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| Magnetic Geometry | ||
| =============================================== | ||
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| We take the magnetic field $\mathbf{B}$ to have the form | ||
| ```math | ||
| \begin{equation} | ||
| \mathbf{B} = B_z \hat{\mathbf{z}} + B_\zeta \hat{\mathbf{\zeta}}, | ||
| \end{equation} | ||
| ``` | ||
| with $B_\zeta = B(r,z) b_\zeta$, $B_z = B(r,z) b_z$ and $b_z$ and $b_\zeta$ the direction cosines of the magnetic field vector. | ||
| Here the basis vectors are those of cylindrical geometry $(r,z,\zeta)$, i.e., | ||
| $\hat{\mathbf{r}} = \nabla r $, $\hat{\mathbf{z}} = \nabla z$, | ||
| and $\hat{\mathbf{\zeta}} = r \nabla \zeta$. The unit vectors $\hat{\mathbf{r}}$, $\hat{\mathbf{z}}$, and $\hat{\mathbf{\zeta}}$ | ||
| form a right-handed orthonormal basis. | ||
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| Supported options | ||
| =============================================== | ||
|
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| To choose the type of geometry, set the value of "option" in the geometry namelist. The namelist will have the following appearance in the TOML file. | ||
| ``` | ||
| [geometry] | ||
| option="constant-helical" # ( or "1D-mirror" ) | ||
| pitch = 1.0 | ||
| rhostar = 1.0 | ||
| DeltaB = 0.0 | ||
| ``` | ||
| If `rhostar` is not set then it is computed from reference parameters. | ||
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| [geometry] option = "constant-helical" | ||
| =============================================== | ||
| Here $b_\zeta = \sqrt{1 - b_z^2}$ is a constant, $b_z$ is a constant input parameter ("pitch") and $B$ is taken to be 1 with respect to the reference value $B_{\rm ref}$. | ||
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| [geometry] option = "1D-mirror" | ||
| =============================================== | ||
| Here $b_\zeta = \sqrt{1 - b_z^2}$ is a constant, $b_z$ is a constant input parameter ("pitch") and $B = B(z)$ is taken to be | ||
| the function | ||
| ```math | ||
| \begin{equation} | ||
| \frac{B(z)}{B_{\rm ref}} = | ||
| 1 + \Delta B \left( 2\left(\frac{2z}{L_z}\right)^2 - \left(\frac{2z}{L_z}\right)^4\right) | ||
| \end{equation} | ||
| ``` | ||
| where $\Delta B $ is an input parameter ("DeltaB") that must satisfy $\Delta B > -1$. | ||
| Recalling that the coordinate $z$ runs from | ||
| $z = -L_z/2$ to $L_z/2$, | ||
| if $\Delta B > 0$ than the field represents a magnetic mirror which traps particles, | ||
| whereas if $\Delta B < 0$ then the magnetic field accelerates particles | ||
| by the mirror force as they approach the wall. | ||
| Note that this field does not satisfy $\nabla \cdot \mathbf{B} = 0$, and is | ||
| only used to test the implementation of the magnetic mirror terms. 2D simulations with a radial domain and$\mathbf{E}\times\mathbf{B}$ drifts are supported in the "1D-mirror" | ||
| geometry option. | ||
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| Geometric coefficients | ||
| =============================================== | ||
| Here, we write the geometric coefficients appearing in the characteristic equations | ||
| explicitly. | ||
|
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| The $z$ component of the $\mathbf{E}\times\mathbf{B}$ drift is given by | ||
| ```math | ||
| \begin{equation} \frac{\mathbf{E}\times\mathbf{B}}{B^2} \cdot \nabla z = \frac{E_r B_\zeta}{B^2} \nabla r \times \hat{\mathbf{\zeta}} \cdot \nabla z | ||
| = - J \frac{E_r B_\zeta}{B^2}, | ||
| \end{equation} | ||
| ``` | ||
| where we have defined $J = r \nabla r \times \nabla z \cdot \nabla \zeta$. | ||
| Note that $J$ is dimensionless. | ||
| The $r$ component of the $\mathbf{E}\times\mathbf{B}$ drift is given by | ||
| ```math | ||
| \begin{equation} \frac{\mathbf{E}\times\mathbf{B}}{B^2} \cdot \nabla r = \frac{E_z B_\zeta}{B^2} \nabla z \times \hat{\mathbf{\zeta}} \cdot \nabla r | ||
| = J \frac{E_z B_\zeta}{B^2}. | ||
| \end{equation} | ||
| ``` | ||
| Due to the axisymmetry of the system, the differential operator | ||
| $\mathbf{b} \cdot \nabla (\cdot) = b_z \partial {(\cdot)}{\partial z}$, | ||
| and the convective derivative | ||
| ```math | ||
| \begin{equation} | ||
| \frac{d B}{d t} = \frac{d z}{d t} \frac{\partial B}{ \partial z} + \frac{dr}{dt}\frac{\partial B}{\partial r}. | ||
| \end{equation} | ||
| ``` |
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| `geo` | ||
| ===== | ||
|
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| ```@autodocs | ||
| Modules = [moment_kinetics.geo] | ||
| ``` |
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examples/fokker-planck/fokker-planck-relaxation-report-resolutions.toml
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| # cheap input file for a 0D2V relaxation to a collisional Maxwellian distribution with self-ion collisions. | ||
| n_ion_species = 1 | ||
| n_neutral_species = 0 | ||
| electron_physics = "boltzmann_electron_response" | ||
| evolve_moments_density = false | ||
| evolve_moments_parallel_flow = false | ||
| evolve_moments_parallel_pressure = false | ||
| evolve_moments_conservation = false | ||
| T_e = 1.0 | ||
| T_wall = 1.0 | ||
| initial_density1 = 0.5 | ||
| initial_temperature1 = 1.0 | ||
| initial_density2 = 0.5 | ||
| initial_temperature2 = 1.0 | ||
| z_IC_option1 = "sinusoid" | ||
| z_IC_density_amplitude1 = 0.001 | ||
| z_IC_density_phase1 = 0.0 | ||
| z_IC_upar_amplitude1 = 0.0 | ||
| z_IC_upar_phase1 = 0.0 | ||
| z_IC_temperature_amplitude1 = 0.0 | ||
| z_IC_temperature_phase1 = 0.0 | ||
| z_IC_option2 = "sinusoid" | ||
| z_IC_density_amplitude2 = 0.001 | ||
| z_IC_density_phase2 = 0.0 | ||
| z_IC_upar_amplitude2 = 0.0 | ||
| z_IC_upar_phase2 = 0.0 | ||
| z_IC_temperature_amplitude2 = 0.0 | ||
| z_IC_temperature_phase2 = 0.0 | ||
| charge_exchange_frequency = 0.0 | ||
| ionization_frequency = 0.0 | ||
| constant_ionization_rate = false | ||
| # nuii sets the normalised input C[F,F] Fokker-Planck collision frequency | ||
| nuii = 1.0 | ||
| nstep = 200000 | ||
| dt = 1.0e-3 | ||
| nwrite = 50 | ||
| nwrite_dfns = 50 | ||
| use_semi_lagrange = false | ||
| n_rk_stages = 4 | ||
| split_operators = false | ||
| z_ngrid = 1 | ||
| z_nelement = 1 | ||
| z_nelement_local = 1 | ||
| z_bc = "wall" | ||
| z_discretization = "chebyshev_pseudospectral" | ||
| r_ngrid = 1 | ||
| r_nelement = 1 | ||
| r_nelement_local = 1 | ||
| r_bc = "periodic" | ||
| r_discretization = "chebyshev_pseudospectral" | ||
| vpa_ngrid = 5 | ||
| vpa_nelement = 8 | ||
| vpa_L = 6.0 | ||
| vpa_bc = "zero" | ||
| vpa_discretization = "gausslegendre_pseudospectral" | ||
| vperp_ngrid = 5 | ||
| vperp_nelement = 4 | ||
| vperp_L = 3.0 | ||
| vperp_discretization = "gausslegendre_pseudospectral" | ||
| # Fokker-Planck operator requires the "gausslegendre_pseudospectral | ||
| # options for the vpa and vperp grids | ||
|
|
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| Original file line number | Diff line number | Diff line change |
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| @@ -0,0 +1,84 @@ | ||
| n_ion_species = 1 | ||
| n_neutral_species = 0 | ||
| boltzmann_electron_response = true | ||
| evolve_moments_density = false | ||
| evolve_moments_parallel_flow = false | ||
| evolve_moments_parallel_pressure = false | ||
| evolve_moments_conservation = false | ||
| T_e = 1.0 | ||
| T_wall = 1.0 | ||
| initial_density1 = 1.0 | ||
| initial_temperature1 = 1.0 | ||
| z_IC_option1 = "gaussian" | ||
| z_IC_density_amplitude1 = 0.001 | ||
| z_IC_density_phase1 = 0.0 | ||
| z_IC_upar_amplitude1 = 1.0 | ||
| z_IC_upar_phase1 = 0.0 | ||
| z_IC_temperature_amplitude1 = 0.0 | ||
| z_IC_temperature_phase1 = 0.0 | ||
| vpa_IC_option1 = "gaussian" | ||
| vpa_IC_density_amplitude1 = 1.0 | ||
| vpa_IC_density_phase1 = 0.0 | ||
| vpa_IC_upar_amplitude1 = 0.0 | ||
| vpa_IC_upar_phase1 = 0.0 | ||
| vpa_IC_temperature_amplitude1 = 0.0 | ||
| vpa_IC_temperature_phase1 = 0.0 | ||
| initial_density2 = 1.0 | ||
| initial_temperature2 = 1.0 | ||
| z_IC_option2 = "gaussian" | ||
| z_IC_density_amplitude2 = 0.001 | ||
| z_IC_density_phase2 = 0.0 | ||
| z_IC_upar_amplitude2 = -1.0 | ||
| z_IC_upar_phase2 = 0.0 | ||
| z_IC_temperature_amplitude2 = 0.0 | ||
| z_IC_temperature_phase2 = 0.0 | ||
| vpa_IC_option2 = "gaussian" | ||
| vpa_IC_density_amplitude2 = 1.0 | ||
| vpa_IC_density_phase2 = 0.0 | ||
| vpa_IC_upar_amplitude2 = 0.0 | ||
| vpa_IC_upar_phase2 = 0.0 | ||
| vpa_IC_temperature_amplitude2 = 0.0 | ||
| vpa_IC_temperature_phase2 = 0.0 | ||
| charge_exchange_frequency = 0.5 | ||
| ionization_frequency = 0.05 | ||
| constant_ionization_rate = true | ||
| nstep = 10000 | ||
| dt = 1.0e-3 | ||
| nwrite = 50 | ||
| nwrite_dfns = 50 | ||
| use_semi_lagrange = false | ||
| n_rk_stages = 4 | ||
| split_operators = false | ||
| r_ngrid = 1 | ||
| r_nelement = 1 | ||
| z_ngrid = 5 | ||
| z_nelement = 16 | ||
| z_bc = "wall" | ||
| z_discretization = "chebyshev_pseudospectral" | ||
| vpa_ngrid = 5 | ||
| vpa_nelement = 16 | ||
| vpa_L = 6.0 | ||
| vpa_bc = "zero" | ||
| vpa_discretization = "chebyshev_pseudospectral" | ||
| vperp_ngrid = 5 | ||
| vperp_nelement = 8 | ||
| vperp_L = 3.0 | ||
| vperp_bc = "zero" | ||
| vperp_discretization = "chebyshev_pseudospectral" | ||
| vz_ngrid = 9 | ||
| vz_nelement = 64 | ||
| vz_L = 18.0 | ||
| vz_bc = "both_zero" | ||
| vz_discretization = "chebyshev_pseudospectral" | ||
|
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| [numerical_dissipation] | ||
| vpa_dissipation_coefficient = 1.0e-3 #1.0e-2 #1.0e-1 | ||
| vperp_dissipation_coefficient = 1.0e-3 #1.0e-2 #1.0e-1 | ||
| force_minimum_pdf_value = 0.0 | ||
|
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| [geometry] | ||
| option="1D-mirror" | ||
| DeltaB=0.5 | ||
| #option="constant-helical" | ||
| pitch=1.0 | ||
| rhostar= 1.0 |
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