On the Distribution of Heat in Fibered Magnetic Fields

Abstract

We study the equilibrium temperature distribution in a model for strongly magnetized plasmas in dimensions two and three. 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 is related to the non-integrable structure of the magnetic field, confirming a physical conjecture of Paul et al (J Plasma Phys 88(1):905880107, 2022). Our proof crucially uses a certain quantitative ergodicity condition for the magnetic field lines on a 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.

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 $\psi :D\rightarrow {\mathbb {R}}$ such that $|\nabla \psi |\ne 0$ 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

$$\begin{aligned} \int _{D} u\, \textrm{d}\mu = \int _{\psi _-}^{\psi _+}\int _{S_\psi } \frac{u}{|\nabla \psi |} \, \textrm{d}{\mathscr {H}}^{(d-1)}\textrm{d} \psi , \end{aligned}$$

(A.1)

see e.g. [20]. We start with a simple result that generalizes Lemma E.2 from [21].

Lemma A.1

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

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} \psi } \int _{S_\psi } F \, \textrm{d}{\mathscr {H}}^{(d-1)} = \int _{S_{\psi }} {{\,\textrm{div}\,}}\left( \frac{\nabla \psi }{|\nabla \psi |} F\right) \,{\frac{ \textrm{d}{\mathscr {H}}^{(d-1)}}{|\nabla \psi |}}. \end{aligned}$$

(A.2)

Proof

We start with the observation that

$$\begin{aligned} \int _{S_{\psi _2}} F \, \textrm{d}{\mathscr {H}}^{(d-1)} - \int _{S_{\psi _1}} F\, \textrm{d}{\mathscr {H}}^{(d-1)} = \int _{D_{\psi _1, \psi _2}} {{\,\textrm{div}\,}}\left( \frac{\nabla \psi }{|\nabla \psi |} F\right) \, \textrm{d}\mu , \end{aligned}$$

(A.3)

where $D_{\psi _1, \psi _2} = \cup _{\psi _1 \le \psi ' \le \psi _2} S_{\psi '}$ denotes the region bounded by the surfaces $S_{\psi _1}, S_{\psi _2}$. Indeed, by the divergence theorem,

$$\begin{aligned} \int _{D_{\psi _1, \psi _2}} {{\,\textrm{div}\,}}\left( \frac{\nabla \psi }{|\nabla \psi |} F\right) \, \textrm{d}\mu= & {} \int _{S_{\psi _2}} n^{S_{\psi _2}}\cdot \frac{\nabla \psi }{|\nabla \psi |} F \, \textrm{d}{\mathscr {H}}^{(d-1)} \nonumber \\{} & {} + \int _{S_{\psi _1}} n^{S_{\psi _1}}\cdot \frac{\nabla \psi }{|\nabla \psi |} F \, \textrm{d}{\mathscr {H}}^{(d-1)}, \end{aligned}$$

(A.4)

where $n^{S_{\psi }}$ denotes the outward-pointing unit normal to $S_\psi$. Then

$$\begin{aligned} n^{S_{\psi _2}} = \frac{\nabla \psi }{|\nabla \psi |}\Big |_{S_{\psi _2}} \quad \text {and} \quad n^{S_{\psi _1}} = -\frac{\nabla \psi }{|\nabla \psi |}\Big |_{S_{\psi _1}}, \end{aligned}$$

(A.5)

so (A.4) gives (A.3). Dividing (A.3) by $\psi _2 - \psi _1$ and taking the limit gives (A.2). $\square$

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

$$\begin{aligned} \int _{S_\psi } \Delta F{ \frac{\textrm{d}{\mathscr {H}}^{(d-1)}}{|\nabla \psi |}}= & {} \frac{\textrm{d}}{\textrm{d}\psi } \left( \int _{S_{\psi }} |\nabla \psi | F' \textrm{d}{\mathscr {H}}^{(d-1)}\right) \nonumber \\= & {} \frac{\textrm{d}}{\textrm{d}\psi } \left( \left[ \int _{S_\psi } |\nabla \psi | \textrm{d}{\mathscr {H}}^{(d-1)}\right] F'\right) . \end{aligned}$$

(A.6)

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 ${{\,\textrm{div}\,}}X$ in D is related to the divergence operator ${{\,\textrm{div}\,}}_{S_\psi }$ on $S_\psi$ by

$$\begin{aligned} \frac{{{\,\textrm{div}\,}}X}{|\nabla \psi |} \Big |_{S_\psi } = {{\,\textrm{div}\,}}_{S_\psi }\left( \left[ \frac{ X}{|\nabla \psi |}\right] \Big |_{S_\psi }\right) . \end{aligned}$$

(A.7)

In particular, if X is divergence-free in D, then $\varrho := {|\nabla \psi |^{-1}}\big |_{S_\psi }$ 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

$$\begin{aligned} -\int _D {{\,\textrm{div}\,}}X u\, \textrm{d}\mu&= \int _D (X\cdot \nabla ) u\, \textrm{d}\mu = \int _{\psi _-}^{\psi _+} \int _{S_\psi } (X\cdot \nabla ) u\,\frac{ \textrm{d}{\mathscr {H}}^{(d-1)}}{|\nabla \psi |}\\&= \int _{\psi _-}^{\psi _+} \int _{S_\psi } (X\cdot \nabla ^T) u \frac{ \textrm{d}{\mathscr {H}}^{(d-1)}}{|\nabla \psi |} \\&= -\int _{\psi _-}^{\psi _+} \int _{S_\psi } {{\,\textrm{div}\,}}_{S_\psi }\left( \frac{ X}{|\nabla \psi |} \right) u \, \textrm{d}{\mathscr {H}}^{(d-1)}, \end{aligned}$$

where we used that $X\cdot \nabla = X\cdot \nabla ^T$ on $S_\psi$, where $\nabla ^T$ denotes the tangential gradient on $S_\psi$,

$$\begin{aligned} \nabla ^T U = \left( \nabla - \frac{\nabla \psi }{|\nabla \psi |^2} \nabla \psi \cdot \nabla \right) u \end{aligned}$$

(A.8)

whenever u is an extension of U from $S_\psi$ to a neighborhood of $S_\psi$. By (A.1), the left-hand side is

$$\begin{aligned} -\int _{\psi _-}^{\psi _+} \int _{S_\psi } \frac{{{\,\textrm{div}\,}}X}{|\nabla \psi |} u\, \textrm{d}{\mathscr {H}}^{(d-1)}. \end{aligned}$$

(A.9)

Then (A.7) follows since u is arbitrary. $\square$