Joglekar 2014 reconstruction and hydro coupling

This page is both a benchmark specification and an implementation boundary. The target is A. S. Joglekar et al., Physical Review Letters 112, 105004 (2014), doi:10.1103/PhysRevLett.112.105004. The reduced heating configuration in configs/vfp-2d/joglekar-2014-prl.yaml exercises the parts that are implemented today. It is not yet the fully implicit PRL reproduction.

What the benchmark requires

The paper retained the Cartesian-tensor equivalent of all spherical harmonics through \(\ell=2\). Its generalized kinetic Ohm law was

\[ \mathbf E=\bar\eta\mathbf j+\frac{\mathbf j\times\mathbf B}{e n_e} -\mathbf v_T\times\mathbf B -\frac{\nabla(n_em_e\langle v^5\rangle)}{6en_e\langle v^3\rangle} -\frac{\nabla\cdot(n_em_e\langle\mathbf{vv}v^3\rangle)}{2en_e\langle v^3\rangle}, \]

with

\[ \mathbf v_T=\frac{\langle\mathbf v v^3\rangle}{2\langle v^3\rangle} +\frac{\mathbf j}{e n_e}. \]

The last term is an \(f_2\) pressure-anisotropy contribution and is essential at the X point; an \(f_0+f_1\) model is therefore insufficient. VFP2D now emits the exact scalar, vector, and traceless tensor moments in these equations, together with \(\mathbf j\), \(T_e\), and \(\mathbf v_T\).

The published setup used:

\(T_{e0}=1.6\) keV and \(n_{e0}=2.5\times10^{22}\,\mathrm{cm}^{-3}\);
\(v_{th}/c=0.08\), \(\omega_{pe}\tau_n=125\), and \(\lambda_{mfp}=0.34\,\mu\)m;
\(-1500<x/\lambda_{mfp}<1500\) and \(-100<y/\lambda_{mfp}<100\), with a uniform central region \(-400<x/\lambda_{mfp}<400\) and stretched cells outside it;
\(\Delta x=13.3333\lambda_{mfp}\), \(\Delta y=3.125\lambda_{mfp}\), and \(\Delta v=0.0625v_{th}\);
two Gaussian inverse-bremsstrahlung heating spots with \(r_0=50\lambda_{mfp}\), \(H_0=0.5\), laser intensity \(2.5\times10^{14}\,\mathrm{W/cm^2}\), and centers at the two \(y\) boundaries;
a prescribed hidden density derivative \(\partial_z n=(n_0/L_n)\) times the same spatial envelope, with \(L_n=50\lambda_{mfp}\), switched off at \(800\tau_n\);
stationary ions, with the main comparison at \(19000\tau_n\) and the final state at \(27000\tau_n\simeq0.6\) ns.

The boundary-centered source fixes the spot separation from the published \(y\) extent. The Letter does not state the mean ionization in its five pages; that value must come from an original input deck, a longer manuscript, or a controlled inference and must not silently become a “reference” value in ADEPT.

Long-timescale field modes

The current explicit Vlasov–Maxwell integrator is useful for equation tests and kinetic wave problems. It is not a viable route to \(27000\tau_n\): its timestep is constrained by the light-wave CFL condition and electron plasma oscillations, neither of which is part of the slow collisional physics being studied. The PRL used a fully implicit VFP code.

The production deck uses Chang–Cooper differencing for \(f_{00}\) because laser heating drives the distribution away from a Maxwellian. The nominal 2 fs Strang step applies the collision operator in 1 fs half-steps, about \(0.045\tau_n\) for the published normalization. The alternative log-mean flux has a zero semidiscrete energy derivative at the frozen state, not exact energy conservation for a finite nonlinear implicit step. Both the collision timestep and velocity grid must therefore be included in convergence scans.

The implemented kinetic-ohm mode takes the following quasineutral path:

enforce \(n_e=Z n_i\);
obtain the required current from Ampere’s law without displacement current, \(\mathbf j=\nabla\times\mathbf B/\mu_0\);
evaluate the electric field from the complete moment equation above;
advance \(\mathbf B\) with Faraday’s law;
minimally project the bulk-current moment of \(f_1\) onto the Ampere current while retaining its velocity-dependent heat-flux structure;
retain all five moment contributions as output diagnostics.

This removes the light-wave and plasma-frequency timestep restrictions and is an appropriate first long-timescale benchmark model. It assumes negligible electron inertia and the small contracted \(\langle\mathbf{vvv}\rangle:\mathbf E\) term used to derive the displayed Ohm law. The current projection also exchanges a small amount of kinetic energy with the unresolved constraint system. Both effects must be measured in convergence tests.

The implemented oshun-implicit alternative solves a local discrete \(3\times3\) kinetic-current response for the electric field enforcing Ampere’s current, without projecting f1 or using the algebraic Ohm law. Faraday and non-electric transport remain explicit, so this is not a fully implicit Maxwell solve. It currently supports stationary ions without spatial sharding. ampere provides a second alternative: an explicit Ampere residual divided by a configurable relative permittivity, with correspondingly slower light and plasma frequencies. See the field-mode configuration. A fully implicit field/transport solve and long-time validation of the full published geometry remain future work.

The 2.5D Biermann source

For gradients lying in the evolved \(x\)–\(y\) plane, the Biermann field should emerge from the quasineutral electric solve. The PRL geometry is different: it evolves \(x,y\) but prescribes \(\partial_z n\). The implementation therefore uses an explicit hidden_density_gradient field in the pressure-gradient part of the same Ohm residual. The kinetic equation receives the matching \(\partial_z f=(\partial_z n/n)f\) Tzoufras streaming term; this balances the pressure electric field for an isothermal Maxwellian and prevents the current projection from turning that equilibrium force into numerical heating. It must not be implemented as an arbitrary magnetic source because doing so would lose the corresponding electric field and electron response.

The published benchmark deliberately gives \(\partial_z n\) the same two-spot Gaussian envelope as the laser source. A uniform in-plane hidden gradient is also tested as a null problem: it must leave an isothermal state spatially uniform and generate no Biermann field.

The source gate is smooth and differentiable during optimization but can reproduce the published switch-off at \(800\tau_n\) in validation mode. A direct \(\nabla n\times\nabla T\) diagnostic should be saved in both modes.

Recommended ion-fluid coupling

Use a quasineutral ion fluid and keep electrons kinetic in the local ion center-of-mass frame. The minimum useful fluid state is \((n_i,n_i\mathbf u_i,\mathcal E_i)\):

\[ \partial_t n_i+\nabla\cdot(n_i\mathbf u_i)=0, \]

\[ \rho_i D_t\mathbf u_i=\mathbf j\times\mathbf B -\nabla\cdot\mathbf P_e-\nabla p_i+\mathbf R_{ext}, \]

with electron pressure and heat transport taken directly from \(f_{\ell m}\), not from a second electron-energy equation. The electron distribution must receive the equal and opposite work and momentum exchange. Its update splits naturally into:

conservative bulk advection by \(\mathbf u_i\);
relative electron streaming and electromagnetic harmonic coupling;
velocity-space compression/inertial terms caused by \(\nabla\mathbf u_i\) and acceleration of the ion frame;
implicit collisions and laser heating.

The coupled source half-steps include \(\mathbf j\times\mathbf B\), with \(\mathbf j=c^2\nabla\times\mathbf B\) in code units, and midpoint ion mechanical work \(\mathbf u_i\cdot(\mathbf j\times\mathbf B)\). The first half-step uses the initial magnetic field and the second uses the field after the kinetic/Faraday step. Each source update remaps the electrons into the updated ion frame. Electron pressure contributes ion mechanical work \(-\mathbf u_i\cdot\nabla\cdot\mathbf P_e\); electron peculiar pressure work \(-\mathbf P_e:\nabla\mathbf u_i\) is supplied once, by the moving-frame deformation operator. Their sum is the pressure-flux divergence \(-\nabla\cdot(\mathbf P_e\cdot\mathbf u_i)\), not pointwise cancellation of thermal and mechanical work. No additional pressure source is applied to electron f00. Collision and inverse-bremsstrahlung inputs use the midpoint ion number density \(n_i=\rho_i/m_i\). Setting ion_fluid.frozen: true suppresses both hydro and coupled ion source updates. Coupled ion runs require grid.lmax >= 1 and grid.mmax >= 1 so frame remaps can preserve all three momentum components. They reject active hidden density gradients: the ion continuity and pressure operators only include x/y derivatives. The hidden-gradient workflow remains available with ion_fluid.active: false.

For the first coupled milestone, an isothermal or adiabatic ion pressure law is adequate. The acceptance tests should verify total particle number, total momentum, total energy, and \(\nabla\cdot\mathbf B\) across the kinetic–fluid exchange. A prescribed bulk velocity is a useful operator test, but it should not be presented as the physical coupling.

Delivery sequence

Frozen-ion, moment-projected kinetic-Ohm solve on a periodic uniform mesh. Implemented.
Published 2.5D hidden density gradient, IB profile, five-term diagnostics, and x-sharded wide-box pilot. Implemented; long-time validation remains. Reproduce magnetic generation, Nernst inflow, the five Ohm-law traces, gated reconnection-rate history, and the local \(B_z\) quadrupole score.
Fully implicit kinetic-current response and mapped/stretched \(x\) mesh with non-periodic thermal boundaries for the full published box.
Conservative ion-fluid coupling. Implemented with local and nonlinear refinement tests; production-scale moving-ion comparisons remain.
Higher \(\ell_{max}\), convergence scans, and differentiable parameter inference.

The benchmark should first lock the published \(\ell_{max}=2\) result, then demonstrate that increasing \(\ell_{max}\) changes observables by less than a stated tolerance. That distinguishes reproduction from extension.