ForceFree::AnalyticData::MagnetosphericWald Class Reference

The magnetospheric Wald problem proposed in [114]. More...

#include <MagnetosphericWald.hpp>

Classes
struct	Spin

Public Types
using	options = implementation defined

Public Member Functions
	MagnetosphericWald (const MagnetosphericWald &)=default
MagnetosphericWald &	operator= (const MagnetosphericWald &)=default
	MagnetosphericWald (MagnetosphericWald &&)=default
MagnetosphericWald &	operator= (MagnetosphericWald &&)=default
	MagnetosphericWald (double spin, const Options::Context &context={})
auto	get_clone () const -> std::unique_ptr< evolution::initial_data::InitialData > override
void	pup (PUP::er &p) override
template<typename... Tags>
tuples::TaggedTuple< Tags... >	variables (const tnsr::I< DataVector, 3 > &x, tmpl::list< Tags... >) const
	Retrieve a collection of EM variables at position x.
template<typename Tag >
tuples::TaggedTuple< Tag >	variables (const tnsr::I< DataVector, 3 > &x, tmpl::list< Tag >) const
	Retrieve the metric variables.
virtual auto	get_clone () const -> std::unique_ptr< InitialData >=0

Static Public Attributes
static constexpr Options::String	help

Friends
bool	operator== (const MagnetosphericWald &lhs, const MagnetosphericWald &rhs)
auto	variables (const tnsr::I< DataVector, 3 > &x, tmpl::list< Tags::TildeB >) const -> tuples::TaggedTuple< Tags::TildeB >
	Retrieve the EM variables at (x,t).
static auto	variables (const tnsr::I< DataVector, 3 > &x, tmpl::list< Tags::TildeE >) -> tuples::TaggedTuple< Tags::TildeE >
	Retrieve the EM variables at (x,t).
static auto	variables (const tnsr::I< DataVector, 3 > &x, tmpl::list< Tags::TildePsi >) -> tuples::TaggedTuple< Tags::TildePsi >
	Retrieve the EM variables at (x,t).
static auto	variables (const tnsr::I< DataVector, 3 > &x, tmpl::list< Tags::TildePhi >) -> tuples::TaggedTuple< Tags::TildePhi >
	Retrieve the EM variables at (x,t).
static auto	variables (const tnsr::I< DataVector, 3 > &x, tmpl::list< Tags::TildeQ >) -> tuples::TaggedTuple< Tags::TildeQ >
	Retrieve the EM variables at (x,t).

Detailed Description

The magnetospheric Wald problem proposed in [114].

This is an initial value problem that evolves the magnetosphere of a rotating black hole. The initial condition is given as same as the exact Wald solution [204] (see also documentation of ForceFree::Solutions::ExactWald)

$$\begin{array}{}\text{(1)}& A_{\mu }=\frac{B_0}{2}({\varphi }_{\mu }+2a{t}_{\mu }),\end{array}$$

but electric field is set to zero at $t=0$.

In the cartesian projection of the spherical Kerr-Schild coordinates (which we use in the code for representing tensors), initial magnetic fields is given as

$$\begin{array}{}\text{(2)}& {\tilde{B}}^x& =a{B}_{0}z[(ax-ry)\{\frac{1}{r^4}+\frac{2Mr(r^2-a^2)}{(r^4+a^2{z}^2{)}^2}\}+aMrx\{\frac{r^2-z^2}{r^4(r^4+a^2{z}^2)}-\frac{4(r^2+z^2)}{(r^4+a^2{z}^2{)}^2}\}]\text{(3)}& {\tilde{B}}^y& =a{B}_{0}z[(rx+ay)\{\frac{1}{r^4}+\frac{2Mr(r^2-a^2)}{(r^4+a^2{z}^2{)}^2}\}+aMry\{\frac{r^2-z^2}{r^4(r^4+a^2{z}^2)}-\frac{4(r^2+z^2)}{(r^4+a^2{z}^2{)}^2}\}]\text{(4)}& {\tilde{B}}^z& =B_0[1+\frac{a^2{z}^2}{r^4}+\frac{M{a}^2}{r^3}\{1-\frac{z^2(a^2+z^2)(5r^4+a^2{z}^2)}{(r^4+a^2{z}^2{)}^2}\}].\end{array}$$

where $M$ and $a$ are mass and (dimensionless) spin of the Kerr black hole, $B_0$ is the amplitude of magnetic field, and $r$ is the radial coordinate defined in the spherical Kerr-Schild coordinate system (see the documentation of gr::Solutions::SphericalKerrSchild). All other variables are set to zero at $t=0$.

There is no known exact solution to this problem, but numerical simulations [114] [161] [68] report that the system converges to a steady state with an equatorial current sheet inside the ergosphere.

Note

We set $M=1$ and $B_0=1$ in the initial data to fix scales.

Member Function Documentation

get_clone()

auto ForceFree::AnalyticData::MagnetosphericWald::get_clone ( ) const -> std::unique_ptr< evolution::initial_data::InitialData >

overridevirtual

Implements evolution::initial_data::InitialData.

Member Data Documentation

help

constexpr Options::String ForceFree::AnalyticData::MagnetosphericWald::help

staticconstexpr

Initial value:

{
      "Magnetospheric Wald initial value problem"}

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

• src/PointwiseFunctions/AnalyticData/ForceFree/MagnetosphericWald.hpp