SpECTRE
v2024.05.11
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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). | |
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] [149] [62] report that the system converges to a steady state with an equatorial current sheet inside the ergosphere.
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overridevirtual |
Implements evolution::initial_data::InitialData.
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staticconstexpr |