Circularly polarized Alfvén wave solution in Minkowski spacetime travelling along a background magnetic field. More...

#include <AlfvenWave.hpp>

Classes
struct	AdiabaticIndex
	The adiabatic index for the ideal fluid. More...

struct	BackgroundMagneticField
	The background static magnetic field vector. More...

struct	ElectronFraction
	The constant electron fraction throughout the fluid. More...

struct	Pressure
	The constant pressure throughout the fluid. More...

struct	RestMassDensity
	The constant rest mass density throughout the fluid. More...

struct	WaveMagneticField
	The sinusoidal magnetic field vector associated with the Alfvén wave, perpendicular to the background magnetic field vector. More...

struct	WaveNumber
	The wave number of the profile. More...

Public Types
using	equation_of_state_type = EquationsOfState::IdealFluid< true >

using	options = implementation defined

Public Types inherited from grmhd::AnalyticSolution
template<typename DataType >
using	tags = implementation defined

Public Member Functions
	AlfvenWave (const AlfvenWave &)=default

AlfvenWave &	operator= (const AlfvenWave &)=default

	AlfvenWave (AlfvenWave &&)=default

AlfvenWave &	operator= (AlfvenWave &&)=default

	AlfvenWave (double wavenumber, double pressure, double rest_mass_density, double electron_fraction, double adiabatic_index, const std::array< double, 3 > &background_magnetic_field, const std::array< double, 3 > &wave_magnetic_field)

auto	get_clone () const -> std::unique_ptr< evolution::initial_data::InitialData > override

template<typename DataType , typename Tag1 , typename Tag2 , typename... Tags>
tuples::TaggedTuple< Tag1, Tag2, Tags... >	variables (const tnsr::I< DataType, 3 > &x, double t, tmpl::list< Tag1, Tag2, Tags... >) const
	Retrieve a collection of hydro variables at `(x, t)`

template<typename DataType , typename Tag , Requires< tmpl::list_contains_v< gr::analytic_solution_tags< 3, DataType >, Tag > > = nullptr>
tuples::TaggedTuple< Tag >	variables (const tnsr::I< DataType, 3 > &x, double t, tmpl::list< Tag >) const
	Retrieve the metric variables.

void	pup (PUP::er &) override

const EquationsOfState::IdealFluid< true > &	equation_of_state () const


template<typename DataType >
auto	variables (const tnsr::I< DataType, 3 > &x, double t, tmpl::list< hydro::Tags::RestMassDensity< DataType > >) const -> tuples::TaggedTuple< hydro::Tags::RestMassDensity< DataType > >
	Retrieve hydro variable at `(x, t)`

template<typename DataType >
auto	variables (const tnsr::I< DataType, 3 > &x, double t, tmpl::list< hydro::Tags::ElectronFraction< DataType > >) const -> tuples::TaggedTuple< hydro::Tags::ElectronFraction< DataType > >
	Retrieve hydro variable at `(x, t)`

template<typename DataType >
auto	variables (const tnsr::I< DataType, 3 > &x, double t, tmpl::list< hydro::Tags::SpecificInternalEnergy< DataType > >) const -> tuples::TaggedTuple< hydro::Tags::SpecificInternalEnergy< DataType > >
	Retrieve hydro variable at `(x, t)`

template<typename DataType >
auto	variables (const tnsr::I< DataType, 3 > &x, double, tmpl::list< hydro::Tags::Pressure< DataType > >) const -> tuples::TaggedTuple< hydro::Tags::Pressure< DataType > >
	Retrieve hydro variable at `(x, t)`

template<typename DataType >
auto	variables (const tnsr::I< DataType, 3 > &x, double, tmpl::list< hydro::Tags::SpatialVelocity< DataType, 3 > >) const -> tuples::TaggedTuple< hydro::Tags::SpatialVelocity< DataType, 3 > >
	Retrieve hydro variable at `(x, t)`

template<typename DataType >
auto	variables (const tnsr::I< DataType, 3 > &x, double, tmpl::list< hydro::Tags::MagneticField< DataType, 3 > >) const -> tuples::TaggedTuple< hydro::Tags::MagneticField< DataType, 3 > >
	Retrieve hydro variable at `(x, t)`

template<typename DataType >
auto	variables (const tnsr::I< DataType, 3 > &x, double, tmpl::list< hydro::Tags::DivergenceCleaningField< DataType > >) const -> tuples::TaggedTuple< hydro::Tags::DivergenceCleaningField< DataType > >
	Retrieve hydro variable at `(x, t)`

template<typename DataType >
auto	variables (const tnsr::I< DataType, 3 > &x, double, tmpl::list< hydro::Tags::LorentzFactor< DataType > >) const -> tuples::TaggedTuple< hydro::Tags::LorentzFactor< DataType > >
	Retrieve hydro variable at `(x, t)`

template<typename DataType >
auto	variables (const tnsr::I< DataType, 3 > &x, double t, tmpl::list< hydro::Tags::SpecificEnthalpy< DataType > >) const -> tuples::TaggedTuple< hydro::Tags::SpecificEnthalpy< DataType > >
	Retrieve hydro variable at `(x, t)`

template<typename DataType >
auto	variables (const tnsr::I< DataType, 3 > &x, double t, tmpl::list< hydro::Tags::Temperature< DataType > >) const -> tuples::TaggedTuple< hydro::Tags::Temperature< DataType > >
	Retrieve hydro variable at `(x, t)`

virtual auto	get_clone () const -> std::unique_ptr< InitialData >=0

Public Member Functions inherited from hydro::TemperatureInitialization< AlfvenWave >
auto	variables (const tnsr::I< DataType, Dim > &x, tmpl::list< hydro::Tags::Temperature< DataType > >) const -> tuples::TaggedTuple< hydro::Tags::Temperature< DataType > >

auto	variables (const tnsr::I< DataType, Dim > &x, const double t, tmpl::list< hydro::Tags::Temperature< DataType > >) const -> tuples::TaggedTuple< hydro::Tags::Temperature< DataType > >

auto	variables (ExtraVars &extra_variables, const tnsr::I< DataType, Dim > &x, Args &... extra_args, tmpl::list< hydro::Tags::Temperature< DataType > >) const -> tuples::TaggedTuple< hydro::Tags::Temperature< DataType > >

Static Public Attributes
static constexpr Options::String	help

Static Public Attributes inherited from grmhd::AnalyticSolution
static constexpr size_t	volume_dim = 3_st

Protected Member Functions
template<typename DataType >
DataType	k_dot_x_minus_vt (const tnsr::I< DataType, 3 > &x, double t) const

Protected Attributes
double	wavenumber_ = std::numeric_limits<double>::signaling_NaN()

double	pressure_ = std::numeric_limits<double>::signaling_NaN()

double	rest_mass_density_ = std::numeric_limits<double>::signaling_NaN()

double	electron_fraction_ = std::numeric_limits<double>::signaling_NaN()

double	adiabatic_index_ = std::numeric_limits<double>::signaling_NaN()

std::array< double, 3 >	background_magnetic_field_

std::array< double, 3 >	wave_magnetic_field_

EquationsOfState::IdealFluid< true >	equation_of_state_ {}

tnsr::I< double, 3 >	initial_unit_vector_along_background_magnetic_field_ {}

tnsr::I< double, 3 >	initial_unit_vector_along_wave_magnetic_field_ {}

tnsr::I< double, 3 >	initial_unit_vector_along_wave_electric_field_ {}

double	magnitude_B0_ = std::numeric_limits<double>::signaling_NaN()

double	magnitude_B1_ = std::numeric_limits<double>::signaling_NaN()

double	magnitude_E_ = std::numeric_limits<double>::signaling_NaN()

double	alfven_speed_ = std::numeric_limits<double>::signaling_NaN()

double	fluid_speed_ = std::numeric_limits<double>::signaling_NaN()

gr::Solutions::Minkowski< 3 >	background_spacetime_ {}

Friends
bool	operator== (const AlfvenWave &lhs, const AlfvenWave &rhs)

Detailed Description

Circularly polarized Alfvén wave solution in Minkowski spacetime travelling along a background magnetic field.

An analytic solution to the 3-D GRMHD system. The user specifies the wavenumber $k$ of the Alfvén wave, the constant pressure throughout the fluid $P$ , the constant rest mass density throughout the fluid $ρ_{0}$ , the adiabatic index for the ideal fluid equation of state $γ$ , the magnetic field parallel to the wavevector ${\vec{B}}_{0}$ , and the transverse magnetic field vector ${\vec{B}}_{1}$ at $x = y = z = t = 0$ .

We define the auxiliary velocities:

$v_{B 0}^{2} = \frac{B_{0}^{2}}{ρ_{0} h + B_{0}^{2} + B_{1}^{2}}$

$v_{B 1}^{2} = \frac{B_{1}^{2}}{ρ_{0} h + B_{0}^{2} + B_{1}^{2}}$

The Alfvén wave phase speed that solves the GRMHD equations, even for finite amplitudes [49], is given by:

$v_{A}^{2} = \frac{2 v_{B 0}^{2}}{1 + \sqrt{1 - 4 v_{B 0}^{2} v_{B 1}^{2}}}$

The amplitude of the fluid velocity is given by:

$v_{f}^{2} = \frac{2 v_{B 1}^{2}}{1 + \sqrt{1 - 4 v_{B 0}^{2} v_{B 1}^{2}}}$

The electromagnetic field vectors define a set of basis vectors:

$\begin{aligned} {\hat{b}}_{0} & = \vec{B_{0}} / B_{0} \\ {\hat{b}}_{1} & = \vec{B_{1}} / B_{1} \\ \hat{e} & = {\hat{b}}_{1} \times {\hat{b}}_{0} \end{aligned}$

We also define the auxiliary variable for the phase $ϕ$ :

$ϕ = k (\vec{x} \cdot {\hat{b}}_{0} - v_{A} t)$

In Cartesian coordinates $(x, y, z)$ , and using dimensionless units, the primitive quantities at a given time $t$ are then

$\begin{aligned} ρ (\vec{x}, t) & = ρ_{0} \\ \vec{v} (\vec{x}, t) & = v_{f} (- {\hat{b}}_{1} \cos ϕ + \hat{e} \sin ϕ) \\ P (\vec{x}, t) & = P, \\ ϵ (\vec{x}, t) & = \frac{P}{(γ - 1) ρ_{0}} \\ \vec{B} (\vec{x}, t) & = B_{1} ({\hat{b}}_{1} \cos ϕ - \hat{e} \sin ϕ) + \vec{B_{0}} \end{aligned}$

Note that the phase speed is not the characteristic Alfvén speed $c_{A}$ , which is the speed in the limiting case where the total magnetic field is parallel to the direction of propagation [49] :

$c_{A}^{2} = \frac{b^{2}}{ρ_{0} h + b^{2}}$

Where $b^{2}$ is the invariant quantity $B^{2} - E^{2}$ , given by:

$b^{2} = B_{0}^{2} + B_{1}^{2} - B_{0}^{2} v_{f}^{2}$

Member Function Documentation

◆ get_clone()

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

overridevirtual

Implements evolution::initial_data::InitialData.

Member Data Documentation

◆ background_magnetic_field_

std::array<double, 3> grmhd::Solutions::AlfvenWave::background_magnetic_field_

protected

Initial value:

{
      {std::numeric_limits<double>::signaling_NaN(),
       std::numeric_limits<double>::signaling_NaN(),
       std::numeric_limits<double>::signaling_NaN()}}

◆ help

constexpr Options::String grmhd::Solutions::AlfvenWave::help

staticconstexpr

Initial value:

= {

"Circularly polarized Alfven wave in Minkowski spacetime."}

◆ wave_magnetic_field_

std::array<double, 3> grmhd::Solutions::AlfvenWave::wave_magnetic_field_

protected

Initial value:

{
      {std::numeric_limits<double>::signaling_NaN(),
       std::numeric_limits<double>::signaling_NaN(),
       std::numeric_limits<double>::signaling_NaN()}}

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

src/PointwiseFunctions/AnalyticSolutions/GrMhd/AlfvenWave.hpp

Classes

Public Types

Public Member Functions

Static Public Attributes

Protected Member Functions

Protected Attributes

Friends

Detailed Description

Member Function Documentation

◆ get_clone()

Member Data Documentation

◆ background_magnetic_field_

◆ help

◆ wave_magnetic_field_