 

VEHICLE PROPULSION AND RELATED EXPERIMENTS 

Parameter Propellant 

Hydrogen Helium 

Mass, m, AMU 1 4 

Number density (breech), nb, cm−3 1.0·1015 1.0·1015

Number density (throat), nt, cm−3 0.65·1015 0.65·1015

Temperature (breech), Tb, eV 50 100 

Temperature (throat), Tt, eV 38 75 

Magnetic field,a B, T 0.20 0.35 

Electron cyclotron frequency, ωc e, s−1 3.2·1010 5.6·1010

Electrical conductivity (throat),bσt, mho/m 2.6·105

 3.7·105

Resistive diffusivity (throat), Dt, m

2

/s 3.2 2.3 

Electron collision frequency (throat), υe t, s−1 0.8·108

 1.0·108

Flow velocity (throat), Vt, m/s 1.1·105

 1.0·105

Exit velocity,c Vex, m/s 2.1·105

 2.0·105

a

Local ratio of thermal to magnetic pressure β = 1 in breech. 

b

ln Λ = 10, where Λ is length of breech to throat divided by ion 

gyroradius. c

Also, specific impulse (10–1 s). 

resistive-drag losses. The addition of such trim coils then 

becomes a factor in mission considerations dealing with size 

and weight of the vehicle. It may be that the occurrence of 

enhanced resistivity in the downstream region would further 

ease the detachment problem. 

The above Bernoulli results on nozzle flow are now 

applied to hydrogen propellant, with parameters similar to 

those relevant to interplanetary travel. Also, the Bernoulli 

results are applied to helium propellant with parameters 

similar to those proposed by Turchi (Ref. 2) for a 

magnetic nozzle experiment at Ohio State University using 

the 1-MJ, 1-GW pulsed-power source. Table I presents 

approximate round-number values of nominal reference 

parameters assumed to be inserted into the breech of the 

nozzle. The quantities are derived according to the Bernoulli 

equation for the throat and exit regions of the nozzle. 

It is noted that the assumed high-enthalpy plasmas provide a desired specific impulse of about 20 000 s. Also, only 

moderate magnetic field strength is required to effect an 

initial balance between plasma pressure and magnetic pressure. (Downstream of the breech, one expects the magnetic 

flux to press inward in radius as the plasma’s internal pressure becomes converted into longitudinal flow energy.) 

Hydrogen propellant is assumed to be axially inserted into 

the breech of the nozzle and preheated to the desired temperature by drawing off edge plasma into the nozzle from the 

fusion power reactor. 

From Table I, a feature that proves to be very important 

with regard to the Hall effect is observed. The electron 

cyclotron frequency in the edge plasma ωc e largely exceeds 

the electron coulomb collision frequency, υe, by more than 2 

orders of magnitude. In the throat, magnetic field B is still 

0.7 of its breech value, hence, also ωc e. Even a severely 

anomalous electron collision frequency still could be dominated by the electron cyclotron frequency. 

At a fixed temperature in the breech, the classical collision frequency there is proportional to 3 2 n T whereas the 

electron cyclotron frequency there is proportional to nT

via pressure balance. Thus, for the given temperature, the 

reference density would have to be increased by orders of 

magnitude and out of range in order to bring the electron’s 

coulomb collision frequency into the neighborhood of the 

electron cyclotron frequency. For these reasons, the Hall 

effect in the mixing layer is regarded as sufficiently important to merit a separate treatment below. 

Finally, we note that the velocity in the throat region, Vt

, 

together with the envisioned meter-scale length of the 

nozzle, can be used to provide a rough estimate of the 

transit-time duration available for diffusive spreading of the 

plasma-field mixing layer. For a meter-length nozzle, this 

longitudinal transit time is on the order of 10 μs. (In Appendix B it is shown that if we allow for the transit time to 

increase from zero velocity in the breech, there is an amplification of the transit time by a factor of ~2.) 

2.2 Diffusive Intermixing in Plasma-Field Interface 

This subsection addresses diffusive intermixing of plasma 

into magnetic field and also resistive diffusion of magnetic 

field into plasma. Both mixing processes occur together. We 

turn now to Equation (3) for crossfield diffusion of plasma. 

2.2.1 Diffusion of Wall-Confined Plasma 

Equation (3) is discussed as an illustration for the worst 

case. This case is one where hot plasma, initially inserted 

into the breech of the nozzle, inadvertently immediately 

spreads over all upstream flux surfaces (plasma-wall interactions are ignored). The plasma profile is assumed to extend 

uniformly from the symmetry axis out to the wall, so that the 

transverse pressure gradient is now neglected relative to the 

centrifugal force. Effective gravity from centrifugal force 

with adverse curvature would cause wall-confined plasma to 

sink against the outer wall, creating a pressure gradient that 

is positive outwards. By neglecting this pressure gradient, 

we will overestimate the crossfield velocity of plasma and 

the concomitant resistive deviation of the streamline from 

the field line. But even with uniform pressure, the plasma is 

still wall confined. Then Equation (3) reads 

2

// ˆ V

R

  ρ − =×      

x JB (12) 

At the same time, assume that Ohm’s law holds in its 

simplest resistive form, namely 
NASA/TP—2009-213439 10 

σ× = VB J (13) 

where σ is the electrical conductivity of plasma (σ = 1 η)

and V B× is the effective electric field in the moving reference frame of the plasma, driving the current density. 

Equation (13) implies that 

2 JB V × = −σB ⊥ (14) 

It should be observed that this MHD force density, as 

obtained from the simple Ohm’s law, is directed oppositely 

to the crossfield flow velocity. This phenomenon is known 

as “eddy current braking” of the plasma crossfield flow. The 

same phenomenon creates the detachment problem at the 

nozzle’s exit. 

Use of Equation (14) in Equation (12) presents a steadyflow balance between the outward centrifugal force resulting 

from parallel flow along a curved field line and the MHD 

braking force. As a concrete example, we shall evaluate this 

balance just upstream of the throat, where the field lines 

might be curving radially inwards towards the throat with an 

adverse curvature (of course, it also is possible to imagine a 

nozzle field with good curvature everywhere). This force 

balance can be expressed as a ratio of transverse to longitudinal velocities in the following form: 

































= ⊥

mag //

kin

// RV

D

W

W

V

V (15) 

Here, 2 2 Wkin = ρV is the kinetic energy density of 

parallel flow approaching the throat region; and 

0

2

mag W = B 2μ is the magnetic energy density near that 

region. Also, 1

0 ( )

− Dcl = σclμ is the classical resistive diffusivity there (Russian or Soviet papers sometimes refer to D

as “magnetic viscosity”). The magnetic field is chosen to 

have β of order unity, and the flow speed is on the order of 

the speed of sound. (Even though β is of order unity, it is 

assumed that the wall takes up the plasma pressure in the 

present example.) Therefore, the first nondimensional 

ratio, the ratio of energy densities in Equation (15), is of 

order unity. The principal determining factor for the velocity 

ratio V⊥ V// is the second nondimensional ratio in 

Equation (15). For R = 1 m, the hydrogen parameters yield

5

// 3 10− D RV ≈ ⋅ cl , and the helium parameters yield 

5

// 2 10− D RV ≈ ⋅ cl . 

These very small numbers for Dcl R V// signify that 

flowing plasma having a diffuse profile remains very well 

attached to field lines (V⊥ << V//), assuming classical resistivity. This situation would still hold true even for R = 0.1 m. 

It can therefore be expected that with the simple Ohm’s law 

with classical resistivity and for the area of each incremental 

annular flux tube having a converging-diverging dependence 

on axial distance, nozzle-based acceleration still can occur 

for diffuse attached plasma. Moreover, it is worth noting that 

the small ratio V⊥ V// is so small indeed that the plasma 

resistivity could even be anomalously large by several orders 

of magnitude and yet have the plasma remain fairly well 

guided by the field lines. 

To summarize Section 2.2.1, it can be stated that diffuse 

wall-confined plasma preattached to the magnetic flux can 

usefully acquire nozzle-based acceleration provided that the 

consequent detachment problem is solved. Reference 7 has 

described an approach to solving the detachment problem, 

which involves implementation of a weakly diverging 

magnetic nozzle field in the exhaust region. 

2.2.2 Diffusion of Magnetically Confined Plasma 

We return now to Equation (3), assuming that the edge 

plasma at the interface has such a sharp boundary that the 

transverse pressure gradient (negative outwards) completely 

dominates the centrifugal force term. Use of the simple 

Ohm’s law, Equations (13) and (14), then yields an equation 

for the transverse velocity in terms of the transverse pressure 

gradient, tot tot P P( /)ˆ ∇ ≈− δ ⊥ x : 

tot 2 ˆ 2

D P

B

⊥ ⊥

  η β =− ∇ ≈     δ

V x (16) 

Here, η is the resistivity, η = 1/σ, and D is again resistive 

diffusivity D = η μ0 . Also, β is the ratio of thermal to 

magnetic pressure within the interface; 2

0 tot β= μ2 P B , 

where Ptot is the sum of electron and ion pressures, 

P PP tot = +e i . Finally, δ represents the thickness of the 

transition layer of magnetically confined plasma. The pressure gradient of confined plasma is noted to be inward, along 

− xˆ , where V⊥ = Vx . 

From Equation (16), one then obtains the estimate 

















δ

β ≈ ⊥

// // 2 V

D

V

V (17) 

Equation (17) for the magnetically confined plasma has the 

same structure as Equation (15) for the wall-confined 

plasma, but now with a very small plasma-field mixing layer 

width in the denominator. However, even if δ were 4 orders 

of magnitude smaller than R, for example δ = 0.01 cm, one 

would still have a relatively small transverse velocity, namely V⊥ ≈ 0.1 V//. This can be seen by utilizing the same parameters as for Section 2.2.1. Moreover, if δ ≈ 1 cm, 

comparable to an ion gyroradius, then V⊥ ≈ 0.001V//. Thus, 
NASA/TP—2009-213439 11 

qualitatively it appears that classical resistivity, with the 

given parameters, does not destroy the small ratio of transverse to longitudinal velocity, even for rather thin plasmafield mixing layers. 

The analysis leading to Equation (17) can be extended. 

We take into account that the plasma-field mixing layer 

width δ is related to the transverse diffusion velocity V⊥. The 

two really should be treated together in a self-consistent 

manner. To exploit this point of view, we cancel V// from 

each side of Equation (17), and make the reasonable identification within the plasma-field mixing layer: 

t

V

d

dδ

⊥ = (18) 

This equation represents the rate at which the plasma-field 

mixing layer thickens with time as the flow is followed 

downstream. Use of Equation (18) in Equation (17) and 

multiplying through by δ then yields 

d 2

2 d

2

D

t

β ≈





 



δ

 (19) 

Assuming that the initial boundary-layer width of attached 

plasma in the breech is negligibly small (corrected in 

Sec. 3.0, “Resistivity From Gradient-Driven Microinstabilities,”), this differential equation suggests the estimate 

δ ≈ βDtb t (20) 

where tb t is the time for longitudinal flow to traverse the 

nozzle, say from breech to throat. (The integration of 

Eq. (19) over time is addressed in Appendix B.) For a propellant whose transverse pressure is balanced by confining 

magnetic pressure, the quantity β will be of order unity 

within the plasma-field mixing layer. A more rigorous and 

detailed analytic procedure that includes a non-MHDderived initial layer thickness is considered beyond the scope 

of this report (see Sec. 3.0, “Resistivity From GradientDriven Microinstabilities”). 

Note that Equation (20) assumes that there is no significant initial thickness of the plasma-field transition layer in 

the breech of the nozzle. It is appropriate for comparison to 

MHD simulations that make the same assumption. This 

issue, which cannot be fully addressed within the resistive 

MHD model, is considered later in this report in Section 3.1, 

“Interface Width in Breech of Nozzle.” It is assumed here 

that the field coils and the injection process are such as to 

allow a macroscopically gentle injection, such that there are 

no anomalous or extreme dynamical macroscopic processes 

that produce an initial width of the transition layer in the breech 

of the nozzle. Otherwise, such a macroscopic initial layer 

thickness certainly would have to be included in the contribution to the layer thickness downstream. See Reference 1. 

The terse derivation of Equation (20) will now be supplemented with a more rigorous derivation, which provides a 

more precise interpretation of the symbol β. It turns out that 

β is neither a local nor a global concept here, but has aspects 

of both. 

For slow, noninertial resistive flow in the interface, one 

has a situation of quasi-equilibrium. 

∇P = J×B (21)

The quasi-radial component of this equation, outwards 

across the flux surfaces, involves only the azimuthal component of the current density. However, the azimuthal electric 

field vanishes in an axially symmetric steady state, so that 

the azimuthal current density can be represented in Ohm's 

law by 

ηJ = V ×B (22) 

which involves the outward cross-flux velocity of plasma. 

Use of the latter equation in the former yields 

V

















μ

∇ = −

0

2

2

2 B D P (23) 

in which the gradient and the velocity are directed outwards 

across the flux surfaces, in the quasi-radial x-direction. Here, 

D is the resistive diffusivity η/μ 0. 

In the absence of a detailed model of the interface, one 

must have recourse to reasonable but expeditious assumptions to go forward with the theory. In this instance, such an 

assumption is that the plasma outward velocity corresponds 

to the rate of thickening of the interface width δ. 

t

Vx d

dδ = (24) 

Integrating the quasi-equilibrium equation from x = 0 at the 

core edge to x = δ at the vacuum end of the interface, we 

then have 

t

B D P

d

d

2

2

0

2

core

δ δ

μ − = − (25) 

In the above equation, Pcore represents the plasma pressure at 

the edge of the core region, and the brackets represent an 

average of the enclosed quantity over the interface width, 

0 ≤ x ≤ δ. This equation also can be written as 
NASA/TP—2009-213439 12 

2 d

d

D

t

δ = β (26) 

in which β is now given by 

2

2 0 core

B

μ P β = (27) 

In subsequent equations that represent resistive thickening of 

the interface by resistive diffusion, β should be accorded this 

interpretation. 

For the transit time t we invoke b t Lb t Vt t = , where 

Lb t is regarded as a characteristic length of the nozzle from 

breech to throat (similarly, Lb ex is the characteristic length 

of the nozzle from breech to exit plane). Also, Vt

 represents 

velocity in the throat region, which is known to be on the 

order of the ion thermal velocity. Then, from Equation (20), 

it can be shown that the square of the boundary-layer width 

is 

i

e

e

bt

i pi

e

e

bt

i

m

c L m

m

L m

a 















 λ















ω = β 















λ δ ≈

2

2 2 2 (28) 

 

Here, we have retained β in Equation (20) and have set the 

electron and ion temperatures equal, Te = Ti

. Retention of 

beta in Equation (28) is practically just a formality. The 

square root of the local beta is near 1 within the interface and 

so is the square root of the “combination beta” described 

above. The model of the interface invoked here is just not 

sufficiently detailed as to provide more accuracy than is 

available with such numerical factors. The ion thermal gyroradius is connected to the ion inertia length by the square 

root of the local beta. So, the factor that is being suppressed 

in delta, as given by the second part of Equation (28), is the 

square root of the ratio of the combination beta to the local 

beta. The square root of that ratio is expected to be near 1. 

Also, the resistivity η in D in Equation (20) has been reexpressed in terms of its fundamental factors for hydrogen 

plasma; 2 η= υ m nq ee e , where me is electron mass, ne is 

electron number density, q is the magnitude of the electron 

charge, and υe is the momentum-transfer collision frequency 

of an average electron with background ions or with microturbulent fluctuating fields. The negative charge of an electron will be explicitly indicated as –q; thus q is a positive 

quantity. 

 Recall that the resistive diffusivity D = η μ0 in mks 

units, and = η 4π 2 D c in cgs units. Here, μ0 is the 

magnetic permeability of vacuum, and c is the speed of light 

in vacuum. It is useful to note that 0 0

2 c = 1 ε μ in mks 

units, where ε0 is the electrical permittivity of vacuum. 

In Equation (28), ai

 is a representative ion gyroradius in 

the attached plasma within the plasma-field mixing layer, 

[ ] 2 2 2 i Ti mi c i a = ω , with ωc i being the ion gyrofrequency in 

the boundary-layer magnetic field. Moreover, λe is a representative mean free path for collisions of electrons (possibly 

anomalous) with ions; hence 1 2 ee e e T m − λ =υ , and me/mi

is the electron-to-ion mass ratio. Furthermore, ωp i is the ion 

plasma frequency, with 2 4 pi i i ω= πnq m in cgs units and 

2

i i 0 nq m ε in mks units. The ratio c/ωp i is called the ion 

inertia length. 

The ion gyroradius ai comprises a principal feature of the 

boundary-layer width. Other highly significant factors also 

codetermine the resistive MHD boundary-layer width in Equation (28). Thus, δ ai depends upon the dimensionless longitudinal distance traveled, in the form ( ) ( ) 1 2 1 4 L λe me mi . 

Suppose L = 1 m. The throat velocity is Vt

 = 105 m/s; 

hence the transit time is about t = 10−5

 s. Assuming β ≈ 1, and 

taking classical resistive diffusivity Dcl ≈ 3 m2

/s for hydrogen, 

one finds from Equation (20) that δ ≈ 0.6 · 10−2

 m = 0.6 cm, 

with a slightly smaller value for helium propellant. This 

value of δ constitutes an estimate for the width attained by 

the plasma-field mixing layer after following propellant up 

to and just through the nozzle throat. It is generated by resistive diffusion of plasma into the magnetic nozzle field. It 

overestimates the diffusion, since Dcl is the throat value with 

temperature reduced from its value in the breech. It underestimates the layer thickness, however, by neglecting any initial 

thickness. This neglect is appropriate for comparison of layer 

thickness δ with that from resistive MHD simulations. 

In the assumed case of classical resistivity, with β of the 

order one and taking parameter values listed in Table I 

above, broadening of the plasma-field mixing layer after 

passage of propellant up to and just through the nozzle throat 

thus proves to be rather small compared to system dimensions. This would be so even for a modest plasma throat 

radius of 10 cm. Any simulation (or experiment) of magnetic 

nozzle flow of nominally confined plasma having those 

parameters and giving results with substantially larger diffusion of the plasma-field mixing layer may be violating one or 

more of the model assumptions set forth above. Also, the 

simulation may contain some numerical diffusion. 

In this simple presentation of radial resistive diffusion, we 

have used a constant characteristic nozzle flow velocity for 

estimating δ, which has been taken to be the velocity in the 

throat region of the nozzle. This procedure serves to define a 

characteristic dwell time as a characteristic axial length 
NASA/TP—2009-213439 13 

divided by the characteristic longitudinal velocity. However, 

the dwell time of the propellant between breech and throat 

actually is somewhat longer due to the time needed for the 

propellant to accelerate up to its characteristic velocity. This 

increased dwell time then gives the resistive plasma-field 

mixing layer more time to grow. The effect of the more 

accurate dwell time is discussed in detail later, in Section 2.4, “Summary,” and in Appendix B. It is estimated that 

the time needed for acceleration engenders a factor of ~2 in 

the increased thickness of the plasma-field mixing layer. 

2.2.3 Diffusion of Confining Magnetic Field 

In the preceding case, diffusion of plasma into the confining field was considered. There also is another process for 

attaching plasma to the confining field. After the ambient 

magnetic field has been pressed against the outer conducting 

wall by the initial pressure of injected hot plasma, the field 

can rebound spatially downstream, by undergoing inward 

diffusion into the core plasma regarded as a resistive 

medium. Turchi analytically treated this view of plasma 

attachment (diffusion of a pulsed magnetic field into a resistive medium), using a well-known model of time-dependent 

magnetic diffusion (Ref. 8, personal communication), based 

upon a discussion of time-dependent magnetic diffusion. We 

now analyze this magnetic diffusion process by a slightly 

different method than that used by Turchi. The treatment 

here of the diffusion of a magnetic field into a conductor 

agrees (to within a trivial numerical factor) with that presented in Section 2.2.2 of plasma diffusion across a magnetic 

field. The reason is that the square root of beta is near 1 for 

the special beta described. 

The point of view adopted here is that the exact time 

dependence of the magnetic field experienced at the boundary of the moving plasma is regarded to be less important 

than the overall time scale for that time variation. The reason 

is that the extent of the field diffusion into the plasma is 

obtained by time integration. This point of view thus motivates us to choose a simple but expeditious time dependence, 

containing a certain relevant characteristic time, for the 

magnetic field experienced by the moving plasma at its 

boundary. 

In a reference frame moving downstream with the plasma 

velocity comparable to the velocity in the throat ~Vt

, where 

that velocity is assumed constant and uniform, Maxwell’s 

equations are written as follows. Here, E′ = E + Vt

×B is the 

azimuthal electric field in the moving frame. 

 

() ( )

B B

E J B

D t

D

= − ∇ = −∂

∇× ′ = ∇× η = ∇× ∇×

2 (29) 

In an axially symmetric configuration, one can note the 

following property of this set of equations. The longitudinal 

component (along B) involves the spatial variation—in the 

direction normal to the flux surfaces—of azimuthal vector 

components of electric field E′ and current density J. 

Here, it is important to note two other key features to be 

used in Equation (29). First, in the moving frame, the azimuthal electric field E′ will not necessarily vanish, even in a 

situation of steady flow and steady fields in the lab frame. 

Second, the azimuthal current density vector J is assumed 

related to the azimuthal electric field E′ (in the moving 

frame) by the simple Ohm’s law, as before. Since J involves 

the difference between two vector velocities, those of electrons and ions, its evaluation in any frame produces the same 

result for quasi-neutral nonrelativistic plasma. Also, the 

magnetic field is the same in the moving frame, when the 

velocity is nonrelativistic. The resistive diffusivity D in 

Equation (29) is assumed here to be a scalar constant representative of the throat region of the nozzle. 

Cylindrical geometry can be suppressed when examining 

magnetic diffusion in the vicinity of the plasma’s sharp 

boundary, as long as the plasma-field mixing layer width is 

small compared to its radial distance from the axis of symmetry. Accordingly, if we let the x-coordinate represent the 

direction normal to the flux surfaces (so that the distance x

increases towards the wall, see Fig. 2), then the last part of 

Equation (29) reduces to a simple diffusion equation within 

the plasma, considered as a resistive medium: 

D

t x

B B ∂ ∂ = 2 (30) 

There is no loss of generality in setting x = 0 at the plasma 

edge. This situation is effectively the same as a magnetic 

field diffusing into a conducting half-space bounded by a 

planar surface. 

Following the line of Turchi’s argument in a general way, 

we shall represent the time variation of the magnetic field 

experienced by the longitudinally moving plasma with simple exponential time dependence. The method is a form of 

the “separation-of-variables” technique for solving partial 

differential equations. Thus, 

( ) () ( ) , bt t t B xt G x e = (31) 

The incoming conducting plasma in the region upstream of 

the throat presses the external field against the metallic wall 

(coils), thus increasing the field strength in the local neighborhood. By assuming this monotonic increase with time of 

the magnetic field strength seen by the moving edge plasma 

during its traverse up to the throat region, the penetration of 

field into plasma shall surely be overestimated. (Using the 

plasma temperature in the throat, one overestimates the 

prethroat classical resistive diffusivity D.) Then, using Equation (31), Equation (30) is reduced to 
NASA/TP—2009-213439 14 

( ) 1 ( ) G x DtG x

b t

′′ = (32) 

Here, G′( ) x = dG dx . The solution of Equation (32) within the plasma region (x < 0) is 

( ) 1

I

b t x Dt

G x Ce− = (33) 

where x is the positive (decreasing plasma density) or negative (increasing plasma density) distance from the plasma 

boundary (Fig. 2). This spatial profile within the plasma, 

engendered by the time-dependent magnetic field imposed at 

the plasma boundary, clearly exhibits the character of a 

boundary layer of magnetic field protruding into the interior 

of the plasma. 

The moving plasma would generally experience significant changes in field strength of a time on the order of the 

plasma’s transit time up to the nozzle throat. Therefore, it is 

reasonable to regard the time constant for imposed field 

growth as tb t , which is the plasma’s transit time through the 

converging part of the nozzle. Then, the plasma-field mixing 

layer thickness implied by Equation (33) can be expressed as 

δ = Dtb t (34) 

where tb t is the nominal transit time through the converging 

part of the nozzle, b t Lb t Vt t ≈ . Thus, essentially the same 

answer is reached for field diffusion into plasma as in Subsection 2.2.2 (assuming β is of order 1) when the plasma 

diffuses into the field. In both cases, for the given parameters 

and classical resistivity, the plasma can be expected to 

intermix with the field to a distance of at most about 1 cm 

during transit through the converging part of the nozzle. This 

distance is to be compared to a system length on the order of 

1 m and a plasma throat radius perhaps on the order of 

10 cm. 

For comparison, the steady-state magnetic-field diffusion 

problem is solved in the lab frame in Section 2.3.3, “Diffusion of Field Into Plasma With Hall Effect,” and in Appendix C. In the limit of simple resistive diffusion, a similarity 

solution is found from which a characteristic penetration 

depth evolves with increasing axial distance downstream. 

That result proves to be in very good agreement with the 

present treatment in the moving frame. 

2.3 Hall Effect, Plasma Rotation, and Electron 

Pressure in Ohm’s Law 

We consider the foundations of the simple form of Ohm’s 

law within the context of magnetic nozzle physics. Certain 

global conditions prove to be required in order for the simple 

Ohm’s law to provide a valid description. This circumstance 

may influence the nature of the computed plasma-field mixing layer and its spatial rate of broadening. 

The momentum equation of the electron fluid constitutes 

the complete Ohm’s law. Let Ve be the electron macroscopic 

(fluid) velocity, whereas Vi

 represents the ion macroscopic 

velocity. Therefore, the current density is 

( ) ei e J VV = − n q (35) 

The electron number density is ne (the same as the ion number density for atomic number Ξ = 1, which is assumed 

here), the electron’s charge is −q, and the mass of a single 

electron is me. The fluid-momentum equation of electrons in 

the lab frame then reads, in mks units, 

( )

1

( )

te e e e e e

ee i e e

P m n

q

m

  ∂ + ⋅∇ + ∇    

= −υ − − + ×

VV V

V V EV B

(36) 

Here, Pe is the electron pressure, and υe is the momentumtransfer collision frequency of an average electron with the 

ions. As usual, the latter effect is represented as a friction. 

The electron collision frequency can be either the classical 

coulomb collision frequency, or it can be anomalous because 

of the activity of microturbulence. The leftmost term in 

Equation (36) vanishes in the assumed steady state, and the 

inertial [Ve ⋅∇Ve] term generally can be neglected in comparison to the electron-pressure gradient when the electron 

macroscopic velocity Ve is small compared to the electron 

thermal velocity [ ]1 2 2Te me . That inequality is well satisfied at the postulated temperatures. 

Accordingly, by neglecting the two leftmost terms, multiplying Equation (36) by me/qe, and using Equation (35), 

Equation (36) becomes 

e e 1

i e

m

P

q nq nq nq

       υ + ×= + ×− ∇              

J J EV B B (37a) 

A more familiar form is 

1 1

i eP

nq nq

  + × =η + × − ∇    

EV B J JB (37b) 

The left-hand side is the electric field in the moving frame 

(E is the electric field in the lab frame), and η is the fundamental resistivity of plasma, η = σ−1

. The electrical conductivity of plasma is 2 e e σ= υ nq m . The second term on the 

right, involving J × B, is the magnetic Hall term. Other terms 
NASA/TP—2009-213439 15 

in Equation (37b) that go beyond the simple Ohm’s law are 

the motional electromotive force due to plasma rotation, 

which is part of the V × B term, and the thermal Hall term, 

which is the electron-pressure gradient term. The thermal 

force term arising from the specific nature of coulomb collisions of charged particles (Ref. 9) is neglected here in the 

interest of simplicity. It can easily be included if so desired. 

In this section, we want to emphasize that the effective 

electrical conductivity is not necessarily the fundamental 

conductivity σ. Rather, the effective electrical conductivity 

generally depends on the Hall terms, on whether rotation 

occurs, and especially on the global electrical boundary 

conditions that are present (either in a simulation or an experiment). Concomitantly, we shall illustrate that the nature of 

the plasma-field mixing layer and its spatial rate of broadening depends on the possibility of plasma rotation, on the 

presence of an electron pressure gradient, and on the global 

electrical boundary conditions. In order to demonstrate the 

role of the Hall effect in a simple manner, we first neglect 

rotation and electron pressure. They are reinserted later. 

2.3.1 No Plasma Rotation, No Electron Pressure 

It is apparent that 

c e

e

B

nq

  σ ω

  = =Ω

  υ

 (38) 

wherein ωc e is the electron cyclotron frequency, and Ω is 

the conventional Hall parameter. Recall that Ω >> 1 in the 

plasma’s edge layer with magnetic field in the parameter 

regime of interest. Then multiplying Equation (37b) by the 

conductivity 2 ee ee σ= υ nq m is seen to be equivalent to 

J + ΩJ ×b

ˆ = σ( ) E + Vi ×B (39) 

where bˆ is the unit vector along B. The Hall term is potentially very important in the considered applications, since Ω

is very large in the plasma-field mixing layer, on the order of 

400 for hydrogen propellant and 560 for helium propellant in 

Table I. Equation (39) is now solved for the current density 

to exhibit the effective electrical conductivity. 

The azimuthal component of Equation (39) is 

J θ − ΩJ x = −σVxB (40) 

where it is noted that, in an axially symmetric steady state in 

the lab frame, the azimuthal electric field Eθ = 0. The condition on Eθ results from the integral form of Faraday’s law of 

induction, namely ∇×E = −∂t

B. The component of Equation (39) normal to the flux surfaces (the xˆ -component) is 

x Ex ΩJ θ + J = σ (41) 

It has been noted that rotation is precluded here. It also has 

been implicitly assumed that there is no azimuthal magnetic 

field. (The relevance of the azimuthal magnetic field to the 

considered magnetic nozzle configuration will be brought 

out later. See Appendix D.) 

Upon multiplication of Equation (41) by Ω and adding to 

Equation (40), one finds 

( ) 2 1 x x Jθ E VB   σ = Ω−     + Ω

 (42) 

Now in the limit Ω << 1 (the electron gyrofrequency much 

smaller than the electron collision frequency), Equation (42) 

reduces to a form of Equation (13), the simple Ohm’s law: 

J θ = −σVxB (43) 

 

The remainder of this subsection shows how the simple 

Ohm’s law for azimuthal current conduction still can be 

valid in spite of the presence of a strong longitudinal magnetic field, Ω >> 1. Two extreme examples will be discussed 

to illustrate the importance of the Hall effect together with 

global boundary conditions. The first example is that the 

Hall current is absolutely not allowed to flow because of 

global electrical boundary conditions imposed upon the 

system. The second one is at the opposite extreme: namely, 

the Hall voltage is completely shorted out by another kind of 

those boundary conditions, and the Hall current then flows 

unimpeded. (Of course, there could be a number of intermediate possibilities, any one of which may be relevant to a 

particular experiment or simulation.) 

2.3.1.1 No Hall current allowed.—If absolutely no current is allowed to flow across the flux surfaces (no quasiradial currents) because of insulating boundary conditions at 

the wall (and the backplate in the breech of the nozzle), then 

Jx = 0 and Equation (40) again yields the simple Ohm’s law, 

exactly in the form of Equation (43) above. This result is 

obtained in spite of a formally huge magnetic Hall term in 

Equation (39). On the other hand, motivated by the huge 

Hall parameter Ω, it is prudent to be mindful, both in 

experiment and numerical simulation, as to whether any 

quasi-radial Hall currents are inadvertently allowed to flow. 

Of course, the coaxial wall (which is the field coil set) has 

to be metallic so as to support and shape the magnetic nozzle 

field. However, that requirement need not conflict with the 

suppression of Hall currents because the conducting wall can 

be covered with a thin layer of insulating material. (Such a 

composite boundary condition then should be properly 

represented in numerical simulations.) Nevertheless, the 

presence of the insulating wall layer does not necessarily 

suppress the Hall current. A conducting plasma layer could 
NASA/TP—2009-213439 16 

form along the wall in a real experiment and thus could 

support a closure path for the Hall current. 

In the present subcase it is instructive to examine the Hall 

electric field that corresponds to complete suppression of the 

Hall current. If no quasi-radial currents can flow, Equation (41) indicates that the associated space-charge field (the 

Hall field) is given by Hall E ( )J = Ωσ θ . It is instructive to 

write this in reverse, utilizing Equation (38), as 

Hall

Hall

nqE J E

B θ

  σ = =     Ω (44) 

In the given circumstance of no Hall current, a Hall electric 

field builds up across the flux surfaces to just such a value 

that the azimuthal “E/B” (ratio of of electric to magnetic 

field strengths) drift velocity exactly satisfies the simple 

Ohm’s law. The effective electrical conductivity then is just 

the fundamental conductivity σ, which is the same as if there 

were no magnetic field. The Hall electric field is a spacecharge field associated with surface charges that are built up 

on insulating surfaces that stop the flow of the Hall current. 

The E/B drift velocity referred to above must be the azimuthal guiding-center drift velocity of the electrons. 

Although it is true that the ions have exactly the same E/B

guiding-center drift velocity, such an ion drift is not effective 

in canceling the electron current for the following reason: it 

has been assumed here at the outset that the ion fluid has no 

macroscopic rotational velocity; hence, within the present 

model the ion fluid is incapable of providing an azimuthal 

current to cancel that of the electrons. The physics that 

makes this consistent is that there are other ion drifts besides 

the E/B guiding center drift. For example, there is the ∇B

drift, and there is the diamagnetic drift resulting from a 

density gradient, which means that there is a lack of cancellation of neighboring gyro-orbits. 

In order that the ion fluid not rotate, which was the 

present assumption, it must be the case that the azimuthal 

E/B guiding center drift velocity of the ions is canceled by 

other effects, such as ion ∇B drifts or ion magnetization 

currents. It does not really matter what the other effects are. 

All that matters is the assumption that such individual particle mechanisms add up to constitute an ion fluid that does 

not rotate. It will be seen later that the assumption of no ionfluid rotation is indeed consistent with the companion 

assumption that the Hall current cannot flow. 

2.3.1.2 Hall current allowed (voltage shorted out).—

Suppose the Hall electric field is completely shorted out 

(EHall = 0), and the Hall current (the quasi-radial current 

JHall) is perfectly free to flow. Note that plasma rotation is still 

assumed to be absent. The assumption of no macroscopic 

mass rotation is not generally consistent with the presence of 

Hall current in the axially symmetric configuration of the 

magnetic nozzle, but could be made consistent in a simulation. Also, it could represent a physically realistic situation 

within a limited region upstream, before the new ion fluid 

has had a chance to be spun up by the JHall × B magnetic 

body force of the Hall current. 

A concrete example of shorting out the Hall electric field 

would be the use of a noninsulated metallic backplate in the 

breech of the nozzle. Conduction in the metallic backplate, 

for example a copper plate, is unaffected by the embedded 

magnetic nozzle field because of the very large collision 

frequencies (with phonons) of the metallic conduction electrons, which, however, also have a very high density that 

produces a high conductivity. If high-conductivity longitudinal electron currents (along the magnetic field lines) were 

to be allowed access to the backplate, then it would have the 

effect of setting all flux surfaces at practically the same 

potential, thus shorting out the radial space-charge field. 

Such a condition, which allows the radial Hall current to 

flow unimpeded, implies that either the uninsulated coaxial 

wall or an edge layer of conducting plasma is available at 

large radius to provide a longitudinal current return path to 

the backplate. If there were no such return path, then either 

net charge would be continually removed from the system or 

bipolar space-charge electric fields would continually build 

up, neither of which is tenable within the steady-state context. 

In the present subcase, in which the Hall electric field 

EHall is assumed to be completely shorted out, Equation (42) 

reduces to 

2 1+ Ω

σ

θ = − V B J x (45) 

The Hall effect, in conjunction with the assumed boundary 

conditions and in the absence of plasma rotation, has reduced 

the effective electrical conductivity by a huge factor; 

( ) 2 σeff = σ 1+ Ω . For the parameters of interest, this reduction factor is on the order of 105

. 

The physical basis for this reduced conductivity, termed 

“magnetoresistance,” has an origin that is somewhat complicated. In view of the present assumption that the ions have 

no fluid velocity in the azimuthal direction, the burden of 

carrying the azimuthal current falls on the electrons. Without 

the “EHall /B” drift velocity induced by the Hall field (as in 

the preceding subcase), an electron can only undergo a 

displacement at each collision by approximately a tiny gyroradius step across the magnetic field. This displacement is in 

the azimuthal direction, in response to the azimuthal applied 

electric field Eθ app = −VxB. It can be shown that the electron’s 

effective gyroradius must be small, on the order of Eθ app/Bωc e. 

This particular gyroradius is much smaller than the electron’s 

thermal gyroradius. Hence, the effective step size involved in 

the azimuthal electrical conductivity of electrons across the 

magnetic field is even smaller than the electron’s thermal 

gyroradius. In contrast, the null-magnetic-field conductivity is 

based upon a much larger step size; namely λe. 
NASA/TP—2009-213439 17 

When the Hall voltage is shorted out, thereby precluding 

the EHall/B drift, it is the very small step size in the azimuthal direction at a collision of an electron that proves to be 

responsible for the decreased electrical conductivity. This 

phenomenon is termed “magnetoresistance.” The details of 

the three-dimensional scattering of the cycloidal electron 

orbits underlying magnetoresistance lie beyond the purview 

of this report. 

The minimal intermixing of plasma and magnetic field, 

found earlier for the case of classical resistivity, was based 

upon the applicability of the simple Ohm’s law. In the 

present context, that result requires that no Hall current be 

allowed to flow. However, if the Hall electric field were 

even partially shorted out, Hall current would flow, and a 

concomitant drastic reduction in effective electrical conductivity would ensue, thereby enhancing the plasma-field 

intermixing. 

Clearly, it is important for the elucidation of plasma-field 

intermixing in the magnetic nozzle to ascertain the boundary 

conditions and Hall-return-current paths in the envisioned 

magnetic nozzle device or experiment, as well as in numerical simulations of plasma flow through magnetic nozzles. 

Attention also should be directed to whether plasma rotation 

is allowed in the simulations. Also, computer simulations are 

vulnerable to subtle numerical effects that impact the computed physics. It is conceivable that these might inadvertently allow such Hall currents to intrude. Thus, any observed 

deviation in the magnetic nozzle device or in its numerical 

simulation that allows even some Hall current may signify a 

partial shorting of the Hall voltage. The question should then 

be raised regarding the effective electrical conductivity that 

governs diffusive intermixing of plasma and magnetic field. 

It is instructive at this point to examine the Hall current 

with completely shorted Hall field. From Equations (41) and 

(42), with EHall = 0 (Hall field shorted out), and in the limit 

of large Ω, one finds with the help of Equation (38) that 

 

Hall e x J = n qV (46) 

Note that the velocity V represents the massive plasma 

velocity, not the electron velocity. Thus, the Hall current is 

carried across flux surfaces by ions. Electrons remain relatively attached to the flux surfaces in regard to their radial 

motion, in the limit of large Ω and concomitantly small 

electron collision frequency υe. Since the space-charge field 

is assumed to be globally shorted out, quasi-neutrality is not 

enforced in the conventional manner. Electrons vacate the 

local region at the same rate as the ions by flowing away 

along the longitudinal magnetic field. 

Note also that the above result for JHall can be written as 

follows: 

Hall ( ) 2 en q

B

= ××

B J VB (radial x-component) (47) 

This form of expression for the quasi-radial Hall current 

indicates that it is associated with a guiding-center drift 

velocity of the ions Eθ app/B in the applied electric field, 

Eθ app = V × B. A stationary magnetoplasma placed in an 

externally applied electric field would acquire a Hall current 

of this form when the Hall voltage is shorted out. 

Obviously, electrons connecting longitudinally along 

magnetic flux to a conducting backplate could serve to provide part of a current-return path for ions that connect transversely across flux surfaces to the conducting coaxial wall or 

to a conducting low-density plasma layer along the wall. If 

such a partial current path were fully completed along the 

wall, the space-charge field (Hall field) that otherwise 

enforces quasi-neutrality in the radial direction would be 

shorted out. Instead, quasi-neutrality would then be maintained by having the electrons and the ions vacate a given 

position at the same rate; however, they would flow out in 

their different directions, longitudinal versus radial. 

With regard to such a closure path for the Hall current, in 

terms of classical (Spitzer) resistivity, the longitudinal electrical conductivity in a plasma-wall layer would be rather 

insensitive to the density of plasma in that layer (Ref. 10). It 

is important to note that such a path-closure process with 

longitudinal electron flow is implicitly assumed in the above 

result for the ions as carriers of the radial Hall current. Otherwise, powerful radial space-charge electric fields would 

cause electrons to move across flux surfaces with the ions to 

preserve quasi-neutrality. Examples of such powerful spacecharge fields are given in a later discussion in the beginning 

of Section 3.0, “Resistivity From Gradient-Driven 

Microinstabilities.” 

As already mentioned, a highly conducting backplate 

would practically set all flux surfaces at the same potential, 

thus shorting out the Hall electric field. But even if the conducting backplate were rendered inaccessible to electrons by 

a coating of insulating material, an alternative Hall-current 

return path might become available, such as a layer of lowtemperature plasma next to the insulated backplate. A lowtemperature plasma-wall layer would raise the electron 

collision frequency up to near the electron cyclotron frequency. A 50 eV in hydrogen plasma dropping to 1 or 2 eV 

in the plasma layer at the backplate would accomplish similar results. The possibility then arises of electron conduction 

currents flowing radially outwards across flux surfaces that 

are intersecting the backplate. Although the electrical conductivity of such an electron-shorting path (~103

 mho/m) 

would be quite small in comparison to metallic conductivity 

(~107

 mho/m), it would still be many orders of magnitude 

larger than the conductivity of any insulator. Since the full 

effect on the conductivity with unimpeded Hall current is a 

huge effect, even an imperfect shorting path may make a 

meaningful difference. Moreover, ions within the plasmawall layer next to the backplate could also contribute to path 

closure by crossing the flux surfaces there, traveling radially 

inwards. 
NASA/TP—2009-213439 18 

It is important to remark that having longitudinal Hallcurrent closure along magnetic nozzle field lines necessitates 

allowing for azimuthal magnetic field components. Longitudinal currents in axially symmetric configurations generate 

azimuthal magnetic fields according to the integral form of 

Ampere’s law: ∇×B = μ0J. In Appendix D the subject is 

briefly addressed, while also allowing for rotation of plasma. 

It is shown there that the consequent azimuthal magnetic 

field component Bθ has the order-of-magnitude βi

B, where βi

is the ratio of ion thermal pressure to magnetic pressure of 

the nozzle magnetic field, B. 

A complete model of propellant flow in a nozzle magnetic 

field, which embodies longitudinal currents and azimuthal 

magnetic fields as well as plasma rotation and electron pressure, is very complicated. It would be accessible to simulations that include Hall effects and that carefully treat global 

boundary conditions. The model is not extended in such a 

manner here, although some discussion of the extended 

model is offered in Appendixes D and E. However, a partial 

extension of the present model that is tractable and instructive is introduced in Section 2.3.2.2, “Hall current allowed 

(voltage shorted out).” 

Thus, we shall complete this section on the Hall effect by 

including electron pressure and plasma rotation in the extension of Ohm’s law. In the axially symmetric configuration, it 

will be seen that Hall current and plasma rotation must occur 

together. The partially extended model presented below, 

however, does preclude an azimuthal magnetic field, Bθ. 

Nevertheless, it is found that allowing for the possibility of 

plasma rotation proves to have a profound effect upon the 

nature of the plasma-field mixing layer, when the Hall electric field is shorted out. 

2.3.2 Plasma Rotation and Electron Pressure 

In the above discussion, the azimuthal component of the 

plasma fluid momentum equation has been ignored. This 

restriction would apply to a simulation that had provision 

only for radial and axial components of propellant velocity. 

To be consistent, as shown below, such a code also should 

have hard-wired into it the condition that the Hall current 

automatically be forbidden. In the above discussion, 

however, rotation was automatically excluded but the Hall 

current was not automatically excluded. In a real magnetic 

nozzle experiment or device, the occurrence of propellant 

rotation and Hall current would be coupled. 

These remarks are illustrated by the azimuthal component 

of the plasma momentum equation for the case of an axially 

symmetric steady-state configuration. 

( )( ) θ θ ρ V ⋅∇V = J ×B (48) 

Upon working out the details, one finds 

( ) ( ) ( ) ( ) //

l Hall

V rlV rlV J B

rl rl θ θ

  ρ ρ   ⋅∇ = ∂ = −      

V (49) 

Here Vθ is the rotational velocity of plasma, V// is the 

velocity of plasma along the field line, l measures length 

along a field line, and the x-direction (normal to the flux 

surfaces) pertains to JHall. The magnetic field component Bθ

has been ignored. 

It is important to note that the radius variable r(l) is a 

function of l that characterizes the geometry of the magnetic 

nozzle field. Thus r(l) constitutes the distance from the axis 

of symmetry to a considered point a distance l along a given 

field line, as measured from the breech. Strictly speaking, the 

spatial derivative in Equation (49) is along the r,Z-projection 

of a propellant streamline, since the θ-coordinate is in the 

ignorable direction. In principle, the shape of the field line 

r(l) will be somewhat distorted from its vacuum-field shape 

by the plasma propellant and must be determined selfconsistently with the propellant flow, as is done in MHD 

simulations such as MACH2. However, the Hall effects were 

excluded from the MACH2 simulation in Reference 2. 

It is important to remark that if the Hall current density 

JHall is everywhere forbidden by external boundary conditions, then Equation (49) implies that there can be no rotation of plasma, provided that injected plasma is inserted with 

axial symmetry into the breech and has no net injected 

macroscopic rotation at any radius. Conversely, if it is 

known that there is no macroscopic rotation anywhere, then 

Equation (49) implies that there also can be no Hall current. 

Simulations that are without access to the azimuthal momentum equation of the propellant then lose that means of 

enforcing consistency between the presence of the Hall 

current and the presence of rotation. 

It is also important to note that even in the event of Hall 

currents and concomitant rotation of plasma, the rotational 

kinetic energy can be recovered downstream for conversion 

into axially directed thrust. This conversion can be effected 

in the diverging region of the nozzle where presumably 

JHall approaches 0. Reduction of temperature downstream 

increases υe relative to ωc e, making Hall effects less important downstream. From Equation (49) above, the rotational 

velocity then vanishes as [r(l)]−1

. The conversion of rotational kinetic energy into longitudinal kinetic energy is a 

consequence of conservation of total energy. This feature 

can be demonstrated for axially symmetric steady flow by 

means of the generalized Bernoulli equation including rotation, which describes energy balance along the streamline 

projections in the meridian plane. 

We now invoke Ohm’s law including rotation and electron pressure. Define an effective electric field Eeff such that 

eff

1 Pe nq

  =+×+ ∇    

E EVB (50) 

Here, E is the electric field in the lab frame. Multiplying 

Equation (37b) by σ and combining with Equation (38), the 

result is 
NASA/TP—2009-213439 19 

eff J + ΩJ ×b

ˆ = σE (51) 

The azimuthal and Hall components of Equation (51) are, 

respectively, 

θ −Ω Hall = σEeff θ J J (52a) 

and 

E x J J Ω θ + Hall = σ eff (52b) 

Now Eθ = 0 in an axially symmetric steady state in the lab 

frame, and there is likewise no azimuthal pressure gradient 

∇θPe = 0. Therefore, from Equation (50), 

E VB eff θ = − x (53a) 

Also, 

eff Hall

1 E E VB P x x e nq

θ

  = ++ ∂    

 (53b) 

2.3.2.1 No Hall current allowed.—In this case, JHall = 0, 

so that Equations (52a) and (53a) produce Jθ = − σVxB, 

which is again the simple form of Ohm’s law. Thus, the 

results obtained earlier for the plasma-field mixing layer width 

are again realized even when plasma rotation and electron 

pressure are not precluded from the model, as long as the 

Hall current is strictly forbidden by the global electrical boundary conditions. Moreover, when the Hall current is forbidden, so is plasma rotation, as discussed above. 

Electron pressure plays a new role, though, as can be seen 

from Equations (52b) and (53b), which now yield, with the 

help of Equation (38) 

 

Hall 1 x e

E

J nq P B B θ

    = +∂         (54) 

In this case, the electron-fluid pressure-gradient term 

B xPe ∂ −1 contributes to the azimuthal current density as 

well as the E/B drift. The second term of Equation (54) is 

partially caused by a lack of cancellation of azimuthal 

current elements from neighboring electron gyro-orbits, due 

to the spatial gradient of electron density in the x-direction 

(normal to flux surfaces). From the manner in which the 

fluid equations are derived from more detailed kinetic models, the electron pressure gradient in the fluid model also 

automatically incorporates other individual-particle contributions, namely, guiding center drifts. For brevity, the 

B xPe ∂ −1 term is henceforth referred to as “magnetization 

current.” 

Although the E/B guiding-center drift velocity of the ions 

equals that of the electrons, seemingly canceling that 

contribution to the electron current, it is also true—in this 

subcase—that there is no net rotation of the ion fluid. The 

“ion-fluid momentum” is the same as the “plasma-fluid 

momentum.” The ion fluid cannot produce any macroscopic 

contribution to azimuthal current in the absence of 

propellant-fluid rotation, as pointed out earlier. Thus, on the 

basis of the azimuthal momentum equation of the plasma 

when the Hall current is forbidden, the E/B drift velocity of 

the ions must be canceled by other individual-particle ion 

effects, such as other guiding-center drifts and magnetization 

currents. Therefore, the E/B drift velocity in Equation (54) 

must refer to the guiding-center drift velocity of the electrons 

only, it being understood that there is no net rotational contribution from the ions when JHall = 0. 

2.3.2.2 Hall current allowed (voltage shorted out).—In 

the extreme case in which EHall = 0, concomitant with the 

existence of current-return paths sufficient to provide a 

complete circuit for the Hall current, the solution for the 

azimuthal current density Jθ from Equations (52a) and (52b) 

reads as follows: Using Equations (53a) and (53b), one finds 

( ) 2

1

1 x xe J V B VB P

nq

θ θ

σ         = − +Ω + ∂      

+ Ω        

 (55) 

In the limit of large Ω, with help of Equation (38), this 

expression reduces to 

x 1 x e

nqV J nqV P B θ θ

    =− + + ∂       Ω   (56) 

It will be seen shortly that over most of the length of the 

nozzle, Vθ >> Vx in the plasma-field mixing layer between 

plasma and magnetic field. Moreover, Ω >> 1 there. Hence 

the azimuthal current carried by the plasma in the plasmafield mixing layer essentially contains two principal contributions, one from the ion-fluid rotation, and the other from 

the electron magnetization current. 

Next, we calculate the Hall current density for large Ω, 

from Equations (52b), (53b), and (56): 

Hall eff = 

1

x

x xe xe

x

J JE

nqV nqV P V B P B nq

nqV

θ

θ θ

−Ω +σ

  Ω     = −Ω − ∂ +σ + ∂            

=

 (57) 

The last step is obtained by invoking Equation (38). Thus, in 

the approximation Ω2

 >> 1, it would appear that ions still 

carry the Hall current across flux surfaces, when rotation and 

electron pressure are included in the model. However, this 
NASA/TP—2009-213439 20 

result is misleading because Equation (56) is not completely 

accurate, being only the leading approximation (for small 

Ω−1). The leading approximation to Jθ, in terms of Ω−1, is 

prone to error if Jθ is multiplied by large Ω, as in the above 

equation for JHall. The exact answer, using the exact Equation (55) instead of Equation (56), and again invoking Equations (38) and (53b), proves to be 

2

Hall 2 2

1

1 1 x x e J nqV nqV P B θ

  Ω Ω      = + +∂          +Ω +Ω       

 (58) 

Now finally taking the limit of large Ω, this reduces to 

Hall 2

111 1 x x e J nqV nqV P B θ

       =− + + ∂          Ω    Ω  

 (59) 

Thus, the quasi-radial Hall current is only partly carried 

directly by ions across flux surfaces. Because of subtle magnetic effects, it also contains contributions from the ion-fluid 

rotation and from the azimuthal electron magnetization 

current. This is true only in regions that the plasma has been 

able to reach. The final results will indicate that Vθ/Ω is 

indeed of the same order of magnitude as Vx. 

It will be seen that the ions do not travel very far radially 

from the core plasma during times of interest. Although the 

Hall current circuit could be completed near the core's edge, 

the presence of an inadvertent halo of low-density plasma 

could as well facilitate a more distant closure of the Hall 

path. Nevertheless, the inadvertent presence of a pervasive 

low-density halo plasma might allow longitudinal completion of the Hall current circuit out near the wall, and across a 

conducting backplate. 

Now safely invoking Equation (56) within this higher order 

calculation in Ω−1

, Equation (59) becomes, in leading order, 

Hall x

J

J nqV θ = +

Ω (60) 

This result also is in direct agreement with Equation (52a), 

whose meaning is now clarified. 

With the help of Equations (56) for Jθ and (60) for JHall, 

both transverse components of the plasma momentum equation can be utilized to find Vθ and Vx. An inversion in the 

procedure proves convenient relative to the simple case of a 

resistive plasma-field mixing layer. Now the quasi-radial 

momentum equation determines the rotational velocity, 

whereas the azimuthal momentum equation determines the 

quasi-radial velocity. These results are then used to examine 

the evolution of the plasma-field interface layer thickness.

 

The qualitative conclusion follows. If the Hall voltage is 

shorted to allow Hall current in the presence of selfconsistent rotation and electron pressure, the plasma-field 

mixing layer thickness between propellant and magnetic 

field becomes very different from that obtained using the 

simple resistive Ohm’s law. 

To carry out this calculation, we first utilize the plasma 

momentum equation in the direction normal to the flux 

surfaces, the x-direction (as usual, it is assumed that the 

transverse pressure gradient dominates the transverse inertial 

effects). This equation reads 

xP J B nqV B P tot θ θ x e ∂ = ≈ +∂ (61) 

In the second part of Equation (61), we made use of Equation (56), wherein the small Vx/Ω term was neglected in 

comparison to Vθ. Since the total plasma pressure is Ptot = Pi

+ Pe, the second part of Equation (61) is equivalent to 

1

x i P nqV B θ

 ∂ ≈     (62) 

Therefore, the macroscopic ion-fluid rotation is actually 

seen to be effected by the ion magnetization current, to 

leading order in Ω−1

. 

Next, Equations (61) and (62) are input to Equation (49), 

the azimuthal component of the plasma momentum equation. 

This is done as follows: Integrate Equation (49) along a 

given representative field line, from the breech region over 

an indefinite length L, which is a significant fraction of the 

length of the nozzle. The present model is valid while Hall 

effects are dominant, which means that the temperature is yet 

large enough that ωc e > υe. The result is 

( ) Hall

0 //

d

L

L

rJ B rV l

V θ

  = −  

ρ    (63) 

Quantitative numerical results would require carrying out the 

indicated integration along a selection of representative field 

lines near and in the plasma-field mixing layer, with their 

self-consistent field-line shapes characterized via the function r(l). Instead, we make the following approximation, in 

the spirit of obtaining an order-of-magnitude scaling result 

for the plasma-field mixing layer width: The integral is 

approximated as the average value of the integrand, say, near 

the throat, multiplied by the distance L. Then, suppressing 

moderate numerical distinctions between the values of quantities at l = L and their representative average values, we 

multiply Equation (63) by the quasi-neutral ion number 

density multiplied by charge, ni

q, to obtain 
NASA/TP—2009-213439 21 

Hall

//

i

nqB n qV L J V θ

    ≈ −       ρ    

 (64) 

Here, for example, the numerical ratio of the average 

value of r along a field line to r(L) has been suppressed. 

Note that the composite ratio i n qB ρ is just the ion 

cyclotron frequency ωc i = i qB m . Accordingly, we define a 

large parameter Λ >> 1 by 

//

ciL

V

ω

Λ ≡ (65) 

Now V// is closely coupled by the Bernoulli equation to 

( ) 1 2 TT m ie i   +   γ , the speed of sound, which also is near 

the ion thermal velocity. It follows that the parameter Λ has 

the order of magnitude of the ratio of nozzle length to ion 

gyroradius. To within a trivial numerical factor, 

ai

L Λ ≈ (66) 

Therefore, Equation (64) is rewritten, using Equations (60) 

and (65), as 

Hall x

J nqV J nqV θ

θ

  ≈ −Λ = −Λ +     Ω (67) 

Here, it is important to note, in view of Equation (62), that 

the Hall current JHall exists in virtue of the ion pressure 

gradient, which creates the ion-fluid rotation. This is provided that the Hall electric field EHall is shorted out. Recall 

that Ω = ωc e/υe, where ωc e is the electron gyrofrequency, and 

υe is the collision frequency of an average electron with ions. 

The quasi-radial velocity Vx is solved as follows: Combining the left-hand side of Equation (67) with Equation (62), 

and substituting Equation (61) for Jθ on the right-hand side, 

the resulting equation for Vx can be written as 

tot

1 1

x xi x nqV P P B B

    ≈− ∂ + ∂         Λ Ω (68) 

In the parameter regime of interest, Λ and Ω are large and of 

the same order, and the two pressure gradients also are of the 

same order of magnitude. It is implicitly assumed that electrons and ions enter the breech with practically the same 

temperature, and also the same density profiles because of 

the quasi-neutrality condition. Therefore, this equation 

essentially constitutes a verification that Vx is indeed very 

small compared with Vθ. (See Eq. (62).) Now referring back 

to Equation (67), note that all terms therein are of the same 

order of magnitude when Λ is multiplied through on the 

right-hand side. 

Although not actually necessary, the following discussion 

is simplified by setting the electron pressure approximately 

equal to the ion pressure, within the plasma and within the 

plasma-field mixing layer: Pe ≈ Pi

. Then Equation (68) reads 





 



  ∂



 



 



 



 



 





Ω

Λ

+

Λ x ≈ − xPi B

neV 2 1 1 1 (69) 

It is understood that Equation (69) is not necessarily an 

accurate quantitative statement, but rather represents a 

qualitative scaling result. In that sense, all quantities therein 

are meant to be representative average values when following the flow of a piece of propellant along a given magnetic 

field line. 

Equation (69) can be utilized to estimate the plasma-field 

mixing layer width δ at the plasma-field interface. The basic 

modeling approximation, which also was employed earlier in 

the case of resistive diffusion, is to associate Vx with dδ/dt, 

the rate of increase of the plasma-field mixing layer width as 

the longitudinal flow is followed downstream: 

t

Vx d

dδ ≈ (70) 

The other part of the basic idea, also employed earlier, is to 

represent the ion pressure gradient within the plasma-field 

mixing layer by its value within the core plasma divided by 

the width of the plasma-field mixing layer: 

δ

∂ ≈ − i x i

P P (71) 

Then, Equation (69) can be expressed as follows, where Ti

refers to the ion temperature and where 2

th 2 V Tm i ii = is the 

square of the ion thermal velocity: 

( ) 2 2

th d 2 1 1 12 12

d 2

i i

ci

nT V

t nqB

δ       Λ Λ

=+ =+       Λ Ω Λ Ωω        

 (72) 

It can be shown that the right-most factor ci i V ω 2

th is 

insensitive to position along the nozzle. 

One now integrates Equation (72) over time, while 

following the propellant downstream, assuming that the 

initial plasma-field mixing layer width of attached plasma in 

the breech is negligibly small. On the right-hand side, one 
NASA/TP—2009-213439 22 

simply takes average or representative values during transit 

of the length of the nozzle from breech to throat Lb t and then 

finds

t

V

ci

i





















ω 



 



 



 





Ω

Λ

+

Λ

δ ≈

2

2 th 1 2 1 (73) 

A rigorous approach requires integration over Lb t: 

b t d dt = V// d dL , noting that Λ depends on Lb t V// and 

that V// cancels in Equation (72). When Λ < Ω, δ ai 

increases slowly with L, as [ ] ( ) ( ) 1 2 2 ln b t b b i L L + δ a , 

where Lb is a small starting length in the breech, and δb is an 

initial layer thickness derived later as δb ≈ 2ai

. 

For the transit time t, we invoke L V// tb t = b t , where 

V// is a flow velocity in transit, and also note Equation (65) 

for Λ. Then Equation (73) becomes simply 

 

2

2

2

2 th 1 2 1 2 i

ci

i a

V





 



 



 





Ω

Λ = +

ω





 



 



 





Ω

Λ δ ≈ + (74) 

It is therefore seen from Equation (74) that when Λ < Ω, the 

plasma-field mixing layer width is on the order of just a few 

ion gyroradii. This is a very favorable result in the sense that 

the width of the plasma-field mixing layer appears to be independent of the electron collision frequency, if the condition 

Λ < Ω is satisfied. On the other hand, when the opposite condition is satisfied, namely when Λ >> Ω, then a fundamental 

change occurs in the scaling of the boundary-layer width. 

To examine this second possibility, note that 2(Λ/Ω) has 

the following order of magnitude: 

i

e

e

b t

m

L m

















λ  ≈



 





Ω

Λ 2 2 (75) 

where λe is recalled as the mean free path of electrons due to 

collisions (possibly anomalous) with ions. Also, me/mi

 is the 

electron-to-ion mass ratio. Thus, from Equation (74), if there 

are fewer electron mean free paths along Lb t than mi me , 

then the boundary-layer width at the plasma edge is just one 

ion gyroradius. However, if the characteristic length of the 

nozzle is substantially greater than that relative to λe, then 

from Equation (74) with Λ >> Ω, 

i

e

e

b t

i

m

L m

a 















λ δ ≈ 2 2 2 (76) 

It is satisfying to note that Equation (76) exactly reproduces 

Equation (28) for the case of simple resistive diffusion of 

plasma into field, without Hall current and without plasma 

rotation. It also is of interest to record the following equivalent form of Equation (76), representing the case of simple 

resistive diffusion: 

Ω

δ ≈ 2 2aiLb t (77) 

which is to be compared with δ

2 ≈ 2

i a when the nozzle’s 

length is subcritical. Here, Ω = ωc e/υe, where ωc e = qB/me

is the electron gyrofrequency, and υe is the electron collision 

frequency. 

To summarize, in a model that incorporates rotation, electron pressure, and Hall current unimpeded by a Hall voltage, 

we found a transition in the behavior of the boundary-layer 

width from the ion gyroradius to the width associated with 

simple resistive diffusion. The transition is realized provided 

that the nozzle is long enough. The critical length Lcrit is on 

the order of λe mi me . In the case of classical resistivity, it 

can be shown that this quantity is essentially the same as λi

mi me , which relates to λi

, the mean free path for ion-ion 

coulomb collisions. The physical reason is that the plasma 

has spun up enough along its length in the edge layer so that 

the rotational velocity in the edge layer represents a motional 

electric field in the rotating frame, which is strong enough to 

suppress the outward quasi-radial flow of Hall current (simple resistive Ohm’s law). To see this, note that the outward 

flow of Hall current produces a JHall×B force (for a magnetic 

field that points downstream), which is counterclockwise 

(looking downstream). Therefore, the Vrot×B component of 

the electric field in the rotating frame produces a quasi-radial 

component of the electric field that points inward, towards 

the plasma. If Vrot is large enough, this effect can suppress 

Hall current.

2.3.3 Diffusion of Field Into Plasma With Hall Effect 

This section on the basic physics shall conclude with an 

account of external magnetic field diffusion into the interior 

plasma, when unimpeded Hall current and consequent 

plasma rotation are present, in a straight cylindrical geometry model of the core propellant. This complements the 

preceding treatment of protrusion of plasma into magnetic 

field, with Hall effect, as discussed in Section 2.3.2.2, “Hall 

current allowed (voltage shorted out).” Also, this treatment 

will thus generalize the earlier one that dealt with a planar 

interface, in Section 2.2.3, “Diffusion of Confining Magnetic 

Field.” Such a magnetic-field diffusion process into the core 

plasma occurs at the same time as that of diffusion of the 

plasma into the external magnetic field. When the Hall current is totally suppressed by boundary conditions, the simple 
NASA/TP—2009-213439 23 

Ohm’s law governs plasma-field intermixing, and both types 

of resistive diffusion act simultaneously to broaden the 

plasma-field mixing layer. 

With Hall current (and rotation) present, a more complex 

version of Ohm’s law is in force, together with the presence 

of an azimuthal magnetic field. This azimuthal field surrounds the longitudinal return current that replenishes the 

quasi-radial Hall current in a steady state. Under such circumstances, the diffusion of the nozzle magnetic field into 

plasma then becomes rather involved. Nevertheless, it is 

possible to develop a model of this process—diffusion of 

field into plasma—to the point that some of the principal 

features can be identified. In so doing, we shall assume axial 

symmetry and a steady state. Appendix D shows that the 

azimuthal field should be relatively small compared with the 

other components of the field. 

The development begins with Faraday’s law of induction 

in the lab frame: 

∇ ×E = −∂tB (78) 

According to the complete Ohm’s law, this can be written as 

1 1 0 Pe t

nq nq

     ∇× η + × − ∇× × + ∇ = −∂ =          

J JB VB B

 (79) 

The right-hand side is zero in a steady state in the lab frame 

of reference. 

For the J × B term in Equation (79), we use the momentum equation of the plasma. It is assumed that the electron 

and ion temperatures are uniform. Then, the plasma pressure 

gradient from the momentum equation is parallel to the 

plasma density gradient, so that the pressure gradient term 

does not survive the curl operator. Hence, only the inertial 

term appears from the momentum equation in connection 

with the J × B term. Similarly, the electron pressure gradient 

is parallel to the plasma density gradient. Then the electron 

pressure gradient term disappears from Equation (79). 

We shall consider here only the case of classical Spitzer 

resistivity, which is insensitive to the electron number density and depends essentially only on the electron temperature, assumed uniform. Whether the final conclusions still 

would be valid for any anomalous (microturbulent) 

resistivity ηa is an open question. In the case of resistivity 

arising from the LHD instability, an approximate scaling 

argument will be made later. The result is that after allowing 

for adiabatic longitudinal flow and for radial pressure 

balance in the plasma-field interface layer, ηa is seen to 

depend only very weakly on density alone, as n

−1/6. This 

circumstance suggests that the arguments to be given here 

for classical resistivity would still hold for the LHD 

resistivity. This suggestion cannot be considered a rigorous 

conclusion, however, because ηa actually proves to be 

profile dependent through factors B/n and (ai

/xn)

2

, where ai

is the local thermal ion gyroradius and xn is the radial gradient length at the plasma-field interface. 

Returning to the case of classical resistivity, the uniformity 

of electrical conductivity follows from the assumption of 

uniform electron temperature. Then, it is expeditious to multiply Equation (79) by σμ0 = 1/D, where D is the resistive 

diffusivity. Also note the identity σ/nq = Ω/B, where Ω is the 

Hall parameter Ω=ω υ ce e , and B therein is the magnitude 

of the magnetic field (neglecting the azimuthal field component). Note that the electron gyrofrequency ωc e ∝ B, whereas 

the electron plasma frequency ωp e ∝ n . Also, note that the 

electron collision frequency is υe ∝ 3 2 Te n . 

We also invoke a vector identity for the ∇×(V×B) term 

and recall that ∇ ⋅ B = 0. Then, upon multiplying through by 

σμ0, Equation (79) reads 

( ) 0 1

0 0

− − ∇⋅ + ⋅∇ − ⋅∇ =





 



 μ ρ ⋅∇ 

 



 Ω ∇× μ +

B V B V V B

J V V

D

B (80) 

We shall preclude radial velocity from the plasma model, 

so that Vr = 0. The inward-radial velocity associated with 

resistive dissipation (by the longitudinal return current) of 

magnetic energy in the azimuthal magnetic field is estimated 

in Appendix E, and is found to be very small. Then, a divergenceless flow condition is satisfied by uniform axially 

directed flow of plasma, Vz, that is assumed to be constant in 

time. Thus, for the purpose of examining penetration of 

magnetic fields into the plasma core, the plasma core is here 

modeled as a straight cylinder moving rigidly downstream 

with a constant axial velocity. The possibility of rotation is 

allowed, however, so the inertial term does not vanish, even 

if the density is uniform. 

In this situation, the plasma density still is allowed to have 

a radial gradient, which it must in any case at the boundary 

of the core plasma. The plasma continuity equation in a 

steady state reads ∇ ⋅ (ρV) = 0 = ρ∇ ⋅ V + V ⋅∇ρ . Because of 

the assumption about the plasma velocity, and the uniformity 

of the plasma density along the axial direction, each of the 

two terms on the right-hand side of the continuity equation 

vanishes, yet the plasma mass density ρ can have a nonuniform radial profile. 

It is important to note that in this case of uniform temperature and classical resistivity, the combination (Ω/B)ρ 

depends only on the electron temperature and hence is uniform; that is, Ω=ω υ ce e and υe ~ 3 2 n Te . This feature is 

used in carrying out the details in Equation (80). 
NASA/TP—2009-213439 24 

In the term ( ) − B∇ ⋅ V + B ⋅∇V − V ⋅∇B of Equation (80), 

those terms that contain both Bθ and Vθ cancel one another 

because the configuration is axially symmetric. It therefore is 

expedient to define projections of the velocity and magnetic 

field vectors in the r,Z-plane. Thus, the projection of the 

flow velocity vector is Vproj ≡ (V⋅Z) zˆ = (0, 0, VZ). 

One might believe that we could also assume that Br = 0; 

that is, that there should be no radial magnetic field component in a straight cylindrical configuration. However, this is 

not the case. In the lab frame, one has a steady-state 

boundary-value situation, in which the nozzle magnetic field 

generally obeys restrictions in or near the breech region of 

the nozzle. For example, if the backplate is highly conducting, then the nozzle field lines are frozen into the backplate. 

This constraint implies that when the BZ-component tries to 

diffuse into the resistive core plasma, the nozzle field lines 

must be stretched inwards and thereby necessarily must 

produce a Br-component. This would happen even if the 

backplate were not a conductor, for the nozzle field lines 

also are somewhat constrained by the very conductive rings 

that generate the nozzle field itself. The mathematical relation between axial and radial components of magnetic field 

follows from ∇ ⋅ B = 0. It reads r r zBz r ∂ rB = −∂ − ( ) 1 . 

Therefore, we must retain Br in principle, and the 

magnetic field projection vector then becomes Bproj = (Br, 0, 

BZ). Then the term ( ) − B∇ ⋅ V + B ⋅∇V − V ⋅∇B of Equation (80) becomes 

( )

( )( ) ( )

( )( ) ( )

proj proj proj proj proj proj proj proj

ˆ

ˆ ˆ ˆ

ZZ Z r rr ZZ r ZZ

Z Zr rr Z Z r Z Z Z ZZ

V

V B B BV r

V

V B B B V BV B V B r

θ θθ θ

θ

θθ θ

θ

θ θ

⋅∇ − ⋅∇ + ⋅∇ + ⋅∇ − ⋅∇ − ⋅∇

        = − ∂ + ++ ∂+ ∂ − ∂ − ∂              

        = − ∂ + ∂+ ∂ − − ∂ + − ∂              

=

B V V B B V B VV B V B

0 BB 0 V r B

r θ Z

( ) ( ) ˆ ˆ ˆ Z Zr r r Z Z Z Z Z ZZ

V

V B Br B V V B V B r

θ

θ θ

          −∂ + ∂ + ∂ − ∂ +−∂            

r θ Z

 (81) 

Here, the obvious unit vectors for the cylindrical coordinate 

system have been introduced. The relation ∂θ rˆ = ˆ

θ was 

used. Also, it is worth noting that the operator VZ ∂ Z would 

play the role of a time derivative in the moving frame. 

The next calculation is to write out the fluid-inertial term 

in Equation (80). The occurrence of the inertial term is a 

direct consequence of the Hall term in Ohm’s law. In evaluating the inertial term, the following vector identity is 

used: 

 

= V ⋅∇V + V ×∇× V 















∇

2

2 V (82) 

The gradient term on the left-hand side does not survive the 

curl operator in Equation (80), when the other factors within 

that term are uniform. Therefore, effectively, 

V ⋅∇V = −V ×π (83) 

where π = ∇ × V is the vorticity. Operating on the inertial 

term with the curl in Equation (80), recalling that (Ω/B)ρ is 

uniform and using Equation (83), yields 

( )( )

Z Z

V V

r

θ

θ θ

∇× ⋅∇ =− ∇⋅ − ∇⋅ + ⋅∇ − ⋅∇

    =− ⋅∇ − ∂ − ∂        

V V V V VV

V

ππ π π

π ππ

 (84) 

Now, the cylindrical components of vorticity (πr, πθ, πZ) 

are as follows, when noting V = (0, Vθ, VZ), with uniform VZ: 

( )

1 , , ()

1 , 0, ( ) , ,

Z rZ r

Z r rZ

V V rV r

V rV r

θ θ

θ θθ

    = −∂ − ∂ ∂        

    = −∂ ∂ = π π π        

π

 (85) 

Using this expression for the vorticity in the previous equation and working out the details, one finds 
NASA/TP—2009-213439 25 

( )( )

( )

ˆ

ˆ ˆ

ZZr

r r ZZ ZZZ

V

V r VV r

θ

θ

∇× ⋅∇ = ∂ π

    − π ∂ +π ∂ + ∂ π        

VV r

θ Z (86) 

This result is entirely analogous to Equation (81), except that 

before in Equation (81), Bθ ≠ 0, whereas now in Equations 

(85) and (86), πθ = 0. 

Finally, we display the first term of Equation (80), using 

μ0J = ∇×B. Assuming the usual axial symmetry, and in 

terms of cylindrical coordinates, one finds 

( )

( )

2 2

0

1

Zr Z rZ Zr r

r rr

B BB

rB r

  ∇ × μ = −∂ + ∂ ∂ = −∂  

    −∂ ∂        

J

(87a) 

( ) ( ) 2

0

1

Zr r B rB r θ θ θ

           ∇× μ = − ∂ +∂ ∂            

J

 (87b) 

( ) 0 ( ) 1

r Zr rZ Z

rB B r

    ∇× μ = ∂ ∂ −∂           J

 (87c) 

Equations (81), (86), and (87) are used in Equation (80). To 

simplify the presentation, long-thin ordering for the nozzlemagnetic field geometry is invoked, namely 

Br << BZ (88a) 

∂Z << ∂r (88b)

 

~ r r

Z Z

B

B

δ

δ (88c) 

Here, δr << δZ and symbolize gradient scale lengths in the 

r- and Z-directions, respectively, for the special dependence 

of the magnetic field. The third of these conditions is 

required by the restriction that the magnetic field have no 

divergence. Using Equations (81), (86), and (87), the following three component equations are then obtained from 

Equation (80): 

( )

0

1

1 0

r rr

ZZ r r

rB r

V B

B D

    −∂ ∂        

        Ω + ∂     μ ρ π+ =          

 (89a) 

( )

0

1

1

0

r r

r r zz

r r ZZ ZZ

rB r

V

r V

B r

V

Br B V V B

D r

θ

θ

θ

θ

θ θ

    −∂ ∂        

    Ω     − μ ρ π ∂ +π ∂                

    − ∂ + ∂ −∂        

=

 (89b) 

( )

0

1

1 0

r rZ

ZZ Z Z

r B r

V B

B D

  − ∂∂    

        Ω + ∂     μ ρ π+ =          

 (89c) 

These equations, which describe penetration of fields into 

core plasma, are evidently very complicated. For example, 

the azimuthal component equation for Bθ depends on both Br

and BZ as obtained from the other two component equations. 

Moreover, all three component equations depend on the 

rotational velocity Vθ, as calculated from a simultaneous 

solution of plasma penetrating into field (why computer 

simulations such as MACH2, extended to include Hall current and rotation, are needed). Nevertheless, it proves possible to obtain some information regarding penetration of 

fields into core plasma, including cylindrical geometry, Hall 

current, return current, and plasma rotation. 

The first thing that can be done is to find out what this set 

of equations reduces to when the Hall current is absent. In 

that case, there is no longitudinal return current; hence 

Bθ = 0. Moreover, without Hall current, there is no mechanism to drive up the fluid rotational velocity of the plasma; 

hence Vθ = 0. Then the vorticity also vanishes; thus π = 0

(see Eq. (85)). Under these restrictions, the above three 

equations reduce to the following: 

( ) 1 Z

r r r Zr

V rB B

r D

      ∂ ∂ =∂             (90a) 

0 0 = (90b) 

( ) 1 Z

r rZ ZZ

V rB B

r D

    ∂∂ = ∂         (90c) 

Clearly, these reduced equations for steady-state disposition 

of the r- and Z-components of magnetic field exhibit the 

character of resistive diffusion of the magnetic field, which 

was the subject of a previous section based upon the simple 
NASA/TP—2009-213439 26 

Ohm’s law. When the boundary-layer width is small compared to the radius of core plasma, the relative variation of 

the r-coordinate within the plasma-field mixing layer can be 

neglected, and the equations reduce to a description of diffusion in planar geometry. The only essential difference from 

the previous section just referred to is that now Equation (90) 

with respect to the lab frame, is not the moving frame. Thus, 

the operator VZ ∂ Z plays the role of a time derivative in a 

frame of reference moving with plasma propellant. 

In the lab frame of reference, with ancillary boundary 

conditions in force at the breech of the nozzle, inward diffusion of the BZ component can occur only with the simultaneous appearance and inward diffusion of a Br component. 

This must be so whether or not the Hall effect is present. The 

fundamental requirement that the magnetic field be divergence free, ∇⋅ B = 0, forces the appearance of Br with a nonuniform radial profile. This must happen whenever the term 

V B Z Z Z ∂ is significant. Moreover, although the present 

discussion has been cast in terms of a straight cylindrical 

plasma core, its conclusions, with allowance for obvious 

geometric modifications, have meaning as well for a 

converging-diverging nozzle-shaped geometry. 

In Appendix C, a similarity solution is derived for BZ in 

Equation (90) above, assuming a thin plasma-field mixing 

layer. The solution reads 

pe 1 erf

2 Z

X B B uτ

    = −            

 (91) 

where Bpe is the assumed-uniform value of BZ at the plasma 

edge, X is the inward radial distance from the plasma edge, 

and uτ = ZD/VZ is a time-like similarity variable. In a more 

realistic scenario, the edge field would be nonuniform along 

Z, since the starting flux between the plasma and the wall 

would be preserved because that flux spread out downstream 

during the inward diffusion of the magnetic field. However, 

such a scenario is not accessible to a similarity solution. We 

ask for the value of X where the field is reduced to, say, onethird of its edge value. For the value of τ, a characteristic 

length Z = 1 m, D = 3 m2

/s, and Vz = 105 m/s are input, in 

accordance with Table I for parameters of hydrogen propellant. The value of X is then chosen so that the value of the 

error function is 2/3. The result is that the characteristic 

length of protrusion of magnetic field into the plasma—

considered as a simple resistive medium—is X = 0.65 cm, in 

excellent agreement with the earlier calculation based upon a 

different approach. 

We return now to the opposite case, where the Hall current, its return current, and plasma rotation are present. A 

complete discussion is beyond the purview of this report. It 

is sufficient here to just focus upon Equation (89c), which 

describes the steady-state disposition of the longitudinal 

magnetic field relative to core plasma. For convenience, that 

equation is reproduced here: 

( )

0

1

1 0

r rZ

ZZ Z Z

r B r

V B

B D

  − ∂∂    

        Ω + ∂     μ ρ π+ =          

 (89c) 

 

Let us compare the first term with the second term, the Hall 

term. Equation (62) is used in connection with the vorticity 

πz. That equation states that the ion rotational current 

density is given by the ion magnetization current density; 

that is, 1

r i nqV B P − θ ≈ ∂ . It is not significantly material 

whether B is regarded as BZ or as the total magnetic field 

including Bθ. Then one can easily estimate, to order of magnitude, the ratio of the second term of Equation (89c) to the 

first term. The ratio proves to be βi

(λe/Lπ) mi me . Here, 

V VL ZZ i ~ th π∂ , where Vth i is a representative ion thermal 

velocity and Lπ is a characteristic axial length 

related to the development of vorticity in the plasma-field 

mixing layer. For βi

 on the order of 1, when the quantity 

Lπ/λe me mi exceeds 1, the Hall term is unimportant and 

consequently ordinary resistive diffusion of field into plasma 

is obtained. This condition essentially agrees with that 

needed for resistive diffusion of plasma into field in the 

presence of Hall current and rotation, as described in connection with Equations (74) to (76) (see Lcrit discussion in Section 2.3.2.2, “Hall current allowed (voltage shorted out)”. 

Therefore, 

crit

i e e

m

L L

m

π ≈λ ≈ (92) 

In the opposite extreme, if the Hall term were very large, 

one might expect that the radial gradients of BZ would have 

to become very large in the first term of Equation (89c) to 

keep that equation in balance. The “sharp boundary” picture 

emerging from such a scenario implies that BZ has not diffused very much into the core plasma. 

It is instructive to take a somewhat different approach to 

the issue of resistive diffusion of field into plasma as regards 

the alteration of this process by the Hall effect and rotation. 

Note that 

0 mi

B qD

  Ωμ ρ     =      

 (93) 
NASA/TP—2009-213439 27 

Here, as usual, mi is the mass of an ion, q is the magnitude of 

the electron’s charge, and D is the resistive diffusivity. In 

this section, we are assuming singly charged ions. Then 

Equation (89c) can be rewritten as 

 

( ) 1 Z i Z

r rZ Z Z

V m

rB B

r Dq

      π

 ∂∂ = ∂ +          

 (94) 

Recall Equation (85), in particular that the longitudinal component of vorticity πZ is related to the ion-fluid rotational 

velocity by 1 ( ) Z r r rV − π= ∂ θ . Utilizing again the ion magnetization current density B r Pi ∂ −1 to represent the ion-fluid 

azimuthal current density nqi

Vθ, it can then be qualitatively 

estimated that the vorticity term (mi

/qi)πZ in Equation (94), 

can become significant in comparison to the BZ term when the 

plasma-field mixing layer width (for diffusion of plasma into 

field) is small on the order of an ion gyroradius, ai

. 

A more quantitative estimate proceeds as follows. Factoring out mi

/qi, one has in Equation (94) a comparison of the 

ion gyrofrequency, ωc i with the vorticity component πZ. 

Then for a radially uniform ion temperature, a radially uniform magnetic field, and invoking the magnetization current 

density, one can then estimate the vorticity component for a 

given density profile. The simplest case is a linearly decaying density profile of the form n(x) = ncore[1 − x/δ], where x

is the radial distance outwards from the nominal plasma edge 

and ncore is the number density in the core propellant. Then 

one finds 

2 2

core

1

1 1

2

Z x x xi

i i x x ci

nqV P

nq nqB

T an

n

qB n n

θ      π ≈∂ =∂ ∂          

         = ∂ ∂ =− ω                   δ

 (95) 

Thus, the vorticity πZ is comparable to or larger than the ion 

gyrofrequency ωc i when the plasma-into-field layer width δ

is small on the order of the ion gyroradius ai

 and particularly 

in the low-density edge of the layer. This admittedly 

non-self-consistent argument indicates that when the plasmainto-field layer width δ is on the order of ai

, then also under 

the same condition, a large Hall effect manifested as ionfluid rotation acts to dominate resistive penetration of field 

into plasma. 

Assuming that the Hall-vorticity term does indeed dominate the right-hand side of Equation (94), one can then proceed to estimate the field-into-plasma penetration depth δB

as follows. According to the remarks in the previous paragraph |πZ| ≈ |CI|ωc i, where |CI| is a number exceeding 1 (see 

Eq. (95)). Then, equating the left side of Equation (94) to the 

Hall term on the right side one has the following: 

 

π

=

δ DL

CIVthi

B

2

1 (96a) 

2

2

I th I th

2

th

I th th

1 e B

i pe i

e e e

i pi e i

LD L c

CV C V

m c V L

Cm V V

π π

π

  υ δ≈ =     ω 

  υ =     ω 

 (96b) 

Here, the expression (1/σμ0) is inserted in place of D and set 

in terms of the electron plasma frequency, 

2

pe e 0 ω= ε nq m . Also, the corresponding ion plasma 

frequency ωp i is introduced. Then the relation ε0μ0 = 1/c

2

 is 

invoked, where c is the speed of light in mks units. Finally, 

the electron thermal velocity Vth e is also introduced. For 

simplicity, electron and ion temperatures are assumed equal. 

Recalling that the electron collision mean free path is 

given by λe = Vth e/υe, the above equation for the square of 

the field-penetration depth becomes the following: 

2

2

I

1 e B

i pi e

m L c

C m

π  

δ ≈     ω λ  

 (96c) 

In the present example of a dominant Hall effect, the product 

(Lπ/λe) me mi is smaller than unity, in contrast to the 

situation with resistive diffusion. Hence, the field protrusion 

depth is δB < (c/ωp i), which is known as the “ion inertia 

length.” Since ai

 = (c/ωp i) βi , the ion inertia length is on the 

order of the ion gyroradius whenever βi is on the order of 1. 

We conclude that the field protrusion depth into the 

plasma is on the order of the ion inertia length (the ion gyroradius), provided that the characteristic axial length does not 

exceed a certain critical value. This is essentially the same 

critical length found for the Hall effect to dominate resistive 

diffusion of plasma into field. It is important to emphasize 

that the Hall effect and plasma rotation had to be included 

together in order to reach Equation (96c). 

2.4 Summary 

 Since Section 2.0 is lengthy, we pause here for an interim 

summary and discussion of its contents and a few recommendations for future work. 
NASA/TP—2009-213439 28 

First, a simple model of quasi-one-dimensional nozzle 

flow was presented based upon the Bernoulli equation. This 

well-known model affords useful estimates of propellant 

flow speeds and other quantities as related to the injected 

propellant properties; for example, the propellant speed was 

derived at the nozzle throat and at the exit plane. 

Then, we indicated how the longitudinal flow becomes 

distorted by the presence of curved magnetic field lines and 

edge pressure gradients when the plasma propellant is not a 

perfect conductor. Using the simplest form of Ohm’s law, 

we displayed the eddy current braking of propellant flow 

across flux surfaces. For the tabulated parameters of interest, 

which include an assumption of classical resistivity, we 

showed that eddy current braking in the region upstream of 

the nozzle’s throat severely limits the growth of the resistive 

boundary layer of edge plasma, as generated by pressuregradient-driven plasma diffusion into the confining magnetic 

nozzle field. The growth of this plasma-field mixing layer is 

relevant to the amount of attached plasma that evolves during 

transit of the nozzle. A more detailed estimate of the layer 

thickness is presented at the end of this summary section. 

The implicit assumption in our treatment was that, within 

the resistive MHD model, there has occurred no sudden 

initial penetration of magnetic field and plasma. Such behavior might be caused by dynamical resistive MHD effects 

induced by rapid spatial convergence of field lines downstream as induced by upstream plasma impact. Early lateral 

displacement of field lines by the highly conducting injected 

plasma, together with the unavoidable crowding of flux into 

the nozzle’s throat, could enhance the spatial rate of convergence of flux beyond the breech of the nozzle, instigating 

strong curvature and thereby spoiling the gentle convergence 

of propellant flow. Coil discreteness can exacerbate that 

effect. Dynamical resistive MHD processes that may be 

engendered by such rapid spatial changes need to be 

understood and avoided, if possible. References 1 to 3 have 

identified such behavior in an axisymmetric simulation 

and has pointed out its importance. A non-MHD effect that 

causes instant penetration of field and injected plasma is 

demonstrated in Section 3.0, “Resistivity From GradientDriven Microinstabilities.” In other words, the injection 

process is here assumed to be sufficiently smooth and 

gradual as to be benign. A systematic investigation of the 

effects of coil discreteness and the rapidity of spatial convergence, within the axially symmetric resistive MHD model, 

would be a worthwhile and instructive project. 

Although the propellant temperature drops precipitously 

after passage through the throat of the nozzle (see Ref. 6), it 

hardly drops at all between breech and throat. Therefore, 

classical resistivity should still be small up to the throat. This 

leads to the expectation that for 1 m or less of travel (from 

breech to throat), there should be very little growth of the 

resistive plasma-field mixing layer separating core-plasma 

from magnetic nozzle field, in the absence of anomalous 

resistivity, when the simple Ohm’s law applies. 

A separate companion process, which also generates 

attached plasma, is the diffusion of external magnetic field 

into the plasma—considered as a fixed resistive medium. 

This process also was analyzed on the basis of the simple 

form of Ohm’s law with classical resistivity and was found 

to produce essentially the same width of plasma-field mixing 

layer as previously estimated for the other process. This 

result was demonstrated by two independent methods, which 

agreed very well. The first was from the point of view of the 

moving propellant, with an assumed growing timedependent field applied at the boundary. The second was 

from the lab frame point of view, assuming a steady-state 

configuration with a constant external magnetic field. 

It is convenient to encapsulate these results in terms of a 

magnetic Reynolds number, defined as A Dcl Remag = V r . 

Here, the Alfven speed is VA = 2 B μ ρ0 , with B being the 

magnitude of the nozzle magnetic field, and ρ the propellant 

mass density. Also, r is a representative radius of the confined plasma in the nozzle, and Dcl = ηcl μ0 is a representative resistive diffusivity of plasma propellant. For the 

purposes of making approximate estimates, the magnetic 

Reynolds number may be evaluated at the throat of the nozzle. When β ≈ 1, that is, V ≈ Vth i ≈ VA, with Vth i being 

the ion thermal speed, the normalized width δ of the plasmafield mixing layer may be characterized as 

mag

1 ~ Re

Lbt

r r

δ (97) 

where Lb t is a characteristic nozzle axial length. As a numerical 

example, in the region just upstream of the throat Remag ≈ 104

, 

and Lb t/r ≈ 10 for Lb t ≈ 1 m and r ≈ 0.1 m. Then δ/r ~ 0.03. 

This ratio will grow further downstream. The analytical 

description of the entire evolution of δ over the length of the 

nozzle is beyond the purview of this report. Equation (97) 

thus is relevant to the plasma attachment process. 

Turchi (Ref. 8, private communication) points out that if 

the angle φB V between field and flow were regarded as an 

independent parameter, and if the transverse pressure gradient were neglected, then the boundary-layer width would 

scale rather differently, namely as 

mag

1 ~ sin r Re

  δ   φ    

BV (98) 

This result can be derived from the inertial component of the 

plasma-momentum equation across the flux surfaces (the 

x-component) while incorporating the simple form of Ohm’s 

law for J within the J × B force. In that manner, one finds 

that the incident “core” velocity component across the flux 
NASA/TP—2009-213439 29 

surfaces, core sin V V x = φB V , experiences a slowing down 

due to eddy current braking, with a characteristic time 

2 txchar = D VA . The boundary-layer width then can be 

estimated as ~ sin Vtx char δ φB V . Assuming that β ≈ 1 —

that is, V ≈ Vth i ≈ VA —one immediately arrives at the 

expression in Equation (98). The latter constitutes a much 

stronger dependence on Remag. 

The extremely short slowing-down time for inertially driven cross-flux motion, which is the result of eddy current 

braking for the upstream parameters of interest, suggests 

that the angle between field and flow may not be independently specified in a highly conducting plasma. Section 4.0, 

“Rayleigh-Taylor- (RT-) Type Instabilities” quantitatively 

estimates the crossfield braking time within the single-fluid 

resistive MHD model and the consequent shape of the propellant streamlines. The flow would very quickly become 

practically field aligned, with a very small remnant of crossfield flow driven by the edge pressure gradient. Therefore, it 

would appear that the model that underlies Equation (98) is 

more likely to be valid in the low-pressure high-resistivity 

region that exists downstream of the throat, where detachment is of interest rather than attachment. There, the magnetic Reynolds number is much smaller, so that the flow angle 

(relative to the magnetic field) would be less vulnerable to 

alteration by eddy current braking and also would be less 

influenced by the broader transverse pressure gradients. The 

latter point of view was, in fact, adopted for the detachment 

calculation in Reference 7. It is therefore recommended that 

the scaling of the boundary-layer thickness be studied as a 

function of position along the nozzle in order to elucidate the 

transition between Equations (97) and (98). 

The small amount of plasma-field intermixing estimated 

upstream of the throat, for the parameters of interest and 

using classical resistivity, serves as a prompt to consider 

possible causes why simulations or experiments might nevertheless exhibit substantially greater boundary-layer widths 

at the nozzle’s throat. Such causes could be as follows: 

(1) numerical diffusion 

(2) degree of processes allowed (Hall effect, rotation, ion 

viscosity, classical vs. anomalous resistivity) 

(3) global geometrical configurations (number and 

placement of discrete field coils, types of global electrical boundary conditions including plasma effects) 

(4) a subtle combination thereof 

A feature of the boundary-layer physics notably absent 

from this report, so far, is ion viscosity. This feature is introduced later in the discussion of Rayleigh-Taylor type instabilities in Section 4.3, “Finite Larmor Radius (FLR) 

Stabilization of RT-Type Flute Modes.” It can be shown that 

the collisionless contribution to the force term arising 

from ion gyroviscosity (see Ref. 9) bears essentially the 

same order-of-magnitude relation to ion inertia in the 

plasma-momentum equation as does the Hall term to the 

motional electric field in Ohm’s law; that ratio is on the 

order of ion gyroradius to the plasma’s radial scale length. 

Therefore, collisionless ion gyroviscosity ideally should be 

included in the plasma model of the interface whenever Hall 

effects are included. 

In this concluding discussion of the general physics topic, 

we turn now to a review of the underlying physics 

represented in the modeling of the edge plasma during 

plasma flow through magnetic nozzles, specifically, the 

physics associated with the complete Ohm’s law (the electron momentum equation). The derivation of the general 

Ohm’s law was first outlined, and then several special cases 

were examined. The emphasis was on the Hall effect and 

plasma rotation. A major concern, however, was the justification for use of the simple Ohm’s law while in the presence 

of a strong magnetic field. The simple Ohm’s law is often 

perceived as being the limit of the complete Ohm’s law when 

the electron collision frequency is very large, υe >> ωc e. 

The principal results were as follows: First, it was pointed 

out that when the Hall current is completely suppressed by 

electrically insulating global boundary conditions, the simple 

form of Ohm’s law is restored by means of the quasi-radial 

Hall voltage. This result is obtained even though, in the case 

of interest, υe << ωc e. This justifies the resistive diffusion 

result obtained with the simple Ohm’s law, which governs 

the spatial growth of the plasma-field mixing layer. Practically the same result for the plasma-field mixing layer 

growth was obtained from two distinct points of view, 

plasma diffusion into magnetic field and field diffusion into 

plasma. 

Next, in simulations that may short out the Hall voltage 

and thereby allow Hall current to flow—but inconsistently 

without plasma rotation—it was shown that there would be a 

very large reduction in the effective electrical conductivity of 

plasma. This is the phenomenon of “magnetoresistance,” 

which would enhance edge-plasma diffusion across the flux 

surfaces. Magnetoresistance associated with the Hall effect 

has been observed in gas-plasma and solid-state-plasma 

experiments configured so that rotation is impossible. 

Finally, self-consistently allowing for flow of Hall current 

and plasma rotation, a characteristic length Lπ along the 

nozzle was identified beyond which the above-mentioned 

resistive spatial growth of the plasma-field mixing layer 

would occur. However, for lengths smaller than Lπ, the 

plasma-field mixing layer width proved to be insensitive to 

the electron collision frequency; instead, it was on the order 

of the ion gyroradius. Again, essentially the same results 

were obtained from the two points of view: penetration of 

plasma into field and penetration of field into plasma. The 

characteristic length Lπ (≈Lcrit) was identified approximately 

as Lπ ≈ λe mi me , where λe is the collision mean free path 

of a representative electron, mi

 is the mass of an ion, and me

is the mass of an electron. Because the insensitivity of layer 
NASA/TP—2009-213439 30 

thickness to electron collisionality is a favorable result, it is 

recommended that Hall-MHD simulations be performed in 

which the Hall space-charge electric field is shorted out and 

rotation of plasma is allowed, while systematically varying 

the length of the nozzle configuration. Beyond the length Lπ, 

plasma spinup produces a quasi-radial motional electrical 

field that suppresses the flow of Hall current. We then 

recover the preceding results where the Hall current was not 

allowed to flow.

Lastly, in this concluding discussion of Section 2.0, on 

general physics of the magnetic nozzle, we turn to the topic 

of the dynamical dwell time available for resistive growth of 

the plasma-field mixing layer. The model that was employed 

earlier in this report simply assumed that the dwell time is on 

the order of Lb t/Vt

, where Lb t is a characteristic axial length 

of the nozzle and Vt is the velocity of propellant in the throat. 

Now, we shall refine that model by recognizing that the 

dwell time (say, from breech to throat) must be longer 

because of the time required for acceleration of the propellant up to speed. The plasma-field mixing layer thickness at 

the throat will be calculated using this refined model. 

From Equation (19) with β ≈ 1 in the transition layer, the 

algorithm that is used to investigate the effect of increased 

dwell time between breech and throat is 

2

cl 0

d b t tδ = D t  (99) 

where δ is the thickness of the plasma-field mixing layer, tb t

is the time required for the propellant to traverse the distance 

from breech to throat, and Dcl is the classical (Spitzer) resistive diffusivity. It is known that Dcl is insensitive to plasma 

density and essentially depends only on electron temperature. Equation (99) is analyzed in Appendix B, using a 

generic shape for the nozzle. There, the time variable is 

related to the flow velocity, the longitudinal distance along 

the nozzle Lb t, and the nozzle’s geometry (i.e., the contraction ratio). The nominal plasma edge layer width at the 

throat Dt clLbt Vt is found to be enhanced by the longer 

dwell time, but by less than a factor of about 2 from the 

original estimate. Specifically, for a radius-contraction ratio 

of 2 the bounding numerical factor proved to be 1.7, whereas 

for a radius-contraction ratio of 3 it proved to be 2.3. 

From these results, an approximate overestimate of the 

fraction of attached hydrogen plasma propellant at the throat 

can now be made for the sample parameters, with the 

assumptions of classical resistivity and no Hall current. First, 

the 0.6-cm estimate of δ is doubled to allow for bidirectional 

diffusion (plasma into field and field into plasma). Then we 

multiply by another factor of 2 to allow for the effect on the 

layer thickness of the increase in dwell time from its reference value Dt clLbt Vt . This adjustment is for the time 

required during acceleration of propellant from breech to 

throat, over a length of 1 m. The final result is a plasma-field 

mixing layer at the throat of about 2.5 cm. Taking a nozzle’s 

nominal plasma radius at the throat, r = 10 cm, this leads to an 

upper bound of the attached fraction 2δ/r of about 50 percent. 

The factor 2 comes from the geometrically enlarged area of an 

outer annulus of thickness δ in cylindrical geometry. To the 

extent that this result is even approximately realistic, it suggests a search for mitigating effects. Therefore, an example 

of operation at higher stagnation temperature and lower 

resistivity with a heavier propellant ion, to achieve the same 

specific impulse, could be pursued at a future time. 

Since the result for the layer thickness is insensitive to the 

radius-contraction ratio, essentially three options for remediation are allowed, assuming classical resistivity: The first, 

suggested by Turchi, is to shorten the length Lb t from breech 

to throat. However, the thickness of the plasma-field mixing 

layer only scales as Lb t , assuming that the scaling in 

Equation (97) is valid. The second is to enlarge the (plasma) 

throat, which has the disadvantage of increasing the massflow rate of propellant (for fixed stagnation density and 

stagnation temperature), with consequences for vehicle 

design and mission logistics. Hence, the second option 

would entail a smaller propellant density to preserve the 

mass flow rate m . The consequent enlargement of the ionion collision mean free path may require extending the MHD 

flow model into the kinetic regime. The third option is to 

work at a higher stagnation temperature in order to decrease 

the resistive diffusivity D (to increase the magnetic Reynolds 

number) and also to work with heavier propellant atoms to 

maintain the desired exhaust velocity (specific impulse). See 

the discussion on constraints at the end of this discussion 

section. 

The above estimate of plasma-field mixing layer thickness, applied to hydrogen propellant with Lb t ≈ 1 m from 

breech to throat, can as well be applied to the envisioned 

”Godzilla” experiment (see Sec. 2.1, “Bernoulli Equation for 

Nozzle Flow”) with fully stripped helium propellant and 

Lb t ≈ 0.3 m (see Ref. 2). According to the tabulated parameters, taking into account the higher stagnation temperature in 

helium and that the atomic number Ξ is 2, D is about 2/3 of 

the hydrogen value; hence, D is about 0.8 of the hydrogen 

value. Also Lb t is 1/3 of the previous value, so Lb t is 

about 0.6 of the previous value. The velocity near the throat, 

Vt

, is about the same for both propellants. Consequently, the 

previous answer for the boundary-layer width δ, namely 

Dt clLb t Vt , is to be multiplied by 0.6 × 0.8 ≈ 0.5. Since 

the previous δ for hydrogen propellant was about 2.5 cm, the 

new estimate for helium now is δ ≈ 1.3 cm. 

Figure 6 of Reference 2 displays results relevant to the 

planned helium experiment, as obtained by the MACH2 

 
NASA/TP—2009-213439 31 

simulation using classical resistivity. It indicates that, at the 

throat, the radius of core plasma is only about 2 cm. From 

there, the magnetic field drops off from its nominal edge 

value of 0.75 to 0.28 T on axis over only a radial distance of 

about 2 cm. This drop-off ratio is close to a factor 1/e; hence, 

2 cm would appear to constitute a fair representation of the 

characteristic boundary-layer width obtained by MACH2 at 

the throat (compared with the above overestimate of 1.3 cm). 

The plasma density or pressure profiles (not shown in the 

figure) might have indicated some additional outward penetration of plasma into field, leading to a somewhat thicker 

effective plasma-field mixing layer. Since both field penetration into plasma and plasma penetration into field engender 

plasma attachment, it is recommended that the type of 

information in the figure henceforth be made more complete 

by providing the radial density and pressure profiles of 

plasma corresponding to the magnetic profiles therein. The 

He parameters and coil configuration employed in Reference 2 leading to a 2-cm plasma throat signify that 

essentially all of the He propellant has become attached to 

magnetic flux at the throat position. 

Because the plasma’s corresponding density profile or 

pressure profile was absent from Figure 6 in Reference 2, 

our assessment of what transpired in that simulation between 

breech and throat is necessarily limited. What can be said is 

that the model of classical resistive diffusion developed in 

Section 2.0 of this report is in fair agreement with the 

MACH2 simulation of Reference 2, being too small by 

perhaps a factor of about 2 or 3. This is in spite of the fact 

that the derivation carried out in Appendix B was biased in 

several respects towards too much interdiffusion. 

Figure 8 of Reference 3 shows radial profiles in the plasmanozzle throat for a larger system. The system length is 1 m, 

and the system radius is 18 cm. The radial plasma pressure 

profile is shown, along with the radial magnetic pressure 

profile. At the throat of the nozzle-shaped plasma, the interface width extends from about 3 to about 7 cm; thus, the 

width is about 4 cm. For comparison, a rough estimate of the 

resistively diffused width can be obtained from the expression Dt . For helium at 100 eV, D ≈ 2.3 m2

/s. Also, t ≈ 10 μs, 

from a quasi-one-dimensional isentropic flow model 

(velocity in the throat ≈ 10

5

 m/s). Then, the resistively diffused width proves to be about 0.5 cm, which is an order of 

magnitude smaller than the width observed in the MHD 

simulation. (A factor of 2 or 3 enhancement due to the dwell 

time during speed-up does not resolve this discrepancy. See 

Appendix B.) An anomalous process within the resistive 

MHD model ultimately is held responsible for the 50 percent 

mass loss, which is tentatively ascribed to the discrete magnetic geometry. In the present report, no attempt is made to 

address this anomalous process, which appears to lack a 

complete and definitive understanding. The authors of Reference 3 and this author are in agreement that the observed 

anomalous mass loss can probably be mitigated by 

employing a more gradual and smoother magnetic geometry 

transition between the nozzle inlet and the plasma-nozzle 

throat. Such an adjustment may require elongated inlet geometry. This speculation ought to be tested with more numerical MHD simulations. 

The influence of the nozzle shape of core plasma on the 

accrued diffusive edge-layer thickness yet remains to be 

thoroughly investigated. Measured magnetic nozzle fieldline shapes in a real experiment, albeit in coaxial geometry 

over part of the effective length of the thruster, are displayed 

in Figure 1 of Reference 11. Therein, resistive drag by the 

plasma substantially modifies the field-line shape. 

Although the occurrence of substantial diffusive plasma 

attachment motivates exploration of improvements in design, 

it should be recognized that there are fundamental constraints arising out of mission logistics. For example, the two 

quantities dm/dt = m and the specific impulse Ve/g ought to 

remain fixed for a given mission when searching for 

improved parameters. The latter constraint implies that the 

stagnation temperature is fixed for a given propellant; the 

former constraint implies that the quantity ρr

2

 is fixed (say 

at the throat). Here, ρ is the mass density of propellant and r

is the throat radius of the confined plasma core. 

However, there is a third constraint associated with the 

magnetic nozzle: the external magnetic pressure should 

balance the internal plasma pressure transversely. The consequent condition on the global beta, namely that β ≈ 1, 

along with fixed stagnation temperature, implies that ρ

scales as B2

. Therefore, the product Br must be fixed at the 

throat. A fixed temperature then implies that the ratio r/ai

 is 

fixed; that is, that the ratio of ion gyroradius ai

 to the effective plasma-throat radius r is fixed—for a given propellant. 

This result was shown in Reference 2. In terms of different 

propellants, the ratio ai

/r scales approximately as mi/(Ξ)

3/2

for fixed m and specific impulse, where mi

 is the ion mass 

and (Ξ)(1.6×10–19 C) = qi

 is the ion charge. In this report, |qi| 

= |qe|, and only hydrogen propellant is of interest. 

This constraint on ai

/r should be remembered when contemplating the use of Hall current and rotation along with 

axial lengths shorter than λe mi me in order to limit the 

plasma-field mixing layer thickness to the ion gyroradius. 

This constraint also is important when considering the activity of certain microinstabilities at the plasma-field interface, 

such as the LHD instability. The linear and nonlinear properties of the latter depend fundamentally on the size of the ion 

gyroradius relative to transverse plasma gradient lengths 

such as the thickness of the plasma-field interface. 

Finally, it should be emphasized that in this report the 

concept of axially symmetric resistive MHD flow with a 

relatively small rate of resistive interdiffusion at the plasmafield interface has been taken for granted. If ill-understood 

resistive MHD dynamical mixing processes were to exist in 

the breech because of the severity of the plasma injection 

process, and perhaps as exacerbated by the discreteness 
NASA/TP—2009-213439 32 

of field coils, those processes would certainly have to be 

identified and eliminated. Otherwise, supposedly educated 

attempts at systematic engineering designs of magnetic 

nozzles would be premature. 

3.0 Resistivity From Gradient-Driven 

Microinstabilities 

In this section, we review the research that has been done 

on certain microinstabilities, which can potentially contribute to anomalously rapid broadening of the plasma-field 

interface. Attention is focused primarily on those modes that 

propagate in the azimuthal direction, directly across the 

magnetic field. 

Plasma density gradients that have been formed across a 

confining magnetic field are often regarded, within the context of perfect symmetry, as equilibrium states of ideal 

dynamical force-balance. In a practical sense, however, they 

constitute nonequilibrium configurations precariously suspended away from states of lower energy. That is because 

perfect symmetry is never attained. Therefore, such configurations are actually vulnerable to symmetry-destroying 

plasma instabilities that grow out of slight asymmetries 

and that try to access the lower states. For the magnetic 

nozzle, the macroscopic consequence would be a loss of 

definition of the plasma-field interface, with thickening of 

the plasma-field mixing layer, and concomitant attachment 

of peripheral propellant to magnetic nozzle flux. In fact, 

even a macroscopically smooth resistive fluid model entails 

a graininess and concomitant loss of symmetry at the microscopic level, leading to resistive diffusion. 

Some of these instabilities have very large growth rates 

and are not amenable to descriptions within a resistive magnetofluid MHD model. The LHD instability is one that has 

been identified as a major concern in such situations. It is 

nominally a robust nonresonant instability that does not 

require the extreme conditions needed by some other wellknown microinstabilities. For example, it does not need 

current drift velocities in excess of the electron thermal 

speed as would be required by the Buneman instability. It is 

insensitive to the electron-ion temperature ratio and so does 

not need Te >> Ti

 as would be required for destabilization of 

the ion-acoustic instability. Reference 12 presents simulations of screw-pinch and theta-pinch magnetically driven 

implosions using a hybrid computer simulation (particle ions 

and fluid electrons), which employed the so-called Chodura 

resistivity. This is an empirically parametrized representation 

of electron resistivity arising from the ion-acoustic 

instability. However, they ignored the instability requirement 

that Te >> Ti

. In their model, the current drift velocity correctly is required to exceed the ion-acoustic speed, which 

is a much more stringent requirement than in the LHD 

instability. Then, in the empirical model, the electron 

collision frequency is scaled by the ion plasma frequency. 

(Chodura resistivity is an option in the MACH2 resistiveMHD computer simulations.) Sgro and Nielson had some 

limited success with this approach. They computed the 

structure of the magnetic field in magnetically imploded 

plasmas and generated reflected ions in front of the magnetic 

piston; nevertheless, the empirical resistivity lacked a firm 

theoretical foundation, and there was difficulty in treating 

the late phases of those implosions. Regarding the validity 

of the empirical resistivity, it should also be remarked that 

Chodura himself had a similar degree of success in simulations of magnetically compressed plasma, but employed a 

rather different electron collision frequency: namely, the 

constant 0.025 multiplied by the electron gyrofrequency in 

the external magnetic field (see Fig. 1 in Ref. 13). Also, the 

LHD instability can be destabilized by weak plasma gradients, having gradient lengths that are large compared to the 

ion gyroradius. Moreover, although nominally stabilized by 

finite β, it persists into the ion-cyclotron regime even where 

the local β is substantially larger than 1. (For a brief review 

of the various properties of this instability, see the introductory discussion in Ref. 14 and the other references therein.) 

In the magnetic nozzle, the linear phase of the LHD instability would manifest itself as a fine-scale rotating azimuthal 

ripple (flute-type mode) in the plasma-field interface region. 

In this report, the linear and nonlinear theories of this instability will be briefly reviewed. The nonlinear phase evolves 

into a form of plasma turbulence that would be macroscopically evident as an enhanced resistivity. Whether such 

enhancement dominates over classical resistivity depends 

upon parameters such as the electron temperature and the 

local width of the plasma-field mixing layer. 

In this report, the nonlinearly evolved microturbulent 

LHD resistivity will be compared with the classical resistivity 

that arises from coulomb collisions of electrons on ions. 

Although the influence of finite β on the linear phase of the 

instability has been clarified (Ref. 15), this writer believes 

that the β dependence of the evolved microturbulence 

(Ref. 14) is not yet definitively understood. Therefore, in the 

following discussion β will simply be taken to be of order 1 

within the plasma-field mixing layer. 

The width of the plasma-field mixing layer relative to the 

size of the ion gyroradius proves to be a key parameter for 

driving the LHD instability in its linear phase. This ratio also 

characterizes the microturbulent resistivity in the nonlinearly 

evolved turbulent phase. For these reasons, it is important to 

ascertain whether there is an initial plasma-field mixing layer 

width that might be immediately formed during the propellant injection process, and if so, to ascertain its characteristic 

thickness. Moreover, this initial plasma-field mixing layer 

width defines the initial condition for further resistive 

broadening as the propellant moves downstream, no matter 

whether that diffusion be classical or anomalous. Thus, 

the first order of business is to identify the width of the 

plasma-field mixing layer that naturally occurs when 

injected plasma first impacts magnetic flux. 
NASA/TP—2009-213439 33 

3.1 Interface Width in Breech of Nozzle 

We preview the results by emphasizing that there must be 

an irreducible initial plasma-field mixing layer thickness 

engendered at the very injection of plasma into the external 

magnetic field. Moreover, it cannot be understood from 

within the resistive MHD model. Hall effects and kinetic 

(individual particle) effects are responsible for its formation, 

which therefore would be missed in standard resistive MHD 

simulations such as those performed by the MACH2 code. 

Modeling of the initial mixing layer in the breech of the 

nozzle therefore comprises the topic of this section. 

Older theories of the interface width separating an impacting plasma from an external magnetic field, self-consistently 

taking into account the magnetic shielding by the plasma 

particles themselves, predicted that the width would be on 

the order of the hybrid electron-ion gyroradius aiae . This 

collisionless model is known as the Ferraro-Rosenbluth 

sheath. (See Ref. 16 and references therein.) However, Peter, 

Ron, and Rostoker demonstrated in Reference 16 that the 

sheath would rapidly disintegrate as an entity because of 

flute-type instabilities with wavelengths somewhat larger 

than the sheath thickness. Moreover, the disintegration 

would occur very quickly. The growth rate can be inferred to 

be ~ ω ωci ce , the lower hybrid gyrofrequency. The action 

of these fast flute instabilities is believed to be the reason 

that the Ferraro-Rosenbluth sheath has never been observed. 

In this subsection, we develop a model with early turbulent 

collisions for the electrons and later classical or anomalous 

electron-ion collisions, for a much larger interface width, 

~ pi c ω . Such a sheath width of impacting plasma has been 

observed by Ripin et al. (Ref. 17). 

A heuristic picture will be developed of the initial plasmafield mixing layer thickness produced by injected propellant 

in the breech of the nozzle. The result indicates that, on the 

basis of fundamental physical processes, the initial thickness 

is on the order of the ion inertia length pi c ω . Here, c is the 

speed of light in vacuum and ωp i is the ion plasma frequency. 

For conciseness, no detailed attempt will be made to derive 

exact numerical factors, but the length scale of the initial 

quasi-radial plasma-field transition will be established (where 

“initial” refers to conditions in the breech of the nozzle that 

are immediately experienced by a given element of injected 

plasma, but a macroscopic stationary flow is allowed). 

The interaction between newly injected propellant plasma 

and the magnetic nozzle field may be characterized to first 

order as the inability of “new” electrons to freely cross the 

nozzle’s magnetic flux surfaces because of their diminutive 

electron mass. That is, the electrons have not yet had time to 

collisionally diffuse any significant quasi-radial distance 

across the magnetic field in the manner discussed earlier. So 

in this simplified model, the electrons are, at first, treated as 

collisionless during the short initial ion gyrotime of interest. 

The individual ions, on the other hand, make sizeable 

quasi-radial excursions across the flux surfaces into the 

region of strong magnetic field because of their much greater 

mass. Thus, a quasi-radial space-charge electric field Ex, 

which limits the ion excursions, is established in the mixing 

layer. It points radially inwards towards the core plasma. The 

nozzle longitudinal magnetic field Blong is screened out of 

the core plasma according to a characteristic spatial transition of width dx. By Ampere’s law, it arises from the net 

azimuthal current layer J dx y carried by charged particle 

species in the interface region. 

Our model focuses on a short time interval after injection 

of a given plasma element. This is a time interval so short 

that only electrons can carry the azimuthal current—for the 

reasons explicated below. A motional azimuthal electric field 

associated with the quasi-radial fluid velocity Vx expanding 

across a longitudinal magnetic field Blong (local coordinates) 

is neglected. This simplification proves to correspond to a 

highly sub-Alfvenic expansion across the magnetic field, Vx

<< VA. See the justifying remarks in Section 3.1, “Interface 

Width in Breech of Nozzle.” Also, see Reference 17. 

We consider a planar interface model having the nozzle 

magnetic field B in the z-direction (into the page), the quasiradial x-direction pointing vertically upwards across flux 

surfaces towards the region of strong magnetic field, and the 

y-direction (the azimuthal direction) pointing horizontally 

rightwards. The nozzle magnetic field becomes negligible 

towards the negative x-direction, into the core plasma. Also, 

the space-charge electric field E points in the negative 

x-direction, and is perpendicular to the magnetic field (see Fig. 3). 

The ions cross the magnetic field more easily than the 

electrons, so the outer part of the boundary layer is ion rich, 

whereas the inner part of the boundary layer is electron rich. 

These conditions set up a strong space-charge electric field, 

which serves to give the electrons a guiding center drift, in 

conjunction with the presence of the magnetic field. 

During one ion gyroperiod, only the electrons experience 

a guiding-center drift, because of their relatively rapid gyrations. This drift is to the right, in the direction of E×B

(clockwise in terms of the azimuthal angle). This is equivalent to a current flow to the left, which is in the direction to 

weaken the internal magnetic field and to strengthen the 

external magnetic field. 

We consider only very early times while following newly 

injected plasma, during which “new” ions within the magnetic field interface can barely complete a significant fraction of a gyroperiod. (For hydrogen ions entering into a 

magnetic field of 0.5 T, one gyroperiod is 0.1 μs, whereas 

the transit time through the meter-long nozzle is tens of 

microseconds for V// ≈ 105

 m/s.) 
NASA/TP—2009-213439 34 

During this short time interval, the new ions cannot 

express a macroscopic azimuthal current—in the absence of 

an azimuthal electric field (again, see Sec. 3.1). The reason is 

that a macroscopic azimuthal ion current would be the net 

cooperative effect of individual ion guiding-center drifts and 

magnetization current loops (due to partially uncancelled 

gyro-orbits in a density gradient). These individual particle 

mechanisms, by their very nature, require many ion 

gyroperiods to become macroscopically evident. They are 

therefore not available to the ions during an early time 

increment lasting less than one ion gyrocycle after injection 

while following a given plasma element downstream. By the 

way, this view of the situation implicitly defines the local 

reference frame with respect to azimuthal velocity, in that 

the ions are assumed injected without any macroscopic rotational motion. In other words, at injection there is no azimuthal beam of ions in the lab frame of reference. The 

electrons, on the other hand, do exhibit very many gyrocycles during this first ion gyration and so can indeed manifest macroscopic azimuthal currents early on. 

To simplify the model, we suppose that the electrons are 

collisionless (a restriction removed later) and have negligible 

pressure (also removed later). Then, the electrons carry an 

azimuthal macroscopic current only as the result of an E/B

azimuthal guiding-center drift velocity (in mks units). This 

drift, unlike the other current generating processes, does not 

need significant electron pressure. E/B drift results from nonuniform curvature of a gyro-orbit. During energy gain from 

–E, the electron’s orbit acquires a greater radius of curvature. 

Over the part of the orbit against –E, energy is lost producing 

a smaller radius. The trajectory is a cycloid, a gyrocircle with 

a moving center having velocity E/B. If the temperature Te

were very small then the gyrocircle would be very small, but 

the description still applies. 

This temporary exclusion of electron pressure precludes 

other current-generating individual particle effects, namely 

magnetization currents and grad-B guiding-center drifts. The 

simplifying assumption of no field-line curvature in the 

breech also precludes “curvature drifts” of the electron’s 

guiding center. At any rate, the latter drift also vanishes in 

the zero-electron-pressure approximation. (From the way 

the fluid equations are derived from more detailed models, 

we know that the electron fluid equations—with electron 

temperature included—automatically incorporate the various 

individual-particle effects, as regards production of 

macroscopic currents.) 

In view of the zero-electron-pressure assumption, the 

electron macroscopic momentum equation in mks units is 

simply 

0 = −neq( ) E + Ve ×B (100) 

where Ve = E × B/B2 is the guiding-center (macroscopic 

fluid) drift velocity vector of the electrons, of number 

density ne. The above equation, in component form, where 

Bz = Blong, is equivalent to
NASA/TP—2009-213439 35 

n q

J B

E

e

y z x = (101) 

where y e y e J = −n qV is the azimuthal current density carried by the electrons in the plasma-field mixing layer, and 

Vy e is positive to the right. This current is driven such as to 

cancel the vacuum magnetic flux within the plasma and to 

increase it outside of the plasma. Since the longitudinal flux 

cannot be created or destroyed over a short time interval 

within the volume enclosed by the conducting wall (the coil), 

the magnetic flux is pushed aside or displaced by the injected 

plasma. 

By means of Ampere’s law, which reads μ0 J = ∇× B (or 

also μ0Jy = −∂xBz), Equation (101) for the space-charge field 

can be written as 

0

2

2

1

μ

∂













 = − z x e x

B

n q

E (102) 

This result proves to be key to estimating the thickness of 

the plasma-field mixing layer in the nozzle magnetic field. It 

is based on the azimuthal current being carried only by the 

electrons. 

We now have to consider the time-dependent position, 

that is, the trajectory r(t), of an individual ion that is incident 

on the plasma-field mixing layer. It can be shown that the 

inertial force on the individual ion from radial deceleration 

of the plasma fluid is small compared to the force on the 

individual ion from the space-charge electric field, provided 

that the radial fluid expansion velocity is highly subAlfvenic. See Appendix F for the derivation. The single 

particle’s velocity vector is t vi = d ri d . The momentum 

balance equation of an ion, in the plane perpendicular to B, 

reads 

= ( ) E + v⊥ ×B ⊥

i

i

i q

t

m

d

dv

 (103) 

Here, the velocity vector of the ion is partitioned as 

vi = v⊥i + vziB B . The space-charge electric field vector E

is in the local x-direction, and the magnetic-field vector B is 

in the local z-direction. The scalar product (“dot product”) of 

Equation (103) with v⊥ i

 then leads to a statement of energy 

balance in the form 

( ) x xi

i i qE v

t

m v = ⊥

d

d 2

2

1

 (104) 

Equation (104) shows that the magnetic field does no work 

on the ion, and that the work done on the particle by the 

x-directed electric field affects the total perpendicular energy 

of the particle, not merely the energy associated with the 

quasi-radial velocity vx i. That is, in Equation (104) 

2 2 2

i xi yi v⊥ = v + v . Therefore, to make use of this energy balance equation we also need an independent equation for the 

azimuthal velocity component vy i, furnished by the azimuthal component of the momentum balance Equation (103): 

xi z y ii qv B

t

v

m = − d

d

 (105) 

It now proves useful to relinquish the time variable, utilizing instead the x-coordinate of the particle’s quasi-radial 

position as the independent variable. Noting that d d xi v xt = , 

the energy balance Equation (104) becomes 

2 2 d

d2 2

i xi i yi

x

mv mv qE x

    + =    

 (106) 

Likewise, the azimuthal momentum Equation (105) 

becomes 

z y ii qB

x

v

m = − d

d

 (107) 

Multiplying through by vy i in Equation (107) shows that 

Equation (106) can also be written as 

( ) x y i z i xi

q E v B

m v

x = + 



















d 2

d 2

 (108) 

Hence, energy balance does indeed apply to just the quasiradial x-component of ion motion, but only from within the 

instantaneous azimuthal-moving frame of the ion. Note the 

appearance of the radial electric field in the moving frame. 

The best way to convey our approach to the problem of 

the initial thickness of the edge-plasma boundary layer is to 

begin by integrating the azimuthal momentum equation, 

Equation (107). The injected plasma has no directed azimuthal velocity. Thermal ions incident on the plasma-field 

mixing layer then will be equally likely to have positive and 

negative values of vy i. As a representative ion, therefore, we 

choose an intermediate example for which the azimuthal 

velocity at the core region is vy i core = 0. The maximum 

radial extent of this particular trajectory is called Δ, which is 

the quantity to be determined. Presumably, this particular Δ

lies in the midst of the distribution of Δ-values generated by 

internal ions incident on the plasma-field mixing layer at all 

possible angles. 
NASA/TP—2009-213439 36 

Equation (107 ) is integrated from x = 0 to x = Δ. The final 

azimuthal velocity is written as vy i Δ. The result of this 

integration then reads 

yi ci v Δ =− ω Δ (109) 

wherein ci z i ω = qB m is the radial average of the ion 

gyrofrequency over the considered trajectory segment. Note 

that the appearance of the radial average magnetic field is 

mathematically correct and required. The average magnetic 

field within the plasma-field mixing layer is identified as 

roughly half of the outer magnetic field: 2 ω =ω ci ci . 

Thus Bz and ωc i without brackets now indicate the stronger 

magnetic field out beyond the plasma-field mixing layer. 

Then (109) becomes 

2

ci

y i v Δ

ω Δ = − (110) 

Now the energy balance Equation (106) can be integrated 

from x = 0 to x = Δ. The impacting radial velocity is denoted 

by vx i core, and the final radial velocity at the outermost 

extent of the trajectory segment is clearly vx i Δ = 0. Integrated energy balance for an individual ion over the entire 

specified trajectory segment then can be expressed as 

2 2 core

2 2

i xi i yi

x

mv mv q E Δ − + =Δ (111a) 

Equation (111a) slightly rearranged, 

2 2 core

2 2

i xi i yi

x

m v mv q E Δ =− Δ+ (111b) 

states that the incident kinetic energy of the outward moving 

ion, 1 2

2m vi xi core , is partly depleted by doing work 

− q Ex Δ against the inward-directed space-charge electric 

field (recall that Ex is negative). The leftover kinetic energy 

at the end of the considered trajectory segment is just 

2

2

1 mivyiΔ . Again, the appearance of the radial average of 

the electric field Ex is indeed mathematically correct and 

required. 

The expression for the electric field in Equation (102) is 

used in Equation (111b), which then becomes 

2 2 2 core

0

1

2 22

i xi i yi

x e

mv mv B

q

n q

Δ    

=Δ ∂ +       μ    

 (112) 

In the further interpretation of Equation (112), the 

extremely powerful drive to quasi-neutrality is recognized at 

the envisioned plasma density 1015

 cm−3 (provided that the 

space-charge field does not get shorted out). 

Accordingly, we assume quasi-neutrality in the transition 

layer, namely ne ≈ ni = n (for hydrogen, with Ξ = 1). 

Thus,ni

 − ne<< ni. This assumption does not preclude a 

significant (but not unrealistically huge) space-charge electric field that maintains quasi-neutrality. A specific knowledge of the anomalous microturbulence that instigates the 

early transition to quasi-neutrality in the interface region is 

not needed for the estimate of the early width of the interface. 

Note: A preliminary transient phase of virulent microturbulence must exist, driven by a huge space-charge electric 

field, which facilitates rapid transport of electrons across 

flux surfaces as they follow ions in an attempt to preserve 

neutrality. The main text assumes that this preliminary phase 

is already completed so that quasi-neutrality is preserved. 

Without this preliminary adjustment, the space-charge field 

would remain huge. For example, if an ion density of 

1015

 cm−3 were displaced a distance of only 0.001 cm 

beyond the electron boundary, it would create an electric 

field of 2 MV/cm, with severe consequences. An alternative 

possibility is that the electron boundary remains absolutely 

fixed and that the ions are allowed to extend only a thermal 

Debye length (0.00016 cm at 50 eV) beyond the electron 

boundary into the magnetic field. This also seems unrealistic 

and contradicts experimental observations. It seems to be 

necessary that there is a preliminary draconian adjustment of 

the electrons’ distribution upon plasma injection. This anomalous electron turbulence is shown to be necessary in 

Appendix F, where classical mobility and collisional drift 

cannot cause electrons to traverse the required ion penetration depth of an order of a fraction of an ion gyroperiod. 

An approximate estimate of the average term 

( ) ( ) 2

0 1 2 e x nq B ∂ μ in Equation (112) can be made by 

simply using intermediate values of the quantities therein, 

which does not call upon any detailed profiles within the 

plasma-field mixing layer. The value of ne ≈ n in the denominator of Equation (112) ranges from its core value to a 

much smaller value over the plasma-field mixing layer width 

Δ. Its average, therefore, is simply interpreted as half of the 

number density ncore in the core plasma. Also, we interpret 

the average spatial derivative of magnetic pressure in Equation (112) as the outer value of magnetic pressure divided by 

Δ. (The neglected inner value of magnetic pressure is significantly smaller.) 
NASA/TP—2009-213439 37 

The characteristic radial spatial decrease of B2 into the 

plasma should be governed approximately by the same layer 

width Δ as governs the characteristic radial outward decrease 

of plasma density (associated with the radial protrusion of 

the ion trajectories). This behavior of density and magnetic 

field profiles is consonant with the constancy of total 

(plasma plus magnetic) pressure across the plasma-field 

mixing layer (see Appendixes G and H). 

Such rudimentary partitioning of the whole quantity 

( ) ( ) 2

0 1 2 e x nq B ∂ μ in Equation (112) proves to be not 

far numerically (within a factor of 2) from the exact answer 

obtained with specific sample profiles. The detailed profiles 

within the plasma-field mixing layer are, of course, 

unknown. However, for example, with exponentially 

decreasing n(x) and exponentially increasing [B(x)]2

 over 

width Δ (qualitatively consistent with pressure balance 

across the layer), one obtains a numerical factor 1.17 instead 

of 2 in Equation (113) below. Moreover, a square root is 

eventually extracted to find the width Δ, which finally yields 

a comparative error of only about 30 percent. 

Using just the above rudimentary procedure then, after 

multiplying by 2/mi

 Equation (112) can be reduced to 

2 22 core 2 xi A yi v Vv = + Δ (113) 

Here, the square of the Alfven speed VA B 0nmi

2 2 = μ contains the outer magnetic field beyond the plasma-field mixing layer, and the inner number density of core plasma. We 

suppress the subscripts that would have supplied those 

meanings. 

Equation (110) for vy Δ i

 is now used in Equation (113), 

together with the key identity 

A

ci pi

V c = ω ω

 (114) 

The result reads 

( )

2

22 2

core 2 2 1

8

pi

xi A v V

c

  ω

= +Δ  

   

 (115) 

This result is based upon energy and momentum balance for 

an individual ion’s trajectory, together with the relation of 

the radial space-charge electric field to the azimuthal electron current, obtained from the fluid electron’s radial 

momentum equation. No overt connection is invoked here or 

below with the actual temperature of the ions or with the 

thermal gyroradius of an ion. This salient point will be 

emphasized again below. 

Note that the Δ estimated here depends, in principle, on 

the particular slice of the ion’s incident velocity distribution 

used to make the calculation. Although this particular Δ is 

assumed to be representative of the plasma-field mixing 

layer width, that assumption is not rigorously tested in this 

report. 

We now heuristically consider the trajectory of an individual ion as the ion enters at perpendicular incidence and 

proceeds into the plasma-field mixing layer. Incident azimuthal velocity vy core i

 = 0, but incident parallel (to B) velocity 

vz core i

 is arbitrary. If it undergoes roughly one-fourth of a 

gyrocircle in the average magnetic field in the plasma-field 

mixing layer, such a trajectory segment is expected to be 

approximately commensurate with the time needed for that 

ion to reach its maximum radial protrusion beyond the core 

plasma, denoted by Δ. Since the final radial velocity 

vanishes, the average radial velocity over the quarter circle is 

taken to be half of the incident radial velocity. Then one 

obviously has 

1

4

1 1 core core core 2 2

/2 /2

xi x i x i i

ci c i

vv v

π π Δ≈ τ ≈ =

ω ω

 (116) 

In Equation (116), time 

4

1i τ is taken as one-quarter of the 

ion gyroperiod in the average magnetic field within the 

plasma-field mixing layer. Again, the average magnetic field 

is set to half the outer magnetic field, so the ion gyrofrequency ωc i without brackets in Equation (116) is understood 

to contain the outer magnetic field. 

Next, we calculate Δ2

 from Equation (116) and use Equation (115) for 2

x i core v and the identity in Equation (114). In 

that manner we easily obtain the following equation for 

( ) 2 2 norm c Δ = Δωp i where Δnorm denotes the nondimensional version of Δ measured in units of the ion inertia length 

c/ωp i (and ωp i refers to ncore): 















 Δ

+

π Δ = 8

1

2

2

norm 2

2

norm (117) 

The term 8 2 Δnorm represents the final azimuthal kinetic 

energy that ultimately remains of the ion’s incident radial 

kinetic energy, after subtracting the work done by the ion 

against the space-charge field. The term 8 2 Δnorm in Equation (117) proves to be comparable to the first term (i.e., ≈1). 

The solution of Equation (117) is 2Δnorm ≈ 13; then, 

Δnorm ≈ 3.6 = p i Δω c . 
NASA/TP—2009-213439 38 

Use of the specific sample profiles mentioned above in 

evaluating the average space-charge electric field suggest 

the possibility that the coefficient of the first term in Equation (113) (i.e., 2) should perhaps have been “1.” Then, 

instead of Equation (117), 















 Δ

+

π Δ = 4

1

4

2

norm 2

2

norm (118) 

with the solution Δnorm ≈ 2.6. 

These approximate model results paint a definitive picture. Consider a newly injected plasma element near the 

interface between the plasma and the magnetic nozzle flux, 

which is followed downstream at early times not exceeding 

about one-fourth of an ion gyroperiod. The extent of a representative ion protrusion out of the plasma element into the 

(displaced) magnetic flux is a few ion inertia lengths as 

characterized by the core-plasma number density; that is, 

2.6 3.6

pi pi

c c     <Δ<

ω ω  

  (119) 

Our heuristic model cannot guarantee the exact numerical 

factor. Except possibly for the numerical factor, this result 

for the plasma’s edge-layer thickness is dependent neither on 

the incident radial velocity of the ion nor on the strength of 

the external confining magnetic field. It depends solely on 

the ion number density in the bulk plasma. 

The constraints built into the model that yields this result 

are as follows: First, quasi-neutrality within the plasma-field 

mixing layer is achieved immediately in a preliminary transient adjustment involving microturbulence and is maintained thereafter. The almost-massless electrons pulled into 

the magnetic flux by quasi-neutrality have become attached 

to the magnetic flux. Second, the electron current layer 

shields the interior plasma from the magnetic field practically from the outset of injection of the considered plasma 

element (after several electron gyroperiods). Third, the ion 

component makes practically no contribution to the azimuthal current layer. Finally, a cautionary note is that the calculated Δ was obtained here by considering a special restricted 

class of individual ion trajectories selected out of the 

thermal-ion distribution, namely those that enter the plasmafield mixing layer without azimuthal velocity. A rigorous 

calculation of Δ requires an average over the entire incidention velocity distribution. 

Note that the edge-plasma boundary layer thickness Δ as 

calculated here also characterizes the inward-radial decay of 

B2

, according to Equation (112) and the associated discussion. Hence, the characteristic decay distance of B itself into 

the plasma must be 2Δ, and Δ ≈ 2 p i c ω to 3 p i c ω 

(comparable to Eq. (119)). The ion inertia length is about 1 cm 

for a hydrogen-plasma having number density 1015

 cm−3. 

There is indeed experimental evidence (Ref. 17) for 

pi c ω to be the length scale for the plasma-field transition 

layer produced when injected plasma first impacts upon and 

displaces the ambient magnetic field. The account in Reference 17, however, does not provide a clear and definitive 

understanding of this feature. Although Reference 17 

addresses laser-produced plasma expansion into an existing 

magnetic field, the physics issues therein are generic and 

relevant to injection of plasma into a magnetic nozzle. 

The above results for the width of the plasma-field interface of newly injected plasma can be very simply obtained 

from a reduced model, within about a factor of 2. The 

reduced model begins by assuming that ions of only a single 

outward quasi-radial velocity are initially incident on the 

interface, vx core i ≠ 0 and vy core i

 = 0, with vz core i being 

arbitrary but following the plasma in the reference frame of 

vz core i

. It also assumes, as before, that the electron current 

layer is responsible for magnetic shielding of the interior 

plasma. A key assumption is that there is pressure balance 

across the interface. The electron pressure is still neglected 

in this reduced model. 

Consider the rate at which outward ion momentum flows 

into the interface, per unit area, during the first quarter to half 

gyroperiod, before the ions have been fully reflected by the 

outer magnetic field: It is the particle flux multiplied by the 

momentum of a particle; namely, (ncore vx core i

)(mi vx core i) 

= 2 core core x i ρ v . Balancing this outflux of initial radial ion 

momentum against the inward magnetic pressure immediately yields core 2 xiA v V= , in which the Alfven velocity 

is evaluated with the outer magnetic field and the inner mass 

density. The outward fluid velocity would be smaller with a 

full angular velocity distribution. 

The distance beyond the core where an ion starts to turn 

around (radial velocity → 0) during one-quarter of a gyroperiod in the average magnetic field in the plasma-field mixing 

layer is 

1

4

1 core 2

core

2

2 8 8

xi x i i

ci

x i A

ci ci pi

v v

v V c

π Δ= τ ≈

ω

π ππ ≈ == ω ωω

 (120) 

Here as before, the average radial velocity in the layer is 

taken as half the incident velocity, and the average magnetic 

field in the layer has been taken as half the outer magnetic 

field. Because electrons have been pulled out into this layer 

by the forces that maintain quasi-neutrality, they will have 

become attached to magnetic flux in the layer. Thus, the 
NASA/TP—2009-213439 39 

evolution of layer width resembles a ratchet process that 

cannot be reversed. 

Recall the more detailed model that dealt with the energy 

and momentum balance of an ion’s trajectory in the spacecharge electric field and the shielded magnetic field. It 

apparently gives a somewhat larger value for the width of the 

interface: namely 2.6 pi c ω to pi 3.6c ω (comparable to 

Eq. (119)). Also recall that the corresponding decay length 

of the magnetic field into the core plasma is predicted to be 

twice this value. 

It is important to make a clarification regarding the above 

model. A motional azimuthal electric field has been neglected at the outset that otherwise would have been produced 

by a radial fluid velocity of injected plasma expanding 

across the longitudinal magnetic flux. In principle, the ions 

in the interface region can respond to this azimuthal electric 

field during their initial quarter gyroperiod. Thereby they can 

contribute to the azimuthal macroscopic current provisionally assumed carried only by the electrons. It can be shown, 

however, that the ion contribution would be minor, provided 

that the plasma-fluid’s initial quasi-radial expansion velocity 

across B is highly sub-Alfvenic. 

Specifically, it can be demonstrated that the azimuthal 

fluid velocity component Vy e of the current-carrying electrons is of order VA. In contrast, the azimuthal fluid velocity 

Vy i of the ions in response to the motional electric field 

Ey = −Vx iBz during the first quarter gyroperiod is much 

smaller. In particular, Vy i proves to be merely on the order 

of the radial ion-fluid velocity Vx i. See Appendix F for the 

derivation of these results. The massless electron response to 

Ey merely would be a radial E × B guiding center drift. The 

electron contribution to the azimuthal current would not be 

directly affected by the azimuthal electric field. Therefore, if 

the plasma’s radial expansion velocity Vx i is highly subAlfvenic, that is, if Vx i << VA, then Vy i << Vy e and the ions 

cannot make a significant contribution to the macroscopic 

azimuthal current. The electrons carry practically all the 

azimuthal current. 

The reason that it is possible in principle for the fluid 

plasma’s radial expansion velocity to be sub-Alfvenic at the 

plasma interface with the magnetic field is as follows. After a 

brief sub-Alfvenic dynamical adjustment in the strong magnetic field limit, described in Appendix H, further radial 

expansion of the ion fluid involves the ions having to drag 

reluctant electrons across longitudinal magnetic flux. The 

resulting expansion velocity principally depends on the nature 

of the resistivity or the microturbulence that facilitates crossfield electron transport, even right after injection. If, on the 

other hand, the plasma’s initial quasi-radial-expansion fluid 

velocity (macroscopic velocity) were close to the Alfven 

speed in the plasma-field interface region, then it would be 

important to incorporate into the model the corresponding 

motional azimuthal electric field and the ion response to it. 

That has not been done in the present report, which definitely 

assumes a sub-Alfvenic expansion across the magnetic flux. 

We now remove the two restrictive assumptions that were 

made on the electrons. These were the neglect of electronion collisions and the neglect of electron pressure. 

If electron-ion collisions are included (but still with neglect of electron pressure), then the electron-fluid momentum 

equation can easily be solved for the electron-fluid velocity 

Ve, which then reads 

2 2

2

2 1

1

1

1

B B e

E E B V ×

















+ Ω 

 + Ω













+ Ω = −Ω (121) 

Here, Ω = ωc e/υe is the ratio of the electron gyrofrequency 

qB/me to the collision frequency (possibly anomalous) of a 

representative electron with the ions. In the limit Ω → 0, the 

second term, the azimuthal E/B drift velocity term, disappears. The remaining first term reduces to the usual electron 

drift mobility term along the electric field, without a magnetic field; namely, −(q/meυe)E. 

For parameters of interest to us, however, Ω2

 >> 1. For 

example, with the hydrogen plasma particle number density 

1015

 cm−3, a plasma temperature of 50 eV, and a magnetic 

field strength of order 0.5 T (5000 G), one has ωc e ≈

1011 rad/s. In contrast, the classical Spitzer collision frequency is υe ≈ 10 8

 s−1. Then Ω = 1000 and Ω2 = 106. It is 

immaterial to the argument whether 0.2 or 0.5 T is used here. 

Even if the electron collision frequency were anomalously 

large by a factor of 10 or 100, it would still be the case that 

Ω2

 >> 1. For large Ω2, Equation (121) reduces to 

2

1

B B e

E E B V ×  + 

 





Ω = − (122) 

With large Ω, the drift velocity of the electrons along the 

quasi-radial electric field (first term) becomes very small 

compared with the azimuthal E/B drift velocity (second 

term). This result is just what is necessary to carry out the 

above simplified derivation and estimate of the thickness Δ

of the edge-plasma transition layer. 

The radial diffusion distance of electrons due to their 

coulomb collisions with ions during one-fourth of an ion 

gyroperiod can easily be calculated, as can the radial mobilitydrift-distance during that same time interval from the first 

term of Equation (122). The former distance is approximately Ω−1/2

c/ωp i; the latter is Ω−1c/ωp i (see Appendix F 

for the derivation). It is still assumed that the transient subphase that preserves quasi-neutrality has already occurred. 

The quarter ion-gyroperiod diffusion result Ω−1/2(c/ωp i) 

compares with the resistive-MHD boundary-layer diffusion 
NASA/TP—2009-213439 40 

as evolved along flow distance Lb t (Eq. (77)), which is 

approximately δ ≈ Ω− 1⁄2(Lb tc/ωp i)

 1⁄2. 

Both of the above distances are small in comparison to the 

layer width c/ωp i when Ω >> 1. Thus, the effects of collisional diffusion and mobility-drift of electrons in the radial 

direction are not important, for the electrons then would 

travel radially only a small fraction of the layer thickness 

during one-fourth of an ion gyroperiod. This is the case 

although there are many collisions of an average electron 

during one ion gyroperiod. Therefore, the dominant motion 

of electrons still is governed by the azimuthal E/B drift 

velocity, which is what enabled the estimate of c/ωp i for the 

transition layer thickness. 

Moreover, the same qualitative picture would remain 

marginally valid even if the electron collision frequency υe

were several hundred times larger and on the order of the 

electron gyrofrequency ωc e, thus making Ω ≈ 1. Hence we 

reach a different conclusion with respect to the effect of 

electron collisions than did Reference 17, which failed to 

emphasize the importance of the electron’s E/B drift in 

determining the thickness of the plasma-field transition 

layer. 

The above remarks addressing the effects of electron collisions can easily be extended to include electron pressure. 

Instead of Equation (122) one finds 

2

1

B B

eff eff e

E E B

V

×  + 

 





Ω = − (123) 

where Eeff = E + (nqe)

−1

∇Pe. Again, large Ω ensures the 

dominance of the azimuthal electron-fluid velocity, the 

second term in Equation (123). 

However, the second term in (123) now contains Eeff 

instead of just E, so there are some differences in detail as 

regards both the derivation and the result for the thickness of 

the plasma-field transition layer. For example, the negative 

electron pressure gradient adds its effect to the negative 

space-charge electric field so as to enhance the azimuthal 

electron-fluid velocity, augmenting magnetic shielding by 

the electron current layer. One expects that this current 

enhancement will act to decrease the thickness of the 

plasma-field mixing layer. Such calculations are performed 

in Appendixes G and H. 

The derivations in Appendixes G and H include the fluidelectron pressure gradient and for simplicity allow the electrons to have the same temperature as the ions. Also, these 

temperatures are assumed to be uniform within the plasmafield mixing layer. In addition, Appendix H explicitly 

includes the fluid-ion pressure gradient, and moreover, the 

assumption of a static plasma-field mixing layer is relaxed.

(Assuming that the expanding plasma element has made 

the initial adjustment to quasi-neutrality when it first 

encounters the external magnetic field region, then there is 

no inconsistency of quasi-neutrality with the small mobility 

and drift displacements of the electrons.) 

A more rigorous treatment of radial averaging of the electric field is carried out in Appendix G, a somewhat different 

averaging treatment independent from those described 

above. Again, specific profile shapes within the plasma-field 

mixing layer are not essential restrictions to make the calculations. Then, instead of Equation (117) or (118), 

[ ] ( ) 2

2 norm 1

2

2

norm 8 −β + Δ π Δ = e G (124) 

Here e is the base of natural logarithms (≈2.7183) and βG is 

the global β across the plasma-field mixing layer, namely, 

the inner plasma pressure divided by the outer magnetic 

pressure. The new effect of electron pressure is now manifested by the square bracket containing the effect of the 

global β. As βG increases, the solution Δnorm decreases, and 

so then does the thickness of the plasma-field mixing layer. 

However, there must be overall pressure balance across a 

static plasma-field mixing layer, so that βG = 1 (see Appendix H). The solution of Equation (124) then is Δnorm ≈ 2.35, 

which compares well with the earlier estimates of 3.6 and 

2.6. Thus our earlier conclusion retains validity; namely, that 

the initial thickness of the plasma-field boundary layer (i.e., 

in the injection region) is a few ion inertia lengths c/ωp i. 

Early quasi-radial deceleration of injected expanding 

plasma is included in the calculations in Appendix H. Quasiradial deceleration is found to have no major effect on the 

width of the plasma-field mixing layer, provided that the 

quasiradial macroscopic expansion velocity of the fluid 

plasma is highly sub-Alfvenic (Ref. 17). This limit corresponds to a strong magnetic nozzle field. 

Several important concluding remarks can be made 

regarding the initial width of the plasma-field mixing layer. 

The plasma-field mixing layer width at the interface 

between plasma and magnetic field is formed in the injection 

region by processes not accessible to the standard resistive 

MHD model. It serves as the initial condition for further 

broadening of the layer downstream. The radially protruding 

collisionless ion trajectories do not at first become trapped 

on magnetic nozzle flux; rather, those ion trajectories return 

to the field-free core plasma. However, their positive spacecharge halo beyond the core pulls some edge electrons 

radially out of the core so as to preserve quasi-neutrality. 

This space-charge effect constitutes a one-way irreversible 

process facilitated by microturbulence driven by the initially 

large space-charge electric field. A detailed knowledge of 

that initial microturbulence is unnecessary. Since electrons 

are practically massless, some form of electron collisionality, 

be it coulomb collisions with ions or with microturbulent 

fluctuations, is necessary to scatter the edge electrons onto 

the external magnetic flux. 
NASA/TP—2009-213439 41 

There is no physical mechanism of collisions, or effective 

collisions, available for pulling those practically massless 

electrons radially back off of the magnetic flux so as to return 

them, macroscopically, into the core. (Such a mechanism 

would have to work “uphill” against the electron density 

gradient.) They then effectively become trapped on an external annular increment of previously expelled magnetic flux. 

The overall process is like a ratchet, in which a continuous 

supply of radially protruding ion trajectories furnishes space 

charge to keep the ratchet expansion process going, during 

the early times following injection of any given plasma 

element within the interface region. Note, however, that this 

heuristic microphysical description of interface broadening 

has general validity not limited to the injection region. 

If the quasi-radial space-charge field were shorted out at 

early times after injection, by the unimpeded flow of Hall 

current, the edge-plasma situation would be fundamentally 

altered. Then, some of the edge electrons would preserve 

charge balance by flowing longitudinally to the backplate (as 

the ions partially vacate the plasma’s edge layer due to the 

radial extent of their trajectories). The ratchet effect thereby 

would be circumvented. Even if the associated Hall-current 

circuit were not completely closed, such a global redistribution of edge-plasma electrons would drastically reduce the 

otherwise huge local space-charge electric field in the edge 

region and would at least partially mitigate the ratchet effect 

at the plasma-field interface. 

The plasma-field mixing layer in the breech of the nozzle 

(sometimes called the initial boundary-layer width in this 

report) is characterized by the ion inertia length c/ωp i, 

according to the models employed here. This initial broadening of the interface is facilitated by the above-mentioned 

ratchet effect. The results of these several models contain 

numerical multipliers suggesting that the mixing layer in the 

breech of the nozzle probably is a few times thicker than 

p i c ω . This result has important consequences for the 

attachment of propellant to magnetic flux. For example, if 

the initial interface were 2 cm thick and the nozzle breech of 

injected plasma were 30 cm in radius, then 15 percent of the 

injected plasma propellant would be immediately affected. 

This would occur before the onset of additional adverse 

effects along the flow. 

The width δ of the plasma-field interface—just after 

injection—is not only on the order of a few p i c ω , but is also 

on the order of the ion gyroradius ai

. This follows from the 

relation ( ) i p i a c ω = βi , with βi ~ 1 in the interface. 

For example, ai

/δ ≈ 0.5 βi ≈ 1/ 8 = 0.35, when δ ≈ p i 2c ω

and βi ≈ 0.5. (For a more rigorous derivation of this numerical 

result, see Appendix I.) 

The interface itself is expected to broaden during downstream flow. An important question then arises as to whether 

the width of the interface exceeds ai

 downstream, as 

the flow approaches the nozzle’s throat. This question is 

important because both the linear and nonlinear behavior of 

the LHD instability are affected by the plasma-field mixinglayer width relative to the ion gyroradius in the layer. 

We will now estimate the ratio ai

/δ along the downstream 

flow. In so doing, we now formally assume the initial conditions 

i

ai ≈ β δ

0.5 where δ ≈ 2c/ωp i (125) 

to hold in the breech (beginning) of the nozzle, where 

variables such as n, δ, t, and D are labeled with subscript b. 

First, to illustrate the method, suppose that the interface is 

broadened just due to the action of classical resistivity, 

ηcl ∼ 1/T 3/2. (Here, T = Te = Ti

.) Using the broadening algorithm in simplified form (avoiding the time integral), we take 

δ

2

 ~ Dclt, where δ is the width of the interface and t is the 

longitudinal transit time for the propellant flow. Also, Dcl is 

the classical resistive diffusivity arising from coulomb collisions of electrons on ions, Dcl = ηcl μ0 in mks units and 

Dcl = (c

2

/4π)ηcl in cgs units. Then δ

2

 ~ t/T 3/2. The variation 

of the squared ratio of ion gyroradius ai

 to interface width δ

can be written as 

nt t

T

nT t

T

t

T

B

ai T 1 ~ ~ ~ ~

3/ 2 5 / 2 3/ 2

2 2

2

















δ

 (126) 

Here, we utililized radial magnetic pressure balance of 

plasma pressure, and the adiabatic relation T ~ nƔ −1

 with Ɣ

= 5/3) between density and temperature along the flow. 

From this very simple calculation, one would conclude that 

ai

/δ ~ t

−1/2 along the flow. That is, the ion gyroradius apparently becomes smaller in relation to the boundary-layer 

width as the flow progresses downstream, as t increases. A 

more careful calculation, however, shows that this conclusion is too hasty. 

The above argument is deficient in that the initial plasmafield mixing-layer width in the breech δb was ignored. Also, 

there is no indication of a natural time scale for t. Both deficiencies can easily be corrected by allowing for the initial 

condition in the diffusive-broadening algorithm. (A more 

generalized plasma-field interface thickness calculation at 

the throat is performed in Appendix J. Also included is an 

example calculation of the fraction of attached plasma.) The 

algorithm now reads D t b cl

2 2 δ = δ + . Here, δb represents the 

initial ion penetration depth into the magnetic field. Carrying 

out the same derivation as in the above paragraph, one then 

finds that 
NASA/TP—2009-213439 42 

b

b

b

i

i

t

t

n

n

a n n

+

= β δ 1

1/ 2

2

1 (127) 

where the time scale tb is defined in terms of parameter 

values in the breech, b b Db t 2 = δ . Also, we have invoked 

the relation between ion gyroradius ai

 and ion skin depth 

c/ωp i, in the form ( ) ai b = βi δ 0.5 for δb = 2(c/ωp i)b. 

Now, relevant conditions are such that as the nozzle’s throat 

is approached, nb/n ≈ 1.5 and t << tb. The characteristic time 

tb is several hundred microseconds, for both classical and 

anomalous resistivity; whereas the time for longitudinal flow 

is only about 10 μs. The time tb scales as T n 3 2 . We infer 

that δ i a is practically unchanged between breech and 

throat, when the interface broadens because of the action of 

classical resistivity. 

Next, the same argument is carried out with an anomalous 

resistivity, ηa, that arises from LHD microturbulence. The 

expression given below for this resistivity is derived later in 

this report. It applies to the case where the plasma-field 

interface width extends over several ion gyroradii. This 

resistivity is expected to dominate over classical resistivity at 

sufficiently high temperatures (see Sec. 3.5, “Comparison of 

Anomalous and Classical Resistivities”). One then has 

2 2

Brack 2 th

c e e d 1 a i LH i p e

m V C

m V

  ω         η ≈         ω   ω      

 

 where 0.1 ≤ CBrack ≤ 0.4 (128) 

The constant contour factor CBrack is not precisely known, 

but spans 0.1 < CBrack < 0.4. These values are based on a 

numerically obtained expression for anomalous resistivity in 

the Brackbill et al. computer simulations (Ref. 14) of the 

resistivity in a magnetoplasma gradient confined by 

magnetic pressure. Here, ωc e is the electron gyrofrequency, 

ωp e is the electron plasma frequency, and ωLH = ω ω ce ci is 

the lower hybrid gyrofrequency in the case of usual interest 

where 2 2 ω >> ω pe ce . In the above expression for anomalous 

resistivity ηa, in accordance with the notation introduced 

later in Section 3.2, “Linear Theory of Lower Hybrid Drift 

(LHD) Instability,” Vd represents the azimuthal drift velocity 

of electrons in the equilibrium current layer that separates 

plasma from the confining magnetic field. This notation 

is meant to apply in a reference frame in which the ions 

have no azimuthal drift velocity. Also, Vth i represents the 

ion thermal velocity, (2Ti

/mi)

1/2.The anomalous resistive 

diffusivity in cgs units, Da = (c

2

/4π)ηa, therefore scales 

along the flow as 

1/ 6

2 / 3

2

2

th

1 ~ ~

~ ~ 1 ~ ( / )

n n

n

n

T

n

nT

n B

B

V V

D

d i

a

















 (129) 

Again, we have invoked radial pressure balance for B2

 and 

the adiabatic relation between n and T along the flow. It is 

important to note that the anomalous resistive diffusivity Da

scales with the square of the current-drift velocity. 

At first the initial plasma-field mixing-layer width as well 

as the factor (Vd/Vth i)

2

 are neglected. Then, using the simple 

algorithm for variation of the boundary-layer width along the 

flow, δ

2

 ~ Dat, we can estimate the variation along the flow 

of the squared ratio of ai

 to δ (where Vth i is the ion thermal 

velocity): 

n t t

n

nT

T

t

n

B

T

D t

V a

ci a

thi i

5/ 6

1/ 6 1/ 6

2 2

2

2

2 1 ~ ~ ~ 1 ~ 



 



 



































δ ω

 (130) 

The ratio itself then would vary along the flow as 

5/12 1/ 2

1 ~

n t

ai

δ (131) 

For the reasons mentioned earlier, however, it would be 

too hasty to accept this result for the variation of ai

/δ along 

the flow. Instead, taking into account the initial layer thickness as in the earlier calculation with classical resistivity, and 

now also including the factor (Vd/Vth i)

2

 = (ai/δ)

2

 in the 

anomalous resistive diffusivity Da, we find 

















β

δ 



 



 +

= β δ

i b

b i

b

i

i

t

a t

n

n

a n n

/ 4

( / ) 1

1/ 6 2

1/ 2

2

1 (132) 

Although the ratio of interest still appears on the right-hand 

side in the denominator, that is immaterial in the parameter 

range of interest here because the transit time tb t

 is small: 

tb t << tb. As before, the characteristic time tb is defined by 

tb = δb

2

/(Da)b. The time tb scales as (nT)

−1/2. Also as before, 

nb/n ≈ 1.5 between breech and throat. 
NASA/TP—2009-213439 43 

Thus, we again infer that the ratio of the ion gyroradius to 

the interface layer width is practically unchanged between 

breech and throat. In the breech, this ratio is estimated to be 

about 1/3. Hence our review of the LHD instability and its 

nonlinear evolution will be performed within the context of a 

plasma-field interface width that extends over several ion 

gyroradii, at any point along the flow between breech and 

throat. The overall conclusion is that the emphasis on the 

anomalous LHD resistivity should be in the small-iongyroradius regime (ai

<δ). 

Avoiding the time integral of resistive diffusivity D

(whether classical or anomalous) really means that the average value of D between breech and throat has been used. 

This method makes sense here because the plasma parameters do not vary much between breech and throat. The 

density at the throat drops to about two-thirds of its breech 

value, and the throat temperature drops to three-fourths of its 

breech value. 

A final remark is that further downstream beyond the 

throat of the nozzle, there is a rapid spatial drop in plasma 

temperature with increased coulomb scattering of electrons. 

One would then expect that beyond the throat, classical 

resistivity (which is almost independent of density) would 

become more important than the LHD-based resistivity. The 

latter is basically independent of temperature and almost 

independent of plasma density along the flow (see the above 

scaling of Da). Thus, the discussion of microturbulent LHD 

resistivity ηa will be set within the context of edge plasma 

that is situated between the breech and the throat, where ηa

is most likely to matter. 

3.2 Linear Theory of Lower Hybrid Drift (LHD) 

Instability 

This subsection briefly outlines the linear theory of the 

small amplitude phase of the azimuthal mode LHD instability and summarizes the principal results for the frequency, 

growth rate, and wave number of this mode. These results 

then are utilized in the following Subsection 3.3, “QuasiLinear Evolution of LHD Instability and Anomalous Resistivity” on the nonlinear evolution of the mode and the development of an anomalous resistivity due to the mode. It is 

understood that all calculations are being done in the local 

longitudinal reference frame of the edge plasma. Along the 

way, we point out the limitations of the simplified models 

that are often utilized and the results of certain papers that 

attempted to overcome these limitations. There are three 

principal limitations, which can be described as follows: 

(1) The Local Approximation: Most of the papers dealing 

with the linear theory have assumed that the magnetic field 

is in the z-direction and that the wave propagates in the 

azimuthal (y-) direction, with the mode structure e[i(k y − ωt)]

. 

This is then of the “flute-mode” type. The mode frequency ω

may be complex, signifying temporal mode growth. Notably, 

the radial structure of the mode, along the plasma density 

gradient in the x-direction, is neglected. Such neglect of the 

radial structure of the mode is called the local approximation. 

(2) The Electrostatic Approximation: It is commonly 

assumed that the charged particle density fluctuations both 

produce and respond only to fluctuations in the electric field. 

Fluctuations in the confining magnetic field are ignored. 

This is known as “the electrostatic approximation.” Therefore, the fluctuating electric field {δE} is in the azimuthal 

y-direction, along the azimuthal wave vector k; the electric 

field fluctuation then is called longitudinal. 

(3) The Unmagnetized Ion Approximation: It is assumed 

that the important frequencies and growth rates are so large, 

and the mode wavelengths are so short, that the effect of the 

direct current (dc) magnetic field on the ion orbits can be 

ignored. That is, during a mode-fluctuation time, the ions 

cannot nearly complete a gyro-orbit. This is called the 

unmagnetized ion approximation. Therefore, the unperturbed 

ion orbits, which appear in the linear theory, are just straightline orbits. 

Attempts to relax these three restrictions will be summarized at the end of this subsection. With these three restrictions in force, the dispersion relation for the mode is derived 

as follows. One starts with one of the Maxwell equations, 

Ampere’s law in the electrostatic approximation, in which 

the fluctuating magnetic field {δB} is ignored (we shall use 

cgs units here, with c being the speed of light): 

{ } { } 0 1

∂ = 

 



 ∇× B = δD

c

δ t (133) 

Instead of setting ∇×{δB} = 0, the desired result could be 

achieved just by taking the divergence of Ampere’s law. The 

vector {δD} is the electric displacement vector. In terms of 

Fourier-mode amplitudes, this equation becomes 

( ) , ( , ) ( , ) 0 (1) (1) ε ω ω = 

 



 − ω  ω = 

 



 − ω k E k

c

i D k

c

i

y y (134) 

The superscript “(1)” means that the Fourier-mode amplitude 

corresponds to a first-order small perturbation. The {δD} is 

analyzed into its Fourier amplitudes ( , ) (1) Dy k ω , and the 

fluctuating electric field {δE} is analyzed into its Fourier 

components ( , ) (1) Ey k ω . We have used the fact that the 

electric displacement component ( , ) (1) Dy k ω is related to the 

electric field component ( , ) (1) Ey k ω through the dielectric 

function ε(k, ω). 
NASA/TP—2009-213439 44 

In order that the mode amplitude in Equation (134) be 

nonzero, the dielectric function must vanish: 

ε(k,ω) = 0 (135) 

This equation constitutes the dispersion relation that 

determines the real part of the mode’s frequency Real(ω) and 

growth rate, which is the imaginary part, Imaginary(ω) = γ, 

in terms of the azimuthal mode wave number k. It is found 

that the growth rate of the LHD mode is maximized at a 

characteristic value of k, which is near the reciprocal of the 

electron thermal gyroradius 1/ae. The specific form of the 

dielectric function is to be obtained from the linearized 

small-amplitude dynamics of the electrons and the ions. 

It is convenient to express Equation (135) in terms of the 

electric susceptibilities of each charge species, χe(k, ω) and 

χi(k, ω). Then Equation (135) reads 

1+ χe (k,ω) + χi (k,ω) = 0 (136) 

The χe and χ i

 can be formally obtained from the particle 

species conservation equations and the relation of species 

current densities to the fluctuating electric field. Ultimately, 

these formal relations must be given substance by means of 

the Vlasov dynamical equations for the velocity distribution 

functions of each charge species; this shall be done in the 

next Subsection 3.3, but only for χi

(k, ω). It will be seen that, 

because of Equation (136), only χi

 needs to be used to obtain 

the anomalous resistivity in the quasi-linear version of the 

nonlinear evolution of the mode. 

 Of course χe also must be known to obtain the dispersion 

relation Equation (136) in the linear theory. In the unmagnetized ion model χe is much more complicated than χi

 

because the electrons sense the dc magnetic field while the 

ions do not. Therefore χe will not be derived here. The 

detailed expression for χe may be found in the paper of 

Davidson and Gladd (Ref. 18). Reference 18 also contains 

background references to the LHD instability. 

The electric susceptibilities, χe and χi

, are now related to 

the species polarizabilities, αe and αi

, in order to indicate how 

the Vlasov equations for the species dynamics enter the picture. From the continuity equation for each charge species “s,” 

{ } {δ } 0 s t s s

n

q

∂ δ +∇⋅ = J

 (137) 

Here, {δns} is the number density fluctuation of a species, 

{δJs} is the current density fluctuation of the species, and 

the particle charge of the species is q (q = q for ions and 

q = –q for electrons). The negative charge of the electron 

will be explicitly indicated –qe; thus qe itself is a positive 

quantity. In terms of Fourier amplitudes, the above charge 

conservation equation can be written as 

( ) ( ) (1)

(1) , , 0 y s

s s

ikJ k

in k

q

ω

−ω ω + = (138) 

but the Fourier amplitude of the current density fluctuation 

is related to the Fourier amplitude of the electric field 

fluctuation by the electrical conductivity σs for that species, 

namely () () () ,ω = σ ,ω ,ω (1) (1) J k k E k y s s y . The charge-density 

Fourier amplitude of each species is then given by 

( ) () () (1) (1) (1) , , (, ) , s s sy s y

k nk q Ek kEk   ω = σ ω =α ω ω     ω

 (139) 

where αs = (k/ω)σs is the polarizability of species “s.” That 

is, polarizability is the ratio of the charge density fluctuation 

of species “s” to the electric field fluctuation. This ratio can 

be calculated by solving the Vlasov equation, as will be 

carried out for ions in the next subsection. 

The Vlasov equation (the collisionless Boltzmann equation), is merely a statement of conservation of particles in 

position-velocity phase space. 

We can now return to Ampere’s law expressed in terms of 

explicit currents and electric fields, instead of using the 

electric displacement vector. This allows the derivation of 

another equivalent form of Equation (136). Ampere’s law in 

the electrostatic approximation is 

{ } B { }J { } δE 1 δ 4 δ 0 t

c c

∂



 



  + 

 



 π ∇× = = (140) 

The divergence of this equation, even keeping ∇ × {δB}, 

reduces to Poisson’s equation relating the divergence of the 

electric field to charge density when charge conservation is 

invoked. It is actually unnecessary to set ∇ × {δB} = 0, 

provided that only the electrostatic fluctuations are allowed 

to have an effect within the model. 

The equation for the Fourier amplitudes then reads 

( ) ( )  ( ) ω 

 



 ω  ω − 

 



 π  ω +  



 π = , , 4 , 4 0 (1) (1) (1) E k

c

i J k

c

J k

c y e y i y (141) 

Use of the conductivity relation ( ,ω) =σ () () ,ω ,ω (1) (1) J k k E k y s s y , 

and multiplication by ik/ω and then dividing by k yields 

( ) e i

e i

k

i

k

k

k i

k

i

 α + α 

 



 π = +





 





ω

σ 



 



 π  +



 





ω

σ 



 



 π = +

4 1

4 4 0 1

 (142) 
NASA/TP—2009-213439 45 

Comparing Equations (142) and (136), the susceptibility is 

related to the polarizability of species “s” by 

s s k

i

α



 



 π

χ = 4 (143) 

recalling that αs is the ratio of the charge density fluctuation 

of the mode to the electric field fluctuation in the mode. Use 

in Equation (143) of the expressions for αs (s = e, i) obtained 

from the Vlasov equation, and then use of χs in Equation 

(136), yields the specific dispersion relation. This dispersion 

relation provides values of complex ω for given azimuthal 

wave number k. 

The results of this dispersion relation must be obtained 

numerically and presented graphically, as is done in the 

paper of Davidson and Gladd (Ref. 18). All three of the 

approximations mentioned above were invoked in Reference 18. However, analytical results for growth rate maximized over k do become available when Te << Ti

. These 

results of the linear theory are summarized below, and are 

utilized in the nonlinear (quasi-linear) theory in the next 

subsection. Moreover, it conveniently turns out that the 

numerical results are not greatly changed when Te = Ti

, as is 

apparent from the numerical results in Figures 2 and 3 in the 

paper of Davidson and Gladd (see also Refs. 19 and 22). 

Thus, the results presented here are not greatly sensitive to 

the electron-ion temperature ratio. 

At the wave number for which the LHD-mode growth rate 

γ is maximized, and when Te << Ti, one has 

LH

i

d

V

V

ω 













π 

γ =

2

th 8

2 (144a) 

i

e e T

T ka 2 =

 

or also kVthi = 2ωLH (144b) 

 LH d

i

d kV

V

V

2

1

2 th

1 Real( ) ω = 































ω = (144c) 

Here, Vd is the azimuthal current drift velocity in the 

macroscopic rest frame of the ions, Vth i is the ion thermal 

velocity 2Ti mi , and ae is the thermal electron gyroradius. 

Also, ωLH is the lower hybrid gyrofrequency ω ω ce ci . The 

results in Equation (144) were obtained in the regime 

Vd < Vth i. This proves to be the regime of interest, based on 

the earlier calculations of the ratio δ <1 i a , presented at the 

end of the subsection on the initial width of the plasma-field 

mixing layer. 

Note that in this regime the LHD mode is destabilized 

(γ > 0) at arbitrarily small current drift velocities Vd. This 

small drift-velocity condition corresponds to arbitrarily 

broad density gradients. That is because when Te << Ti

, 

the drift-to-ion thermal velocity ratio becomes 

d i i n V V 0.5a x th = , where ai

 is the thermal ion gyroradius 

and x n n n x = ∂ −1 1 is the reciprocal gradient length (xn is 

the quasi-radial length of decay of density). However, the 

apparent ease with which the mode is destabilized proves to 

be an artifact of the unmagnetized ion approximation, as 

shown in Reference 19. Nevertheless, within the context of 

plasma propellant flow in the magnetic nozzle, Equation (144a) yields an effective cutoff at which the mode loses 

importance for the production of microturbulent resistivity. 

To make such an estimate relevant to space-vehicle 

propulsion, we note the following parameter values. A 

hydrogen ion gyroperiod would be of order 0.1 μs, whereas a 

characteristic longitudinal transit time for propellant to 

traverse a 1-m nozzle would be of order 10 μs. On that basis, 

a growth rate smaller than about 0.01 ωc i should be ignored 

for practical purposes. 

For hydrogen plasma propellant with Te << Ti, the effective limiting width of the plasma-field mixing layer is estimated from Equation (144a) to be about 20 ion gyroradii, 

which actually proves to be in good semiquantitative agreement with the Freidberg-Gerwin cutoff (Ref. 19). For 

Te = Ti, Reference 19 would predict about 12 ion gyroradii in 

hydrogen plasma. This degree of agreement comes from the 

circumstance that, in the solutions of the LHD dispersion 

relation allowing for magnetized ion gyroresonances, the 

growth-rate envelope of the individual ion gyroresonances is 

in good agreement with the theory having unmagnetized 

ions. Unfortunately, this condition for effective LHD-mode 

stabilization would imply at least about 10 cm of radius 

having plasma attached to magnetic flux. It should be 

remembered that this result is burdened with the three 

approximations mentioned earlier. 

We end this subsection with an account of attempts to 

relax the three modeling restrictions. Gladd, Sgro, and 

Hewett investigated the local approximation both analytically and with computer simulation in Reference 20. The 

simulation was based upon a so-called hybrid model using 

particle ions and fluid electrons. Self-consistently treating 

the radial structure of the eigenmodes in a nonlocal 

model, they found that, while not perfect, the local 

approximation still provided fairly accurate quantitative 

results. These are graphically compared with the exact 

results. While their model was comprehensive, their 

investigation had only a limited scope and left untreated 

other important aspects of edge-plasma stability (such as 

β effects and gradient-width effects) that are relevant to 

the magnetic nozzle application. 
NASA/TP—2009-213439 46 

Drake, Huba, and Gladd addressed the accuracy of the 

electrostatic approximation for the LHD instability within a 

full electromagnetic treatment in Reference 15. In a linear 

theory, they found that gradually increasing local β had the 

effect of relegating the LHD instability into the ion-cyclotron 

realm. For a local β value below about 1.5, the growth rate 

of the mode is still somewhat larger than 0.01 ωc i, and so it 

would still have some relevance to the space-propulsion 

applications of the magnetic nozzle considered in this report. 

As the local β is further increased up to 2.0, however, the 

growth rate suddenly takes a steep dive down towards very 

small values that have absolutely no relevance. These results 

from a full electromagnetic treatment are important for 

magnetic nozzle applications because in the plasma-field 

mixing layer the local β ranges from very small out in the 

magnetic field region to very large in the plasma region. 

A few remarks are in order here, to provide a perspective 

on the finite-β results of Reference 15. First, the finite-β

mechanism that reduces the growth rate of the mode is the 

absorption of wave energy by those electrons resonant with 

the wave in virtue of their grad-B guiding-center drift. This 

means that those particular electrons can be knocked 

out of resonance by collisions, and the wave can then be 

collisionally destabilized. This possibility deserves further 

examination within the context of the magnetic nozzle application. Second, Reference 15 utilized the local approximation. However, an understanding of the edge plasma in the 

magnetic nozzle requires a nonlocal treatment of the radial 

mode structure all across the edge plasma that also takes into 

account the presence of a large radial variation of β. Third, 

the calculation in Reference 15 was not self-consistent 

because the ions were modeled as unmagnetized even though 

the LHD mode was degraded into the ion-cyclotron regime 

by β exceeding 1. Thus, it seems that there is still some 

relevant work to be done in the area of linear theory—

oriented to the magnetic nozzle application—that includes 

electromagnetic effects, collisions of electrons with ions, 

nonlocal mode structure, and ion orbits that sense the 

ambient magnetic field. E.Y. Choueiri (Ref. 21) studied a 

type of instability closely related to the LHD but that has a 

component of wave vector along the magnetic field. This is 

called the Modified Two-Stream instability in its lowtemperature form, and it is called the Electron Acoustic 

instability (in Soviet journals) in its high-temperature form. 

In his thesis, the instability is modeled as drift driven by 

external electric fields without the presence of plasma gradients. However, in his development of the theory for application to coaxial plasma thrusters, Choueiri included 

electromagnetic effects (hence non-zero β) and electron 

collisions. His treatment is analogous to the local approximation of the LHD instability and also is limited to unmagnetized ions. 

Finally, Freidberg and Gerwin (Ref. 19) addressed the 

unmagnetized ion approximation. By taking into account the 

effect of the ambient magnetic field on the ion orbits, it was 

found that there is a continuous transformation of the LHD 

instability into the Ion Cyclotron Drift instability. This transformation occurs as the ratio ( ) d i e i i n V V = 0.5 1+T T a x th

decreases, hence as the gradient broadens. The absence of an 

instability cutoff at low drift velocities, found in the unmagnetized ion approximation, is thereby corrected. For deuterium plasma, the instability is stabilized when the gradient 

width exceeds 17ai

, and it is stabilized at about 12ai for 

hydrogen plasma, which implies, as earlier, an undesirably 

large fraction of plasma attached to magnetic nozzle flux. A 

note of caution is that Reference 19 still relied on both the 

local approximation and the electrostatic approximation. 

Hence, as mentioned above, a comprehensive linear theory 

free of approximations and oriented to the magnetic nozzle 

application would still be useful. 

Huba and Ossakow (Ref. 22) clarified the role of collisions in the LHD instability, especially as regards its transition into the Ion Cyclotron Drift instability at low drift 

velocities. They showed that even a very small amount of 

ion collisionality, me/mi

 < νi/ωc i < me mi , destroyed the 

ion-cyclotron resonance features. The parameters of interest 

lie near the low-collisionality end of the indicated range. The 

ion-resonance features become smoothed out by occasional 

ion collisions into behaviors that make the instability appear 

as if it were the LHD instability with unmagnetized ions. In 

this manner, the LHD instability acquires more significance 

than previously anticipated, within the low-drift regime. It 

should be emphasized that the linearized model of Huba and 

Ossakow is electrostatic, hence strictly valid only for β = 0. 

One can speculate that if the same results proved to be true 

for the electromagnetic case, then the finite-β, linearized 

theory of Drake, Huba, and Gladd (Ref. 15), with unmagnetized ions, would also accrue more validity. 

 3.3 Quasi-Linear Evolution of LHD Instability and 

Anomalous Resistivity 

Several nonlinear hypotheses have been advanced for the 

saturation level of microturbulent fluctuations arising from 

the LHD instability in magnetoplasma gradients. The saturation level of those fluctuations, in turn, determines the magnitude and parameter dependence of the anomalous 

resistivity ηa within those gradients. The various saturation 

hypotheses have been enumerated, with references, in the 

paper of Gladd, Sgro, and Hewett (Ref. 20), and in Brackbill 

et al. (Ref. 14). Those hypotheses will be briefly summarized 

in the subsection following this one, 3.4 “Alternative Models 

for Saturation of LHD Microturbulence.” 

The main purpose of this subsection, however, is to 

review the second-order, single-mode, quasi-linear hypothesis, as the development of this particular model is easy to 

understand. Moreover, an expression is thereby arrived at for 

ηa, which—surprisingly—is essentially the same as the 

results of other models, as well as those of some simulations. 
NASA/TP—2009-213439 47 

It is understood that all calculations are being done in the 

local longitudinal reference frame of the edge plasma. The 

result of the quasi-linear model for ηa, which is in essential 

agreement with an electromagnetic simulation described in 

Reference 14 except for the β dependence, will be compared 

with classical resistivity. (There is a difficulty of interpreting 

the simulation in (Ref. 14) with regard to the β dependence 

of ηa, which is provisionally obviated by simply taking β of 

order 1 in the plasma-field mixing layer.) 

We begin with the Vlasov equation for the electron component of plasma. The effect of the ambient static magnetic 

field on the electron orbits must be taken into account. (In 

the electrostatic approximation, the electrons do not respond 

to fluctuations in the magnetic field.) Coulomb collisions of 

electrons are neglected for simplicity. This equation for the 

electron distribution function fe(re, ve, t) reads as follows: 

d f f0

d te e t

    ∂ +∇ ⋅ =         Η

Η (145) 

Here, the state of a particle is represented by a point in sixdimensional phase space H, symbolized as follows by the set 

of mutually independent position and velocity variables: 

(, ) = e e Η r v (146) 

In Equation (145), the formal time-rate-of-change of those 

variables, as if along an orbit (although not), is defined here 

by 

d (,) d e e t = v aΗ (147) 

The acceleration ae of an electron particle is given in terms 

of electric field E and magnetic field B, in cgs units, by 

e e e

q

m c

  −   ×

= +      

v B

a E (148) 

Note that electric and magnetic fields E and B in Equation (148) depend on particle position re but not on particle 

velocity ve. In Equation (145), re and ve constitute mutually 

independent variables in phase space, and they also are 

independent of the time variable t. 

For purposes of the particle-continuity interpretation of 

Equation (145), which involves interpretation of the phasespace divergence operator, ve in the first component of 

Equation (147) is considered to be a vector directed within 

r-space, and ae in the second component is considered to be 

directed within v-space. Accordingly, the phase-space divergence operator in Equation (145), namely ∇H ⋅ [(dH/dt)fe] = 

∇r ⋅ [(dH/dt)fe] + ∇v ⋅ [(dH/dt)fe], is to be applied to the 

respective components of dH/dt, as defined in Equation (147). This procedure is described in detail below. 

The distribution function fe, which is to be determined by 

the partial differential Equation (145), is the phase-space 

distribution function of the electrons, fe(H, t). This function 

is the number of electrons per unit phase volume at the point 

H = (re, ve) at time t. That is, fe(H, t) is the number density 

of electron particles in phase space. The flux of electrons 

through phase space is just [(dH/dt)fe], the first component 

of which (see Eq. (147)) is the particle flux through ordinary 

space, and the second of which is regarded as the flux 

through “velocity” space. 

Now consider a fixed element of phase volume, r v 3 3 d d , 

at phase point H. Consistent with the above description of 

electron flux through phase space, the spatial divergence 

3 3 ( f )d d e e r v ∇ ⋅    r v signifies the rate at which electrons 

leave “ordinary-volume” element d3

r; likewise, the velocity 

divergence 3 3 ( f )d d e e r v ∇ ⋅    v a signifies the rate at which 

they leave “velocity-volume” element d3

v. The sum of these 

rates constitutes the total rate of egress of electrons, −∂t

 fe

d r d v 3 3 , away from the combined volume element, 

d r d v 3 3 . This rate balance is encapsulated in Equation (145), 

which thus signifies the local continuity of a gas of particles 

in phase space. Equation (145) closely resembles the equation of local mass continuity in ordinary fluid dynamics. 

Since re and ve are regarded as mutually independent 

variables, the first part of the divergence operator reads as 

( ) f f ∇ ⋅ = ⋅∇ r r ee e e v v (149a) 

The second part of the divergence operator is 

 ( ) f f ∇ ⋅ = ⋅∇ v v ee e e a a (149b) 

Equation (148) for the particle acceleration, ae, was utilized 

in Equation (149b). A detailed justification of Equation (149b) uses the fact that E and B in Equation (148) 

depend only upon re and not on ve; and also uses the fact that 

∇v ⋅ (ve × Bfe) = ve × B ⋅ ∇v fe + fe ∇v ⋅ (ve × B) = ve × B ⋅ ∇v fe + 0.
NASA/TP—2009-213439 48 

Now using Equation (149) and ∇H⋅[(dH/dt)fe] = ∇r ⋅ [(dH/dt)fe] 

+ ∇v ⋅ [(dH/dt)fe], the Vlasov Equation (145) for the electron 

distribution function can be written as follows: 

 

( )

d f ff f

d

f 0

te e te e e

e e e

t

q

m c

    ∂ +∇ ⋅ = ∂ + ⋅∇        

−   × + + ⋅∇ =    

H r

v

H

v

v B

E

 (150) 

Equation (150) is the conventional form of the Vlasov 

equation for the electron phase-space distribution function. 

Collisions of electrons with electrons and with ions are 

neglected. Replacing e − q m by i q m , it becomes the 

Vlasov equation for the ion phase-space distribution function. We now present the quasi-linear formulation of the 

microturbulent resistivity of the magnetoplasma, called 

anomalous resistivity, as it cannot be described in terms of 

coulomb collisions of electrons with ions. 

First, the anomalous resistivity of the plasma, ηa, will be 

formulated in terms of the electric-field fluctuation energy 

by taking the statistical ensemble average of Equation (150) 

averaged over a large number of systems, identical but for 

the indeterminate phase of the LHD wave. The indeterminate 

phase of the wave is due to the fact that there is no preferred 

point in azimuth from which the wave should originate, nor 

is there a preferred point in time at which the wave should 

suddenly appear. That is, the unperturbed configuration is 

uniform in the y-direction and uniform in its static behavior 

in time. The phase of the wave is here regarded as a random 

variable, which is present in the form of a complex exponential phase factor implicitly attached to the Fourier amplitudes 

associated with the wave. 

Second, the saturation level of the electric field fluctuations, which determines ηa, will be estimated from energy 

balance as in the paper of Davidson and Gladd (Ref. 18). A 

very important refinement of that early energy balance 

argument then becomes necessary, as was later pointed out 

by Davidson (Ref. 23). 

Ensemble-averaged quantities are represented by brackets: 

for example, f = f +{ } δf and nn n = +δ{ } . The ensemble average of the Vlasov Equation (150) can then be written 

as follows: 

{ } { }

f f

f

f

te e e

e e e

e

e e

e

q

m c

q

m

∂ + ⋅∇

  − + × ⋅∇    

= δ ⋅∇ δ

r

v

v

v

v E B

E

(151) 

In the electrostatic approximation, the magnetic fluctuation 

{δB} has been ignored. Note that the ensemble-averaged 

distribution function f depends upon the averaged product of the linear fluctuations on the right-hand side. We will 

now define n = “ensemble averaged number density,” and 

e v = “ensemble-averaged electron velocity.” Also Pe

will represent the ensemble-averaged electron pressure. 

Multiplying Equation (151) by meve and integrating over 

all velocity space,  ev

3 d , that equation becomes 

( ) ( )

{ }{ }

te e e e e e

e

mn mn P

nq q n c

∂ +∇⋅ +∇

  ×

=− + − δ δ  

 

v vv

v B

E E

 (152) 

The second term is the divergence of a symmetric tensor. 

We have used the velocity moments of the distribution 

function, namely 3

n v = f d  , 3 f d e e n v =  v v , 

and { }  δn = δ v e e 3 f d . Note the ensemble average of ne is 

quasi-neutral. We have also used the moment that defines the 

electron pressure tensor; that is not written down. The pressure tensor is assumed here to be a scalar (neglect of electron 

viscosity). These moment integrals extend over all of velocity space. 

It is important to realize that the quantities n and e v

are not simply the macroscopic density and electron velocity 

of the static, perturbation-free configuration, but that they 

embody as well the additional average effects of the secondorder products of the fluctuations. The macroscopically 

smooth electron pressure Pe also includes both thermal and 

fluctuation contributions that have been averaged over, but it 

turns out that these detailed considerations regarding the 

pressure do not complicate the following discussion. 

We suppose that the macroscopic ensemble-averaged system only has radial gradients ∂/∂x perpendicular to the 

z-directed magnetic field, and that e v , the electron current 

velocity, is in the azimuthal (y-) direction. Then the 

y-component of Equation (152), which is the macroscopic 

y-momentum equation, reduces to the following: 

t( e y e ) y { }e { } E y ∂ m n v + n q E = −q δn δ (153) 

Here, {δne} is a real electron density fluctuation, which is 

related to the complex wave-Fourier amplitude ( ) ,ω (1) n k e

and its complex conjugate ( ) ,ω (1)* n k e as follows: 
NASA/TP—2009-213439 49 

{ } ( ) (1) (1)

2

1 ∗ δne = ne + ne (154a) 

Likewise, 

{ } ( ) (1) (1)*

2

1 Ey = Ey + Ey δ (154b) 

As mentioned earlier, a random phase factor Ψran i

e is 

implicitly assumed to be incorporated in the complex Fourier 

amplitudes ( ) ,ω (1) n k e and ( ) ,ω (1) E k y in Equation (154). 

Consequently, in the phase-averaged second-order product 

{ } δne { } δEy in Equation (153), only the cross terms 

( ) (1) (1)* (1)* (1) e y e Ey n E + n survive the average because the factor 

Ψran i

e cancels. The remaining terms are ( ) (1) (1) (1)* (1)* e y e E y n E +n , 

which after averaging, produce the factor cos( ) 2 0 ψran = . 

This description of the averaged contribution to { } δne { } δEy

is correct even when allowing for the fact that the wave frequency ω = Real(ω) + iγ is a complex quantity.

Therefore, the right-hand side of Equation (153) can be 

written as follows: 

{ }{ } ( )

( ) ( ) (1) (1)* (1) (1)*

(1) (1)* (1)* (1)

Real

2

1 Real

2

1

4

1

e y e y

e y e y e y

q n E qn E

q n E q n E n E

= − = −

− δ δ = − +

 (155) 

However, from the equations for the electron polarizability 

αe and electron susceptibility χe, from Equations (139) and 

(143), respectively, ( ) (1) (1) (1) 4 e e e y eEy − q n = α E = k πi χ . 

Therefore, 

{ }{ }

( ) (1) (1)*

(1) (1)*

Imaginary 2 4

1

4

Real

2

1

e y y

e y e y y

E E k

E E

i

k

q n E

 χ 

 





π =





 



 χ



 





π − δ δ =

 (156) 

The averaged fluctuation energy density in the real electric 

field fluctuations may be calculated in terms of its complex Fourier amplitude in the same manner as described 

above. One easily finds { } (1) (1)*

2

2 1 Ey = Ey Ey δ . Then 

Equation (156) can be reexpressed in terms of the real electric field fluctuation energy density. Equation (156) becomes 

{ }{ } ( ) { }

( ) E k e W

E

e q n E k y

e y

fl 2 Imaginary

8

2 Imaginary

2

= χ





















π

δ

− δ δ = χ

 (157) 

The averaged energy density in the fluctuating electric field 

has here been defined by = { { } δ 8π} 2

fl y

E W E . 

Now, in the LHD fluctuations, the electron susceptibility 

χe(k, ω) is a very complicated object because the electron’s 

orbit senses the magnetic field. Fortunately, in the quasilinear theory, ω and k of a mode are still regarded as being 

related by the dispersion relation. It is expressed in Equation (136), which implies 

( ) () χe = − χi Imaginary Imaginary (158) 

The ion susceptibility χi

 is easy to calculate when the ions 

are unmagnetized because of the short time scales (so ions 

make straight-line (degenerate) orbits only). Use of Equation (158) in Equation (157) and the latter in Equation (153) 

yields the macroscopic azimuthal electron momentum equation in the form 

( )

( ) fl 2 Imaginary

t e ye y

E

i

m n v nqE

k W

∂ +

=− χ

 (159) 

We will now calculate χi from the Vlasov equation for the 

ions. 

In the macroscopic rest frame of the ions, neglecting the 

influence of the static ambient magnetic field as well as of 

magnetic fluctuations, the Vlasov equation for the ion phasespace distribution function fi

(r, v, t) reads 

f f f0 ti i i

i

q

m

  ∂ + ⋅∇ + ⋅∇ =     r v

E

v (160) 

In Equation (160), fi(r, v, t) is the ion phase-space distribution function described earlier in this section. It represents 

the time-dependent number density of ions at position ri, and 

the time-dependent number density per velocity volume at 

velocity vi. 

In the presence of plasma fluctuations, fi naturally splits 

into a time-independent part that is not directly dependent 

upon the fluctuations, and a part {δfi} that directly participates in those fluctuations. Moreover, {δfi} can be Fourier 

analyzed (expanded in microscopic plane waves) as follows: 

 
NASA/TP—2009-213439 50 

{ }  ⋅ −ω δ = ω

k

k r k r v k k (1) ( ( ) ) ( , , ) ( , ( )) i t

i i f t f e (161) 

The quantity ( , ( )) (1) k ω k i f is called the Fourier amplitude 

or the Fourier coefficient of {δfi(k,ω(k))}. In this Fourier 

expansion, the complex frequency ω(k) is related to the 

wave vector k via the dispersion relation. (The dispersion 

relation is the outcome of self-consistently applying the 

plasma dynamical equations together with Maxwell's electromagnetic equations.) In the present instance, we focus on 

those microscopic waves that possess only an azimuthal 

dependence (dependence on the y-coordinate). 

The linearized equation for the fluctuations then reads 

{ } { } { }

{ } ( )

eq

(0)

ff f

,

ti i i

i

i

i

q

m

q f k

m

  ∂ δ + ⋅∇ δ + ⋅∇ δ  

 

  δ

= − ⋅∇ ω  

 

r v

v

E

v

E

 (162) 

In the macroscopic rest frame of the ions the ion pressure 

is electrostatically confined, so the equilibrium electric field 

in the quasi-radial (x-) direction is 1

eq ( )( ) E Tqn n i x − = ∂ . 

Moreover, the third term in Equation (162) is on the order of 

(qEeq/miVth i){δfi} with Vth i the ion thermal velocity, Vth i = 

2Ti mi . The second term in Equation (162) is on the order 

of (k Vth i){δfi

,} where k is the mode wave number. Therefore, ratio of the third term to the second term proves to be 

~1/(2kxn), where the gradient length is 1 ( ) 1

xn = n ∂ xn − . 

For the LHD modes of interest, kai

 > 1, where ai is the thermal 

ion gyroradius. Actually, k is rather larger than 1/ai

; kae ~1. 

Therefore, kxn >> 1; and so the ratio of the third term to the 

second term in Equation (162), 1/(2kxn), is a very small 

number. 

Accordingly, we neglect the third term in Equation (162) 

and also use the local approximation for an electrostatic 

mode with wave propagation in the azimuthal y-direction. 

Equation (162) then becomes 

{} {} { } ( ) (0) , f f i

t i yi y i y

i yi

q f k v E

m v

    ∂ ω ∂ δ + ∂ δ =− δ       ∂    

 (163) 

For a Fourier-mode e

(iky − iωt)

, this equation relates the 

Fourier amplitudes (corresponding to a first-order small 

perturbation) ( ) ,ω (1) f k i and ( ) ,ω (1) E k y . 

(0) (1) (1) (1) i

i i yi y

i yi

q f i f ikv f E

m v

    ∂

−ω + =−       ∂    

 (164)

The solution for the Fourier amplitude ( ) ,ω (1) f k i of the 

ion’s perturbed distribution function {δfi

} is then 

( )

(1) (0)

(1) , y i yi

i

i yi

q E f v

fk i mkv k

∂ ∂

ω =

− ω

 (165) 

Notice that for an unstable mode, ω has a positive imaginary 

part and therefore the denominator does not vanish. A nearresonant ion-particle response to the fluctuating fields arises 

when the unperturbed particle velocity vy i has a value such 

that the denominator in Equation (165) almost vanishes. 

Such resonances also can occur for the electrons, although 

the electron denominator is more complicated, including the 

electron gyrofrequency along with various electron guidingcenter drifts in the unperturbed orbits. The electron resonances can be nonlinearly broadened and are then thought to 

provide a mechanism of saturation of the microturbulence, as 

will be discussed below. 

By calculating the Fourier amplitude of the ion number 

density, as given by 

( ) (1) (1) 3 , d i i nk f v ω =  (166) 

we can calculate the polarizability ratio (1) (1) i y i α = qn E , 

and then the susceptibility i k χi = 4π αi from Equation (143). The integration in Equation (166) extends over all 

of velocity space. 

In carrying out the calculation indicated by Equations (165) and (166), we shall assume that the equilibrium 

distribution function of the ions is Maxwellian: 

( )( )

3

2 2 2

th

2

(0) th

eq , i i i v V

i i

V

f k vn e

− −  

ω =  

  π  

 (167) 

where i i i V m 2T 2

th = − . We have used and will be using 

2

d u e u ∞ −

−∞ π =  . The constant in front of the gaussian 

exponential in Equation (167) is such that 

( ) eq

(0) 3 f k, (v)d v n i ω =  . 
NASA/TP—2009-213439 51 

The calculation of the Fourier amplitude of the ion density 

fluctuation in Equation (166), from Equations (165) and 

(167), is uneventful and a bit tedious. The result only is 

written here: 

( )

(1) 2

(1) eq

ph

, d

u y

i

i

q ue E n

nk i u

T k uV

∞ −

−∞ ω =−

π −  (168) 

The complex normalized phase velocity is given by 

2

ph th ( ) V kV i

− = ω , where i i i V m 2T 2

th = − with Vth i being 

the ion thermal velocity. By subtracting and adding Vph in 

the numerator of the integrand, the expression in Equation (168) can be put into a more convenient form: 

( )

(1)

(1)

eq

th th

, 1Z y

i

i ii

q E

nk i n

T k kV kV

     ω ω

ω =− +     

    

(169) 

where the so-called plasma dispersion function Z(ζ) is 

defined by 



∞

−∞

−

π − ζ ζ = u

u

e u

d 1 Z( )

2

 (170) 

in which ζ is taken to have a positive imaginary part. Note 

that i = ω kVth ζ is the independent variable of Z(ζ) and 

that Vth i in Equation (169) is the ion thermal velocity, 

2Ti mi . 

The ratio (1) (1) i y i α = qn E is available from Equation (169). Then i k χi = 4π αi can be written as follows: 

2

2 2

th th th

21 Z pi

i

i i i k V kV kV

ω      ω ω

χ= +     

    

 (171) 

The square of the ion plasma frequency has been introduced; 

thus, 2 2

eq 4 pi i ω =πnq m in cgs units. 

Returning now to the phase-averaged azimuthal momentum equation of the electrons (Eq. (159)), we suppose that 

there is a quasi-steady state relative to the rapid microscopic 

fluctuations. The reason for this is discussed below after 

Equation (174). Furthermore, we also set 

E qn v J y a ye a y =η − =η ( ) (172) 

which serves to define anomalous resistivity ηa. Then, from 

Equation (171), Equation (159) reduces to 

2

2 2 th th th

fl

2 2

4 Imaginary Z pi

a i i i

E

e e ye

k

k V kV kV

W

q nv

ω      ω ω

η =     

    

×

 (173) 

Here, average electron velocity y e v is identified with the 

azimuthal current drift velocity Vd of the electrons, in the 

macroscopic rest frame of the ions. Also, recall that the 

second-order-small quantity E Wfl is the average energy 

density in the fluctuating electric field. Thus, E Wfl is a 

second-order-small factor in the expression for the 

anomalous resistivity. The second-order distinction between 

the equilibrium number density neq in 2 ωpi in the numerator 

and the ensemble-averaged n in the denominator is then a 

correction of no consequence for the leading second-order 

evaluation of the right-hand side of Equation (173). 

An anomalous collision frequency υa e of electrons can be 

defined as usual, for singly ionized ions, by 

2 η= υ a ea m nq . We then find from Equation (173) that 

υa e is given by an expression that agrees exactly with Equation (33) in the paper of Davidson and Gladd (Ref. 18). Our 

derivation, however, differs somewhat from that in 

Reference 18. 

Equation (173) is applied the to the small drift velocity 

regime, Vd << vi

, which is equivalent to the case in which the 

gradient length xn is rather larger than a thermal ion gyroradius. This regime is consistent with our estimate of the 

initial width of the plasma-magnetic-field mixing layer. At 

the same time, we follow Reference 18 by using an energy 

argument to obtain an upper bound for the fluctuating 

electric-field energy density E Wfl . This procedure, with 

Equation (173), then yields an upper bound to the anomalous 

resistivity in the small-drift regime. 

Referring to Equation (144) for properties of the most 

rapidly growing linear mode in the low drift velocity regime, 

the real frequency Real(ω) approaches zero with Vd, and the 

growth rate γ approaches zero with 2

d V . The wave number at 

maximum growth, however, stays fixed as Vd → 0. Thus, 

ζ = ω/kVth i → 0 as Vd → 0. A simple contour integration 

then can be used to show that as ω → 0 from above the real 

axis, the Z-function in Equation (173) approaches the imaginary constant value, i π . The imaginary quantity in Equation (173) Imaginary[ζ Z(ζ)] can then be evaluated to 

leading order in the low-drift regime as 
NASA/TP—2009-213439 52 

( )

th th th

Real

Imaginary Z

ii i kV kV kV

       ω ω ω      = π  

        

 (174) 

In order to obtain an upper bound on anomalous resistivity 

ηa, it is noted by Davidson and Gladd (Ref. 18) (after Fowler) that the energy in the fluctuating electric field can only 

be supplied by the drift kinetic energy of the electron current 

(in the rest frame of the ions). This is thought by them to be 

the only source of energy available to be converted into 

fluctuation energy. We are supposing that this drift current is 

at least momentarily sustained by the local average azimuthal electric field, Ey , which, in turn, is induced by the 

momentary local radial expansion of the plasma across the 

longitudinal magnetic nozzle field. Therefore, the electricfield fluctuation energy can at most be 

2

fl 2

1 e y e E W = m n v (175) 

Now, Equations (174) and (175), the expression for 2 ωpi , as 

well as the properties of real frequency Real(ω) and azimuthal wave number k are used in Equation (144). Then, after 

some cancellation, the expression for anomalous resistivity 

in Equation (173) reduces to the following simple form in 

cgs units (with ηa in seconds): 

2

th

1

2

4 















ω

π

η = π

i

d

i LH

e a V

V

m

m (Te << Ti

) (176) 

Here Vth i is the ion thermal velocity, and ω = ωω LH ci ce

is the lower hybrid gyrofrequency. This expression for anomalous resistivity in the magnetoplasma gradient’s current 

layer constitutes an upper bound, which is limited by the 

available drift kinetic energy in the electron’s azimuthal drift 

current in the macroscopic rest frame of the ions. In view of 

our somewhat different derivation, it is satisfying that Equations (173) and consequently (176) for ηa agree exactly with 

Equation (40) in the paper of Davidson and Gladd 

(Ref. 18). The upper bound on the fluctuation-electric-field 

energy density is called the Fowler bound as discussed in 

that paper. 

It is sometimes convenient to carry out basic calculations in 

cgs units rather than mks units. When applied to resistivity, 

the cgs unit is seconds. Thus, the anomalous resistivity in 

Equations (176), (179), (196), and (198) all manifestly have 

identical dimensions of inverse frequency and hence are 

immediately recognized as being in terms of cgs units. The 

same is true as well of Equation (193) in the next subsection. 

To convert any one of these to resistivities measured in the 

mks unit, ohm-meters, multiply these expressions by 9 · 109

. 

The electron collision frequency associated with ηa in 

Equation (176) proves to be 

2 2

th 2

pi d a LH i

V

V

ω   π

υ =  

ω    

 (177) 

wherein 2 4 ip i ω=π nq m is the ion plasma frequency. 

However, a critical refinement of the Fowler upper bound 

is required, as observed by Davidson (Ref. 23). In a regime 

often encountered in magnetoplasmas, which includes magnetic nozzle parameters of interest here, the averaged waveenergy density Wwave far exceeds the energy density in the 

fluctuating electric field fl

E W . This regime of interest has 

2 2 ω >> ω pe ce (or 3 · 109

n >> 3 ·1014B2 in cgs units, with n in 

cm−3

 and B in gauss). 

The energy density in the fluctuating electric field is only 

part of the total wave energy. It is related to the total waveenergy density as follows, according to Reference 23: 

2

fl wave wave 2

E ce

pe

W WW

  ω

= <<  

  ω 

 (178) 

The reason the wave energy Wwave far exceeds the energy in 

the fluctuating electric field fl

E W is that the totality of wave 

energy includes contributions not only from fl

E W , but also 

from energy involved in the fluctuating particle motions 

themselves. The latter contribution actually proves much 

larger when 2 2 ω >> ω pe ce . In turn, the electron-drift kinetic 

energy constitutes the source for the total wave energy. In 

other words, the “Fowler bound” should be supplanted by 

the “wave-energy bound.” 

Following Davidson (Ref. 23), who states (without a clear 

physical justification) that only half the electron-drift 

energy goes into the wave energy, it is apparent from Equation (178) that a factor 1 2 2

2 ce pe

 ω ω   

 has to be inserted 

into Equation (176). Furthermore, Davidson chooses to 

express electron-drift velocity in terms of ion gyroradius in 

the limit Te << Ti

; namely, Vd/Vth i = 2

1 ai

/xn. Here, 

1 | | 1xn = n ∂ xn − is the reciprocal radial gradient length. 

Equation (176) thereby becomes modified so as to yield a 

much smaller upper bound on the anomalous resistivity: 
NASA/TP—2009-213439 53 

3/2 2 2

2

1

2

ce e i

a

p e i LH n

m a

m x

  π ω  

η =       ω ω  

 (Te << Ti) (179) 

This is essentially the result stated by Davidson in the first 

section of his paper (Ref. 23). It can be shown to correspond 

to an electron collision frequency 

2

th

1

8 2

d

a LH

i

V

V

π  

υ= ω      

 (180) 

scaling as ωLH, whereas υa obtained from the Fowler bound 

(Eq. (177)) scales as (1/ωLH). 

Later, the result Equation (179) will be compared with 

two published computer simulations of nonlinear evolution 

and concomitant radial diffusive transport of plasma particles. In this context, it should be noted that Davidson’s 

derivation of Equation (179) had to rely upon the three 

approximations mentioned earlier: the local approximation, 

the electrostatic approximation, and the approximation of 

unmagnetized ions. Moreover, the derivation of Equation (179) in Reference 23 does not include some other 

features that are automatically incorporated in the computer 

simulations, such as wave-wave interactions. 

3.4 Alternative Models for Saturation of LHD 

Microturbulence 

Equation (173) indicates that the microturbulent resistivity 

ηa depends fundamentally on the quantity of fluctuations 

squared in the electric field; specifically, ηa depends upon 

the average energy density (second-order small) of electric 

field fluctuations, namely, = { { } δ 8π} 2

fl y

E W E . More 

complete models of quasi-linear theory include many modes, 

not just one. Alternative hypotheses have been proposed for 

saturation mechanisms for fl

E W , which include trapping of 

ions in the electrostatic wave-potential, electron resonance 

broadening, wave-wave coupling of growing waves to 

damped waves, and individual particle-orbit modification by 

the growing wave (see discussions in Refs. 14, 20, and 24 

to 27). Each mechanism has had its advocates. Within the 

realm of plasma theory, the final arbiter of the various 

nonlinear-saturation hypotheses would be a well-done computer simulation with examples covering broad ranges of the 

relevant parameters. 

In this section, we provide an overview of two iontrapping models, one using an energy method and another 

based on a time-scale method. Ion trapping, being intuitively 

obvious, was one of the first saturation mechanisms to 

be suggested. We also summarize the results of two other 

models, which proved to be in surprising agreement with 

the wave-energy-bound model. It is understood that all 

calculations are being done in the local longitudinal reference frame of the edge plasma. Also, ions are assumed to be 

singly charged in this subsection. We then conclude this 

subsection by summarizing the results of two nonlinear 

computer simulations of LHD microturbulence, one of which 

avoids two of the approximations mentioned above and the 

other of which avoids all three. Their results for the parameter dependencies of ηa bear a striking resemblance to the 

anomalous-resistivity prediction of the quasi-linear model 

with the wave-energy bound. Unfortunately, there apparently 

is as yet no definitive agreement on the value of the numerical coefficient. Moreover, this numerical information is 

needed for optimization of magnetic nozzle design, so as to 

minimize attachment. 

The idea behind the energy-based ion-trapping saturation 

mechanism is that if the traveling drift-wave potential 

accrues too much mass, the wave will be slowed and the 

synchronous dynamics that facilitates its growth will eventually be spoiled. The possible importance of the iontrapping mechanism can be estimated as follows: If the 

electric field fluctuation energy density, evaluated according 

to the wave-energy bound wave Wfl is smaller than the electric 

field fluctuation energy from ion trapping it Wfl then the 

wave-energy-bound level of electric field fluctuations will be 

reached first and will saturate. The fluctuation level for ion 

trapping then would not be energetically accessible. 

The kinetic energy of a thermal ion relative to a traveling 

drift wave of phase velocity Vph is 

2 2 2

2 th

2 1

2 th ph 1 ( − ) = (1−Φ) = (1−Φ) i i mi V i V mi

V T (181) 

where Ti is the ion temperature, and the ratio of wave phase 

velocity to ion thermal velocity is Φ = Vph Vth i . If the 

electrostatic potential-energy pulse experienced by a representative co-moving ion q{δφ} reaches the above level of 

kinetic energy, then the traveling wave potential can begin to 

drag ions along with it. The required electrostatic potential 

fluctuation {δφ} is therefore 

{ } ( )2 δφ ≈ 1− Φ

q

Ti (182) 

The electric field fluctuation is related to the fluctuation of 

electrostatic potential by {δE} ≈ k{δφ}, where k is essentially the azimuthal mode wave number of the fluctuation. 

Also, the field fluctuation energy density from ion trapping 

is on the order of ≈ { } δ /8π it 2

fl W E . Therefore, the electric 

field energy-density level associated with ion trapping is 

2 2

4

2

it

fl (1 ) 8

Ti k W

q

≈ −Φ

π

 (183) 
NASA/TP—2009-213439 54 

On the other hand, the electric field fluctuation energy 

density from the wave-energy bound is (suppressing Davidson’s factor of 1⁄2) 

2 2

wave 2 2

fl 2 2

1 1

2 2

ce ci

ed id

pe pi

W m nV m nV

ω ω ≈ = ω ω

 (184) 

The ratio of field fluctuation energy density from the waveenergy bound (Eq. (184)) to that from the ion trapping 

Equation (183) can be written as follows, after a bit of manipulation that utilizes the expressions for ωc i and ωp i: 

wave 2

fl

it 2 2 2 4

fl th

4 1

(1 )

d

i i

W V

W ka V ≈ − Φ (185) 

As the LHD wave of interest has its phase velocity Vph = 

Real(ω)/k on the order of the drift velocity Vd to within a 

constant CII < 1 (for example, see Ref. 19, nonresonant 

case), Equation (185) will be written as 

wave 2

fl

it 2 2 2 4

fl th II th

4 1

(1 )

d

ii di

W V

W ka V CV V ≈ − where CII < 1 (186) 

In the low-drift-velocity regime of interest (but still the 

nonresonant case), 2

th

2

d i V << V in Equation (186). This 

strong inequality signifies a gradient length xn larger than an 

ion gyroradius ai

. Moreover, recall Equation (144b), which 

gives the wave number of the most rapidly growing LHD 

mode. It has the order of magnitude of k ~ 1/ae. Then, in 

Equation (186), the product 2 2

a i k becomes on the order of 

mi

/me >> 1. Hence, the numerator in Equation (186) is 

somewhat smaller than 1, and the denominator is very large. 

It appears, therefore, that it

fl

wave Wfl << W , so that the iontrapping level of electric field fluctuations is not energetically 

possible in the regime of low drifts and broad gradients. 

Moreover, even in the opposite limit of a sharp gradient, 

Vd >> Vth i, ion trapping according to Equation (186) is again 

not possible. It may be that energy-based ion trapping 

becomes marginally competitive in a narrow velocity interval centered at Vd ≈ Vth i. It is clear that this energy-based 

model of ion trapping requires a single dominant wave. 

There also is a different and simpler criterion for ion trapping, which will be aluded to below. It is based upon a time 

scale argument rather than an energy argument. The idea is 

that if a linearly small growing wave exists with growth rate 

γ that is dominant over all other waves, then wave saturation 

occurs when the bounce frequency of an ion in the dominant 

wave potential ωbnce exceeds the linear-phase growth rate; 

thus, ωbnce > γ is the criterion. From Newton’s equation of 

motion (F = ma) of an ion in the dominant wave potential, 

one easily estimates { } 2

bnce (/ ) i i ω≈ δ q mk E , where k is the 

wave number for the azimuthal direction and {δE} is the 

fluctuating electric field in the azimuthal direction. The 

critical level for ion trapping then is {δE} ≈ γ

2

mi/qk, where γ

is the dominant linear-phase growth rate as given, for example, in the collisionless case by Equation (144). A narrow 

wave spectrum is again required for this form of ion trapping. The anomalous resistivity discussed in the present 

report will not be related to this particular criterion, as 

the latter may require a critical damping condition made 

possible by frequent electron collisions with ions (see discussion below regarding Refs. 25 to 27). Nevertheless, the 

time-scale-based mechanism of ion trapping cannot be definitively ruled out. 

Gary invoked electron resonance broadening as a secondorder mechanism of LHD-mode saturation. See the discussion in Reference 24, and references therein where it is 

believed that, nonlinearly, a large fraction of the electron 

distribution function is brought into near-resonance with the 

waves and extracts energy from them. In the same paper, he 

also applied that technique to several other kinds of microinstabilities that could be expected in magnetoplasma gradients confined by magnetic pressure, notably including the 

“universal instability,” an oblique mode. His calculations are 

carried out analytically, and his result for electron resistivity 

from LHD microinstabilities proves to be practically the 

same as that obtained from the wave-energy bound in 

Reference 23. 

Gary’s calculations on the LHD instability in Reference 24 are restricted within the local approximation, the 

electrostatic approximation, and the unmagnetized ion 

approximation. Also, those calculations ignore wave-wave 

coupling, as does Reference 23. Gary’s employment of 

electron resonance broadening to calculate saturation of 

turbulence and anomalous resistivity has been criticized by 

Drake et al. (Ref. 26) on the grounds that the model is 

restricted to being electrostatic. 

Gary actually finds that anomalous resistivity from the 

oblique “universal mode” (in which the wave vector is not 

perpendicular to the magnetic field) exceeds that from the 

LHD instability. However, Gary’s theory is electrostatic and 

therefore is restricted to zero β. Other work on the universal 

mode has shown that this mode is stable when β exceeds a 

few percent (possibly 14 percent). Electromagnetic theory of 

the universal instability was addressed in References 28 

to 30. We infer that the universal mode should not be of 

principal importance in the edge layer of the propellant 

wherein β is not small compared to 1 and moreover in which 

β has a large range of values. Whether the universal mode 

would be of some secondary importance in a low-density 

plasma halo extending out into the magnetic nozzle field is 

an open question. 
NASA/TP—2009-213439 55 

Drake et al. (Ref. 26) followed the progression of “nearest 

neighbor” wave-wave couplings by means of numerical 

computation. Nearest neighbor modes are separated by the 

wave vector of the fastest growing linearized instability, 

known as the “pump” wave. The LHD waves were driven by 

resonant ions. Saturation of the resulting turbulence (k ⋅ B = 

0) was ultimately obtained by electron damping at short 

wavelengths, either by ∇B drift-resonant electrons or by 

electron collisions (electron viscosity and electron-ion collisions). 

Unlike the simulation in Reference 14 that is described 

below, that in Reference 26 is not a first-principles simulation. Rather, it is based upon reduced-model equations 

obtained from prior physical reasoning. Thus, there are two 

simplifying restrictions: to Te = 0 and to unmagnetized ions. 

There is also a simplification to nonlinear coupling of just 

nearest neighbor modes, which ultimately requires justification (in their appendix) by the presence of electron damping. 

Finally, the model features the spreading of energy in modespace, implying the generation of a broad spectrum of modes 

that is not suitable for the phenomenon of ion trapping. On 

the other hand, the model in Reference 26 is electromagnetic; 

hence, it is valid for nonzero β. Furthermore, the authors do 

simulate examples in which Vd < Vth i, unlike the examples 

presented in Reference 14, which are restricted to 

Vd ≈ Vth i. 

From their electromagnetic standpoint in Reference 26, 

Drake et al. criticize the electrostatic electron-resonance 

broadening theory of Gary (Ref. 24) by claiming that such a 

process in an electrostatic model can only shuffle energy 

back and forth between modes without producing true dissipation and concomitant saturation of turbulence. They claim 

that to produce saturation, an electromagnetic model is 

required with specific wave-particle resonances. They also 

point out that, within the electromagnetic model, magnetic 

energy effectively constitutes a very large reservoir of free 

energy, thereby invalidating the wave-energy-bound concept 

invoked in quasi-linear theory that the wave energy is 

limited by the available electron-drift kinetic energy. (Thus, 

it is implied in Ref. 26 that a resistively broadened plasma 

edge gradient can be steepened anew by the continual application of external magnetic pressure.) 

In view of this remark, it is most interesting that the anomalous resistivity implied in Reference 26 has the same 

parameter dependencies obtained by Davidson (Ref. 23), 

albeit with a larger numerical factor. This result of Drake 

et al., which is not explicitly provided by them, will be 

derived now, based upon Equation (48) in Reference 26. 

The crossfield particle diffusivity D⊥ can be set within the 

context of the usual density-gradient-driven diffusion equation, namely 

nV⊥ = −D⊥∇n (187) 

At the same time, one has radial force balance within the 

plasma gradient, neglecting inertia (hence, slow diffusion). 

This can be written as 

1 P c tot− ∇= ×J B (188) 

wherein Ptot is the total plasma pressure. (It is expeditious to 

use cgs units here, with c the speed of light in vacuum.) 

Invoking the simple form of Ohm’s law, the azimuthal current density is related to the azimuthal motional electric field 

by 

J = σ( V⊥ ×B) −1c (189) 

These three macroscopic equations above may be easily 

combined to yield the relation between the particle diffusion 

coefficient D ⊥ and the resistivity η = 1/σ. For uniform 

temperatures of electrons and ions, that relation is 

















⊥ = β +

i

e i T

T

D D 1

2

1 (190) 

where the resistive diffusivity (in cgs units) is 

η

















π η = 4

2

c D (191) 

Now, Drake et al. (Ref. 26) extract the following expression for the crossfield particle diffusivity from their simulations, notably including the value of the numerical factor 

CDrake: 

2

th

2

Drake es 















⊥ = ω

i

d

LH V

V D C a (where CDrake = 2.4) (192)

Here, from their Equation (48) (in Ref. 26) as extracted from 

the simulations, CDrake = 2.4. The “strange” gyroradius aes

(“es” represents “electron-sound”) is defined by 

aes = ai me 2mi , where ai

 is the thermal ion gyroradius, 

ai

 = Vth i/ωc i. Although the Drake et al. model is electromagnetic, they claim no explicit β dependence for this D⊥. 

One can directly solve Equation (190) for the resistivity, 

using Equations (191) and (192) and the definition of aes. In 

turn, that result can be manipulated further, using the following identities: 
NASA/TP—2009-213439 56 

2

2

th

A

i

i

V

V

β = (193a)

 2 2

2 2

A

pi ci

c V = ω ω

(193b)

 2 2

2 2

ci ce i

pi pe e

m

m

ω ω = ω ω

(193c) 

2 ω =ω ω LH ce ci (193d)

Also, in cgs units, we have to use the following definitions (with Ptot = nTe + nTi

): 

2

8

B

Pi

i

π β = (194a) 

i

i

i m

T V 2 2

th = (194b) 

2

2

4 A

i

B V

n m = π

(194c) 

2

2 4

p

s

n q

m

π

ω = (194d) 

c

s

qB

m c

ω = (194e)

Here, T is the temperature in ergs. 

The final result for the resistivity, now noted as the 

anomalous resistivity, η = ηa, can then be written exactly as 

follows: 

2 2

Drake

2

th

4 1

(1 / )

c e e d a

e i i LH i p e

C mV

TT m V

π ω  

η =  

+ ω ω    

 (195) 

Drake et al. (Ref. 26) sets Te = 0 in their model, as does 

Davidson in Reference 23. Except for the numerical factor, 

this expression is then identical to the zero-β expression of 

Davidson, Equation (179). After converting Equation (179) 

in terms of the current drift velocity Vd, the numerical factor 

in Equation (179) is about 7. In Equation (195) above, for 

CDrake = 2.4, the numerical factor 4πCDrake is ~30. 

The anomalous resistivity inferred in the Brackbill et al. 

paper (Ref. 14) discussed below again proves to be given by 

the identical expression to the above except for the numerical coefficient, provided that the ion-β factor is neglected. In 

the Brackbill et al. paper, the largest value of the numerical 

coefficient in ηa, as extracted from the simulations, is stated 

to be 0.4. Thus, the numerical coefficient in Drake et al. is 

almost 2 orders of magnitude larger than that in Brackbill 

et al. This is a significant discrepancy in view of the fact that 

the results of Drake et al. compare somewhat favorably with 

measurements on a theta pinch implosion, as will be discussed next. 

The output of Drake et al. is compared with measurements 

on theta pinch implosions performed by Fahrbach et al. 

(Ref. 27). In so doing, Drake et al. remark that it is important 

to include electron-ion collisions in the theory, which they 

do. At the same time, they emphatically maintain that their 

Equation (48) (in Ref. 26) should still be interpreted as the 

collisionless expression for D⊥ because their simulations 

show that crossfield particle transport is very insensitive to 

the magnitude of electron dissipation. 

The actual comparisons made with the experiments of 

Fahrbach et al. are not the particle diffusion coefficient D⊥

or the resistively broadened theta pinch profiles; rather, they 

are the shape and position of the mode spectrum and the 

amplitude of the plasma density fluctuations. The spectral 

comparisons agree very well; for example, the predicted 

down-shift in the peak of the wave number spectrum differ 

by only about a factor of 2. The fractional amplitude of the 

density fluctuations differ more, but still within an order of 

magnitude: 0.034 versus 0.014 at the lowest filling pressure 

and 0.014 versus 0.0023 at the intermediate filling pressure 

(where the greater values represent the theoreticals). (Note 

that anomalous resistivity depends on the squares of those 

fluctuation values.) At the highest filling pressure in the 

experiments, the theoretical model finds that the system is 

very close to marginal stability for LHD modes because of 

electron collisions with ions; hence it should be dominated 

by a single mode, and ion trapping should probably be 

responsible for the saturated amplitude. In fact, the experimentalists invoke the time-scale-based ion trapping mechanism to explain their results at all three filling pressures. 

In view of the several competing mechanisms proposed 

for nonlinear saturation, recourse to first-principles simulations of the evolution of magnetoplasma gradients is useful. 

Such simulations are unbiased as to mechanisms. Brackbill 

et al. (Ref. 14) developed full-particle (both electron and ion) 

electromagnetic simulations (hence, capable of addressing 

nonzero β). The considered equilibria had Te/Ti

 < 1, corresponding to our Equation (144) and Davidson’s case 

(Ref. 23), but the electron and ion temperatures were 

allowed to evolve self-consistently with the microturbulence. 

As in Reference 26, a major restrictive assumption was that 

 
NASA/TP—2009-213439 57 

only modes with wave vectors perpendicular to the magnetic 

field were allowed. Unlike Reference 26, however, only 

examples with Vd ≈ Vth i were simulated, although this 

restriction was not required in principle. 

In Reference 14, the microturbulent behavior was simulated in a self-consistent manner over the entire width of the 

plasma density gradient, so there was no local approximation. Particle orbits were self-consistently calculated in the 

ambient zero-order magnetic and electric fields as well as in 

the nonlinearly fluctuating fields produced by the particle 

motions themselves. Fluid equations were utilized in an 

intermediate time step to advance the fields. There was no 

“unmagnetized ion” approximation. The magnetic field 

profile ranged from a large external value to zero deep in the 

plasma, at the center of the current layer. 

To our knowledge, this is the only published firstprinciples electromagnetic full-particle simulation of LHDrelated microturbulence in a global-configuration magnetoplasma gradient, from which a practical formula for the internal anomalous resistivity ηa has been extracted. A paper was 

recently published that included oblique modes, but the 

principal interest there was three-dimensional magnetic 

reconnection across the current layer; see Reference 31. The 

configuration addressed in the Brackbill et al. paper 

(Ref. 14) is relevant to the magnetic nozzle device because 

the internal plasma is confined by external magnetic pressure. If no such practical formula for ηa were forthcoming, 

then simulations of microturbulence in the magnetic nozzle 

would have to be carried out in an iterative manner at many 

locations along the propellant’s flow field within a resistive 

MHD simulation, which seems impractical within normally 

available resources. 

A note of caution, however, is that the paper of Brackbill 

et al., as in related work dealing with theta-pinch implosions, 

differs in a fundamental way from the magnetic nozzle. As 

emphasized by Turchi, in virtue of the longitudinal flow of 

propellant the magnetic nozzle has “new” plasma coming 

from upstream, which is continually entering previously 

disturbed regions of the plasma gradient. Thus, broadening 

of the plasma-field interface region in the magnetic nozzle 

may differ from the results of Reference 14. 

Since Vd/Vth i ≈ 1 in the initial equilibria in Reference 14, 

it is somewhat surprising that the same parameter dependencies (except for the β dependence) are inferred for the anomalous resistivity as are found in References 23 (Davidson) 

and 26 (Drake). However, both Davidson’s and Drake’s 

numerical coefficients in ηa are substantially larger than 

observed in the Brackbill et al. simulations. 

Data from the simulations in Reference 14 is somewhat 

sparse and exhibits significant scatter. Nevertheless, the 

overall trends of the data points from Reference 14 (which 

also include the results of earlier simulations) are consistent 

with the ηa expressions in the papers of Davidson (Ref. 23), 

Gary (Ref. 24), and Drake et al. (Ref. 26). In fact, the practical formula for anomalous resistivity in the Brackbill paper 

was inferred (with an adjustable constant) by comparing the 

parameter dependencies of the particle heating rates in the 

simulation with the second-order heating rates obtained 

analytically by Gary. (Note, however, that Gary employed an 

electrostatic zero-β model.) 

We shall write down the simulation-based formula for ηa

as set forth in the Brackbill paper (Ref. 14), except that a 

factor 1/βi

 shall be suppressed. We believe that this inverseβ factor is not well founded. The ion β is defined by βi

 = 

8πPi/B2. The reasons for our reluctance to accept this β

factor, besides its absence in equation (48) of Drake et al. 

(Ref. 26), is that the Brackbill simulation results are compared with a template based upon a zero-beta electrostatic 

model, so that an inference about the beta dependence is not 

logically possible. In any event, the ion β would be of order 

1 within the magnetically confined plasma gradient. 

The formula suggested in Reference 14 for the effective 

anomalous resistivity then can be written without βi

 as 

2 2

Brack 2 th

ce de e 1 a i LH i pe

m V

C

m V

ω  

η =  

ω   ω  

 

 where 0.1 < CBrack < 0.4 (196) 

This symbol CBrack corresponds to 4πCDrake in the Drake 

et al. discussion above. The largest value of CBrack obtained 

in the Brackbill et al. simulations was CBrack = 0.4, and the 

smallest was CBrack = 0.1. The range of CBrack was related to 

the stage of evolution of the microturbulence and concomitantly of the evolving plasma gradient. When comparing this 

ηa to classical resistivity ηcl, we shall refer to the largest value 

of CBrack, 0.4. It is an open question as to what stage of evolution, in terms of Reference 14, is appropriate in the magnetic 

nozzle, as a given plasma element travels along the magnetic 

nozzle field in a macroscopic state of stationary flow. 

Reverting from the velocity ratio to the ratio of ion gyroradius to gradient scale length, and in the limit of small Te/Ti

, 

the simulation result, Equation (196), is compared with 

Davidson’s quasi-linear result, Equation (179). Since 

(Vd/Vth i)

2

 = 1⁄4(ai/xn)

2

 when Te = 0, the constant CBrack in the 

first-principles simulation result Equation (196) is about an 

order of magnitude smaller than that produced by the quasilinear wave-energy-bound model represented in Equation (179). The simulations in Reference 14 had Ti

 somewhat 

larger than Te (see Table I in Ref. 14), as in Davidson’s 

analytical case (Ref. 23), so the temperature ratio probably 

cannot account for this discrepancy. The discrepancy might 

have been understandable from the expectation that spreading of available energy within wave-vector space by wave 

coupling and energy loss to damped modes in the simulation 

would lead to a reduced level of fluctuations. However, 
NASA/TP—2009-213439 58 

Drake et al. include wave coupling, yet find a much larger 

numerical coefficient for anomalous resistivity ηa than does 

Brackbill et al. 

Drake et al. (Ref. 26) pointed out that if there were a 

channel for magnetic energy to be continually converted to 

plasma energy, then the wave-energy bound used in quasilinear theory, as limited by the electron-drift kinetic energy in 

the azimuthal current layer, would not be applicable. In 

principle, the level of microturbulent fluctuations could then 

increase beyond that value from the wave-energy bound, due 

to the essentially infinite supply of magnetic energy. In fact, 

the simulations in Reference 14 did find that magnetic 

energy was being continually converted into plasma energy. 

However, it was apparently being converted directly into 

thermal energy of the particles, rather than going into an 

increased level of microturbulent fluctuations. This result is 

understandable from the point of view that continually 

applied external magnetic pressure just keeps squeezing and 

heating the plasma. 

3.5 Comparison of Anomalous and Classical 

Resistivities 

The simulation result Equation (196) will now be compared with classical resistivity. However, we should be 

mindful that this may constitute an underestimate of anomalous resistivity in view of the results of Drake et al., which 

have the same parameter dependence but with a significantly 

larger numerical coefficient. Moreover, the latter result 

compares favorably with some experimental measurements. 

It is understood that all calculations are being done in the 

local longitudinal reference frame of the edge plasma. 

For comparison to classical resistivity, it proves convenient to rewrite the anomalous resistivity simulation result, 

Equation (196). The following identities are utilized: 

2 2

2 2

ce ci e

i pe pi

m

m

   ω ω        =

ω ω     

 (197a) 

1

2

i

LH ci e

m

m

 

ω =ω    

 (197b) 

A

pi ci

c V = ω ω

 (197c) 

1

2

i

pi pe

e

m

m

 

ω =ω    

 (197d) 

Here, VA is the Alfven velocity. Invoking these identities, 

after some manipulation, Equation (196) can be written as 

2

Brack

th

A 1 d a p e i

V V C

c V

 

η =  

ω    

 (198) 

Here, c is the speed of light in vacuum, in centimeters per 

second. Numerical calculations of this anomalous resistivity 

shall be performed at a representative position within the 

plasma gradient. 

To evaluate the factor (VA/c) in Equation (198), we utilize 

the cgs expression, VA = B 4πnimi . Mass mi

 is assumed 

to be that of the hydrogen ion. We wish to evaluate ηa in 

Equation (198) at a representative position within the density 

gradient that constitutes the plasma-field interface layer. 

Midway within the gradient, magnetic pressure (B2

/8π) is set 

to half the external magnetic (Bext ) pressure; hence, B2

 = 0.5

2 Bext . Also, ion number density ni

 is set to half the internal 

(core) number density ncore; hence, ni

 = 0.5ncore. Then pressure equilibrium 2 Bext = 16πncoreT is invoked across the 

entire layer width for external magnetic confinement of the 

internal plasma pressure. A uniform temperature T(ergs) 

is assumed, with T = Te = Ti

. The result is VA/c = 0.67 · 10−4

T ( ) eV , with the temperature T(eV) now measured in 

electron volts. Similarly, the electron plasma frequency in 

Equation (198) is given in cgs units by 

2 4 pe p e ω= π = nq m 5.6 · 104

p n , and the plasma 

number density np in the gradient is again set to half the core 

density. Thus np = 0.5ncore. Then, from Equation (198), the 

anomalous resistivity ηa at a representative position within 

the magnetically confined plasma gradient reads as follows: 

( ) 2

9

Brack

core th

eV

[1.7 ](10 ) d a i

T V C

n V

−  

η =      

 (199) 

The ratio of current drift velocity Vd to ion thermal 

velocity Vth i actually is profile dependent. The treatment of 

this quanity is discussed below. Of course, use of Equations (196) or (198) is preferred for ηa in a resistive MHD 

simulation, so as to provide better resolution within the 

edge-plasma gradient. 

The well-known classical resistivity ηcl across the magnetic field (Ref. 9) due to coulomb collisions of electrons 

with ions is practically independent of plasma number 

density. If we set lnΛ ≈10 , which is typical, then ηcl will 

read as follows: 

( ) 2

3

1.1 10 eV 13

cl − − η = ⋅ T (200) 
NASA/TP—2009-213439 59 

Both resistivities ηa and ηcl are measured in seconds in cgs 

units. Multiplication by the number 9 · 109

 converts them 

into mks values measured in ohm-meters. 

Either classical or anomalous resistivity can be employed 

in a simple Ohm’s law, to relate the azimuthal current 

density in the edge-plasma gradient layer to the motional 

azimuthal electric field V × B (the cross product of quasiradial diffusion velocity with longitudinal magnetic field). 

Their use in the simple Ohm’s law, however, requires that

the radial Hall voltage does not become shorted out by a 

flow of Hall current. As a practical matter, a resistive MHD 

simulation such as the MACH2 code can be set to run with 

the sum of the two resistivities. Then the dominant one will 

automatically be the effective resistivity. 

For a more systematic impression of the integrity of the 

plasma-field interface, it is of interest to describe conditions 

under which microturbulent (anomalous) resistivity is the 

larger of the two. From Equations (200) and (199), the ratio 

of classical to anomalous resistivity, where temperature T is 

in electron volts, is 

( )

1/2

cl core 4

2 Brack th

1 1 (10 ) 1.5 a d i

n

C T V V

η − = η

 (201) 

Classical resistivity clearly dominates at lower temperatures 

and higher densities. The two are equal, however (again, 

when temperature T is in electron volts), when 

1/4

2 core

Brack th

1 (10 ) 1.22 d i

n

T

C V V

− = (202) 

Numerical examples for which ηcl = ηa are tabulated 

below for CBrack = 0.4 and T = Te = Ti

. Our earlier analysis 

of the ion gyroradius relative to the initial interface width 

also confirmed the approximate constancy of that ratio (i.e., 

ai

/xn) from breech to throat. Note xn is the characteristic 

density gradient scale length, which is defined in terms of 

the ratio of the local ion number density to the local gradient 

of the ion number density. A special derivation is needed to 

relate that ratio, ai

/xn to the velocity ratio Vd/Vth i by virtue 

of the MHD equilibrium relation in the quasi-radial direction 

across the flux surfaces, within the interface layer. In this 

derivation it is assumed that the electron and ion temperatures are equal and uniform. Accordingly, we have chosen 

the example Vd/Vth i = ai

/xn = 1/3. 

There is a lack of precision, however, because the current 

drift velocity Vd (or also ion gyroradius ai

) depends upon 

location within the plasma-field mixing layer, and also 

because its thickness δ (=xn) as a multiple of (c/ωp i) is not 

precisely known (see Appendix I). Hence, a slightly greater 

value of Vd/Vth i = ai

/xn = 1/(2.5) is also considered. 

Figure 4 presents temperatures (eV) below which ηcl

(with lnΛ =10 ) is larger than ηa (Ref. 14 with CBrack = 0.4), 

for a given ion number density (cm–3). The two values for 

the ratio Vd/Vth i are plotted. The critical temperature according to Equation (202) is rounded to the nearest 10 eV. 

The results in Figure 4 ought to be insensitive to position 

along the nozzle between breech and throat. That is because 

density n at the throat is still about 2/3 of its breech value 

(and n

1/4 is used), and temperature T at the throat is still 3⁄4 of 

its breech value. Thus, at a hydrogen-ion number density 

1015

 cm−3, an envisioned stagnation temperature of 50 eV in 

the breech would represent a condition of classical resistivity, whereas 180 to 220 eV would represent a marginal 

condition where the two resistivities are about equal. The 

220 eV range is, in fact, envisioned in a recent space-mission 

systems study (see Ref. 4). 

In view of the results of Drake et al. (Ref. 26), the reader 

should be warned of the possibility that Figure 4 may 
NASA/TP—2009-213439 60 

wrongly downgrade the importance of anomalous resistivity. 

The critical temperatures in this figure actually depend on 

1 CBrack . If CBrack should have been 2 orders of magnitude larger, then the critical temperatures would be smaller 

by a factor 10. If such were the case, then the anomalous 

resistivity ηa would be totally dominant over the classical 

resistivity ηcl for most cases of interest. However, the fluctuation levels predicted by Drake et al. were somewhat too 

large compared with the experimental measurements, and it 

is the squares of those fluctuation levels that determine the 

anomalous resistivity. 

4.0 Rayleigh-Taylor- (RT-) Type 

Instabilities 

In this section, we primarily examine MHD surface instabilities of the flute-mode type, associated with adverse longitudinal curvature of the plasma boundary. To begin with, 

however, brief consideration is also accorded the unique 

conditions in the injection region associated with the initial 

plasma impact with the confining magnetic field. 

Two distinct regions involve instabilities of the RT type. 

The usual one involves propellant flow along magnetic-field 

lines of gradual adverse curvature. An effective gravitational 

force field arises from centrifugal deceleration. In the same 

region there are separate pressure-driven modes with similar 

characteristics, but for which there is no effective gravity 

because there is no macroscopic centrifugal force field. The 

pressure-driven modes require a separate treatment, which is 

supplied below. In magnetic nozzle flow, a combined instability occurs driven both by centrifugal effects and pressure. 

Therefore, growth rates for modes of the combined type will 

also be derived. The treatment is based on surveying the 

forces acting on a plasma element as it is displaced outwards 

in a region of adverse magnetic curvature. Treatments in 

the literature appear to be more mathematical and less 

accessible. 

For these instabilities, the vulnerable regions are where 

propellant approaches the throat of the nozzle and also near 

the nozzle’s exit, where the diverging flow is redirected 

axially by straightening out the diverging magnetic field 

lines (Fig. 2). The most dangerous RT-type instabilities in 

these two regions are usually considered to be flute modes, 

with azimuthal wave vectors, because these modes do not 

bend magnetic field lines; hence, they are energetically 

easier to destabilize than oblique modes directed partially 

along B. 

The other region to consider is the nozzle entrance region 

where plasma first impacts the magnetic field. That process 

involves quasi-radial expansion of injected hot plasma (in 

the direction ( ) ˆ xˆ = ×θ B B ) across the nozzle magnetic 

field. The plasma’s subsequent sudden deceleration and 

redirection along B necessarily creates a propellant-flow 

streamline with sharp adverse curvature. The problem is first 

addressed in the nozzle entrance, to identify the degree of 

adverse curvature inherent in the injected edge-plasma 

streamlines. After that, the RT-type instabilities occurring 

further downstream are addressed. 

4.1 RT-Type Instabilities at Initial Impact With 

Magnetic Field 

Plasma impact at approximately normal incidence to a 

surrounding magnetic field has been studied experimentally, 

with theoretical validation, by Ripin et al. (Ref. 17). Laserproduced plasma was generated at an aluminum surface, and 

the plasma puff then expanded across an ambient magnetic 

field. Fine-scale radial protrusions of plasma were observed 

at the plasma radius where deceleration of the plasma front 

by the magnetic field began to occur, and they were attributed to RT unstable conditions of deceleration at the 

plasma-field interface. Extensions of the MHD model to 

include Hall effects (Huba, Ref. 32) improved the relation of 

observations to theory. Evolution of RT structures was followed into the nonlinear regime, both experimentally and 

theoretically. Finite Larmor radius (FLR) effects (Huba, 

Ref. 33) were not invoked in Reference 17, possibly because 

the plasma temperature was too low for them to be relevant. 

Also, early experiments on theta-pinch implosions were 

observed in the implosion-onset phase to contain fine-scale 

radial protrusions around the circumference (McKenna, 

Ref. 34), but no detailed identification was attempted. 

In the magnetic nozzle, one might hope to mitigate RT 

instabilities by inserting propellant at grazing incidence to 

magnetic field lines, thus minimizing adverse curvature of 

the edge streamlines in the injection region. This point of 

view is adopted below. It will be seen, however, that the 

transition from oblique incidence of flow (relative to B) over 

to flow approximately along B is so sudden that the usual 

MHD model is not adequate. A more detailed model is 

needed to describe (and enable a more rigorous investigation 

of) propellant flow in the plasma-field interface layer at 

the breech region of the nozzle, which would enable RT 

instabilities. 

From the point of view of the single-fluid MHD model 

with Te = Ti

, the equation governing crossfield motion of 

plasma is, in mks units, 

2

d

d

y z x z x J B V B

x

P

t

V = = −σ ∂

∂

ρ + (203a) 
NASA/TP—2009-213439 61 

This can be rewritten schematically as 

n

i x x A

x

V

V

D

V

t

V

2

th 2

d

d

+ = (203b) 

The negative pressure gradient on the left in Equation (203a) 

has reappeared on the right in Equation (203b) as a positive 

term, and the negative resistive-drag term on the right has 

reappeared as a positive term on the left. The quasi-radial 

fluid velocity is Vx, D is resistive diffusivity, VA is Alfven 

velocity in the gradient layer, and the reciprocal gradient 

length is −1 −1 −1 xn = n ∂ xn = δ . When the inertial term, 

dVx/dt, is neglected, one recovers the quasi-radial resistive 

diffusion velocity that was identified near the 

beginning of Section 2.0. The magnetic flux is assumed to 

have no give because it has already been squeezed against 

the metallic wall (coil) by the plasma, and a state of 

stationary flow has already been reached. Equation (203b) is 

to be interpreted in a Lagrangian sense of following a given 

plasma element. 

The solution of Equation (203b) is 

trans trans core (1 ) tt tt VV e V e xx x − − = +− ∞ (204) 

The subscript “core” now refers to the initial value of the 

crossfield velocity when the considered element of plasma 

first encounters the magnetic field at the nominal plasmafield interface. The subscript “∞” signifies the eventual 

crossfield resistive-diffusion velocity attained at large times 

after the inertial transient has dissipated. Vx ∞ is realistically 

assumed to be very small compared with Vx core. The time 

interval over which the inertial transient lasts, ttrans, is the 

time during which there is appreciable crossfield velocity, in 

terms of the incident velocity of the plasma element. 

trans 2

A

D

t

V

= (205) 

For parameters of interest, this transient lasts only a fraction 

of a nanosecond, which signifies that the macroscopic 

single-fluid MHD model is intruding into the purview of 

microphysical processes. This cautionary conclusion is 

further verified when one considers the radius of curvature 

of the trajectory of the considered edge-plasma element, 

which ordinarily would provide information relevant to 

growth rates of RT instabilities. Pursuant to calculating the 

longitudinal radius of curvature R, the distance the considered plasma element travels along B during time ttrans is 

approximately 

core trans core trans 2

A

V D z Vt

V ≈ = (206) 

valid for a shallow (i.e., grazing) angle of incidence. 

Let φB V(rad) denote the incident angle between B and V 

at the nominal plasma-field interface. Then, sin φB V ≈ φB V

≈ Vx core/Vcore << 1. Simple geometrical considerations show 

that the radius of curvature R of the trajectory can be 

expressed approximately as 

trans core A

A

z VV D R

V ≈ = φ φ BV BV

 (207) 

Equation (207) shows that R is very small for parameters of 

interest in this report, although the angle φB V is 

moderately small (for grazing incidence). For example, 

R ~ 0.01 – 0.001 cm. This follows from typical values of the 

resistive diffusivity D (~104

 cm

2

/s), and of the Alfven velocity VA (~107

 cm/s), jointly producing a very small length 

D/VA. (The impacting velocity Vcore is only smaller than Vth i

by a moderate area ratio, and Vth i ~ VA upstream of the 

throat. Hence, Vcore/VA is only moderately small.) In reality, 

“microscopic” lengths properly incorporated into a more 

complete plasma model, such as the ion gyroradius ai

 or the 

ion skin depth c/ωp i, both ~1 cm, would provide lower 

bounds to R. 

The single-fluid MHD model has proven inappropriate for 

a quantitative description of the initial trajectory of an edgeplasma fluid element at “magnetic impact,” although it is 

incident at a shallow angle with magnetic flux at the interface. Nevertheless, the message is clear. From a macroscopic 

point of view, the plasma element’s incident trajectory onto 

the external magnetic flux undergoes an instantaneous transition into motion practically along B, due to eddy current 

braking from electron resistivity. 

A large RT growth rate, γRT, is implied by the small 

radius of curvature of the incident trajectory, at its point of 

impact with magnetic field. But the importance of RT instabilities in that region yet depends upon the time tcur during 

which the plasma element resides in the highly localized 

region of adverse curvature. The product γRTtcur determines 

the number of e-folds available for the RT instability in that 

region. The RT instabilities in the plasma-field impact region 

therefore should be examined with the use of appropriately 

detailed plasma models beyond the ordinary MHD fluidbased model (see, for example, Ref. 35). The incident internal streamlines that impact the external magnetic flux at the 

plasma-field interface in the injection region (breech) 

unavoidably acquire a segment of severe adverse curvature 

not treatable under ordinary MHD. 
NASA/TP—2009-213439 62 

4.2 RT-Type Instabilities Downstream From 

Injection Region 

We turn now to the topic of RT instabilities downstream 

from the injection region but upstream of the throat, where 

the cross section of the plasma shrinks to form the throat. 

There, the transition to field-aligned flow has already been 

made, and the plasma propellant streamlines near the 

plasma-field interface closely adhere to the magnetic-field 

line shape because of the high electrical conductivity of the 

plasma. Section 3.0 addressed gradient-driven microinstabilities localized within the plasma-field interface, which 

cause broadening of the interface layer by resistive diffusion. 

Here, Section 4.0 addresses macroscopic instabilities that 

cause the interface to deform (to wrinkle) as an entity. This 

process of macroscopic interface deformation is a consequence of instability wavelengths that are large compared 

with the width of the interface layer. The practical consequence is a loss of nozzle-based control of the flow of 

propellant. 

The usual fluid-based MHD model can be applied to 

describe RT flute instabilities, provided thermal excursions 

of individual particles are short compared to characteristic 

macroscopic lengths in the nozzle. Examples of such 

macroscopic lengths are the plasma radius r, the radius of 

curvature of field lines and/or streamlines R, and long flutemode wavelengths in the azimuthal direction 2π/k, with k as 

the azimuthal wave number. It is similarly required that 

times for those thermal excursions be short compared to 

macroscopic times such as longitudinal transit times and 

instability growth times (e.g., see Ref. 9). 

An example of a thermal excursion length across the 

magnetic field is the nominally small ion gyroradius, which 

takes place during a nominally short ion gyroperiod. An 

example along the field is the mean free path for ion-ion 

coulomb collisions, which transpires during a mean free time 

between ion-ion collisions. The electron mean free path is 

about the same as the ion mean free path. For cases of interest here, mean free paths λ along B are several centimeters 

(for n ~ 1015

 cm−3 and T ~ 50 to 100 eV) and are generally 

much smaller than meters. 

Within the core plasma, from which the nozzle magnetic 

field has been excluded by the azimuthal current layer, the 

thermal excursions are governed by just the mean free paths 

and mean free times for coulomb collisions of charged particles. Thermal excursions in the core plasma are unrestricted 

by the magnetic field, and so they take place in all directions. 

If λ is several centimeters, then not only the characteristic 

axial dimension should be at least tens of centimeters, but 

the radial dimension should be as well. 

These restrictions on thermal excursions of individual particles are meant to ensure that any given fluid element retains 

its identity (undergoes very little random thermal dispersal) 

over the relevant macroscopic lengths and times of interest 

in the device. Then the use of a fluid model makes sense. 

Sometimes, though, fluid-MHD simulations are applied to 

hot plasma configurations without raising cautions that the 

above restrictions on thermal excursions may not always be 

well satisfied. At lower densities and higher temperatures 

relative to the nominal working values of interest for this 

report, a kinetic multispecies model may have to supplant the 

basic single-fluid model of MHD. Such models have been 

developed in the magnetic confinement fusion (MCF) community, but for different purposes; nevertheless there is some 

expertise in the MCF community in global-kinetic modeling. 

In the discussion here of RT instabilities downstream 

from the injection region, we shall assume that the conditions for a single-fluid model are fulfilled. An exception is 

made in Section 4.4, “Short-Wavelength Gravity-Driven 

Instabilities,” where short-wavelength instabilities that are 

related to RT physics are discussed. 

The present subsection first derives in a heuristic manner 

the RT-type instabilities in a static field-free plasma (without 

flow), confined by pressure equilibrium with the external 

magnetic field and situated within a region of adverse magnetic curvature. The derivation is carried out by considering 

the forces acting on a plasma element under the region of 

adverse curvature. Then the modification of the derivation to 

incorporate the longitudinal flow of propellant is presented. 

The instantaneous RT growth rate is derived at any point of 

adverse curvature along the flow, due to the combined 

effects of plasma pressure and propellant flow. Then it is 

converted to a practical expression in terms of specific 

impulse. In the course of the derivation, it will become evident that the growth rates from the two effects acting separately actually represent two distinct instabilities. 

Finally, finite gyroradius stabilization due to collisionless 

ion gyroviscosity is used to estimate the shortest azimuthal 

wavelengths that are RT unstable. Their growth rates are 

compared with axial transit times through hypothetical 

regions of adverse curvature. On the basis of these calculations, it is recommended that the magnetic nozzle be (selfconsistently) shaped so as to have no regions of adverse 

magnetic curvature, to the extent possible. 

In the following presentation, we do not attempt to consider longitudinal global effects on the structure and growth 

rates of flute instabilities, which would be induced by finitelength regions of adverse curvature. The treatment of such 

effects actually constitutes the subject of ballooning modes, 

which is sufficiently complicated as to deserve its own special presentation. The subject of ballooning modes in longitudinally flowing plasma therefore is beyond the purview of 

the present report. However, such work was carried out 

within the quest for magnetic fusion energy by Nagornyj, 

Ryutov, and Stupakov (Ref. 36) for plasmas confined longitudinally by magnetic mirrors. That configuration is closely 

related to the magnetic nozzle concept. They found that 

ballooning instabilities originating in regions of adverse 

curvature could be stabilized by the presence of nearby 

regions of strong good curvature, if such regions were sufficiently loaded with plasma. (The nozzle throat and the mirror 
NASA/TP—2009-213439 63 

throat are regions of good curvature.) Even so, they found 

that there remained residual instabilities. The results of 

Nagornyj, Ryutov, and Stupakov are not easily utilized in a 

practical manner for the problem of plasma flow in magnetic 

nozzles. That work ought to be revisited within the present 

context. Of course the entire problem could be avoided by 

eliminating regions of adverse curvature. Due to the selfconsistent modification of the vacuum magnetic nozzle field 

by the injection of hot plasma, this approach seems to imply 

that the nozzle be rather long and gradual. 

4.2.1 Surface Instability of Static Plasma in Region of 

Adverse Curvature 

We consider a static field-free plasma configuration of uniform pressure Ptot and mass density ρ, confined by the pressure of external magnetic field B, 0

2 B 2μ , in a region of 

equilibrium adverse curvature of radius Req (the center of 

curvature lies within the plasma). This is illustrated in 

Figure 5. The plasma pressure and magnetic pressure is 

acting on the rim element of width dr (<<Req), subtending a 

very small meridian angle dφ. Those radial forces must sum 

to zero in equilibrium. The propellant velocity flow vector is 

V. Centrifugal force density (not shown) is outward, with 

magnitude ρV2

/R, and R ≈ Req. Outward radial displacement 

ξ of the rim element as a whole (with dR and dφ held fixed) 

is monitored for instability growth rate. 

The forces are the pressures multiplied by their respective 

surface areas. In calculating the various surface areas of the 

rim element, the element is assumed to extend an azimuthal 

distance dy perpendicular to the plane of Figure 5. For brevity, 

dy will be suppressed in the equations. For example, the 

surface area associated with the outer arc of the rim element 

in equilibrium is just written as Reqdφ, instead of Reqdφdy. 
NASA/TP—2009-213439 64 

The elongated flute mode analyzed here is energetically 

favorable because, unlike oblique modes, it does not bend 

field lines. Also, it is basically a surface perturbation, which 

has only a limited influence beneath the surface. The width 

of the rim element in Figure 5, dR, is chosen to encompass 

the range of influence under the surface. Farther below the 

surface than dR, the fluid remains practically undisturbed by 

the presence of the surface perturbation. The specific interpretation of dR will be made clear below. 

In terms of Figure 5, radial forces are also vertical forces, 

to lowest order in small d. Therefore, in equilibrium, the 

total radial force dFR acting vertically on the element as a 

whole (see Fig. 5), to lowest order in d, then reads 

 (208) 

in mks units. The first term in Equation (208) is the outward 

radial (vertical) force acting on the inner arc (i.e., inner 

surface) of the rim element. The second is the radial (vertical) force due to lateral pressure acting on the canted end 

surfaces of the element. The effective radial component, or 

vertical component, of lateral pressure acting on each end of 

the rim element, which is experienced by the rim element as 

a whole, is just ; the surface area of each end 

is dR (see Fig. 5). The third term in Equation (208) is the 

inward radial (vertical) force due to external magnetic pressure on the outer arc (outer surface) of the rim element. The 

notation used is that B at radius R = Req is denoted as Beq. 

After canceling some terms in Equation (208), the equilibrium condition reduces to 

 (209) 

This condition will be utilized in the following discussion of 

the dynamical behavior of the rim element as it undergoes a 

slight displacement. 

Now consider the rim element as being displaced outward 

radially by a small distance  (see Fig. 5). Both the inner and 

outer arcs are displaced outward by distance , while keeping a fixed separation dR and holding fixed the angle d. The 

plasma fluid both upstream and downstream of the considered rim element is likewise displaced in the meridional 

plane, in the elongated flute mode being discussed. In this 

connection, it is important to note that the external magnetic 

field in the vacuum region, into which the displaced plasma 

will protrude, is completely determined in the local neighborhood. In particular, its R-dependence is determined. 

The local dependence of external B on R can be demonstrated with the aid of a stream function , with B =   ( ) 

=  , and 2

 = 0 from   B = 0 in vacuum. Here, the 

symbol 2

 represents the two-dimensional Laplacian operator in the plane of Figure 5, and is a locally constant unit 

azimuthal vector out of that plane. The Laplacian operator is 

expressed in local cylindrical coordinates based upon the 

local radius of curvature, and the boundary condition on B is 

applied at the equilibrium plasma surface. The salient feature 

of the vacuum magnetic field just above the plasma surface 

is that it depends locally on the radius (for small ) as 1/R, 

with R = Req + . It is important to note that the linearly 

small amplitude  is much smaller than the flute mode’s 

azimuthal wavelength out of the plane of Figure 5. 

Concomitantly, the external magnetic flux is not constrained to accumulate over the plasma’s outward-moving 

conducting protrusion, because the protrusion actually is of 

limited extent in azimuth (perpendicular to the plane of 

Fig. 5). The external longitudinal magnetic field lines are 

shunted azimuthally to either side of the vertically extended 

conducting surface instead of piling up on top of it, thereby 

maintaining a state of lower magnetic energy. That is why 

the external magnetic field magnitude B, just above the crest 

of the protrusion, can be taken as being undistorted by the 

presence of the flute mode. 

Consequently, the field magnitude B just above the crest 

of the protrusion is related to the field Beq at the equilibrium 

surface by 

 (210) 

The local spatial behavior of the longitudinal magnetic 

field just outside of the plasma constitutes the principal 

reason for the adverse curvature instability. As displayed in 

Equation (210), the field strength becomes locally weaker 

going away from the plasma, in the region of adverse curvature. Consequently, a plasma element that is displaced outwards is subjected to a smaller magnetic backpressure. Thus, 

the plasma displacement continues to grow outwards. 

This critical behavior of the spatial variation of the longitudinal magnetic nozzle field can be directly exhibited by 

utilizing Ampere's law, which is in mks units. 

By means of Stokes's theorem, Ampere's law can be converted to an integral representation, which reads 

. Here, the indicated loop integral is taken 

around a closed contour, with I being the net current through 

the contour. In the case being considered, the contour is 

around a wedge-shaped rectangle similar to that shown in 

Figure 5. In the present situation, however, the inner leg of 
NASA/TP—2009-213439 65 

the rectangle lies just below the perfectly conducting sharp 

boundary, where B = 0. The outer leg of the rectangle lies a 

displacement distance ξ above the boundary, say at R = Req

+ ξ, in the notation of Figure 5. Then the integral form of 

Ampere's law reduces exactly to BRdφ = μ0KReqdφ, where K

is the azimuthal surface current density on the boundary. 

Thus, the field strength just beyond the boundary is B = 

μ0KReq/R, which is exactly equivalent to Equation (210). 

Although an equilibrium boundary was assumed in deriving Equation (210), the identical result for B(R) is also 

obtained when taking into account the presence of flute 

deformations of the boundary. This result will be shown 

explicitly in Appendix K. 

Our simplified physical derivation of the RT type of 

instability adapts the viewpoint of a plasma element defined 

in the meridional r,Z-plane. The forces operating on the 

plasma element are identified and summed within this plane. 

This approach highlights the role of adverse longitudinal 

curvature, which clearly must influence a mode that is 

extended along the magnetic field (flute mode). 

On the other hand, we suppress physical effects from 

cylindrical geometry associated with curvature the short way 

around. This approximate treatment is valid for small azimuthal wavelengths. By looking downstream at the r,θ-plane, 

one sees that a very localized (in azimuth) flute-mode crest 

can scarcely be “aware” of the cylindrical curvature. The 

only reminder of cylindrical geometry is that there be an 

integer number of wavelengths around the circumference, to 

avoid discontinuous behavior. This integer is the so-called 

azimuthal mode number M θ, and the associated azimuthal 

wave number is then 

 k = M θ/r (211) 

where r is the cylindrical radius. 

Knowing the radial dependence of the external magnetic 

field allows us to write the equation of motion of the rim 

element in terms of its displacement ξ. The equation of 

motion reads 

( )

2

eq 2

d dd d

d

RR FR

t

  ξ ρφ =      

 (212) 

Here, ρReq dRdφ is the mass of the rim element (suppressing 

dθ), to zero order in ξ. Higher order corrections in ξ to this 

mass element are not relevant in a linear theory for small ξ

because the acceleration in Equation (212), 2 2 d ξ dt , is 

already first order in ξ. Similarly, on the right side of Equation (212), the total radial (vertical) force on the displaced 

rim element, dFR(ξ), needs to be computed only to first order 

in ξ. That force is 

( ) ( )

( ) ( )

tot eq tot

2 2

eq eq

eq 2 0

eq

d d d dd

d

2

F PR R PR R

B R

R

R

ξ = +ξ− φ+ φ

− + ξ φ

μ + ξ

 (213) 

The first term in Equation (213) is the outward force on the 

displaced inner arc (surface) of the rim element. The second 

term is the outward force on the rim element as a whole due 

to the lateral pressure at its canted ends (still of radial width 

dR and subtending angle dφ). The last term is the inward 

magnetic force on the displaced outer arc. Equation (210) 

has been used. 

Actually, the effective lateral pressure (averaged over dR) 

is smaller than indicated in Equation (213), second term. 

Continuity of pressure at the displaced outer surface (to 

avoid unphysically large acceleration of the outer surface) 

requires that the internal plasma pressure should spatially 

decrease so as to approach the reduced external magnetic 

pressure there. Taking this feature into account, however, 

only proves to introduce a small correction of order dR/Req. 

This remark will be quantified below. 

After some cancellations in Equation (213), and expanding to first order in ξ and invoking the equilibrium condition, 

Equation (209), the force in Equation (213) becomes simply 

( ) tot d 2d F P R ξ = ξ φ (214) 

We substitute the first-order force equation, Equation (214), 

into the equation of motion, Equation (212) and look for a 

temporally growing displacement dξ/dt = γRTξ having RT 

growth rate γRT. After further cancellations, the equation of 

motion reduces to an expression for γRT: 

2 tot

RT

eq

2

d

P

R R

ρ γ = (215) 

At this point, it is important to recognize that the most 

energetically favorable motions of plasma in producing the 

flute instability are those that do no compressive work; that 

is, the internal fluid motions in the plane perpendicular to B

are incompressible. For such incompressible motions, a flute 

mode at the plasma surface with the azimuthal dependence 

e

iky has only a limited influence beneath the surface, with 

the radial dependence e

−k x , where x (>0) is the distance 

beneath the surface (e.g., Ref. 37). To derive this feature, 

write the fluid momentum equation for small perturbations 

in the plane perpendicular to B as ργRT{δV} + ∇{δPtot} = 0. 

Then the condition of incompressibility ∇ ⋅ {δV} = 0 implies 

that ∇2

{δPtot} = 0. Inserting the azimuthal mode dependence 

e

iky into Laplace’s equation for the pressure perturbation 

δPtot confirms the stated result. 
NASA/TP—2009-213439 66 

Therefore, the effective depth of the rim element influenced by the surface mode is dR = 1/k, with k being the 

azimuthal wave number. Plasma fluid at lower depths is 

practically undisturbed by flute-mode perturbations at the 

plasma surface. Thus, we interpret dR as 1/k in Equation (215). The RT-type growth rate then becomes 

tot

RT

eq

2P k

R

ρ γ = (216) 

Note that 2Ptot/ρ ≈ S

2

, where S is the speed of sound in the 

plasma. Equation (216) thereby has the form of a gravitational instability of the RT type; namely γ = gk RT , in 

which g ≈ S

2

/Req. The effective gravitational acceleration g

might therefore be interpreted as the centrifugal force of 

charged particles (of small gyroradius) that move approximately with thermal velocity in either direction along the 

curved magnetic field at the plasma surface. Nevertheless, 

this flute mode is not a true gravitational instability because 

no effective gravitational field permeates the plasma. 

To check the assumption of incompressible motion, note 

that the plasma’s velocity ξ = γ ξ d d RT t , as well as the 

effective wave velocity γRT k , should both be small compared to the speed of sound S. The first condition is satisfied 

because ξ is arbitrarily small. The second condition reduces 

to kReq >> 1. The azimuthal wave number is k = Mθ/r 

(Eq. (211)), where r (distinct from Req) is the cylindrical 

radius of the propellant, and Mθ is the azimuthal mode number (i.e., e

iky = i Mθ θ

e ). Therefore, the second condition for 

incompressible motion is satisfied whenever the radius of 

curvature Req is large compared with r, the propellant’s 

radius, or when Mθ >> 1. 

Finally, we return to the requirement of continuity of 

pressure at the outer displaced surface of the rim element. 

The above model, taken literally, yields the following 

expression for the fractional discontinuity of pressure there 

(we omit the details): 

( )

eq

2

tot 0

tot eq

2 d 1

d

R R

P B R

P RR

= +ξ

  − μ       ξ = <<               

 (217) 

expanded to first order in ξ and using the equilibrium relation, Equation (209). Since dR is comparable to an azimuthal 

wavelength, linear theory for small amplitude motions 

requires that ξ << dR. Hence, the first factor on the right has 

to be small in the linearized dynamical model. The second 

factor is also small, assuming that azimuthal wavelengths of 

the considered flute modes are small compared to the radius 

of curvature of the magnetic nozzle field. Consequently, in 

this limit, the error in continuity of pressure at the displaced 

interface is negligible. 

4.2.2 Surface Instability of Flowing Propellant in Region 

of Adverse Curvature 

The previous example serves to illustrate the methods to 

be applied to flowing propellant in the magnetic nozzle. It 

will be assumed in this section that the external magnetic 

field diffuses only a small distance into the plasma compared 

with the thickness of the rim element. The previously 

derived pressure-driven contribution to the growth rate 

persists, but proves to be augmented by the centrifugal force 

effect of the curved flow. The practical expression for the 

growth rate, derived below, would be useful offline, in postprocessing of axially symmetric resistive MHD simulations 

of magnetic nozzle flow. In this manner one could follow the 

flow and the self-consistent nozzle geometry and thereby 

estimate the extent of e-folding of flute instabilities within 

regions of adverse curvature, without the necessity of running three-dimensional simulations. 

The centrifugal force density due to propellant flow with 

velocity V// is here denoted by Fg and is locally defined as 

 

2

//

g

V F

R

ρ = (218) 

where ρ is propellant mass density, and radius R (≈ Req) lies 

within the rim element depicted in Figure 5. This is equivalent to gravitational force density ρg with 2

// g =V R . 

In this case, there are two relevant equilibrium conditions: 

a local one and a global one. Local internal equilibrium 

becomes nonuniform, due to the effective gravitational field 

that now permeates the plasma. The condition of internal 

equilibrium is 

 R g P F tot ∂ = (219) 

Here, as usual, Ptot is the local, total plasma pressure due 

to both electrons and ions. 

The condition Equation (219) holds not only for the equilibrium configuration, but also for the time-dependent perturbed state with growing displacement ξ. As long as the 

radial transit time for sound propagation through depth dR is 

short compared to the growth time of the mode, Equation 

(219) holds. The latter condition ultimately can be expressed 

as γRT/k << S, which is the same as the condition for incompressible motion mentioned in the previous example. 

We turn now to the condition of global equilibrium, which 

applies to the rim element as a whole. The notation P0 refers 

to plasma pressure at the inner arc of the element, and 
NASA/TP—2009-213439 67 

1

2

1 0 tot 2 P P RP ≈+ ∂ d R represents pressure halfway up on 

the width dR and is representative of the average lateral 

pressure on dR. The centrifugal force density Fg will be 

regarded as an average value within the volume of the rim 

element. Then, equilibrium force balance on the rim element 

of Figure 5, radially (or vertically), reads

( ) 1

2

eq

0 eq

2

eq

eq eq

0

d d dd

1 d dd d

2 2 g

R R

PR R P R

B

FR RR R

=

− φ+ φ

      + − φ = φ     μ     

 (220) 

The first term in Equation (220) is the outward force acting 

on the lower surface of the rim element, and the second term 

is the outward force on the element due to lateral pressure 

acting on its canted ends. The third term, analogous to the 

effect of gravity on the element, is the outward force due to 

the centrifugal force density Fg multiplied by the volume of 

the element ( ) 1

eq 2 R RR − d ddφ . An average (midvalue) has 

been used for the arc-length in the volume element. The right 

side of Equation (220) constitutes the inward force acting on 

the element due to the external magnetic pressure acting on 

its outer surface. Dividing through by Req dφ using 

1

2

1 0 tot 2 P P RP =+ ∂ d R and using Equation (219), Equation (220) then can be written exactly as 

2

eq

0

0

d

2 g

B

P FR

 

+ =  

  μ  

 (221) 

Equation (221) is the consequence of global equilibrium of 

the rim element in Figure 5. It is as if tot d d (F R) P R g R = ∂ is 

the pressure increment to be added to the inner pressure P0

to ensure pressure continuity across the outer surface of the 

rim element. 

The linear extrapolation of pressure within dR, 

1

2

1 0 tot 2 P P RP =+ ∂ d R , assumes that the first two terms in 

the Taylor series for the pressure profile are sufficient; that 

is, the pressure gradient term is assumed to constitute a small 

correction. This formula, used to obtain Equation (221), 

proves to require the strong inequality (dR/Req)V2

 << S

2

(refer to Eq. (219)). Because a large radius of curvature is 

assumed in the sense dR << Req, it follows that the equilibrium model allows transonic and supersonic flows within dR. 

Let us define dR = 1/k for the flute-mode wave number. 

Since k = Mθ/r (Eq. (211)), the above inequality then can be 

put in the form (V2

/S

2

) << MθReq/r. In this form, transonic 

and supersonic flows can be handled within the dynamical 

model. The required geometric conditions are that the radius 

of curvature Req be large compared with the cylindrical 

radius r or that the azimuthal mode number Mθ is large 

compared to 1. The same linear extrapolation of the pressure 

profile (with its same consequence) is relevant to the next 

problem. 

We turn now to the dynamical problem, which is to calculate the net force on the plasma element as it is displaced 

outwards by the amount ξ, as in the previous example. As 

before, that force will be inserted into the equation of motion 

(Eq. (212)) of the element, so as to obtain the growth rate of 

the instability. Superscript ξ will refer to the state of the 

superscripted quantity as it exists in the displaced element; 

for example, ξ P0 refers to the pressure on the displaced inner 

surface of the rim element, and ξ

2

P1 refers to the pressure 

halfway up on dr in the displaced element. 

The force on the rim element is written as 

( )

1

2

0 eq

1

eq 2

2 2

eq eq

eq 2 0 eq

d ( d )d d d

d dd

( )d 2 ( )

R

g

F PR R P R

FR R R

B R

R

R

ξ ξ

ξ

= − +ξ φ+ φ

+ − +ξ φ

 

− +   ξ φ   μ + ξ  

 (222) 

Here, the first term is the outward pressure-force on the 

displaced lower surface, and the second term is the outward 

force on the element from lateral pressures at the canted ends 

(still of width dR and subtending angle dφ). The third term is 

the outward gravitational force density multiplied by the 

displaced volume using its average arc length. Finally, the 

last term is the inward magnetic-pressure force on the displaced outer surface of the element. The 1/R dependence of 

the external magnetic field, with R = Req + ξ, has been 

invoked. 

The above-mentioned linear extrapolation within dR for 

ξ

2

P1 , and the local equilibrium relation Equation (219) are 

used, as well as the global equilibrium Equation (221) as it 

regards 2 Beq . Expanding expressions to first order in ξ, 

Equation (222) then becomes 

( ) ( )

( )



















+ + + ξ +

− + −

= φ ξ ξ

ξ ξ

P F R P F R

R

P P F F R

F R

g g

g g

R d d

d

d d

0 0

eq

0 0

eq (223) 

To first order in ξ, the final square bracket in Equation (223) 

may as well be written (P F R) g 2 d 0 + . 

The first term on the right in Equation (223) can be written as ξ∂RPtot (= ξFg). The physical reason for being able to 
NASA/TP—2009-213439 68 

write it this way is that the displaced fluid element climbs up 

the preexisting equilibrium pressure profile, because rising 

fluid at the displaced lower surface of the element is replaced 

at that level by azimuthal inflow of fluid from either side. 

The replacement fluid brings with it the original equilibrium 

pressure-profile at that level. (Note that the fluid below dR

remains practically undisturbed.) 

The second term on the right of Equation (223), due to the 

rising fluid, involves spatial variation of density ρ and position R (supposing zero-order flow velocity V// is constant). It 

can be written as ξ∂RFg dR ( ) ( ) 1 1 d F RL R g − − =ξ − ρ . Here, 

1 1 L R − −

ρ = ρ ∂ ρ is the inverse scale length for radial variation 

of propellant density in the core plasma (not in the edge 

layer). The core plasma is assumed to be fairly uniform, with 

Lρ ~ R >> dR. Then the entire second term, on the order of 

ξFg dR R , can be neglected compared with the first term, ξFg. 

The third term is 2 d ( ) P F R R 0 eq + ξ g ( ) . The gravitational 

contribution (from Fg) is again small compared with the first 

term in Equation (223) because dR << Req. 

Therefore, neglecting terms on the order of dR/Req in 

Equation (223) compared to the first term, Equation (223) 

can be written simply as 

eq 0

eq

d d2 FR F P R g R

    ξ = φξ +      

 (224) 

Using this force in the equation of motion (Eq. (212)) of the 

rim element together with the temporally growing representation dξ/dt = γRTξ yields the following result for γRT: 

2

2 tot eq // eq tot eq

RT

2 / ( / ) (2 / )

d d

F P R VR P R g

R R

+ +ρ

γ= =

ρ

(225) 

Here, V// refers to the velocity of propellant flow parallel to 

the magnetic field B. It does not really matter in Equation (225) whether P0 or Ptot is used anywhere in dR, since the 

difference is small, on the order of dR/R compared with Fg. 

The previous discussion about incompressible perturbation flow in flute-mode dynamics (in the plane perpendicular 

to B as before) provided that the plasma properties (ρ, Ptot, 

and V//) vary only weakly within dR. Then, the periodic 

azimuthal mode structure e

iky again implies that the radial 

mode structure e

−k x decreases exponentially beneath the 

surface. Thus, we again have the relation dR = k

−1

 in terms 

of the azimuthal wave number k. The growth rate, which 

now includes the effects of both flow and pressure in the 

adverse-curvature region, finally can be written as follows: 

2

// tot

RT

eq

V P2 / k

R

  + ρ γ =  

 

 (226) 

The effective gravitational accelerations in regions of 

adverse curvature, which arise both from thermal motions of 

particles and from propellant flow, just appear in an additive 

manner within the growth rate. This is what would be 

expected intuitively. Bidirectional diffusion of plasma and 

magnetic field across the fluted surface will then create a 

plasma-field mixing layer. 

As before in the pressure-driven case, the model used to 

derive this growth rate suffers from a pressure discontinuity 

at the outer displaced surface of the plasma. Nevertheless, it 

can again be demonstrated that the fractional pressure discontinuity reduces to (ξ/dR)(dR/Req), which is very small for 

the two conditions: linearly small amplitudes ξ << dr, and 

dr << Req. We shall omit the details of the demonstration. 

The purpose of this section is to provide rough guidance 

as to whether the idealized RT instability has a chance to 

grow. Larc is the complete length of arc in the adverse curvature region at the plasma edge (convex outwards), and is not 

necessarily small. Concomitantly the subtended longitudinal 

angle in radians is = Larc Req δφ . This finite longitudinal 

angle is to be distinguished from the “small” longitudinal 

angle used in the simplified model, dφ. Nevertheless, as a 

simplification, we shall assume that the growth rate, Equation (226), is still valid. Part of this growth rate contains the 

expression 2P ρ , but the speed of sound is S

2

 = ƔP/ρ, and 

we shall take Ɣ = 5/3. Then, 2

5

6 2P / ρ= S . The condition for 

growth is 1 2

cur 2γRT t > , with tcur denoting the time the edge 

plasma spends in the adverse curvature region. This time is 

cur arc // t = L /V . We use Equation (226) in the growth condition 1 2

cur 2γRT t > , but with the azimuthal wave number k

replaced by Mθ/r, where Mθ is the number of azimuthal wavelengths that fit into the circumference, 2πr, the short way 

around. Then, the condition for growth can be expressed as 

1

2

//

2

5

arc 6 ( ) [1 ] θ − δφ > +

V

S

r

M L (227) 

The quantities 2 S and 2 V// should be taken as average values along Larc, which can be determined from axi-symmetric 

ideal MHD simulations. Ion magneto-viscosity sets an upper 

 
NASA/TP—2009-213439 69 

limit to Mθ (see Eq. (234)). Except for a complete treatment, the Hall effect (which is destabilizing) should also be 

included. This is examined in Section 4.3, “Finite Larmor 

Radius (FLR) Stabilization of RT-Type Flute Modes,” and in 

Section 4.3.1, “Example: Estimate of Hall Term in Coaxial 

MHD Thruster.” 

Equation (226) can be converted to a practical formula for 

the growth rate that is characterized by Vex, the velocity of 

propellant at the nozzle’s exit plane (the specific impulse is 

given by Vex m/s ÷ 9.8 m/s2

). The practical formula could be 

utilized as a postprocessing tool with axially symmetric 

MHD simulations of propellant flow in magnetic nozzles. 

To derive the practical formula, note that the local speed 

of sound, squared, is given by S

2

 = ƔPtot/ρ. Then, by means 

of the Bernoulli equation with the choking condition at the 

throat, using the adiabatic relations along the longitudinal 

flow, and using Equation (211), Equation (226) can be 

written as 

2/3

RT ex

eq

3 1

5 b

M

V

R r

θ

      ρ   γ= −       ρ        

 (228) 

The adiabatic index (ratio of specific heats) is taken to be 

Ɣ = 5/3 (i.e., there are occasional collisions), and relations 

based upon infinite contraction and expansion ratios in the 

nozzle geometry are utilized. However, the Bernoulli flow 

solutions are such that the results are hardly changed by 

using realistic values such as 2 for radius contraction and 3 

for radius expansion. 

Mass density ρ varies along the flow, starting with its 

breech value ρb and finally vanishing at the exit plane. After 

taking the square root in Equation (228), the numerical factor 

due to this density variation ranges from 0.63 in the breech 

to 1.0 at the exit. Therefore, the density expression can be 

omitted by instead inserting an average numerical factor of 

0.8, which is within ≈30 percent of the exact numerical value 

at any point along the flow. Then Equation (228) reads 

RT ex

eq

0.8

M

V

R r

θ  

γ ≈      

 (229) 

The radius of curvature Req and the cylindrical radius of 

propellant r also vary along the flow, as could be monitored 

in MHD simulations. The Req is interpreted as positive in 

regions of adverse curvature and negative in regions of good 

curvature. In the latter case, the RT-type flute mode is a 

gravity wave that just oscillates without growing. The azimuthal 

mode number Mθ remains to be discussed. 

The final issue of interest here, for macroscopic RT-type 

instabilities, involves the maximum azimuthal mode number 

Mθ to be used in the expression for the growth rate in Equations (228) or (229). Arbitrarily large mode numbers would 

produce arbitrarily large growth rates, within the context of 

the present sharp-boundary model of the propellant. The 

question as to the maximum effective mode number can be 

answered in terms of FLR stabilization of the RT-type flute 

modes. For the hot plasmas (~100 eV) contemplated for 

magnetic nozzle applications in space vehicle propulsion, 

FLR stabilization can be a noticeable effect. 

4.3 Finite Larmor Radius (FLR) Stabilization of 

RT-Type Flute Modes 

In the preceding section, we explored plasma particle trajectories at their initial impact with the confining magnetic 

field. The concern was brought up that their sharp adverse 

longitudinal curvatures could potentially be a source of 

instabilities in the injection region. In the following sections, 

on the other hand, we shall look at adverse longitudinal 

curvature instabilities further downstream, wherein a steady 

flow along the confining magnetic field has already been 

established at the plasma boundary. 

In the plane perpendicular to magnetic field B, it is well 

known from plasma transport theory that ions within the 

plasma-field mixing layer would possess a collisionless 

gyroviscosity when the plasma is of sufficiently high temperature that ion collisions are infrequent (see Refs. 9 and 32). 

Even though there is no collisional dissipation as is usually 

connected to viscosity, the ion gyroviscosity can exert a 

stabilizing influence on RT flute instabilities by modifying 

the dynamical processes that have enabled the modes to 

grow. Moreover, the flute-mode amplitude is largest at the 

plasma-field interface with the magnetic field, decaying 

exponentially into the plasma. Therefore, the gyroviscosity is 

effective just where the mode amplitude is the largest. 

Viscous diffusivity for ion gyroviscosity can be 

represented approximately as (Ref. 9) 

D aV vis th i ii = (230) 

where ai is the thermal ion gyroradius. This expression is 

analogous to the ordinary kinematic viscosity, λi

Vth i, 

wherein λi

 is the mean free path for ion-ion coulomb collisions. Kinematic viscosity is the viscosity coefficient divided 

by nmi

. A numerical coefficient of order 1 in Equation (230) 

proves to be almost irrelevant since it only enters the final 

expression for the growth rate as raised to the one-third power. 

In Equation (230) a V i i ci = th ω , the ion thermal velocity 

is Vthi = 2T mi , and ci i ω = qB m is the ion gyrofrequency within the plasma-field interface (using mks units, 

with T in joules). The gyroviscosity obviously vanishes for 

very small ai

, which is associated with very large magnetic 
NASA/TP—2009-213439 70 

fields. It is important to realize that the flute perturbation 

possesses an inverse time scale associated with this gyrodiffusivity, which is k

2

Dvis i, where kMr = θ is the azimuthal 

wave number. In the case of ordinary diffusivity, k

2

Dvis i

would signify a dissipative decay rate of a macroscopic 

dynamical mode, due to ion-ion collisions. 

For mode numbers sufficiently large that the gyrodiffusive 

inverse-time becomes comparable to the RT growth rate, 

those modes would be influenced by FLR stabilization. 

2

vis RT i kD ≈ γ (231) 

The azimuthal mode number corresponding to the condition 

in Equation (231) will be calculated and regarded as approximating the maximum mode number for viable growth of 

RT-type flute instabilities associated with adverse curvature 

of magnetic field lines. 

For Te = Ti

 = T, and Ɣ = 5/3, the square of the speed 

of sound in the breech can be expressed as 

2 2 5

3 th / b b b bi SP V = ρ= γ . The propellant’s exit velocity can 

then be written 

b Vb i V S ex th = 3 = 5 (232) 

Although this relation strictly is for infinite contraction and 

expansion, it also is a good approximation for realistic 

converging-diverging nozzle shapes. Use of Equation (232) 

in Equation (228) for γRT allows the solution to Equation (231) for the upper azimuthal mode number with the 

following result: 

1

1 2 3

3 3 1

3

1 2 3

eq

3 5 1

5 ( )

b

b i

r T

M

T R a

θ

      ρ   ≈ −         ρ     

 (233) 

Since this expression is nondimensional, any consistent set 

of units can now be used. 

Increasing the radius of adverse curvature Req decreases 

the driving force for the flute instability and hence decreases 

the upper mode number that can go unstable in the presence 

of ion gyroviscosity. Increasing the ion gyroradius ai

 also 

enhances the spatial phase mixing of the mode, spoiling the 

growth dynamics, which likewise acts to decrease the upper 

mode number that can go unstable. 

A lower bound for this upper limit on mode number is 

obtained by raising the temperature T up to its breech value 

in Equation (233) and by raising the mass density ρ up to its 

breech value. These approximations actually are fairly harmless between breech and throat because T at the throat only 

drops to three-fourths of its breech value, and ρ at the throat 

only drops to about two-thirds of its breech value. Then 

Equation (233) becomes approximately 

( ) 3

1 2

eq

1.5

R ai

r Mθ ≈ (234) 

Equation (234) may also be written in terms of the azimuthal 

wave number k M r = θ as 

3

1

eq

1.5 













 ≈ R

a ka i

i (235) 

For parameters of interest here, relevant examples are 

ai ≈ 1 cm and Req ≈ 30 cm, in which case kai ≈ 0.5. In other 

words, in ordinary MHD kai

 << 1, but in gyroviscous MHD 

FLR stabilization sets in about when kai

 is on the order of 1. 

Returning to Equation (234), the gyroradius within the 

plasma-field mixing layer is expressed as 2 22 1 i pi 2 a c = β ω . 

Here, β is the local ratio of total plasma pressure of electrons 

and ions to magnetic pressure at a representative position 

within the plasma-field interface, and ωp i is the ion plasma 

frequency at that representative position within the interface. 

Then setting the average density in the interface to half the 

density in the core plasma and reverting to the ion plasma 

frequency expressed in terms of the core density, Equation (234) becomes 

( )

1 1

3 3

θ

2 2

eq

1.5

pi

r M

R c

 

≈  

    β   ω  

 (236) 

Since the value of β is not precisely known, but is near 1 

within the interface and appears only to the one-third power, 

the prefactor in this expression is ignored. 

For hydrogen ion number density 1015

 cm−3, c/ωp i = 1 cm. 

Examples of other relevant parameters are r = 15 cm and Req

= 30 cm. From Equation (236) one sees that Mθ ≈ 5 is about 

the upper limit of flute-mode azimuthal numbers that can 

be unstable, with respect to avoiding FLR stabilization. 

In any event, the RT growth rate only depends on Mθ . 

Moreover, to the extent that the prefactor in Equation (236) 

can be ignored, the upper limit on flute-mode number is 

independent of temperature, depending just on local plasma 

number density in the core, local cylindrical radius of plasma, 

and local radius of curvature at the plasma-field interface. 

From Equation (229) with Vex = 2 · 107

 cm/s (specific impulse 

is 20 000 s), the growth rate for Mθ = 5: γRT ≈ 1.6 · 106

 s−1. 

For example, suppose that the longitudinal extent of the 

region of adverse curvature is only 10 cm, and V// ≈ 107

 cm/s 
NASA/TP—2009-213439 71 

within the nozzle. Then the transit time through that adversecurvature region would be about 10−6

 s. From the product of 

growth rate and transit time, we then infer that there would 

occur only one or two e-foldings of the most unstable 

RT mode. In this particular example, the RT flute instability 

therefore appears to be harmless. 

An expression for the growth rate of the most-unstable 

mode, as determined by the presence of ion gyroviscosity, 

can be obtained without the above-mentioned approximations (i.e., replacing density and temperature by their breech 

values). The resulting expression is valid anywhere along the 

nozzle. The procedure employed is that the maximum k or 

Mθ allowed by ion gyroviscosity according to Equation (231) is used in the RT growth rate, Equation (228). As 

before, the ion gyroradius at a representative point within the 

plasma-field interface is expressed in terms of the ion inertia 

length of the neighboring core plasma, by 2 22 ( ) i pi a c =β ω . 

Here, β (≈1) is the local total β at a representative point 

within the plasma-field mixing layer, but ωp i is the ion 

plasma frequency evaluated within the core plasma. The 

average ion number density within the interface has been set 

to half the core density. 

Taking the maximum wave number k from Equation (231) 

into the growth rate Equation (228) and using the explicit 

expression for the ion plasma frequency, one then finds that 

the growth rate of the most unstable mode, as limited by ion 

gyroviscosity, can be written as follows: 

( ) 2 1

3 3

2

1 1 23 1

6 6 3 6

ex RT

eq /

5 3 1

5

pi b

b

b b

V

R c

n n T

nT n

 

 

γ ≈  

  ω  

              × −         β                

(237) 

Subscript “b” refers to the value of the quantity in the 

breech or beginning of the nozzle. This result is valid at any 

longitudinal position along the nozzle, provided that ion 

gyroviscosity constitutes the dominant modification to the 

high-conductivity plasma fluid model. 

It is clear from Equation (237) that the principal dependence of the RT growth rate, as limited by ion gyroviscosity, 

occurs just in the first bracketed quantity. That quantity 

contains the propellant’s exit velocity (specific impulse) in 

the numerator, the breech value of the ion inertia length in 

the denominator, and the local adverse radius of curvature 

(Req > 0) in the denominator. 

( ) 2 1

3 3

ex RT

eq pi b

V

R c

γ ≈

ω

 (238) 

The Req varies along the flow at the self-consistent plasmafield interface. In regions of good curvature, Req is negative 

and there is no instability. The other two factors in Equation (238) have their positions fixed at the exit and breech. 

The quantity in braces in Equation (237) is of order unity 

and is insensitive to temperature and density along the flow 

field. Using the adiabatic relation between T and n along the 

flow, the product of the density ratio and the inverse temperature ratio in Equation (237) reduces just to (n/nb)

1/18, which 

is essentially just 1.0. In Equation (237), the braced quantity 

also is very insensitive to the value of local β within the 

plasma-field interface. Even the last internal factor in brackets is always between 0.5 and 1.0, but it can be explicitly 

taken into account if desired. 

The one quantity in Equation (237) whose variation with 

longitudinal position along the flow is certainly needed is the 

self-consistent radius of curvature Req of the magnetic field 

at the edge of the plasma. This quantity can be extracted 

from axisymmetric MHD simulations of flow through magnetic nozzles. The cylindrical radius r of the propellant is 

notably absent from the growth rate in Equation (237). 

Thus, for design purposes the reduced version—

Equation (238)—is probably sufficient. One could try to 

apply it to design the self-consistent magnetic nozzle field to 

be such that, in a region of adverse curvature, the product of 

growth rate and transit time of propellant flow through that 

region is less than 1. A fallback position is to totally avoid 

the presence of adverse magnetic curvature. However, 

achievment of that goal may be hindered by adverse selfconsistent deformation of the original vacuum magnetic 

field. Such adverse deformation may be difficult to avoid in 

the injection of hot plasma into a premagnetized nozzle tube 

that is burdened by constrained dimensions or by the discreteness and separation of magnetic source coils. 

Equation (238) constitutes a simple practical formula for 

the RT growth rate at any longitudinal position in the 

magnetic nozzle. It should nevertheless be regarded as 

provisional. For example, one might be concerned that 

downstream of the nozzle throat the propellant temperature 

eventually decays to the point that ion-ion collisions become 

significant, implying that ion gyroviscosity should be 

replaced by ion collisional viscosity. However, almost the 

opposite proves to be the case. The ion-ion collision frequency υi

 changes as n/T 3/2, so is constant along the propellant’s flow, since adiabatically, T ~ n

2/3. The ion-ion mean 

free path, however, varies as λi

 ~ Vth i/υi ~ n

1/3. Thus, the 
NASA/TP—2009-213439 72 

kinematic viscosity varies as λiVth i ~ n

2/3, which decreases 

towards the exit plane. In contrast, taking magnetic pressure 

balance into account, the gyroviscous diffusivity ai

Vth i

varies as n

−1/6, slightly increasing towards the exit plane. 

Thus, gyroviscosity remains the dominant viscous effect as 

long as ωc i > υi

. While the condition ωc i > υi remains in 

force within the plasma-field interface, flute-mode dynamics 

in the plane perpendicular to B are governed by ion gyroviscosity rather than collisional viscosity. Furthermore, for 

relevant injected-plasma parameters, the collisional situation 

ωc i < υi

 would require plasma density to decrease substantially from its injected value, by at least an order of magnitude. The υi

 is practically constant along the adiabatic flow, 

whereas ωc i decreases as n

5/6. But, at that point, the propellant has practically reached the nozzle exit anyway. In a 

more detailed model, there would also be an additional 

intermediate viscous diffusivity (Ref. 9) of the form 

Dvis i

υi/ωc i, but this amount of detail lies beyond the present 

purview. 

On the other hand, the Hall effect has been neglected in 

this modeling of the RT instability. Huba (Ref. 33) has pursued the subject of Hall MHD, and has found that at shorter 

wavelengths ~c/ωp i, the Hall effect, taken alone, increases 

the growth rates of those flute modes, and changes their 

nonlinear structure as well. Actually, there is a competition 

between enhancement of growth by the Hall effect and mode 

stabilization by ion gyroviscosity, a subject addressed later 

by Huba and Winske (Ref. 38). The Hall effect should be 

dominant in cold plasmas, such as in the beginning phase of 

theta-pinch implosions exemplified by Reference 34 and 

in the expansion of laser-produced plasmas, Reference 17. 

Ripin et al. experimentally confirmed the features of the HallMHD model (Ref. 17). MHD modeling of flow through magnetic nozzles should be revisited in this regard, so that Hall 

effects, together with ion gyroviscosity, can be applied in a 

knowledgeable manner to “adverse curvature” instabilities in 

propellant flow through magnetic nozzles. 

That Hall effects can occur in a cold plasma is correct and 

can be understood by considering that the generalized Ohm's 

law (mks units) does not explicitly contain thermal effects; 

yet, it contains the Hall term (as the second term on the 

right-hand side) as a contribution to the Hall electric field 

(see Eq. (37b)). Nevertheless, in specific applications to 

plasma-nozzle flow, thermal effects can sometimes creep in 

when estimating the importance of the Hall term. 

 ( )

nq

×

+ × =η +

J B E VB J (239) 

 

 The first term on the right in Ohm's law is the resistive 

term. Here, η is the resistivity. Equation (239) is just the 

electron-fluid momentum equation without the electron 

inertia and without the electron pressure gradient. The 

absence of the latter emphasizes the circumstance that here 

we are considering “cold plasma.” 

 A necessary condition for the Hall term to be important is 

that it not be dominated by the resistive term. This requirement is easily shown to be equivalent to having the electron 

gyrofrequency be large compared to the electron collision 

frequency with the ions (or with plasma fluctuations in the 

case of microturbulence). Such a large gyrofrequency and 

relatively small collision frequency is to be expected in the 

plasma-field interface of the magnetic nozzle for the parameters of interest. The size of the system in the RT problem is 

represented by the azimuthal wavelength of modes. 

 If the above requirement is satisfied (small enough electron collision frequency), then the next step in estimating the 

importance of the Hall term is to compare it with the 

motional electric field term, V × B. Here, V is the local fluid 

velocity of the plasma, which is the same as the ion-fluid 

velocity. Also, B is the local magnetic field vector. This 

particular comparison will be made in the following example 

of the coaxial MHD thruster, where it will be shown that the 

Hall effect is important when the ion gyroradius is not too 

small compared with the size of the system. 

4.3.1 Example: Estimate of Hall Term in Coaxial MHD 

Thruster 

The salient feature of the following example is that the 

Hall effect becomes important when the ion gyroradius 

becomes comparable to the macroscopic size of the system. 

In the RT problem, the size of the system is represented by 

the azimuthal wavelength of modes. 

In the coaxial plasma thruster, there are inner and outer 

coaxial electrodes, which form an annular nozzle; plasma 

flow velocity V is in the longitudinal direction, magnetic 

field B is azimuthal, and current density J is radial between 

the electrodes. Hence the strength of B decreases in the 

longitudinal direction, as required by Ampere's law 

∇ ×Β = μ0J. 

From Equation (239) we want to compare the magnitudes 

(mks units) of the motional electric field term VB and the 

Hall term JB/nq: 

VB versus JB/nq

Suppose that V is about the same as the ion thermal velocity, 

Vth i, which, at any rate, is a desired feature of nozzle flow. 

When the electric field term is comparable to the Hall term, 

then the Hall effect becomes significant. 

The following relationships are used: 

2 2

pi i 0 ω= ε nq m , where ωp i is the ion plasma frequency 

2 ε0μ0 = 1 c , where c is the speed of light 
NASA/TP—2009-213439 73 

V c A ω=ω ci pi , where VA is the Alfven speed and ωc i is 

the ion gyrofrequency 

Vthi VA = βi

Then the above comparison can be reduced as follows: 

coax versus ( / )/

versus /

c L

VB JB nq

βi ωp i

↓ ↓

Here, the quantity βi signifies the local ratio of ion pressure 

to magnetic pressure, and Lcoax is the coaxial longitudinal 

distance over which the azimuthal magnetic field becomes 

depleted by pushing the plasma downstream. Thus Lcoax is 

the length of the annular nozzle. 

The thermal ion gyroradius ai is related to the ion inertia 

length c/ωp i by a c i i pi =β ω ( ) , so the above comparison 

finally reads 

coax

coax

versus /

versus ( / )/

versus /

a L

c L

VB JB nq

i i

i p i

β

↓ ↓

β ω

↓ ↓

In realistic situations in the coaxial thruster, βi can be of 

order 1. As the ions flow downstream they make gyrations in 

the r,Z-plane, around the azimuthal magnetic field lines. 

Thus, if the ion gyroradius ai is not too small compared with 

Lcoax, so that the ions undergo only a few magnetogyrations 

as they flow through the coaxial nozzle, the term on the right 

can be important, and concomitantly, the Hall effect is 

important. In the case of flute modes, the azimuthal wavelength is the “size of the system.” 

4.3.2 Concluding Thoughts 

In the flute-mode discussion of the RT instability of this 

report (in Sec. 4.2.2, “Surface Instability of Flowing Propellant in a Region of Adverse Curvature”), a similar effect is 

noted in the literature, since the ion skin depth (c/ωp i) is nearly 

the same as the thermal ion gyroradius when β is near 1. A 

paper by Huba and Winske (Ref. 38) contains a detailed 

account of the RT instability when both Hall effects and ion 

gyroviscosity (FLR or FLR stabilization) are active. Linear 

stability analyses and nonlinear numerical simulations based 

on both fluid and kinetic models were carried out in that 

paper. It is instructive to consider why this pioneering paper is 

not yet applicable to the magnetic nozzle configuration. 

First, in Reference 38, the disparate light and heavy fluids, 

which constitute the RT configuration, are separated by a 

planar interface in the presence of an artificial gravitational 

force. Thus, geometric effects are missing. There is the effect 

of lateral pressure forcing a wedge-shaped element radially 

outwards (see Fig. 5). A concomitant effect is the outward 

spatial decay of external field strength in the region of 

adverse curvature. These geometric effects of adverse curvature are essential for the derivation that produces the pressure contribution to the growth rate in Equation (226). They 

are especially required since artificial gravity is not invoked 

in our model. 

Second, the models used in Reference 38 are “low β” 

models. This assumption has two consequences. The magnetic field changes only slightly in traversing the interface, in 

contrast to the magnetic field profile in the interface of the 

magnetic nozzle. Moreover, the ion gyroradius ai

, important 

for FLR stabilization, is small against the ion inertia length, 

c/ωp i, in Reference 38. The latter is important for Hall destabilization. In contrast, both of these lengths are comparable 

in the magnetic nozzle interface, because β is of order 1 

there. Thus, short azimuthal wavelengths would be simultaneously influenced to roughly equal extents by both 

processes in the magnetic nozzle. In Reference 38, however, 

the Hall effect was dominant because the ion inertia length 

was much larger than the ion gyroradius. 

Since Equation (238) is based only upon FLR stabilization, one may question whether it provides meaningful 

engineering guidance. It does so by eliminating from consideration those plasma profiles that are RT unstable. That is, if 

a configuration were found to be RT unstable in spite of 

including FLR stabilization effects in the model, then surely 

it would prove to be yet more unstable had Hall destabilization effects also been included. On the other hand, if a stable 

configuration is found in the sense that γRTt < 1, there is then 

no certainty that such a configuration would remain benign if 

Hall effects had been present. Thus, further work is needed 

in this regard. 

 Thus, in the flute-mode RT instability, it has been found 

by Huba and Winske (Ref. 38) that the Hall effect becomes 

important and destabilizing when the macroscopic wavelength is on the order of the ion gyroradius, but also it is 

known that the FLR stabilization effect becomes important 

under the same conditions. Thus the two effects work against 

one another. This process constitutes a delicate balance that 

deserves a careful calculation oriented to the magnetic 

nozzle. The paper of Huba and Winske shows that the two 

effects are in opposing directions, but those special calculations are artificial and constructed to clearly show both 

effects by turning one on after the other is on. That paper is 

not directly applicable to the magnetic nozzle, however; a 

special calculation focussed on the magnetic nozzle problem 

is therefore needed. 

 In conclusion, it is indeed the case that the model of the 

flute-mode RT instability in this report, which includes FLR 
NASA/TP—2009-213439 74 

stabilization but omits the Hall effect, may be giving incorrect results by not including both effects at the same time. 

That is to say, a traditional approach was followed, consisting of subjecting the ideal MHD results to FLR stabilization, 

without regard to any additional effects. That is exactly why 

it is suggested that there exists a need for the more complete 

calculation in Section 4.2.2, since if the RT instability 

appears threatening even when including FLR stabilization 

effects, it would appear even worse if the Hall destabilization 

effect were also included. In this sense, the formula Equation (238) is a meaningful engineering design criterion. 

4.4 Short-Wavelength Gravity-Driven Instabilities 

In this section, we shall examine ion gyroviscous stabilization of the unstable MHD flute modes associated with 

adverse longitudinal curvature. It is also pointed out that the 

destabilizing Hall effect should ultimately be included selfconsistently in the model so as to obtain a more complete 

picture of the behavior of these modes. 

Up to this point, discussion of RT-type instabilities has 

been carried out within the fluid description and has been 

directed to flute modes of long wavelengths compared with 

the thickness of the plasma-field interface. The manifestation 

of these long wavelength modes is the azimuthal deformation of the interface as an entity. 

However, it was recognized by Davidson and Gladd in 

Reference 18, and by Gary and Thomsen in Reference 35, 

that ion inertial force, or effective gravity, can drive instabilities of the LHD type within the interface layer itself. 

These unstable flute modes have very short wavelengths, 

which range between thermal ion gyroradius ai

 and thermal 

electron gyroradius ae. They are not stabilized by finite iongyroradius effects, except at very small current drift velocities where the unstable wavelengths become as large as the 

ion gyroradius (see Ref. 19). 

The salient observation regarding the short-wavelength 

LHD flute modes, as connected to the presence of effective 

gravitational acceleration, is that the treatment given in 

Section 3.0 of this report, “Resistivity From Gradient-Driven 

Microinstabilities,” applies without modification (see 

Ref. 18). The reason is that the treatment is set in terms of 

the current-drift velocity Vd. The reader will recall that the 

ratio Vd/Vth i is the characteristic driving factor for the real 

part of the frequency of this drift mode, and its square is the 

characteristic driving factor for the growth rate of the 

instability (see Eq. (144)). Moreover, the square of this 

velocity ratio also is a principal feature of the anomalous 

resistivity arising from LHD microturbulence, as was discussed in Section 3.0. Thus, the theory in terms of Vd/Vth i

remains intact. 

The role of gravity in the LHD modes only becomes 

apparent when the expression for the current drift-velocity 

itself is examined. This expression will now be developed 

within the context of propellant flow in the magnetic nozzle. 

For simplicity of discussion electron temperature will be 

neglected, but this is not an essential restriction. The ion 

macroscopic momentum equation is written in a frame of 

reference that is instantaneously at rest in the ion fluid; 

nevertheless, ion inertial effects (ion-fluid acceleration) can 

appear in this frame. 

 

xP g nqE ix x eff ∂ − ρ = (240) 

The x-direction is the quasi-radial direction, outwards across 

the external magnetic flux. The effective gravitational field 

is eq

2

eff // g x = V R , with V// the propellant flow 

velocity along B and Req the local equilibrium radius of 

adverse curvature. The magnetic field vector defines the 

local z-direction. The electron drift velocity Vd y e = 

[(E×B)/B2

]y in the y- (azimuthal) direction is obtained by 

dividing Equation (240) by nqB. The result is (in mks units) 

2

//

eq

1

yde

n ci

T V V

qB x R

= +

ω

 (241) 

Here, we have assumed uniform ion temperature and have 

defined the density gradient length within the interface by 

(1/xn) = n

−1 ∂xn . Earlier in this report, the interface width 

was described as δ, so xn = δ. 

Now, throughout most of the nozzle, the flow velocity is 

somewhat comparable to the local sound speed, which, in 

turn, is comparable to the ion thermal velocity. Therefore, 

Equation (241) says that, in general, the gravitational term is 

not important unless Req is small enough to be comparable to 

xn. Such is clearly not the case throughout most of the 

nozzle: Req is likely several tens of centimeters, whereas xn

is on the order of just a few centimeters. We conclude that 

gravitational acceleration due to adverse curvature generally 

should constitute only a minor modification to the edgelayer gradient that drives the short-wavelength LHD flute 

instabilities. 

The exception to this conclusion occurs at the point of 

impact of injected plasma with magnetic flux. It was seen 

earlier that even at near-grazing incidence of injected flow 

streamlines with the ambient magnetic field, the local radius 

of curvature at the point of impact would be so small as to lie 

beyond the purview of a fluid model. Its scale of smallness 

would probably be set by “microscopic” lengths, such as ai

, 

or by c/ωp i. Only then would the two contributions to Vd

become comparable in Equation (241). Thus, it is indicated 

that the initial point of impact of plasma with field in the 

injection region be scrutinized with respect to modification 

of LHD flute modes by gravitational effects. 

Near the exit plane, the second (gravitational) term in 

Equation (241) would increase (as n

−5/6) faster than the first 
NASA/TP—2009-213439 75 

term (as n

−1/6), unless the adverse radius of curvature Req is 

maintained at a large value by judicious arrangement of 

downstream magnetic coils. 

5.0 Summary and Conclusions 

The principal objective of this report was to ascertain the 

degree of attachment of propellant plasma to magnetic nozzle flux, as represented by the spatially evolving width of the 

plasma-field mixing layer, and to explore ways of mitigating 

that attachment. However, it was emphasized at the outset 

that even attached plasma undergoes converging-diverging 

nozzle acceleration along with core plasma. Moreover, 

reference was made to extant calculations showing that 

attached plasma can be resistively detached downstream 

with little loss of efficiency. This is provided that the necessary weak divergence of the nozzle in a long detachment 

region supported by trim coils is compatible with the overall 

mission design of the vehicle. 

The four principal results from this report are as follows: 

(1) The initial thickness δ of the plasma-magnetic field 

interface in the nozzle breech b (where the hot plasma is 

injected) is derived. It is δb ≈ 2c/ωp i, which serves as the 

initial condition for resistive broadening of the interface 

along the flow. Here, c is the speed of light in vacuum and 

ωp i, the ion plasma frequency in the breech. For the parameters of interest here, δb is a few centimeters. 

(2) The subject of gradient-driven Lower Hybrid Drift 

(LHD) microturbulent (anomalous) resistivity ηa is reviewed 

and compared with the classical resistivity ηcl that arises 

from electron coulomb scattering on the ions. The results 

from the smaller of two predictions for ηa are summarized. 

This subject bears on the spatial rate of resistive broadening 

of the interface, for which a simple algorithm is derived in 

terms of a general resistivity. The parameter dependencies of 

ηa appear to be agreed upon in various models, but the large 

discrepancies in the numerical coefficient need to be 

resolved. 

(3) The subject of flute instabilities of the magnetically 

confined interface in regions of adverse magnetic curvature 

is addressed. A physical derivation of the growth rate is 

performed for curved geometry without artificial gravity, 

which includes the effects of both pressure and flow. A 

practical formula for the growth rate is obtained that takes 

into account flute-mode stabilization by ion gyroviscosity 

(finite Larmor radius (FLR) stabilization). The formula can 

be used in magnetic nozzle design, by monitoring the 

self-consistent plasma-nozzle shape and flow velocity in 

axisymmetric simulations of propellant flow through magnetic nozzles. In this manner, one can estimate the extent of 

e-folding of flute instabilities in regions of adverse 

curvature. 

(4) The relevance of the Hall effect to Ohm’s law is clarified. The simple Ohm’s law used to connect azimuthal current density with azimuthal motional electric field in 

calculating resistive interface broadening depends on the 

Hall voltage not being shorted out by a flow of Hall current. 

This subject is important because the electron gyrofrequency 

ωc e far exceeds the electron collision frequency υe within 

the plasma-magnetic field interface. A complementary 

example is provided whereby the full Hall current is allowed 

to flow; and concomitantly the plasma is allowed to rotate. A 

critical nozzle length is identified below which the interface 

thickness is limited to about one ion gyroradius ai

. The 

critical length is e mi me λ / , where λe is a representative 

electron mean free path and mi

 and me are the ion and electron masses, respectively. 

 In this study, the basic features of converging-diverging 

nozzle flow were reviewed. An algorithm was identified to 

represent resistive broadening along the flow, of the plasmafield mixing layer. The results for the evolving width of the 

mixing layer proved to be essentially the same for two distinct models: diffusion of plasma into magnetic field and 

diffusion of magnetic field into plasma. Assuming a zerowidth starting condition for the mixing layer, so as to compare with some reported resistive magnetohydrodynamics 

(MHD) simulations, the simulations were found to be rather 

more effectively diffusive than would be predicted by classical resistivity. 

Resistive MHD simulations sometimes take for granted 

the simple form of Ohm’s law, despite the presence of a 

strong longitudinal magnetic field in the mixing layer. Thus, 

the magnetic field produces the condition, ωc e >> υe. Invoking the simple form of a local Ohm’s law that incorporates 

unmagnetized electrical conductivity, to relate azimuthal 

current density to azimuthal motional electric field, actually 

requires that the Hall voltage is not shorted out. In other 

words, it is required that no Hall current be allowed to flow. 

Therefore, global electrophysical boundary conditions in the 

device ought to be kept in mind when assessing the validity 

of this local Ohm’s law. In order to suppress the Hall current, resistive MHD simulations (and experiments) should 

have an insulating wall layer inserted between the metallic 

field coils and the plasma. Such a wall layer also aids in 

protecting the field coils. (However, plasma near the wall 

might longitudinally short out the insulating wall layer and 

thereby enhance a closure path for the Hall current.) 

The opposite condition, zero Hall voltage, therefore was 

also explored in this report. If the Hall current were indeed 

allowed to flow, as well as to self-consistently spinup the 

plasma propellant, then it might be possible to limit plasma 

attachment to magnetic nozzle flux. Specifically, we found 

that if the nozzle’s length were held smaller than a critical 

length λe mi me , then the width of the plasma-field interface layer could be held to about one ion gyroradius (here, 
NASA/TP—2009-213439 76 

the initial interface-layer thickness was neglected, except in 

one previously discussed exception). If Hall current could be 

maximized in a magnetic nozzle of subcritical length, then 

the detachment problem at the nozzle’s exit would be 

reduced to a secondary issue. It was also pointed out that the 

associated rotational spinup energy would be recovered in 

the diverging portion of the flow field. 

It is necessary to specify a spatial initial condition in the 

injection region, for calculating the downstream evolution of 

the width of the plasma-field mixing layer. Moreover, its 

width characterizes the anomalous resistivity that influences 

that very evolution. Therefore, special attention was given to 

the mixing-layer’s width δ = xn in the injection (breech) 

region of the magnetic nozzle. From first-principles physics 

considerations, it was inferred that it would be somewhat 

more than twice the ion inertia length, c/ωp i (in terms of the 

internal ion number density). Concomitantly, it was estimated 

that the ion gyroradius ai

 would be only about one-half to onethird of δ. In addition, it was shown that the ratio ai

/δ is insensitive to position between breech and throat of the nozzle. 

This result for ai

/δ determines the ratio of electron current 

drift-velocity to ion thermal velocity. That velocity ratio, in turn, 

is a principal factor in the growth rate of LHD modes as well 

as their subsequent evolution into microturbulent resistivity. 

The LHD instability was singled out for explaining the 

evolution of edge gradients of magnetically confined plasmas, because it is an instability that is easier to excite than 

other possible microinstabilities. For example, in comparison 

to the ion-acoustic instability, the current drift-velocity need 

not be large, and the electron-ion temperature ratio also need 

not be large. The ion-acoustic instability and an associated 

empirical resistivity had been previously invoked to explain 

the structure of magnetically imploded theta-pinches. Moreover, because the LHD mode is basically a non-cyclotronresonant mode in regards to the ions, it is robust enough to 

saturate at a significant level of fluctuations. This behavior 

presents a contrast with the well-studied electron-cyclotron 

drift instability, which has a very high growth rate but is 

basically a cyclotron-resonant electron mode easily 

destroyed by nonlinearity. 

The derivation of the small-amplitude linear LHD mode 

was outlined and its properties at maximal growth rate were 

summarized. The properties show a drift-wave behavior, 

with a wave phase velocity Vph = Real(ω)/k, proportional to 

current-drift velocity Vd. The characteristic wave number k is 

on the order of the reciprocal electron gyroradius ae. The 

wave’s temporal growth rate is a fraction of the lower hybrid 

gyrofrequency ω ω ce ci . Moreover, the growth rate is 

proportional to (Vd/Vth i)

2

. In the ion-cyclotron drift regime, 

with magnetized ions and very small Vd, the growth rate is 

so small as to have little relevance to the transit time in the 

magnetic nozzle application. 

The linear characteristics of the LHD wave were then 

invoked in a derivation of the quasi-linear theory of wave 

saturation. The saturated level of electric field fluctuations is 

a principal factor in the size of the anomalous resistivity 

produced by the wave. Use of the wave-energy bound, 

in which the total wave-fluctuation energy (not just 

electric-field-fluctuation energy) is obtained from electron 

drift inetic energy, produced a much smaller resistivity than 

did the Fowler bound. The resistivity from the wave-energy 

bound has parameter dependence identical to some alternative models of wave saturation and is notably similar to a 

first-principles electromagnetic particle simulation (Ref. 14) 

by Brackbill et al. The latter, however, produced a numerical 

factor in the resistivity that is about one order of magnitude 

smaller than in the quasi-linear model with the wave-energy 

bound. The explanation may lie in wave-wave coupling to 

damped modes in the simulation, except that Drake et al. 

(Ref. 26) also consider wave-wave coupling and yet find a 

much larger numerical coefficient. 

One principal observation of this report is that several 

alternative models of the anomalous resistivity in the interface layer do agree on the multiparameter dependencies of 

ηa. The second principal observation is that there are large 

discrepancies in the numerical value of the coefficient ηa

predicted by these models. For example, the model of Drake 

et al. (Ref. 26) has a numerical coefficient almost 2 orders of 

magnitude larger than the largest value from the simulations 

of Brackbill et al. (Ref. 14). This disagreement in the numerical coefficient needs to be resolved in order for the subject 

of anomalous resistivity to be applied with confidence to 

engineering design of the magnetic nozzle. 

The anomalous resistivity ηa extracted from the simulation runs of Brackbill et al. was expressed in terms relevant 

to the plasma-field interface in the magnetic nozzle concept, 

facilitating a comparison to classical resistivity ηcl arising 

from coulomb collisions of electrons with ions. The results 

are displayed in Figure 4. For example, at hydrogen-ion 

densities near 1015

 cm−3, ηa is dominant over ηcl at injectedplasma temperatures exceeding about 200 eV. However, if 

the results of Drake et al. are correct, then a breech temperature exceeding only 20 eV would be sufficient for anomalous 

resistivity to become dominant. 

Also, an open question remains as to the correctness of the 

β dependence of ηa, which purportedly was extracted from 

the Brackbill et al. simulations by comparison with the 

plasma heating results of Gary (Ref. 24). This is a very 

important question because the local β of the magnetically 

confined plasma varies from zero outside the plasma to >> 1 

inside the plasma, in traversing the edge-plasma gradient. 

Thus, resolution of this question affects the ability of resistive MHD simulations to properly represent the internal 

evolution of the plasma’s edge gradient, hence to represent 

the evolution of the width of the plasma-field interface. The 

difficulty with the β dependence proposed in Reference 14 is 
NASA/TP—2009-213439 77 

outside the scope of this report. We note the contrasting result 

of Drake et al., in which no β dependence is claimed even 

though their simulations employ an electromagnetic model. 

We conclude this summary with a few remarks on the 

Rayleigh-Taylor- (RT-) type instability. The presence of 

RT-type flute modes in regions of adverse curvature 

represents a loss of axisymmetric nozzle-based control of the 

flow of propellant. This loss of control can be mitigated by 

having the growth of the mode with the maximum mode 

number be limited by the available transit time of propellant 

through the adverse region. 

A physical derivation was given of the growth rate of 

RT-type flute modes downstream from the injection region 

in magnetically confined plasma in regions of adverse 

magnetic curvature. As is appropriate to the magnetic 

nozzle, the derivation incorporated the joint effects of 

propellant pressure and propellant flow (see Eq. (226)). A 

practical formula was obtained for the growth rate 

(Eq. (238)), which takes into account the stabilization of 

RT-type flutes of the higher mode numbers by the effects 

of collisionless ion gyroviscosity FLR stabilization). 

Within the injection region, the point of initial oblique 

impact of propellant with external nozzle magnetic field was 

singled out as having a localized region of very severe 

adverse curvature in the flow streamline. The extreme curvature there is due to the strong eddy current braking associated with the crossfield motion of injected hot plasma of 

high electrical conductivity. It was recognized that the structure of this region and the instabilities therein are inaccessible to a fluid model. It was suggested that the region of 

initial meeting of plasma with excluded flux receive special 

consideration regarding gravitational effects on shortwavelength flute modes that are closely related to the LHD 

instability. 

Lastly, there are effects of hot-plasma physics not covered 

in the fluid-based modeling of Huba on the FLR and Hall 

effects. These hot plasma kinetic effects should be taken into 

account with regard to modes driven by adverse curvature, 

even in modes that are not limited to short wavelengths, and 

are, therefore, also additional to the short wavelength effects 

existing in the LHD-type modes. Here, we exclude the ballooning modes that were already mentioned with regard to 

Reference 36. 

We were careful to base our physical derivation of elongated flute instabilities on the presence of adverse magnetic 

curvature rather than on artificial gravity. It is nevertheless 

expeditious for purposes of incorporating more plasma 

physics to relinquish the curved geometry in favor of a plane 

interface with uniform artificial gravity. With this artifice, 

Migliuolo (Ref. 39) generalized the famous FLR-flute mode 

theory of Rosenbluth et al. (first reference in Ref. 39) to 

extend to more plasma-physical effects. In particular 

Migliuolo included electromagnetic effects, nonzero β effects, 

and the effects of particle drift-resonances azimuthally 

across the magnetic field. Thus, Reference 39 shows how to 

obtain more accurate and more comprehensive properties of 

gravity-driven flute modes. He also explains that the short 

wavelength modes treated in Reference 35 do not contain 

electromagnetic effects. 

Also, Freidberg and Wesson (Ref. 40) used the same artifice to show that conditions could arise wherein flute modes 

are essentially stabilized by FLR, but at the same time, gravitydriven oblique modes are unstable. The latter are driven by 

resonant ion motion along the magnetic field, which is sometimes able to supply the extra energy needed to bend 

magnetic field lines. In magnetic nozzles containing regions 

of self-consistent adverse curvature traversed by hot plasma, 

it therefore would be important to apply appropriately 

detailed plasma-physical models to accurately describe and 

deal with the resulting instabilities. 

It is possible to include more detailed plasma-physical 

models in the analysis of surface perturbations of curved 

plasma boundaries by neglecting the surface curvature and 

instead introducing an artificial gravity. The relevance, 

however, of these artificial models to the integrity of the 

magnetic nozzle has not yet been demonstrated. 

In 2005, Arefiev and Breizman published a paper 

(Arefiev, A.V.; and Breizman, B.N.: Magnetohydrodynamic 

Scenario of Plasma Detachment in a Magnetic Nozzle. Phys. 

Plasmas, vol. 12, 2005.) in which they show that in a highly 

conductive plasma—where the flow kinetic energy density 

exceeds the ambient magnetic energy density—the plasma 

can stretch the magnetic field lines to infinity and thereby 

escape. (A follow-on paper published in 2008 (Breizman, 

B.N.; Tushentsov, M.R.; and Arefiev, A.V.: Magnetic Nozzle and Plasma Detachment Model for a Steady-State Flow. 

Phys. Plasmas, vol. 15, 2008.) presents a model that includes 

kinetic ions, but not electron-ion collisions, so resistivity was 

still absent.) However, here in the present report, it is found 

that the temperature of the plasma drops downstream of the 

throat of the nozzle. This is also shown by George Marklin’s 

numerical calculation in Reference 6 (Gerwin, Richard A., 

et al.: Characterization of Plasma Flow Through Magnetic 

Nozzles. AL–TR–89–092 (LA–UR–89–4212), 1989.). Hence, 

resistivity becomes important, precluding the stretching of 

field lines. Thus we conclude that the detachment of plasma 

from magnetic field lines remains an important process. 

In summary, there are many fundamental aspects left to be 

studied regarding the physics of high-temperature, highnumber-density plasma acceleration by means of the magnetic nozzle. The derivations in this report on selected topics, 

building on the work of others, can guide further theory 

development and experimental research in this emerging 

area of study. Only then can fast, piloted interplanetary 

travel utilizing magnetic nozzles begin to become viable. 

Glenn Research Center 

National Aeronautics and Space Administration 

Cleveland, Ohio, December 11, 2009

NASA/TP—2009-213439 79 

Appendix A.—Symbols and Expressions

A plasma cross-sectional area 

a local particle acceleration 

a thermal gyroradius 

B magnetic field 

{δB} fluctuating magnetic field

B magnitude of magnetic field B 

b

ˆ unit vector along B

C constant 

c speed of light in vacuum 

{δD} electric displacement 

D diffusion coefficient, arising from resistive 

diffusivity 

( ) ,ω (1) D k y Fourier-mode amplitude of {δD} 

E electric field (or generalized electric field) 

{δE} electric field fluctuation 

E magnitude of electric field E

( ) ,ω (1) E k y Fourier-mode amplitude of azimuthal component of electric field E

e base of natural logarithm, (2.7183) 

F force density 

f(u) generic function of a similarity variable in 

Appendix C 

f function 

{δf} perturbed distribution function 

( ) ,ω (0) f k i ion phase-space distribution function in the 

equilibrium situation 

( ) ,ω (1) f k i Fourier-mode amplitude corresponding to ion’s 

{δf} 

G(x) resistively diffused magnetic field’s 

x-dependence within plasma 

g gravitational field 

g(u) first derivative of f(u) with respect to u in 

Appendix C 

H 6-dimensional phase space [r, v] 

h fraction of radial fluid velocity lost by deceleration (h ≤ 1) 

I electric current 

J current density 

{δJ} current density fluctuation 

J magnitude of current density J 

K azimuthal surface current density 

K magnitude of azimuthal surface current density 

K

k azimuthal wave number

k azimuthal wave vector 

L length 

l vector distance along magnetic field line (or 

streamline in field-free region of core plasma) 

l magnitude of distance l

M mode number 

m mass 

m mass flow rate 

n number density 

( ) ,ω (1) n k e Fourier-mode amplitude of electron density 

fluctuation 

{δn} number density fluctuation 

P pressure 

{δP} pressure perturbation 

Q degrees of freedom 

q fundamental electric charge (1.6×10–19 C) 

R local longitudinal radius of curvature of field 

line 

Re Reynolds number 

r particle position 

rˆ unit vector in radial r-direction 

r radial coordinate (positive toward wall); origin 

at throat on nozzle centerline 

S speed of sound 

T temperature 

t time 

u similarity variable 

V macroscopic (fluid) velocity 
NASA/TP—2009-213439 80 

{δV} velocity perturbation 

V magnitude of fluid velocity V

v particle velocity 

v magnitude of particle velocity v

W energy density 

X inward radial distance from plasma edge 

xˆ unit vector in quasi-radial x-direction 

x quasi-radial coordinate (positive toward wall); 

origin at plasma edge in plasma-field mixing 

layer 

yˆ unit vector in azimuthal y-direction 

y azimuthal coordinate (positive clockwise, 

facing downstream); origin at plasma edge in 

plasma-field mixing layer 

Zˆ unit vector in axial Z-direction 

Z axial coordinate (positive downstream); origin 

at throat on nozzle centerline 

Z(ζ) plasma dispersion function 

zˆ unit vector in longitudinal z-direction 

z longitudinal coordinate (positive downstream, 

along magnetic field); origin at plasma edge in 

plasma-field mixing layer 

α polarizability 

β local ratio of thermal to magnetic pressure 

within the interface 

Ɣ adiabatic index 

γ wave growth rate 

Δ penetration depth of ions into confining magnetic field in breech 

δ characteristic resistive diffusive width of 

plasma-field mixing layer 

ε(k, ω) dielectric function 

ε0 permittivity of empty space 

ζ independent variable of plasma dispersion 

function 

η plasma resistivity 

Θ longitudinal magnetic flux enclosed by path 

integral of electric field taken around moving 

(expanding) plasma boundary 

ˆ

θ unit vector in azimuthal θ-direction 

θ azimuthal coordinate (positive clockwise, 

facing downstream); origin at throat on nozzle 

centerline 

Λ length from breech to throat L divided by ion 

gyroradius ai. 

λ mean free path 

μ0 magnetic permeability of empty space 

Ξ atomic number 

ξ (small outward virtual) displacement of rim 

element 

ρ mass density of plasma 

σ electrical conductivity 

ς longitudinal distance measured from throat 

τ gyroperiod 

υ collision frequency 

Φ ratio of wave phase velocity to ion thermal 

velocity 

φ angle 

φ electrostatic potential 

{δφ} fluctuation of electrostatic potential 

χ electric susceptibility 

Ψ stream function 

Ω conventional Hall parameter 

ω frequency 

π vorticity 

Subscripts 

A Alfven 

a anomalous 

app applied 

area area 

arc complete length of arc in region of adverse 

curvature 

B magnetic field 

B magnitude of magnetic field 

b breech 

back backwards 

bnce bounce 
NASA/TP—2009-213439 81 

Brack Brackbill 

c cyclotron 

char characteristic 

cl classical 

coax coaxial 

core core 

crit critical 

cur (adverse) curvature 

D diffusion 

d drift 

Drake Drake 

e electron 

eff effective 

eq equilibrium 

es electron sound 

ex exit 

ext external 

fl fluctuating 

G global 

g effective gravitational field due to centrifugal 

force 

Hall Hall 

i ion 

in incoming 

kin kinetic 

LH Lower Hybrid 

long longitudinal 

mag magnetic 

max1 maximum bound 

max2 maximum bound (even larger) 

mid middle of transition layer 

n number density 

norm normalized 

out outgoing 

p plasma 

proj projection 

pe plasma edge 

ph wave phase 

R longitudinal radius of curvature 

r radial vector component 

ran random 

RT Rayleigh-Taylor 

rot rotating 

s species 

t throat

tot total 

th thermal 

trans transient 

V velocity 

vis viscous 

wave wave 

x quasi-radial coordinate (positive toward wall); 

origin at plasma edge in plasma-field mixing 

layer 

y azimuthal coordinate (positive clockwise, 

facing downstream); origin at plasma edge in 

plasma-field mixing layer 

Z axial coordinate (positive downstream); origin 

at throat on nozzle centerline 

z longitudinal coordinate (positive downstream, 

along magnetic field); origin at plasma edge in 

plasma-field mixing layer 

1⁄4 one quarter 

1⁄2 radial location halfway through width dR 

I constant from Section 2.3.3, Equation (96a) 

II constant from Section 3.4, Equation (186) 

III denoting constant of integration from 

Equation (C4) 

IV denoting constant of integration from 

Equation (C7) 

V denoting constant from Equation (G8b) 

VI denoting constant from Equation (G14) 

VII denoting constant from Equation (G14) 

β pertaining to global beta 

Δ maximum radial extent 

η resistive 

δ just beyond diffusive mixing layer
NASA/TP—2009-213439 82 

μ mobility 

ρ density 

θ azimuthal 

τ time 

π vorticity 

⊥ perpendicular to local flux surfaces 

// parallel to local longitudinal field 

∞ asymptotic 

Superscripts 

E electric field 

it ion trapping 

k azimuthal wave number

Mθ azimuthal mode number 

ran random 

wave wave-energy bound 

x quasi-radial coordinate (positive toward wall); 

origin at plasma edge in plasma-field mixing 

layer 

y azimuthal coordinate (positive clockwise, 

facing downstream); origin at plasma edge in 

plasma-field mixing layer 

Δ variation radially outward 

ξ state of quantity as it exists in displaced 

element 

ψ phase factor exponent 

(0) Fourier-mode amplitude corresponding to a 

zero-order small perturbation (equilibrium) 

(1) Fourier-mode amplitude corresponding to a 

first-order small perturbation 

′ moving reference frame of plasma 
NASA/TP—2009-213439 83 

Appendix B.—Influence of Dwell Time on Plasma-Field Mixing Layer Width

Starting from Equation (99), and within the context of a 

quasi-one-dimensional model, it can be shown without further approximation that the square of the interface width, 

undergoing resistive diffusion along the flow, can be 

expressed as follows: 

back ( ) ( ) ( ) 2 back

0 back

b t L d

t

t t tbt

D L D A

V D AL

ρ ς ρς ς ς δ =

ρ ρ  (B1) 

It has been assumed that the initial interface width is zero. If 

not, one can add 2

bδ to the right-hand side of the above 

equation. In this equation, subscript t refers to the position of 

the throat, and the integration proceeds back a distance, 

Lback, to the breech of the nozzle. The breech position is 

represented by subscript b. The symbols D, ρ, V, and A

represent, respectively, the resistive diffusivity, plasma mass 

density, plasma flow velocity, and core plasma crosssectional area at position ς measured upstream from the 

throat at the position ς = 0. 

If the resistivity were classical, then the quasi-onedimensional temperature dependence of D on position ς

would be known along the flow. Likewise, the quasi-onedimensional density dependence ρ(ς) is known along the 

flow. For a given nozzle shape (actually the shape of the 

confined plasma), the area dependence A(ς) is known as 

well. All of these quantities can be obtained from an ideal 

magnetohydrodynamic (MHD) simulation or a quasi-onedimensional model. Their numerical representations can then 

be inserted into the above integral, which can subsequently 

be performed numerically to provide an estimate of the 

resistively broadened interface width at the position of the 

throat. Thus, an ideal MHD simulation or quasi-onedimensional model can be employed to make a preliminary 

estimate of the resistive interface broadening, by numerically 

evaluating a single integral. 

In this appendix, however, we want to show that fairly 

close upper bounds to the above expression can be obtained 

in a simple manner, without having to perform the numerical 

integration. 

It is assumed here that the relative-area variation of an 

annular ring containing the plasma-field mixing layer is the 

same as that of the main nozzle shape, a function of longitudinal distance along the flow. Conservation of mass flow is 

assumed (ρ(ς)V(ς)A(ς) = a constant), where A(ς) is the crosssectional area at position ς and velocity V(ς) is primarily 

parallel to the core plasma Z-axis. From Equation (99), we 

see that if the resistive diffusivity is replaced by its larger 

value Dt

 in the throat (since temperature drops along the 

flow and there is classical resistivity), the square of the 

maximum value of the mixing layer 2

max1 δ can be bound at 

the throat by 

back 0 2

max1 0 0

1 dd d

( )

bt

bt

t L

t tt

t

D t D tD

V

δ = =− = ς

ς   (B2) 

Here, we integrate backwards from the throat t to the breech 

Lback using a quasi-one-dimensional model. The longitudinal 

coordinate ς runs from 0 in the throat to Lback in the breech. 

The outgoing longitudinal velocity V(ς) > 0 is assumed to be 

essentially the same in the plasma-field mixing layer as in 

the core plasma, in virtue of the above statement about relative area variation in the annulus containing the plasma-field 

mixing layer. 

Now, by quasi-one-dimensional mass conservation, 

V VA A () () () ς =   ρ ρς ς tt t [ ]   . A still larger bound on δmax2

can be obtained by replacing V in the above integral by a 

lower bound. This is obtained by replacing ρ(ς) by the larger 

density in the breech ρb in the above expression for the 

longitudinal flow velocity V(ς). Then the expression in 

Equation (D2) is bounded above by 

2 back

max 2 0

( ) d

L b

t

tt t

A D

V A

  ρ ς δ =     ς

ρ   (B3) 

For the relative area variation, we take the following 

generic model. The radius of the nozzle as a function of 

longitudinal position measured from the throat at position 

ς = 0 is taken as 

1

2 2

area

() 1 t r r

    ς   ς= +    ς     

 (B4) 

where ςarea is a characteristic axial length over which significant radial variation of the nozzle area occurs. (The length 

ςarea is not exactly the same as the distance Lback from throat 

to breech.) Obviously, dr/dς = 0 at ς = 0, which is appropriate for the shape of the throat. Therefore, the relative area 

variation is 

2

area

( ) 1

t

A

A

ς ς  

= +  

ς  

 (B5) 

Use of Equation (B5) in Equation (B3) yields 
NASA/TP—2009-213439 84 

3

2 back back

max 2 area area area

1

3

b

t

t t

L L

D

V

       ρ δ= ς +       

ρ ςς       

 (B6) 

A radius contraction by a factor 2 from breech to throat is 

reasonable, producing density and velocity functions of ς up 

to the throat that differ only by a few percent from what 

would be produced by an infinite contraction ratio. From 

Equation (B5), an area contraction of 4 implies that 

Lback/ςarea = 3 . Then Equation (B6) becomes 

( ) 2 back

max 2

back

2 3

3

2

b

t

t t

b

t

t t

L

D

V

D L

V

  ρ   δ =      ρ  

  ρ =  

ρ 

 (B7) 

Now, ρb/ρt ≈ 3/2 constitutes a very good approximation to 

the nozzle flow results (see Eq. (10)). Then the upper bound 

becomes 

2

max 2 back

3

t

t

D L

V

  δ ≈    

 (B8) 

Thus, for a radius-contraction ratio of 2, the resistive layer 

width at the throat is bounded above by a quantity that is 

larger than the original estimate, [ ] Dt Lback Vt , by only a 

factor of about 3 = 1.7. For a radius-contraction ratio of 3, 

the numerical factor proves to be 2.3. 
NASA/TP—2009-213439 85 

Appendix C.—Planar Similarity Solution for Resistive Diffusion of Field Into Plasma

When cylindrical effects are not important (thin plasmafield mixing layer), Equation (90c) can be written as follows: 

∂ X B = ∂τB 2 (C1) 

Here, B = Bz; X is the inward radial distance from the plasma 

edge; and [ ] ( ) Vz τ = zD , where z is the local axial coordinate, D is the uniform resistive diffusivity of plasma, and Vz

is the uniform axial velocity of plasma. Note that Equations (90) and (C1) take the point of view of describing the 

process of magnetic-field diffusion into the plasma within 

the lab frame, in which a steady state (with flow) exists. 

Boundary conditions on the system are envisioned to be 

applied at the breech end and will be discussed below. 

One-dimensional diffusion equations of this type permit 

similarity solutions having the form B(X,τ) = f(u), wherein u

is the similarity variable, = τ 2 u X . The equation for f(u) 

reads 

( ) ( ) 



 



  ′ +  



 ′′ = − u

f u f u

2 1

4

1 (C2) 

where prime symbols denote derivatives with respect to the 

argument of the function. 

Defining g(u) = f′(u), the equation for g(u) is 

[ ] ( )

( ) ( ) 0

d

4

ln

4

d

d

d ln =

















+

+

4

u

u u

4

u

g u (C3) 

The solution of Equation (C3) for g(u) is immediate. One 

then has 

( ) 4 III

1 f u

uC e

u

  − ′ =    

 (C4) 

where CIII is a constant of integration. Integrating Equation (C4) from u = ∞ to u, one has 

() ( )

4 4

III 0 0

d d f f

u u u eu eu u C u u

− −  ∞ 

− ∞= −         (C5) 

though f(∞) corresponds to a field in the plasma at z = 0. A 

boundary condition is that the fields have not yet diffused 

into the plasma at the breech of the nozzle. Thus, f(∞) = 0. It 

is also observed that 

( ) u u

u e u u

2 erf 0.5 d

0

4

= π  −

 (C6) 

Then Equation (C5) becomes the following: 

( ) 





























 = = − uτ

X

u B C 0.5 f 1 erf IV (C7) 

Here, CIV is just another modified constant of integration, 

and the original variables have been reinserted. Also, recall 

that z u zD V τ = . Note that the latest constant CIV represents 

the uniform value of B at the edge of the plasma, where 

X = 0. 

NASA/TP—2009-213439 87 

Appendix D.—Azimuthal Magnetic Field When Hall Voltage Is Shorted Out

This appendix constitutes an estimate of the azimuthal 

magnetic field that is necessitated by the incoming longitudinal return current, which must balance the outgoing transverse Hall current if the Hall current circuit is fully closed. 

The estimate is admittedly non-self-consistent in the sense 

that results obtained by neglecting the azimuthal magnetic 

field are used to estimate the size of that field. The effects of 

plasma rotation and electron pressure are included here. 

From Equations (62) and (67) the Hall current density can 

be written as 

1 11

Hall x i J nqV B P − −− = −Λ = −Λ ∂ θ (D1) 

All quantities herein are, in principle, functions of the distance along a given field line. In practice, they shall signify 

representative or average values near or in the throat region 

of the nozzle and near or in the plasma-field mixing layer. 

Now, let Iin represent the incoming longitudinal current, 

which will be presumed to exist inside the plasma core of 

radius r(l). The region of the plasma-field mixing layer then 

begins at r(l), and extends outwards by a small increment 

δ(l). 

The total outgoing average Hall current Iout can be roughly 

estimated as 

Iout ≈ JHall2πrL (D2) 

where r signifies the radius of the plasma core. The righthand side of Equation (D2) is intended to signify only average or representative values. 

By setting Iin = Iout, the azimuthal magnetic field in or 

near the plasma-field mixing layer then can be estimated as 

follows: 

L L Br Z xPi J

r

I B = μ = −μ Λ ∂

π

μ = − − θ

1 1

0 Hall 0

0 in

2 (D3) 

Here, Br Z represents the original longitudinal magnetic 

nozzle field magnitude without any azimuthal component. 

This is equivalent to the projection of the total magnetic field 

vector into the r,Z-plane, Now, by making the convenient 

approximation that ∂ xPi ≈ −Pi δ , the ratio r Z Bθ B can be 

estimated. Also recall the definition of the ion βi

, namely 

that 2 βi = 2μ0Pi B . Dividing Equation (D3) by the magnetic field Br Z one then finds 

1

2

0 2

2

θ −δ



















 μ

Λ =

r Z

i

r Z B

L P

B

B (D4) 

Recalling Equation (65) for Λ, this ratio of fields becomes 

//

2

i

rZ ci

B V

B

θ β   =  

  ω δ  

 (D5) 

Since the longitudinal flow velocity V// is on the order of the 

ion thermal velocity, this result is approximately 

2

i i

r Z

B a

B

θ β   =     δ (D6)

 

Suppose δ is scaled by the ion thermal gyroradius ai

, as in 

Equation (74). For example, suppose that δ is one ion gyrodiameter. The azimuthal magnetic field ratio is then on the 

order of 0.25βi

, whereas with the resistive plasma-field 

mixing layer, as in Equation (77), δ could be somewhat 

larger, and hence the azimuthal field would be somewhat 

smaller. 

It is worth noticing that when the azimuthal magnetic field 

is somewhat comparable to the magnetic nozzle field, as is 

indicated here, then it is possible that substantial magnetic 

shear can arise within the plasma-field mixing layer. The 

magnetic nozzle field increases from a relatively small value 

within the plasma core to its ambient value in a rather short 

radial distance, which is the plasma-field mixing layer. It is 

well known that magnetic shear has a stabilizing influence 

on some microinstabilities as well as on flute-type instabilities driven in regions of adverse curvature of the magnetic 

field lines. For this reason among others it appears worth 

exploring in some detail whether the full Hall current mode 

of operation (including plasma rotation) may enhance the 

integrity of the plasma-field mixing layer. This is further 

developed in Appendix E. 

NASA/TP—2009-213439 89 

Appendix E.—Inward Radial Drift of Propellant

Although the main concern of this report is loss of propellant outward onto the nozzle magnetic field lines, it is worth 

pointing out that there also is a mechanism for inward drift, 

which is activated by the presence of the azimuthal magnetic 

field in a resistive plasma medium. For simplicity we assume 

a straight, cylindrical geometry. 

The azimuthal magnetic field lines must enclose the core 

where, in a steady state, the incoming longitudinal current 

balances the transversely outgoing Hall current. The resulting small inward drift may be somewhat beneficial in maintaining the integrity of the core plasma. For simplicity, it was 

not included in the model of field diffusion into the plasma. 

It must arise, however, from a steady-state balance between 

the rate at which magnetic energy is resistively dissipated by 

the longitudinal current and the rate at which magnetic 

energy is brought into the core. This balance can be 

expressed as follows: 

2

2 2

0

2d d

2 r Z

B

V rZ J r Z θ     π =η π   μ  

 (E1) 

Here, it is known that in highly conducting plasma, magnetic 

flux can only be brought towards the core by the 

inward velocity of plasma; that is, the flux lines move 

inwards approximately with the plasma. Equation (E1) is 

essentially an integral form of Poynting’s theorem. 

Now, from Ampere’s law, Bθ = (μ0I)/(2πr), where I is the 

longitudinal current within the core of radius r. Also, we 

have assumed uniform resistivity and uniform density of 

longitudinal current, 2 J I r Z = π . As a result, one finds 

from Equation (E1) that the radial inward velocity at the 

edge of the core is 

r

D Vr4 = (E2) 

where D = η/μ0 is the resistive diffusivity of plasma. 

Consider an example with classical resistivity (Spitzer 

resistivity) in hydrogen plasma with the parameters in 

Table I. The resistive diffusivity then is Dcl = 3 m2

/s. Also, 

taking the core radius to be r = 0.1 m, the inward drift 

velocity then is 120 m/s, which is 3 orders of magnitude 

smaller than the longitudinal flow velocity of propellant. 

This inward drift velocity is somewhat comparable to the 

outward diffusion velocity found earlier (see the discussion 

following Eq. (17)). 

NASA/TP—2009-213439 91 

Appendix F.—Derivation of Some Results Noted in Presentation 

of Initial Boundary-Layer Width

The results presented below in Sections F.1 and F.2 refer 

to the small electron displacement distances treated in the 

discussion around Equations (121) and (122) in Section 3.1, 

“Interface Width in Breech of Nozzle.” Numerical factors of 

order 1 will be suppressed in the derivations represented 

here. Section F.3 pertains to small inertial forces on individual ions (see Eq. (103)). Section F.4 validates the assumption 

when only electrons carry azimuthal current. 

F.1 Radial Mobility Drift Distance of Electron 

Versus c/ωp i

In Equation (122), the first term gives the outward quasiradial mobility drift-velocity of the electron fluid in the 

electric field Ex in the limit of large Ω= ω υ ( ) ce e . This 

quasi-radial velocity term will be multiplied by about onefourth of an ion gyroperiod ( ) 1 ≈ ωci so as to obtain the 

outward quasi-radial distance xμ drifted by electrons during 

that time increment. 

0

0

11 1 1

1 1

x i ce y z

z ce ci z ce e

y i

ce e

E m J B

x

B nqB m

J m

nq m

μ

ω = = Ω ωω Ω ω

μ = Ω μω

(F1) 

Ampere’s law is used in the numerator of the last expression 

above, to replace μ0Jy by [∇ × B]y. Then, the radial length 

increment in the definition of [∇ × B]y is identified as c/ωp i. 

After substituting the definition of ωc e = qB/me, a resulting 

combination of factors is identified as the square of the 

electron plasma frequency 2 2

p e 0 e ω= ε nq m in mks units. 

Finally, recall the relation ε0μ0 = 2 1 c , where c is the speed 

of light. After some cancellation, and using 

pe pi i e ω =ω m m , the above expression reduces to 

1

pi

c

xμ = Ω ω (F2) 

Thus the classical outward mobility drift distance traversed 

by an electron in the space-charge quasi-radial electric field 

is small compared with the initial outward ion penetration 

depth. 

F.2 Radial Diffusion Distance of Electron Versus 

c/ωp i

We consider the quasi-radial diffusion distance xD traversed by an electron undergoing a random walk (of gyroradius steps) across the magnetic field, during one-fourth of an 

ion gyroperiod. Recall that the diffusion coefficient of an 

electron in a magnetic field is given by the expression 

2 2 Da a e e e ee = τ= υ , wherein υe is the collision frequency 

of an average electron (having ωc e >> υe) and ae is the 

electron thermal gyroradius. Then 

2 2

2 e ei i i ce

De e

ce ci e

am a a

x Dt a

m

υ ω = = = == ωω Ω Ω Ω (F3) 

but the ion thermal gyroradius ai

 is comparable to the ion 

inertia length p i c ω whenever the ion β is on the order of 1 

(as expected according to Appendix H). Hence, 

1

2 D p i x c ≈Ω ω (F4) 

Thus the classical outward collisional diffusion traversed by 

an electron across the local magnetic field is small compared 

with the initial outward ion penetration depth. 

F.3 Fluid Deceleration Neglected in Ion Trajectory 

Equation 

An equivalent gravitational deceleration field was neglected in the trajectory equation of an ion, which was 

employed in Section 3.2, “Linear Theory of Lower Hybrid 

Drift (LHD) Instability.” This force field is to be compared to 

the force on the ion from the radial space-charge electric 

field. 

From Equation (H11) and by recalling that the square of 

the Alfven velocity is 2 2 V B A = μ ρ0 , we estimate that 

geffδ = hVxVA. Here Vx is the radial fluid expansion velocity 

of the plasma during the first one-fourth ion gyroperiod, and 

h is the fraction of that velocity that is lost to deceleration 

during that time increment. Setting p i A ci δ= ω = ω c V (the 

second equation is an identity), the effective gravitational 

acceleration becomes 

eff x ci g hV = ω (F5) 
NASA/TP—2009-213439 92 

which is approximately the same as Equation (H11). Recall 

that c is the speed of light, ωp i is the ion plasma frequency, 

and ωc i is the ion gyrofrequency in the plasma-field mixing 

layer. Because ci i ω = qB m , the equivalent gravitational 

force on the ion is 

m g hqV B i x eff = (F6) 

The radial momentum equation for fluid electrons (neglecting electron pressure for simplicity) is nqEx = JyBz. With 

help of Ampere’s law, this becomes

2 2

0

1

2

i A x x pi

B m V qE n c

 

=− ∂ ≈     μ ω  

 (F7) 

We have ignored a sign to just get the order of magnitude 

and have set the length scale for magnetic pressure change to 

p i c ω . Using the identity mentioned above, radial force on 

the ion (magnitude only) due to the space-charge field is 

found to be 

qE qV B x = A (F8) 

This electric force exceeds migeff when the plasma-fluid’s 

radial expansion velocity is sub-Alfvenic. 

F.4 Azimuthal Current Carried by the Ions During 

One-Fourth of a Gyroperiod 

A fundamental assumption in Section 3.1 has been that 

only the electrons carry azimuthal current during the short 

time (1/ωc i) when a volume element of plasma first runs into 

the confining external magnetic field. In this subsection, we 

test that assumption. Again neglecting electron pressure for 

simplicity, it is seen from the above estimate of the radial 

space-charge electric field that the azimuthal E × B drift 

velocity of the electrons is just the Alfven velocity. If the 

azimuthal fluid velocity of the ions were much smaller than 

that in the early-time increment of interest, then the ion 

contribution to the azimuthal current also would be small. 

The azimuthal electric field from the radial fluid expansion 

is Ey = VxBz. The Faraday phase due to time-dependent 

interior flux expulsion by the instantaneous electron current 

layer is assumed already completed. The azimuthal velocity 

acquired by an ion during time 1 i ci τ≈ ω is then 

V qE m qE m E B V y i y i i y i ci y z xi ≈ τ = ω= = (F9) 

This is small compared with the electron-current velocity 

when the plasma fluid crossfield expansion is sub-Alfvenic.
NASA/TP—2009-213439 93 

Appendix G.—Effect of Electron Pressure on the Initial 

Plasma-Field Mixing Layer Width

This appendix examines the effect of electron pressure on 

the initial width of the plasma-field mixing layer. The effect 

of electron collisions has been sufficiently covered in the 

main text. For conciseness, we will ignore electron collisions 

here. However, electron (and ion) pressure (Pe ≈ Pi

) will be 

included in this calculation of the initial width of the plasmafield mixing layer. Moreover, we attempt to carry out a more 

rigorous and detailed evaluation of the radial average electric 

field needed in this calculation, yet without invoking specific 

shapes for the profiles of density and magnetic field within 

the plasma-field mixing layer. 

Consider the azimuthal frame of reference in which newly 

injected ions carry no macroscopic azimuthal current. This is 

presumably the lab frame. In that frame, the quasi-radial 

component (i.e., directed across flux surfaces) of the 

electron-fluid momentum equation then reads as follows: 

∂ =− + xP nq E V B e x yz ( ) (G1) 

A key feature of the model is that, for times less than an ion 

gyroperiod the azimuthal current density Jy = −nqVy is carried only by the electrons. As discussed in the main text, we 

shall assume that quasi-neutrality has been achieved in an 

early transient subphase and is maintained (see Section 3.1, 

“Interface Width in Breech of Nozzle”). We also assume that 

the electron and ion temperatures are equal and uniform 

within the plasma-field mixing layer. 

Solving Equation (G1) for the space-charge electric field 

and using Ampere’s law on Jy, 

2

tot

0

1

2 2 x x e

B P

E

n q

 

=− ∂ +  

μ  

 (G2) 

Here, Ptot = Pe + Pi

 = 2Pi is the total plasma pressure. Therefore, with mass density ρ = mi

n, it is obvious that 

2

tot

0

2 e 1 x x iq B E P

m

Δ  

− =Δ ∂ +  

ρ μ   

 (G3) 

where Δ is the penetration depth of ions into confining magnetic field in the breech. The brackets ... signify the radial 

average of the enclosed quantity over the width of the 

plasma-field mixing layer. 

Use of azimuthal momentum balance Equation (110) for 

the velocity of an individual ion at the farthest radial extent 

of the ion’s trajectory, vy i Δ = −1⁄2ωc iΔ, allows the total 

energy balance of that ion, from Equation (111b) to be 

expressed as follows (the ion gyrofrequency, ωc i without 

brackets, refers to the outer magnetic field beyond the 

plasma-field mixing layer): 

2 22

core

2 1

4

e x i x ci

i

q v E

m

Δ =− + ω Δ (G4) 

Here, vx core i

 is the radial velocity with which the ion is 

incident upon the edge boundary layer. It has been assumed 

that the incident azimuthal velocity vy core i

 = 0 for a representative ion in the midst of a thermal distribution. 

The intent is to utilize Equation (G3) in Equation (G4). 

Therefore, the two radial averages indicated in Equation (G4) are evaluated. We begin with the second term 

because it entails the most straightforward evaluation. Using 

Ptot = 2nT with uniform T, the second term is 

tot

1 22 ln x x x i ii

T T n

P n

nm m n m

∂ Δ ∂ =Δ = Δ ∂ (G5) 

The notation 2 2 V Tm thi i = is introduced for the square of 

the ion thermal velocity. The radial average of the partial 

derivative of the logarithm of number density is evaluated as 

follows: 

core core

core

core

1 1 ln ln d [ln ln ]

1 1 ln ln

x x n nx n n

n n

n n

Δ Δ

Δ

Δ

∂= ∂ = −

Δ Δ

= =− Δ Δ



 (G6) 

This is to be multiplied by Δ in Equation (G5). It is reasonable to associate the characteristic width Δ of the plasma-field 

mixing layer with the 1/e spatial decay length of the plasma 

density (where e ≈ 2.71828, the base of natural logarithms). 

The characteristic width Δ originally was associated with the 

protrusion of an average ion’s trajectory beyond the core 

plasma into the magnetic field region. Its quantitative connection with the density profile’s spatial decay, as well as its 

connection with the spatial increase of the magnetic pressure, requires more calculation than can be accommodated 

here. This clarification would require following the full 

thermal velocity distribution of trajectories of outgoing ions 

into the (unknown) nonuniform magnetic field profile. But 

using a few Δs for the 1/e spatial decay of the plasma 

density, say n = ncoreexp(−x/CVΔ), makes little difference, as 

shown later. Then ln(ncore/n

Δ

) = 1/CV, where n

Δ

 = ncore/e is 
NASA/TP—2009-213439 94 

the variation of number density radially outward and CV is 

an unknown free constant. Note, however, that the use of this 

specific decaying exponential density profile is only an 

example; it is not essential to the argument. Consequently, 

1 ln

Δ x ∂ =− n (G7a) 

If the spatial decay length of the density were taken as CVΔ, 

then instead of Equation (G7a) one would find 

 Δ

∂ = −

V

1 ln

C

n x (G7b) 

Use of Equation (G7a) in Equation (G5) yields 

2

tot th

1 2

x i

i

T P V m Δ ∂ =− =−

ρ

 (G8a) 

Retaining the flexibility of CVΔ for the characteristic distance of spatial decay of the density profile, Equation (G8a) 

becomes instead 

2

th

tot

V

1 i

x

V

P

C

Δ ∂ =−

ρ

 (G8b) 

We have assumed that the characteristic length for the 

increase of magnetic pressure in the plasma-field mixing 

layer 0

2 B 2μ is Δ. The quantitative connection of the 

density decay length to Δ has not been established, so a few 

values CV ≠ 1 are considered.

Now the second term on the right-hand side in 

Equation (G3) has been accounted for. Next, we address the 

first term in Equation (G3). A simple calculation in terms of 

the local Alfven velocity VA yields 

n

n V V B x x x A A

∂ = Δ ∂ +

μ

∂

ρ

Δ 2 2

0

2 1 (G9) 

Recall that the local Alfven speed squared is = μ0ρ 2 2 VA B . 

The first term of Equation (G9) can be written 

2 2

core

22 2 2 core

1 d xA xA

AA A A

V Vx

V V V eV

Δ

Δ Δ

Δ ∂ =Δ ∂

Δ

=− ≈=

 (G10a) 

Subscript “Δ” refers to local evaluation at the outer edge of 

the plasma-field mixing layer, and subscript “core” refers to 

evaluation at its inner (plasma core) edge. But the final VA

without those subscripts is to be evaluated with the outer B

and the inner ρ. The inner ρ is taken as e multiplied by the 

outer ρ. The density’s spatial decay length is taken as the 

original Δ. 

If the spatial decay length of the density were instead taken 

to be CVΔ, then instead of Equation (G10a) one would find 

1 2 2 CV Δ∂ = xV eV A A (G10b) 

Next, the second term of Equation (G9) is evaluated just by 

using an intermediate value of the local 2 VA . Since B 

increases going outwards and ρ decreases, 2 VA increases 

strongly going outwards. Therefore, its intermediate value is 

simply taken as half the outer value. For the density decay 

scale of Δ, one finds 

2 2 1

2

x A A

n V eV n∂ Δ =− (G11a) 

The local 2 VA appears in Equation (G11a), and the final 2 VA

in Equation (G10a) has been evaluated at the outer B and the 

inner ρ. 

If the spatial decay length of the density were CVΔ, one 

would find instead 

1 2 2 V

V

1 1

2

x C

A A

n V eV

n C

∂ Δ =− (G11b) 

Use of Equations (G7a) to (G11a) in Equation (G3) yields 

2 2 1

2

G

x A

i

q e E V m eΔ   β − =−     (G12a) 

The global beta, βG, is the ratio of core plasma pressure 

(electron pressure plus ion pressure) to outer magnetic pressure. Since βG = 1 for pressure balance across a static layer, 

the right-hand side is guaranteed to be positive, in agreement 

with Ex being negative. In case of radial deceleration of the 

macroscopic plasma fluid, it will be shown that βG < 1, 

making the right-hand side of Equation (G12a) still more 

positive. 

Equation (G12a) applies to the case that the density decay 

length in the plasma-field mixing layer is only Δ. If instead it 

were CVΔ, then Equation (G12a) would be replaced by 

1

V

1

V

2 V

V

V

2

2 1 1 1

2 (2 1)

C

C

x i

A G

q E m

C e V

C e C

Δ −

  −   = −β    −   

(G12b) 
NASA/TP—2009-213439 95 

For the negative space-charge field Ex it is clearly necessary 

that CV > 1⁄2, but this is not sufficient because the 

expression in brackets in Equation (G12b) also must be 

positive. The latter condition amounts to 

1

V V 1 1 2

C C e G

  − > +β    

 (G13) 

It will be shown below that βG ≤ 1. Then this condition is 

satisfied for CV ≥ 0.5925. The lower bound 0.5925 for CV is 

an artifact of shortening the electron pressure drop for fixed 

magnetic pressure gradient. The radial electron momentum 

equation reads: neqEx = −∂xPe − ∂x(B2/2μ0). Thus, an artificially sharp negative electron pressure gradient, for fixed 

positive magnetic pressure gradient, annihilates the spacecharge electric field, an unphysical consequence. 

Now Equations (G12a) and (G12b) are used to calculate 

the plasma-field mixing layer thickness Δ, following the 

same procedures as are utilized in the main text. Both Equations (G12a) and (G12b) can be expressed as 

( ) 2

VI VII

2 1 x A G

i

q E CV C

m

Δ − =− β (G14) 

where CVI and CVII are constants: 

1

V V

VI

V

2 1

2

C C C e

C

− = (G15a) 

1

V

VII

V

1

(2 1) C

C

e C

=

−

 (G15b) 

The Δ represents the physical penetration distance of a 

representative ion at the edge of the plasma into the confining magnetic field, when that element of plasma first 

encounters the interface region. The nondimensional version 

of Δ, Δnorm, is represented as 

norm

pi

c

Δω

Δ = (G16) 

in which ω p i is the ion plasma frequency referred to the ion 

number density in the core. Using the same procedures as in 

the main text, namely Equation (117), ( )2 norm c pi

2Δ = Δω , 

and the subsequent discussion, one finds 

( )

2

2

norm VI VII 2

4 1

1 16

C C G

π Δ= − β

− π

 (G17) 

Inserting the constants from Equation (G15) into Equation (G17), we then find 

1

V

2

2

norm V 2 V

4 1 (2 1)

1 16 2

C e C G C

π   Δ = −−   β

− π  

 (G18) 

Setting the global beta to its upper bound βG = 1 in Equation (G18), the lower bounds can be found for the numerical 

values of Δnorm. We say that these results are “lower 

bounds” for the following reason: In Appendix H, we show 

that βG = 1 in the absence of radial deceleration of the 

plasma by the magnetic field (i.e., no fluid-plasma motion 

across magnetic flux), but βG < 1 in the presence of radial 

deceleration (“radial” always means “quasi-radial” across the 

flux surfaces). Hence, according to Equation (G18), setting 

βG = 1 produces a value of Δnorm that constitutes an underestimate in the presence of radial deceleration. 

Figure 6 illustrates the dependence of nondimensional ion 

penetration depth Δnorm into the external magnetic field on 

the characteristic electron density decay length into that 

field, (CVΔ), when CV is regarded as an unknown free parameter. Figure 6 indicates that the (underestimated) width of 

the initial plasma-field mixing layer, as defined by the radial 

protrusion of a representative ion into the magnetic field, is 

quite insensitive to the exact numerical decay length 

assumed for the density halo. Here, βG has been replaced by 

its upper bound. Appendix H relates the numerical value of 

βG to the presence or absence of macroscopic radial deceleration of newly injected, radially expanding plasma. The 

effects of both electron and ion pressure gradients are explicitly included in Appendix H. 

NASA/TP—2009-213439 97 

Appendix H.—Relation Between Global Beta and Radial Deceleration

Initially, displaced magnetic flux between the plasma and 

the metallic wall (behind the insulating layer) builds up 

external magnetic pressure to match the radially expanding 

internal plasma pressure and slow down the radial expansion. This phase constitutes a dynamical adjustment 

described below in more detail. It is to be distinguished from 

the subsequent resistive phase that consists of radially 

expanding ions attempting to drag reluctant electrons across 

magnetic flux. The latter phase may involve either classical 

resistivity or anomalous resistivity from microturbulence. 

Newly injected high-temperature plasma tries to expand 

across the ambient magnetic field, if it is not injected right 

along the magnetic field lines. The preexisting nozzle magnetic field, however, counters that tendency early on. In the 

simplest MHD model, the high-conductivity Ohm’s law is 

E + V  B = 0, which has some validity at short times during 

the setting-up phase. Here, E is the electric field in the lab 

frame of reference. The symbol V is employed for the fluid 

(macroscopic) velocity vector of the plasma ions in order to 

be clearly distinct from the velocity vector vi of an individual 

ion. As the plasma’s lateral boundary crosses outwards 

through downstream-directed longitudinal magnetic flux (in 

the local z-direction), with quasi-radial fluid velocity Vx (in 

the local x-direction, x ˆ = ˆ  B/B), an azimuthal electric 

field (VxBz) is generated in the clockwise direction (looking 

downstream). Then, by Faraday’s law (or Lenz’s law), 

 (H1) 

where the closed-line integral is taken around the plasma 

circumference in the clockwise direction, and is the 

enclosed longitudinal magnetic flux. 

The downstream-directed magnetic flux thereby becomes 

diminished within the plasma and displaced to the annular 

volume outside of the plasma. Longitudinal magnetic flux 

can be laterally repositioned, but once established it cannot 

be created or destroyed within the volume defined by the 

highly conducting wall, during the short time interval of 

interest. The shifted flux, in addition to the original external 

flux, is squeezed between the plasma boundary and the 

conducting wall or coil, which thereby results in an increased 

magnetic field strength in the annulus outside of the plasma. 

The reconfigured flux therefore exerts an increased magnetic 

backpressure impeding the further radial expansion of 

plasma. The squeezed longitudinal magnetic flux acts like a 

compressible gas having an adiabatic index of 2. The 

expanding plasma therefore experiences a radial deceleration. 

Although the above brief description was based upon the 

simplest MHD model of a high-conductivity Ohm’s law, it 

can be extended to more detailed plasma models. This model 

envisions neutral plasma motion across the magnetic field. 

The kinetic ion intrusion  in the breech into the confining 

magnetic field takes place in a few hundredths of a microsecond in a field of thousands of Gauss. This takes place in a 

very short time span (on the order of one-quarter ion gyroperiod) compared with the transit time from breech to throat 

(~10 μs at a velocity of 105

 m/s). We assume this kinetic 

penetration has already occurred, and so we go to the next 

phase, which is described by a fluid model. For example, as 

a steady-flow configuration is approached, with axial symmetry, the azimuthal electric field in the lab frame tends to 

vanish; however, resistive currents are driven so as to continue to shield the plasma interior from penetration by the 

external magnetic field. To see this, use the next level of 

plasma model, namely, a resistive medium of resistivity . In 

this case the perfect conductivity model E + V  B = 0 

evolves over time into V  B = J, where J is the current 

density in the edge plasma (where V represents the resistive leakage of fluid velocity of the plasma in the quasiradial direction across the local magnetic field. The 

resistive currents generated according to this model are 

driven in the counterclockwise direction (looking downstream). They generate magnetic fields that have the net 

effect of excluding magnetic flux from the interior plasma 

but reenforcing it in the exterior annulus. Magnetic shielding by an edge layer of plasma current is according to 

Ampere’s law,   B = μ0J. 

Still more detailed plasma models can be taken into 

account, in which collisions of electrons with ions are rare and 

the Hall effect is important. It is then necessary to 

include the effects of electric fields due to charge separation 

and to employ separate models for the electron and ion 

motions (see, for example, the discussion after Equation (101) 

in Section 3.1, “Interface Width in Breech of Nozzle”). Magnetic shielding of the interior plasma is again realized. 

The purpose of this appendix is to focus on the initial 

dynamical adjustment phase, by furnishing an account of the 

influence of the radial deceleration of plasma on the G. We 

work in a reference frame that follows a plasma element in 

the main longitudinal flow, and a planar model of the 

plasma-field interface is employed. Curvature of the 

magnetic field lines is neglected for the short time interval of 

interest. The ion and electron quasi-radial momentum equations then can be expressed as follows: 

 (H2) 

The first equation is the ion quasiradial momentum balance 

equation, wherein the ions carry no azimuthal current at 

early times; the second equation, with the help of Ampere’s 

law, is the electron quasiradial momentum balance equation. 
NASA/TP—2009-213439 98 

With the help of the ion mass conservation equation, 

∂tρ + ∂ x ( ) ρVx = 0 , the equations in Equation (H2) may be 

combined as follows: 

 

2

2

tot

0

( ) ( ) ( )0 2 tx xx x

B ∂ ρV VP + ∂ ρ +∂ + =

μ

 (H3) 

As mentioned above, Vx is the radial component of the ionfluid velocity, and the equations are written in the frame that 

tracks the longitudinal flow. In Equation (H3), the total 

plasma pressure is Ptot = Pe + Pi

. This equation would have 

had the same form were the ions to carry some azimuthal 

current. 

Equation (H3) is integrated across the evolving interface 

plasma-field mixing layer of width δ. The integration is 

performed from a point at the edge of the core plasma where 

the density is large (subscript “core”) and the magnetic field 

is small, to just beyond the mixing layer where the density is 

small and the magnetic field is large (subscript “δ”). These 

inner and outer points are not fixed, but move radially with 

the plasma fluid. 

The integral of the first term in Equation (H3) is 

rearranged with the time derivative moved in front of the 

whole integral. It then is important to note that contributions 

to that integral arise that are associated with the timedependent limits of integration. These exactly cancel the 

contributions to the integration that originate from the 

second term in Equation (H3). In view of these cancellations, 

the integrated equation then can be written as 

2

eff core

0 2

B

g P δ  

ρ δ= −     μ  

 (H4) 

In Equation (H4), ρ is the average mass density in the 

layer of width δ, and the effective gravitational acceleration 

geff is 

t

V

g x

d

d

eff = − (H5) 

where Vx is the density-weighted average fluid velocity in 

the layer. Specifically, 

core

core

d

d

x

x

V x

V

x

δ

δ

ρ

=

ρ





 (H6a) 

core

1 dx

δ

ρ = ρ δ  (H6b) 

Now dividing through in Equation (H4) by the external 

magnetic pressure, the global beta is 

0

2

eff

2

1

μ

ρ δ β = −

Bδ

g

G (H7) 

If there were no radial deceleration (g = 0) or radial 

macroscopic expansion (Vx = 0), then one would have 

βG = 1. With radial deceleration geff > 0, one must have 

βG < 1. We shall now estimate the deviation ( ) 1− βG due to 

geff. In order to do so, the following three assumptions are 

made: 

(1) The average density in the plasma-field mixing layer is 

roughly half of the density in the bulk plasma: 

core

2

ρ ρ ≈ (H8) 

(2) Consistent with the estimates made earlier, the width 

of the edge-plasma boundary layer still is at least roughly 

given by 

A

pi ci

c V δ≈ =

ω ω

 (H9) 

The second part of this equation is an identity. Here, the 

inner plasma density and the outer magnetic field are used. 

(3) A fraction h of the radial fluid velocity is lost by 

deceleration during the short time increment of interest, 

dt ≈ 1/ωc i and h ≤ 1. The effective gravitational acceleration 

g V t eff = − d x d , can then be represented as follows. The 

numerator, − d Vx = h Vx , is to be divided by dt, roughly 

one-fourth of an ion gyroperiod. Then, 

eff ci x g ≈ ω h V (H10) 

Using the above three assumptions and referring to Equation (H7), a simple calculation shows that the deviation of 

the βG from 1 is given by 

A

x

V

V

h

B

g ≈ μρ δ

δ 0

2

eff

2

 (H11) 

Our basic model has been based on the concept of a 

highly sub-Alfvenic expansion of injected plasma across the 

flux surfaces, Vx << VA. This approximation corresponds to a 

very strong vacuum magnetic field and/or injection of propellant at grazing incidence to B, so that the displaced magnetic flux very quickly exerts a backpressure that prevents 
NASA/TP—2009-213439 99 

the realization of a large radial expansion velocity. As the 

plasma expands radially, the plasma pressure falls and the 

external magnetic pressure rises, so that the net force on the 

edge layer is inwards. After this initial dynamical adjustment, 

the plasma may still expand slowly across the magnetic flux, 

but only by the ions dragging reluctant electrons across the 

flux. This second phase corresponds to resistive diffusion. 

Since h ≤ 1, it then follows that βG is very close to 1. 

 

NASA/TP—2009-213439 101 

Appendix I.—Initial Ratio of Ion Gyroradius to Plasma-Field Interface Width

Starting with the basic local identity 

= β

ωp i

i

c

a (I1) 

we wish to evaluate this quantity first at a representative 

position in the middle of the plasma’s edge gradient layer, 

the interface region, and then convert that result to the 

ratio [ai]mid/δ, where δ is the width of the interface; 

δ ≈ 2[c/ωp i]core. Here, “core” means the internal number 

density is used to evaluate the ion plasma frequency. In 

Section 3.1, “Interface Width in Breech of Nozzle” the 

interface width was estimated to have a somewhat larger 

value. 

When evaluating the local 2 βi = 8πnTi B , the local n in 

the numerator is set to half the core density, and the local 

magnetic pressure in the denominator is set to half the external magnetic pressure. The result is βi

 = βG i, where the 

subscript “G” refers to the global ion beta, which is defined 

by the ratio of core ion pressure to external magnetic pressure. Assuming that the uniform temperatures are equal, 

Te = Ti

, the total βG must be 1 across a static plasma-field 

mixing layer. Therefore, the ion contribution to βG is onehalf; βG i = 1/2. Equation (I1) then reads 

mid

1

2

i

p i

a

c

 

  =

  ω  

 (I2) 

However, mid core [] [] 2 ω =ω pi pi , when the mid-density 

is taken as half the core density. Then, Equation (I2) 

becomes

mid

core

1

2 2

i

pi

r

c

    =   ω  

 (I3) 

Dividing Equation (I3) by 2 and for core 2 pi δ≈ ω   c

  , 

we then obtain 

mid 1

2

i   a  = δ (I4) 

Because δ actually was calculated to have a somewhat 

larger value, we chose a somewhat smaller final working 

ratio: 

mid 1

3

i   a  = δ (I5) 

The distinction is not overly significant (see Fig. 4). This 

then characterizes the ratio of electron current-drift velocity 

to ion thermal velocity at a representative position within the 

edge-plasma gradient. Of course, different positions within 

the gradient layer will produce different numerical results for 

ai/δ. We do not attempt to obtain a profile-dependent result 

in this report. 

NASA/TP—2009-213439 103 

Appendix J.—Generalized Plasma-Field Interface Thickness at the 

Throat Calculation and Example

This appendix presents a more generalized derivation of 

the plasma-field interface thickness at the throat—where 

both the initial interface thickness at the breech and its 

growth from breech to throat are taken into account. The 

related issue of the ion gyroradius to interface thickness ratio 

equation is also discussed. 

J.1 Calculation of Plasma-Field Interface Thickness 

and Fraction of Attached Plasma Discussion 

and Example 

Given a desired specific impulse, which is essentially the 

desired exit velocity Vex at the effective exit of the thruster’s 

nozzle, the velocity of propellant at the throat is, to a good 

approximation, 

5 ex Vt = 0. V (J1) 

The provisionally estimated travel time tb t of propellant 

from breech to throat may be expressed as 

t

bt

bt V

L

t = (J2) 

where Lb t is the axial distance from breech to throat. This 

time estimate is subject to correction as described later. 

If the interface thickness in the breech were zero, it would 

thicken with time as 

t

bt

V

DL

δ = Dt = (J3) 

so as to acquire this value in the throat, provided that we take 

beta to be 1 as a representative magnitude within the interface. We provisionally regard this δ as the width of the radial 

density profile in the interface, but subject to correction as 

described later. 

If, instead, the boundary layer thickness in the breech 

were δb , then the above formula would generalize to 

t

bt

b b V

DL

δ = δ + Dt = δ + 2 2 (J4) 

This result is based upon the solution of a differential equation in time along the flow. 

In this rendering of Equation (J4), a time-integration has 

been artificially avoided by regarding D as the average value 

of the resistive diffusivity between breech and throat. However, D does not vary much between breech and throat, 

whether classical or anomalous due to Lower Hybrid Drift 

(LHD) microturbulence. Using D at the position of the throat 

provides a reasonable estimate of the interface thickness for 

conditions of interest, because the diffusive contribution to 

Equation (J4) generally does not greatly dominate the initial 

(breech) contribution 2

b δ , and moreover, a square root is 

taken to get the final result. 

When modified as described below, Equation (J4) can be 

utilized to estimate the fraction of attached plasma at the 

position of the throat. 

As shown in Section 3.1, “Interface Width in Breech of 

Nozzle,” the interface thickness in the breech, which appears 

in Equation (J4), can be estimated as 

p i

b

c

ω

δ = 2 (J5) 

in which c is the speed of light in free space and ωp i is the 

ion plasma frequency. 

Equation (J4), moreover, provides the basis for analytic 

estimates of the ratio of ion gyroradius to interface thickness, 

ai/δ, describing that ratio as it varies along the flow. That 

work is presented in Section 3.1. A particular relevance of 

that ratio is that it plays a central role in analytic theories of 

LHD microturbulence, and especially in the anomalous 

magnitude of the associated resistivity. In this regard, it is 

important to remark that these analytic theories (for example, 

Refs. 18, 23, 24, and 35) are usually limited to the electrostatic (zero-beta) model, in which the gradient under 

consideration is only the density gradient. Thus, for the 

purpose of referring to the seminal analytic results of these 

theories, the equations for ai/δ should be viewed within the 

context of a resistively diffused density gradient. The δ

therein thus represents the density gradient width, without 

the effect of bidirectional diffusion. 

In the application to space vehicle thrusters with magnetic 

nozzles, however, we need to estimate the fraction of plasma 

that has become attached to the magnetic nozzle field lines. 

For that application, inward magnetic diffusion into the 

plasma is just as important as outward diffusion of plasma 

into the external magnetic field. 

In that case, two corrections ought to be made to Equation (J4). The first is trivial; namely, to expand the travel 

time by about a factor 2 in order to allow for the time needed 

to accelerate the interface plasma up to speed. Thus, 

t = Lb t/Vt becomes 

t

bt

V

L

t

2 = (J6) 
NASA/TP—2009-213439 104 

Equation (J6) is the approximate result of an analytic investigation with realistic nozzle geometry for the plasma edge. 

(This modification with the factor 2, in principle, should also 

be made to the above-mentioned equations for the ratio ai/δ, 

but that will have only a minor effect on the general conclusion that this ratio varies little between breech and throat.) 

The second correction is the more subtle, namely, to allow 

for simultaneous resistive diffusion of plasma into nozzle 

magnetic field and of field into plasma; the so-called bidirectional diffusion. Each of these diffusive processes separately 

(in the absence of Hall current and plasma rotation) would 

accrue a diffusive distance of about √(D t), starting from 

zero thickness in the breech. (For D = 3 m2/s and t = 10 μs, 

this diffusive distance is about 0.5 cm.) Heuristically adding 

the two effects, we could reasonably estimate the total interface thickness as 

δ = Dt = 4Dt (J7) 

The estimate that constitutes Equation (J7) is in need of a 

careful numerical investigation, within the context of a 

slowly convergent breech-to-throat geometry, as mediated by 

a continuously distributed and continuously graded magnet 

coil. Such a gradual configuration geometry would likely 

preclude the anomalous cross-field loss of plasma observed 

in a prior resistive MHD simulation, which employed discrete coils and axially limited distances appropriate to an 

envisioned lab experiment. 

Accounting for both of the above-mentioned corrections, 

Equation (J4) now reads 

 

t

bt

b V

8DL 2 δ = δ + (J8) 

At sufficiently low temperatures, D would be the classical 

resistive diffusivity associated with coulomb scattering of 

electrons on ions. 

The classical resistivity itself appears in Equation (200) in 

CGS units. The coulomb logarithm therein (the logarithm of 

the ratio of the maximum-to-minimum impact parameter in 

an electron collision with an ion) has been set to 10, which is 

sufficient at the present level of development of our investigation. (See Ref. 9.) Multiplication by (c

2

/4π) converts this 

resistivity to D in CGS units. Multiplication of the CGS 

resistivity by 9⋅109

 converts the resistivity to MKS units. 

Dividing the latter by μ 0, which is the magnetic permeability 

of free space, then yields D in MKS units. Alternatively, one 

can convert D (CGS) into D (MKS) by just dividing the 

former by 104

. 

At sufficiently high temperatures, D could result from 

LHD microturbulence evolving from edge-gradient instabilities. At intermediate temperatures, D would be appropriately 

regarded as arising from the sum of both resistivities; that is, 

the two types of electron collision frequencies, classical and 

anomalous, would be additive. 

We shall now present a numerical example of the calculation of the fraction of attached plasma, based upon Equation (J8). Of course, the parameters to be assumed here can 

be changed as needed, concomitantly with changes of the 

resistive diffusivity D, and the interface width in the breech 

δb. 

J.2 Calculation of the Fraction of Attached 

Plasma—An Example 

Consider hydrogen propellant, breech temperature 50 eV, 

and breech number density 1.0⋅10 15 cm

−3. The specific 

impulse is then about 20 000 s, and Vt = 1.0⋅107

 cm/s. (Here, 

we employ approximate numbers, but they can be made 

more accurate in a final calculation. Of course, other parameters can also be assumed.) 

The interface thickness in the breech is then δb = 1.7 cm. 

More to the point, 

2

b 2.9 cm 2 δ = (J9) 

For the given parameters, classical resistivity by far dominates over anomalous resistivity due to the LHD instability 

(see Fig. 4) and one finds approximately 

D 3 10 cm s

4 = ⋅ (J10) 

which is the value in the throat and which thus allows for the 

temperature drop relative to the breech. 

With an axial breech to throat distance of one meter, Lb t = 

100 cm, then Equation (J8) becomes 

δ = 3.0 = 2.4 = 2.3 cm (J11) 

Because we included bidirectional diffusion, the diffusive 

contribution is about the same as that from the initial 

(breech) contribution. Without the correction for bidirectional diffusion, the second term would be rather small in 

comparison to the breech contribution to the interface thickness. That result for density diffusion alone then would be 

relevant to the gyro-radius ratio to the thickness of the density gradient, as discussed earlier. 

In order to calculate the fraction of attached plasma, one 

must already know the rate of mass ejection, m . This quantity is supposed to be predetermined by the logistics of the 

given mission. With a hydrogen plasma radius in the throat, 

rt = 10 cm, one has m = 3.5 g/s, which already accounts for 

the drop in density from breech to throat. Different desired 

values of the mass ejection rate would lead to different values of the plasma radius in the throat, at the assumed breech 
NASA/TP—2009-213439 105 

density. (Of course, the plasma radius in the throat must be 

the result of a self-consistent steady-flow interaction 

between the injected plasma and the shaped vacuum 

magnetic nozzle field. Prior simulations have already demonstrated the self-consistent formation of a plasma throat, 

within the time-dependent resistive MHD model.) 

If the core density were to extend throughout the width of 

the interface, then the fraction of attached plasma would be, 

to sufficient approximation, 

46 percent

2 = δ

tr (J12) 

However, the average density in the interface will, of course, 

be somewhat less than the core density. A reasonable provisional estimate, subject to the results of a future resistive 

MHD simulation without ill understood anomalies, is that 

the average density in the interface is roughly half of the 

core value. (This would be the case for a linear spatial 

decrease to zero in the radial edge density profile.) Then one 

finds the fraction of attached plasma to be 23 percent. This 

is, after all, a significant loss of plasma, which portends that 

there will be a significant detachment problem to be solved 

downstream, if the idealized thruster results are to be 

approached. 

At first glance, the above example points to the advantages 

of working at higher temperatures to slow down resistive 

diffusion. Then, higher mass propellant ions would be 

employed to preserve the desired specific impulse. Furthermore, in order to mitigate the initial interface width in the 

breech, it would be advantageous as well to work with a 

higher number density of ions, so as to increase the ion 

plasma frequency there. But note that the higher mass ion 

decreases the ion plasma frequency, whereas the higher ion 

charge increases it! The presence of a higher ion charge also 

tends to increase resistive diffusion due to coulomb scattering of electrons by ions. Obviously, there are several competing effects that need scrutiny. A scenario involving higher 

stagnation number density and higher stagnation temperature 

in a propellant consisting of higher mass (and charge) ions, 

becomes of interest for future investigations. 

NASA/TP—2009-213439 107 

Appendix K.—Raleigh-Taylor Instability: Further Considerations

In this appendix, we furnish the details leading to the spatial dependence of the external magnetic field when flute 

distortions of the surface are present. In the main report, we 

presented a derivation of the growth rate of this flute-mode 

instability in the magnetic nozzle-propellant interface. The 

derivation given suppressed some details; specifically, the 

property that the external magnetic field B has a 1/R dependence is crucial, but was not given a full discussion. Here, R

is the local longitudinal radius of curvature of the surface, 

which can be extended beyond the equilibrium surface to 

signify a spatial variable in the external volume outside of 

the propellant. 

In the equilibrium situation involving only the unperturbed boundary surface of propellant, the above-mentioned 

spatial dependence of the external magnetic field can readily 

be demonstrated. The integral form of Ampere’s law is 

employed, using an Ampere circuit around the highly conducting, sharp-boundary propellant configuration, which is 

assumed to diamagnetically exclude the magnetic flux. 

Therefore, the integral form of Ampere’s law immediately 

yields the result that B = μ0KReq/R, where Req is the equilibrium radius of local longitudinal curvature and K is the local 

transverse surface current per unit longitudinal arc length. 

The question remains as to the spatial dependence of the 

external magnetic field under the nonequilibrium condition 

associated with the distortion of the surface from Req 

(denoted by δR) into flutes. Then R = Req + δR represents the 

distance of a point on the distorted interface from the local 

center of longitudinal curvature of the interface. (The equilibrium shape of the interface still has longitudinal radius of 

curvature Req.) 

Figure 7 illustrates a longitudinal local section of a 

converging-nozzle surface with “adverse” longitudinal curvature and flutes. The center of longitudinal curvature is at C. 

The equilibrium surface possesses a radius of longitudinal 

curvature Req. The crests and troughs of the flutes on that 

surface are indicated by R = (Req + δR). The flutes are 

observed by rotating the r,Z-plane of the figure around the 

Z-axis, while C is fixed (in the rotating plane) but the radius 

of longitudinal curvature fluctuates. The subtended longitudinal angle is dφ, and the length of arc shown (solid) is Req dφ. 

In the equilibrium situation, without surface distortions, one 

may regard R as a radial position variable in the space outside of the unperturbed surface. Since the magnetic field B

only exists in the external volume, one may apply the 

integral form of Ampere’s law to the circuit indicated by 

(→), to show that B = μ0 K (Req/R), where K is the surface 

current per unit longitudinal arc length. 

In this appendix we furnish the details leading to the spatial dependence of the external magnetic field when flute 

distortions of the surface are present. The proof will be 

carried out primarily verbally and pictorially to facilitate its 

accessibility, but we believe that it is nevertheless a rigorous 

proof when based upon the following assumptions and their 

corollaries: 

(1) The propellant core is highly conducting (a “perfect” 

conductor) and is bounded by a sharply defined surface (the 

interface) separating the internal field-free plasma from the 

external longitudinal magnetic nozzle field (Br, BZ). These 

are the only components of B after the original axially symmetric configuration becomes distorted by flute modes. 

(2) According to the integral form of Ampere’s law, 

∇×B = μ0J, there must necessarily exist a surface current 

that separates the null-field interior plasma from the nozzle 

magnetic field in the external region. We denote this surface 

current density by K. 

(3) The longitudinal flute-mode displacement of each 

point on the sharp-boundary plasma surface occurs only in 

the r,Z-plane corresponding to that equilibrium point and is a 

periodic function of the azimuthal angle θ. (The cylindrical 

coordinates are (r, θ, Z).) We caution that the flute-distortion 

motion of the surface itself is not identical to the motion of 

the underlying fluid. Nevertheless, only the surface displacement is relevant to the disposition of the external magnetic field. 

(4) Our simple model of the flute distortion is that the 

local distorted surface remains “parallel” to the equilibrium 

surface, in the sense depicted in Figure 7(a). This assumption 

is meant to reproduce, as closely as possible, the fluting of a 

plane surface, in which the longitudinal generators of the 

distortions remain parallel to the original plane. This particular model of the flute distortions is realized by having the 

center of curvature remain fixed in the r,Z-plane as it rotates 

around the axis of initial symmetry. Then the longitudinal 

radius of curvature of the surface varies as the plane rotates, 

thus generating the crests and troughs of the flutes. (Note: 

Restriction of the present model to address only a short 

longitudinal increment of the surface means that the phenomena of global ballooning modes are neglected. Localized 

g-modes are the objects of the present investigation. Eventually, however, the ballooning modes should be studied to 

provide a more comprehensive picture of processes that 

affect regions of adverse curvature. Such regions might exist 

throughout the nozzle: either upstream of the nozzle-throat 

(where the nozzle narrows down into the throat) or also 

downstream of the throat (where the diverging field lines are 

turned so as to straighten out the flow). 

(5) The magnetic field must remain external to the distorted fluid surface because magnetic field lines cannot 

penetrate a perfect conductor if they initially lie outside. 

Moreover, the field lines must remain longitudinal (Bθ = 0) 

during the flute distortion, because there exists no conductingsurface motion restricted to longitudinal flutes that can 

initiate any transverse magnetic field component. (There is 
NASA/TP—2009-213439 108 
NASA/TP—2009-213439 109 

no preferred transverse field direction under the given surface displacement.) 

(6) The longitudinal field lines just outside the surface 

must remain tangential to the distorted surface: first, because 

they cannot enter or leave the surface and second, because 

any vacated external vacuum volume would become instantly 

filled with magnetic field (at the speed of light). 

(7) The vector field that describes the current density K

must (of course) lie on the evolving surface during the distortion, and it also must remain locally perpendicular to the 

contiguous longitudinal magnetic field vectors. Otherwise, if 

K had acquired a component along B, then B would necessarily acquire a transverse component, but this is impossible 

under the conditions restricted to flute distortions of the 

surface (see above). 

Based upon the above assumptions, we can now show that 

the magnetic field at the distorted surface varies as 1/R, 

where R is the local longitudinal radius of curvature on the 

distorted surface. This is done by means of the integral form 

of Ampere’s law. 

It can be inferred from Figure 7(a) that as the r,Z-plane is 

rotated to register a sequence of flute distortions, lines of the 

vector field K cannot cross the boundaries of the sector dφ. 

For if they did, the surface would then necessarily acquire a 

component along longitudinal B, which cannot happen in the 

considered flute distortions (see above). Therefore, the total 

surface current in dφ is conserved. That is, KR dφ is constant. 

(R dφ is the longitudinal arc length crossed by the surface 

current.) 

Figure 7(b) shows an Ampere circuit of vanishing width, 

indicated by (→). The thin Ampere circuit contains the 

propellant’s sharply defined surface. The model considered 

here is delimited by a longitudinal angle increment dφ, 

which is measured from the center of longitudinal curvature C shown in Figure 7(a). The symbol “xxxxx” schematically indicates the surface current per unit longitudinal 

length, denoted as K. The plane of Figure 7(b) (and 7(a)) is 

the r,Z-plane, but the plane of the Ampere circuit is perpendicular to the distorted (fluted) surface. Hence the 

Ampere circuit is in the r,Z-plane only at the crests and 

troughs of the distorted surface. Figure 7(c) illustrates this 

end-on view of a fluted surface, showing three Ampere 

circuit planes. 

Considering Figure 7(b), we can apply the integral form 

of Ampere’s law to the indicated circuit. The result reads BR

dφ = μ0KRdφ; also, BRdφ = constant. Hence, B varies as 1/R

even when the surface possesses flute distortions. This result 

was specifically utilized in Section 4.2.1, “Surface Instability 

of Static Plasma in Region of Adverse Curvature,” in the 

discussion of the exponential growth rate of the linearized 

flute-mode fluctuations. 

Aside from the spatial variation of magnetic field B near 

the surface of the propellant, there is a concern that B could 

suffer an overall shift because of a change in surface current 

K as the flutes evolve. However, the large inductance of the 

global currents provides very large impedance against shifts 

in K on the time scales for growth of the Raleigh-Taylor 

instabilities. 

NASA/TP—2009-213439 111 

References

 1. Mikellides, I.G., et al.: Theoretical Design of a MagneticNozzle Acceleration System for Fusion Propulsion. AIAA–

2000–3367, 2000. 

 2. Turchi, P.J., et al.: Numerical Simulation of Magnetic Nozzle 

Flow for Nuclear-Fusion Space Propulsion. AIAA–1999–

2701, 1999. 

 3. Mikellides, I.G., et al.: Design of a Fusion Propulsion System—

Part 2: Numerical Simulation of Magnetic-Nozzle Flows. J. 

Propul. P., vol. 18, no. 1, 2002. 

 4 Williams, Craig H., et al.: Realizing “2001: A Space 

Odyssey”—Piloted Spherical Torus Nuclear Fusion Propulsion. NASA/TM—2005-213559, 2001. 

 5. Williams, Craig H., et al.: A Sperical Torus Nuclear Fusion 

Reactor Space Propulsion Vehicle Concept for Fast Interplanetary Piloted and Robotic Missions. AIAA–1999–2704, 1999. 

 6. Gerwin, Richard A., et al.: Characterization of Plasma Flow 

Through Magnetic Nozzles. AL–TR–89–092 (LA–UR–89–

4212), 1989. 

 7. Moses, R.W.J.; Gerwin, R.A.; and Schoenberg, K.F.: Resistive 

Plasma Detachment in Nozzle Based Coaxial Thrusters. AIP 

Conference Proceedings, pt. 3, 1992, pp. 1293–1303. 

 8. Peter Turchi, 1999, Ohio State University, Columbus, OH, 

personal communication. 

 9. Braginskii, S.I.: Transport Processes in a Plasma. Rev. Plasma 

Phys., vol. I, M.A. Leontovich, ed., Consultants Bureau, New 

York, NY, 1965, p. 205. 

10. Spitzer, Lyman: Physics of Fully Ionized Gases. Interscience 

Publishers, New York, NY, 1956. 

11. Black, D.C., et al.: Two-Dimensional Magnetic Field Evolution 

Measurements and Plasma Flow Speed Estimates From the 

Coaxial Thruster. Phys. Plasmas, vol. 1, issue 9, 1994, pp. 

3115–3131. 

12. Sgro, A.G.; and Nielson, C.W.: Hybrid Model Studies of Ion 

Dynamics and Magnetic Field Diffusion During Pinch Implosions. Phys. Fluids, vol. 19, 1976, pp. 126−133. 

13. Chodura, R.: A Hybrid Fluid-Particle Model of Ion-Heating in 

High Mach Number Shock Waves. Nuclear Fusion Journal, 

vol. 15, no. 7, 1975, p. 55. 

14. Brackbill, J.U., et al.: Nonlinear Evolution of the Lower-Hybrid 

Drift Instability. Phys. Fluids, vol. 27, no. 11, 1984, pp. 2682–

2693. 

15. Drake, J.F.; Huba, J.D.; and Gladd, N.T.: Stabilization of the 

Lower-Hybrid-Drift Instability in Finite-Beta Plasmas. Phys. 

Fluids, vol. 26, no. 8, 1983, pp. 2247–2249. 

16. Peter, W.; Ron, A.; and Rostoker, N.: Instability of the 

Boundary-Layer Between a Streaming Plasma and a Vacuum 

Magnetic-Field. Phys. Fluids, vol. 26, no. 8, 1983, pp. 2276–2280. 

17. Ripin, B.H., et al.: Sub-Alfvenic Plasma Expansion. Phys. 

Fluids B, vol. 5, no. 10, 1993, pp. 3491–3506. 

18. Davidson, R.C.; and Gladd, N.T.: Anomalous Transport Properties Associated With Lower-Hybrid-Drift Instability. Phys. 

Fluids, vol. 18, no. 10, 1975, pp. 1327–1335. 

19. Freidberg, J.P.; and Gerwin, R.A.: Lower Hybrid Drift Instability at Low Drift Velocities. Phys. Fluids, vol. 20, no. 8, 

1977, pp. 1311–1315. 

20. Gladd, N.T.; Sgro, A.G.; and Hewett, D.W.: Microstability 

Properties of the Sheath Region of a Field-Reversed Configuration. Phys. Fluids, vol. 28, no. 7, 1985, pp. 2222–2234. 

21. Choueiri, E.Y.: Electron—Ion Streaming Instabilities of an 

Electromagnetically Accelerated Plasma. Ph.D. Thesis, Princeton Univ., 1991. 

22. Huba, J.D.; and Ossakow, S.L.: Destruction of Cyclotron 

Resonances in Weakly Collisional, Inhomogeneous Plasmas. 

Phys. Fluids, vol. 22, no. 7, 1979, pp. 1349–1354. 

23. Davidson, R.C.: Quasilinear Stabilization of Lower-HybridDrift Instability. Phys. Fluids, vol. 21, no. 8, 1978, pp. 1375–

1380. 

24. Gary, S. Peter: Wave-Particle Transport From Electrostatic 

Instabilities. Phys. Fluids, vol. 23, issue 6, 1980, pp. 1193–1204. 

25. Drake, J.F.; Guzdar, P.N.; and Huba, J.D.: Saturation of the 

Lower-Hybrid-Drift Instability by Mode-Coupling. Phys. Fluids, vol. 26, no. 3, 1983, pp. 601–604. 

26. Drake, J.F., et al.: Nonlinear Mode-Coupling Theory of the 

Lower-Hybrid-Drift Instability. Phys. Fluids, vol. 27, no. 5, 

1984, pp. 1148–1159. 

 27. Fahrbach, H.U., et al.: Measurement of Lower Hybrid Drift 

Fluctuations in the Boundary-Layer of a High-Beta Plasma by 

Collective CO2-Laser Light-Scattering. Nucl. Fusion, vol. 21, 

no. 2, 1981, pp. 257–270. 

28. Berk, H.L.; and Dominguez, R.R.: Variational Method for 

Electromagnetic Waves in a Magneto-Plasma. J. Plas. Phys., 

vol. 18, 1977, pp. 31–48. 

29. Hastings, D.E.; and McCune, J.E.: The High-β Universal Drift 

Mode. Phys. Fluids, vol. 25, 1982, pp. 509−517. 

30. Huba, J.D.; and Gary, S.P.: Finite-β Stabilization of the Universal Drift Instability: Revisited. Phys. Fluids, vol. 25, 1982, pp. 

1821−1823. 

31. Lapenta, G.; and Brackbill, J.U.: 3D Reconnection Due to 

Oblique Modes: A Simulation of Harris Current Sheets. Nonlinear Processes in Geophysics, vol. 7, no. 3/4, 2000, 

pp. 151−158. 

32. Huba, J.D.: Finite Larmor Radius Magnetohydrodynamics of 

the Rayleigh-Taylor Instability. Phys. Plasmas, vol. 3, no. 7, 

1996, pp. 2523–2532. 

33. Huba, J.D.: Hall Magnetohydrodynamics In-Space and Laboratory Plasmas. Phys. Plasmas, vol. 2, no. 6, pt. 2, 1995, 

pp. 2504–2513. 

34. McKenna, K.F., et al.: Radial Plasma Structure During a ThetaPinch Implosion—Flute Instabilities. Phys. Fluids, vol. 18, no. 

10, 1975, pp. 1371–1373. 

35. Gary, S. Peter; and Thomsen, Michelle F.: Collisionless Electrostatic Interchange Instabilities. J. Plasma Phys., vol. 28, 

pt. 3, 1982, pp. 551–564. 

36. Nagornyj, V.P.; Ryutov, D.D.; and Stupakov, G.V.: Flute 

Instability of Plasma in a Gas-Dynamic Trap. Nucl. Fusion, 

vol. 24, no. 11, 1984, pp. 1421–1438. 

37. Landau, L.D.; and Lifshitz, E.M.: Fluid Mechanics. Second ed., 

Pergamon Books Ltd., Oxford, 1987. 

38. Huba, J.D.; and Winske, D.: Rayleigh-Taylor Instability: Comparison of Hybrid and Nonideal Magnetohydrodynamic Simulations. Phys. Plasmas, vol. 5, no. 6, 1998, pp. 2305–2316. 

39. Migliuolo, S.: Stability Theory of Drift-Type Flute Modes in 

Finite-Beta Plasmas. Phys. Fluids, vol. 25, no. 12, 1982, 

pp. 2289–2294. 

40. Freidberg, J.P.; and Wesson, J.A.: Ion Drift Wave Instability. 

Phys. Fluids, vol. 13, issue 4, 1970, pp. 1009–1016. 
REPORT DOCUMENTATION PAGE Form Approved 

OMB No. 0704-0188 

The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the 

data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this 

burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. 

Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB 

control number. 

PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 

1. REPORT DATE (DD-MM-YYYY)

01-12-2009 

2. REPORT TYPE

Technical Paper 

3. DATES COVERED (From - To)

4. TITLE AND SUBTITLE

Integrity of the Plasma Magnetic Nozzle 

5a. CONTRACT NUMBER

C-32057-G 

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S)

Gerwin, Richard, A. 

5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

WBS 22-973-90-01 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration 

John H. Glenn Research Center at Lewis Field 

Cleveland, Ohio 44135-3191 

8. PERFORMING ORGANIZATION

 REPORT NUMBER

E-14974 

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration 

Washington, DC 20546-0001 

10. SPONSORING/MONITOR'S

 ACRONYM(S)

NASA 

11. SPONSORING/MONITORING

 REPORT NUMBER

NASA/TP-2009-213439 

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified-Unlimited 

Subject Category: 15 

Available electronically at http://gltrs.grc.nasa.gov 

This publication is available from the NASA Center for AeroSpace Information, 443-757-5802

13. SUPPLEMENTARY NOTES

14. ABSTRACT

This report examines the physics governing certain aspects of plasma propellant flow through a magnetic nozzle, specifically the integrity of 

the interface between the plasma and the nozzle’s magnetic field. The injection of 100s of eV plasma into a magnetic flux nozzle that 

converts thermal energy into directed thrust is fundamental to enabling 10 000s of seconds specific impulse and 10s of kW/kg specific 

power piloted interplanetary propulsion. An expression for the initial thickness of the interface is derived and found to be ~10-2 m. An 

algorithm is reviewed and applied to compare classical resistivity to gradient-driven microturbulent (anomalous) resistivity, in terms of the 

spatial rate and time integral of resistive interface broadening, which can then be related to the geometry of the nozzle. An algorithm 

characterizing plasma temperature, density, and velocity dependencies is derived and found to be comparable to classical resistivity at local 

plasma temperatures of ~200 eV. Macroscopic flute-mode instabilities in regions of “adverse magnetic curvature” are discussed; a growth 

rate formula is derived and found to be one to two e-foldings of the most unstable Rayleigh-Taylor (RT) mode. After establishing the 

necessity of incorporating the Hall effect into Ohm’s law (allowing full Hall current to flow and concomitant plasma rotation), a critical 

nozzle length expression is derived in which the interface thickness is limited to about 1 ion gyroradius.

15. SUBJECT TERMS

Magnetic nozzle; Nuclear fusion; Space propulsion; Plasma physics 

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF

 ABSTRACT

UU 

18. NUMBER

 OF 

 PAGES

102 

19a. NAME OF RESPONSIBLE PERSON

STI Help Desk (email:help@sti.nasa.gov) 

a. REPORT

U 

b. ABSTRACT

U 

c. THIS 

PAGE

U 

19b. TELEPHONE NUMBER (include area code)

443-757-5802 

Standard Form 298 (Rev. 8-98)

Prescribed by ANSI Std. Z39-18