Energy and current balance

The electrons exchange energy with the electric field, the ions, the radiation and each other. Their currents set the field through the induction equations. Each term of these balances is a building block of an Equation, and EquationSystem checks that the pieces match.

The energy of the cold electrons

The energy of the cold electrons is an unknown of its own, as in DREAM:

\[W_\mathrm{cold} = \tfrac{3}{2}\, e\, n_\mathrm{cold} T_\mathrm{cold},\]

the algebraic row of thermal_energy_equation with the factor THERMAL_ENERGY_FACTOR. The row of the temperature holds the time derivative of this energy, TimeDerivative(:cold_thermal_energy) (TimeDerivative). Its terms are powers in W/m³, and the temperature follows from the algebraic row. The energy takes its initial value from the temperature and the density (set_thermal_energy!).

The channels of this row:

IonEnergyExchangeTerm with side = :ions is the same exchange in the row of the ion energies, so the two sides are antisymmetric. RunawayFluidEnergyTransferTerm is DREAM's heating by runaways without a runaway grid, the power $e c\, E_c\, n_\mathrm{re}$: runaways at the speed of light against the friction of the critical field $E_c$, with the Coulomb logarithm at $p = 20\, m_e c$. With a runaway grid, free_electron_energy_transfer_term takes its place: the energy moment of the electron-electron collisions of that grid.

The channels other than SecondaryHeatingTerm are those of DREAM's equation of $T_\mathrm{cold}$ (src/Settings/Equations/T_cold.cpp, DREAM v26.5). With radiation = :dream and bremsstrahlung_corrections = :dream, RadiatedPowerTerm takes DREAM's radiation. DREAM adds the collisional transfer from a runaway grid in every mode, and from a hot-tail grid in every mode but one:

FrictionEnergyTransferTerm is that transfer from the hot-tail distribution: the moment $\frac{1}{V'} \sum_{ij} m_e c\, v_i p_i\, \nu_s(p_i) f_{ij} \mathcal{V}'_{ij} \Delta p_i \Delta\xi_j$ at the cell centres, with the superthermal frequencies on the cold electrons alone. As a moment of f it takes any scheme of the operator, and it counts the energy of an electron until it leaves the grid, also through $p = 0$ under p_zero_boundary = :outflow. For a runaway grid it is DREAM's form of the transfer, the option next to Runes2P's free_electron_energy_transfer_term (Comparison with DREAM). test/test_cold_energy_dream.jl compares the row with ohmic heating, radiation and heat diffusion with DREAM in its fluid mode.

The total energy in the fully kinetic mode

In the fully kinetic mode the hot-tail distribution f covers the bulk. The electrons above the threshold of the hot region (a MomentumRegion with a ThermalMomentum limit) hold the energy

\[W_\mathrm{hot} = \frac{1}{V'} \sum_{ij} w_i\, m_e c^2 (\gamma_i - 1)\, f_{ij}\, \mathcal{V}'_{ij} \Delta p_i\, \Delta\xi_j,\]

the EnergyMomentTerm over that region, with the envelope $w$ of the region above the threshold $p_0$. total_energy_equation adds the unknown

\[W_\mathrm{tot} = W_\mathrm{cold} + W_\mathrm{hot}\]

with its algebraic row. The row of the temperature then holds $\mathrm{d}W_\mathrm{tot}/\mathrm{d}t$, TimeDerivative(:total_electron_energy), and the temperature follows from $W_\mathrm{cold} = W_\mathrm{tot} - W_\mathrm{hot}$.

The row lists only the channels by which energy enters or leaves the electrons of the hot-tail grid:

• The work of the field is $E j_\mathrm{ohm} + E j_\mathrm{hot}$ (two ProductTerms), where the ohmic current already leaves out the hot current, so each electron is counted once.
• The radiation, the heat transport and the exchange with the ions are as in the row of the cold energy.
• The heating by the secondaries of a kinetic ionization (SecondaryHeatingTerm) moves their energy from the sinks to the cold electrons.
• The sinks of the collisions of f are KineticEnergyLossTerm with part = :sinks and the hot region.
• The outflow at $p_\mathrm{max}$ is MomentumBoundaryFluxTerm with weight = :energy and coefficient = -1.
• The channel of the electrons that cross to a runaway grid is InterfaceEnergyFluxTerm.

The collisional transfer from the hot electrons to the cold ones does not appear in the row. Neither do the electrons that cross the threshold, nor the threshold that moves with the temperature. The time derivative acts on the energy of the hot region itself, so these exchanges inside $W_\mathrm{tot}$ are exact for any time stepper.

What the system enforces

Every change of $W_\mathrm{hot}$ that the row does not list is charged to the cold electrons. EquationSystem therefore checks the row against the equation of f. The docstring of EquationSystem gives the conditions.

• Collisions with sinks (synchrotron loss, bremsstrahlung stopping power, partial or no screening) remove energy from f, so the row needs the sinks term.
• The electrons that leave a Fokker-Planck term with collisions at $p_\mathrm{max}$, or cross its interface to a runaway grid, take their energy with them. The row needs the energy flux of that boundary or interface.
• Radial transport of f and RadialEdgeLossTerm are refused, since no term carries the energy moment of radial transport.
• A KnockOnTerm on the grid is refused, since no term carries the energy of the secondaries.
• A RosenbluthPutvinskiSourceTerm is refused, since its primaries have no energy unknown.

Each refused channel would take its energy from the cold electrons of the radius where the electrons are moved or created.

Why the hot-tail grid only

The runaway grid stays outside $W_\mathrm{tot}$. A relative error $\varepsilon$ of $W_\mathrm{tot}$ becomes a relative error of the cold energy of

\[\varepsilon\, \frac{W_\mathrm{tot}}{W_\mathrm{cold}},\]

so the temperature is lost to cancellation when $W_\mathrm{tot}$ is much larger than $W_\mathrm{cold}$. Keeping the runaways out keeps this ratio small. The runaway grid enters the row through two channels only: free_electron_energy_transfer_term for its collisions with the cold electrons, and InterfaceEnergyFluxTerm for the electrons that it receives.

The field work on the hot region

The row charges $E j_\mathrm{hot}$ for the work of the field on the hot region. The energy moment of the discrete field term over that region differs from it. The difference includes the energy $E_k(p_0)\, \Gamma$, with $E_k(p_0) = m_e c^2 (\gamma(p_0) - 1)$ and $\Gamma$ the flux of electrons that the field moves across the threshold $p_0$, which the total energy takes from the cold electrons. test/test_energy_budget.jl compares the two in a driven fully kinetic plasma, next to the transfer from a slowing-down tail and the checks of the row.

Energy transfer between the distributions

KineticEnergyLossTerm is the energy moment of a part of the collisions of a FokkerPlanckTerm:

\[L = -\frac{1}{V'} \sum_{ij} w_i\, m_e c^2 (\gamma_i - 1)\, \left[C_a f - C_b f\right]_{ij} \mathcal{V}'_{ij} \Delta p_i \Delta\xi_j,\]

with the envelope $w$ of the region.

• part = :free_electrons: $C_a$ the electron-electron collisions (electron_electron_operator), $C_b = 0$. This is the energy that f gives to the free cold electrons, free_electron_energy_transfer_term.
• part = :sinks: $C_a$ all collisions without the field, $C_b$ their electron-electron part. This is the energy that leaves the electrons by the friction on bound electrons, bremsstrahlung and synchrotron radiation.

The sinks take no flux through $p_\mathrm{max}$: the energy of the electrons that leave there belongs to MomentumBoundaryFluxTerm alone, so it is not counted twice. With the sinks charged to $W_\mathrm{tot}$, the cold electrons receive exactly the free-electron part of what the hot electrons lose.

MomentumBoundaryFluxTerm with weight = :energy and InterfaceEnergyFluxTerm weight the flux at the top of the grid with the kinetic energy of the last momentum cell. That is the energy that the discrete energy moment of the grid loses with the electrons. The electrons that cross to the runaway grid take their energy with them and do not heat the cold ones. test/test_energy_transfer.jl and test/test_energy_moment.jl test these terms and the energy moment, and test/test_friction_energy_transfer.jl tests FrictionEnergyTransferTerm.

The energy of the kinetic ionization

A kinetic ionization (KineticIonizationTerm) has no energy of its own: the friction on bound electrons of the Fokker-Planck term takes the energy of the ionization from f. With bound_friction = :ionization_split of CollisionSettings, that friction charges f the energy $I_k + \langle E_s \rangle_k$ of each ionization of the charge state $k$, with $I_k$ its ionization potential and $\langle E_s \rangle_k$ the mean energy of its secondary. The potential is stored in the ions and released at recombination (RadiatedPowerTerm with radiation = :recombination). The secondaries join the cold electrons through the closure of the cold density, and SecondaryHeatingTerm with cut = :none gives them their energy. ionization_energy_flows separates the two parts.

EquationSystem requires the heating by secondaries where the sinks of f are booked with this friction, with the same spectrum and cut = :none. With DREAM's bound_friction = :bethe, the friction does not hold the energy of each ionization, so the requirement does not apply (KineticEnergyLossTerm). Next to a KnockOnTerm on the bound electrons, SecondaryHeatingTerm with cut = :none counts the energy of the secondaries above the knock-on cutoff twice, since the knock-on term also puts them into f. Under the total energy the system refuses the knock-on term. In the row of the cold energy the system does not check the pair.

Currents and conductivity

The total current density is the sum

\[j_\mathrm{tot} = j_\mathrm{ohm} + j_\mathrm{hot} + j_\mathrm{re},\]

each an algebraic unknown. The hot current is the CurrentMomentTerm of f over the hot region. The runaway current is the current moment of the runaway grid, or $e c\, n_\mathrm{re}$ of a fluid runaway density.

The ohmic current takes a ConductivityTerm with a ConductivityModel:

• fluid electrons: $j_\mathrm{ohm} = \sigma E$, with the conductivity of Braams and Karney (braams_karney_conductivity), which is DREAM's conductivity of Sauter and Redl in a cylinder.
• fully kinetic: $j_\mathrm{ohm} = j(f) - j_\mathrm{hot} + E (\sigma - \sigma_\mathrm{kin})$, DREAM's corrected conductivity.

The conductivity of Braams and Karney: B. J. Braams, C. F. F. Karney, Phys. Fluids B 1, 1355 (1989), doi:10.1063/1.858966. The conductivity of Sauter: O. Sauter, C. Angioni, Y. R. Lin-Liu, Phys. Plasmas 6, 2834 (1999), doi:10.1063/1.873240. The formulae of Redl: A. Redl et al., Phys. Plasmas 28, 022502 (2021), doi:10.1063/5.0012664.

The test-particle operator does not conserve the momentum of the electron-electron collisions, so its conductivity is below $\sigma$. The correction adds the difference. Two models of $\sigma_\mathrm{kin}$:

• kinetic = :dream_fit: DREAM's fit of the conductivity of its kinetic operator (test_particle_conductivity_fraction).
• kinetic = :discrete_operator: the conductivity of the discrete operator of the run (kinetic_conductivity), by its linear response to a small field.

With :discrete_operator, the steady state of a small field has the reference conductivity whatever the grid and the collision settings are. The correction is a statement on the steady state: for some collision times after a change of the field, the current stays below $\sigma E$. With its heat-flux modes, the momentum-conserving FieldParticleTerm brings the conductivity of the distribution close to that of Spitzer and Härm, for a nonrelativistic bulk. The conductivity of Spitzer and Härm: L. Spitzer, R. Härm, Phys. Rev. 89, 977 (1953), doi:10.1103/PhysRev.89.977. The system refuses it next to a correction. The system also refuses a :discrete_operator model whose pitch rule differs from that of the CurrentMomentTerm in the same equation.

The ohmic heating ohmic_heating_term reads the same ohmic current, so the heating and the current agree by construction. test/test_conductivity.jl tests the conductivity against Spitzer and Härm and the Lorentz gas, the corrected conductivity in steady state and transient, and the resistive decay of a kinetic run against a fluid one.

Induction

induction_unknowns and induction_equations give the induction equations in cylindrical geometry, with DREAM's pairing of rows and unknowns. The poloidal flux $\psi$ is divided by the major radius, as in DREAM.

The row of the field holds the time derivative of the flux (TimeDerivative with of). Ampère's law is PoloidalFluxDiffusionTerm, with the gradient at the edge from $\psi_\mathrm{edge}$. The plasma current is CrossSectionIntegralTerm, and the inductance between the edge and the wall edge_to_wall_inductance. A loop voltage of zero at the wall is a perfectly conducting wall. The caller gives the row of $j_\mathrm{tot}$, the sum of the currents of the run. initial_poloidal_flux solves Ampère's law for the initial current density, as DREAM does.

test/test_induction.jl tests Ampère's law on analytic profiles and the resistive decay with the wall at the edge and at a distance. test/test_self_consistent_dream.jl compares a self-consistent field on both kinetic grids with DREAM. test/test_first_step_field.jl checks the field of one short step of a system that holds its current against Braginskii's, independently of DREAM (Collision models).