@@ -4,7 +4,7 @@ Braginskii module
44===================
55
66Equations solved and methods
7- ---------------------------
7+ ----------------------------
88
99The ``Braginskii `` module implements the anisotropic heat and momentum fluxes specific
1010to weakly collisional, magnetised plasma like the intracluster medium
@@ -37,7 +37,7 @@ though adapted to vector quantities.
3737 cell interface by a simple arithmetic average
3838 (Eq. (6)-(7) from Sharma & Hammett 2007).
3939 However in the same paper, the authors showed that this implementation can lead to
40- unphysical heat flux from high to low temperature regions.
40+ unphysical heat flux from low to high temperature regions.
4141 So we implemented slope limiters for the computation of these transverse heat fluxes,
4242 as described in Eq. (17) from Sharma & Hammett (2007).
4343 Only the van Leer and the Monotonized Central (MC) limiters are available
@@ -72,18 +72,68 @@ of the Braginskii heat flux and viscosity.
7272
7373.. _braginskiiParameterSection :
7474
75+
76+ Saturation with collisionless heat flux
77+ ---------------------------------------
78+
79+ The ``Braginskii `` module can include a collisionless saturation of the Braginskii heat flux, typically due to supra-thermal electrons.
80+ The heat flux is then computed as follows:
81+
82+ :math: `q = \alpha (q_B + q_\perp ) + (1 -\alpha )\beta *p*v`,
83+
84+ where :math: `\alpha \in [0 ,1 ]` controls the transition between the Braginskii heat flux and the collisionless heat flux
85+ and :math: `\beta ` controls the amplitude of the collisionless heat flux (typically :math: `\beta \in [1 ,4 ]`, see Hollweg 1976).
86+
87+ .. note ::
88+ As a result, even with :math: `\kappa _\perp = 0 `, the heat flux is no longer necessarilly strictly aligned with the magnetic field.
89+ .. note ::
90+ The collisionless heat flux is a hyperbolic term and the diffusion coefficient is set proportional to :math: `\alpha `.
91+ .. note ::
92+ If selected, slope limiters are also used in the collisionless flux, where an upwind scheme has been implemented for stability.
93+ .. note ::
94+ This saturation has been thought to be used mostly using the userdef function that takes four userdef arrays as input.
95+
7596Main parameters of the module
7697-----------------------------
7798
7899The ``Braginskii `` module can be enabled adding one or two lines in the ``[Hydro] `` section
79- starting with the keyword
80- `bragTDiffusion ` or/and *bragViscosity *. The following table summarises the different options
100+ starting with the keyword `bragTDiffusion ` or/and *bragViscosity *. The following tables summarise the different options
81101associated to the activation of the Braginskii module:
82102
83103+--------+-----------------------+-------------------------+---------------------------------------------------------------------------------------+
84104| Column | Entry name | Parameter type | Comment |
85105+========+=======================+=========================+=======================================================================================+
86- | 0 | bragModule | string | | Activates Braginskii diffusion. Can be ``bragTDiffusion `` or ``bragViscosity ``. |
106+ | 0 | bragTDiffusion | string | | Activates Braginskii thermal diffusion. |
107+ +--------+-----------------------+-------------------------+---------------------------------------------------------------------------------------+
108+ | 1 | integration | string | | Specifies the type of scheme to be used to integrate the parabolic term. |
109+ | | | | | Can be ``rkl `` or ``explicit ``. |
110+ +--------+-----------------------+-------------------------+---------------------------------------------------------------------------------------+
111+ | 2 | slope limiter | string | | Choose the type of limiter to be used to compute anisotropic transverse flux terms. |
112+ | | | | | Can be ``mc ``, ``vanleer `` or ``nolimiter ``. |
113+ +--------+-----------------------+-------------------------+---------------------------------------------------------------------------------------+
114+ | 3 | saturation mode | string | | Include or not collisionless saturation. Can be ``nosat `` or ``wcless ``. |
115+ +--------+-----------------------+-------------------------+---------------------------------------------------------------------------------------+
116+ | 4 | diffusivity type | string | | Specifies the type of diffusivity wanted. Can be ``constant `` or ``userdef ``. |
117+ +--------+-----------------------+-------------------------+---------------------------------------------------------------------------------------+
118+ | 5 | parallel diffusivity | real | | Mandatory if the diffusivity type is ``constant ``. Not needed otherwise. |
119+ | | | | | Value of the parallel diffusivity. Should be a real number. |
120+ +--------+-----------------------+-------------------------+---------------------------------------------------------------------------------------+
121+ | 6 | normal diffusivity | real | | When bragModule ``bragTDiffusion `` and diffusivity type ``constant ``, |
122+ | | | | | value of the normal diffusivity. Should be a real number. |
123+ +--------+-----------------------+-------------------------+---------------------------------------------------------------------------------------+
124+ | 7 | alpha collisionless | real | | If the diffusivity type is ``constant `` and saturation is ``wcless ``. |
125+ | | | | | Set to 1 if not provided. |
126+ +--------+-----------------------+-------------------------+---------------------------------------------------------------------------------------+
127+ | 8 | beta collisionless | real | | If the diffusivity type is ``constant `` and saturation is ``wcless ``. |
128+ | | | | | Set to 0 if not provided. |
129+ +--------+-----------------------+-------------------------+---------------------------------------------------------------------------------------+
130+
131+ for the *bragViscosity *:
132+
133+ +--------+-----------------------+-------------------------+---------------------------------------------------------------------------------------+
134+ | Column | Entry name | Parameter type | Comment |
135+ +========+=======================+=========================+=======================================================================================+
136+ | 0 | bragViscosity | string | | Activates Braginskii viscosity. |
87137+--------+-----------------------+-------------------------+---------------------------------------------------------------------------------------+
88138| 1 | integration | string | | Specifies the type of scheme to be used to integrate the parabolic term. |
89139| | | | | Can be ``rkl `` or ``explicit ``. |
@@ -101,7 +151,7 @@ associated to the activation of the Braginskii module:
101151+--------+-----------------------+-------------------------+---------------------------------------------------------------------------------------+
102152
103153Numerical checks
104- ---------------
154+ ----------------
105155
106156In Cartesian geometry, the ``Braginskii `` module has been tested with many setups
107157and in all configurations of magnetic polarisation:
@@ -119,3 +169,5 @@ The same goes for the anisotropic heat flux in Cylindrical/Polar geometry while
119169the anisotropic viscosity has *never * been tested in this geometry.
120170In spherical geometry, both ``Braginskii `` operators have been partially validated
121171(diffusion along the polar axis has not been directly tested).
172+
173+ The collisionless saturation has been tested in 1D and 2D spherical geometry.
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