Reduced-MHD spectral smoke benchmark¶

The first physical workflow in the rebuilt MHX is a deliberately small periodic 2D reduced-MHD benchmark. It is intended to validate package architecture, spectral operators, time integration, diagnostics, and automatic differentiation. It is not yet a calibrated FKR tearing-growth benchmark.

Equations¶

MHX evolves magnetic flux \(\psi(x,y,t)\) and vorticity \(\omega(x,y,t)\):

\[ \partial_t \psi + [\phi,\psi] = \eta \nabla^2 \psi, \]

\[ \partial_t \omega + [\phi,\omega] = [\psi,\nabla^2\psi] + \nu \nabla^2 \omega, \]

with

\[ \nabla^2 \phi = \omega, \]

and periodic boundary conditions. The 2D Poisson bracket is

\[ [a,b] = \partial_x a\,\partial_y b - \partial_y a\,\partial_x b. \]

The implementation uses FFT derivatives and an inverse Laplacian with the zero Fourier mode fixed to zero:

\[ \widehat{\nabla^{-2} f}_{\mathbf{k}} = -\frac{\hat f_{\mathbf{k}}}{|\mathbf{k}|^2}, \qquad \widehat{\nabla^{-2} f}_{\mathbf{0}} = 0. \]

FAST initial condition¶

The current smoke initial condition is

\[ \psi_0 = \cos(y) + \epsilon \cos(x)\cos(y), \qquad \omega_0 = 0. \]

It is chosen because it is periodic, deterministic, smooth, and inexpensive. Dedicated tearing validation now lives in mhx benchmark fkr-growth, mhx benchmark linear-tearing-eigenvalue, and mhx benchmark linear-tearing-timedomain; the smoke initial condition remains an IO and differentiability fixture.

Diagnostics¶

The active diagnostics are

\[ E_B = \frac{1}{2}\langle |\nabla\psi|^2\rangle, \qquad E_K = \frac{1}{2}\langle |\nabla\phi|^2\rangle. \]

mhx run examples/linear_tearing.toml --outdir outputs/smoke writes a JSON manifest and scalar energy diagnostics for this benchmark.

Differentiability check¶

The test suite includes a finite-difference check of

\[ \frac{\partial E_B(t_f)}{\partial \eta} \]

through a three-step RK4 solve using jax.grad. This is intentionally tiny but establishes the contract that future inverse-design objectives must remain differentiable unless explicitly documented otherwise.

Mode amplitude and growth rate¶

The first tearing-oriented diagnostic tracks the normalized Fourier amplitude

\[ A_{k_x,k_y}(t) = |\hat{\psi}_{k_x,k_y}(t)| \]

for a configured mode through the diagnostics registry, currently (1, 1) in the FAST smoke benchmark. MHX fits

\[ A(t) \approx A_0 e^{\gamma t} \]

by least squares on \(\log A(t)\). This gamma_fit is useful for plumbing and regression tests, but it should not be interpreted as an FKR tearing rate until the eigenfunction, equilibrium, fit window, and parameter regime are validated.

The fit window is an explicit config field:

[diagnostics]
quantities = ["energy", "mode_growth", "divergence_error"]
mode = [1, 1]
fit_time_window = [0.02, 0.1]

MHX records fit_time_window and fit_sample_count in diagnostics.json, so growth-rate comparisons are auditable and reproducible.