On the distribution of heat in bered magnetic elds Theodore D. Drivas Daniel Ginsberg and Hezekiah Grayer II Abstract. We study the equilibrium temperature distribution in a model for strongly magnetized

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On the distribution of heat in ﬁbered magnetic ﬁelds

Theodore D. Drivas, Daniel Ginsberg, and Hezekiah Grayer II

Abstract. We study the equilibrium temperature distribution in a model for strongly magnetized

plasmas in dimension two and higher. Provided the magnetic ﬁeld is suﬃciently structured (inte-

grable in the sense that it is ﬁbered by co-dimension one invariant tori, on most of which the ﬁeld

lines ergodically wander) and the eﬀective thermal diﬀusivity transverse to the tori is small, it is

proved that the temperature distribution is well approximated by a function that only varies across

the invariant surfaces. The same result holds for “nearly integrable” magnetic ﬁelds up to a “criti-

cal” size. In this case, a volume of non-integrability is deﬁned in terms of the temperature defect

distribution and related the non-integrable structure of the magnetic ﬁeld, conﬁrming a physical

conjecture of Paul–Hudson–Helander [6]. Our proof crucially uses a certain quantitative ergodicity

condition for the magnetic ﬁeld lines on full measure set of invariant tori, which is automatic in

two dimensions for magnetic ﬁelds without null points and, in higher dimensions, is guaranteed by

a Diophantine condition on the rotational transform of the magnetic ﬁeld.

1. Introduction

The heat conduction in strongly magnetized plasmas is inﬂuenced locally by the direction of

the magnetic ﬁeld B:Rd→Rd.Braginskii [3] (see also [4, 12]) derived an eﬀective anisotropic

diﬀusion equation for the temperature Tin such an environment which, in steady state and free of

heat sources, reads

div(b∇bT+ε∇⊥

bT) = 0 (1.1)

where, assuming the magnetic ﬁeld has no null points |B| 6= 0, we introduced

b=B

|B|∇b=b· ∇,∇⊥

b=∇ − b∇b.(1.2)

This equilibrium equation captures macroscopically the phenomenon that charged particle dynam-

ics strongly inﬂuenced by Bfavors collisions aligned with b. In (1.1), the parameter ε > 0 represents

the ratio κ⊥/κkof the transverse diﬀusion coeﬃcient to the longitudinal. In general it is a scalar

function of local density and ﬁeld magnitude |B|, however its magnitude is small in many appli-

cations of interest where |B|is large. In our work, we treat εas a constant and study the limit

ε→0.

For arbitrary B, it is not immediate what emerges in the limit ε→0 of (1.1), given some ﬁxed

boundary conditions. We focus on toroidal “Arnold ﬁbered” ﬁelds B. These are solenoidal vector

ﬁelds Bhaving the property that there is a smooth function ψ:D→Rdeﬁned in a bounded

region D⊂Rdwith |∇ψ| 6= 0 in D, whose level sets Sψare (d−1)–dimensional tori such that ψis

a ﬁrst integral

B· ∇ψ= 0.(1.3)

We shall term these ﬁelds (toroidally) ﬁbered. In two dimensions, if Bis divergence-free and

suﬃciently smooth, then B=∇⊥Afor a “streamfunction” A:D→Rwhere ∇⊥= (−∂y, ∂x). If B

has no nulls, then |∇A|>0, so any non-vanishing divergence-free ﬁeld in two dimensions is ﬁbered

by its streamfunction, e.g. ψ=A. See Figure 1 (a). In three dimensions, its straightforward to

arXiv:2210.09968v1 [math.AP] 18 Oct 2022

2 THEODORE D. DRIVAS, DANIEL GINSBERG, AND HEZEKIAH GRAYER II

write down explicit ﬁbered ﬁelds, see (1.13) and Figure 1 (b). Moreover, as we will later discuss,

non-degenerate magnetohydrostatic equilbria have this property.

Figure 1. Examples of ﬁbered magnetic ﬁelds. Left: a 2d magnetic ﬁeld without

null points on a (topologically) annular domain – the periodic channel. Integral

curves are levels of the streamfunction. Right: a 3d toroidal magnetic ﬁeld; depicted

in grayscale are distinct level surfaces of the ﬁrst integral, ψ, the ﬂux function.

The temperature equation (1.1) is to be solved in a toroidal shell Dwith boundaries S±that are

level sets of the ﬁrst integral ψ. Call ψ−:= infDψand ψ+:= supDψ. Since ψis non-degenerate by

assumption, S±are the levels corresponding to the values ψ±. To complete the problem, we impose

Dirichlet boundary conditions for the temperature ﬁeld T:D→Ron these surfaces. Overall, we

consider the system

div(b∇bT+ε∇⊥

bT) = 0 in D,

T=T±,on S±,(1.4)

for constants T−, T+. The 1.4 system is used in practice as an eﬃcient method to visualize the ﬂux

surfaces of the magnetic ﬁeld [4, 5, 6].

To state our main result concerning the convergence of Tε, we use some notions from mixing to

characterize the behavior of Bvia its trajectories on the ﬂux surfaces Sψ. Denoting I= [ψ−, ψ+],

we say that Sψis an “ergodic” surface if ψis in the set

E(γ, M) := nψ∈I:kuk˙

H−γ(Sψ)≤Mk∇BukL2(Sψ),for all u∈H1(Sψ)o(1.5)

for some nonnegative γand M, where ˙

Hγ(Sψ) denotes the homogeneous Sobolev space of index γ

on Sψ. The sets E(γ, M ) of ergodic values may be empty or may have full measure, depending on

B. The deﬁnition of these sets is motivated by a Diophantine condition, see (4.1). We then deﬁne

the collection

N(γ, M) = I\E(γ, M),(1.6)

of “non-ergodic” values of ψ. Note that if M > M0then N(γ, M)⊆N(γ, M0).

Definition 1.We say that Bsatisﬁes the “ergodicity condition” if, with N(γ, M ) deﬁned as

in (1.6), for some c, γ > 0, we have

lim

M→∞ Mcµ(N(γ, M)) = 0,(1.7)

where µdenotes the one-dimensional Lebesgue measure.

Our main result below roughly states that, provided Bis ergodic on almost all of the surfaces Sψ

such that the ergodicity condition holds, the temperatures proﬁles Tεindeed converge (in H1(D))

to the eﬀective temperature T0. A consequence of our theorem is that the limiting temperature

proﬁle T0itself ﬁbers B. This fact partially motivated the work of Paul–Hudson–Helander [6].

ON THE DISTRIBUTION OF HEAT IN FIBERED MAGNETIC FIELDS 3

Theorem 1.1.Let d≥2and let Bbe toroidally ﬁbered by ψ, and let Dbe the region bounded

by two level sets S±. For ε > 0, let Tε:D→Rbe the solution of system (1.4) for constants T−

and T+. If the ergodicity condition from Deﬁnition 1 holds, then

Tε→T0:= Θ(ψ)in H1(D) (1.8)

where Θ(ψ)is the solution of the one-dimensional boundary-value problem on ψ∈[ψ−, ψ+]:

dψ dΘ

dψZSψ|∇ψ|dH(d−1)!= 0,Θ(ψ±) = T±.(1.9)

In fact, there is C:= C(D, B)>0such that

kTε−T0kH1(D)≤Cε c

2+c,(1.10)

where cis the largest so that there is a γ > 0making condition (1.7) of Deﬁnition 1 hold.

The proof is found in §2. Brieﬂy, if each integral curve of B|Sψcovers Sψdensely for some ψ,

(that is, if Sψis an “irrational torus”), then B“nearly” spans the tangent space at each point. On

such a torus, one encounters a small divisors problem; the operator ∇Bis bounded below on Sψ

but the lower bound may be arbitrarily small. However, for ψ∈E(γ, M), this lower bound cannot

be less than 1/M . On the complement N(γ, M), the operator ∇Bis not bounded below, but the

ergodicity condition (1.7) ensures that the measures of the sets N(γ, M) go to zero as Mincreases.

The net result is one of homogenization to a one-dimensional limit proﬁle adapted to the geometry

of the invariant tori that satisﬁes an eﬀective diﬀusion equation. See §2 for further discussion.

In the upcoming Corollaries 1.1, 1.2, we show that this condition holds for a large family of

physically-relevant vector ﬁelds B. Whenever d= 2, the sets N(γ, M ) are empty for large enough

M; that is, every surface Sψis ergodic in this setting (in three and higher-dimensions, the ergodicity

condition need not be true in general). Thus cin bound (1.10) may be taken to ∞for any γ≥0.

It follows from our main theorem that, in this case, we have convergence of Tεto the eﬀective

temperature T0. More quantitatively:

Corollary 1.1.Let d= 2 and let Bbe a non-vanishing divergence-free vector ﬁeld. Then

kTε−T0kH1(D)≤Cε, (1.11)

where T0= Θ(ψ)where Θis given by (2.6).

In three dimensions, an important example of ﬁbered ﬁelds are the smooth solutions of the

magnetohydrostatic equations

(curl B)×B=∇p, div B= 0,in D⊂R3,(1.12)

having the property that the pressure satisﬁes ∇p6= 0. As noted by Arnold [1, 2] since |∇p|is

nonvanishing by assumption, each surface Spis a smooth two-dimensional surface which admits

two everywhere transverse non-vanishing tangent vector ﬁelds (curl Band B) and are thus two-

dimensional tori or cylinders. In this setting Bis ﬁbered by its pressure, ψ=p. It is straightforward

to construct ﬁelds Bof this type which are axisymmetric, see e.g. [11] and [14]. It is an open

problem (see [14], [15]) to construct such smooth magnetohydrostatic equilibria with |∇p|>0

outside of Euclidean symmetry.

More generally, in three dimensions, given a non-degenerate function ψ:D→Rwhose level

sets are tori along with functions θ, φ :D→R, any vector ﬁeld of the form

B=∇ψ× ∇θ+∇φ× ∇χ(1.13)

4 THEODORE D. DRIVAS, DANIEL GINSBERG, AND HEZEKIAH GRAYER II

is divergence-free. If χis chosen so that χ=χ(ψ, φ), Bis ﬁbered by ψ, and known as “integrable”

because the integral curves of Bobey a Hamiltonian system with Hamiltonian χ, and this Hamil-

tonian is integrable in the usual sense1when ∂θχ= 0 (see (1.15)). See Figure 1, right panel. Fields

of this form play an important role in the problem of conﬁning a plasma with a magnetic ﬁeld [13].

Such ﬁelds may sometimes be regarded as MHS solutions held steady by external forcing (e.g. by

current carrying coils in some particular geometry) [9, 10].

Suppose θ, φ form a coordinate system on Sψand so we write u=u(ψ, θ, φ). Then if Bis as in

(1.13) with ∂θχ=∂φχ= 0, it follows after writing ι(ψ) = χ0(ψ),

(B· ∇)u= [∂φu+ι(ψ)∂θu]J, (1.16)

where J=∇ψ× ∇θ· ∇φ. Generally, by a theorem of Sternberg [21], if Bis any nonvanishing

divergence-free vector ﬁeld ﬁbered by a function ψ, (in particular, this includes the case χ=χ(ψ, φ)

of (1.13)) then on each Sψthere are coordinates θ, φ and a number ι=ι(ψ) so that, expressed in

these coordinates, Btakes the form (1.16) for a function J=J(ψ, θ, φ)>0. We call the function

ιfrom (1.16) the rotational transform. Our main result in three dimensions, proven in §4, is that

provided ιis invertible with Lipschitz inverse, we have convergence Tε→T0in H1(D).

Corollary 1.2.Suppose that Bis a nonvanishing divergence-free vector ﬁeld ﬁbered by a

function ψ. Suppose that the rotational transform ιfrom (1.16) is invertible and for some L > 0

|ψ1−ψ2| ≤ L|ι(ψ1)−ι(ψ2)|(1.17)

holds for all ψ1, ψ2∈I. Then condition (1.7) holds for any γ > 2with c= 1. Consequently,

kTε−T0kH1(D)≤Cε1

3,(1.18)

where T0= Θ(ψ)where Θis given by (2.6).

In other words, we show that the ergodic condition holds for integrable Arnol’d ﬁbered ﬁelds

Bwith monotone rotational transform. Such ﬁelds are of speciﬁc interest in the plasma physics

community, see the discussion in [9]. However such plasma equilibria, if they exist, may be unstable.

Thus, it is important to also understand the behavior of non-integrable ﬁelds ∂θχ6= 0. There is an

obstruction: the behavior of particle transport (and thus of heat) in non-integrable ﬁelds can be

quite complicated because non-integrable Hamiltonian systems may exhibit chaos.

In [6], the authors consider non-integrable magnetic ﬁelds taking the form (1.13) where

χε(ψ, θ, φ) = χ0(ψ) + εaχ1(ψ, θ, φ),(1.19)

and a≥1/2. A modiﬁcation of the proof of Theorem 1.1 (see Section 5) gives the following

generalization of Corollary 1.2 to ﬁelds of this type which are “weakly nonintegrable.” We require

1We suppose that with Bas in (1.13), the functions θ, φ, ψ together form a coordinate system in D. Then for

any smooth u:D→R, we have ∇u=∂ψu∇ψ+∂φu∇φ+∂θu∇θand so, writing J=∇ψ× ∇θ· ∇φ, which is

nonvanishing by our assumption, we have the formula

(B· ∇)u= [∂φu+ι(ψ, θ, φ)∂θu+τ(ψ, θ, φ)∂ψu]J, (1.14)

where τ(ψ, θ, φ) := −∂θχ(ψ, θ, φ) and where we have introduced the rotational transform ι(ψ, θ, φ) := ∂ψχ(ψ, θ, φ).

There is a simple interpretation of the function χ. Consider any integral curve of B, parametrized by φ. That is, we

consider Ψ(φ), ϑ(φ) deﬁned by

dφΨ = B· ∇ψ

B· ∇φ=−∂θχ, d

dφϑ=B· ∇θ

B· ∇φ=∂ψχ, (1.15)

with the understanding that the quantities on the right-hand sides are evaluated at (ψ, θ, φ) = (Ψ(φ), ϑ(φ), φ). Thus

the integral curves of Bsatisfy a Hamiltonian system with Hamiltonian χ. Note that if ∂θχ= 0, the above system is

integrable (has a conserved quantity) since ψis constant along the ﬂow. This also be seen from the formula (1.14).

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OnthedistributionofheatinberedmagneticeldsTheodoreD.Drivas,DanielGinsberg,andHezekiahGrayerIIAbstract.Westudytheequilibriumtemperaturedistributioninamodelforstronglymagnetizedplasmasindimensiontwoandhigher.Providedthemagneticeldissucientlystructured(inte-grableinthesensethatitisberedbyco-dimens...

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On the distribution of heat in bered magnetic elds Theodore D. Drivas Daniel Ginsberg and Hezekiah Grayer II Abstract. We study the equilibrium temperature distribution in a model for strongly magnetized

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