Closure

Closure#

Hermes-3 currently uses the standard Braginskii closure, but with a selection of collision frequencies that can be used: the formulas use all solved collisions by default in order to have some accounting of multiple ion species, which are not considered in standard Braginskii. See the next section for how this can be changed to reproduce the exact Braginskii closure.

The different parts of the closures are currently implemented in different components, which are described in the Equations section and on this page. They all use collision frequencies calculated in the Collisions component.

Conduction: Parallel conduction for all species is implemented in the braginskii_conduction component.

Viscosity: Parallel and perpendicular viscosity for ions is in the braginskii_ion_viscosity top-level component. The parallel viscosity for electrons is in braginskii_electron_viscosity.

Thermal force: Thermal force is implemented here: braginskii_thermal_force.

Friction force/heating: The frictional momentum and energy sources are calculated in the top-level Braginskii Friction component.

Thermal equilibration: Thermal transfers are calculated in the Braginskii Heat Exchange top-level component.

Neutral diffusion: The parallel projection of diffusion from the wall in 1D is captured in the 1D: neutral_parallel_diffusion top-level component, while both parallel Braginskii transport and perpendicular pressure-diffusion for 2D/3D are captured in the 2D/3D: neutral_mixed species-level component.

Collision frequency selection#

When configured in multispecies mode (default), all collision frequencies for solved collisions of a particular species are counted, e.g. electron conduction would include ee, ei and en collisions, and ion conduction would include ii, ie, in and CX. In this way, the closure is similar to that used in UEDGE and has a way to account for multiple ion species. Note that any of the collisions can be disabled (see next section).

For code comparison purposes, it can be useful to reproduce exact Braginskii closure, e.g. featuring only self-collisions for conduction and viscosity. This is enabled by changing the mode to braginskii. Note that this will be invalid for simulations with multiple ion species.

The collision frequency choice for conduction is done under the species header, e.g.:

[d+]  # Deuterium ions
type = (evolve_density, evolve_pressure, evolve_momentum,
      noflow_boundary, upstream_density_feedback)

conduction_collisions_mode = braginskii

The choice for viscosity is done under the [braginskii_ion_viscosity] header due to it being a top level component, e.g.:

[braginskii_ion_viscosity]
viscosity_collisions_mode = braginskii

In addition to the parallel closure, there is also a collision frequency choice for neutral diffusion, which is present both in neutral_parallel_diffusion (1D) and neutral_mixed (2D). In this case, there are again two choices: multispecies features all enabled collisions, while afn selects only ionisation and charge exchange, consistent with the AFN (Advanced Fluid Neutral) work in N. Horsten Nucl. Fusion 57 (11) 116043 (2017).

neutral_mixed is a species level component and should be set under the species header:

[d]
type = neutral_mixed
diffusion_collisions_mode = afn

While neutral_parallel_diffusion is a top level component and must be set under its own header, e.g.:

[neutral_parallel_diffusion]
diffusion_collisions_mode = afn

Braginskii Collisions component#

Inputs and ouputs#

This top-level component calculates the collision frequencies of all collisional processes in Hermes-3. These frequencies are then used to calculate the closure terms. By default, the following collisions are enabled:

[braginskii_collisions]
electron_ion = true
electron_electron = true
electron_neutral = false
ion_ion = true
ion_neutral = false
neutral_neutral = true

electon_neutral collisions are disabled as they are are typically a very minor contributor, while ion_neutral collisions are disabled as they are already accounted for by charge exchange which is enabled by default.

All of the collision frequencies are added to the state in species["collision_frequencies"]. They are also available as diagnostics, e.g. Kd+e_coll is the ion-electron collision frequency.

Theory#

For collisions between charged particles. In the following all quantities are in SI units except the temperatures: \(T\) is in eV, so \(eT\) has units of Joules.

Debye length \(\lambda_D\)

\[\lambda_D = \sqrt{\frac{\epsilon_0 T_e}{n_e e}}\]

Coulomb logarithm, from [NRL formulary 2019], adapted to SI units

  • For thermal electron-electron collisions

    \[\ln \lambda_{ee} = 30.4 - \frac{1}{2} \ln\left(n_e\right) + \frac{5}{4}\ln\left(T_e\right) - \sqrt{10^{-5} + \left(\ln T_e - 2\right)^2 / 16}\]

    where the coefficient (30.4) differs from the NRL value due to converting density from cgs to SI units (\(30.4 = 23.5 - 0.5\ln\left(10^{-6}\right)\)).

  • Electron-ion collisions

    \[\begin{split}\ln \lambda_{ei} = \left\{\begin{array}{ll} 10 & \textrm{if } T_e < 0.1 \textrm{eV or } n_e < 10^{10}m^{-3} \\ 30 - \frac{1}{2}\ln\left(n_e\right) - \ln(Z) + \frac{3}{2}\ln\left(T_e\right) & \textrm{if } T_im_e/m_i < T_e < 10Z^2 \\ 31 - \frac{1}{2}\ln\left(n_e\right) + \ln\left(T_e\right) & \textrm{if } T_im_e/m_i < 10Z^2 < T_e \\ 23 - \frac{1}{2}\ln\left(n_i\right) + \frac{3}{2}\ln\left(T_i\right) - \ln\left(Z^2\mu\right) & \textrm{if } T_e < T_im_e/m_i \\ \end{array}\right.\end{split}\]

  • Mixed ion-ion collisions

    \[\ln \lambda_{ii'} = 29.91 - ln\left[\frac{ZZ'\left(\mu + \mu'\right)}{\mu T_{i'} + \mu'T_i}\left(\frac{n_iZ^2}{T_i} + \frac{n_{i'} Z'^2}{T_{i'}}\right)^{1/2}\right]\]

    where like the other expressions the different constant is due to converting from cgs to SI units: \(29.91 = 23 - 0.5\ln\left(10^{-6}\right)\).

The frequency of charged species a colliding with charged species b is

\[\nu_{ab} = \frac{1}{3\pi^{3/2}\epsilon_0^2}\frac{Z_a^2 Z_b^2 n_b \ln\Lambda}{\left(v_a^2 + v_b^2\right)^{3/2}}\frac{\left(1 + m_a / m_b\right)}{m_a^2}\]

Note that the cgs expression in Hinton is divided by \(\left(4\pi\epsilon_0\right)^2\) to get the expression in SI units. The thermal speeds in this expression are defined as:

\[v_a^2 = 2 e T_a / m_a\]

Note that with this definition we recover the Braginskii expressions for e-i and i-i collision times.

The electron-electron collision time definition follows Braginskii (note that Fitzpatrick uses a different definition in his notes, these are not consistent with Braginskii):

\[\nu_{ee} = \frac{ln \Lambda e^4 n_e} { 12 \pi^{3/2} \varepsilon_0^2 m_{e}^{1/2} T_{e}^{3/2} }\]

For conservation of momentum, the collision frequencies \(\nu_{ab}\) and \(\nu_{ba}\) are related by:

\[m_a n_a \nu_{ab} = m_b n_b \nu_{ba}\]

  • Ion-neutral and electron-neutral collisions

    Note: These are disabled by default. If enabled, care is needed to avoid double-counting collisions in atomic reactions e.g charge-exchange reactions.

    The cross-section for elastic collisions between charged and neutral particles can vary significantly. Here for simplicity we just take a value of \(5\times 10^{-19}m^2\) from the NRL formulary.

  • Neutral-neutral collisions

    Note This is enabled by default.

    The cross-section is given by

\[\sigma = \pi \left(\frac{d_1 + d_2}{2}\right)^2\]

where \(d_1\) and \(d_2\) are the kinetic diameters of the two species. Typical values are [Wikipedia] for H2 2.89e-10m, He 2.60e-10m, Ne 2.75e-10m.

The mean relative velocity of the two species is

\[v_{rel} = \sqrt{\frac{eT_1}{m_1} + \frac{eT_2}{m_2}}\]

and so the collision rate of species 1 on species 2 is:

\[\nu_{12} = v_{rel} n_2 \sigma\]

The implementation is in BraginskiiCollisions:

struct BraginskiiCollisions : public Component

Calculates the collision rate of each species with all other species, using the Braginskii equation.

Important: Be careful when including both ion_neutral collisions and reactions such as charge exchange, since that may result in double counting. Similarly for electron_neutral collisions and ionization reactions.

Public Functions

BraginskiiCollisions(const std::string &name, Options &alloptions, Solver*)

The following boolean options under alloptions[name] control which collisions are calculated:

  • electron_electron

  • electron_ion

  • electron_neutral

  • ion_ion

  • ion_neutral

  • neutral_neutral

The user can also set

  • ei_multiplier arbitrary multiplier on electron-ion collision rate

Parameters:

alloptions – Settings, which should include:

  • units

    • eV

    • inv_meters_cubed

    • meters

    • seconds

virtual void outputVars(Options &state) override

Add extra fields for output, or set attributes e.g docstrings.

Braginskii Friction#

Input#

This top-level component calculates the frictional forces between each pair of species for which collisional frequencies have been calculated (see Braginskii Collisions component). As such, it must be run after braginskii_collisions. If the option frictional_heating is enabled then it will also calculate the energy source arising from friction.

Theory#

Momentum exchange, force on species a due to collisions with species b:

\[F_{ab} = C_m \nu_{ab} m_a n_a \left( u_b - u_a \right)\]

Where the coefficient \(C_m\) for parallel flows depends on the species: For most combinations of species this is set to 1, but for electron-ion collisions the Braginskii coefficients are used: \(C_m = 0.51\) if ion charge \(Z_i = 1\); 0.44 for \(Z_i = 2\); 0.40 for \(Z_i = 3\); and 0.38 is used for \(Z_i \ge 4\). Note that this coefficient should decline further with increasing ion charge, tending to 0.29 as \(Z_i \rightarrow \infty\).

Frictional heating is included by default, but can be disabled by setting the frictional_heating option to false. When enabled it adds a source of thermal energy corresponding to the resistive heating term:

\[Q_{ab,F} = \frac{m_b}{m_a + m_b} \left( u_b - u_a \right) F_{ab}\]

This term has some important properties:

  1. It is always positive: Collisions of two species with the same temperature never leads to cooling.

  2. It is Galilean invariant: Shifting both species’ velocity by the same amount leaves \(Q_{ab,F}\) unchanged.

  3. If both species have the same mass, the thermal energy change due to slowing down is shared equally between them.

  4. If one species is much heavier than the other, for example electron-ion collisions, the lighter species is preferentially heated. This recovers e.g. Braginskii expressions for \(Q_{ei}\) and \(Q_{ie}\).

This can be derived by considering the exchange of energy \(W_{ab,F}\) between two species at the same temperature but different velocities. If the pressure is evolved then it contains a term that balances the change in kinetic energy due to changes in velocity:

\[\begin{split}\begin{aligned} \frac{\partial}{\partial t}\left(m_a n_a u_a\right) =& \ldots + F_{ab} \\ \frac{\partial}{\partial t}\left(\frac{3}{2}p_a\right) =& \ldots - F_{ab} u_a + W_{ab, F} \end{aligned}\end{split}\]

For momentum and energy conservation we must have \(F_{ab}=-F_{ba}\) and \(W_{ab,F} = -W_{ba,F}\). Comparing the above to the Braginskii expression we see that for ion-electron collisions the term \(- F_{ab}u_a + W_{ab, F}\) goes to zero, so \(W_{ab, F} \sim u_aF_{ab}\) for \(m_a \gg m_b\). An expression that has all these desired properties is

\[W_{ab,F} = \left(\frac{m_a u_a + m_b u_a}{m_a + m_b}\right)F_{ab}\]

which is not Galilean invariant but when combined with the \(- F_{ab} u_a\) term gives a change in pressure that is invariant, as required.

The implementation is in BraginskiiFriction:

struct BraginskiiFriction : public Component

Calculates the friction force applied to each species due to collisions.

Public Functions

BraginskiiFriction(const std::string &name, Options &alloptions, Solver*)

  • frictional_heating Include R dot v heating term as energy source? (includes Ohmic heating)

Parameters:

alloptions – Settings, which has switches for additional terms:

virtual void outputVars(Options &state) override

Add extra fields for output, or set attributes e.g docstrings.

Braginskii Heat Exchange#

Input#

This top-level component calculates the heat exchange between species due to collisions (see Braginskii Collisions component). As such, it must be run after braginskii_collisions. There are no configurations for this component.

Theory#

Thermal energy exchange, heat transferred to species \(a\) from species \(b\) due to temperature differences, is given by:

\[Q_{ab,T} = \nu_{ab}\frac{3n_a m_a\left(T_b - T_a\right)}{m_a + m_b}\]

The implementation is in BraginskiiHeatExchange:

struct BraginskiiHeatExchange : public Component

Calculates the heat exchange between species due to collisions

Public Functions

BraginskiiHeatExchange(const std::string &name, Options &alloptions, Solver*)
Parameters:

alloptions – Settings. There is nothing to be configured.

virtual void outputVars(Options &state) override

Add extra fields for output, or set attributes e.g docstrings.