7th Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe Nagoya

On the Grad Conjecture in Anisotropic MHD

Naoki Sato1 and Michio Yamada2
1Graduate School of Frontier Sciences, The University of Tokyo,
2
Research Institute for Mathematical Sciences, Kyoto University
e-mail (speaker): sato_naoki@edu.k.u-tokyo.ac.jp
The Grad conjecture represents a fundamental unsolved
mathematical problem in the theory of PDEs: whether
one can find isotropic magnetohydrodynamic equilibria
or steady Euler flows that are not invariant under somecontinuous Euclidean isometry (according to thestandard formulation of the Grad conjecture, the answer
should be negative). This problem has broad implications
for plasma physics and fluid mechanics, especially forthe design of stellarators, where the existence ofasymmetric confining magnetic fields remains a core
theoretical issue.
In this study, we consider a weaker version of thismathematical problem: can one find a vector field suchthat both the vector field and its curl are tangential to agiven set of nested toroidal surfaces? We show that theanswer is always positive, and that these configurations
can be identified with magnetohydrodynamic equilibria
with anisotropic pressure [1].
In formulae, the problem considered here amounts tosolving the system of PDEs,[(∇ × 𝒘) × 𝒘] × ∇Ψ = 𝟎,

∇ ⋅ 𝒘 = 0,(1)

where Ψ(𝒙) is a given (flux) function whose level sets
define a family of nested toroidal surfaces. Notice thatstandard isotropic MHD equilibria are solutions of (1),although the converse is not true.
By using a tailored Clebsch parametrization of thesolution, we show that system (1) can be translated into
the problem of determining a periodic solution withperiodic derivatives of a two-dimensional linear elliptic
second-order partial differential equation on eachtoroidal surface and prove the existence of smoothsolutions. Examples of smooth solutions foliated bytoroidal surfaces that are not invariant under continuous
Euclidean isometries are also constructed explicitly, andthey are identified as equilibria of anisotropic
magnetohydrodynamics. These solutions have thefollowing form:𝒘 = 𝑓(Ψ𝜖 )𝝃𝜖 ,𝑃(𝑥, 𝑦) and 𝑄(𝑥, 𝑦) harmonic conjugate functions in
two dimensions, and 𝑆(𝑧) any function of 𝑧. Weremark that the family of vector fields (2) includes
steady solutions of anisotropic magnetohydrodynamics(∇ × 𝒘) × 𝒘 = ∇ ⋅ Π,

∇ ⋅ 𝒘 = 0,(5)which are not invariant under continuous (as well asdiscrete) Euclidean isometries, i.e. combinations oftranslations, rotations, and reflections. Here, Π is theanisotropic pressure tensor
1
Π 𝑖𝑗 = (𝑃 − 𝛾𝒘2 ) 𝛿 𝑖𝑗 + 𝛾𝑤 𝑖 𝑤 𝑗 , 𝑖, 𝑗 = 1,2,3.
2(6)In particular, notice that (2) solves (5) with 𝑃 = 0 and𝛾 = 1 − 𝑓 −2 . An example of anisotropic MHD
equilibrium without continuous Euclidean isometries isgiven in figure 1 below.
In conclusion, this work shows that as long as pressure
anisotropy is allowed steady smooth solutions ofmagnetohydrodynamics exist without continuous ordiscrete Euclidean isometries.
References[1] N. Sato and M. Yamada, “Nested invariant tori
foliating a vector field and its curl: toward MHD
equilibria and steady Euler flows in toroidal domains
without continuous Euclidean isometries,”
arXiv:2211.10757 (2022).

(2)

where 𝑓 is any function of the flux functionΨ𝜖 =12{[𝑟𝑒 −𝜖𝑄(𝑥,𝑦) − 𝑟0 ] + 𝑧 2 𝑒 −𝜖𝑆(𝑧) },2(3)and 𝝃𝜖 the vector field𝝃𝜖 = ∇(𝜑 + 𝜖𝑃(𝑥, 𝑦))(4)

with (𝑟, 𝜑, 𝑧) cylindrical coordinates, 𝜖 a real constant,Figure 1: Contour plot (a) and vector plot (b) of ananisotropic MHD equilibrium 𝒘 without continuous
Euclidean isometries. Contour plot (c) and vector plot (d)of ∇ × 𝒘. See [1] for additional details.