Understand the Zhao Stationary Closure

surface_current_model.model="zhao_stationary" assumes a stationary sheath outside the BEACH domain and connects its currents and velocity-space barriers to the top of the box. It does not evolve the outer sheath in time. BEACH solves the zero-current root once at run startup, then applies the fixed total currents to hit and return distributions tracked inside the box.

This page explains how the root, the 0 V reservoir, and photoelectron (PE) return and escape fit together. See the input-parameter reference for every key and constraint, and the output-format reference for the complete receipt list.

Problem solved at startup

The Zhao stationary closure assumes a planar, collisionless, unmagnetized outer sheath. It uses ambient electrons, cold ions, and photoelectrons when PE is active to solve a Zhao Type A, B, or C zero-current root. The solution provides the branch, surface potential $\phi_0$, potential minimum $\phi_m$, ambient-electron density, and species current densities.

BEACH does not resolve the root again during the run. The branch, $\phi_0$, $\phi_m$, and current targets remain fixed as the surface charge changes between batches. This differs from quasistatic matching-plane coupling, which iterates the outer response against the outward fluxes in every batch.

No PE means Type C

With photoelectron_source_scale=0.0, omit the PE species and PE-specific inputs. zhao_branch="auto" or "c" selects Type C and solves

$$J_e + J_i = 0.$$

The result contains only electron and ion absorption targets and the z-high kinetic map. It creates no PE particles and no PE emission, return, or escape targets.

With PE, close the large channels separately

The PE source density follows from solar elevation $\alpha$, reference density $n_{pe,ref}$, and scale $s_{UV}$:

$$n_{pe,0}=s_{UV}n_{pe,ref}\sin\alpha.$$

The ion-species density is the ion density at infinity. The configured electron density and PE emit_current_density_a_m2 sample the raw particle distributions; the Zhao root determines the fixed current targets. Current signs below denote contributions to surface charging.

Channel	Sign	BEACH treatment
Ambient-electron absorption $J_e$	negative	Surface absorption target
Ion absorption $J_i$	positive	Surface absorption target
PE emission $J_{emit}$	positive	Emission-reaction target
PE return $J_{return}$	nonpositive	Surface reabsorption target
PE escape $J_{escape}$	positive	External-boundary target, not deposited on the surface

The Zhao root supplies two balances:

$$J_{return}=J_{escape}-J_{emit}\le0, \qquad J_e+J_i+J_{escape}=0.$$

The PE channels and surface channels therefore close independently:

$$J_{return}+J_{emit}-J_{escape}=0, \qquad J_e+J_i+J_{return}+J_{emit}=0.$$

BEACH rescales emission and return separately instead of dividing by their small net PE current. The signed current carried outward by escaping PE particles is $-A J_{escape}$ and is not added to a surface element.

Map the 0 V reservoir to the current box top

The configured ambient Maxwell VDF represents a reservoir at the plasma potential at infinity, defined as 0 V. At the start of each batch, BEACH evaluates the mean z-high potential $\phi_f$ from the current surface charge. It selects an inflow tail that can reach both the outer access bottleneck and $\phi_f$, then maps the normal speed to the injection face by

$$\frac12 m v_{n,f}^{2}=\frac12 m v_{n,\infty}^{2}-q(\phi_f-0).$$

Inspect the potential variation across z-high when deciding whether this face-mean approximation is adequate.

Branch	Ambient-electron access	Electron / PE outward barrier	Ion
Type A	$\phi_m$	$\phi_m$	0 V
Type B / C	0 V	0 V	0 V

For an outward z-high crossing, BEACH uses the local crossing potential rather than the face mean, together with the particle’s normal kinetic energy. An electron or PE that cannot reach the fixed barrier is specularly reflected at z-high; only a particle with enough energy is classified as escape. The PE launch VDF remains the configured surface half-Maxwellian.

Fixed quantities and batch-dependent quantities

Fixed at run startup	Re-evaluated or resampled each batch
Branch, $\phi_0$, $\phi_m$, ambient-electron density	Surface charge and field inside the BEACH domain
Signed species current densities and reference area	Mean z-high potential $\phi_f$ and selected inflow tail
Absorption, emission, and escape current targets	Trajectories, hit positions, and local return / escape classification
Branch-dependent access and barrier potentials	Target charge $I\Delta t$ and scaling of the raw distribution

This separation enforces the stationary total currents, while the elementwise charging pattern remains a Monte Carlo result of the trajectories in each batch.

Validity domain

• BEACH does not solve the outer electric field, space charge, Debye shielding, turning-point distance, or flight time.
• Reflection at z-high is an adiabatic boundary contraction of the outer return trajectory.
• The uniform magnetic field must be zero because the closure is unmagnetized.
• A planar stationary solution does not respond self-consistently to curvature, collisions, outer transients, or plasma and illumination conditions that change during the run.
• A small zero-current residual does not establish convergence of the finite-particle hit and return maps.

See the input-parameter reference for the complete species, boundary, charge, and temperature contract. See Photoelectron emission and return for the raycast reaction charge and VDF.

Run an example

The complete PE case is examples/periodic2_zhao_fixed_current.toml.

beach examples/periodic2_zhao_fixed_current.toml

Use examples/periodic2_zhao_no_photo_fixed_current.toml for the no-PE Type C case. Both examples include the required reservoir, open z-high face, and fixed-current species in addition to [surface_current_model].

After completion, inspect summary.txt for the model, selected branch, and two budget residuals. Then use charge_ledger.csv to compare raw, target, and applied charge, hit counts, and fixed_*_weight_scale. For a PE case, vary ray count, batch width, and RNG seed until the elementwise return distribution converges. A completed run and a closed zero-current budget do not by themselves validate the outer-sheath approximation or the spatial distribution.

The implementation-level definitions are in SPEC section 7.7 and bem_surface_current_model.f90.