Quasi-geostrophic MHD equations: Hamiltonian formulation and nonlinear stability

Abstract

Magnetic fields in stars and planets are generated by a dynamo process that results from multi-scale interactions of the flows in conducting fluids. On the large scales, these flows are dominated by a strong zonal component, while the magnetic fields exhibit a strong toroidal/zonal character. Although dissipation certainly acts on these flows, the kinematic and magnetic viscosities associated with these large-scale flows are small, so that, over the timescale of several years and beyond, the system may be modelled as a conservative one. In this context, the Hamiltonian formulation may give several insights, providing a systematic way to relate the symmetries of the system with conservation laws. In the present article, we introduce the Hamiltonian formulation for a model that reasonably describes the dynamics of large-scale flows in stars and planets: the two-dimensional magnetohydrodynamic quasi-geostrophic equations. In this context, we find the invariants of the system, which are of two kinds: the Casimirs, related to the particle relabelling symmetry, and the zonal momentum, which is related to the translational invariance in the zonal direction. We then use these invariants to study the stability of some stationary solutions that are relevant for geophysical and astrophysical applications.

Appendix: Differential forms representation of magnetic fields

The representation of magnetic fields in terms of differential form is given in terms of 2-forms. First introduce the following operations, first the isomorphism that takes vector field on ${\mathbb {R}}^3$ to 1-forms $^{\flat }:{T^{{\mathbb {R}}^{3}}}\rightarrow T^{*}{\mathbb {R}^{3}}$ defined by

$$\begin{aligned} (e_i)^{\flat }=\textrm{d}x_i \end{aligned}$$

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$i=1,2,3$, where $\{e_1,e_2,e_3\}$ is the basis of $\mathbb {R}^3$, the inverse isomorphism,$^{\flat }:T\mathbb {R}^3\rightarrow T^{*}\mathbb {R}^3$, is defined by

$$\begin{aligned} (e_i)^{\#}=\textrm{d}x_i \end{aligned}$$

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Finally, introduce the Hodge star operator that takes l forms to $n-l$ forms in $\mathbb {R}^n$. In $\mathbb {R}^3$,

$$\begin{aligned} \begin{aligned} *(1)=\textrm{d}x_1\wedge \textrm{d}x_2\wedge \textrm{d}x_3;\quad *(\textrm{d}x_i)=\textrm{d}x_j\wedge \textrm{d}x_k\\ *(\textrm{d}x_j\wedge \textrm{d}x_k)=\textrm{d}x_i;\quad *(\textrm{d}x_1\wedge \textrm{d}x_2\wedge \textrm{d}x_k)=1 \end{aligned} \end{aligned}$$

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With this, a given magnetic field, ${B}=B_1 e_1 + B_2 e_2+B_1 e_3$, is expressed by the following differential form

$$\begin{aligned} \beta =*{B}^{\flat }=B_1 \textrm{d}y\wedge \textrm{d}z + B_2 \textrm{d}x \wedge \textrm{d}z+B_3\textrm{d}x \wedge \textrm{d}y \end{aligned}$$

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the law on non existence of magnetic monopoles $\nabla .{\textbf {B}}=0$ is them expressed as

$$\begin{aligned} *d(*{{B}}^b)=0 \iff d\beta =0 \end{aligned}$$

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which implies, by Poincare’s Lemma in the existence (at least locally) in the existence of a 1-form such that $\beta =d \alpha$, from which we define the vector potential ${\textbf {A}}$

$$\begin{aligned} {\textbf {B}}=[*(d{\textbf {A}}^{\flat })]^{\#}=(*d\alpha )^{\#} ;\quad {\textbf {A}}= \alpha ^{\#} \end{aligned}$$

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For an Euler–Poincaré equations with advected quantities (Holm et al. 2009) the variations in the action with respect with the advected quantity (${\textbf {A}}$ or ${\textbf {B}}$) is of the form

$$\begin{aligned} \begin{aligned} \delta {\textbf {A}}={\mathscr {L}}_{\delta {\textbf {u}}}d\alpha ;\quad \delta {\textbf {B}}={\mathscr {L}}_{\delta {\textbf {u}}}d\beta \end{aligned} \end{aligned}$$

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We need to calculate the Lie derivatives of these forms

$$\begin{aligned} \begin{aligned} {{\mathscr {L}}}_{{\textbf {u}}}\alpha =i_{{\textbf {u}}} d\alpha + di_{{\textbf {u}}} \alpha =({\textbf {u}}.\nabla {\textbf {A}}+{\textbf {A}}.\nabla {\textbf {u}})^{\flat }= [{\textbf {u}}\times \nabla \times {\textbf {A}}+\nabla ({\textbf {A}}. {\textbf {u}})]^{\flat } \end{aligned} \end{aligned}$$

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and

$$\begin{aligned} \begin{aligned} {{\mathscr {L}}}_{{\textbf {u}}}\beta =d({\mathscr {L}}_{{\textbf {u}}}\alpha )={\mathscr {L}}_{{\textbf {u}}}d\alpha =i_{{\textbf {u}}} d\alpha + di_{{\textbf {u}}} \alpha = *[(\nabla \times ({\textbf {u}}\times {\textbf {B}}))^{\flat }] \end{aligned} \end{aligned}$$

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