Proceedings of ITC/ISHW2007

The response of toroidal plasmas to error fields
Jong-kyu Park1) , Allen H. Boozer2) and Jonathan E. Menard1)
1)

2)

Princeton Plasma Physics Laboratory, Princeton, NJ 08543, U.S.A.
Department of Applied Physics and Applied Mathematics, Columbia University, NY 10027, U.S.A.
October 14, 2007

The response of toroidal plasmas to three-dimensional magnetic error fields is studied by using Ideal
Perturbed Equilibrium Code (IPEC). Since the toroidal plasmas are highly sensitive to small error fields, theperturbation theory is efficient and useful to describe the equilibrium ∇P = J × B in the presence of error
fields. The perturbed force balance equation in ideal MagnetoHydroDynamics (MHD) is solved by augmenting
a stability code and by constructing the interface between plasma and external system. In an ideally perturbed
equilibrium, a shielding current arises on a rational surface to prevent an island from opening. When error
fields reach a critical magnitude, the shielding current will be dissipated and an island will open. This effectof error fields can be greatly mitigated by adjusting currents in auxiliary coils to reduce the shielding current,or equivalently the resonant field. From the coupling between the resonant field and error fields, the effectsof various error fields on toroidal plasmas were studied. The most important external field is almost always
localized on the outboard midplane, which gives an important implication to the study and control of the responseof toroidal plasmas to error fields.
Keywords: toroidal plasma, MHD, perturbed equilibrium, error field, magnetic island
The magnetically confined toroidal plasmas, such as
tokamaks and stellarators, are highly sensitive to externallydriven magnetic perturbations. This sensitivity implies
that the response of plasmas to small magnetic perturbations is a critical issue in design and control of equilibrium[1, 2, 3, 4, 5]. The small magnetic perturbations always
exist in toroidal devices due to error fields, such as, imperfections of primary magnets and other conducting components. When an external perturbation occurs, plasma responds to it and relaxes to a new equilibrium state. Sincethe perturbations are in practice very small compared withthe field of the original equilibrium, the perturbation theoryis effective to describe the response of plasmas.
The perturbed force balance equation is given byf (ξ) = −∇δP + J × δB + δJ × B = 0(1)in ideal MagnetoHydroDynamics (MHD). Using Maxwell
relations and adiabatic plasma response, δJ = (∇ ×δB)/µ0 , δB = ∇ × (ξ × B) and δP = −ξ · ∇P − γP(∇ · ξ),
the force balance equation becomes a vector differential
equation for the plasma displacement ξ. Assuming that theplasma conserves its two independent profiles, rotationaltransform ι(ψ) and pressure p(ψ), then the same equation
can be derived from the theory of ideal, linear MHD stability. Through Rminimization of a perturbed potential energy
δW = −1/2 dx3 ξ · f (ξ), the exact same equation for ξ
can be obtained. Therefore, an existing code in the stability analysis can be used to solve the problem of perturbed
equilibria. Only the interface between plasma and external
system is required to obtain an perturbed equilibrium given
author’s e-mail: jpark@pppl.gov

an external error field.
The Ideal Perturbed Equilibrium Code (IPEC) [6]modifies and augments the DCON ideal MHD stability
code [7]. For a given axisymmetric equilibrium, the use of
DCON gives a set of M plasma displacements ξi (ψ, θ, ϕ),which is the set of M ideal MHD eigenmodes for a given
toroidal harmonic number n, where 1 ≤ i ≤ M andM is the number of poloidal harmonics retained. Here(ψ, θ, ϕ) are magnetic coordinates which are straight on
the field line. Each of these displacements ξi is associated with a certain deformation of the plasma boundary,
ξi · nb ≡ (ξi · n)(ψb , θ, ϕ), where ξi is evaluated on theunperturbed plasma boundary at ψ = ψb and nb is the normal to the unperturbed plasma boundary. Each of these
M displacements of the plasma boundary ξi · nb defines
a perturbed equilibrium if an external magnetic field produces a required force to support it. That is, the set of M
ideal MHD eigenmodes found by DCON defines a set of
M neighboring perturbed equilibria. Each of the neighboring equilibria is supported by an external magnetic field
and has the same profiles of ι(ψ) and p(ψ) as the unperturbed equilibrium; only the shape of the plasma has beenchanged.
A plasma displacement determines a magnetic perturbation δB = ∇ × (ξ × B), so IPEC uses the displacementof the plasma boundary ξ · nb to determine a part of theperturbed magnetic field that is normal to the unperturbed
plasma boundary, δB · nb , and a part that is tangential tothe plasma boundary, nb × δB (p) . Since the normal fieldδB · nb is continuous across the plasma boundary and thecontrol surface, δB · nb then gives a unique vacuum fieldProceedings of ITC/ISHW2007
outside the plasma, δB (vo) , that vanishes at infinity. The
difference between the tangential field infinitesimally outside the control surface nb × δB (vo) and the tangential fieldon the plasma side of the control surface nb × δB (p) determines an external surface current on the control surface,µ0 K x = nb × δB (vo) − nb × δB (p) . Once K x is known, theexternally produced normal magnetic field δB x · nb can befound by ∇ × δB x = µ0 J x in vacuum. Note that the external field δB x designated by superscript x is a vacuum field
without plasma response, compared with the total field δB.
Each of the M neighboring equilibria calculated by
DCON has a unique distribution of the external normalmagnetic field δBix · nb , where 1 ≤ i ≤ M, that must
be produced by currents outside the plasma to sustain thatequilibrium. If an external magnetic perturbation, such asthat due to a magnetic field error δB x · nb , is specified on
the unperturbed plasma boundary, this perturbation can bePM
expanded as δB x · nb = i=1
ci δBix · nb , with expansion
coefficients ci . If this is done, the plasma displacement thatgives the perturbed equilibrium produced by the field error
PMis ξ(ψ, θ, ϕ) = i=1ci ξi (ψ, θ, ϕ). This is the method usedby IPEC to find the perturbed equilibrium associated witha given magnetic field error [6].
The normal total magnetic fields of the M neighboringequilibria can be represented by
Xi(mθ−nϕ)(2)(δB · nb )(θ, ϕ) = Re  Φm w(θ)e ,min Fourier space, where the weight function w(θ) =1/(J(θ)|∇ψ|(θ)) with the Jacobian J(θ)
is used for an orH
thogonal basis, by the definition of w fm fm0 da = δmm0 on
the boundary surface, with fm = ei(mθ−nϕ) .
The jump in the tangential field across the control
surface just outside the plasma gives a surface current
J = Kδ(ψ − ψb ). The surface current can also be expressed as K = ∇κ(θ, ϕ) × ∇ψ with a surface current potential κ(θ, ϕ). The potential κ(θ, ϕ) can be used for representing the surface current K byX(3)
κ(θ, ϕ) = Re  Im ei(mθ−nϕ) mwith the vector I having units of current. Combining thetotal fluxes Φi and external currents I ix of the M neighboring equilibria, one can obtain a plasma inductance matrix
Λ on the Fourier space, where the M poloidal harmonics
are retained. Λ gives the relation between an total flux andan external current by Φ = Λ · I x . Similarly, the external
normal magnetic perturbation δB x · nb producing the surface current K x can be expanded and related to the currentby Φ x = L · I x , where L is a surface inductance matrix
since it depends only on the shape of the boundary surface.
The linear relation between an total flux Φ and an external flux Φ x can then be written asΦ = P · Φx

(4)Fig. 1 The deformed plasma boundary of typical (a) NSTX (b)DIII-D tokamak plasmas due to each intrinsic error field.
The scale is arbitrary.

with a permeability matrix P = Λ · L−1 . If a magnetic
field error Φ x is specified on the boundary, one can expand
PMPM
ci P −1 · Φi , or equivalently byci Φix = i=1it by Φ x = i=1PMx
Φ = P ·Φ = i=1 ci Φi to obtain the perturbed equilibrium
PMby ξ(ψ, θ, ϕ) = i=1ci ξi (ψ, θ, ϕ). Each actual flux Φi is
associated with a plasma displacement ξi through Eq. (2)and δB = ∇ × (ξ × B). This is how IPEC constructs theinterface and solve the perturbed equilibrium from a given
error field [6]. Fig. 1 shows the computational examples,
deformed plasma boundary of typical (a) NSTX [12] (b)DIII-D [13] plasmas due to each intrinsic error field.
An important consequence when plasma is ideallyperturbed is a shielding current on a rational surface ι =n/m to prevent a magnetic island from opening. This indicates mathematically that the normal component of theresonant field has to be vanished on the rational surface,that is, (δB · ∇ψ)mn = 0. This constraint in ideal MHD enforces inner boundary condition and discontinuous tangential field across the rational surface. The jump of tangential
field is [8]#"∂ δB · ∇ψ(5)∆mn ≡∂ψ B · ∇ϕ mn
The shielding current j s is related to the jump as
js =∆mn iei(mθ−nϕ)H
δ(ψ − ψmn )B,µ0 m( dS B2 /|∇ψ|3 )(6)

where ψ is a toroidal flux. A total resonant field driving

Proceedings of ITC/ISHW2007

B = C · Φx .

(7)

The ith important mode can be also defined by the ith singular mode in the SVD (Singular Value Decomposition)
analysis of C. Note again that each ith important mode represents the external field on the plasma boundary, not thetotal field including plasma response.
A practical way to describe the important modes is togive the amplitude of the external field (δB x · nb ) relative
to the plasma boundary in real space. If the most importantmode, or the first mode is highly dependent on equilibria
when it is mapped in real space, the correction of the mode
will be very difficult in practice. Fig. 3 shows the mostimportant external field for n = 1 in the (a) DIII-D and (b)NSTX. The three-dimensional field can be constructed as(δB x · nb )(θ, φ) = C(θ)cos(nφ) + S (θ)sin(nφ), where φ isthe polar toroidal angle. This has to be distinguished froma magnetic toroidal angle, ϕ.

0.002021(per 1 Guass)31

0.00150.0010mn

magnetic islands, (δB · n)mn , can be defined by the field
produced by the shielding current, ∇ × δB = µ0 j s . Notethat n is normal to a magnetic surface at ι = n/m, differently from nb normal to the plasma boundary.
The sustainment of the shielding current is important
to improve plasma performance since otherwise an island
would open and destruct flux surfaces. This happens whenerror fields are larger than a critical magnitude. The importance of the shielding current, or equivalently the resonant
field has been recently verified in tokamak experiments oflocked modes [9]. The use of IPEC has shown that the external field driving the resonant field (δB ·n)mn can be welldescribed through perturbed equilibria.
A long standing supposition, which was supported bycylindrical theory [10, 11], is that the total resonant field
driving islands, (δB · n)mn , is proportional to the resonant
component of the external field, namely, the external resonant field, (δB x · n)mn . When this supposition was applied
to mode locking experiments in DIII-D and NSTX, theresults were paradoxical. When the control coil currents
were optimized empirically, the external resonant field wasoften increased–not decreased as the standard supposition
required. However, the IPEC calculation has shown thatthe total resonant fields were indeed decreased consistently
when the control coil currents were optimized.
The failure in the previous method is due to the strong
coupling between the resonant field (δB · n)mn and theexternal field (δB x · nb ) specified here on the plasmaboundary. The poloidal harmonic coupling is very broadand shifted to higher poloidal harmonics than expected, as
shown in Fig. 2 [6, 9].
The most important external field driving the total resonant field has to be defined through the coupling, that is,by the first singular vector when decomposing the coupling
matrix C between the total resonant field (δB ·n)mn and theexternal field on the boundary (δB x · nb )mn . If one defines
B with the total resonant field on R rational surfaces, thisis written as

0.00050.000001020

Poloidal harmonic number of external perturbation

Fig. 2 (a) The typical poloidal harmonic coupling spectrum between the Fourier components of the external error fieldon the plasma boundary (δB x · nb )mn and the total resonant field, which is here measured by ∆21 and ∆31 .

Fig. 3 The most important external field driving the total resonant field on rational surfaces, in the (a) DIII-D and (b)NSTX. The three dimensional distribution can be constructed by δB x · nb = A(θ)cos(φ) + B(θ)sin(φ) relative
to the plasma boundary (black line).

An important implication is that the most important
external field is localized on the outboard midplane. The
localizations are very robust and almost regardless of various characteristics in equilibria [9]. For example, Fig.
4 (a) shows very little dependency of the localization on
the plasma density, which is represented by the normalized
beta, βn = hβt iI p /aBt0 , where βt is the toroidal β, I p is theplasma current, a is the minor radius and Bt0 is the toroidal
magnetic field at the magnetic axis. This explains well thatthe error-field control coils located on the outboard midplane could effectively mitigate the effect of error fields
despite the limitation of poloidal harmonic controllability.
The mitigation of error fields, therefore, can be optimized
by developing the method to null the most important external field by designing proper control coils.
The detailed information of the coupling between theexternal field and the total resonant field can be used tomany different purposes. For instance, one can find theexternal field only driving one magnetic island on a partic-

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Fig. 4 (a) The most important n = 1 external field and (b) then = 1 external field driving only ι = 1/2 as a function ofβn in NSTX. The external fields are represented relativelyto the plasma boundary as in Fig. 3.

ular rational surface, as shown in Fig. 4 (b). The external
field beyond the most important part is typically not localized and may be too difficult to make relevant corrections
in reality. Nonetheless, the specific pattern in the external
field can be used to suppress islands in a particular region,
for instance, the edge region. This is another important
application of IPEC to the suppression of Edge Localized
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