Hydrostatic MHD-Wave System

• Hydrostatic MHD-wave system is a model that filters out vertical acoustic waves via hydrostatic balance, focusing on horizontal magneto-gravity and Alfvén modes.
• The mathematical formulation features a degenerate hyperbolic magnetic field equation and employs Gevrey 7/6 regularity to establish local well-posedness.
• Applications include enhancing numerical schemes for astrophysical simulations, accurately capturing wave energy distribution in stellar and planetary atmospheres.

A hydrostatic MHD-wave system refers to a class of magnetohydrodynamic (MHD) models and wave equations formulated under the condition of vertical hydrostatic balance, in which the fast vertically-propagating magneto-acoustic waves (vertical sound waves) are filtered out. This regime is particularly relevant in gravitationally stratified fluids such as stellar atmospheres, radiative zones, and planetary atmospheres, where horizontal length scales dominate over vertical ones and magnetic fields strongly influence the dynamics. The hydrostatic approximation, together with MHD, yields a system in which horizontal and certain magneto-gravity wave modes persist, but vertical acoustic branches are absent. Recent mathematical advances have established well-posedness for a hydrostatic MHD-wave system featuring a degenerate hyperbolic magnetic field equation, introducing substantial analytic challenges and necessitating function spaces such as Gevrey $\frac{7}{6}$ regularity for control of solutions (Li et al., 6 Dec 2025, Braithwaite et al., 2012). Such systems are directly relevant for physically-motivated numerical schemes in astrophysics, and for the theoretical description of wave generation by drivers such as logarithmic spiral motions in the solar photosphere (Mumford et al., 2015).

1. Foundational Hydrostatic MHD Equations

The hydrostatic MHD-wave system replaces the vertical component of the momentum equation with the condition of exact hydrostatic balance, suppressing vertical sound waves and reformulating the dynamics to focus on slower, horizontally or weakly-vertically-propagating modes (Braithwaite et al., 2012). For a domain $\Omega = \mathbb{T}_x \times (0,1)$, the unknowns are the tangential and normal velocities $u(t,x,y), v(t,x,y)$, tangential and normal magnetic fields $f(t,x,y), g(t,x,y)$, and pressure $p(t,x)$ (independent of $y$). The core system (with fixed $\eta=1$) is:

$$\begin{cases} \partial_t u + u\,\partial_xu + v\,\partial_yu + \partial_x p - \partial_y^2u = f\,\partial_x f + g\,\partial_yf, \ \partial_y p = 0, \ \partial_t^2 f + \partial_t f + u\,\partial_x f + v\,\partial_y f - \partial_y^2f = f\,\partial_x u + g\,\partial_y u, \ \partial_t^2 g + \partial_t g + u\,\partial_x g + v\,\partial_y g - \partial_y^2g = f\,\partial_x v + g\,\partial_y v, \ \partial_x u + \partial_y v =0, \qquad \partial_x f + \partial_y g =0, \end{cases}$$

with $(u,v,\partial_y f,g)|_{y=0,1} = 0$ and specified initial conditions (Li et al., 6 Dec 2025). The vertical hydrostatic approximation emerges by enforcing $\partial_z P = -\rho g$, eliminating vertical acceleration and, thus, filtering vertically-propagating acoustic waves from the solution space (Braithwaite et al., 2012).

2. Mathematical Structure and Well-posedness

A defining feature of the hydrostatic MHD-wave system is the degenerate hyperbolic nature of the tangential magnetic field equation: unlike the purely parabolic magnetic diffusion equation of classical hydrostatic MHD, the inclusion of a second time derivative leads to mixed hyperbolic-parabolic character. This structure complicates the analytic landscape by destroying certain cancellation mechanisms between parabolic and hyperbolic contributions.

Li and Xu (Li et al., 6 Dec 2025) proved local well-posedness for convex initial data in Gevrey $\frac{7}{6}$ class, introducing weighted norms,

$$\|h\|_{X_{\rho,\sigma,r}}^2 = \sum_{m=0}^\infty \left( \frac{\rho^{m+1}(m+1)^r}{(m!)^\sigma} \right)^2 \|\partial_x^m h\|_{L^2(\Omega)}^2,$$

with $1 \leq \sigma \leq 7/6$, $r \geq 10$, and initial data regularized accordingly. The proof leverages:

• A priori Gevrey energy hierarchies controlling tangential derivatives and time derivatives.
• Maximum/minimum principles to ensure preservation of strict convexity, critical for denominator control in nonlinear terms.
• A boundary decomposition method to resolve the stream function $\varphi$ in Gevrey spaces, decomposing into "soft," heat, transport, and remainder components to track boundary effects analytically.
• Careful absorption of boundary and remainder terms via parameter choices in the weighted energy hierarchy.

These techniques collectively enable control over the mixed character of the system and provide guarantees of uniqueness, existence, and continued regularity in the allowable time interval, provided convexity of the initial profile is maintained (Li et al., 6 Dec 2025).

3. Physical Wave Modes and Spectral Filtering

Linearization about a static, horizontally uniform hydrostatic background (with horizontal magnetic field $\mathbf{B}_0$) produces wave modes constrained by vertical hydrostatic equilibrium. Vertical acceleration is excluded via $\partial_z P = -\rho g$, precluding the existence of vertically-propagating acoustic waves (filtered from the mode spectrum) (Braithwaite et al., 2012). The two principal classes of waves are:

• Alfvén waves: Propagate horizontally, polarized perpendicular to both the magnetic field and stratification plane, obeying

$$\omega = \pm V_A k, \quad V_A = \frac{B_0}{\sqrt{4\pi \rho_0}}.$$

• Magneto-gravity waves: Analogous to shallow-water waves with magnetic corrections, exhibiting the dispersion relation

$\omega^2 = (gH + V_A^2) k^2,$

where $H$ is the effective vertical scale of stratification.

A key consequence is the absence of vertical acoustic branches $\omega^2 = c_s^2 k_z^2$—vertical sound waves are eliminated by the hydrostatic balance (Braithwaite et al., 2012).

4. Energy Flux Decomposition and Mode Amplitudes

In gravitationally stratified (hydrostatic) atmospheres such as the solar photosphere, logarithmic spiral drivers inject perturbations into a self-similar flux tube. The ensuing velocity and field perturbations are decomposed along coordinate axes defined by the local magnetic field: parallel ($\parallel$), perpendicular ($\perp$), and azimuthal (torsional, $\phi$). For any perturbation $\delta \mathbf{v}$,

$$\delta v_\parallel = \delta\mathbf{v}\cdot\hat{\mathbf{e}}_\parallel, \quad \delta v_\perp = \delta\mathbf{v}\cdot\hat{\mathbf{e}}_\perp, \quad \delta v_\phi = \delta\mathbf{v}\cdot\hat{\mathbf{e}}_\phi,$$

where the basis is constructed from the magnetic topology (Mumford et al., 2015). The normalized energy flux in these directions, computed as

$$\frac{\langle F^2_{\parallel,\perp,\phi} \rangle }{ \langle F^2_\parallel + F^2_\perp + F^2_\phi \rangle }$$

as a function of the driver expansion parameter $B_L$, allows quantification of the dominant wave mode:

$B_L$	$$\langle F^2_\phi \rangle / \langle F^2_{\rm tot} \rangle$$	$$\langle F^2_\parallel \rangle / \langle F^2_{\rm tot} \rangle$$
$0.015$	0.6	0.2
$1.5$	0.1	0.6

This shows that for low $B_L$, the energy is predominantly torsional (Alfvén); increasing $B_L$ shifts the energy into the parallel (sausage/compressive) component (Mumford et al., 2015).

5. Applications in Solar and Astrophysical MHD

Hydrostatic MHD-wave models underlie numerical schemes for stratified astrophysical environments such as radiative stellar zones, compact objects, and planetary atmospheres, as reviewed in Braithwaite & Cavecchi (Braithwaite et al., 2012). The hydrostatic approximation is particularly apt when the vertical scale is much less than the horizontal, and the field configuration is quasi-horizontal or organized into flux tubes.

Physically, high-resolution observations of the solar photosphere reveal abundant logarithmic spiral motions in locations with enhanced magnetic field. Numerical simulations demonstrate that such motions, when analyzed in a hydrostatically stratified atmosphere, drive a spectrum of MHD waves whose partition is sensitively dependent on observable driver parameters. Crucially, at observed values of the spiral expansion factor $B_L \simeq 0.15 \pm 0.04$, the system lies near a regime where both Alfvén and compressive sausage modes are significantly excited. This implies that realistic solar drivers are a source of mixed-mode wave spectra, relevant to coronal heating and MHD wave transport modeling (Mumford et al., 2015).

6. Hyperbolic-Parabolic Dichotomy and Analytical Challenges

Unlike classical hydrostatic MHD (without wave terms), in which tangential magnetic field evolution is governed by a purely parabolic equation, the hydrostatic MHD-wave system introduces a degenerate hyperbolic equation for the magnetic field variable. This hybridization destroys the cancellation available in the purely parabolic regime and necessitates analytic techniques blending parabolic, hyperbolic, and transport estimates, and the warrant for Gevrey $\sigma< 3/2$ spaces rather than only Sobolev or classical analytic frameworks (Li et al., 6 Dec 2025). The preservation of initial convexity ($\partial_y^2 u_0 > 0$) is foundational: failure to maintain strict convexity leads to ill-posedness by loss of denominator estimates and breakdown of maximum principles for second derivatives.

A plausible implication is that further generalizations of hydrostatic MHD systems requiring less stringent convexity, or handling more general boundary/topological conditions, remain mathematically challenging due to the inherent lack of smoothing and the complex wave-transport coupling.

7. Numerical Schemes and Spectral Implications

Implementations of numerical schemes utilizing the hydrostatic MHD-wave system benefit from the elimination of rapid vertical acoustic oscillations, permitting larger timesteps and more efficient resolution of long-timescale MHD phenomena in stratified media (Braithwaite et al., 2012). The allowed wave spectrum, as confirmed analytically and numerically, consists of Alfvén and magneto-gravity modes only; vertical acoustic branches are excluded, resulting in explicit spectral filtering. This has direct consequences for the modeling and interpretation of MHD wave spectra in simulations of stellar interiors and solar atmospheric dynamics.

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Key References:

• "Gevrey well-posedness of the hydrostatic MHD-wave system" (Li et al., 6 Dec 2025)
• "A numerical magnetohydrodynamic scheme using the hydrostatic approximation" (Braithwaite et al., 2012)
• "Photospheric Logarithmic Velocity Spirals as MHD Wave Generation Mechanisms" (Mumford et al., 2015)