# 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](diagnostics.md),
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:

```toml
[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.