MAGPIE reference scales and flow analysis

VFP-2D can analyze a prescribed reconnection region using ion-flow, magnetic, and electron-distribution diagnostics. The intended first comparison is the carbon MAGPIE platform. A reduced calculation can test transport, energy partition and sensitivity to imposed upstream conditions alongside GORGON. It does not yet predict wire ablation, the pulsed-power circuit, radiative transfer, evolving charge states, or a validated experiment outcome.

A reproducible carbon reference

The defaults in adept.vfp2d.analysis.MagpieReference are upstream values from Hare et al., Physical Review Letters 118, 085001 (2017), Table I:

Quantity	Upstream reference
Electron density	$3\times10^{17}\ \mathrm{cm}^{-3}$
Mean ion charge	4
Electron temperature	15 eV
Ion temperature	50 eV (the reported upstream upper bound)
Reconnecting magnetic field	3 T
Inflow speed	50 km/s
Harris sheet half-width $\delta$	0.6 mm

The same paper reports sheet values $n_e=6\times10^{17}$ cm$^{-3}$, $\bar Z=6$, $T_e=100$ eV, $T_i=600$ eV, and an outflow of 130 km/s. Those are comparison targets, not imposed upstream conditions. The measured heating exceeds the classical dissipation estimate; matching input scales does not establish that a fluid-ion model can reproduce it.

The characteristic sheet half-length $L=7$ mm follows the definition in Hare et al., Physics of Plasmas 25, 055703 (2018), section III. This is a layer scale, not the box size or wire-array radius. That study also shows that changing electrode placement changes density, magnetic field, and layer geometry together. A local parameter scan does not establish which combinations are experimentally accessible; a global model or measurement must supply that mapping.

Generate the SI scale report without launching a simulation:

python -m adept.vfp2d.analysis

To evaluate another upstream state, provide a JSON object with any of the MagpieReference field names. All lengths, densities, fields and speeds use SI; temperatures use eV; ion_mass_u uses unified atomic mass units.

python -m adept.vfp2d.analysis --parameters carbon_upstream.json

For example:

{"electron_density_m3": 4e23, "magnetic_field_t": 3.5}

The reference calculation uses carbon mass $12\,u$ and produces:

Derived quantity	Value
Ion mass / electron mass	21874.7
Alfvén speed $B/\sqrt{\mu_0\rho_i}$	69.2 km/s
Alfvén Mach number	0.722
Isothermal ion-acoustic speed $\sqrt{e(\bar ZT_e+T_i)/m_i}$	29.7 km/s
Isothermal sonic Mach number	1.68
Magnetic pressure $B^2/(2\mu_0)$	3.58 MPa
Ram pressure $\rho_i u_{in}^2$	3.74 MPa
Scalar thermal pressure $e(n_eT_e+n_iT_i)$	1.32 MPa
Dynamic beta	1.04
Thermal beta	0.369
Ion inertial length $\sqrt{m_i/(\mu_0n_i\bar Z^2e^2)}$	0.717 mm
$d_i/\delta$	1.20
$L/\delta$	11.7
Inflow crossing time $\delta/u_{in}$	12 ns
Alfvén transit time $L/v_A$	101 ns

These are recomputed from the listed inputs. For example, the thermal beta agrees with the approximately 0.4 value in the 2018 platform table; it need not match every rounded value in the earlier article’s prose. The sonic speed above is isothermal and must not be substituted for an adiabatic CFL wave speed. Dynamic beta uses ram pressure, so $\beta_{dyn}=2M_A^2$; it is twice the directed kinetic-energy-density / magnetic-pressure ratio.

No resistivity is selected by this report. In particular, the reported $S\simeq120$ is a layer-temperature estimate, not the result of putting the 15 eV upstream temperature into a transport formula. Cooling, Coulomb mean free paths, ion viscosity and ionization require specified closures and local state. They are not silently inferred from these reference inputs.

Diagnostics in `moments.nc`

The geometry convention is x along the sheet/outflow and y across the sheet/inflow, interchanged relative to the experimental papers’ labels. The topology diagnostic samples a fixed center near the coordinate origin (or box midpoint when the origin is outside the domain). It requires opposite upstream $B_x$, reasonable two-sided balance, a small central in-plane field, an $A_z$ saddle and a central current peak. It is not a search for displaced X-points or multiple plasmoids. For asymmetric or migrating sheets, inspect the full fields or supply a separate topology analysis.

The upstream locations are the two peaks of $|B_x|$ on this central x line. Their saved coordinates, upstream_y_lower and upstream_y_upper, make the sampling reproducible and reveal motion or switches between peaks. The existing normalized_reconnection_rate retains its Nernst normalization. A zero-Nernst run now correctly marks that normalization invalid while still allowing the independent bulk and Alfvén diagnostics.

Variable	Meaning
`upstream_ion_inflow_y`	Mean signed inward ion speed; negative contributions remain negative
`upstream_ion_inflow_y_lower`, `upstream_ion_inflow_y_upper`	Separate sides, each positive inward
`upstream_alfven_speed`	Mean of the two reconnecting-field Alfvén speeds
`normalized_reconnection_rate_bulk`	Central $E_z/\langle
`normalized_reconnection_rate_alfven`	Central $E_z/\langle
`bulk_rate_normalization_valid`, `alfven_rate_normalization_valid`	Independent validity masks, unrelated to the Nernst run maximum
`centerline_ion_outflow_speed`	Mean of each side’s peak outward speed on the full center y line
`upstream_ram_pressure`, `upstream_magnetic_pressure`, `upstream_thermal_pressure`	Two-side mean pressure scales
`upstream_dynamic_beta`, `upstream_thermal_beta`	Ratios to total magnetic pressure, including guide field
`upstream_electric_flux_inflow`	Signed inward magnetic-flux transport from the full $E_z$
`upstream_bulk_flux_inflow`	Same measurement using only $-(\mathbf u_i\times\mathbf B)_z$
`current_sheet_gradient_half_width`	$\langle

The electric-flux diagnostic uses $\langle s\,\mathrm{sign}(B_x) E_z\rangle$, where $s=+1$ below and $-1$ above the sheet. It separates total transport from bulk-ion transport without pretending that $E_z/B_x$ is a unique magnetic advection velocity. Its sign is positive for flux delivered towards the sheet; the central signed reconnection rates follow the stored $E_z$ sign. Neither a finite inflow speed nor a finite flux budget independently proves reconnection.

In normalized solver units, $p_B=c_{norm}^2|B|^2/2$ and $v_A=c_{norm}|B_x|/\sqrt{\rho_i}$. Missing light_speed_normalized metadata makes these diagnostics NaN rather than assuming a normalization. temperature_energy_normalized supplies $T_0/(m_ev_0^2)$ for electron pressure. Saved temperatures and pressures must not be divided without this factor. The saved electron temperature is a scalar second moment in the ion frame; the reported thermal beta is a weak-drift thermal-pressure proxy and includes any resolved relative electron-drift contribution. New MLflow rate metrics omit invalid values and retain explicit valid-fraction metrics. The legacy Nernst metrics retain their existing zero-on-invalid convention.

The X-point history and Ohm-term plots include the ion bulk term. plots/reconnection/flow_history.png separates velocity, pressure ratios, gated rates and flux transport. topology_ion_final.png overlays ion-flow arrows; the existing Nernst topology plot remains available. The current_sheet_rms_width remains a current-weighted RMS over a finite central window; it is not interchangeable with the Harris half-width. Peak outflow is an easily inspected diagnostic, not a mass-flux-weighted measurement at an experimental observation chord.

A fixed-line Faraday budget

The centerline_lower_* and centerline_upper_* variables integrate signed $B_x$ from the fixed lowest saved y cell center to the sheet center, and from there to the highest saved y cell center. For either interval $[a,b]$:

\[ \Phi=\int_a^b B_x\,dy,\qquad \dot\Phi=E_z(a)-E_z(b). \]

bx_flux, faraday_rate, and faraday_residual save the inventory, electric flux difference and their mismatch. The raw residual includes external magnetic drive and must not be treated as a numerical conservation error in a driven run. The residual uses finite differences between saved times and trapezoidal spatial integration; refine the save cadence and grid before interpreting it as an evolution error. A single snapshot has a NaN residual. The integrals retain mean magnetic flux that the periodic $A_z$ reconstruction excludes. They do not assume moving upstream peaks are fixed integration boundaries and do not cover the unsaved half-cell beyond either edge. They are local Faraday consistency checks, not full-domain energy or mass conservation claims.

When reservoir_magnetic_field_change(t,x,y,component) is present, the analysis also saves each side’s source_bx_flux and source_accounted_faraday_residual. The former integrates the cumulative measured magnetic increment of the reservoir on the same fixed interval; the latter is

\[ \frac{d}{dt}\int_a^b(B_x-\Delta B_{x,R}^{cumulative})\,dy - [E_z(a)-E_z(b)]. \]

This subtracts the implemented source, including spatial-envelope derivatives, without constructing an artificial source electric field. It retains errors from time sampling, spatial quadrature and the physical update. Missing source history leaves the original diagnostics unchanged; a single snapshot still has no evaluable time derivative.

What a comparison can establish

A useful sequence is to verify magnetic-pressure acceleration and energy transfer, establish converged colliding-flow transport, and then compare field reversal, compression, flow, width and electron/ion heating with a specified experimental state or GORGON extraction. Keep each imposed profile and observation chord with the result. A short periodic-box run alone cannot establish sustained experiment-scale reconnection: boundary reservoirs, open exhausts, drive history, box-size independence and adequate runtime must be demonstrated separately.

The carbon reference has $d_i/\delta$ of order unity. Collisional ions can justify a fluid moment description in some regions without removing Hall physics or proving an isotropic scalar-pressure ion closure adequate. Counterstreaming ion populations, ion viscosity and anomalous heating need additional validation. Fixed $\bar Z=4$ conserves one chosen charge state; it cannot reproduce the measured increase to 6 in the sheet. The platform paper’s carbon layer cooling time (about 600 ns) is longer than a 12 ns crossing time, but this does not make radiation negligible everywhere or for an entire drive. Aluminium has much stronger cooling and requires a different validation program. Use these limitations to choose an analysis question and comparison observables, rather than treating agreement in one diagnostic as experimental design validation.

Review trigger: this page defines the analysis contract for MAGPIE-scale comparisons.