ForceFree::System Struct Reference

General relativistic force-free electrodynamics (GRFFE) system with divergence cleaning. More...

#include <System.hpp>

Public Types
using	boundary_conditions_base = BoundaryConditions::BoundaryCondition
using	boundary_correction_base = BoundaryCorrections::BoundaryCorrection
using	variables_tag = ::Tags::Variables< tmpl::list< Tags::TildeE, Tags::TildeB, Tags::TildePsi, Tags::TildePhi, Tags::TildeQ > >
using	flux_variables = implementation defined
using	non_conservative_variables = implementation defined
using	gradient_variables = implementation defined
using	spacetime_variables_tag = ::Tags::Variables< gr::tags_for_hydro< volume_dim, DataVector > >
using	flux_spacetime_variables_tag = ::Tags::Variables< tmpl::list< gr::Tags::Lapse< DataVector >, gr::Tags::Shift< DataVector, 3 >, gr::Tags::SqrtDetSpatialMetric< DataVector >, gr::Tags::SpatialMetric< DataVector, 3 >, gr::Tags::InverseSpatialMetric< DataVector, 3 > > >
using	compute_volume_time_derivative_terms = TimeDerivativeTerms
using	compute_largest_characteristic_speed = Tags::LargestCharacteristicSpeedCompute
using	inverse_spatial_metric_tag = gr::Tags::InverseSpatialMetric< DataVector, volume_dim >

Static Public Attributes
static constexpr bool	is_in_flux_conservative_form = true
static constexpr bool	has_primitive_and_conservative_vars = false
static constexpr size_t	volume_dim = 3

Detailed Description

General relativistic force-free electrodynamics (GRFFE) system with divergence cleaning.

For electromagnetism in a curved spacetime, Maxwell equations are given as

$$\begin{array}{rl}{\mathrm{\nabla }}_a{F}^{ab}& =-J^b,\\ {\mathrm{\nabla }}_a^{\ast }{F}^{ab}& =0.\end{array}$$

where $F^{ab}$ is the electromagnetic field tensor, ${}^{\ast }{F}^{ab}$ is its dual ${}^{\ast }{F}^{ab}={ϵ}^{abcd}{F}_{cd}/2$, and $J^a$ is the 4-current.

Note

• We are using the electromagnetic variables with the scaling convention that the factor $4\pi$ does not appear in Maxwell equations and the stress-energy tensor of the EM fields (geometrized Heaviside-Lorentz units).
• We adopt following definition of the Levi-Civita tensor by [140]

  $$\begin{array}{rl}{ϵ}_{abcd}& =\sqrt{-g}[abcd],\\ {ϵ}^{abcd}& =-\frac{1}{\sqrt{-g}}[abcd],\end{array}$$

  where $g$ is the determinant of spacetime metric, and $[abcd]=\pm 1$ is the antisymmetric symbol with $[0123]=+1$.

In SpECTRE, we evolve 'extended' (or augmented) version of Maxwell equations with two divergence cleaning scalar fields $\psi$ and $\varphi$ :

$$\begin{array}{rl}{\mathrm{\nabla }}_a(F^{ab}+g^{ab}\psi )& =-J^b+{\kappa }_{\psi }{n}^b\psi \\ {\mathrm{\nabla }}_a{(}^{\ast }{F}^{ab}+g^{ab}\varphi )& ={\kappa }_{\varphi }{n}^b\varphi \end{array}$$

which reduce to the original Maxwell equations when $\psi =\varphi =0$. For damping constants ${\kappa }_{\psi ,\varphi }>0$, Gauss constraint violations are damped with timescales ${\kappa }_{\psi ,\varphi }^{-1}$ and propagated away.

We decompose the EM field tensor as follows

$$\begin{array}{r}{F}^{ab}=n^a{E}^b-n^b{E}^a-{ϵ}^{abcd}{B}_c{n}_d,\end{array}$$

where $n^a$ is the normal to spatial hypersurface, $E^a$ and $B^a$ are electric and magnetic fields.

Evolved variables are

$$\begin{array}{r}\mathbf{U}=\sqrt{\gamma }\left[\begin{array}{c}{E}^i\\ B^i\\ \psi \\ \varphi \\ q\end{array}\right]\equiv \left[\begin{array}{c}{\tilde{E}}^i\\ {\tilde{B}}^i\\ \tilde{\psi }\\ \tilde{\varphi }\\ \tilde{q}\end{array}\right]\end{array}$$

where $E^i$ is electric field, $B^i$ is magnetic field, $\psi$ is electric divergence cleaning field, $\varphi$ is magnetic divergence cleaning field, $q\equiv -n_a{J}^a$ is electric charge density, and $\gamma$ is the determinant of spatial metric.

Corresponding fluxes ${\mathbf{F}}^j$ are

$$\begin{array}{rl}{F}^j({\tilde{E}}^i)& =-{\beta }^j{\tilde{E}}^i+\alpha ({\gamma }^{ij}\tilde{\psi }-{ϵ}_{(3)}^{ijk}{\tilde{B}}_k)\\ F^j({\tilde{B}}^i)& =-{\beta }^j{\tilde{B}}^i+\alpha ({\gamma }^{ij}\tilde{\varphi }+{ϵ}_{(3)}^{ijk}{\tilde{E}}_k)\\ F^j(\tilde{\psi })& =-{\beta }^j\tilde{\psi }+\alpha {\tilde{E}}^j\\ F^j(\tilde{\varphi })& =-{\beta }^j\tilde{\varphi }+\alpha {\tilde{B}}^j\\ F^j(\tilde{q})& ={\tilde{J}}^j-{\beta }^j\tilde{q}\end{array}$$

and source terms are

$$\begin{array}{rl}S({\tilde{E}}^i)& =-{\tilde{J}}^i-{\tilde{E}}^j{\partial }_j{\beta }^i+\tilde{\psi }({\gamma }^{ij}{\partial }_j\alpha -\alpha {\gamma }^{jk}{\mathrm{\Gamma }}_{jk}^i)\\ S({\tilde{B}}^i)& =-{\tilde{B}}^j{\partial }_j{\beta }^i+\tilde{\varphi }({\gamma }^{ij}{\partial }_j\alpha -\alpha {\gamma }^{jk}{\mathrm{\Gamma }}_{jk}^i)\\ S(\tilde{\psi })& ={\tilde{E}}^k{\partial }_k\alpha +\alpha \tilde{q}-\alpha \tilde{\varphi }(K+{\kappa }_{\varphi })\\ S(\tilde{\varphi })& ={\tilde{B}}^k{\partial }_k\alpha -\alpha \tilde{\varphi }(K+{\kappa }_{\varphi })\\ S(\tilde{q})& =0\end{array}$$

where ${\tilde{J}}^i\equiv \alpha \sqrt{\gamma }{J}^i$.

See the documentation of Fluxes and Sources for further details.

In addition to Maxwell equations, general relativistic force-free electrodynamics (GRFFE) assumes the following which are called the force-free (FF) conditions.

$$\begin{array}{rl}{F}^{ab}{J}_b& =0,\\ {}^{\ast }{F}^{ab}{F}_{ab}& =0,\\ F^{ab}{F}_{ab}& >0.\end{array}$$

In terms of electric and magnetic fields, the FF conditions above read

$$\begin{array}{rl}{E}_i{J}^i& =0,\\ q{E}^i+{ϵ}_{(3)}^{ijk}{J}_j{B}_k& =0,\\ B_i{E}^i& =0,\\ B^2-E^2& >0.\end{array}$$

where $B^2=B^a{B}_a$ and $E^2=E^a{E}_a$. Also, ${ϵ}_{(3)}^{ijk}$ is the spatial Levi-Civita tensor defined as

$$\begin{array}{r}{ϵ}_{(3)}^{ijk}\equiv n_{\mu }{ϵ}^{\mu ijk}=-\frac{1}{\sqrt{-g}}{n}_{\mu }[\mu ijk]=\frac{1}{\sqrt{\gamma }}[ijk]\end{array}$$

where $n^{\mu }$ is the normal to spatial hypersurface and $[ijk]$ is the antisymmetric symbol with $[123]=+1$.

There are a number of different ways in literature to numerically treat the FF conditions. For the constraint $B_i{E}^i=0$, cleaning of the parallel electric field after every time step (e.g. [158]) or adopting analytically determined parallel current density [116] were explored. On the magnetic dominance condition $B^2-E^2>0$, there have been approaches with modification of the drift current [115] or manual rescaling of the electric field [158].

We take the strategy that introduces special driver terms in the electric current density $J^i$ following [3] :

$$\begin{array}{}\text{(1)}& J^i=J_{\mathrm{d}\mathrm{r}\mathrm{i}\mathrm{f}\mathrm{t}}^i+J_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{l}}^i\end{array}$$

with

$$\begin{array}{}\text{(2)}& J_{\mathrm{d}\mathrm{r}\mathrm{i}\mathrm{f}\mathrm{t}}^i& =q\frac{{ϵ}_{(3)}^{ijk}{E}_j{B}_k}{B_l{B}^l},\text{(3)}& J_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{l}}^i& =\eta [\frac{E_j{B}^j}{B_l{B}^l}{B}^i+\frac{\mathcal{R}(E_l{E}^l-B_l{B}^l)}{B_l{B}^l}{E}^i].\end{array}$$

where $\eta$ is the parallel conductivity and $\eta \to \mathrm{\infty }$ corresponds to the ideal force-free limit. $\mathcal{R}(x)$ is the ramp (or rectifier) function defined as

$$\begin{array}{r}\mathcal{R}(x)=\left\{\begin{array}{lc}x,& \text{if }x\ge 0\\ 0,& \text{if }x<0\end{array}\right\}=max(x,0).\end{array}$$

Internally we handle each pieces ${\tilde{J}}_{\mathrm{d}\mathrm{r}\mathrm{i}\mathrm{f}\mathrm{t}}^i\equiv \alpha \sqrt{\gamma }{J}_{\mathrm{d}\mathrm{r}\mathrm{i}\mathrm{f}\mathrm{t}}^i$ and ${\tilde{J}}_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{l}}^i\equiv \alpha \sqrt{\gamma }{J}_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{l}}^i$ as two separate Tags since the latter term is stiff and needs to be evolved in conjunction with implicit time steppers.

---

The documentation for this struct was generated from the following file:

• src/Evolution/Systems/ForceFree/System.hpp