ForceFree::AnalyticData::MagnetosphericWald Class Reference

The magnetospheric Wald problem proposed in [123]. More...

#include <MagnetosphericWald.hpp>

Classes
struct	Spin

Public Types
using	options = tmpl::list<Spin>

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.
Public Member Functions inherited from evolution::initial_data::InitialData
virtual const InitialData &	unwrap () const
	Unwrap wrapper types (e.g. WithNoise, which wraps analytic initial data and adds random noise) to reach the underlying analytic solution or data.

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 [123].

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 [218] (see also documentation of ForceFree::Solutions::ExactWald)

\begin{equation} A_\mu = \frac{B_0}{2}(\phi_\mu + 2a t_\mu) , \end{equation}

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{align}\tilde{B}^{x} &= a B_0 z \left[ (ax-ry) \left\{ \frac{1}{r^4} + \frac{2M r (r^2-a^2)}{(r^4+a^2z^2)^2} \right\} + a M r x \left\{ \frac{r^2-z^2}{r^4(r^4+a^2z^2)} - \frac{4(r^2+z^2)}{(r^4+a^2z^2)^2} \right\} \right] \\ \tilde{B}^{y} &= a B_0 z \left[ (rx+ay) \left\{ \frac{1}{r^4} + \frac{2M r (r^2-a^2)}{(r^4+a^2z^2)^2} \right\} + a M r y \left\{ \frac{r^2-z^2}{r^4(r^4+a^2z^2)} - \frac{4(r^2+z^2)}{(r^4+a^2z^2)^2} \right\} \right] \\ \tilde{B}^{z} &= B_0 \left[ 1 + \frac{a^2z^2}{r^4} + \frac{M a^2}{r^3}\left\{ 1 - \frac{z^2(a^2+z^2)(5r^4+a^2z^2)}{(r^4+a^2z^2)^2} \right\} \right] . \end{align}

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 [123] [173] [74] 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

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