LPSE 2D (Envelope-2D) Solver

Example decks live in configs/envelope-2d/. To run one:

uv run run.py --cfg configs/envelope-2d/tpd

Equations and Quantities

These equations model the evolution and interaction of the complex envelopes of light waves and plasma waves. This is faster than modeling the plasma waves using a fluid or kinetic solver along with modeling the light waves.

Note on Pump Depletion

One can solve these equations with or without “pump depletion”. “Pump depletion” is the effect of the plasma waves on the light waves. By default the pump is prescribed analytically (an external driver for the plasma waves), which is adequate below the absolute instability threshold; above it the plasma-wave and Raman fields grow without bound because nothing depletes the pump.

Setting terms.light.pump_depletion: true evolves the pump E0 with the finite-difference envelope solver and a two-point boundary injector at the low-density side. Every active instability contributes its reciprocal pump term:

SRS contributes \(-i e/(4 \omega_1 m_e) (\nabla^2 \phi) \mathbf{E}_1\).
TPD contributes \(i e/(4 \omega_0 m_e) e^{i(\omega_0-2\omega_{p0})t} E_{h,y}\nabla\!\cdot\!\mathbf{E}_h\) to the y-polarized pump. This is the discrete reciprocal of the existing TPD source and conserves \(\int[|E_0|^2+(\omega_{p0}/\omega_0)|\mathbf{E}_h|^2]\) for the coupling terms alone.

Both terms are included when TPD and SRS are enabled together. Dynamic pump evolution enables true transmission/absorption diagnostics and provides the physical saturation channel needed above threshold.

SRS diagnostics

With terms.epw.source.srs on, the default time series includes OSIRIS-comparable channels (fields converted to \(m_e c \omega_0/e\), lengths to \(c/\omega_0\), fluxes normalized to the nominal incident flux):

series	meaning	OSIRIS scan2 counterpart
`epw_energy`	\(\frac{1}{4}\,dx \sum_x \langle\|E_{epw}\|^2\rangle_y\) (cycle-averaged field energy)	`W(t) = 1/2 dx sum e1^2`
`epw_dissipation`	total EPW energy handed to electrons per ps (Landau + collisional, the solver’s own rates; includes the kinetic half)	absorbed fraction / hot-electron source
`epw_boundary_loss`	total EPW energy removed by the absorbing boundaries per ps	—
`incident_flux`, `transmitted_flux`	pump flux at probes near each edge	`incident_t`, `T_t`
`reflected_flux`, `backrefl_flux`	Raman flux leaving left / right	`R_t`
`reflectivity`, `e1_sq`	legacy probes (unchanged)	—

post_process logs scalar metrics with the same names and definitions as osiris_lpi (laser_reflectivity, laser_transmissivity, laser_absorbed_frac, per-quarter _seg{i}of4 variants, epw_growth_rate per \(\omega_0\) with the identical automated fit window, electron_energy_frac_final, and the laser_incident_flux_ratio health check).

Electron Plasma Waves

\[ \nabla \cdot \left[ i \left(\frac{\partial}{\partial t} + \nu_e^{\circ} \right) + \frac{3 v_{te}^2}{2 \omega_{p0}} \nabla^2 + \frac{\omega_{p0}}{2}\left(1-\frac{n_b(x)}{n_0}\right) \right] \textbf{E}_h = S_{TPD} + S_h \]

Two Plasmon Decay

\[ S_{\text{TPD}} \equiv \frac{e}{8 \omega_{p0} m_e} \frac{n_b(x)}{n_0} \nabla \cdot [\nabla (\textbf{E}_0 \cdot \textbf{E}_h^*) - \textbf{E}_0 \nabla\cdot \textbf{E}_h^*] e^{-i (\omega_0 - 2 \omega_{p0})t} \]

Laser Driver

We only have a plane wave implementation for now:

\[ E_0(t, x, y) = \sum_j^{N_c} A_j ~ \exp(-i k_0 x - i \omega_0 \Delta\omega_j ~ t + \phi_j) \]

Boundary Conditions

Set per axis under terms.epw.boundary:

Value	Behaviour
`absorbing`	A damping layer of width `grid.boundary_width` is applied at both ends of that axis. The mask is \(\exp(-\alpha \, dt \, (1 - \text{env}_x \text{env}_y))\) with \(\alpha =\) `grid.boundary_abs_coeff` and a tanh ramp over `boundary_width / 5`, so the envelope is attenuated smoothly rather than clipped.
anything else	Periodic — the underlying representation is spectral, so this is the default behaviour when no absorbing layer is requested.

x and y are configured independently, which is the usual arrangement for TPD: absorbing along the density gradient, periodic transverse to it. The \(k=0\) mode is masked out of the spectral operators.

Forcing and Drivers

With terms.light.pump_depletion: false, the laser is prescribed forcing and is therefore appropriate only while depletion is negligible. With depletion enabled, the injected pump propagates through the box and receives the reciprocal TPD and/or SRS feedback described above.

Term	Role
\(E_0\) (laser driver)	A sum of \(N_c\) plane-wave components, each with amplitude \(A_j\), frequency offset \(\Delta\omega_j\), and phase \(\phi_j\). Bandwidth models (SSD, CPP speckle) are built by populating these components — see the example decks `tpd-static-cpp-speckle.yaml` and `tpd-dynamic-ssd-speckle.yaml`.
\(S_\text{TPD}\)	The two-plasmon-decay source coupling \(E_0\) to the plasma-wave envelope.
\(S_h\)	A noise/seed source for the plasma waves.
\(\nu_e^\circ\)	Landau damping of the plasma-wave envelope.

Ion-acoustic waves

terms.iaw.active: true evolves the fractional ion-density perturbation and ion-velocity divergence using the MATLAB LPSE split step. The pressure contains acoustic restoring force plus ponderomotive drive from the EPW, pump, and Raman fields. The resulting density perturbation feeds back into the detuning of all three envelope equations. IAW boundary damping defaults to the EPW boundary configuration; collisional and Landau damping are independently configurable.

Hybrid particles and the 2D boundary

The HPE particle tracker supplies self-consistent Landau-damping feedback and hot-electron diagnostics in both quasi-1D and 2D runs. For ny > 1, a single box-wide ensemble carries (x, y, p_x, p_y) and samples both electric-field components; angle-resolved projected-velocity histograms provide the resonant distribution for every (k_x, k_y) mode without assigning particles to individual grid cells. Periodic boundaries wrap particles, while reservoir walls use flux-weighted thermal-tail reinjection. HPE can be combined with TPD, SRS, pump depletion, and IAWs, as demonstrated by configs/envelope-2d/tpd-srs-iaw.yaml; srs-hpe.yaml remains the smaller quasi-1D example.

What Gets Saved

binary/:

File	Contents
`fields.xr`	The real-space envelope fields over time
`k-fields.xr`	The same in wavenumber space — this is where TPD growth is measured
`series.xr`	Scalar time series

When IAWs are active, fields.xr and k-fields.xr also contain iaw_density and iaw_velocity_divergence; series.xr includes iaw_density_sq and iaw_density_abs_max.

plots/, per field k:

File	Contents
`<k>_x.png`, `<k>_x_r.png`	Spatial profiles, magnitude and real part
`<k>.png`, `log-<k>.png`, `real-<k>.png`	Slices, linear / log / real part
`spacetime-<k>.png`, `spacetime-log-<k>.png`, `spacetime-real-<k>.png`	Space-time plots

The log-scaled space-time plots are the ones to read for instability growth rates.

Configuration Reference

See the Configuration Reference for complete YAML schema documentation.