MURaM Simulation: Coronal Seismology Insights

• MURaM Simulation is a state-of-the-art 3D radiative MHD model that replicates solar photosphere, chromosphere, and low corona conditions for testing coronal seismology.
• It generates synthetic spectral observables—such as Fe XIII line intensities, Doppler velocities, and polarization signals—to validate magnetoseismological inversion techniques.
• The simulation framework quantifies parameter sensitivities, achieving magnetic field estimates with typical uncertainties of 20-30%, thereby enhancing diagnostic robustness.

The MURaM simulation is a state-of-the-art three-dimensional radiative magnetohydrodynamics (MHD) model employed for forward modeling of the solar atmosphere, particularly the photosphere, chromosphere, and low corona. In recent coronal seismology studies, such as the forward-modeling validation of two-dimensional (2D) coronal seismology with CoMP/UCoMP-like spectral imaging, the MURaM simulation provides the essential physical testbed and synthetic observables to assess, calibrate, and validate magnetoseismological inversion techniques for reconstructing the coronal magnetic field strength and direction (Yang et al., 15 Jan 2026).

1. Theoretical and Numerical Framework

The MURaM code solves the compressible MHD equations, incorporating radiative energy exchange, partial ionization, and a realistic equation of state on a uniform Cartesian grid. In the configuration pertinent to coronal seismology validation (Yang et al., 15 Jan 2026), the simulation domain is 295 × 49 × 197 Mm³ discretized at 384 × 384 × 96 km, capturing the upper convection zone up to the open corona. The magnetic field incorporates both mean and periodically varying vertical components, e.g., $B_z=5+50\,\sin(2\pi x/295\,\mathrm{Mm})$ G at the base, then evolves via self-consistent convective relaxation and coronal heating for $\sim 72$ hr.

Boundary conditions include a photosphere at $z \approx 8$ Mm and an upper boundary with an open, potential field. The model achieves a magnetostatic equilibrium with spontaneous excitation of transverse (kink) MHD waves in both open and closed structures due to the injection of energy at the lower boundary and internal instabilities. The simulation tracks full vector fields $(\rho, \mathbf{v}, \mathbf{B}, T)$ as a function of space and time, forming the “ground truth” for forward-modeled diagnostics.

2. Synthesis of Spectral Observables

To enable direct validation against spectroscopic observations (e.g., from CoMP/UCoMP), physical variables from the MURaM simulation are post-processed to generate synthetic spectral observables:

• Fe XIII 10747 Å and 10798 Å forbidden line intensities ($I_1$, $I_2$), using CHIANTI v11 atomic data to compute emissivities $\varepsilon = 0.83\,N_e^2\,G(T, N_e, h)$ under coronal conditions.
• Synthetic Doppler velocities, $v_{\mathrm{LOS}}$, derived from plasma velocities along the line-of-sight, convolved with the spectral response.
• Linear polarization signals (Stokes Q, U), enabling retrieval of the magnetic azimuth via $\phi = \frac{1}{2}\arctan(U/Q)$.

All synthetic observables are computed as line-of-sight integrals, weighted by emission, to reproduce the LOS-averaged (emissivity-weighted) nature of ground-based coronal observations.

3. Validation of 2D Coronal Seismology

The synthesized observables from the MURaM cube are subjected to the same analysis procedures as real CoMP/UCoMP data. The pipeline includes:

1. Wave-Tracking: Cross-correlation analysis on the time series of synthetic Doppler velocity $v_{\rm Doppler}(x,z,t)$ isolates the coherent, prograde (outward) branch of kink waves, extracting local phase speed $v_{\rm ph}(x,z)$ and propagation direction $\psi(x,z)$.
2. Electron Density Inference: The pixelwise intensity ratio $R = I_2 / I_1$ is inverted using a CHIANTI-derived lookup table to yield $n_e(x,z)$, which is converted to mass density $\bar{\rho}(x,z)$ (assuming a mean molecular weight $\bar{\mu} \approx 1.27$).
3. Magnetic Field Inversion: Using the measured $v_{\rm ph}$ and $\bar{\rho}$, the POS magnetic field magnitude is inverted via $B = v_{\rm ph}\sqrt{\mu_0\bar{\rho}}$.
4. Magnetic Orientation: The field direction is recovered from wave-propagation angles, and, for comparison, from the linear polarization azimuth.

Emissivity-weighted line-of-sight averaged “ground-truth” parameters $B_{\rm true}(x,z), \psi_{\rm true}(x,z)$ are computed from the native simulation cube for quantitative comparison with seismological inversions.

4. Parameter-Space Sensitivity and Error Characterization

Critical for the validation is the exploration of parameter space controlling the reliability of phase speed measurements and, hence, the inferred $B$. The key dimensionless parameter is

$$\alpha = \frac{t}{T} = \frac{L/v_{\rm ph}}{1/f} = \frac{n_{\rm pt}\,dx\,f}{v_{\rm ph}}$$

where $n_{\rm pt}$ is the number of pixels along the propagation path, $dx$ is the spatial sampling, $f$ the central bandpass frequency, and $v_{\rm ph}$ the local kink phase speed.

The analysis demonstrates:

• $\alpha \lesssim 0.3$: Time-distance ridges are nearly vertical, yielding large errors (RMSE $>50\%$).
• $\alpha \approx 0.5$: The RMSE plateaus to $\sim 20\%$.
• $\alpha \gg 0.5$: Excessive wave-path averaging can increase local errors in regions of strong phase-speed gradients.

The MURaM-based synthetic tests show root-mean-squared errors in seismologically-inferred $B$ of $\sim 22\%$ for $n_{\rm pt}=55$, $f=0.05$ Hz (with 55 path points and a high observational cadence), with near-unity correspondence between retrieved and true LOS-averaged $B$ and propagation direction.

5. Best Practices for Robust Seismological Diagnostics

Forward modeling with MURaM rigorously establishes optimal strategies for real observations:

• Use sufficiently long wave paths and high bandpass frequencies to ensure $\alpha \gtrsim 0.5$; adjust $n_{\rm pt}$ adaptively across the FOV.
• Employ the Fe XIII 10798/10747 Å line ratio, with CHIANTI lookups at the correct height and photo-excitation parameters, for robust density estimation.
• Perform cross-validation of wave-tracked direction and phase speed, as well as with polarization-based azimuth, to resolve directional ambiguities introduced by the Hanle effect (Van Vleck ambiguity).
• All diagnosed quantities are strictly LOS, emission-weighted averages; no strict geometric altitude is attributed to any value.

6. Limitations and Future Developments

Systematic limitations highlighted by forward-modeling include:

• Emissivity-weighted averaging along extended LOS hampers unambiguous height attribution, especially in regions of multi-thermal or tangled structure.
• Counter-propagating waves and overlapping wave trains introduce artefacts in phase speed measurement, although this is less significant in actual CoMP data where outward power is dominant.
• Photon noise, not natively included in the forward-model test, is a further source of uncertainty; next-generation tests must include synthetic noise injection.
• Complex multi-layer geometry may require advanced multi-component or tomographic inversion schemes beyond current single-path analyses.

The MURaM forward-modeling framework demonstrates that, under optimal parameter regimes, 2D coronal seismology recovers the LOS, emissivity-weighted coronal magnetic field with typical uncertainties of $20-30\%$. The simulation-based validation provides direct, quantitative justification for routine application of magnetoseismological inversions to distributed imaging spectropolarimetry, supporting the routine construction of large-FOV coronal magnetograms with upcoming instrumentation (Yang et al., 15 Jan 2026).