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Produced by the NASA Center for Aerospace Information (CASI)
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MAGNETOHYDRODYNAMIC A14D GASLYNAMIC THEORIES

FOR PLANETARY BOW WAVES

by

John R. Spreiter

Division of Applied Mechanics

Stanford University

Stanford, CA 94305

and

Stephen S. Stahara

RM Associates, Inc.

3738 Mt. Diablo Blvd., Suite 200

Lafayette, CA 94549
C

ABSTRACT

A bow wave has been observed in the solar wind upstream of each of

the six planets visited by spacecraft. The observed properties of ;these

bow waves and the associated plasma flows, are outlined, and those

features identified that can be described by a continuum

magnetohydrodynamic flow theory as opposed to a more detailed

multicomponent particle and field plasma theory. The primary objectives

of this paper are to provide an account of the fundamental concepts and

current status of the magnetohydrodynamic and gas dynamic theories for

solar wind flow past planetary bodies. This includes a crit-ical

examination of (a) the fundamental assumiptions of the theories, (b) the

various simplifying approximations introduced to obtain tractable

mathematical problems, (c) the limitations they impose on the results,

and (d) the relationship between the results of the simpler gas

dynamic-frozen field theory and the more accurate but less completely

worked out magnetohydrodynamic theory.

RepresenL;ative results of the various theories are presented and

compared. A number of deficiencies, ambiguities, and suggestions for

improvements are discussed, and several significant extensions of the

theory required to provide comparable results for all planets, their

satellites, and comets are noted. The paper concludes with some remarks

about anticipated trends in the future development and application of

the theory and a description of a number of applications and extensions

currently under development.
INTRODUCTION

Direct observations have established the presence of a bow shock

wave upstream of each of the innermost six planets of the Solar

System. These bow waves form in the solar wind plasma, which flows past

the planets at supersonic speeds carrying solar plasma and magnetic

fields away from the Sun, and define with certain exceptions the

upstream boundary of the region influenced by the presence of the

planetary body. It is the purpose of this paper to review the

quantitative magnetohydrodynamic and gasdynamic theories for calculating

the properties of these bow waves and their associated flow and magnetic

fields.

The principal features of these flows are as follows. Hot plasma

of the solar corona accelerates to supersonic speeds within a few solar

radii of the Sun and flows outward through the Solar System. Because of

its large scale and high electrical conductivity, this plasma carries

the embedded solar magnetic field with it, distorting it as it proceeds,

and generally forming a spiral magnetic field configuration as a result

of the combined action of the outward flow of the plasma and the

rotation of the Sun. At the orbit of Earth, the free-stream velocity v.,

may be less than 300 km s-' or greater than 1500 km s- 1 , the proton

number den.l ity NP

may occasionally be less than 1 cm- 3 or as high as

00

100 cm-3 , the proton temperature Tp may be less than 50,000 K or as

much as 500,000 K, and the magnetic field BCO may be less than 1 nT or as

much as 50 nT. The ion composition is primarily protons, i.e., ionized

hydrogen atoms; with from 5 to 20 percent of helium and other ions.

There is also an equal number of electrons, and their temperatures are

s 1
I

usually several times greater than that of the ions. The solar wind

flow approaches the Earth within a few degrees of the direction to the

Sun, with the aborration due to the planet's orbital motion providing a

substantial part of the deviation. The interplanetary magnetic field

direction is particularly variable, with the greatest predominance being

approximately in the plane of the ecliptic with an angle to the

Sun-Earth line of approximately 45 0 , and directed about equally either

toward or away from the Sun.

Figure 1 illustrates the nature of the resulting flow about the

Earth and other major obstacles in the Solar System in barest outline.

The Earth's magnetic field, being relatively strong, dominates the

plasma in an extended region surrounding the Earth to form the

magnetosphere. This region is bounded by the magnetopause, a relatively

thin layer in which electrical currents flow in such a way as to bound

the magnetospheric magnetic field on one side and the rapidly flowing

solar wind plasma and magnetic field on the other„ The solar wind

plasma experiences considerable difficulty in trying to cross the

magnetopause, and with some exceptions is excluded from the

magnetosphere and forced to flow around the magnetopause much as though

it were an impenetrable obstacle. The solar wind flow approaches the

planet at speeds of the order of 5 to 10 times the fastest wave

propagation speeds involving the ion motions, and decelerates abruptly

through a bow shock wave much as air does as it approaches a blunt-nosed

obstacle at supersonic speeds. Near the nose of this bow wave, the flow

is slowed to subsonic speeds, and turned to begin its flow around the

magnetopause. It accele-ates as it flows, and attains supersonic speeds

2
as it proceeds toward the flanks about as sketched. For representative

conditions in the solar wind, the geocentric distance to the

magnetopause nose is about 10 Earth radii, and that to the bow wave nose

is about 3.5 Earth radii more.

It is apparent that many features of this flow resemble those of

familiar supersonic gasdynamic flow past a blunt-nosed obstacle such as

a round-nosed bullet or a re-entry space vehicle. But gasdynamics alone

is not sufficient to describe other important features of the flow, such

as the formation of the magnetospheric obstacle itself, that depend on

the effects of a magnetic field in a highly conducting plasma. The

simplest theory that is capable of representing the main features of -

these flows is thus magnetohydrodynamics of a compressible highly

conducting plasma.

Although it is clearly established that ma netohydrodynamics can

provide a remarkably good prediction of many features of these flows, it

should be noted that this theory is not only complicated and far from

completely worked out, but also fundamentally incapable of describing a

number of other important features of the interaction. These include

waves and energetic particles that sometimes propagate upstream from the

bow wave along the interplanetary magnetic field, the differing nature

of the magnetic field jumps across the bow wave depending on the angle

between the incident interplanetary magnetic field and the shock normal,

and the nonMaxwellian character of the proton velocity distributions.

Analysis and prediction of such properties of these flows requires use

of multi-component plasma theory as opposed to the single-fluid

3
magnetohydrodynamic theory under discussion in this paper; and is the

subject of a number of other papers at this conference.'

The Earth may serve as a prototype for discussing planetary bow

shock waves and their associated flow fields, but there are also notable

differences amongst the planets as outlined in Figure 1. These stem

from differences in the planetary magnetic fields and atmosphere, and

from changes in the solar wind with distance from the Sun. As for the

solar wind, the velocity remains virtually the same from the orbit of

Mercury to beyond Saturn, the temperature, density, and-magnetic field

all decrease with distance from the Sun, and the sonic and Alfven Mach

numbers of the flow incident upon the planets remain much greater than

one; but the planetary properties differ quite markedly. Compared to

the Earth, which is essentially a magnetic obstacle in the solar wind,

Mercury has such a weak magnetic field and tenuous atmosphere that the

distance ro to the nose of the magnetopause is only about 1.5 times the

radius rp of the planet; whereas Jupiter and Saturn have such large

magnetic fields that the ratio ro /rp has values of about 70 and 22

respectively. The Earth's magneto pause is approximately axisymmetric

about a line through the Earth.'s center and parallel to the free-stream

direction; whereas the magnetopause of Jupiter, and perhaps of Saturn,

is flattened significantly toward the equatorial plane of the planet,

probably because of its rapid rotation and large size. Venus, which

formerly seemed so much like a sister planet of Earth but is now known

to be very different in many respects, has virtually no detectable

magnetic field, but does have a sufficientl y dense ionosphere to

withstand the solar wind and form an obstacle having a shape as

4
illustrated. In recognition that the boundary of the flowing solar wind

bounds the ionosphere and not the magnetic field, it is called the

ionopause. Its nose is only a few hundred kilometers above the planet's

c

surface, and the nose of the bow wave is about a third of a planetary

r radius above that. The case of Mars remains uncertain because of the

scarcity of observational evidence and the possibility that the location

of the magneto ionopause can be accounted for by either a magnetic field

or an ionospheric interaction. It is probably impossible to reach a

k definitive conclusion with the available magnetometer data, but other

evidence indicates that there is only a small range of possibilities for

the magnetic field that is compatible with the interpretation that Mars

acts like a magnetic obstacle. If the magnetic field were substantially larger than the proposed values, the magnetosphere would be noticeably larger. If the magnetic field were substantially smaller than suggested, it would have negligible effect on the location of the magnetnionopause and the bow wave, and the obstacle would be clearly ionospheric.

While not a planetary body, it is of interest to comment on the

case of the Moon, whose environment has been explored extensively with

plasma probes and magnetometers. Its interaction with the solar wind is

quite different from that of the planets. Having neither an atmosphere

nor significant magnetic, field, the Moon is unable to stop the approach

of the solar wind until it is absorbed at the lunar surface. The solar

wind thus flows unimpeded into the sunward surface of the Moon, and

leaves a cavity in the solar wind extending downstream from the night

side. Instead of forming a bow shock wave upstream of the Moon, a

trailing expansion Mach wave fan extends from the vicinity of the lunar

5
.

terminator as i Ulustrated through which the solar wind plasma is first

turned toward the axis to begin to fill in the cavity.

Still another class of related objects in the Solar System for

which similar questions are being addressed are comets. At this

writing, no measurements have been made in situ at comets, but several

spacecraft are already launched on their courses to rendezvous with

comets in the near future. There is considerable uncertainty about the

precise nature of the interaction, but the general consensus is that the

situation is as illustrated in Figure 1. The comet nucleus-is a source

of gas and particles that melt or evaporate off as the comet travels

along the more sunward part of its trajectory. These particles, emitted

as neutrals, become ionized as they are exposed to sunlight and the

passing solar wind particles, and in effect appear to suddenly

materialize in the ionized gas flow the moment they become ionized.

This simulates addition (or subtraction) of mass, momentum, and energy

in the flow; thereby requiring further fundamental considerations in

the formulation of the theoretical model. If the ionization occurs

sufficiently rapidly and near the cometary nucleus, the flow field may

resemble that of the planets with an ionopause that separates most of

the cometary material from the solar wind. If the ionizatioa extends

over a substantially larger region, the approaching solar wind flow may

be altered considerably, even to the extent that there might be no bow

wave whatsoever as the mass-momentum-energy addition slows and heats the

oncoming plasma so as to gradually change the supersonic flow to

subsonic.

6
ally, there are the natural satellites or moons of the other

Not a great deal is knower about the flows about these objects,

it is evident that a number of statements can be made on the

basis of their environment, sometimes in the solar wind upstream of the

planetary bow wave, and sometimes in the planetary magnetoionosphere or

the intervening magnetosheath flow; and also on their own atmospheric

properties. From the standpoint of magnetohydrodynamic and gasdynamic

theories of planetary bow waves, the differences among the various

planetary flows are to some degree secondary as they z re concerned more

with the specification of appropriate boundary conditions than with the

governing physical laws and equations. The remainder of this paper will

concentrate on providing an account of the theoretical formulation and -

resu i is of a number of rnatiie►imat i cal model s for these f 1 airs, and on

outlining the various approximations that have been introduced to

surmount the almost intractable computational problems posed by the

magnetohydrodynamic model,

MAGNETOHYDRODYNAMIC THEORY FOR PLANETARY BOW WAVE FLOWS

The objective for theoretical studies of planetary bow wave flows

is to predict the bow wave location and the properties in the

magnetoionosheath region of the associated flow field given the

properties of the incident solar wind plasma and the planetary body.

The simplest theoretical model that can provide the salient features of

these flows is that provided by magnetohydrodynamics of a compressible

highly conducting perfect gas. This theory, which has been gradually

7

l
developed over the last forty or fifty years represents a combination of

single fluid, continuum gasdynamics, and Faraday electricity and

magnetism, i.e., Maxwell's displacement current is omitted on the basis

of its smallness in appropriate app ll r;ations, This theory is most

frequently presented in its perfect dissipationless form, but it is

advantageous for the present discussion to include the dissipative terms

representing effects of viscosity, electrical resistance, and thermal

conduction. These terms will be written in the usual way (see e.g.,

Landau and Lifshitz, 1960 9 or Jeffrey, 1966) with scalar coefficients

for first and second viscosity u and ;, electrical conductivity Q , and

thermal conductivity k; but it should be recognized that the transport

properties of a nearly collisionless plasma with embedded magnetic field

may be highly anisotropic with very different values parallel and

perpendicular to the magnetic field. These considerations lead to the

following set of partial differential equations for the pressure P,

density p, temperature T, velocity », magnetic field B, and entropy s

involving the universal gas constant R = 8.31 x 10 7 ergs gm 1 K-1 , the

mean molecular mass m of the gas in atomic units, and the gravitational

potential ^ and constant G = 6.67 x 10 -1 dyne cm' gm-2 ; and in which

the independent variables are the time t and space coordinates x of an

inertial system.

Perfect gas

P = pRT/m (1)

8
Conservation laws

Mass

a + axk 

(PVk ) 0

Momentum

at

a7Ti k

0

( pv i ) + axk -

where

BiBk

B 9i9k ^_z___

"ik = pv ikv + Pa ik - 7F + "ik 

+ ^ _ - oik (4)

a 'vi avk 2

avQ avQ

g i - ax i

°ik -(ax

k + axi - 7 d ik ax Q +4d ik axR)(5)

Energy

t (Y+ pe+p(P +U) +div q =0 (6)

where

z

q =pv( v +e+ p +^ )+ 1 B (vxB)

c2

- 1-= B x curl B- v Q' - k D T (7;

a

or the Heat Transfer Equation

Ds ay. z

PT 7t

= a ax k + div(kvT) + 16a ( curl B) 2 = div(kOT)

( vi N2 avk, z

+ ax k + ax i - 7 d ik ax Q )2 + ^(div v)2 + ZcT (furl B) 2 (8)

(Z)

(3)

9
and the Faraday (preMaxwell) Electricity and Magnetism Equations

aB 

n curl (v x B) + c2 7 2B , div B - 0

at - a

In these equati11 nik 

and oik are the momemtum flux and viscous

stress tensors, gi is the acceleration of gravity, 6 ik is the Kronecker

delta function, a is the energy flux vector, and e - c vT is the internal

energy. Other relations not explicitly used in the above equations, but

that are useful in the discussion that follows are for the enthalpy h

e + p / R - C 

p 

T and for the difference c - c v - R/m and ratio cp /cv - Y

of the specific heats at constant pressure and volume.

To these equations must be added appropriate boundary conditions to

represent propertiesref the incident solar wind and the planetary body.

These include P,, , T,,, va,, and B,„ in the incident solar wind and Po, To,

v o , B o , ^o , and ion composition at some magnetoionosphere reference

level between the magnetoionopause and the planetary surface.

The difficulties presented in the solution of these equations are

so great that neither analytic nor numerical solutions have yet been

obtained for realistic three-dimensional boundary conditions

representative of a planetary application. These difficulties arise

from several sources. First of all, the equations are nonlinear and the

dissipative terms that contain the highest order derivatives in each of

the differential equations are all multiplied by small coefficients.

The shock waves and boundary and current layers that appear in the

course of the solution are thus finite but thin regions of rapid

variation at locations that cannot be specified in advance, but must be

10

(g)
found as part of the solution. Their adequate treatment requires the

use of either fine spatial grids over extended regions or fine adaptive

grids in the vicinity of the surface; or the acceptance of poor

resolution of the solution in these regions of rapid tariations at best,

and widespread contamination of the solution at worst. Use of fine

spatial grids requires small time steps in unsteady simulations, or very

many iterations to converge a steady flow simulation. All of these

factors contribute to the substantial computational requirements arising

from the sheer magnitude of the problem. Eight quantities P, T, v, and

B must be calculated at every grid point. For reasonable resolution of

the total 3-D interaction region, there would be a minimum'of say 100 x

100 x 100 = 106 grid points not allowing for extra grids 'in the vicinity

of shock waves, boundary layers, etc., and hundreds to thousands of

iterations or time steps required to converge to a steady state solution

from an assumed set of boundary and initial conditions. Obviously, the

calculation of solutions will remain costly for some time, and we

believe it will be many years before solutions of these equations will

become inexpensive enough for extensive routine use in the analysis and

interpretation of space observations.

However, advances in computational capabilities are still

increasing at a rapid pace, approximately doubling every 7 years

according to Chapman (1979), due almost equally to advances in computers

and in numerical methods. We may enquire at this point, therefore, as

to what might be anticipated from the solutions of these equations if

they could be attained. First, we believe these solutions would indeed

provide good predictions of the bow wave location and the large-scale

11
features of the associated flow. Shock waves and magnetoionopause

regions would have finite thickness, and magnetic merging and

reconnection would occur at the magnetoionopause. There would also be

extensive heating and rarefaction at these boundary regions,

particularly those of the ionopause type, with the possible formation of

magnetic barrier regions of high magnetic field separating the solar

wind and ionospheric plasmas.

We believe that these predictions would be correct qualitatively,

but the quantitative predictions would be unreliable because of

'inadequancies of the scalar transport coefficients. As noted previously

(Spreiter, 1976), the expressions developed for nonmagnetized plasmas

are highly inappropriate. Tor example, the widely used expression U =

10-" T 5/2 - 0.469 vt

p 

p Zd g cm 1 s ' for the viscosity of

fully-ionized hydrogen having a representative value of 9 2 for the

Coulomb logarithm, and in which T is the temperature in K, vtp is the

thermal velocity of the protons, p is the density, a: nd kd is the

effective mean free path for cumulative deflection of 90 0 by Coulomb

interactions, leads to a Reynolds number R e = p vD/p of only 0.002 when

the number density of protons N p = 10 cm s , T = 105 K, and the radius of

the Earth 6.37 x 108 cm is taken for D. Such a value for R

e

 is rout at

all indicative of an aerodynamic-like solar-wind flow past the Earth;

but is more typical of a small ball sinking through tar! Use of such a

value for u would lead to the prediction of enormously thick boundary

layers and shock waves, completely different from those observed.

12
There is no dilemma, however, since the particles were as;,umed in

the derivation to travel in straight lines between collisions which,

even in the more conservative sense of cumulative small Coulomb

deflections, are indicated to be separated by mean distances of the

order of half an astronomical unit when the above-stated conditions are

applied. This is obviously grossly inappropriate for planetary

applications which involve phenomena of much smaller scale.

Part of the answer to this apparent deficiency is provided by the

fact that the presence of a magnetic field in the solar wind prevents

the particles from traveling in straight lines between colli§ions, and

causes them to spiral along the moving magnetic field lines. This

reduces the transport transverse to the field lines approximately as the

square of the ratio of the gyroradius of the protons to the distance Qd.

If a representative value of 1.4 x 10 -6 is used for this ratio,

corresponding to a magnetic field of 5 nT, the Reynolds number in the

example cited above would increase to 9 x 10 6 . Such a value is of the

order of that encountered in ordinary aerodynamics, and is consistent

with the presence of relatively thin shock waves and magnetoionopause

surfaces and the generally gord agreement between observations and the

results of dissipationless fluid theories.

However, all is not that simple. The magnetic field does not

reduce the transport coefficients equally in all directions; in fact,

it does not reduce the values for transport parallel to the field lines

at all. The dissipative part of the associated model is thus highly

anisotropic. The direction of the anisotropy, moreover, depends on the

13

A. ,syk '..p . <
magnetic field, and hence also the flow, and cannot be specified in

advance. Since there is at present virtually no theoretical development

for the behavior of such a fluid for any application, the space

scientist studying these features of the flow in terms of an anisotropic

dissipative fluid is faced with they task of achieving major theoretical

advances or, as is more often the case, being satisfied with hopefully

describing what he thinks will happen in qualitative terms based on

analogy with the known behavior of isotropic fluids, In view of the

extreme anisotropy of the solar-wind plasma and the fact that the

Coulomb deflection times upon which the analysis is based are much

longer than the times required for solar-wind particles to traverse the

significant part of a planetary flow field, it is evident that .

considerable caution should be exercised in relying on such

descriptions. It is, moreover, quite possible that this particle

approach to transport coefficients may be only marginally relevant to

the actual dissipative processes; and that the significant mechanisms

are associated more with larger scale eddying and turbulent features of

the flow as in many analogous situations with boundary layers in

ordinary fluid mechanics.

Finally, we may anticipate that the presence of time derivatives

and nonlinear convective terms in equations (2) through (9) almost

certainly leads to fluctuations (turbulence?) in the solutions,

particularly downstream of shock waves and with increasing distance from

the nose along the magnetoionopause. These may, in fact, be correct

14
predictions, but their presence and small scale introduces additional

complications in the numerical solution that probably exceed by a

substantial margin present computational capabilities to resolve .

Actual solution of the dissipative magnetohydrodynamic equations

for planetary flows is still in its infancy (see Walker (1983) for a

recent review). At the present time, about the only solution that has

been worked out by using physically realistic, as opposed to numerical,

dissipation is that of Brackbill (1982) illustrated in Figure 2. This

is for unsteady two-dimensional, as opposed to three-dimensional, flow

about a magnetic dipole, and is intended to demonstrate the sequence d'f

events which follow the southward turning of the interplanetary magnetic

field. The time units are Earth radii divided by the solar wind speed.

As the southward field reaches the dayside magnetopause, reconnection

begins at t = 35. Reconnection in the tail occurs when the rotational

discontinuity reaches the nightside at t = 45. A magnetic bubble forms

in the tail (n) which propagates down the tail. Earthward of the

reconnection region, tail field lines first are stretched out (n i ) and

then snap back to a more dipolar configuration as tail reconnection

begins. To verify that it is physical resistivity and not numerical

resistivity that is responsible for the tail reconnection, Brackbill

also ran his code with the electrical resistivity set to zero. In this

case, reconnection occurs on the dayside because of numerical

resistivity, but none occurs on the nightside. It should be recognized

that these and other unsteady two-dimensional models are fundamentally

deficient in that there can be no return flow around the Earth back to

the dayside. Therefore magnetic flux must be returned to the dayside

I'

15
artificially. In addition, questions of stability may have very

different results in two and three dimensions. To illustrate, nobody

would marvel at a tight-rope walker in a two-dimensional worldl On the

other hand, the appeal of two-dimensional models is great in numerical

simulations because of the substantial reduction in the number of

numerical values involved. For the example cited above with 100 grid

cells in each direction the reduction is from 8 x 10 6 to 6 x 10" values

required to define conditions at any step in the calculation.

DISSIPATIONLESS MAGNETOHYDRODYNAMIC THEORY FOR

PLANETARY BOW WAVE FLOWS

Because of the difficulties noted above, most analyses and

discussions of planetary bow wave flows have been based on the

dissipationless magnetohydrodynamic theory. This avoids altogether the

uncertainties associated with the proper form of the dissipative terms,

and all the mathematical difficulties associated with the thin, but

finite, regions of rapid variations representing the bow wave,

magnetoionopause, etc. The equations defining this theory are just

equations (1) through (9) with all the terms involving u, ^, k, and a

set equal to zero.

Let us examine the consequences of omitting the dissipative terms.

There is no diffusion of vorticity, heat, or magnetic field; there is

no magnetic merging nor recombination, and magnetically separated

ricgions can occur. The magnetic flux through a circuit that moves with

16
the fluid is constant; giving rise to the "frozen field" concept.

Entropy, and hence P/AY , is constant for a fluid element, and thus

constant along a streamline for a steady flow, except when it crosses a

shock wave. Since the amount of entropy increase varies along the bow

wave, the entropy differs from one streamline to the next. Such a flow

is often called isentropic, but not homentropic. This entropy condition

reduces the number of dependent variables from 8 to 7 At each grid

point. Shock waves, boundary layers, and current sheets used to

represent the bow wave, magnetoionopause, and other thin layers become

mathematical discontinuity surfaces of zero thickness across which the

plasma properties change discontinuously in definite ways governed by

the conservation laws. This requires use in the numerical solution of

either shock fitting techniques, probably with adaptive grids; or shock

capturing methods with artificial viscosity, thermal, and electrical

a

F

conductivities, either explicit or numerical, which thicken the

discontinuity urfaces over several y grid spacings. It is anticipated

that most of the differences between these solutions and the corresponding

dissipative ones are confined to the regions near these discontinuities;

and that there is probably little change in most of the large-scale

features of the predicted flow and magnetic field properties. Finally,

we remark that the dissipationless equations are still difficult and

costly to solve; and that very few solutions have actually been

determined for realistic three dimensional flows at this time.

There are, however, a number of definite and exact solutions of

these equations for simpler boundary and initial conditions that are

fundamental to both the qualitative interpretation of planetary bow wave

17

y 
flows and the establishment of quantitative predictive models based on

the dissipationless magnetohydrodynamic theory. Foremost is the normal

propagation speed C n of a plane wave. In gasdynamics, this is equal to

the speed of sound C S and the resulting expression C S = (y RT/m)1/2

follows directly from the dissipationless forms of equations (1) through

(9) with the magnetic field B set to zero. This speed is the sane in

all directions, and can be represented in a planar cross section by the

circle shown in the upper left of Figure ; in which C S for a plane sound

wave is illustrated as a function of angle 0 between the wave normal and

an arbitrary coordinate axis. Sound waves depend on the compressibility

of the medium, and do not occur in an idealized incompressible fluid.

In magnetohydrodynamics, however, wave propagation can occur even in an

incompressible conducting fluid. Such waves are the Alfven waves, and

the diagram in the upper right of Figure 3 illustrates how the normal

propagation speed C nA of such a plane wave varies with direction such

that CnA 

`2

cos 0, where CA = (B 2 / C4Tr p ) ) 1/2 i s the Al fvdn speed and 0

is the angle between the wave normal and the magnetic field. The two,

lobes in this figure are thus circles osculating at the origin with

their line of centers along the direction of the magnetic field. Plane

Alfven waves can propagate in any direction, except exactly normal to

the magnetic field, but it can be deduced from the properties

illustrated that the disturbance created by a point source will

propagate only along the direction of the magnetic field and with the

Alfven speed C A. Values for CS and CA for fully ionized hydrogen plasma

are shown in Figure 3, from which it can be concluded that both C S and

C

A

 have values of about 25 to 50 km s "1 for conditions representative of

those in the solar wind at the orbit of Earth.

18

MZY

F

a .,".. F
The propagation speeds of a magnetohydrodynamic plane wave in a

' dissipationless perfect gas are substantially more complex than either

of the two limiting cases just discussed. There are, first of all,

r three distinct wave modes each with propagation speeds varying with

r

angle E). Of these, the rotational wave is simplest to discuss. It has

exactly the same propagation speed and properties as the Alfven wave for

an incompressible fluid. The other two plane wave modes, with

Cn = * 1{CS + CA2 +- [( C S + CA2 ) 2 - 4CSCA cos 2 ®] 1/2 } 1/2 (10)

are called the fast and slow waves because their propagation speeds are

greater or less than that of the rotational wave. The fast wave, which

has a plus sign before the inner square root; is of greater interest

because it determines the region of influence and dependence in the

solution, and in the flow it represents. Its normal velocity C nf varies

from a maximum of (CS + C A ) 1/2 in the direction perpendicular to the

magnetic field to a minimum equal to the larger of C S or CA in the

direction of the magnetic field. From the two bottom diagrams of Figure

3, it may be concluded that the gasdynamic sound speed C S approximates

reasonably the magnetohydrodynamic fast wave speed for C S > C A, but

poorly for C S < CA.

For steady flows with velocity in excess of the fast wave speed,

the Mach cones from an infinitesimal point disturbance are of prime

interest. Their traces in the plane of the v and B vectors may be

constructed either from the curves at the bottom of Figure 3, which are

called Friedrichs I diagrams, by constructing the envelope curves called

19
20

the Friedrichs II diagrams and drawing tangents from the end of the

velocity vector terminating at the origin (see e.g., Jeffrey (1966));

or directly from the Friedrichs I diagrams by the procedure illustrated

in Figure 4 (Spreiter et al., 1966b). In the latter, a circle is drawn

with the velocity vector as a diameter, and the Mach cone is determined

by the straight line from the end of the velocity vector to the

intersection of the circle and a C n curve of the Friedrichs I diagram.

There are thus, in general, three Mach cones, each associated with one

of the three kinds of waves; and they are appropriately called the

fast, slow, and rotational Mach cones. They correspond to the single

Mach cone of gasdynamics shown in the upper right of Figure 4, which

makes an angle w = aresin (MS -1 ) with the velocity vector. The

magnetohydrodynamic Mach angles depend not only on MS and MA , but on the

angle(). To illustrate the latter effect, four limiting cases are

presented in the lower parts of Figure 4. For a given MS and MA , the

maximum and minimum Mach cone angles are given by wmax = aresin (MS -2 +

MA -2)112 and the larger of wmin = aresin (MS -1) or wmin aresin (MA-1).

For Bparallel to v, w„ = aresin {.[(M S2 + MA2 - 1)/(MS2 + MA)] 1/2 

and 11

for B perpendicular to v, wi- = aresin{( {MS2+ MA2 + I +[(MS2+ P + 1)2

4"1 S 2 MA 2 ] 1/2 }/(2MS2 MA 2 ) ) 1/2 }. Lack of appreciation of these differences is

widespread in the space research literature.

The wave speed and Mach cone diagrams of Figures 3 and 4 are for

infinitesimal disturbances, and are therefore not appro priate for

dealing with the finite disturbances associated with the planetu../ bow

wave and any other shock waves that may develop in the flow. These are

governed by the magnetohydrodynamic discontinuity relations, which
4

follow directly from the conservation laws and auxiliary relations of

equations ( 1) through ( 9). With the unit vectors normal and tangential

to the discontinuity surface defined by n and t. the differences between

any quantity Q downstream and upstream of the discontinuity designated

r

by [Q] = Q z - Q l , and " n = vnflow - vndisc' which for steady flow is "vn

= vnflow, the conservation laws for mass, momentum, and energy are

[Avn] = 0 (11)

z B B

[pvn v + (P + r) n - 47T = 0 (12)

z , z B v B

[On ( e + p + 7 + p i

) - vnd sc (P + Ti) - ^--J = 0 (13)

These must be solved together with the Faraday magnetic jump equations

[B t"v n - B n v t J = 0 [BnJ = 0 (14)

` and the auxiliary condition for entropy [s] ? 0. In gasdynamics, these

conditions with B set to zero completely determine the shock jump

relations; but in magnetohydrodynamics an additional condition is

required. It is the evolutionary condition (see, for example, Jeffrey

and Taniuti (1964)), which states in essence that there must be a

mechanism by which a succession of infinitesimal waves can evolve into a

finite discontinuity. This distinction does riot have to be made in

gasdynamics because the entropy condition implies the evolutionary

condition and conversely; but this is not so in magnetohydrodynamics.

In the latter, the evolutionary condition implies the entropy condition,

but the entropy condition does not imply the evolutionary condition;

21

-me :.- -
and both must be satisfied. An example of a situation in which the

entropy condition, but not the evolutionary condition is satisfied is

the case in Which the shock normal is parallel to the magnetic field and

M S and M A are within certain ranges. Theory indicates that a finite

magnetohydrodynamic shock wave cannot evolve from an accumulation of

infinitesimal waves; and observational evidence indicates that a well

defined shock wave may indeed fail to develop under such conditions.

Solutions of equations (11) through (14) show that five types of

discontinuity surfaces may occur in dissipationless magnetohydrodynamic

flows. They may be classified as two boundary types for which Vn =

vnflow - vndisc

0, namely

tangential discontinuit7es

B n = 0, [ vt] # 0, [ Bt] ^ 0, [P] # 0, [P + B2 /87r] = 0 (15)

contact discontinuities

B n ^ 0, [v] = [B] = [P] = 0, [P] ^ 0 (16)

and three shock wave types for which q ¢ 0, namely

rotational or Alfven shock waves

v" n = vn 

r 

= ± Bn/( 47T P) 1/2 [vt] _ [Bt/(4'rP)'/2] (17)

[P] = [ P ] = [v n ] = [v 2] = [ B 2 ] = [ B n ] = 0

fast shock waves

Pv n > Pvn 

r , 

[P] > 0 , [P] > 0 , [Bn ] = 0 , [B2 ] > 0 (18)

sloe; shock waves

Pv n < Pv n r ! [P] > 0 , IN > 0 , [Bn ] = 0 , (B2

) < 0 (,19 )

'^ 22
All types may be anticipated in planetary flow fields if there are

sufficient fluctuations in the solar wind or planetary flow boundaries.

They may also be improperly induced by growing perturbations in a

numerical solution of the dissipationless equations, unless controlled

by artificial viscosity and conductivities.

More explicitly, a planetary bow wave must normally be a fast shock

wave because pv n of the incident solar wind exceeds pv nr by a

considerable margin over the forward part of the bow wave; and the

magnetoionopause must normally be a tangential discontinuity in order to

separate the solar wind and _planetary magnetic fields and plasmas. The

outermost Mach wave of the Moon must be a fast Mach wave, and the .

boundary between the flowing solar wind plasma and the lunar downstream

cavity might be either a tangential or a contact discontinuity. In

addition, all of these types of discontinuities can occur in the solar

wind as a result of interactions of plasmas with differing speed and

properties.

The most completely worked out solution for dissipationless

magnetohydrodynamic flow past a planetary obstacle is that of Spreiter

and Rizzi (1974) for the special case of an axisymmetric obstacle in a

flow in which B is parallel to v in the incident solar wind (see also

Spreiter et al., 1970) for a related application to the Moon). Under

these conditions B = Xp v everywhere; and the dissipationless

magnetohydrodynamic equations can be transformed into those of

gasdynamics by introduction of new variables

23
v* . v[1-X2p/(4n )] T p* : p[1_X2p/(4Tr)j "1,

a a

P* n P + B2 / ( $n ) , G* : S ,

h* ; h + 

( X 2 p/(4Tr) ] [ 1_X2p/ (41T)l , h : e + P/ p (20)

in which

x2p/(47T) : MA-2 ° ( p/pOO) MA:2

(21)

except that the equation of state P = p RT/m is replaced by a rather

complicated relation that corresponds to no real gas. These

pseudogdsdynamic equations can be solved using the methods developed for

ordinary gasdynamics , to produce the results displayed in Figure 5. They

show, for a given axisymmetric obstacle shape and fixed MS.0= 10 and Y =

5/3, the location of the bow wave for selected MAW 

between 2.5 and 20.

The conclusion to be gained from these results is that the

magnetohydrodynamic bow wave l.r._.tions are virtually independent of MA.

for values greater than about 10, and are practically identical with

those of gasdynamics in which all effects of the magnetic field are

disregarded. They also show that for small MA«,the bow wave flares out

more at the flanks, but approaches the obstacle closer near the nose.

From this, it is evident that attempts to represent the combined effects

of MS. and 

MA. 

by an "equivalent value" for M

S,, to be used in a

gasdynamic model cannot succeed. In gasdynamics there is a monotonic

relationship between Hs c, and distance to the bow wave in all directions

from the obstacle. If MS. 

is diminished, the bow wave moves farther

from the obstacle everywhere. There is no way that an equivalent Mach

number can be found that will simultaneously move the flanks of a

gasdynamics bow wave farther outward and the nose farther inward.

24
Solutions for non-aligned fields are much more difficult to work

out than for the aligned case, and very few results are available at the

present time. Among these is the solution of Wu, Walker, and Dawson

(1951) for three-dimensional, steady, magnetohydrodynamic flow past a

dipole magnetic field to aimulate the Earth, for which constant density

and pressure contours are presented in Figure 6. Although there is no

magnetic field in the interplanetary plasma in their model, the results

display the broad outlines of the observed flow. The results indicate a

moderately defined bow wave across which conditions change approximately

in accordance with the Rankine-Hugoniot jump conditions, and a poorly

defined magnetopause. The locations of the magnetopause and bow wave

resemble those observed in space, although it should be noted that the

Mach number used in the calculations, i.e., MS. 0 2.5, is substantially

less than that normally observed in the solar wind upstream of the

Earth's bow wave. Perhaps the least satisfactory aspect of the theory

is the thickness of the bow wave and the magnetopause. The model

boundaries are 4-5 Earth radii (3-4 grid spaces) thick, whereas the

observed boundaries are a few hundred kilometers thick at most, and

usually considerably thinner. The calculated thickness of these

boundaries results from the numerical dissipation in the simulation,

which erroneously spreads over several grid spacings what should be a

mathematical discontinuity surface in the dissipationless theory on

which the model is based. It may be anticipated on the basis of

continuing rapid improvements of computers and numerical algorithms that

finer grids will be used in future models, thereby providing better

resolution of these regions of rapid variation. As long as numerical

dissipation is used in the calculations, however, caution should be

25
exercised in interpreting the results, particularly those associated

^a

with real dissipative prncesses that may be expected to occur. Although

the prediction of such features may tantalizingly resemble those

actually believed to occur, it should be remembered that the associated

F quantitative values lack proper physical basis, and must be regarded

more as suggestive than quantitative predictions.

THE GASDYNAMIC APPROXIMATION OF MAGNETOHYDRODYNAMIC

PLANETARY FLOWS

In recognition of the great computational demands of full

magnetohydrodynamic solutions, particularly as viewed in terms of the

limited computing ability then available, an approximate theory of

considerable utility was developed about two decades ago. Following on

the success of purely gasdynamic calculations by Axford (1962), Kellogg

(1962), and Spreiter and Jones (1963) of flow past an obstacle shape

defined by the Beard (1960) and Spreiter and Briggs (1961) solutions of

the classical Chapman-Ferraro theory of interaction of a collisionless

stream of protons and electrons with a magnetic dipole, a more

consistent approximate theory based on magnetohydrodynamics was

developed by Spreiter et al. (1966a,b; 1967; 1968; 1969; and

1970a,b,c), Alksne et al. (1967, 1970), and Dryer et al. (1966, 1967). In

this theory, the magnetohydrodynamic tangential discontinuity jump

conditions are retained for the boundary conditions at the magnetopause,

and the bow wave is identified as a fast magnetohydrodynamic shock wave,

which wiuld tend to resemble a gasdynamic shock wave for large M A and

co

26
for MA

.,

> MS... The key approximation is that the magnetic terms may be

disregarded in the momentum and energy conservation equations on the

basis of their smallness for large «A , The resulting equations are thus

CO

Perfect gas

P - p kT/m (22)

Conservation laws

Mass

at + axk ( p v k ) = 0 (23)

Momentum

a(pv•)g•g 2

at + axvi k (p v k + Pa ik + 4 - TrG) = 0 (24)

Energy

at (per +pe+p4))+div[pv (^-+e+P +(D)] =0 (25

and the Faraday Electricity and Magnetism Equations

a6

at - curl (v x B)

(

div B - 0 . 26)

The heat transfer equation degenerates to Ds/Dt = 0, which indicates

that the entropy is constant along a streamline for dissipationless

flow; but this relation does not apply across a shock wave because

dissipation always occurs there in such a way that [s] > 0 with the

amount of entropy augmentation depending on the incident Mach number and

the angle between the shock normal and the free-stream velocity vector.

In most, but not all applications, the gravitat'onal terms involving g

and 0 may be omitted.

27
With this approximation, the conservation equations become exactly

those of gasdynamics. The magnetic field B therefore has no influence

in the solution for the pressure P, the density A, the temperature T, or

the velocity X. The latter quantities can be solved for using the

methods of gasdynamics without further consideration of the magnetic

field. The Faraday magnetic equations are the same as in the

dissipationless magnetohydrodynamic theory, which implies that the

magnetic field is "frozen" into the flow. However, the values for v in

equation (21) are those of the gasdynamic solution. The resulting

prediction for the magnetic field differs from that of an exact

magnetohydrodynamic solution by an unknown amount dependent on the

difference in the velocities in the two theories. This approximate

separation of the magnetohydrodynamic problem into two parts, which can

be solved in succession, leads to a significant reduction in computing

requirements. The resulting component problems are still sufficiently

complex that they have not been solved without further approximation.

These difficulties are associated not only with the nonlinear and mixed

elliptic-hyperbolic type of the differential equations, but also with

the discontinuity surfaces that represent the bow wave and the

magnetoionopause whose locations are not known a priori but must be

found as part of the solution.

Problems associated with the unknown location of the

magnetoionopause can be avoided to an acceptable degree over most that

surface by use of an approximate relation for the pressure of the

flowing solar wind on the magnetoionopause, i.e.,

28

e y ^ 
P = K Poo v im, 

(27) cos t

This serves to decouple the calculation of the magnetoionopause shape

from that of the surrounding flow. In this relation, K is a constant

usually equated to unity and ^ is the angle between the free-stream

velocity and the exterior normal to the magnetoionopause with directions

definPA so that = 0 at the magnetoionopause nose. With the proper

selection of a value for K, this relation provides gooa results for the

pressure over most of the magnetoionopause, but deterioration sets in as

^ approaches 7T /2. For ^ > Tr /2, equation (27) is totally inappropriate,

and should not be used,

Equation (27) has two distinct origins in planetary studies. On

the one hand, there is the original usage in a long series of works

starting with Chapman and Ferraro (1931) and particularly Ferraro

(1952) in which the relation P = K m p n p v cos co ` ^ is derived by

considering the interaction of a collisionless stream of protons of mass

m and number density N and an equal number of electrons, all with

uniform initial velocity v,., with a dipole magnetic field oriented

normal to the flow. Depending on the assumptions, K was usually found

to be either 1 or 2, although 1/2 also appears occasionally in the

literature of the time. That analysis was developed from a cold particle

stream viewpoint to explain how a cavity that we would now call the magnetosphere forms in a collisionless plasma stream or cloud of solar origin that sweeps past the Earth and produces a geomagnetic storm.

29
The second line of argument that leads to equation (27) stems from

the recognition that a highly supersonic solar wind exhibiting continuum

fluid-like properties is present at all times; and that one of its

manifestations is the bow shock wave that forms upstream of the

magnetosphere, which acts much as an impermeable obstacle in the flow.

From this point of view, equation (27) may be regarded as a close

approximation for high Mach number flow to the Newtonian pressure

relation for hypersonic flow P - P. = K p. v; cosl ^. This is usually,

but not always, approximated in planetary applications by equation (27)

in which P.0 is disregarded on the basis of its relative smallness as indicated by Poo =

pOO 

v:/ (YM2). The constant K is determined from the Rayleigh pitot formula for the stagnation pressure for supersonic flow

decelerated to rest after passing through a normal shock wave. Under

these conditions, K is found to be

K _ ( 2 )(Y*1)/(Y-1)

1 (28)

Y[Y -( Y - 1)

/(2Mz")-.1/(Y-1)

For large Mach numbers, K m 0.881 for y = 5/3 and 0.844 for y = 2, the

two most- commonly used values for the ratio of specific heats Y in

planetary flow studies. In using these pressure relations, it should be

observed that equation (27) is not an exact relationship within any

modern formulation of planetary flow theory. The original collisionless

plasma beam theory is inappropriate because it disregardz completely the

deflection of the flow by the bow wave and in the intervening region;

and the gasdynamic derivation is approximate because the Newtonian

30
pressure relation is a semi-empirical relation that is not exact in any

limiting case. That equation (27) does indeed lead to good results

compared with an essentially exact numerical gasdynamic solution for a

number of representative solar wind conditions has been shown by

Spreiter et al. (1966b, 1968). It seems to us equally plausible to

consider for magnetohydrodynamic applications that K p. v. cos2 ^ should

be-equated to the sum of the gas and magnetic pressures P +B 2 /8Tr or (P+

B 2 /8Tr)-(Pn)+B

C*

2

/8Tr) rather than to P or (P-PO. ) alone as is customary.

With the latter interpretation and disregarding P O

+BO2

/87r as small,

the tangential discontinuity boundary condition to be applied at a pianetary magnetoionopause becomes

KpV 2 cos 2 1 = (P + B2/8Tr),Ijp , 

B n - g (29)

where subscript MIP refers to conditions just below the magnetoionopause.

This enables the calculation of (a) the magnetoionopause location, and

(b) conditions in the magnetoionosphere without further consideration of

the solar wind plasma flow; and (c) reduces these problems for an

idealized magnetic planet to those of the classical but physically

outmoded Chapman-Ferraro theory (1931)•

We may summarize the specifications for such an idealized problem,

and also its counterpart for a nonmagnetic planet, as follows. In

either case, the normal component of the magnetic field is zero, and the

solar wind pressure K p. . cos2 ^ on the magnetoionopause is balanced by

the sum of the magnetic and ionospheric gas pressures on the inside of

the magnetoionopause in accordance with equation (29). Inside the

magnetosphere of a magnetic planet such as the Earth, the intrinsic

31
0

planetary field dominates until it is terminated by the currents flowing

in the magnetopause. Other current systems are present and important in

the calculation of magnetospheric processes; but, with the exception of

the magnetotail curroot sheet which flows across the tail in

approximately -the geomagnetic, equatorial plane, they are usually

relatively unknown and unimportant, and disregarded in the calculation

of the magnetopause shape and the surrounding flow. The field inside

the magnetosphere is therefore just that of the intrinsic planetary

field, usually approximated for the Earth by a magnetic dipole as

distorted by the effects of the currents in the magnetoionopause and the

tail plasma sheet. That field is calculated by solving

div B = 0 , curl B = 47rj/c (30)

subject to the boundary conditions

Kp^v^ cos t _ (B ./8Tr)Mp , Bn = 0 (31)

t.

at the magnetopause due to a planetary magnetic dipole field

B p = -(Mp /r 3 )(e sin e + r 2 cos e) (32)

at the origin. In this equation, M is the planetary magnetic moment

and r and a are geomagnetic radius and colatitude. In most applications

j = 0 except at the magnetopause and in the tail current sheet. With

neglect of the latter, the probl'em is exactly the Chapman-Ferraro

problem.

32

k
For a nonmagnetic planet with a sufficiently dense ionosphere to

stop the solar wind, the corresponding problem is specified by assuming

the ionospheric pressure above some reference level Po

can be

approximated by

P = P0f(r,e) , B = 0 (33)

within the ionosphere; and

Kp^v2 cos z ^ = P Ip (34)

at the ionopause. In the original presentation of this model (Spreiter

et al.,1970a,b), the ionospheric pressure was assumed to be simply P = Po

ex,— -(r-ro )/H1 where H = RT/mg = const, i.e., a constant scale height

ionosphere was assumed. Subsequently, Spreiter and Stahara(1980a,b) and

Stahara et al. (1980), included the inverse square variation of the

gravitational acceleration g with r, while retaining the simple

assumption of a constant temperature ionosphere. With improved

knowledge, as has since been acquired for Venus, the only known clearly

nonmagnetic planet, refinements in the specification of f(r,e) can

obviously be made and incorporated into the calculations. We believe

that little change in the ionopause shape will result, however, because

the rate at which P diminishes with r is so great that only small

changes in the ionopause shape will result. In other words, a

nonmagnetic planet with a substantial ionosphere, such as Venus,

presents a rather firm obstacle to the solar wind flow compared with a

strongly magnetic planet as the Earth or Jupiter for which considerable

33

.ati.r.I ti
change in the size of the magnetosphere occurs with variations in the

solar wind momentum flux p0

Figureigure 7 illustrates the resulting shape calculated for the

magnetopause associated with a magnetic dipole of magnitude and range of

orientations in the solar wind resembling that of the Earth. These

M

results are for the first-order approximation of Beard (1960) as solved

by Spreiter and Briggs (1962) and Briggs and Spreiter (1963). A number

t.

of higher-order solutions have been worked out subsequently, but the

results are -virtually indistinguishable on the scale shown and the

differences are not particularly relevant to the present discussion.

(See Wu and Cole, 1982, and Wu, 1903, for recent corentary.) These

results, and all others of the kind, have been based on the

Chapman-Ferraro formulation of the interaction. However, they may

equally be considered to be solutions of the continuum gasdynamic or

even magnetohydrodynamic models to the extent that the solar wind

pressure on the magnetopause can be approximated by the extended

Newtonian relation of equation (29). From the latter point of view,

inaccuracies in the calculated magnetopause shape may be anticipated

along the magnetosphere tail and near the neutral points because the

Newtonian pressure relation becomes increasingly inaccurate as

approaches n12. The neutral points are of particular interest because

they represent points in the idealized theory where the magnetospheric

magnetic field vanishes and particle entry into the magnetosphere may

occur. There is considerable evidence that such entry does indeed

occur, but any quantitative prediction of conditions in the vicinity of

the neutral points that uses the Newtonian pressure relation in the

^a

34
boundary condition as given in equation (31) or (33) must be regarded as

unreliable. The reason is that the condition B = 0 at a neutral point N

leads through the boundary condition . to cos 2^ = 0, or ^ a it/2; but that

is precisely the condition for which the Newtonian approximation clearly

fails. In the Chapman-Ferraro plasma beam approach, a further

deficiency is that effects of double impacts, as would occur near a

neutral point when a particle grazes the magnetopause just forward of

the neutral point and hits the magnetopause again just aft of the

neutral point, are not considered. In the continuum gasdynamic or

magnetohydrodynamic models, the presence of an indented region results

in the formation of an embedded shock attached to the surface (Walters,

1966); but Spreiter and Summers (1967) pointed out that such a shock

could not develop adjacent to the magnetopause. The reason is that the

pressure jump across the shock wave cannot be matched with a

corresponding jump in the magnetic pressure in -the magnetosphere. The

resolution they proposed was that the flow would separate somewhat ahead

of the neutral point and then reattach downstream-of the neutral point,

and that the constant pressure along the intervening free streamline

surface would cause a cusp-shaped plasma filled region to form. Recent

magnetohydrodynamic numerical solutions of Wu (,1983) support this

proposal. With the cusped geometry, it is no longer required for the

magnetic field to vanish at the netural point, but solar wind particles

can still enter the magnetosphere near the tip of. the cusp.

Considerable interest has been attached to observations in these polar

cusp regions in recent years, and also to the conditions in the mantle

region which extends downstream from the cusped regions along the

magnetopause.

35
the present point of view of planetary bow waves and their

I flow fields, perhaps the main point is that there exists a

well worked out body of theory for the calculation of the magnetopause

shape and location, and that it has been found in a great many studies

to provide a reasonably good prediction of conditions actually observed

in space. This is true for both the Earth and for other planets for

which data are available, provided allowance is made for the differing

nature of the planetary obstacles. For example, the ionopause shape for

the nonmagnetic planet ienus is axisymmetric in the theory and nearly so

in observations; and there are of course no neutral point phenomena.

Jupiter's magnetosphere appears to be flattened somewhat toward the

equatorial plane, which is understandable in terms of the effects of its

extremely large trapped particle regions and high spin rate.

To proceed with the solution of our problem, the next step is to

calculate the supersonic gasdynamic flow around the magnetopause, which

can at this stage be considered to be similar to -an impermeable obstacle

of known shape. For a nonmagnetic planet, this obstacle is axisymmetric

about the line through the planet center directed along the free-stream

velocity direction. For a magnetic planet, as the Earth, however, the

obstacle shape is nonaxisymmetric and a decision has to be made whether

or not to include the effects of the moderate departure from

axisymmetry. When the original calculations were made over twenty years

ago, there was no choice. Only the theory for axisymmetric flow had

been developed, and even that was very new at the time. The computers

and methods then available were totally inadequate for dealing with a

fully three-dimensional supersonic flow past a blunt body. As a result,

36
a further simplification was introduced in that the magnetopause was

approximated by an axisymmetric shape. This was usually, but not

always, taken to be the body formed by rotating the equatorial trace of

the magnetopause surface. This choice was motivated partly by that fact

that most space observations for the Earth were made near the equatorial

plane. Today, advances in computers and numerical methods, and recent

motivation provided by the needs of the nonaxisymmetric space shuttle as

opposed to the axisymmetric vehicles of the earlier era, have made

available methods for calculating supersonic gasdynamic flows about

nonaxisymnetric blunt objects. However, the cost in computing time

rises substantially for these 3-D flows, and the errors from assuming an

axisymmetric magnetopause shape are anticipated to be no larger than

those from a number of other sources in the entire procedure.

Even with the assumption of an axisymmetric obstacle, the

determination of the gasdynamic flow properties is the most difficult

and time-consuming portion of the numerical solution of the entire

planetary flow and magnetic field problem, As sketched in Figure 8, two

different methods are used in the calculations; one for the nose

region, where both subsonic and supersonic flows occur, and another

downstream of that region, where a more computationally economical

procedure can be employed since the flow is supersonic everywhere. In

the earlier analyses, an inverse iteration method was used for the nose

region, and the method of characteristics was employed for the remaining

supersonic region. Both of these procedures are inferior from a

computational point of view to more recently developed methods and were

replaced in the modernization of these solutions described by Spreiter

37
and Stahara (1980a,b) and most comprehensively by Stahara et al. (1980).

The nose -region is treated using a new axisymmetric implicit unsteady

Euler equation solver specifically developed for the present

application. That procedure determines the solution in the nose region

by an unsteady asymptotic time-marching procedure which advances the

solution forward in time until the steady state is attained. The

remainder of the flow field is determined by a shock capturing marching

procedure which spatially advances thu solution downstream as far as

required by solving the steady Euler equations. Inasmuch as these

methods are complex, lengthy, and fully described in the above

references, we turn directly to a discussion of the results.

Figures 9 and 10 show typical results for the Earth for

representative solar wind conditions of Y = 5/3 and M S O 8. In addition

to the bow wave and the magnetopause, Figure 9 shows the sonic line

which delinee.tes the boundary between subsonic and supersonic flow, the

streamlines which indicate the local direction of the flow, and the Mach

or characteristic lines which indicate the zones of influence and

dependence of an infinitesimal disturbance in the flow, Figure 10 shows

contour lines of constant density, velocity, and temperature. These are

sharply defined solutions, with much finer resolution than the

magnetohydrodynamic ° ri'rutions presently available, and can be determined

in one to two minutes on a large modern computer as opposed to several

hours for a much less well resolved magnetohydrodynamic solution. An

extensive catalog of results for various interplanetary conditions is

given in Stahara et al. (1980).

38

• ..A^.., .;; ... . s
with the flow field determined by the gasdynamic calculations, the

magnetic field B may be determined by integrating equations (14) and

(26); or alternatively the following equations derived from them

f f S 6 dS - 0 ^ (p) p (B v) v (35 )

in which S is an arbitrary surface moving with the fluid and D/Dt = a/at

* (v • ®) is the material or substantial derivative. As noted

previously, these equations are commonly interpreted as indicating the

field lines move with the fluid. For the steady state in which a/at = 0

and vv" n = v n , these equations lead to a straightforward calculation in .

which the vector distance from each point on an arbitrarily selected

field line to its corresponding point on an adjacent field line in the

downstream direction is determined by numerically integrating fv dt = As

over a fixed time interval At. Once the magnetic field lines are

determined, the magnetic field at any point may be calculated from the

relation

B/Boo = ( p /per ) W/AkQ (.36)

where OQ is the vector length of a small element of a flux tube. In

closing this discussion, one should observe that, in general, both

streamlines and magnetic field lines are three-dimensional curves. For

axisymmeiric flows, the streamlines are planar curves, but the field

39
4 •

lines are three-dimensional curves except for the special case of

aligned field flow in which the B and v vectors are locally parallel

throughout the flow.

The foregoing procedure is valid generally, but for axisymmetric

flows it is much simpler and completely equivalent to compute the

magnetic field by a decomposition theorem developed by A1ksne and

Webster (1970) whereby the magnetic field at P is given by

BP - (— 11 BCO (r)1 B. * @n (^) n B^ (37)

00 1 n

in which

BP

-

P

P BP - ^P

( ) u - P v CO BC,

- 

rp P^

(38)

can be calculated directly from the gasdynamic solution. (B P/Bw)y must

still be calculated using the As - Ak method, but only for the plane of

magnetic symmetry in which B C, ,'- 0 and for which B P is two-dimensional.

n

In this decomposition, the subscripts 11 , i , and n refer to

contributions associated with the component B CO, of B O parallel to vC.;

the component B CO^perpendicular to b in 'the plane that contains th,a

point P, the center of the planet, and the vector v ro; and the component

BOo normal to the latter plane, where e„ is a unit vector in the latter

n

direction. The quantity r in equation (38) is the radial cylindrical

coordinate of the streamline through P. Figure 11 is included to

facilitate understanding of the quantities involved in this analysis, a

40
key point of which is the orientation of the coordinate system so that

the point P at which the magnetic field is to be evaluated is in the xi.

plane.

Representative results for the magnetic field direction and

intensity are shorn in Figure 12, The upper part of the figure presents

the magnetic field lines and contours of equal field magnitude in the

plane of magnetic symmetry, for which the magnetic field lines are all

planar curves. Out 'l'r th y .. plane, the field lines are three-dimensional

curves and two projections are necessary to convey the results. These

magnetic field results vary greatly with changes in the interplanetary

field direction, but are relatively insensitive to moderate variations

in the other flow parameters.

A property of the magnetic field calculated in this way that is

sometimes cited as a deficiency of the theory, but which can actually be

understood in terms of real processes as viewed in the limit of

vanishing magnetic field and dissipation, is that the calculated field

becomes very large near the magnetoionopause. This may be deduced

directly from equation (36) for the general case or equations (37) and

(38) for axisymmetric flows from the fact that BpPJ (B. p p ) approaches

infinity as P approaches the magnetoionopause, because the length AQ/oQ"

of the element of a convecting magneticflux tube directly incident upon the

magnetoionopause nose enlarges indefinitely as it flows past the

magnetoionopause. It should be noted that this is not just for the

gasdynamic-convected- fi^.Id model, but applies equally to the

dissipationless magnetohydrodynamic theory since equation (36) is an

41
exact relation in that theory. The only difference between the

gasdynamic and magnetohydrodynamic procedures is the difference in the

values for v in the calculation of 4 f v dt. Such changes will

invalidate the above: statements only if v is zero in the

magnetohydrodynamic solution over the entire magnetoionopause instead of

just at the stagnation point and that is not the case. The theory thus

indicates that either BP /B,, is infinite or p p is zero. Both

possibilities are meaningful in the limit of vanishing dissipation and

interplanetary field as we explain below.

If the boundary is of the pure magnetopause type with a vacuum

magnetic field in the magnetosphere, the interpretation is that BP

increases to the finite magnetospheric level, which is independent of

the interplanetary field B.0 ; and p diminishes to zero as the point P

moves across the magnetopause from the solar wind plasma to the

magnetosphere. Now the theory does not provide values for B P directly,

but only for the ratio B Pp^ /(BC, p p ) and that strictly only in the limit

of vanishing Bc.. The infinite value for this ratio is thus the theory's

best effort to predict a finite value for B P that is independent of B.

as B., approaches zero. In the dissipationless magnetohydrodynamic

theory for which the limiting process of vanishing BCO would not be

applied, the magnetic field would be expected to rise to the

magnetospheric level in much the same way as the point P app-oaches the

magnetopause. Meanwhile the density would drop in such a solution and

become exactly zero at the idealized magnetopause; so that in that

interpretation of the theory B Pp./(B^pp) becomes infinite at the

magnetopause, but for a different reason. This is, in fact, the

+

42

1 
mechanism by which a magnetopause would form in dissipationless

magnetohydrodynamic theory with distinct magnetic field configurations

on the two sides of the boundary for steady flows.

If, on the other hand, the boundary is of a pure ionopause type for

which there may be no ionospheric magnetic field, the prediction of

infinite Bp pw /(Boo pp) may still be understood, although in a somewhat

different way. As point P approaches the ionopause, the magnetic field

would again rise, but this time the rise would be limited by the

ionospheric pressure, which is independent of the interplanetary field.

Thus the ratio of the finite ionopause magnetic field to the

interplanetary magnetic field is infinite in the limit of vanishing

interplanetary magnetic field. If the magnetic field actually rose to

such a value as to balance the ionospheric pressure, the gas pressure

would be driven toward zero. This would indicate the presence of

magnetic barrier region of high field and little plasma separating the

flowing solar wind and the ionospheric plasmas. From the higher point

of view of dissipative theory, an ionopause is a shear layer in

hypersonic flow in which intensive heating occurs. Since the pressure

on the ionopause is determined primarily by the momentum balance and is

nearly independent of the dissipative processes, a rise in temperature

leads, through the perfect gas law P = P R Tfi , to a fall in density in

the shear layer. in this way, a barrier region of hot rarefied plasma

would form between the solar wind and the ionospheric plasma even in the

absence of all magnetic fields. From several points of view, therefore,

43
it appears plausible to expect these theories to predict enhancement of

the interplanetary magnetic field near a magnetoionopause and a

relatively evacuated magnetic barrier region at an ionopause.

The above account provides an overview of the essential parts of

the gasdynamic-convected-field model, and its relationship to the

dissipative and nondissipative magnetohydrodynamic theories from which

it is derived. Once the flow and magnetic field properties are

determined in the manner described, further extensions such as to the

calculation of the electric field E = -(v x B)/c,.proton velocity

distributions, and many other quantities can be accomplished in a

straightforward way with the introduction of the appropriate formulae.

Because it is the most completely worked out method for calculating

planetary bow waves and their associated flow fields and is gaining

widespread use in the interpretation and analysis of data, we feel it

appropriate in concluding this section to make a number of general

remarks about the present state of this theory as implemented by the

computer model of Stahara et al. (1980), and Spreiter and Stahara

(1980a,b).

First of all, it should be understood that the

gasdynamic-convected-field model is essentially a magnetohydrodynamics

theory in the limit-of vanishing dissipation and vanishing B.., i.e., for

M -} - with M > M . Numerical solutions are obtained after the

A00 A S

introduction of addit;onal approximations, particularly the Newtonian

pressure relation and an axisymmetric obstacle shape. The entire model

is fully implemented with a documented computer code in NASA CR 3182

44

F
(Stahara et al., 1980). An entire calculation requires about 120 seconds for

a single case on a CDC 7600 or about 20 seconds on a Cray XMP. About 90

percent of this time is for the gasdynamic part of the calculation, with

only about 10 percent being required for the magnetic field. 'fhe model

is proving to be a useful tool in the interpretation of data,

particularly in understanding effects of the three-dimensional nature of

the interaction process. Several examples of this may be found in other

articles in this conference proceedings. Extended applications

currently underway include:

• bow shock shape and position for all the terrestrial planets

(Slavin et al., 1981, 1983)

• distant planetary Mach cone and bow shock shapes for Menus,

Earth and Mars (Slavin et al., 1984a)

• magnetospheric source of energetic particles upstream of

Earth's bow shock (Luhmann et al., 1984b)

locations and asymmetries of magnetic field merging sites on

the Earth's magnetopause (Luhmann et al., 1984a; Crooker et

al., 1984a)

• magnetic field draping on the Earth's magnetopause (Crooker

et al., 1984b)

• intrinsic magnetic field of Mars ( . Russell, 1984)

46

", 

70V
• detailed plasma and field studies in the Venusian ionosheath

(Mihalov et al., 1982)

• bow shock shape and position for Jupiter and Saturn (Slavin

et al., 1984b)

CONCLUSIONS

To summarize the preceding discussion of magnetohydrodynamic and

gasdynamic theories for planetary bow waves, we conclude with the

following remarks.-

Magnetohydrodynamic theory provides a good basis for modeling many

aspects of planetary bow waves and their associated flow fields.

Current development of numerical solutions for representative

applications is in the early stages, and presently require extensive

computer times. Even when these solutions are obtained, there remain a

number of important phenomena that cannot be described by the theory.

Examples include upstream waves or any other multi-component plasma process. Moreover, the dissipationless limit of the general MHD theory cannot account for any phenomena that depends on the finite thickness of

shock waves, boundary layers, or current sheets.

The aligned magnetohydrodynamic case in which the magnetic field is

aligned with the velocity can be reduced to a pseudogasdynamics problem

46
and solved by methods of gasdynamics. The results display substantial

effects of variations of the Alfven Mach number as well as the sonic

Mach number, and are of such nature that they cannot be made into a

function of one parameter by any combination of the sonic and Alfvenic

Mach numbers.

The gasdynamic-convected-field theory is a limiting form of

magnetohydrodynamic theory for large Alfven Mach number and for Alfven

Mach numbers in excess of the sonic Mach number. With the aid of

certain additional approximations including the Newtonian pressure

relation for predetermining the magnetoionopause shape and the

introduction of an axisymmetric magnetoionopause shape, thetheory can

be reduced to a form that is both amenable to efficient numerical

solution and satisfactorily accurate for many purposes. This theory has

been fully implemented in a readily available model that has been

demonstrated to give useful results over a wide range of conditions.

Several groups are now actively using the model in their work,

developing further extensions and applications, and generally finding

the results useful for both interpreting data and developing new

insights.

Finally, we note that the dissipationless, gasdynamic-convectedfield theory contains certain properties that are sometimes interpreted

as false and without meaning, but which actually can be understood as

true indicators of real processes when viewed in the limit of vanishing

dissipation and interplanetary magnetic field. A discussion of several

examples is provided.

47
ACKNOWLEDGEMENTS

The preparation of this paper wa' supported in part by the National

Aeronautics and Space Administration under grant NAGW - 2781 to Stanford

University and by RM Associates, Inc., Independent Research and Development Program.

4

48
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49
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51
7

Slavin, J. A. and R. E. Holzer, Solar wind flow about the terrestrial

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52

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54

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55
FIGURE CAPTIONS

Figure 1. Illustration of solar wind flows past major bodies in the

solar system.

Figure 2. Two-dimensional magnetohydrodynamic simulation of evolution

of the geomagnetic field in a substream (Brackbill, 1982).

Figure 3. Magnetohydrodynamic propagation speeds for plane waves.

Figure 4. Magnetohydrodynamic Mach cone trace, delineating regions of

influence of a small disturbance at a fixed point in a

steady flow.

Figure 5. Calculated bow wave positions for various Alfvdn Mach

numbers MA

in a steady aligned magnetohydrodynamic flow

co

with M S =10 and y=5/3 (Spreiter and Rizzi, 1974).

cc

Figure 6. Results of three-dimensional magnetohydrodynamic simulation

of steady solar wind flow past the dipole magnetic field of

the Earth in the absence of an interplanetary magnetic field

(Wu et al., 1981).

t

Figure 7. Shape and size of the Earth's magnetopause according to the

Chapman-Ferraro theory, or- equivalently the Newtonian

gasdynamic model.
Figure 8e Outline of two alternative methods for calculating

axisymmetric supersonic flow past a blunt-nosed body.

Figure 9. Calculated gasdynamic streamlines and Mach lines for steady

supersonic solar wind flow past the Earth's magnetosphere,

M S =8, Y=5/3.

W

Figure 10. Calculated gasdynamic density, velocity, and temperature

contours for steady supersonic solar wind flow past the

Earth's magnetosphere, M S =8, Y=5/3,

co

Figure 11. Diagrams ;o illustrate the xln coordinates and magnetic

field components used in Equations (37) and (38) to

implement the Alksne-Webster, (1970 ) magnetic field

decomposition theorem for axisy mmetric flows.

Figure 12. Magnetosheath magnetic field magnitude and direction for

representative conditions for solar wind flow past the

Earth, M S =8, y=5/3.

00
Ion AdHitinn

Bow S hoc

v -

Cometary Obstacles

Largi variety of sizes

Bow Shock /^ Magnetopause

supersonic

sonic line

v

, — —

.------

subsonic 

r r

o p

Magnetic Obstacles

Planet rp'km ro/rp

Mercury 2400 1.5

Earth 6400 10.0

Mars (?) 3400 1.1

Jupiter 71000 70.0

Saturn 60000 22.0

Bow Shock supersonic

sonic line Ionopause

v .

\

subsonic

r r

o p

Ionospheric Obstacles

Planet rp'km

Venus 6200

Hars (?) 3400

Expansion Mach wave _ —

v

^^ Cav ity

Solid Obstacles

rW

k 

m

[loon 1700

FIGURE 1

("L! -AL -.Ifp
t n 30

•10 0 -10 -20 -30

^Z

t•35

•10 0 -t0 -20 -30

t-40

+10 0 -10 -20 - 30

/ t-45

+10 0 -10 -20 - 30

- t-5

♦10 0 10 20 30

t-55

4+ 0 0 -10 - 20 -30

FIGURE 2
B

50

CA

km/ s

0

B,nT

5

CC +C ^^

.ast

totatic

S1o%,

B - 0 GD Sound Wave p - Const., MHD Alfven (lave

~ C S

YRT/m C A = B/ 4np

'01

Hydrogen

Plasma

CS

v = 5/3

k/ s m m = 1112

0

T,K 105

FIGURE 3
Fast

CS 

+C Al

Rotational

Slow

CA

Lim MHD - GD

B-0

sin wGD = C S /v = f1S-'

Min Angle

B

sin wmin = max (MS 1 f1A 1 )

B it v

v B

General MHD

.. Cnf

nr

v

sin w = vn /v

flax Angle

<

3 CS

ax

sin wmax Sv2 A = 3 MSZA2+fl

MS2+MA2-1

sines = - S2++f^

MS 2 +MA 2 +1±,(MS 2 +MA 2 +1) -4MS2MA2

sin w =

` S A

FIGURE 4
G54

r/r

0

3210

2.5 3

MA

5

10

20

mic

x/r0

FIGURE 5
X

DENSITY CONTOURS PRESSURE CONTOURS

M = 2.5 M`25

jV r

to

Z O -I' ^_— Z

w X to ' P saw w.+a SUN 4rX

10 — > Psoaw wr4

y.1 - --- < P so,w wro Y,. y ---- <PSOLM W"

,P ate' 1° off'

CONDITIONS ALONG SUN-EARTH LINE

I S1.

!0

_ a

H ^0 c 20

!! M!!

h

I

II 4l

j a

10

0 zoo zo

0

X

FIGURE 6
N

MAC

MEF

PL

v r' _m

MAG.

EQUATORIAL PLANE

15 

(

r T TTTT t1 ^nmn 1

10

0. nm.

EARTH PROTONS /CM3

RADI1 100

S

K•I n I

0

100 500 1000

vm, kmAec

EQUINOX • NEUTRAL SOLSTICE

POINTS

7

N '. \N

vm

v

` m

. S ,S

FIGURE 7
OF POuic ^...^..

Former method

Supersonic region met

Of Ch0 ► aChstiCt

solution

Inverse iteration

method solulron

Bow shock

wave

,.w pto/iorusphere boLndory

symm ric)

M » I

i

Planet

Subsonic regicn

Present method

Obstacle nose

solution

Implicit unsteac

Eule ► equation

solver

hock-capturing marching

solution

FIGURE 8
3.0 - --STREAMLINES

---- MACH LINES

SHOCK AE

2.0-

'/D

I 0-

0

1.0 0

X/D

MAGNETOSPHERE

BOUNDARY

EARTH

-1.0

FIGURE 9
A n -

VELOCITY, vv^vm T

TEMPERATURE, T I es

SHOCK WAVE

13.--

1 , 2 P/P^ ::)0-5;\64 d^tT/TW

MAGNETOSPHERE - MAGNETOSPHERE

BOUNDARY BOUNDARY

EARTH

./ 2EARTH

.O,

0 -I.0 10 0 •1.0

X/D X/D

FIGURE 10
ORIG1t'

,i,^ It Y

OF POOk

FIGURE 11
E.

I

._ 1

2.0 Mco- B. Y-5/3 \\ \ \ 8

FIELD LINE FIELD LINE

7

30

4

35 IBI

'

^ R10 IBmI - 15

60 2.0 / 70 r

9010,' EARTH 450 I V

5 EARTH

r/D -+ ^+ —

VC0 I >t `\

V00

30

i 35

_ \ 40 IBI

3.025

2.0

\\ . 3.0 2.5

L i I L 1 I

2 I 0 -I 2 1 0 -1

x/D x/D

Out of Plane of Magnetic Symmetry

PLAN VIEW \ I ELEVATION

\ IBI IBI

I B mI IBmI

\ 1.0

FIELD LINE -1.0 FIELD LINE

\ 2.0

\3.0

2.5

3.5

.5 

4

I ^ 4.0 .0

45 •

EARTH EARTH 'S•0

,F (_ r

V10 5 .0

5.0

\ 4.0 4.0

3.5 II 0

3.0

2.5

2 I 0 -1 0 I 2

x/D

FIGURE 12