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 | 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 = tmpl::list< WaveNumber, Pressure, RestMassDensity, AdiabaticIndex, BackgroundMagneticField, WaveMagneticField > |
 Public Types inherited from grmhd::AnalyticSolution | |
| template<typename DataType > | |
| using | tags = tmpl::push_back< typename gr::AnalyticSolution< 3 >::template tags< DataType >, hydro::Tags::RestMassDensity< DataType >, hydro::Tags::SpecificInternalEnergy< DataType >, hydro::Tags::Pressure< DataType >, hydro::Tags::SpatialVelocity< DataType, 3 >, hydro::Tags::MagneticField< DataType, 3 >, hydro::Tags::DivergenceCleaningField< DataType >, hydro::Tags::LorentzFactor< DataType >, hydro::Tags::SpecificEnthalpy< DataType > > |
Public Member Functions | |
| AlfvenWave (const AlfvenWave &)=delete | |
| AlfvenWave & | operator= (const AlfvenWave &)=delete |
| AlfvenWave (AlfvenWave &&) noexcept=default | |
| AlfvenWave & | operator= (AlfvenWave &&) noexcept=default |
| AlfvenWave (double wavenumber, double pressure, double rest_mass_density, double adiabatic_index, const std::array< double, 3 > &background_magnetic_field, const std::array< double, 3 > &wave_magnetic_field) noexcept | |
| template<typename DataType , typename... Tags> | |
| tuples::TaggedTuple< Tags... > | variables (const tnsr::I< DataType, 3 > &x, double t, tmpl::list< Tags... >) const noexcept |
Retrieve a collection of hydro variables at (x, t) | |
| template<typename DataType , typename Tag > | |
| tuples::TaggedTuple< Tag > | variables (const tnsr::I< DataType, 3 > &x, double t, tmpl::list< Tag >) const noexcept |
| Retrieve the metric variables. | |
| void | pup (PUP::er &) noexcept |
| const EquationsOfState::IdealFluid< true > & | equation_of_state () const noexcept |
| template<typename DataType > | |
| auto | variables (const tnsr::I< DataType, 3 > &x, double t, tmpl::list< hydro::Tags::RestMassDensity< DataType > >) const noexcept -> 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::SpecificInternalEnergy< DataType > >) const noexcept -> 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 noexcept -> 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 noexcept -> 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 noexcept -> 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 noexcept -> 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 noexcept -> 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 noexcept -> tuples::TaggedTuple< hydro::Tags::SpecificEnthalpy< DataType > > |
Retrieve hydro variable at (x, t) | |
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 noexcept |
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 | 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) noexcept |
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 \(\rho_0\), the adiabatic index for the ideal fluid equation of state \(\gamma\), 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:
\[v^2_{B0} = \frac{B_0^2}{\rho_0 h + B_0^2 + B_1^2}\]
\[v^2_{B1} = \frac{B_1^2}{\rho_0 h + B_0^2 + B_1^2}\]
The Alfvén wave phase speed that solves the GRMHD equations, even for finite amplitudes [33], is given by:
\[v_A^2 = \frac{2v^2_{B0}}{1 + \sqrt{1 - 4 v^2_{B0}v^2_{B1}}}\]
The amplitude of the fluid velocity is given by:
\[v_f^2 = \frac{2v^2_{B1}}{1 + \sqrt{1 - 4 v^2_{B0}v^2_{B1}}}\]
The electromagnetic field vectors define a set of basis vectors:
\begin{align*} \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{align*}
We also define the auxiliary variable for the phase \(\phi\):
\[\phi = k(\vec{x}\cdot\hat{b}_0 - v_A t)\]
In Cartesian coordinates \((x, y, z)\), and using dimensionless units, the primitive quantities at a given time \(t\) are then
\begin{align*} \rho(\vec{x},t) &= \rho_0 \\ \vec{v}(\vec{x},t) &= v_f(-\hat{b}_1\cos\phi +\hat{e}\sin\phi)\\ P(\vec{x},t) &= P, \\ \epsilon(\vec{x}, t) &= \frac{P}{(\gamma - 1)\rho_0}\\ \vec{B}(\vec{x},t) &= B_1(\hat{b}_1\cos\phi -\hat{e}\sin\phi) + \vec{B_0} \end{align*}
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 [33] :
\[c_A^2 = \frac{b^2}{\rho_0 h + b^2}\]
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 Data Documentation
◆ background_magnetic_field_
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protected |
◆ help
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staticconstexpr |
◆ wave_magnetic_field_
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protected |