8  Initial conditions and matched modes

8.1 Specify a physical field

A reproducible amplitude statement is

Bcom_rms(z_reference; kmin <= k <= kmax) = value
normalization = total_band_rms | reference_band_rms
window = declared cutoff function; seed = declared integer

A bare “3 pG PMF” leaves the epoch, bandwidth, normalization, and realization undefined. Direct initialization does not supply a pre-start transfer model: MHD_PMF_NORMALIZATION_Z must equal MHD_ZSTART unless a compatible pre-evolved state is supplied. Earlier physical normalization requires transferring density, velocity, magnetic fields, photons, and chemistry consistently to the handoff. ^{pmf_normalization_z}

8.2 Batchelor initializer

The nonhelical Gaussian initializer constructs a random vector potential in Fourier space and forms

\widehat{\mathbf B}(\mathbf k)=i\mathbf s(\mathbf k)\times\widehat{\mathbf A}(\mathbf k), \qquad s_j(k_j)=\frac{\sin(k_j\Delta x)}{\Delta x}. \tag{8.1}

The centered curl uses the same derivative symbol as the centered divergence diagnostic, giving s dot B = 0 to roundoff. A continuum/spectral curl uses s_j=k_j instead and is not automatically divergence-free under the centered grid diagnostic. The low-k magnetic shell energy has Batchelor slope E_B(k) proportional to k^4; this is not a statement that the plotted three-dimensional per-mode P_B(k) also has slope four. ^{pmf_batchelor_field}

kcut controls a smooth cutoff; kmax is the hard band edge. With no explicit kcut, the raw runner scales it with grid Nyquist. That is useful for a grid-scale benchmark, not a fixed-physical-spectrum resolution experiment.

8.3 Lock phases and amplitudes

Keep the box, physical band, seed, window, normalization, realization algorithm, curl symbol, and lock grids fixed when refining:

• mode_lock_n constructs the shared low-mode realization on a fixed grid. The same seed without locking is insufficient when random-array traversal changes with the simulation grid.
• curl_reference_n fixes the centered derivative symbol used to generate shared coefficients. It must not exceed the simulation grid.
• reference_kmax and reference_kcut preserve the declared low band when adding ultraviolet modes. MHD_PMF_HIGH_MODE_LOCK_N can separately lock the high-band realization.
• mode_realization="physical_hash" keys vector-potential coefficients by physical Fourier indices. Shared, non-Nyquist modes in integer-related boxes then retain phases and polarizations. realization_box/box must be a positive integer for this construction.
• physical_hash_zero_phase is a deliberately altered phase ensemble, not another draw from the default random-phase ensemble. fixed_mode_power changes amplitude statistics and must be reported too.

Check actual Fourier coefficients in overlap bands; do not rely on matching configuration labels alone. ^{_physical_hash_vector_potential_k}

8.4 Normalization and resolution

total_band_rms fixes the RMS over all generated modes. reference_band_rms fixes RMS only below the reference cutoff, permitting total RMS to increase when new UV modes are added. Renormalizing every extended band to the same total RMS changes the shared large-scale field.

The initializer reports reference-band and total RMS separately. Record both. PMFSpectrumSpec and pmf_multiband_manifest provide a resolution-independent spectrum definition and minimum even grids for a set of band edges. ^{PMFSpectrumSpec pmf_multiband_manifest}

For a hard cutoff, cells per shortest magnetic wavelength are

n_\lambda=\frac{2\pi N}{k_{\max}L}. \tag{8.2}

Specify minimum_cells_per_wavelength with kmax to make this a runtime requirement. A Nyquist mode always has two cells per wavelength, no matter how large the grid. Quadratic Maxwell stress reaches roughly twice the magnetic wavenumber, so resolving the magnetic wavelength alone does not establish force convergence. A useful first science matrix targets at least 32 cells per shortest B wavelength and 16 per shortest quadratic-force wavelength, then measures convergence. The shipped research example’s looser cutoff is not a certificate that this criterion has been met.

Try it: compare grids without changing the physical problem

Create two copies of examples/pmf-recombination.toml. Keep kcut, kmax, mode_lock_n, curl_reference_n, seed, band normalization, and physical box identical; double only run.n. Adjust Gaussian smoothing cells to preserve its physical length. Run to identical output redshifts and compare the initial spectra, occupied mode counts, and subsequent histories before interpreting the difference as a resolution error.

8.5 Analytic modes and phases

Mode ICs are cell-centered at x_i=(i-1/2)/N, with phase 2*pi*kmode*x_i. The domain is periodic and 0 < kmode < N/2 for matched mode diagnostics.

IC	Magnetic components	What it tests
forcefree / circular	By=b*sin(phase), Bz=b*cos(phase), Bx=0	Constant magnetic pressure
sine / magpressure	By=b*sin(phase), other components zero	Quadratic density/force response at 2k
alfven_linear	Bx=Bguide, By=Bpert*sin(phase)	Small transverse guide-field perturbation
meanfield_linear / sv25_mode	Oblique guide field; cosine in-plane or out-of-plane perturbation	Matched magnetosonic/Alfven reference comparisons

The sine-mode amplitude convention uses peak field, while rms rescales the non-force-free sine to its RMS. For a scalar sine wave, peak/RMS is sqrt(2); the circular two-component mode already has constant magnitude. Report MHD_PMF_SINGLE_MODE_NORMALIZATION rather than silently converting. ^{_init_single_b_k! _init_meanfield_linear_k!}

The default gas starts uniform and at rest; the lattice DM begins at uniform cell centers with zero peculiar velocity. These are controlled PMF forcing experiments, not a general primordial adiabatic matter realization. Photon adiabatic/comoving ICs and optional terminal momenta must be recorded. Changing start redshift changes the pre-start history even with the same PMF seed.

8.6 Importing a handoff

Generate and inspect a magnetic realization

julia --project=. -t auto scripts/generate_ic.jl \
  examples/cosmology-smoke.toml results/initial-field

This standalone Batchelor magnetic-field generator writes bx.f32, by.f32, bz.f32, and ic.toml, including field hashes, dimensions, byte order, normalization statistics, and source fingerprint. Arrays are Float32 in Julia column-major order with x fastest, in MHD code magnetic units at the configured start redshift, not raw gauss or nG. It refuses a nonempty output directory. It does not pre-evolve gas, photons, particles, or chemistry. ^{magnetic_code_amplitude}

To use the exact field at the same epoch and units, configure MHD_INITIAL_B_IMPORT_DIR in [environment] with that directory. Regenerate after changing the normalization epoch. Magnetic-only import is not a full physical handoff and is not checkpoint resume. Use the separate Python matched-state generators for controlled linear guide-field handoffs.

Import a complete pre-evolved state

MHD_INITIAL_STATE_DIR supports the driver’s explicit state import path. Treat field names, dimensions, scalar type, storage convention, normalization epoch, and photon/chemistry payload as a single contract. Use the parser’s actual schema rather than constructing an ad hoc file from a plot. State import is not the same operation as checkpoint resume. ^{initial_state_dir}