ForceFree::AnalyticData::MagnetosphericWald Class Reference

The magnetospheric Wald problem proposed in [106]. 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.
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 [106].

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 [189] (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 [106] [148] [62] 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