.. _sec-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 :ref:`sec-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 :ref:`sec-braginskii_conduction` component. **Viscosity:** Parallel and perpendicular viscosity for ions is in the :ref:`sec-braginskii_ion_viscosity` top-level component. The parallel viscosity for electrons is in :ref:`sec-braginskii_electron_viscosity`. **Thermal force:** Thermal force is implemented here: :ref:`sec-braginskii_thermal_force`. **Friction force/heating:** The frictional momentum and energy sources are calculated in the top-level :ref:`sec-braginskii_friction` component. **Thermal equilibration:** Thermal transfers are calculated in the :ref:`sec-braginskii_heat_exchange` top-level component. **Neutral diffusion:** The parallel projection of diffusion from the wall in 1D is captured in the :ref:`sec-neutral_parallel_diffusion` top-level component, while both parallel Braginskii transport and perpendicular pressure-diffusion for 2D/3D are captured in the :ref:`sec-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.: .. code-block:: ini [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.: .. code-block:: ini [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: .. code-block:: ini [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.: .. code-block:: ini [neutral_parallel_diffusion] diffusion_collisions_mode = afn .. _sec-collisions: 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: .. code-block:: ini [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: :math:`T` is in eV, so :math:`eT` has units of Joules. Debye length :math:`\lambda_D` .. math:: \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 .. math:: \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 (:math:`30.4 = 23.5 - 0.5\ln\left(10^{-6}\right)`). - Electron-ion collisions .. math:: \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. - Mixed ion-ion collisions .. math:: \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: :math:`29.91 = 23 - 0.5\ln\left(10^{-6}\right)`. The frequency of charged species `a` colliding with charged species `b` is .. math:: \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 :math:`\left(4\pi\epsilon_0\right)^2` to get the expression in SI units. The thermal speeds in this expression are defined as: .. math:: 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): .. math:: \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 :math:`\nu_{ab}` and :math:`\nu_{ba}` are related by: .. math:: 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 :math:`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 .. math:: \sigma = \pi \left(\frac{d_1 + d_2}{2}\right)^2 where :math:`d_1` and :math:`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 .. math:: 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: .. math:: \nu_{12} = v_{rel} n_2 \sigma The implementation is in `BraginskiiCollisions`: .. doxygenstruct:: BraginskiiCollisions :members: .. _sec-braginskii_friction: 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`: .. math:: F_{ab} = C_m \nu_{ab} m_a n_a \left( u_b - u_a \right) Where the coefficient :math:`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: :math:`C_m = 0.51` if ion charge :math:`Z_i = 1`; 0.44 for :math:`Z_i = 2`; 0.40 for :math:`Z_i = 3`; and 0.38 is used for :math:`Z_i \ge 4`. Note that this coefficient should decline further with increasing ion charge, tending to 0.29 as :math:`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: .. math:: 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 :math:`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 :math:`Q_{ei}` and :math:`Q_{ie}`. This can be derived by considering the exchange of energy :math:`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: .. math:: \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} For momentum and energy conservation we must have :math:`F_{ab}=-F_{ba}` and :math:`W_{ab,F} = -W_{ba,F}`. Comparing the above to the `Braginskii expression `_ we see that for ion-electron collisions the term :math:`- F_{ab}u_a + W_{ab, F}` goes to zero, so :math:`W_{ab, F} \sim u_aF_{ab}` for :math:`m_a \gg m_b`. An expression that has all these desired properties is .. math:: 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 :math:`- F_{ab} u_a` term gives a change in pressure that is invariant, as required. The implementation is in `BraginskiiFriction`: .. doxygenstruct:: BraginskiiFriction :members: .. _sec-braginskii_heat_exchange: 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 :math:`a` from species :math:`b` due to temperature differences, is given by: .. math:: 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`: .. doxygenstruct:: BraginskiiHeatExchange :members: