# On the distribution of heat in fibered magnetic fields

Theodore D. Drivas  Department of Mathematics, Stony Brook University, Stony
Brook, NY, 11794<EMAIL_ADDRESS>,  Daniel Ginsberg  Department
of Mathematics, Princeton University, Princeton, NJ 08544<EMAIL_ADDRESS>and  Hezekiah Grayer II  Program in Applied and Computational Mathematics,
Princeton University, Princeton, NJ 08544<EMAIL_ADDRESS>
###### 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 (integrable 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 “critical” 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 $B:\mathbb{R}^{d}\to\mathbb{R}^{d}.$
Braginskii [3] (see also [4, 12]) derived an effective anisotropic diffusion
equation for the temperature $T$ in such an environment which, in steady state
and free of heat sources, reads

$\displaystyle\operatorname{div}(b\nabla_{b}T+\varepsilon\nabla_{b}^{\perp}T)$
$\displaystyle=0$ (1.1)

where, assuming the magnetic field has no null points $|B|\neq 0$, we
introduced

$b=\tfrac{B}{|B|}\qquad\nabla_{b}=b\cdot\nabla,\qquad\nabla^{\perp}_{b}=\nabla-b\nabla_{b}.$
(1.2)

This equilibrium equation captures macroscopically the phenomenon that charged
particle dynamics strongly influenced by $B$ favors collisions aligned with
$b$. In (1.1), the parameter $\varepsilon>0$ represents the ratio
$\kappa_{\perp}/\kappa_{\|}$ of 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 applications of
interest where $|B|$ is large. In our work, we treat $\varepsilon$ as a
constant and study the limit $\varepsilon\to 0$.

For arbitrary $B$, it is not immediate what emerges in the limit
$\varepsilon\to 0$ of (1.1), given some fixed boundary conditions. We focus on
toroidal “Arnold fibered” fields $B$. These are solenoidal vector fields $B$
having the property that there is a smooth function $\psi:D\to\mathbb{R}$
defined in a bounded region $D\subset\mathbb{R}^{d}$ with $|\nabla\psi|\not=0$
in $D$, whose level sets $S_{\psi}$ are $(d-1)$–dimensional tori such that
$\psi$ is a first integral

$B\cdot\nabla\psi=0.$ (1.3)

We shall term these fields _(toroidally) fibered_. In two dimensions, if $B$
is divergence-free and sufficiently smooth, then $B=\nabla^{\perp}A$ for a
“streamfunction” $A:D\to\mathbb{R}$ where
$\nabla^{\perp}=(-\partial_{y},\partial_{x})$. If $B$ has no nulls, then
$|\nabla A|>0$, so any non-vanishing divergence-free field in two dimensions
is fibered by its streamfunction, e.g. $\psi=A$. See Figure 1 (a). In three
dimensions, its straightforward to 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, $\psi$, the flux function.

The temperature equation (1.1) is to be solved in a toroidal shell $D$ with
boundaries $S_{\pm}$ that are level sets of the first integral $\psi$. Call
$\psi_{-}:=\inf_{D}\psi$ and $\psi_{+}:=\sup_{D}\psi$. Since $\psi$ is non-
degenerate by assumption, $S_{\pm}$ are the levels corresponding to the values
$\psi_{\pm}$. To complete the problem, we impose Dirichlet boundary conditions
for the temperature field $T:D\to\mathbb{R}$ on these surfaces. Overall, we
consider the system

$\displaystyle\operatorname{div}(b\nabla_{b}T+\varepsilon\nabla_{b}^{\perp}T)$
$\displaystyle=0$ $\displaystyle\qquad\text{ in }D,$ (1.4) $\displaystyle T$
$\displaystyle=T_{\pm},$ $\displaystyle\qquad\text{ on }S_{\pm},$

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 $T_{\varepsilon}$, we
use some notions from mixing to characterize the behavior of $B$ via its
trajectories on the flux surfaces $S_{\psi}$. Denoting
$I=[\psi_{-},\psi_{+}]$, we say that $S_{\psi}$ is an “ergodic” surface if
$\psi$ is in the set

$E(\gamma,M):=\left\\{\psi\in I:\|u\|_{\dot{H}^{-\gamma}(S_{\psi})}\leq
M\|\nabla_{B}u\|_{L^{2}(S_{\psi})},\text{ for all }u\in
H^{1}(S_{\psi})\right\\}$ (1.5)

for some nonnegative $\gamma$ and $M$, where $\dot{H}^{\gamma}(S_{\psi})$
denotes the homogeneous Sobolev space of index $\gamma$ on $S_{\psi}$. The
sets $E(\gamma,M)$ 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

$N(\gamma,M)=I\setminus E(\gamma,M),$ (1.6)

of “non-ergodic” values of $\psi$. Note that if $M>M^{\prime}$ then
$N(\gamma,M)\subseteq N(\gamma,M^{\prime})$.

######  Definition 1.

We say that $B$ satisfies the “ergodicity condition” if, with $N(\gamma,M)$
defined as in (1.6), for some $c,\gamma>0$, we have

$\lim_{M\to\infty}M^{c}\mu(N(\gamma,M))=0,$ (1.7)

where $\mu$ denotes the one-dimensional Lebesgue measure.

Our main result below roughly states that, provided $B$ is ergodic on almost
all of the surfaces $S_{\psi}$ such that the ergodicity condition holds, the
temperatures profiles $T_{\varepsilon}$ indeed converge (in $H^{1}(D)$) to the
effective temperature $T_{0}$. A consequence of our theorem is that the
limiting temperature profile $T_{0}$ itself fibers $B$. This fact partially
motivated the work of Paul–Hudson–Helander [6].

######  Theorem 1.1.

Let $d\geq 2$ and let $B$ be toroidally fibered by $\psi$, and let $D$ be the
region bounded by two level sets $S_{\pm}$. For $\varepsilon>0$, let
$T_{\varepsilon}:D\to\mathbb{R}$ be the solution of system (1.4) for constants
$T_{-}$ and $T_{+}$. If the ergodicity condition from Definition 1 holds, then

$T_{\varepsilon}\to T_{0}:=\Theta(\psi)\qquad\text{ in }H^{1}(D)$ (1.8)

where $\Theta(\psi)$ is the solution of the one-dimensional boundary-value
problem on $\psi\in[\psi_{-},\psi_{+}]$:

$\displaystyle\frac{{\rm d}}{{\rm d}\psi}\left(\frac{{\rm d}\Theta}{{\rm
d}\psi}\int_{S_{\psi}}|\nabla\psi|\,{\rm d}\mathscr{H}^{(d-1)}\right)$
$\displaystyle=0,\qquad\Theta(\psi_{\pm})=T_{\pm}.$ (1.9)

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

$\|T_{\varepsilon}-T_{0}\|_{H^{1}(D)}\leq C\varepsilon^{\frac{c}{2+c}},$
(1.10)

where $c$ is the largest so that there is a $\gamma>0$ making condition (1.7)
of Definition 1 hold.

The proof is found in §2. Briefly, if each integral curve of $B|_{S_{\psi}}$
covers $S_{\psi}$ densely for some $\psi$, (that is, if $S_{\psi}$ is 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
$\nabla_{B}$ is bounded below on $S_{\psi}$ but the lower bound may be
arbitrarily small. However, for $\psi\in E(\gamma,M$), this lower bound cannot
be less than $1/M$. On the complement $N(\gamma,M)$, the operator $\nabla_{B}$
is not bounded below, but the ergodicity condition (1.7) ensures that the
measures of the sets $N(\gamma,M)$ go to zero as $M$ increases. 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 $N(\gamma,M)$ are empty for large enough $M$; that is, every surface
$S_{\psi}$ is ergodic in this setting (in three and higher-dimensions, the
ergodicity condition need not be true in general). Thus $c$ in bound (1.10)
may be taken to $\infty$ for any $\gamma\geq 0$. It follows from our main
theorem that, in this case, we have convergence of $T_{\varepsilon}$ to the
effective temperature $T_{0}$. More quantitatively:

######  Corollary 1.1.

Let $d=2$ and let $B$ be a non-vanishing divergence-free vector field. Then

$\|T_{\varepsilon}-T_{0}\|_{H^{1}(D)}\leq C\varepsilon,$ (1.11)

where $T_{0}=\Theta(\psi)$ where $\Theta$ is given by (2.6).

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

$(\operatorname{curl}B)\times B=\nabla
p,\qquad\operatorname{div}B=0,\qquad\text{ in }D\subset\mathbb{R}^{3},$ (1.12)

having the property that the pressure satisfies $\nabla p\not=0$. As noted by
Arnold [1, 2] since $|\nabla p|$ is nonvanishing by assumption, each surface
$S_{p}$ is a smooth two-dimensional surface which admits two everywhere
transverse non-vanishing tangent vector fields ($\operatorname{curl}B$ and
$B$) and are thus two-dimensional tori or cylinders. In this setting $B$ is
fibered by its pressure, $\psi=p$. It is straightforward to construct fields
$B$ of 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 $|\nabla p|>0$ outside of Euclidean symmetry.

More generally, in three dimensions, given a non-degenerate function
$\psi:D\to\mathbb{R}$ whose level sets are tori along with functions
$\theta,\phi:D\to\mathbb{R}$, any vector field of the form

$B=\nabla\psi\times\nabla\theta+\nabla\phi\times\nabla\chi$ (1.13)

is divergence-free. If $\chi$ is chosen so that $\chi=\chi(\psi,\phi)$, $B$ is
fibered by $\psi$, and known as “integrable” because the integral curves of
$B$ obey a Hamiltonian system with Hamiltonian $\chi$, and this Hamiltonian is
integrable in the usual sense111 We suppose that with $B$ as in (1.13), the
functions $\theta,\phi,\psi$ together form a coordinate system in $D$. Then
for any smooth $u:D\to\mathbb{R}$, we have $\nabla
u=\partial_{\psi}u\nabla\psi+\partial_{\phi}u\nabla\phi+\partial_{\theta}u\nabla\theta$
and so, writing $J=\nabla\psi\times\nabla\theta\cdot\nabla\phi,$ which is
nonvanishing by our assumption, we have the formula
$(B\cdot\nabla)u=\left[\partial_{\phi}u+\iota(\psi,\theta,\phi)\partial_{\theta}u+\tau(\psi,\theta,\phi)\partial_{\psi}u\right]J,$
(1.14) where
$\tau(\psi,\theta,\phi):=-\partial_{\theta}\chi(\psi,\theta,\phi)$ and where
we have introduced the rotational transform
$\iota(\psi,\theta,\phi):=\partial_{\psi}\chi(\psi,\theta,\phi).$ There is a
simple interpretation of the function $\chi$. Consider any integral curve of
$B$, parametrized by $\phi$. That is, we consider $\Psi(\phi),\vartheta(\phi)$
defined by  $\displaystyle\frac{{\rm d}}{{\rm d}\phi}\Psi$
$\displaystyle=\frac{B\cdot\nabla\psi}{B\cdot\nabla\phi}=-\partial_{\theta}\chi,\qquad\frac{{\rm
d}}{{\rm
d}\phi}\vartheta=\frac{B\cdot\nabla\theta}{B\cdot\nabla\phi}=\partial_{\psi}\chi,$
(1.15) with the understanding that the quantities on the right-hand sides are
evaluated at $(\psi,\theta,\phi)=(\Psi(\phi),\vartheta(\phi),\phi)$. Thus the
integral curves of $B$ satisfy a Hamiltonian system with Hamiltonian $\chi$.
Note that if $\partial_{\theta}\chi=0$, the above system is integrable (has a
conserved quantity) since $\psi$ is constant along the flow. This also be seen
from the formula (1.14). when $\partial_{\theta}\chi=0$ (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 $\theta,\phi$ form a coordinate system on $S_{\psi}$ and so we write
$u=u(\psi,\theta,\phi)$. Then if $B$ is as in (1.13) with
$\partial_{\theta}\chi=\partial_{\phi}\chi=0$, it follows after writing
$\iota(\psi)=\chi^{\prime}(\psi)$,

$(B\cdot\nabla)u=\left[\partial_{\phi}u+\iota(\psi)\partial_{\theta}u\right]J,$
(1.16)

where $J=\nabla\psi\times\nabla\theta\cdot\nabla\phi$. Generally, by a theorem
of Sternberg [21], if $B$ is _any_ nonvanishing divergence-free vector field
fibered by a function $\psi$, (in particular, this includes the case
$\chi=\chi(\psi,\phi)$ of (1.13)) then on each $S_{\psi}$ there are
coordinates $\theta,\phi$ and a number $\iota=\iota(\psi)$ so that, expressed
in these coordinates, $B$ takes the form (1.16) for a function
$J=J(\psi,\theta,\phi)>0$. We call the function $\iota$ from (1.16) the
rotational transform. Our main result in three dimensions, proven in §4, is
that provided $\iota$ is invertible with Lipschitz inverse, we have
convergence $T_{\varepsilon}\to T_{0}$ in $H^{1}(D)$.

######  Corollary 1.2.

Suppose that $B$ is a nonvanishing divergence-free vector field fibered by a
function $\psi$. Suppose that the rotational transform $\iota$ from (1.16) is
invertible and for some $L>0$

$|\psi_{1}-\psi_{2}|\leq L|\iota(\psi_{1})-\iota(\psi_{2})|$ (1.17)

holds for all $\psi_{1},\psi_{2}\in I$. Then condition (1.7) holds for any
$\gamma>2$ with $c=1$. Consequently,

$\|T_{\varepsilon}-T_{0}\|_{H^{1}(D)}\leq C\varepsilon^{\frac{1}{3}},$ (1.18)

where $T_{0}=\Theta(\psi)$ where $\Theta$ is given by (2.6).

In other words, we show that the ergodic condition holds for integrable
Arnol’d fibered fields $B$ with 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
$\partial_{\theta}\chi\not=0$. 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

$\chi_{\varepsilon}(\psi,\theta,\phi)=\chi_{0}(\psi)+\varepsilon^{a}\chi_{1}(\psi,\theta,\phi),$
(1.19)

and $a\geq 1/2$. 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 the following anisotropic Sobolev
spaces tailored to the invariant tori: $f\in L^{2}(D)$ which are finite in the
norm

$\|f\|_{H^{(0,\gamma)}}^{2}:=\int_{\psi_{-}}^{\psi_{+}}\|f(\psi,\cdot)\|_{H^{\gamma}(S_{\psi})}^{2}\,{\rm
d}\psi.$ (1.20)

######  Theorem 1.2.

Suppose that $B$ is as in (1.13) where $\chi_{\varepsilon}$ is given by (1.19)
for $a\geq 1/2$ and satisfies
$\|\partial_{\theta}\chi_{1}\|_{L^{\infty}(D)}<1$ and
$\partial_{\theta}\chi_{1}|_{\partial D}=0$. Moreover, assume that
$\iota=\chi_{0}^{\prime}$ satisfies the condition from Corollary 1.2. Then
letting $B_{0}$ be defined by (1.13) with $\varepsilon=0$, there is a constant
$C=C(L)$ such that

$\displaystyle\|\nabla_{b_{0}}^{\perp}(T_{\varepsilon}-T)\|_{L^{2}}+\|T_{\varepsilon}-T\|_{L^{2}}^{2}$
$\displaystyle\leq
C\varepsilon^{2a-1}\|\partial_{\theta}\chi_{1}\|_{L^{2}}^{2}+C\varepsilon^{\frac{1}{3}}\left(\|\Delta
T_{0}\|_{H^{(0,\gamma)}}^{2}+\|\Delta T_{0}\|_{L^{\infty}}^{2}\right),$ (1.21)
$\displaystyle\|\nabla_{b_{0}}(T_{\varepsilon}-T)\|_{L^{2}}$
$\displaystyle\leq
C\varepsilon^{2a}\|\partial_{\theta}\chi_{1}\|_{L^{2}}^{2}+C\varepsilon^{\frac{4}{3}}\left(\|\Delta
T_{0}\|_{H^{(0,\gamma)}}^{2}+\|\Delta T_{0}\|_{L^{\infty}}^{2}\right).$ (1.22)

When $a>1/2$, the estimate (1.21) implies that $T_{\varepsilon}\to T_{0}$
almost everywhere. When $a=1/2$, we show the same result is true, and in this
case, even though $\rho^{*}=\lim_{\varepsilon\to 0}T_{\varepsilon}-T_{0}\in
H^{1}(D)$ may not vanish almost everywhere, we find $\rho^{*}|_{S_{\psi}}=0$
for all $\psi$ except possibly for a family of $\psi$ lying in a set of
measure zero. Here, $\rho^{*}|_{S_{\psi}}$ denotes the trace of the function
$\rho^{*}$ on the surface $S_{\psi}$, and this quantity is well-defined
whenever $\rho^{*}\in H^{1}(D)$, by the trace theorem. In fact, the support of
$\rho^{*}=\lim_{\varepsilon\to 0}\rho_{\varepsilon}$ is contained in the
collection of non-ergodic surfaces $N(\gamma)=I\setminus E(\gamma)$, which has
measure zero, where $E(\gamma)$ denote the family of ergodic surfaces

$E(\gamma)=\bigcup_{M>0}E(\gamma,M).$ (1.23)

######  Corollary 1.3.

Under the hypotheses of Corollary 1.2 with $a=1/2$, the sequence
$\rho_{\varepsilon}=T_{\varepsilon}-T_{0}$ converges weakly in $H^{1}$ to a
distribution $\rho^{*}$ in $H^{1}(D)$ with the property that

$\rho^{*}|_{S\psi}=0,\qquad\text{ whenever }\psi\in E(\gamma).$ (1.24)

That is, the support of $\rho^{*}$ is contained in $N(\gamma)=I\setminus
E(\gamma)$.

Our final result relates directly to the work of [6]. In that paper, the
authors consider the sets

$\mathcal{N}(\varepsilon)=\\{(\psi,\theta,\phi):|\nabla_{b}T(\psi,\theta,\phi)|^{2}\geq\varepsilon|\nabla_{b}^{\perp}T(\psi,\theta,\phi)|^{2}\\},$
(1.25)

where $b=B/|B|$ with $B$ as in (1.13), and study them as a proxy for the “non-
integrability” of the field $B$. Using the estimates from the above section,
we can get an upper bound on the measure set in the limit $\varepsilon\to 0$.

######  Proposition 1.1.

Define $B$ as in (1.13) and $\mathcal{N}(\varepsilon)$ as in (1.25). Suppose
that the boundary values $T_{\pm}$ from (1.9) satisfy $T_{+}\not=T_{-}$. Under
the hypotheses of Corollary 1.2, there is a constant $C$ depending
continuously on $\|T_{0}^{\prime}\|_{L^{\infty}},$
$\|1/T_{0}^{\prime}\|_{L^{\infty}}$, $\|T_{0}^{\prime\prime}\|_{L^{\infty}}$,
$(1-\|\nabla\chi_{1}\|_{L^{\infty}})^{-1}$ and $(2-\gamma)^{-1}$ so that

$\mu(\mathcal{N}(\varepsilon))\leq
C\left(\varepsilon^{2a-1}\|\partial_{\theta}\chi_{1}\|_{L^{\infty}}^{2}+\varepsilon^{1/3}\|\Delta
T_{0}\|_{H^{(0,\gamma)}}^{2}\right).$ (1.26)

Note that the “integrable” case corresponds to taking $\chi_{1}=0$ and it
follows that in this case

$\lim_{\varepsilon\to 0}\mu(\mathcal{N}(\varepsilon))=0.$ (1.27)

Since now $\chi=\chi_{0}$ is integrable, this agrees with the fact that the
effective volume of non-integrability is zero. If $\chi_{1}$ is nonzero, we
get the same result with $a>1/2$ but if $a=1/2$ we instead have

$\lim_{\varepsilon\to 0}\mu(\mathcal{N}(\varepsilon))\leq
C\|\partial_{\theta}\chi_{1}\|_{L^{\infty}}^{2}.$ (1.28)

This exhibits a relationship between the volume of the set (1.25) and the non-
integrability of the Hamiltonian $\chi$, captured by the $\theta$-dependence
of the perturbation $\chi_{1}$. Our proof of the Proposition (see Section 5)
partially confirms a conjecture announced in [6].

##  2\. Proof of Theorem 1.1

The limiting profile $\Theta(\psi)$ is a function of the flux function $\psi$
only, and is determined from the following heuristic. If $B$ is fibered,
expanding $T_{\varepsilon}=T_{0}+\varepsilon T_{1}$ leads to

$\displaystyle\operatorname{div}(b\nabla_{b}T_{0})$ $\displaystyle=0,$ (2.1)
$\displaystyle\operatorname{div}(\nabla_{b}^{\perp}T_{0})$
$\displaystyle=-\operatorname{div}(b\nabla_{b}T_{1}).$ (2.2)

Equation (2.1) is underdetermined for $T_{0}$; indeed noting that
$b\cdot\nabla\psi=0$ one sees that any function

$T_{0}=\Theta(\psi)$ (2.3)

will satisfy (2.1). This arbitrariness of $\Theta$ may be eliminated by
considering the second condition. Indeed, note that since $b$ is tangent to
each surface $S_{\psi}$, we have

$\frac{\operatorname{div}(b\nabla_{b}T_{1})}{|\nabla\psi|}\Big{|}_{S_{\psi}}=\operatorname{div}_{S_{\psi}}\left(\left[\frac{b\nabla_{b}T_{1}}{|\nabla\psi|}\right]\Big{|}_{S_{\psi}}\right)$
(2.4)

where $\operatorname{div}_{S_{\psi}}$ denotes the divergence operator on
$S_{\psi}$. See Lemma A.2. In light of this, equation (2.2) comes with the
following compatibility condition: on each invariant torus $S_{\psi}$,

$\int_{S_{\psi}}\frac{\Delta T_{0}}{|\nabla\psi|}{\rm
d}\mathscr{H}^{(d-1)}=0,$ (2.5)

where ${\rm d}\mathscr{H}^{(d-1)}$ is the $(d-1)$ dimensional Hausdorff
measure on $S_{\psi}$. Here we used that $\nabla_{b}^{\perp}T_{0}=\nabla
T_{0}$ since $b\cdot\nabla T_{0}=0$. Because $T_{0}$ is constant on
$S_{\psi}$, it follows from Lemma (A.1) (see (A.5)) that the solvability
requirement (2.5) and the boundary conditions of (1.4) are satisfied if
$\Theta$ is the unique solution to (1.9). From (1.9), we deduce that the
effective temperature distribution is give explicitly by

$\Theta(\psi)=T_{-}+(T_{+}-T_{-})\frac{H(\psi;\
\psi_{-})}{H(\psi_{+};\psi_{-})},\qquad\text{where}\qquad
H(\psi;\psi_{-}):=\int_{\psi_{-}}^{\psi}\frac{{\rm d}s}{\Gamma(s)}.$ (2.6)

Note that in two-dimensions, $\Gamma(\psi)$ is simply the circulation of the
vector field $B$ on the circle $S_{\psi}$:

$\Gamma(\psi)=\int_{S_{\psi}}|\nabla\psi|\,{\rm
d}\mathscr{H}=\int_{S_{\psi}}B\cdot{\rm d}\ell\qquad\text{in two dimensions}.$
(2.7)

Observing (2.5), we start with the following simple lemma. We first introduce
the homogenous fractional Sobolev seminorms on $S_{\psi}$. For each $S_{\psi}$
we pick coordinates $\theta_{1},...,\theta_{d-1}$ on $S_{\psi}$ such that
$\theta_{j}$ maps $S_{\psi}$ to $[0,2\pi]$. For $k\in\mathbb{Z}^{d-1}$, we
define

$\widehat{u}(k)=\frac{1}{(2\pi)^{d-1}}\int_{[0,2\pi]^{d-1}}\prod_{j=1}^{d-1}e^{ik_{j}\theta_{j}}u(\theta_{1},\cdots,\theta_{d-1}){\rm
d}\theta_{1}\cdots{\rm d}\theta_{d-1},$ (2.8)

and then for $\gamma\in\mathbb{R}$, we define $\|\cdot\|_{\dot{H}^{\gamma}}$
by

$\|u\|_{\dot{H}^{\gamma}(S_{\psi})}^{2}=\sum_{k\in\mathbb{Z}^{d-1}\setminus
0}|k|^{2\gamma}|\widehat{u}(k)|^{2}.$ (2.9)

Recall also that (1.20) defines the anisotropic spaces $H^{(0,\gamma)}$
tailored to the tori. The result is then

######  Lemma 2.1.

Suppose that $F\in H^{(0,\gamma)}(D)\cap L^{\infty}(D)$ for some $\gamma\geq
0$ and that $F$ satisfies

$\int_{S_{\psi}}\frac{F}{|\nabla\psi|}\,{\rm d}\mathscr{H}^{(d-1)}=0,$ (2.10)

for all $\psi\in[\psi_{-},\psi_{+}]$. There is a constant $C$ depending only
on $D$ so that for any $M>0$ we have

$\left|\int_{D}Fu\,{\rm d}\mu\right|\leq
CM\|F\|_{H^{(0,\gamma)}}\|\nabla_{b}u\|_{L^{2}}+C\mu(N(\gamma,M))^{1/2}\|F\|_{L^{\infty}}\|u\|_{L^{2}}.$
(2.11)

###### Proof.

By the co-area formula (A.1), we have

$\int_{D}Fu\,{\rm
d}\mu=\int_{\psi_{-}}^{\psi_{+}}\int_{S_{\psi}}\frac{F}{|\nabla\psi|}u\,{\rm
d}\mathscr{H}^{(d-1)}{\rm d}\psi.$ (2.12)

Now, for each $\psi$, we write

$\int_{S_{\psi}}\frac{F}{|\nabla\psi|}u\,{\rm
d}\mathscr{H}^{(d-1)}=\int_{[0,2\pi]^{d-1}}G(\psi,\theta_{1},...,\theta_{d-1})u(\psi,\theta_{1},...,\theta_{d-1}){\rm
d}\theta_{1}\cdots{\rm d}\theta_{d-1},$ (2.13)

where $G=\frac{F}{|\nabla\psi|}|h|^{1/2}$, where we are writing the metric on
$S_{\psi}$ as $h=h_{\alpha\beta}{\rm d}\theta^{\alpha}{\rm d}\theta^{\beta}$
and $|h|=\det h_{\alpha\beta}$. By Parseval’s theorem,

$\int_{[0,2\pi]^{d-1}}G(\psi,\theta_{1},...,\theta_{d-1})u(\psi,\theta_{1},...,\theta_{d-1}){\rm
d}\theta_{1}\cdots{\rm
d}\theta_{d-1}=(2\pi)^{d-1}\sum_{k\in\mathbb{Z}^{d-1}}\widehat{G}(\psi,k)\widehat{u}(\psi,-k),$
(2.14)

where $\widehat{F},\widehat{u}$ are defined as in (2.8). Now we note that

$\widehat{G}(\psi,0)=\int_{[0,2\pi]^{d-1}}\frac{F}{|\nabla\psi|}|h|^{1/2}{\rm
d}\theta_{1}\cdots{\rm
d}\theta_{d-1}=\int_{S_{\psi}}\frac{F}{|\nabla\psi|}\,{\rm
d}\mathscr{H}^{(d-1)}=0,$ (2.15)

by assumption. We therefore have

$\int_{S_{\psi}}\frac{F}{|\nabla\psi|}u\,{\rm
d}\mathscr{H}^{(d-1)}=(2\pi)^{d-1}\sum_{k\in\mathbb{Z}^{d-1}\setminus
0}\widehat{G}(\psi,k)\widehat{u}(\psi,-k),$ (2.16)

and it follows that

$\left|\int_{D}Fu\,{\rm d}\mu\right|\leq
C\int_{\psi_{-}}^{\psi_{+}}\sum_{|k|\not=0}|\widehat{G}(\psi,k)\widehat{u}(\psi,-k)|\,{\rm
d}\psi.$ (2.17)

Now we split $[\psi_{-},\psi_{+}]=E(\gamma,M)\cup N(\gamma,M)$ and bound

$\int_{\psi\in
E(\gamma,M)}\sum_{|k|\not=0}|\widehat{G}(\psi,k)\widehat{u}(\psi,-k)|\,\,{\rm
d}\psi\\\ \leq\int_{\psi\in
E(\gamma,M)}\|G\|_{H^{\gamma}(S_{\psi})}\|u\|_{\dot{H}^{-\gamma}(S_{\psi})}\,{\rm
d}\psi\leq M\int_{\psi\in
E(\gamma,M)}\|G\|_{H^{\gamma}(S_{\psi})}\|\nabla_{B}u\|_{L^{2}(S_{\psi})}\,{\rm
d}\psi\\\ \leq M\|G\|_{H^{(0,\gamma)}(D)}\|\nabla_{B}u\|_{L^{2}(D)},$ (2.18)

by the definition of $E(\gamma,M)$ and

$\int_{\psi\in
N(\gamma,M)}\sum_{|k|\not=0}|\widehat{G}(\psi,k)\widehat{u}(\psi,-k)|\,\,{\rm
d}\psi\leq\|G\|_{L^{2}(N(\gamma,M))}\|u\|_{L^{2}(D)}\\\
\leq\mu(N(\gamma,M))^{1/2}\|G\|_{L^{\infty}(D)}\|u\|_{L^{2}(D)}.$ (2.19)

This gives the result. ∎

Our main result, Theorem 1.1, will be a direct result of the following
estimate.

######  Proposition 2.1.

Define $T_{0}=\Theta(\psi)$ where $\Theta$ is given by (2.6) and let
$\rho=T_{\varepsilon}-T_{0}$. Under the hypotheses of Theorem 1.1, there is a
constant $C$ depending only on the domain $D$ so that for each $\varepsilon>0$
and $M>0$, we have

$\|\nabla_{b}\rho\|_{L^{2}}^{2}+\varepsilon\|\nabla_{b}^{\perp}\rho\|_{L^{2}}^{2}+\varepsilon\|\rho\|_{L^{2}}^{2}\leq
C\left(M^{2}\varepsilon^{2}+\varepsilon\mu(N(\gamma,M))\right)\left(\|\Delta
T_{0}\|_{H^{(0,\gamma)}}^{2}+\|\Delta T_{0}\|_{L^{\infty}}^{2}\right)$ (2.20)

###### Proof.

The remainder $\rho$ satisfies

$\displaystyle\operatorname{div}(b\nabla_{b}\rho)+\varepsilon\operatorname{div}(\nabla_{b}^{\perp}\rho)$
$\displaystyle=-\varepsilon\Delta T_{0},\qquad\text{ in }D,$ (2.21)
$\displaystyle\rho|_{S_{\pm}}$ $\displaystyle=0,$ (2.22)

where we wrote $\Delta T_{0}=\operatorname{div}(\nabla_{b}^{\perp}T_{0})$
since $b\cdot\nabla T_{0}=0$. If we multiply (2.21) by $\rho$ and integrate
over $D$, using the co-area formula (A.1) we find

$\int_{D}\left(|\nabla_{b}\rho|^{2}+\varepsilon|\nabla_{b}^{\perp}\rho|^{2}\,\right){\rm
d}\mu=\varepsilon\int_{D}\Delta T_{0}\rho\,{\rm
d}\mu=\varepsilon\int_{\psi_{-}}^{\psi_{+}}\int_{S_{\psi}}\frac{\Delta
T_{0}}{|\nabla\psi|}\rho\,{\rm d}\mathscr{H}^{(d-1)}{\rm d}\psi.$ (2.23)

Recall that we have defined $T_{0}$ so that $F=\Delta T_{0}$ satisfies the
condition (2.10). We can therefore apply Lemma 2.1, and by (2.11) we have

$\displaystyle\varepsilon\bigg{|}\int_{\psi_{0}}^{\psi_{1}}\int_{S_{\psi}}\frac{\Delta
T_{0}}{|\nabla\psi|}\rho\,$ $\displaystyle{\rm d}\mathscr{H}^{(d-1)}{\rm
d}\psi\bigg{|}$ $\displaystyle\leq M\varepsilon\|\Delta
T_{0}\|_{H^{(0,\gamma)}}\|\nabla_{b}\rho\|_{L^{2}}+\varepsilon\mu(N(\gamma,M))^{1/2}\|\Delta
T_{0}\|_{L^{\infty}}\|\rho\|_{L^{2}}$
$\displaystyle\leq\frac{1}{2}\left(M^{2}\varepsilon^{2}\|\Delta
T_{0}\|_{H^{(0,\gamma)}}^{2}+\frac{1}{\delta}\varepsilon\mu(N(\gamma,M))\|\Delta
T_{0}\|_{L^{\infty}}^{2}\right)+\frac{1}{2}\|\nabla_{b}\rho\|_{L^{2}}^{2}+\frac{\delta}{2}\varepsilon\|\rho\|_{L^{2}}^{2}$
(2.24)

for any $\delta>0$. Since $\rho|_{\partial D}=0$, by Poincaré’s inequality we
have

$\|\rho\|_{L^{2}}^{2}\leq C_{P}\|\nabla\rho\|_{L^{2}}^{2},$ (2.25)

where $C_{P}$ is the Poincaré constant for $D$. Taking $\delta$ so that
$\delta C_{P}$ is sufficiently small, we see from (2.23) and (2.24) that there
is a constant $C>0$ depending only on $C_{P}$ so that

$\|\nabla_{b}\rho\|_{L^{2}}^{2}+\varepsilon\|\nabla_{b}^{\perp}\rho\|_{L^{2}}^{2}+\varepsilon\|\rho\|_{L^{2}}^{2}\leq
C\left(M^{2}\varepsilon^{2}+\varepsilon\mu(N(\gamma,M))\right)\left(\|\Delta
T_{0}\|_{H^{(0,\gamma)}}^{2}+\|\Delta T_{0}\|_{L^{\infty}}^{2}\right),$ (2.26)

after using (2.25) again to bound $\varepsilon\|\rho\|_{L^{2}}^{2}\leq
C^{\prime}\|\nabla_{b}\rho\|_{L^{2}}^{2}+C^{\prime}\varepsilon\|\nabla_{b}^{\perp}\rho\|_{L^{2}}^{2}$
for another constant $C^{\prime}$.

∎

###### Proof of Theorem 1.1.

If we take $M=\varepsilon^{-\frac{1}{2+c}}$, then writing
$N_{\varepsilon}=N(\gamma,\varepsilon^{-\frac{1}{2+c}})$, (2.20) gives

$\displaystyle\|\nabla_{b}^{\perp}\rho\|_{L^{2}}+\|\rho\|_{L^{2}}^{2}$
$\displaystyle\leq
C\varepsilon^{\frac{c}{2+c}}\left(1+\varepsilon^{-\frac{c}{2+c}}\mu(N_{\varepsilon})\right)\left(\|\Delta
T_{0}\|_{H^{(0,\gamma)}}^{2}+\|\Delta T_{0}\|_{L^{\infty}}^{2}\right),$ (2.27)
$\displaystyle\|\nabla_{b}\rho\|_{L^{2}}$ $\displaystyle\leq
C\varepsilon^{1+\frac{c}{2+c}}\left(1+\varepsilon^{-\frac{c}{2+c}}\mu(N_{\varepsilon})\right)\left(\|\Delta
T_{0}\|_{H^{(0,\gamma)}}^{2}+\|\Delta T_{0}\|_{L^{\infty}}^{2}\right).$ (2.28)

By assumption, $\lim_{\varepsilon\to
0}\varepsilon^{-\frac{c}{2+c}}\mu(N_{\varepsilon})=0$ and the result follows.
∎

##  3\. Proof of Corollary 1.1: The 2d case

If $|B|>0$ in $D$, $B$ is fibered by its streamfunction $\psi$,
$B=\nabla^{\perp}\psi$. We bound

$\|u\|_{\dot{H}^{0}(S_{\psi})}^{2}=\sum_{k\in\mathbb{Z}\setminus
0}|\widehat{u}(k)|^{2}\leq\sum_{k\in\mathbb{Z}}|k|^{2}|\widehat{u}(k)|^{2}\leq\frac{C}{\inf_{S_{\psi}}|B|^{2}}\|\nabla_{B}u\|_{L^{2}(S_{\psi})}^{2}$
(3.1)

for a constant $C>0$. Here we have used that $B$ spans the tangent space to
$S_{\psi}$ at each point. Therefore, $E(0,M)=D$ whenever
$M\geq\frac{C}{\inf_{D}|B|}$, and so $N(0,M)$ is empty in this case. Thus
(1.7) holds for any $c\geq 0$ and the result follows.

##  4\. Proof of Corollary 1.2: The 3d integrable case

We first show that if $\gamma>2$, under the hypotheses of Corollary 1.2, the
condition (1.7) holds with $c=1$. We start by relating this condition to the
“Diophantine” condition.

Let $I=[\psi_{-},\psi_{+}]$. Fix $\iota=\iota(\psi)$ with $\iota\in
L^{\infty}(I)$. Let $|(m,n)|=\sqrt{m^{2}+n^{2}}$. We define

$D(\gamma,M)=\left\\{\psi\in
I:|m+\iota(\psi)n|\geq\frac{1}{M|(m,n)|^{\gamma}}\quad\text{for
all}\quad(m,n)\in\mathbb{Z}^{2}\setminus\\{0\\}\right\\}.$ (4.1)

If $\psi\in D(\gamma,M)$ for some $\gamma,M$, we will say that $\psi$ is
“Diophantine” and that the surface $S_{\psi}$ is a “Diophantine surface”. With
$m+\iota(\psi)n$ replaced by $\omega\cdot(m,n)$ for $\omega\in\mathbb{R}^{2}$,
these sets play a fundamental role in the proof of the celebrated KAM theorem
[19].

At least when $\iota(\psi)=\psi$, these sets are empty if $\gamma<2$ (see
[18]), but it turns out that if $\iota$ is bi-Lipschitz and $\gamma>2$ they
have positive measure; in fact the complement of $\cup_{M>0}D(\gamma,M)$ has
zero measure, as the next result shows.

######  Lemma 4.1.

Let $\iota:I\to\mathbb{R}$ be an invertible function and suppose there is
$L>0$ so that

$\frac{1}{L}|\psi_{1}-\psi_{2}|\leq|\iota(\psi_{1})-\iota(\psi_{2})|\leq
L|\psi_{1}-\psi_{1}|.$ (4.2)

Define $D(\gamma,M)$ as in (4.1) and let $\mu$ denote the one-dimensional
Lebesgue measure. If $\gamma>2$ and $M>0$, there is a constant $K$ depending
only on $\mu(I)$, $L$, and $1/(\gamma-2)$ so that

$\mu(I\setminus D(\gamma,M))\leq
K\left(\frac{1}{M^{\frac{1}{1+\gamma}}}+\frac{1}{M}\right).$ (4.3)

Taking $M\to\infty$ in (4.3) shows that the set of $\psi$ which fails the
condition (4.1) for _all_ $M>0$ has zero measure. Equivalently, the set of
$\psi$ the condition in (4.1) for _some_ $M$ has full measure, though in
general the complement may be nonempty.

###### Proof.

This result follows from a straightforward modification of the argument from
e.g. [18]. We include the details here for the convenience of the reader. For
$(m,n)\in\mathbb{Z}^{2}\setminus(0,0)$, we define

$\Pi_{(m,n)}(\gamma,M)=\left\\{\psi\in
I:|m+\iota(\psi)n|<\frac{1}{M|(m,n)|^{\gamma}}\right\\},$ (4.4)

so that

$I\setminus
D(\gamma,M)\subseteq\bigcup_{(m,n)\not=(0,0)}\Pi_{(m,n)}(\gamma,M).$ (4.5)

If $n\not=0$, $\Pi_{(m,n)}(\gamma,M)$ is contained in the interval
$[\psi_{1},\psi_{2}]\subset I$ where $\psi_{1}$ are such that
$\iota(\psi_{1})=\frac{M}{n}\frac{1}{|(m,n)|^{\gamma}}+\frac{m}{n}$ and
$\iota(\psi_{2})=-\iota(\psi_{1})$ (such $\psi_{1},\psi_{2}$ exist and are
unique since by (4.2) $\iota$ is invertible). These are the maximal and
minimal values of $\psi$ such that (4.4) holds for a given $(m,n)$. Therefore

$\mu(\Pi_{(m,n)}(\gamma,M))\leq|\psi_{1}-\psi_{2}|\leq\frac{L}{2M|n||(m,n)|^{\gamma}}.$
(4.6)

If $n=0$, $\Pi_{(m,0)}(\gamma,M)=I$ when $|m|<M^{-1/(1+\gamma)}$ and it is
empty otherwise. Therefore

$\sum_{(m,n)\not=(0,0)}\mu(\Pi_{(m,n)}(\gamma,M))\leq\sum_{m\not=0}\mu(\Pi_{(m,0)}(\gamma,M))+\sum_{(m,n)\not=(0,0),n\not=0}\mu(\Pi_{(m,n)}(\gamma,M))\\\
\leq|I|\frac{1}{M^{1/(1+\gamma)}}+\frac{L}{M}+\frac{L}{2M}\sum_{|m|,|n|\geq
1}\frac{1}{|n||(m,n)|^{\gamma}},$ (4.7)

since there are two terms in the second sum on the first line with $m=0$. The
last term here is bounded by

$2\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{n}\frac{1}{(m^{2}+n^{2})^{\gamma/2}}\leq
2\sum_{m=1}^{\infty}\int_{1}^{\infty}\frac{{\rm
d}z}{z(m^{2}+z^{2})^{\gamma/2}}=2\sum_{m=1}^{\infty}\frac{1}{m^{\gamma}}\int_{1/m}^{\infty}\frac{{\rm
d}w}{w(1+w^{2})^{\gamma/2}}\\\ \leq
2\sum_{m=1}^{\infty}\frac{1}{m^{\gamma-1}}\int_{0}^{\infty}\frac{{\rm
d}w}{(1+w^{2})^{\gamma/2}}$ (4.8)

which is finite if $\gamma>2$. In this case we therefore have

$\mu\left(I\setminus
D(\gamma,M)\right)\leq\mu\left({\bigcup}_{(m,n)\not=(0,0)}\Pi_{m,n}(\gamma,M)\right)\leq
C_{1}\left(\frac{1}{M^{1/(1+\gamma)}}+\frac{1}{M}\right),$ (4.9)

for a constant $C_{1}$ depending only on $|I|$, $L$ and $1/(\gamma-2)$, which
proves (1.26). ∎

The values of $\psi\in D(\gamma,M)$ are sometimes called “strongly non-
resonant”, and the surfaces $S_{\psi}$ with $\psi\in D(\gamma,M)$ are called
“non-resonant flux surfaces”.

###### Proof of Corollary 1.2.

We first consider a field of the form (1.13) when $\chi=\chi(\psi)$ and where
$\theta,\phi$ form a coordinate system on each $S_{\psi}$. With
$\iota(\psi)=\chi^{\prime}(\psi)$, in this setting we have

$(B\cdot\nabla)u=J\left[\partial_{\phi}+\iota(\psi)\partial_{\theta}\right]u,J=\nabla\psi\times\nabla\theta\cdot\nabla\phi$
(4.10)

where $J\not=0$ by assumption. We claim that the Diophantine surfaces are
ergodic, in the sense that there is a constant $C$ with

$D(\gamma,CM)\subseteq E(\gamma,M)$ (4.11)

for any $M>0$. It follows from this claim that $N(\gamma,M)\subseteq
I\setminus D(\gamma,CM)$ and so the condition (1.7) holds for any $c<1$, since
(4.3) then implies that

$\mu(N(\gamma,M))\leq\frac{C}{M},\qquad M\geq 1.$ (4.12)

We now prove (4.11). Whenever $\psi\in D(\gamma,M)$, for any smooth function
$u:S_{\psi}\to\mathbb{R}$, we have

$\|u\|_{\dot{H}^{-\gamma}(S_{\psi})}^{2}=\sum_{(m,n)\in\mathbb{Z}^{2}\setminus
0}\frac{|\widehat{u}(m,n)|^{2}}{\left(m^{2}+n^{2}\right)^{\gamma}}\leq
M^{2}\sum_{(m,n)\in\mathbb{Z}^{2}\setminus
0}|m+\iota(\psi)n|^{2}|\widehat{u}(m,n)|^{2}.$ (4.13)

Now we note that by (4.10),

$\widehat{\left(\nabla_{B/J}u\right)}(\psi,m,n)=-2\pi
i(m+\iota(\psi)n)\widehat{u}(\psi,m,n).$ (4.14)

It follows that if $\psi\in D(\gamma,M)$, there are constants $C_{1},C_{2}$ so
that

$\|u\|_{\dot{H}^{-\gamma}(S_{\psi})}^{2}\leq
C_{1}M^{2}\|\nabla_{B/J}u\|_{L^{2}(S_{\psi})}^{2}\leq
C_{2}M^{2}\|\nabla_{b}u\|_{L^{2}(S_{\psi})}^{2},$ (4.15)

and this gives (4.11).

We now show how to get the same result for any non-vanishing divergence-free
fibered field provided the rotational transform satisfies the bound (1.17). We
first show that the rotational transform is well-defined in this setting; that
is, that we can find coordinates so that $B$ takes the form (1.16).

In light of (A.6), since $\operatorname{div}B=0$, it follows that
$\frac{1}{|\nabla\psi|}$ is an integral invariant of $B|_{S_{\psi}}$ (namely
$U=\frac{1}{|\nabla\psi|}$ is a conserved density along the flow of
$B|_{S_{\psi}}$ on $S_{\psi}$) and so by Sternberg’s theorem (Theorem 1 of
[21]), $B$ is orbitally conjugate to a linear operator on $S_{\psi}$. That is,
there are coordinates $(\theta,\phi)$ on $S_{\psi}$ so that in these
coordinates, there is a nonvanishing function $J=J(\theta,\phi)$ so that on
$S_{\psi}$, $B$ takes the form

$(B\cdot\nabla^{T})u=J\left(\partial_{\phi}+\iota\partial_{\theta}\right)u,\qquad
u\in C^{\infty}(S_{\psi})$ (4.16)

for a real number $\iota$ (compare with (1.14)), where $\nabla^{T}$ denotes
the tangential gradient on $S_{\psi}$, given by
$\nabla^{T}u=(\nabla-\frac{\nabla\psi}{|\nabla\psi|^{2}}\nabla\psi\cdot\nabla)u$
when $u$ is a function defined in a neighborhood of $S_{\psi}$. Applying this
theorem on each $S_{\psi}$ then gives $\iota=\iota(\psi)$.

The above proof goes through with a minor change, which is that we want to
replace the fractional Sobolev norm $\|u\|_{\dot{H}^{\gamma}}$, which was
defined relative to a fixed coordinate system (because the Fourier
coefficients $\widehat{u}(k)$ depend on the choice of coordinates in (2.8)),
with a fractional Sobolev norm $\|u\|_{\dot{\widetilde{H}}^{\gamma}}$ defined
relative to the coordinates guaranteed by Sternberg’s theorem. This means we
want to modify the definition of $E(\gamma,M)$ and define

$\widetilde{E}(\gamma,M)=\left\\{\psi\in
I:\|u\|_{\dot{\widetilde{H}}^{-\gamma}(S_{\psi})}\leq
M\|\nabla_{B}u\|_{L^{2}(S_{\psi})},\text{ for all }u\in
H^{1}(S_{\psi})\right\\}.$ (4.17)

It is clear that the proof of Theorem 1.1 goes through without change if we
replace the sets $E(\gamma,M)$ with $\widetilde{E}(\gamma,M)$. It is also
clear from the formula (4.16) and the above argument that there is a constant
$C>0$ so that (4.11) holds with $E$ replaced with $\widetilde{E}$, if the
rotational transform $\iota=\iota(\psi)$ from (4.16) satisfies (1.17).

∎

##  5\. Proof of Theorem 1.2, Corollary 1.3 and Proposition 1.1: the non-
integrable case

We now consider the non-integrable case $\chi_{1}\not=0$ of (1.19). With $B$
as in (1.13)-(1.19), we set

$B_{0}=\nabla\psi\times\nabla\theta+\iota(\psi)\nabla\phi\times\nabla\psi,\qquad
B_{1}=\nabla\phi\times\nabla\chi_{1}$ (5.1)

where $\iota(\psi)=\chi_{0}^{\prime}(\psi)$. We then write $b=B/|B|$ in the
form

$b=b_{0}+\varepsilon^{a}b_{1},\qquad b_{0}=\frac{B_{0}}{|B|},\quad
b_{1}=\frac{B_{1}}{|B|}.$ (5.2)

###  5.1. Proof of Theorem 1.2.

We start by recording a simple estimate.

######  Lemma 5.1.

If $\|\nabla\chi_{1}\|_{L^{\infty}}<1$ and $a\geq 1/2$, there is a constant
$C>0$ so that for any function $u\in H^{1}(D)$, we have

$\frac{1}{C}\left(\|\nabla_{b}u\|_{L^{2}}^{2}+\varepsilon\|\nabla_{b}^{\perp}u\|_{L^{2}}^{2}\right)\leq\|\nabla_{b_{0}}u\|_{L^{2}}^{2}+\varepsilon\|\nabla_{b_{0}}^{\perp}u\|_{L^{2}}^{2}\leq
C\left(\|\nabla_{b}u\|_{L^{2}}^{2}+\varepsilon\|\nabla_{b}^{\perp}u\|_{L^{2}}^{2}\right).$
(5.3)

###### Proof.

This follows after writing
$\nabla_{b_{0}}=\nabla_{b}-\varepsilon^{a}\nabla_{b_{1}}$, noting that
$|b_{1}|\leq C|\nabla\chi_{1}|$, and

$|\nabla_{b_{0}}u|\leq|\nabla_{b}u|+\varepsilon^{a}|\nabla\chi_{1}||\nabla
u|,\qquad|\nabla_{b_{0}}^{\perp}u|=|(\nabla-
b_{0}\nabla_{b_{0}})u|\leq|\nabla_{b}^{\perp}u|+\varepsilon^{a}|\nabla\chi_{1}||\nabla
u|$ (5.4)

for smooth $u$. It follows that for $u\in H^{1}(D)$,

$\|\nabla_{b_{0}}u\|_{L^{2}}^{2}+\varepsilon\|\nabla_{b_{0}}^{\perp}u\|_{L^{2}}^{2}\leq\|\nabla_{b}u\|_{L^{2}}^{2}+\varepsilon\|\nabla_{b}^{\perp}u\|_{L^{2}}^{2}+\varepsilon^{2a}\|\nabla\chi_{1}\|_{L^{\infty}}^{2}\|\nabla
u\|_{L^{2}}^{2}.$ (5.5)

Provided $2a\geq 1$, this gives

$(1-\varepsilon\|\nabla\chi_{1}\|_{L^{\infty}})\|\nabla_{b_{0}}u\|_{L^{2}}^{2}+\varepsilon(1-\|\nabla\chi_{1}\|_{L^{\infty}}^{2})\|\nabla_{b_{0}}^{\perp}u\|_{L^{2}}^{2}\leq\|\nabla_{b}u\|_{L^{2}}^{2}+\varepsilon\|\nabla_{b}^{\perp}u\|_{L^{2}}^{2},$
(5.6)

which is the second bound in (5.3). The first bound is nearly identical. ∎

###### Proof of Corollary 1.2.

Since $\nabla_{b_{0}}T_{0}=0$ we have

$\operatorname{div}(b\nabla_{b}T_{0})+\varepsilon\operatorname{div}(\nabla_{b}^{\perp}T_{0})=\varepsilon^{a}\operatorname{div}(b\nabla_{b_{1}}T_{0})+\varepsilon\Delta
T_{0}-\varepsilon\operatorname{div}(b\nabla_{b}T_{0})\\\
=(\varepsilon^{a}-\varepsilon^{1+a})\operatorname{div}(b\nabla_{b_{1}}T_{0})+\varepsilon\operatorname{div}(\nabla
T_{0}).$ (5.7)

Then $\rho=T_{\varepsilon}-T_{0}$ satisfies

$\displaystyle\operatorname{div}(b\nabla_{b}\rho)+\varepsilon\operatorname{div}(\nabla_{b}^{\perp}\rho)$
$\displaystyle=(\varepsilon^{1+a}-\varepsilon^{a})\operatorname{div}(b\nabla_{b_{1}}T_{0})-\varepsilon\operatorname{div}(\nabla
T_{0})\qquad\text{ in }D,$ (5.8) $\displaystyle\rho|_{S_{\pm}}$
$\displaystyle=0.$ (5.9)

Since $\partial_{\theta}\chi_{1}|_{\partial D}=0$ by assumption, $b$ is
tangent to $\partial D$ and so if we multiply this by $\rho$ and integrate
over $D$, we find

$\int_{D}|\nabla_{b}\rho|^{2}+\varepsilon|\nabla_{b}^{\perp}\rho|^{2}\,{\rm
d}\mu=\varepsilon\int_{D}\Delta T_{0}\rho\,{\rm
d}\mu+(\varepsilon^{a}-\varepsilon^{1+a})\int_{D}\nabla_{b_{1}}T_{0}\nabla_{b}\rho\,{\rm
d}\mu,$ (5.10)

after integrating by parts in the second term on the right-hand side. Since
$|\nabla_{b_{1}}T_{0}|\leq|\partial_{\theta}\chi_{1}||T_{0}^{\prime}|$, for
any $\delta>0$ we have

$\|\nabla_{b}\rho\|_{L^{2}}^{2}+\varepsilon\|\nabla_{b}^{\perp}\rho\|_{L^{2}}^{2}\leq\frac{\varepsilon^{2a}}{2\delta}\|\partial_{\theta}\chi_{1}\|_{L^{2}}^{2}\|T_{0}^{\prime}\|_{L^{\infty}}^{2}+\frac{\delta}{2}\|\nabla_{b}\rho\|_{L^{2}}^{2}+\varepsilon\left|\int_{\psi_{-}}^{\psi_{+}}\int_{S_{\psi}}\frac{\operatorname{div}\nabla
T_{0}}{|\nabla\psi|}\rho\,{\rm d}\mathscr{H}^{(d-1)}{\rm d}\psi\right|.$
(5.11)

Arguing as in (2.24)-(2.26) and using (5.3), this implies that there is a
constant $C>0$ so that for any $M>0$,

$\|\nabla_{b_{0}}\rho\|_{L^{2}}^{2}+\varepsilon\|\nabla_{b_{0}}^{\perp}\rho\|_{L^{2}}^{2}+\varepsilon\|\rho\|_{L^{2}}^{2}\\\
\leq
C\varepsilon^{2a}\|\partial_{\theta}\chi_{1}\|_{L^{2}}^{2}\|T_{0}^{\prime}\|_{L^{2}}^{2}+C\left(M^{2}\varepsilon^{2}+\varepsilon\mu(N(\gamma,M))\right)\left(\|\Delta
T_{0}\|_{H^{(0,\gamma)}}^{2}+\|\Delta T_{0}\|_{L^{\infty}}^{2}\right).$ (5.12)

If we take $M=\varepsilon^{-1/3}$ and use the estimate (4.12) for
$\mu(N(\gamma,M))$, we find

$\displaystyle\|\nabla_{b_{0}}^{\perp}\rho\|_{L^{2}}+\|\rho\|_{L^{2}}^{2}$
$\displaystyle\leq
C\varepsilon^{2a-1}\|\partial_{\theta}\chi_{1}\|_{L^{2}}^{2}+C\varepsilon^{\frac{1}{3}}\left(\|\Delta
T_{0}\|_{H^{(0,\gamma)}}^{2}+\|\Delta T_{0}\|_{L^{\infty}}^{2}\right),$ (5.13)
$\displaystyle\|\nabla_{b_{0}}\rho\|_{L^{2}}$ $\displaystyle\leq
C\varepsilon^{2a}\|\partial_{\theta}\chi_{1}\|_{L^{2}}^{2}+C\varepsilon^{\frac{4}{3}}\left(\|\Delta
T_{0}\|_{H^{(0,\gamma)}}^{2}+\|\Delta T_{0}\|_{L^{\infty}}^{2}\right).$ (5.14)

∎

###  5.2. Proof of Corollary 1.3.

Before proving Corollary 1.3, we collect some preliminary results. First, from
the uniform bounds (1.21) and (1.22), it follows that that the sequence
$T_{\varepsilon}-T_{0}$ has weak limit $\rho^{*}$ in $H^{1}$ and that
$\nabla_{b}(T_{\varepsilon}-T_{0})$ converges strongly to $0$ in $L^{2}$.

If we knew that $\nabla_{B}\rho^{*}$ was smooth, it would follow that
$\nabla_{B}\rho^{*}=0$ everywhere. We however only know that
$\nabla_{B}\rho^{*}$ is in $L^{2}$ and in particular the restriction
$\nabla_{B}\rho^{*}|_{S_{\psi}}$ need not be defined. The following result
shows that $\nabla_{B}\rho^{*}|_{S_{\psi}}=0$ in a weak sense.

######  Lemma 5.2.

Let $\rho^{*}=\lim_{\varepsilon\to 0}T_{\varepsilon}-T_{0}$. For any $\psi$
and any $v\in C^{2}(S_{\psi})$, with $B^{\prime}=B|\nabla\psi|^{-1}$,

$\int_{S_{\psi}}\rho^{*}\operatorname{div}_{S_{\psi}}(B^{\prime}v)\,{\rm
d}\mathscr{H}^{(d-1)}=0.$ (5.15)

######  Remark.

The statement of (5.15) holds with $B^{\prime}$ replaced by $B$ (or indeed any
multiple of $B$) but it is more convenient for our purposes to use
$B^{\prime}$ since
$\operatorname{div}_{S_{\psi}}(B^{\prime}v)=|\nabla\psi|\operatorname{div}(Bv)=|\nabla\psi|\nabla_{B}v$,
by (A.6) and the fact that $\operatorname{div}B=0$. See Lemma 5.3.

###### Proof.

The idea is to use Stokes’ theorem to write (5.15) in terms of an integral in
the interior of $D$ and to integrate by parts in the interior to exploit the
fact that $\nabla_{B}\rho^{*}=0$ almost everywhere.

Fix a surface $S_{\psi^{\prime}}$. Define a cutoff function $Q\in
C^{\infty}(\mathbb{R})$ with $Q(z)=1$ when $|z|\leq 1$ and $Q(z)=0$ when
$|z|>2$. Define $Q_{\delta}(\psi)=Q((\psi-\psi^{\prime})/\delta)$, so that
$Q_{\delta}(\psi)$ vanishes when $|\psi-\psi^{\prime}|<2\delta$ and
$Q_{\delta}(\psi)\equiv 1$ when $|\psi-\psi^{\prime}|\leq\delta$. Let
$D_{-}(\psi^{\prime})=D\cap\\{\psi\leq\psi^{\prime}\\}$. Then the boundary of
$D_{-}(\psi^{\prime})$ is $S_{\psi^{\prime}}\cup S_{\psi_{-}}$, and if
$\delta$ is sufficiently small, $Q_{\delta}$ vanishes identically on
$S_{\psi_{-}}$.

Let $V$ denote the constant extension of $v$ to $D_{-}(\psi^{\prime})$,
$V(\psi,\theta,\phi)=v(\psi^{\prime},\theta,\phi)$ for all
$(\psi,\theta,\phi)\in D_{-}(\psi^{\prime})$. By Stokes’ theorem, since the
outer unit normal to $S_{\psi^{\prime}}$ is
$n=\frac{\nabla\psi}{|\nabla\psi|}$, we have

$\int_{S_{\psi^{\prime}}}\rho^{*}\operatorname{div}_{S_{\psi^{\prime}}}(B^{\prime}u)\,{\rm
d}\mathscr{H}^{(d-1)}=\int_{S_{\psi^{\prime}}}(n\cdot
n)\,\rho^{*}\operatorname{div}_{S_{\psi^{\prime}}}(B^{\prime}V)Q_{\delta}\,{\rm
d}\mathscr{H}^{(d-1)}\\\
=\int_{D_{-}}Q_{\delta}\operatorname{div}\left(\rho^{*}\operatorname{div}_{S_{\psi^{\prime}}}(B^{\prime}V)\frac{\nabla\psi}{|\nabla\psi|}\right){\rm
d}\mu+\int_{D_{-}}\rho^{*}\operatorname{div}_{S_{\psi^{\prime}}}(B^{\prime}V)\,\frac{\nabla\psi}{|\nabla\psi|}\cdot(\nabla
Q_{\delta})\,{\rm d}\mu.$ (5.16)

The first term here is bounded by

$\|\operatorname{div}_{S_{\psi^{\prime}}}(B^{\prime}V)\|_{H^{1}(|\psi-\psi^{\prime}|\leq\delta)}\|\rho^{*}\|_{H^{1}(|\psi-\psi^{\prime}|\leq\delta)}\leq
C\delta^{1/2}\|v\|_{C^{2}(S_{\psi^{\prime}})}\|\rho^{*}\|_{H^{1}(D)}.$ (5.17)

As for the second term, we use the identity (A.6) to write

$\operatorname{div}_{S_{\psi}}(B^{\prime}V)=\frac{1}{|\nabla\psi|}\operatorname{div}(BV),$
(5.18)

and since $\frac{\nabla\psi}{|\nabla\psi|}\cdot(\nabla
Q_{\delta})=\frac{1}{\delta}|\nabla\psi|Q_{\delta}^{\prime}$ and $b$ is
tangent to $S_{\psi^{\prime}}$, the second term in (5.16) is

$\frac{1}{\delta}\int_{D_{-}}\rho^{*}\operatorname{div}_{S_{\psi^{\prime}}}(B^{\prime}V)\,|\nabla\psi|Q_{\delta}^{\prime}\,{\rm
d}\mu=\frac{1}{\delta}\int_{D_{-}}\rho^{*}\operatorname{div}(BV)Q_{\delta}^{\prime}\,{\rm
d}\mu=-\frac{1}{\delta}\int_{D_{-}}\nabla_{B}(\rho^{*}Q_{\delta}^{\prime})V\,{\rm
d}\mu,$ (5.19)

after integrating by parts. We have therefore shown that for any $\delta>0$,

$\left|\int_{S_{\psi^{\prime}}}\rho^{*}\operatorname{div}_{S_{\psi}}(B^{\prime}V)\,{\rm
d}\mathscr{H}^{(d-1)}\right|\leq
C\delta^{1/2}\|v\|_{C^{2}(S_{\psi^{\prime}})}\|\rho^{*}\|_{H^{1}(D)}+\frac{1}{\delta}C\left(\|\nabla_{B}\rho^{*}\|_{L^{2}}+\|\nabla_{B}Q_{\delta}^{\prime}\|_{L^{2}}\right)\|V\|_{L^{2}}\\\
=C\delta^{1/2}\|v\|_{C^{2}(S_{\psi^{\prime}})}\|\rho^{*}\|_{H^{1}(D)},$ (5.20)

where we used that $\nabla_{B}u=0$ whenever $u=u(\psi)$ and that
$\nabla_{B}\rho^{*}=0$ in $L^{2}$. Taking $\delta\to 0$ gives the claim. ∎

The condition (5.15) nearly says that $\rho^{*}=0$ on each $S_{\psi}$ in the
sense of distributions, but in order to conclude this we would need to know
that any test function $v$ can be written in the form
$v=\operatorname{div}_{S_{\psi}}(Bw)$ for some test function $w$. This need
not be possible on an arbitrary surface $S_{\psi}$, but by the next Lemma it
is possible provided $S_{\psi}$ is a Diophantine surface. We set

$D(\gamma)=\bigcup_{M>0}D(\gamma,M).$ (5.21)

Note that by (4.11), $D(\gamma)\subseteq E(\gamma)$ where
$E(\gamma)=\cup_{M>0}E(\gamma,M)$ denotes the collection of all ergodic values
of $\psi$. Note also that by Lemma 4.1, the complement $I\setminus D(\gamma)$
has zero measure when $\gamma>2$.

######  Lemma 5.3.

Fix $\gamma>2$ and define $D(\gamma)$ as in (5.21). If $\psi\in D(\gamma)$ and
$v\in H^{s+\gamma}(S_{\psi})$ for some $s\geq 0$, there is $w\in
H^{s}(S_{\psi})$ satisfying

$\operatorname{div}_{S_{\psi}}(B^{\prime}w)=v,\qquad
B^{\prime}=B|\nabla\psi|^{-1}.$ (5.22)

######  Remark.

The equation (5.22) is sometimes known as the “magnetic differential
equation”. Lemma 5.3 simply says that you can solve this equation on good
(sufficiently ergodic) flux surfaces.

###### Proof.

We start by using (A.6) to write

$\operatorname{div}_{S_{\psi}}(B^{\prime}w)=\frac{1}{|\nabla\psi|}\operatorname{div}(Bw)=\frac{1}{|\nabla\psi|}\nabla_{B}w,$
(5.23)

and so (5.22) takes the form

$\nabla_{B/J}w=V,\qquad V=J|\nabla\psi|v,$ (5.24)

where recall $J=|g|^{-1/2}=|(\nabla\psi\times\nabla\theta)\cdot\nabla\phi|$.
We now define

$\widehat{w}(m,n)=\frac{i}{2\pi}\frac{1}{m+\iota(\psi)n}\widehat{V}(m,n).$
(5.25)

Because $\psi\in D(\gamma)$, for some $M>0$, we have the bound

$|\widehat{w}(m,n)|^{2}\leq
M^{2}\left(m^{2}+n^{2}\right)^{\gamma}|\widehat{V}(m,n)|^{2},$ (5.26)

so that in particular $w\in H^{s}(S_{\psi})$ whenever $v=V/(J|\nabla\psi|)\in
H^{s+\gamma}(S_{\psi})$. By the identity (4.14), it follows that $w$ satisfies
(5.22). ∎

###### Proof of Corollary 1.3.

Fix $\gamma>2$ and define $D(\gamma)$ as in (5.21). It follows from Lemma 5.3
that given $v\in C^{\infty}(S_{\psi})$, there is $w\in C^{\infty}(S_{\psi})$
so that with $\operatorname{div}_{S_{\psi}}(B|\nabla\psi|w)=v$. It then
follows from Lemma 5.15 that

$\int_{S_{\psi}}\rho^{*}v\,{\rm
d}\mathscr{H}^{(d-1)}=\int_{S_{\psi}}\rho^{*}\operatorname{div}_{S_{\psi}}(B|\nabla\psi|w)\,{\rm
d}\mathscr{H}^{(d-1)}=0,$ (5.27)

as required. ∎

###  5.3. Proof of Proposition 1.1: effective volume of non-integrability

It follows immediately from the equation (1.9) (or, equivalently, (2.6)) for
$T_{0}=\Theta(\psi)$ that either $T_{0}$ is constant or else $T_{0}^{\prime}$
is nonvanishing. Since $T_{0}|_{S_{+}}=T_{+}\not=T_{-}=T_{0}|_{S_{-}}$ we have
that $T_{0}^{\prime}$ is nonvanishing and in particular $T_{0}^{\prime}$ is
bounded below. With $\lambda=\min_{D}\frac{1}{|T_{0}^{\prime}|}$, for any set
$N\subseteq D$ we therefore have

$\mu(N)\leq\lambda^{2}\int_{N}|T_{0}^{\prime}|^{2}\,{\rm d}\mu\leq
C\left(\lambda^{2}\int_{N}|\nabla_{b}^{\perp}T|^{2}\,{\rm
d}\mu+\lambda^{2}\int_{N}|\nabla_{b}^{\perp}\rho|^{2}\,{\rm d}\mu\right),$
(5.28)

where we used that $T_{0}^{\prime}=|\nabla\psi|^{-2}\nabla
T_{0}\cdot\nabla\psi$ and that $|\nabla
T_{0}|\leq|\nabla_{b}^{\perp}T|\leq|\nabla_{b}^{\perp}T|+|\nabla_{b}^{\perp}\rho|^{2}$.
In particular, with $N=N(\varepsilon)$, since
$\|\nabla_{b}^{\perp}T\|_{L^{2}(N(\varepsilon))}^{2}\leq\varepsilon^{-1}\|\nabla_{b}T\|_{L^{2}(N(\varepsilon))}^{2}$,

$\mu(N(\varepsilon))\leq
C\lambda^{2}\left(\varepsilon^{-1}\int_{N(\varepsilon)}|\nabla_{b}T|^{2}\,{\rm
d}\mu+\int_{N(\varepsilon)}|\nabla_{b}^{\perp}\rho|^{2}\,{\rm d}\mu\right)\\\
\leq
C\lambda^{2}\left(\varepsilon^{-1}\int_{N(\varepsilon)}|\nabla_{b}T_{0}|^{2}\,{\rm
d}\mu+\varepsilon^{-1}\int_{D}|\nabla_{b}\rho|^{2}\,{\rm
d}\mu+\int_{D}|\nabla_{b}^{\perp}\rho|^{2}\,{\rm d}\mu\right)\\\ \leq
C\lambda^{2}\varepsilon^{-1}\left(\int_{N(\varepsilon)}|\nabla_{b}T_{0}|^{2}\,{\rm
d}\mu+\int_{D}|\nabla_{b}\rho|^{2}\,{\rm
d}\mu+\varepsilon\int_{D}|\nabla_{b}^{\perp}\rho|^{2}\,{\rm d}\mu\right).$
(5.29)

Since $b=b_{0}+\varepsilon^{a}b_{1}$ and $\nabla_{b_{0}}T_{0}=0$, we have

$\|\nabla_{b}T_{0}\|_{L^{2}(D)}^{2}\leq\varepsilon^{2a}\|\nabla_{b_{1}}T_{0}\|_{L^{2}(D)}^{2}\leq\varepsilon^{2a}\|\partial_{\theta}\chi_{1}\|_{L^{2}(D)}^{2}\|T_{0}^{\prime}\|_{L^{\infty}(D)}^{2}.$
(5.30)

To handle the second and third terms in (5.29), we use (1.21)-(1.22) combined
with (5.3). By (5.30) we therefore have

$\mu(N(\varepsilon))\leq
C\lambda^{2}\left(\varepsilon^{2a-1}\|\partial_{\theta}\chi_{1}\|_{L^{2}(D)}^{2}\|T_{0}^{\prime}\|_{L^{\infty}(D)}^{2}+\varepsilon^{1/3}\|\Delta
T_{0}\|_{H^{(0,\gamma)}(D)}^{2}\right).$ (5.31)

##  Appendix A Geometric identities from the co-area formula

In this section we collect some geometric formulas that we will use
repeatedly. In what follows, we fix $\psi:D\to\mathbb{R}$ such that
$|\nabla\psi|\neq 0$ on $D$ and so that the level surfaces $S_{\psi}$ are
codimension one manifolds which foliate $D$. Let $\psi_{-}=\inf_{D}\psi$ and
$\psi_{+}=\sup_{D}\psi$. We will use the co-area formula

$\int_{D}u\,{\rm
d}\mu=\int_{\psi_{-}}^{\psi_{+}}\int_{S_{\psi}}\frac{u}{|\nabla\psi|}\,{\rm
d}\mathscr{H}^{(d-1)}{\rm d}\psi,$ (A.1)

see e.g. [22]. We start with a simple result that generalizes Lemma E.2 from
[8].

######  Lemma A.1.

If $F\in H^{2}(D)$, we have

$\frac{{\rm d}}{{\rm d}\psi}\int_{S_{\psi}}F\,{\rm
d}\mathscr{H}^{(d-1)}=\int_{S_{\psi}}\operatorname{div}\left(\frac{\nabla\psi}{|\nabla\psi|}F\right)\,{\rm
d}\mathscr{H}^{(d-1)}.$ (A.2)

###### Proof.

We start with the observation that

$\int_{S_{\psi_{2}}}F\,{\rm d}\mathscr{H}^{(d-1)}-\int_{S_{\psi_{1}}}F\,{\rm
d}\mathscr{H}^{(d-1)}=\int_{D_{\psi_{1},\psi_{2}}}\operatorname{div}\left(\frac{\nabla\psi}{|\nabla\psi|}F\right)\,{\rm
d}\mu,$ (A.3)

where
$D_{\psi_{1},\psi_{2}}=\cup_{\psi_{1}\leq\psi^{\prime}\leq\psi_{2}}S_{\psi^{\prime}}$
denotes the region bounded by the surfaces $S_{\psi_{1}},S_{\psi_{2}}$.
Indeed, by the divergence theorem,

$\int_{D_{\psi_{1},\psi_{2}}}\operatorname{div}\left(\frac{\nabla\psi}{|\nabla\psi|}F\right)\,{\rm
d}\mu=\int_{S_{\psi_{2}}}n^{S_{\psi_{2}}}\cdot\frac{\nabla\psi}{|\nabla\psi|}F\,{\rm
d}\mathscr{H}^{(d-1)}+\int_{S_{\psi_{1}}}n^{S_{\psi_{1}}}\cdot\frac{\nabla\psi}{|\nabla\psi|}F\,{\rm
d}\mathscr{H}^{(d-1)},$ (A.4)

where $n^{S_{\psi}}$ denotes the outward-pointing unit normal to $S_{\psi}$.
Then $n^{S_{\psi_{2}}}=\frac{\nabla\psi}{|\nabla\psi|}\big{|}_{S_{\psi_{2}}}$
and $n^{S_{\psi_{1}}}=-\frac{\nabla\psi}{|\nabla\psi|}\big{|}_{S_{\psi_{1}}}$,
so (A.4) gives (A.3). Dividing (A.3) by $\psi_{2}-\psi_{1}$ and taking the
limit gives (A.2). ∎

In particular, if $F=F(\psi)$ is constant on $S_{\psi}$, writing $\Delta
F=\operatorname{div}(\nabla F)=\operatorname{div}(\nabla\psi F^{\prime})$ we
have

$\int_{S_{\psi}}\Delta F{\rm d}\mathscr{H}^{(d-1)}=\frac{{\rm d}}{{\rm
d}\psi}\left(\int_{S_{\psi}}|\nabla\psi|F^{\prime}{\rm
d}\mathscr{H}^{(d-1)}\right)=\frac{{\rm d}}{{\rm
d}\psi}\left(\left[\int_{S_{\psi}}|\nabla\psi|{\rm
d}\mathscr{H}^{(d-1)}\right]F^{\prime}\right).$ (A.5)

Another consequence of the formula (A.1) is the following

######  Lemma A.2.

Let $X$ be a vector field defined in $D$ with the property that
$X|_{S_{\psi}}$ is tangent to $S_{\psi}$. Then the divergence
$\operatorname{div}X$ in $D$ is related to the divergence operator
$\operatorname{div}_{S_{\psi}}$ on $S_{\psi}$ by

$\frac{\operatorname{div}X}{|\nabla\psi|}\Big{|}_{S_{\psi}}=\operatorname{div}_{S_{\psi}}\left(\left[\frac{X}{|\nabla\psi|}\right]\Big{|}_{S_{\psi}}\right).$
(A.6)

In particular, if $X$ is divergence-free in $D$, then
$\varrho:=\frac{1}{|\nabla\psi|}\big{|}_{S_{\psi}}$ is a density conserved by
$X$ on $S_{\psi}$.

###### Proof.

This can be seen by working in local coordinates but it is simpler to use
(A.1) and note that if $u\in C^{\infty}(D)$ is any test function then

$\displaystyle-\int_{D}\operatorname{div}Xu\,{\rm d}\mu$
$\displaystyle=\int_{D}(X\cdot\nabla)u\,{\rm
d}\mu=\int_{\psi_{-}}^{\psi_{+}}\int_{S_{\psi}}(X\cdot\nabla)u\,\frac{{\rm
d}\mathscr{H}^{(d-1)}}{|\nabla\psi|}$
$\displaystyle=\int_{\psi_{-}}^{\psi_{+}}\int_{S_{\psi}}(X\cdot\nabla^{T})u\frac{{\rm
d}\mathscr{H}^{(d-1)}}{|\nabla\psi|}=-\int_{\psi_{-}}^{\psi_{+}}\int_{S_{\psi}}\operatorname{div}_{S_{\psi}}\left(\frac{X}{|\nabla\psi|}\right)u\,{\rm
d}\mathscr{H}^{(d-1)},$

where we used that $X\cdot\nabla=X\cdot\nabla^{T}$ on $S_{\psi}$, where
$\nabla^{T}$ denotes the tangential gradient on $S_{\psi}$,
$\nabla^{T}U=(\nabla-\frac{\nabla\psi}{|\nabla\psi|^{2}}\nabla\psi\cdot\nabla)u$
whenever $u$ is an extension of $U$ from $S_{\psi}$ to a neighborhood of
$S_{\psi}$. By (A.1), the left-hand side is
$-\int_{\psi_{-}}^{\psi_{+}}\int_{S_{\psi}}\frac{\operatorname{div}X}{|\nabla\psi|}u\,{\rm
d}\mathscr{H}^{(d-1)}$. Then (A.6) follows since $u$ is arbitrary. ∎

### Acknowledgements.

We thank P. Constantin, P. Helander, S. Hudson, E. Paul, and P. Torres de
Lizaur for numerous insightful discussions. The research of TD was partially
supported by NSF-DMS grant 2106233. The work of DG was partially supported by
the Simons Center for Hidden Symmetries and Fusion Energy award # 601960. The
studies of HG are supported by Ford Foundation and the NSF Graduate Research
Fellowship under grant DGE-2039656.

## References

  * [1] Arnold, V. I. (1966). On the topology of three-dimensional steady flows of an ideal fluid. In Vladimir I. Arnold-Collected Works (pp. 25-28). Springer, Berlin, Heidelberg.
  * [2] Arnold, V. I., and Khesin, B. A. (2021). Topological methods in hydrodynamics (Vol. 125). Springer Nature.
  * [3] Braginskii, S. I. (1965). Transport Processes in Plasma, ed. MA Leontovich (New York, USA: Consultants Bureau), 201.
  * [4] Hudson, S. R., and Breslau, J. (2008). Temperature contours and ghost surfaces for chaotic magnetic fields. Physical review letters, 100(9), 095001.
  * [5] Helander, P., Hudson, S. R., and Paul, E. J. (2022). On heat conduction in an irregular magnetic field. Part 1. Journal of Plasma Physics, 88(1).
  * [6] Paul, E. J., Hudson, S. R., and Helander, P. (2022). Heat conduction in an irregular magnetic field. Part 2. Heat transport as a measure of the effective non-integrable volume. Journal of Plasma Physics, 88(1).
  * [7] Braginskii, S.I., 1965. Transport processes in a plasma. Reviews of plasma physics, 1.
  * [8] Constantin, P., Drivas, T. D., and Ginsberg, D. (2021). Flexibility and rigidity in steady fluid motion. Communications in Mathematical Physics, 385(1), 521-563.
  * [9] Constantin, P., Drivas, T. D., and Ginsberg, D. (2021). On quasisymmetric plasma equilibria sustained by small force. Journal of Plasma Physics, 87(1).
  * [10] Constantin, P., Drivas, T. D., and Ginsberg, D. (2022). Flexibility and rigidity of free boundary MHD equilibria. Nonlinearity, 35(5), 2363.
  * [11] Freidberg, J P. 1987. “Ideal magnetohydrodynamics”. United States.
  * [12] Helander, P. and Sigmar, D., Collisional Transport in Magnetized Plasmas. United Kingdom: Cambridge University Press, 2005.
  * [13] Helander, P. Theory of plasma confinement in non-axisymmetric magnetic fields, Reports on Progress in Physics.
  * [14] Grad, H., Toroidal containment of a plasma. The Physics of Fluids, 10(1), 137-154, (1967)
  * [15] Grad, H., Theory and applications of the nonexistence of simple toroidal plasma equilibrium. International Journal of Fusion Energy 3.2, 33-46, (1985)
  * [16] L. Ryzhik, A. Novikov, G. Papanicolaou, Boundary layers for cellular flows at high Peclet numbers, Communications in Pure and Applied Mathematics, 58, 2005, 867-922.
  * [17] Bensoussan, A., Lions, J. L., & Papanicolaou, G. (2011). Asymptotic analysis for periodic structures (Vol. 374). American Mathematical Soc.
  * [18] Treschev, D., Zubelevich, O. (2009). Introduction to the Perturbation Theory of Hamiltonian Systems. Germany: Springer Berlin Heidelberg.
  * [19] Arnold, Vladimir I. ”Proof of a theorem of AN Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian.” Russian Mathematical Surveys 18.5 (1963): 9.
  * [20] Fitzpatrick, R. Helical temperature perturbations associated with tearing modes in tokamak plasmas Cite as: Physics of Plasmas 2, 825 (1995);
  * [21] Sternberg, S., American Journal of Mathematics , Apr., 1957, Vol. 79, No. 2 (Apr., 1957), pp. 397-402
  * [22] Gariepy, R. F., Evans, L. C. (2015). Measure Theory and Fine Properties of Functions, Revised Edition. United Kingdom: CRC Press.