ForceFree::Solutions::AlfvenWave Class Reference

Alfven wave propagating along $x$ direction in flat spacetime with the wave speed $\mu$. More...

#include <AlfvenWave.hpp>

Classes
struct	WaveSpeed
	The wave speed. More...

Public Types
using	options = implementation defined

Public Member Functions
	AlfvenWave (const AlfvenWave &)=default
AlfvenWave &	operator= (const AlfvenWave &)=default
	AlfvenWave (AlfvenWave &&)=default
AlfvenWave &	operator= (AlfvenWave &&)=default
	AlfvenWave (double wave_speed, 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, const double t, tmpl::list< Tags... >) const
	Retrieve a collection of EM variables at (x, t)
template<typename Tag >
tuples::TaggedTuple< Tag >	variables (const tnsr::I< DataVector, 3 > &x, double t, 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 AlfvenWave &lhs, const AlfvenWave &rhs)
auto	variables (const tnsr::I< DataVector, 3 > &x, double t, tmpl::list< Tags::TildeE >) const -> tuples::TaggedTuple< Tags::TildeE >
	Retrieve the EM variables at (x,t).
auto	variables (const tnsr::I< DataVector, 3 > &x, double t, tmpl::list< Tags::TildeB >) const -> tuples::TaggedTuple< Tags::TildeB >
	Retrieve the EM variables at (x,t).
auto	variables (const tnsr::I< DataVector, 3 > &x, double t, tmpl::list< Tags::TildeQ >) const -> tuples::TaggedTuple< Tags::TildeQ >
	Retrieve the EM variables at (x,t).
static auto	variables (const tnsr::I< DataVector, 3 > &x, double t, tmpl::list< Tags::TildePsi >) -> tuples::TaggedTuple< Tags::TildePsi >
	Retrieve the EM variables at (x,t).
static auto	variables (const tnsr::I< DataVector, 3 > &x, double t, tmpl::list< Tags::TildePhi >) -> tuples::TaggedTuple< Tags::TildePhi >
	Retrieve the EM variables at (x,t).

Detailed Description

Alfven wave propagating along $x$ direction in flat spacetime with the wave speed $\mu$.

This test problem was introduced in [114] with $\mu =0$.

In the wave frame (with prime superscript), the stationary solution is given by

$$\begin{array}{rl}{B}_x^{\prime }& =B_y^{\prime }=1.0,\\ B_z^{\prime }(x^{\prime })& =\left\{\begin{array}{ll}1.0& \text{if }{x}^{\prime }<-0.1\\ 1.15+0.15\sin \left(5\pi x^{\prime }\right)& \text{if }-0.1<x^{\prime }<0.1\\ 1.3& \text{if }{x}^{\prime }>0.1\end{array}\right\},\\ E_x^{\prime }& =-B_z^{\prime },\\ E_y^{\prime }& =0\\ E_z^{\prime }& =1.0\end{array}$$

and

$$\begin{array}{rl}{q}^{\prime }& =\left\{\begin{array}{ll}-0.75\pi \cos \left(5\pi x^{\prime }\right)& \text{if }-0.1<x^{\prime }<0.1\\ 0& \text{otherwise}\end{array}\right\},\\ J_x^{\prime }& =0,\\ J_y^{\prime }& =\left\{\begin{array}{ll}-0.75\pi \cos \left(5\pi x^{\prime }\right)& \text{if }-0.1<x^{\prime }<0.1\\ 0& \text{otherwise}\end{array}\right\},\\ J_z^{\prime }& =0.\end{array}$$

Applying the Lorentz transformation, electromagnetic fields and 4-current in the grid frame at $t=0$ are given by

$$\begin{array}{rl}{E}_x(x)& =E_x^{\prime }(\gamma x),\\ E_y(x)& =\gamma [E_y^{\prime }(\gamma x)+\mu B_z^{\prime }(\gamma x)],\\ E_z(x)& =\gamma [E_z^{\prime }(\gamma x)-\mu B_y^{\prime }(\gamma x)],\\ B_x(x)& =B_x^{\prime }(\gamma x),\\ B_y(x)& =\gamma [B_y^{\prime }(\gamma x)-\mu E_z^{\prime }(\gamma x)],\\ B_z(x)& =\gamma [B_z^{\prime }(\gamma x)+\mu E_y^{\prime }(\gamma x)].\end{array}$$

and

$$\begin{array}{rl}q(x)& =\gamma q^{\prime }(\gamma x),\\ J_x(x)& =\gamma \mu q^{\prime }(\gamma x),\\ J_y(x)& =J_y^{\prime }(\gamma x),\\ J_z(x)& =0.\end{array}$$

The wave speed can be chosen any value $-1<\mu <1$, and the solution at time $t$ is $f(x,t)=f(x-\mu t,0)$ for any physical quantities.

Member Function Documentation

get_clone()

auto ForceFree::Solutions::AlfvenWave::get_clone ( ) const -> std::unique_ptr< evolution::initial_data::InitialData >

overridevirtual

Implements evolution::initial_data::InitialData.

Member Data Documentation

help

constexpr Options::String ForceFree::Solutions::AlfvenWave::help

staticconstexpr

Initial value:

{
      "Alfven wave propagating along x direction in flat spacetime with the "
      "wave speed mu"}

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

• src/PointwiseFunctions/AnalyticSolutions/ForceFree/AlfvenWave.hpp