hydro Namespace Reference

Items related to hydrodynamic systems. More...

Namespaces
namespace	OptionTags
	Tags for options of hydrodynamic systems.
namespace	Solutions
	Holds classes implementing common portions of solutions to various different (magneto)hydrodynamical systems.
namespace	Tags
	Tag containing TildeD * SpecificInternalEnergy Useful as a diagnostics tool, as input to volume integral.
namespace	units
	Functions, constants, and classes for converting between different units.

Classes
class	TemperatureInitialization
	Wrapper to add temperature variable to initial data providing only density and or energy_density initialization. More...

Enumerations
enum	MagneticFieldTreatment { AssumeZero , CheckIfZero , AssumeNonZero }
	Used to specify how to handle the magnetic field. More...
enum class	HalfPlaneIntegralMask { None , PositiveXOnly , NegativeXOnly }

Functions
std::ostream &	operator<< (std::ostream &os, MagneticFieldTreatment t)
std::ostream &	operator<< (std::ostream &os, HalfPlaneIntegralMask mask)
std::string	name (HalfPlaneIntegralMask mask)
template<typename DataType , size_t Dim, typename Fr = Frame::Inertial>
void	quadrupole_moment (const gsl::not_null< tnsr::ii< DataType, Dim, Fr > * > result, const Scalar< DataType > &tilde_d, const tnsr::I< DataType, Dim, Fr > &coordinates)
	Function computing the quadrupole moment. More...
template<typename DataType , size_t Dim, typename Fr = Frame::Inertial>
void	quadrupole_moment_derivative (const gsl::not_null< tnsr::ii< DataType, Dim, Fr > * > result, const Scalar< DataType > &tilde_d, const tnsr::I< DataType, Dim, Fr > &coordinates, const tnsr::I< DataType, Dim, Fr > &spatial_velocity)
	Function computing the first time derivative of the quadrupole moment. More...
template<typename DataType >
void	energy_density (gsl::not_null< Scalar< DataType > * > result, const Scalar< DataType > &rest_mass_density, const Scalar< DataType > &specific_enthalpy, const Scalar< DataType > &pressure, const Scalar< DataType > &lorentz_factor, const Scalar< DataType > &magnetic_field_dot_spatial_velocity, const Scalar< DataType > &comoving_magnetic_field_squared)
	The total mass-energy density measured by a normal observer, $E=n_a{n}_b{T}^{ab}$. More...
template<typename DataType >
void	momentum_density (gsl::not_null< tnsr::I< DataType, 3 > * > result, const Scalar< DataType > &rest_mass_density, const Scalar< DataType > &specific_enthalpy, const tnsr::I< DataType, 3 > &spatial_velocity, const Scalar< DataType > &lorentz_factor, const tnsr::I< DataType, 3 > &magnetic_field, const Scalar< DataType > &magnetic_field_dot_spatial_velocity, const Scalar< DataType > &comoving_magnetic_field_squared)
	The spatial momentum density $S^i=-{\gamma }^{ij}{n}^a{T}_{aj}$. More...
template<typename DataType >
void	stress_trace (gsl::not_null< Scalar< DataType > * > result, const Scalar< DataType > &rest_mass_density, const Scalar< DataType > &specific_enthalpy, const Scalar< DataType > &pressure, const Scalar< DataType > &spatial_velocity_squared, const Scalar< DataType > &lorentz_factor, const Scalar< DataType > &magnetic_field_dot_spatial_velocity, const Scalar< DataType > &comoving_magnetic_field_squared)
	The trace of the spatial stress tensor, $S={\gamma }^{ij}{\gamma }_{ia}{\gamma }_{jb}{T}^{ab}$. More...
template<typename DataType >
void	stress_energy_tensor (gsl::not_null< tnsr::AA< DataType, 3 > * > result, const Scalar< DataType > &rest_mass_density, const Scalar< DataType > &specific_internal_energy, const Scalar< DataType > &pressure, const Scalar< DataType > &lorentz_factor, const Scalar< DataType > &lapse, const Scalar< DataType > &comoving_magnetic_field_magnitude, const tnsr::I< DataType, 3 > &spatial_velocity, const tnsr::I< DataType, 3 > &shift, const tnsr::I< DataType, 3 > &magnetic_field, const tnsr::ii< DataType, 3 > &spatial_metric, const tnsr::II< DataType, 3 > &inverse_spatial_metric)
	Stress Energy Tesnor, $T^{ab}=(\rho h{)}^{\ast }{u}^a{u}^b+p^{\ast }{g}^{ab}-b^a{b}^b$,. More...
template<typename DataType , size_t Dim, typename Fr = Frame::Inertial>
void	transport_velocity (gsl::not_null< tnsr::I< DataType, Dim, Fr > * > result, const tnsr::I< DataType, Dim, Fr > &spatial_velocity, const Scalar< DataType > &lapse, const tnsr::I< DataType, Dim, Fr > &shift)
	Function computing the transport velocity. More...
template<typename DataType >
void	comoving_magnetic_field (gsl::not_null< tnsr::A< DataType, 3 > * > result, const tnsr::I< DataType, 3 > &spatial_velocity, const tnsr::I< DataType, 3 > &magnetic_field, const Scalar< DataType > &magnetic_field_dot_spatial_velocity, const Scalar< DataType > &lorentz_factor, const tnsr::I< DataType, 3 > &shift, const Scalar< DataType > &lapse)
	The comoving magnetic field vector $b^{\mu }$ and one-form $b_{\mu }$. More...
template<typename DataType >
tnsr::A< DataType, 3 >	comoving_magnetic_field (const tnsr::I< DataType, 3 > &spatial_velocity, const tnsr::I< DataType, 3 > &magnetic_field, const Scalar< DataType > &magnetic_field_dot_spatial_velocity, const Scalar< DataType > &lorentz_factor, const tnsr::I< DataType, 3 > &shift, const Scalar< DataType > &lapse)
	The comoving magnetic field vector $b^{\mu }$ and one-form $b_{\mu }$. More...
template<typename DataType >
void	comoving_magnetic_field_one_form (gsl::not_null< tnsr::a< DataType, 3 > * > result, const tnsr::i< DataType, 3 > &spatial_velocity_one_form, const tnsr::i< DataType, 3 > &magnetic_field_one_form, const Scalar< DataType > &magnetic_field_dot_spatial_velocity, const Scalar< DataType > &lorentz_factor, const tnsr::I< DataType, 3 > &shift, const Scalar< DataType > &lapse)
	The comoving magnetic field vector $b^{\mu }$ and one-form $b_{\mu }$. More...
template<typename DataType >
tnsr::a< DataType, 3 >	comoving_magnetic_field_one_form (const tnsr::i< DataType, 3 > &spatial_velocity_one_form, const tnsr::i< DataType, 3 > &magnetic_field_one_form, const Scalar< DataType > &magnetic_field_dot_spatial_velocity, const Scalar< DataType > &lorentz_factor, const tnsr::I< DataType, 3 > &shift, const Scalar< DataType > &lapse)
	The comoving magnetic field vector $b^{\mu }$ and one-form $b_{\mu }$. More...
template<typename DataType >
void	comoving_magnetic_field_squared (gsl::not_null< Scalar< DataType > * > result, const Scalar< DataType > &magnetic_field_squared, const Scalar< DataType > &magnetic_field_dot_spatial_velocity, const Scalar< DataType > &lorentz_factor)
	The comoving magnetic field vector $b^{\mu }$ and one-form $b_{\mu }$. More...
template<typename DataType >
Scalar< DataType >	comoving_magnetic_field_squared (const Scalar< DataType > &magnetic_field_squared, const Scalar< DataType > &magnetic_field_dot_spatial_velocity, const Scalar< DataType > &lorentz_factor)
	The comoving magnetic field vector $b^{\mu }$ and one-form $b_{\mu }$. More...
template<typename DataType >
void	inverse_plasma_beta (gsl::not_null< Scalar< DataType > * > result, const Scalar< DataType > &comoving_magnetic_field_magnitude, const Scalar< DataType > &fluid_pressure)
	Computes the inverse plasma beta. More...
template<typename DataType >
Scalar< DataType >	inverse_plasma_beta (const Scalar< DataType > &comoving_magnetic_field_magnitude, const Scalar< DataType > &fluid_pressure)
	Computes the inverse plasma beta. More...
template<typename DataType , size_t Dim, typename Frame >
void	lorentz_factor (gsl::not_null< Scalar< DataType > * > result, const tnsr::I< DataType, Dim, Frame > &spatial_velocity, const tnsr::i< DataType, Dim, Frame > &spatial_velocity_form)
	Computes the Lorentz factor $W=1/\sqrt{1-v^i{v}_i}$.
template<typename DataType , size_t Dim, typename Frame >
Scalar< DataType >	lorentz_factor (const tnsr::I< DataType, Dim, Frame > &spatial_velocity, const tnsr::i< DataType, Dim, Frame > &spatial_velocity_form)
	Computes the Lorentz factor $W=1/\sqrt{1-v^i{v}_i}$.
template<typename DataType >
void	lorentz_factor (gsl::not_null< Scalar< DataType > * > result, const Scalar< DataType > &spatial_velocity_squared)
	Computes the Lorentz factor $W=1/\sqrt{1-v^i{v}_i}$.
template<typename DataType >
Scalar< DataType >	lorentz_factor (const Scalar< DataType > &spatial_velocity_squared)
	Computes the Lorentz factor $W=1/\sqrt{1-v^i{v}_i}$.
template<typename DataType , size_t Dim, typename Frame >
void	mass_flux (gsl::not_null< tnsr::I< DataType, Dim, Frame > * > result, const Scalar< DataType > &rest_mass_density, const tnsr::I< DataType, Dim, Frame > &spatial_velocity, const Scalar< DataType > &lorentz_factor, const Scalar< DataType > &lapse, const tnsr::I< DataType, Dim, Frame > &shift, const Scalar< DataType > &sqrt_det_spatial_metric)
	Computes the vector $J^i$ in $\dot{M}=-\int J^i{s}_i{d}^{2}S$, representing the mass flux through a surface with normal $s_i$. More...
template<typename DataType , size_t Dim, typename Frame >
tnsr::I< DataType, Dim, Frame >	mass_flux (const Scalar< DataType > &rest_mass_density, const tnsr::I< DataType, Dim, Frame > &spatial_velocity, const Scalar< DataType > &lorentz_factor, const Scalar< DataType > &lapse, const tnsr::I< DataType, Dim, Frame > &shift, const Scalar< DataType > &sqrt_det_spatial_metric)
	Computes the vector $J^i$ in $\dot{M}=-\int J^i{s}_i{d}^{2}S$, representing the mass flux through a surface with normal $s_i$. More...
template<typename DataType , size_t Dim, typename Frame >
void	u_lower_t (gsl::not_null< Scalar< DataType > * > result, const Scalar< DataType > &lorentz_factor, const tnsr::I< DataType, Dim, Frame > &spatial_velocity, const tnsr::ii< DataType, Dim, Frame > &spatial_metric, const Scalar< DataType > &lapse, const tnsr::I< DataType, Dim, Frame > &shift)
	Compute $u_t=$.
template<typename DataType , size_t Dim, typename Frame >
Scalar< DataType >	u_lower_t (const Scalar< DataType > &lorentz_factor, const tnsr::I< DataType, Dim, Frame > &spatial_velocity, const tnsr::ii< DataType, Dim, Frame > &spatial_metric, const Scalar< DataType > &lapse, const tnsr::I< DataType, Dim, Frame > &shift)
	Compute $u_t=$.
template<typename DataType >
void	mass_weighted_internal_energy (gsl::not_null< Scalar< DataType > * > result, const Scalar< DataType > &tilde_d, const Scalar< DataType > &specific_internal_energy)
	Compute tilde_d * specific_internal_energy Result of the calculation stored in result.
template<typename DataType >
Scalar< DataType >	mass_weighted_internal_energy (const Scalar< DataType > &tilde_d, const Scalar< DataType > &specific_internal_energy)
	Compute tilde_d * specific_internal_energy Result of the calculation stored in result.
template<typename DataType >
void	mass_weighted_kinetic_energy (gsl::not_null< Scalar< DataType > * > result, const Scalar< DataType > &tilde_d, const Scalar< DataType > &lorentz_factor)
	Compute tilde_d * (lorentz_factor - 1.0) Result of the calculation stored in result.
template<typename DataType >
Scalar< DataType >	mass_weighted_kinetic_energy (const Scalar< DataType > &tilde_d, const Scalar< DataType > &lorentz_factor)
	Compute tilde_d * (lorentz_factor - 1.0) Result of the calculation stored in result.
template<typename DataType , size_t Dim, typename Fr = Frame::Inertial>
void	tilde_d_unbound_ut_criterion (gsl::not_null< Scalar< DataType > * > result, const Scalar< DataType > &tilde_d, const Scalar< DataType > &lorentz_factor, const tnsr::I< DataType, Dim, Fr > &spatial_velocity, const tnsr::ii< DataType, Dim, Fr > &spatial_metric, const Scalar< DataType > &lapse, const tnsr::I< DataType, Dim, Fr > &shift)
	Returns tilde_d in regions where u_t < -1 and 0 in regions where u_t > -1 (approximate criteria for unbound matter, theoretically valid for particles following geodesics of a time-independent metric).
template<typename DataType , size_t Dim, typename Fr = Frame::Inertial>
Scalar< DataType >	tilde_d_unbound_ut_criterion (const Scalar< DataType > &tilde_d, const Scalar< DataType > &lorentz_factor, const tnsr::I< DataType, Dim, Fr > &spatial_velocity, const tnsr::ii< DataType, Dim, Fr > &spatial_metric, const Scalar< DataType > &lapse, const tnsr::I< DataType, Dim, Fr > &shift)
	Returns tilde_d in regions where u_t < -1 and 0 in regions where u_t > -1 (approximate criteria for unbound matter, theoretically valid for particles following geodesics of a time-independent metric).
template<HalfPlaneIntegralMask IntegralMask, typename DataType , size_t Dim>
void	tilde_d_in_half_plane (gsl::not_null< Scalar< DataType > * > result, const Scalar< DataType > &tilde_d, const tnsr::I< DataType, Dim, Frame::Grid > &grid_coords)
	Returns tilde_d in one half plane and zero in the other IntegralMask allows us to restrict the data to the x>0 or x<0 plane in grid coordinates (useful for NSNS).
template<HalfPlaneIntegralMask IntegralMask, typename DataType , size_t Dim>
Scalar< DataType >	tilde_d_in_half_plane (const Scalar< DataType > &tilde_d, const tnsr::I< DataType, Dim, Frame::Grid > &grid_coords)
	Returns tilde_d in one half plane and zero in the other IntegralMask allows us to restrict the data to the x>0 or x<0 plane in grid coordinates (useful for NSNS).
template<HalfPlaneIntegralMask IntegralMask, typename DataType , size_t Dim, typename Fr = Frame::Inertial>
void	mass_weighted_coords (gsl::not_null< tnsr::I< DataType, Dim, Fr > * > result, const Scalar< DataType > &tilde_d, const tnsr::I< DataType, Dim, Frame::Grid > &grid_coords, const tnsr::I< DataType, Dim, Fr > &compute_coords)
	Returns tilde_d * compute_coords IntegralMask allows us to restrict the data to the x>0 or x<0 plane in grid coordinates (useful for NSNS).
template<HalfPlaneIntegralMask IntegralMask, typename DataType , size_t Dim, typename Fr = Frame::Inertial>
tnsr::I< DataType, Dim, Fr >	mass_weighted_coords (const Scalar< DataType > &tilde_d, const tnsr::I< DataType, Dim, Frame::Grid > &grid_coords, const tnsr::I< DataType, Dim, Fr > &compute_coords)
	Returns tilde_d * compute_coords IntegralMask allows us to restrict the data to the x>0 or x<0 plane in grid coordinates (useful for NSNS).
template<typename DataType , size_t ThermodynamicDim>
void	sound_speed_squared (gsl::not_null< Scalar< DataType > * > result, const Scalar< DataType > &rest_mass_density, const Scalar< DataType > &specific_internal_energy, const Scalar< DataType > &specific_enthalpy, const EquationsOfState::EquationOfState< true, ThermodynamicDim > &equation_of_state)
	Computes the relativistic sound speed squared. More...
template<typename DataType , size_t ThermodynamicDim>
Scalar< DataType >	sound_speed_squared (const Scalar< DataType > &rest_mass_density, const Scalar< DataType > &specific_internal_energy, const Scalar< DataType > &specific_enthalpy, const EquationsOfState::EquationOfState< true, ThermodynamicDim > &equation_of_state)
	Computes the relativistic sound speed squared. More...
template<typename DataType >
void	relativistic_specific_enthalpy (gsl::not_null< Scalar< DataType > * > result, const Scalar< DataType > &rest_mass_density, const Scalar< DataType > &specific_internal_energy, const Scalar< DataType > &pressure)
	Computes the relativistic specific enthalpy $h$ as: $h=1+ϵ+\frac{p}{\rho }$ where $ϵ$ is the specific internal energy, $p$ is the pressure, and $\rho$ is the rest mass density.
template<typename DataType >
Scalar< DataType >	relativistic_specific_enthalpy (const Scalar< DataType > &rest_mass_density, const Scalar< DataType > &specific_internal_energy, const Scalar< DataType > &pressure)
	Computes the relativistic specific enthalpy $h$ as: $h=1+ϵ+\frac{p}{\rho }$ where $ϵ$ is the specific internal energy, $p$ is the pressure, and $\rho$ is the rest mass density.

Detailed Description

Items related to hydrodynamic systems.

Enumeration Type Documentation

MagneticFieldTreatment

enum hydro::MagneticFieldTreatment

Used to specify how to handle the magnetic field.

Enumerator
AssumeZero	Assume the magnetic field is zero.
CheckIfZero	Check if the magnetic field is zero.
AssumeNonZero	Assume the magnetic field is non-zero.

Function Documentation

comoving_magnetic_field() [1/2]

template<typename DataType >

tnsr::A< DataType, 3 > hydro::comoving_magnetic_field	(	const tnsr::I< DataType, 3 > &	spatial_velocity,
		const tnsr::I< DataType, 3 > &	magnetic_field,
		const Scalar< DataType > &	magnetic_field_dot_spatial_velocity,
		const Scalar< DataType > &	lorentz_factor,
		const tnsr::I< DataType, 3 > &	shift,
		const Scalar< DataType > &	lapse
	)

The comoving magnetic field vector $b^{\mu }$ and one-form $b_{\mu }$.

The components of the comoving magnetic field vector are:

$$\begin{array}{}\text{(1)}& b^0& =W{B}^j{v}^k{\gamma }_{jk}/\alpha \text{(2)}& b^i& =B^i/W+B^j{v}^k{\gamma }_{jk}{u}^i\end{array}$$

where $u^i=W(v^i-{\beta }^i/\alpha )$.

Using the spacetime metric, the corresponding one-form components are:

$$\begin{array}{}\text{(3)}& b_0& =-\alpha W{v}^i{B}_i+{\beta }^i{b}_i\text{(4)}& b_i& =B_i/W+B^j{v}^k{\gamma }_{jk}W{v}_i\end{array}$$

The square of the vector is:

$$\begin{array}{}\text{(5)}& b^2=B^i{B}^j{\gamma }_{ij}/W^2+(B^i{v}^j{\gamma }_{ij}{)}^2\end{array}$$

See also Eq. (5.173) in [14], with the difference that we work in Heaviside-Lorentz units where the magnetic field is rescaled by $1/\sqrt{4\pi }$, following [144] .

comoving_magnetic_field() [2/2]

template<typename DataType >

void hydro::comoving_magnetic_field	(	gsl::not_null< tnsr::A< DataType, 3 > * >	result,
		const tnsr::I< DataType, 3 > &	spatial_velocity,
		const tnsr::I< DataType, 3 > &	magnetic_field,
		const Scalar< DataType > &	magnetic_field_dot_spatial_velocity,
		const Scalar< DataType > &	lorentz_factor,
		const tnsr::I< DataType, 3 > &	shift,
		const Scalar< DataType > &	lapse
	)

The comoving magnetic field vector $b^{\mu }$ and one-form $b_{\mu }$.

The components of the comoving magnetic field vector are:

$$\begin{array}{}\text{(6)}& b^0& =W{B}^j{v}^k{\gamma }_{jk}/\alpha \text{(7)}& b^i& =B^i/W+B^j{v}^k{\gamma }_{jk}{u}^i\end{array}$$

where $u^i=W(v^i-{\beta }^i/\alpha )$.

Using the spacetime metric, the corresponding one-form components are:

$$\begin{array}{}\text{(8)}& b_0& =-\alpha W{v}^i{B}_i+{\beta }^i{b}_i\text{(9)}& b_i& =B_i/W+B^j{v}^k{\gamma }_{jk}W{v}_i\end{array}$$

The square of the vector is:

$$\begin{array}{}\text{(10)}& b^2=B^i{B}^j{\gamma }_{ij}/W^2+(B^i{v}^j{\gamma }_{ij}{)}^2\end{array}$$

See also Eq. (5.173) in [14], with the difference that we work in Heaviside-Lorentz units where the magnetic field is rescaled by $1/\sqrt{4\pi }$, following [144] .

comoving_magnetic_field_one_form() [1/2]

template<typename DataType >

tnsr::a< DataType, 3 > hydro::comoving_magnetic_field_one_form	(	const tnsr::i< DataType, 3 > &	spatial_velocity_one_form,
		const tnsr::i< DataType, 3 > &	magnetic_field_one_form,
		const Scalar< DataType > &	magnetic_field_dot_spatial_velocity,
		const Scalar< DataType > &	lorentz_factor,
		const tnsr::I< DataType, 3 > &	shift,
		const Scalar< DataType > &	lapse
	)

The comoving magnetic field vector $b^{\mu }$ and one-form $b_{\mu }$.

The components of the comoving magnetic field vector are:

$$\begin{array}{}\text{(11)}& b^0& =W{B}^j{v}^k{\gamma }_{jk}/\alpha \text{(12)}& b^i& =B^i/W+B^j{v}^k{\gamma }_{jk}{u}^i\end{array}$$

where $u^i=W(v^i-{\beta }^i/\alpha )$.

Using the spacetime metric, the corresponding one-form components are:

$$\begin{array}{}\text{(13)}& b_0& =-\alpha W{v}^i{B}_i+{\beta }^i{b}_i\text{(14)}& b_i& =B_i/W+B^j{v}^k{\gamma }_{jk}W{v}_i\end{array}$$

The square of the vector is:

$$\begin{array}{}\text{(15)}& b^2=B^i{B}^j{\gamma }_{ij}/W^2+(B^i{v}^j{\gamma }_{ij}{)}^2\end{array}$$

See also Eq. (5.173) in [14], with the difference that we work in Heaviside-Lorentz units where the magnetic field is rescaled by $1/\sqrt{4\pi }$, following [144] .

comoving_magnetic_field_one_form() [2/2]

template<typename DataType >

void hydro::comoving_magnetic_field_one_form	(	gsl::not_null< tnsr::a< DataType, 3 > * >	result,
		const tnsr::i< DataType, 3 > &	spatial_velocity_one_form,
		const tnsr::i< DataType, 3 > &	magnetic_field_one_form,
		const Scalar< DataType > &	magnetic_field_dot_spatial_velocity,
		const Scalar< DataType > &	lorentz_factor,
		const tnsr::I< DataType, 3 > &	shift,
		const Scalar< DataType > &	lapse
	)

The comoving magnetic field vector $b^{\mu }$ and one-form $b_{\mu }$.

The components of the comoving magnetic field vector are:

$$\begin{array}{}\text{(16)}& b^0& =W{B}^j{v}^k{\gamma }_{jk}/\alpha \text{(17)}& b^i& =B^i/W+B^j{v}^k{\gamma }_{jk}{u}^i\end{array}$$

where $u^i=W(v^i-{\beta }^i/\alpha )$.

Using the spacetime metric, the corresponding one-form components are:

$$\begin{array}{}\text{(18)}& b_0& =-\alpha W{v}^i{B}_i+{\beta }^i{b}_i\text{(19)}& b_i& =B_i/W+B^j{v}^k{\gamma }_{jk}W{v}_i\end{array}$$

The square of the vector is:

$$\begin{array}{}\text{(20)}& b^2=B^i{B}^j{\gamma }_{ij}/W^2+(B^i{v}^j{\gamma }_{ij}{)}^2\end{array}$$

See also Eq. (5.173) in [14], with the difference that we work in Heaviside-Lorentz units where the magnetic field is rescaled by $1/\sqrt{4\pi }$, following [144] .

comoving_magnetic_field_squared() [1/2]

template<typename DataType >

Scalar< DataType > hydro::comoving_magnetic_field_squared	(	const Scalar< DataType > &	magnetic_field_squared,
		const Scalar< DataType > &	magnetic_field_dot_spatial_velocity,
		const Scalar< DataType > &	lorentz_factor
	)

The comoving magnetic field vector $b^{\mu }$ and one-form $b_{\mu }$.

The components of the comoving magnetic field vector are:

$$\begin{array}{}\text{(21)}& b^0& =W{B}^j{v}^k{\gamma }_{jk}/\alpha \text{(22)}& b^i& =B^i/W+B^j{v}^k{\gamma }_{jk}{u}^i\end{array}$$

where $u^i=W(v^i-{\beta }^i/\alpha )$.

Using the spacetime metric, the corresponding one-form components are:

$$\begin{array}{}\text{(23)}& b_0& =-\alpha W{v}^i{B}_i+{\beta }^i{b}_i\text{(24)}& b_i& =B_i/W+B^j{v}^k{\gamma }_{jk}W{v}_i\end{array}$$

The square of the vector is:

$$\begin{array}{}\text{(25)}& b^2=B^i{B}^j{\gamma }_{ij}/W^2+(B^i{v}^j{\gamma }_{ij}{)}^2\end{array}$$

See also Eq. (5.173) in [14], with the difference that we work in Heaviside-Lorentz units where the magnetic field is rescaled by $1/\sqrt{4\pi }$, following [144] .

comoving_magnetic_field_squared() [2/2]

template<typename DataType >

void hydro::comoving_magnetic_field_squared	(	gsl::not_null< Scalar< DataType > * >	result,
		const Scalar< DataType > &	magnetic_field_squared,
		const Scalar< DataType > &	magnetic_field_dot_spatial_velocity,
		const Scalar< DataType > &	lorentz_factor
	)

The comoving magnetic field vector $b^{\mu }$ and one-form $b_{\mu }$.

The components of the comoving magnetic field vector are:

$$\begin{array}{}\text{(26)}& b^0& =W{B}^j{v}^k{\gamma }_{jk}/\alpha \text{(27)}& b^i& =B^i/W+B^j{v}^k{\gamma }_{jk}{u}^i\end{array}$$

where $u^i=W(v^i-{\beta }^i/\alpha )$.

Using the spacetime metric, the corresponding one-form components are:

$$\begin{array}{}\text{(28)}& b_0& =-\alpha W{v}^i{B}_i+{\beta }^i{b}_i\text{(29)}& b_i& =B_i/W+B^j{v}^k{\gamma }_{jk}W{v}_i\end{array}$$

The square of the vector is:

$$\begin{array}{}\text{(30)}& b^2=B^i{B}^j{\gamma }_{ij}/W^2+(B^i{v}^j{\gamma }_{ij}{)}^2\end{array}$$

See also Eq. (5.173) in [14], with the difference that we work in Heaviside-Lorentz units where the magnetic field is rescaled by $1/\sqrt{4\pi }$, following [144] .

energy_density()

template<typename DataType >

void hydro::energy_density	(	gsl::not_null< Scalar< DataType > * >	result,
		const Scalar< DataType > &	rest_mass_density,
		const Scalar< DataType > &	specific_enthalpy,
		const Scalar< DataType > &	pressure,
		const Scalar< DataType > &	lorentz_factor,
		const Scalar< DataType > &	magnetic_field_dot_spatial_velocity,
		const Scalar< DataType > &	comoving_magnetic_field_squared
	)

The total mass-energy density measured by a normal observer, $E=n_a{n}_b{T}^{ab}$.

This quantity sources the gravitational field equations in the 3+1 decomposition (see Eq. (2.138) in [14]).

Perfect fluid contribution (Eq. (5.33) in [14]):

$$\begin{array}{}\text{(31)}& E_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{i}\mathrm{d}}=\rho h{W}^2-p\end{array}$$

Magnetic field contribution (Eq. (5.152) in [14]):

$$\begin{array}{}\text{(32)}& E_{\mathrm{e}\mathrm{m}}=b^2(W^2-\frac{1}{2})-(\alpha b^t{)}^2\end{array}$$

where $\alpha b^t=W{B}^k{v}_k$.

Parameters

result	Output buffer. Will be resized if needed.
rest_mass_density	$\rho$
specific_enthalpy	$h$
pressure	$p$
lorentz_factor	$W$
magnetic_field_dot_spatial_velocity	$B^k{v}_k$
comoving_magnetic_field_squared	$b^2$

See also

gr::Tags::EnergyDensity

inverse_plasma_beta() [1/2]

template<typename DataType >

Scalar< DataType > hydro::inverse_plasma_beta	(	const Scalar< DataType > &	comoving_magnetic_field_magnitude,
		const Scalar< DataType > &	fluid_pressure
	)

Computes the inverse plasma beta.

The inverse plasma beta ${\beta }^{-1}=b^2/(2p)$, where $b^2$ is the square of the comoving magnetic field amplitude and $p$ is the fluid pressure.

inverse_plasma_beta() [2/2]

template<typename DataType >

void hydro::inverse_plasma_beta	(	gsl::not_null< Scalar< DataType > * >	result,
		const Scalar< DataType > &	comoving_magnetic_field_magnitude,
		const Scalar< DataType > &	fluid_pressure
	)

Computes the inverse plasma beta.

The inverse plasma beta ${\beta }^{-1}=b^2/(2p)$, where $b^2$ is the square of the comoving magnetic field amplitude and $p$ is the fluid pressure.

mass_flux() [1/2]

template<typename DataType , size_t Dim, typename Frame >

tnsr::I< DataType, Dim, Frame > hydro::mass_flux	(	const Scalar< DataType > &	rest_mass_density,
		const tnsr::I< DataType, Dim, Frame > &	spatial_velocity,
		const Scalar< DataType > &	lorentz_factor,
		const Scalar< DataType > &	lapse,
		const tnsr::I< DataType, Dim, Frame > &	shift,
		const Scalar< DataType > &	sqrt_det_spatial_metric
	)

Computes the vector $J^i$ in $\dot{M}=-\int J^i{s}_i{d}^{2}S$, representing the mass flux through a surface with normal $s_i$.

Note that the integral is understood as a flat-space integral: all metric factors are included in $J^i$. In particular, if the integral is done over a Strahlkorper, the gr::surfaces::euclidean_area_element of the Strahlkorper should be used, and $s_i$ is the normal one-form to the Strahlkorper normalized with the flat metric, $s_i{s}_j{\delta }^{ij}=1$.

The formula is $J^i=\rho W\sqrt{\gamma }(\alpha v^i-{\beta }^i)$, where $\rho$ is the mass density, $W$ is the Lorentz factor, $v^i$ is the spatial velocity of the fluid, $\gamma$ is the determinant of the 3-metric ${\gamma }_{ij}$, $\alpha$ is the lapse, and ${\beta }^i$ is the shift.

mass_flux() [2/2]

template<typename DataType , size_t Dim, typename Frame >

void hydro::mass_flux	(	gsl::not_null< tnsr::I< DataType, Dim, Frame > * >	result,
		const Scalar< DataType > &	rest_mass_density,
		const tnsr::I< DataType, Dim, Frame > &	spatial_velocity,
		const Scalar< DataType > &	lorentz_factor,
		const Scalar< DataType > &	lapse,
		const tnsr::I< DataType, Dim, Frame > &	shift,
		const Scalar< DataType > &	sqrt_det_spatial_metric
	)

Computes the vector $J^i$ in $\dot{M}=-\int J^i{s}_i{d}^{2}S$, representing the mass flux through a surface with normal $s_i$.

Note that the integral is understood as a flat-space integral: all metric factors are included in $J^i$. In particular, if the integral is done over a Strahlkorper, the gr::surfaces::euclidean_area_element of the Strahlkorper should be used, and $s_i$ is the normal one-form to the Strahlkorper normalized with the flat metric, $s_i{s}_j{\delta }^{ij}=1$.

The formula is $J^i=\rho W\sqrt{\gamma }(\alpha v^i-{\beta }^i)$, where $\rho$ is the mass density, $W$ is the Lorentz factor, $v^i$ is the spatial velocity of the fluid, $\gamma$ is the determinant of the 3-metric ${\gamma }_{ij}$, $\alpha$ is the lapse, and ${\beta }^i$ is the shift.

momentum_density()

template<typename DataType >

void hydro::momentum_density	(	gsl::not_null< tnsr::I< DataType, 3 > * >	result,
		const Scalar< DataType > &	rest_mass_density,
		const Scalar< DataType > &	specific_enthalpy,
		const tnsr::I< DataType, 3 > &	spatial_velocity,
		const Scalar< DataType > &	lorentz_factor,
		const tnsr::I< DataType, 3 > &	magnetic_field,
		const Scalar< DataType > &	magnetic_field_dot_spatial_velocity,
		const Scalar< DataType > &	comoving_magnetic_field_squared
	)

The spatial momentum density $S^i=-{\gamma }^{ij}{n}^a{T}_{aj}$.

This quantity sources the gravitational field equations in the 3+1 decomposition (see Eq. (2.138) in [14]).

Perfect fluid contribution (Eq. (5.34) in [14]):

$$\begin{array}{}\text{(33)}& S_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{i}\mathrm{d}}^i=\rho h{W}^2{v}^i\end{array}$$

Magnetic field contribution (Eq. (5.153) in [14]):

$$\begin{array}{}\text{(34)}& S_{\mathrm{e}\mathrm{m}}^i=b^2{W}^2{v}^i-\alpha b^t{\gamma }^{ij}{b}_j\end{array}$$

where $\alpha b^t{\gamma }^{ij}{b}_j=B^k{v}_k{B}^i+(B^k{v}_k{)}^2{W}^2{v}^i$.

Parameters

result	Output buffer. Will be resized if needed.
rest_mass_density	$\rho$
specific_enthalpy	$h$
spatial_velocity	$v^i$
lorentz_factor	$W$
magnetic_field	$B^i$
magnetic_field_dot_spatial_velocity	$B^k{v}_k$
comoving_magnetic_field_squared	$b^2$

See also

gr::Tags::MomentumDensity

quadrupole_moment()

template<typename DataType , size_t Dim, typename Fr = Frame::Inertial>

void hydro::quadrupole_moment	(	const gsl::not_null< tnsr::ii< DataType, Dim, Fr > * >	result,
		const Scalar< DataType > &	tilde_d,
		const tnsr::I< DataType, Dim, Fr > &	coordinates
	)

Function computing the quadrupole moment.

Computes the quadrupole moment, using $\tilde{D}{x}^i{x}^j$ (equation 21 of [178]), with $\tilde{D}=\sqrt{\gamma }\rho W$, $W$ being the Lorentz factor, $\gamma$ being the determinant of the spatial metric, and $x$ the coordinates in Frame Fr. Result of calculation stored in result.

quadrupole_moment_derivative()

template<typename DataType , size_t Dim, typename Fr = Frame::Inertial>

void hydro::quadrupole_moment_derivative	(	const gsl::not_null< tnsr::ii< DataType, Dim, Fr > * >	result,
		const Scalar< DataType > &	tilde_d,
		const tnsr::I< DataType, Dim, Fr > &	coordinates,
		const tnsr::I< DataType, Dim, Fr > &	spatial_velocity
	)

Function computing the first time derivative of the quadrupole moment.

Computes the first time derivative of the quadrupole moment, using $\tilde{D}(v^i{x}^j+x^i{v}^j)$ (equation 23 of [178]), with $\tilde{D}=\sqrt{\gamma }\rho W$, $W$ being the Lorentz factor, $\gamma$ being the determinant of the spatial metric, $x$ the coordinates in Frame Fr, and $v$ the corresponding spatial velocity. Result of calculation stored in result.

stress_energy_tensor()

template<typename DataType >

void hydro::stress_energy_tensor	(	gsl::not_null< tnsr::AA< DataType, 3 > * >	result,
		const Scalar< DataType > &	rest_mass_density,
		const Scalar< DataType > &	specific_internal_energy,
		const Scalar< DataType > &	pressure,
		const Scalar< DataType > &	lorentz_factor,
		const Scalar< DataType > &	lapse,
		const Scalar< DataType > &	comoving_magnetic_field_magnitude,
		const tnsr::I< DataType, 3 > &	spatial_velocity,
		const tnsr::I< DataType, 3 > &	shift,
		const tnsr::I< DataType, 3 > &	magnetic_field,
		const tnsr::ii< DataType, 3 > &	spatial_metric,
		const tnsr::II< DataType, 3 > &	inverse_spatial_metric
	)

Stress Energy Tesnor, $T^{ab}=(\rho h{)}^{\ast }{u}^a{u}^b+p^{\ast }{g}^{ab}-b^a{b}^b$,.

where $(\rho h{)}^{\ast }=\rho h+b^2$ and $p^{\ast }=p+b^2/2$ are the enthalpy density and fluid pressure augmented by contributions of magnetic pressure $p_{mag}$ = b^{2}/2, respectively.

$b$ refers to magnetic field measured in the comoving frame of the fluid $b^a{=}^{\ast }{F}^{ab}{u}_b$.

stress_trace()

template<typename DataType >

void hydro::stress_trace	(	gsl::not_null< Scalar< DataType > * >	result,
		const Scalar< DataType > &	rest_mass_density,
		const Scalar< DataType > &	specific_enthalpy,
		const Scalar< DataType > &	pressure,
		const Scalar< DataType > &	spatial_velocity_squared,
		const Scalar< DataType > &	lorentz_factor,
		const Scalar< DataType > &	magnetic_field_dot_spatial_velocity,
		const Scalar< DataType > &	comoving_magnetic_field_squared
	)

The trace of the spatial stress tensor, $S={\gamma }^{ij}{\gamma }_{ia}{\gamma }_{jb}{T}^{ab}$.

This quantity sources the gravitational field equations in the 3+1 decomposition (see Eq. (2.138) in [14]).

Perfect fluid contribution (Eq. (5.36) in [14]):

$$\begin{array}{}\text{(35)}& S_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{i}\mathrm{d}}=3p+\rho h(W^2-1)\end{array}$$

Magnetic field contribution (Eq. (5.155) in [14]):

$$\begin{array}{}\text{(36)}& S_{\mathrm{e}\mathrm{m}}=b^2(W^2{v}^2+\frac{3}{2})-{\gamma }^{ij}{b}_i{b}_j\end{array}$$

where ${\gamma }^{ij}{b}_i{b}_j=b^2+(B^k{v}_k{)}^2(W^2{v}^2+1)$.

Parameters

result	Output buffer. Will be resized if needed.
rest_mass_density	$\rho$
specific_enthalpy	$h$
pressure	$p$
spatial_velocity_squared	$v^2={\gamma }_{ij}{v}^i{v}^j$
lorentz_factor	$W$
magnetic_field_dot_spatial_velocity	$B^k{v}_k$
comoving_magnetic_field_squared	$b^2$

See also

gr::Tags::StressTrace

transport_velocity()

template<typename DataType , size_t Dim, typename Fr = Frame::Inertial>

void hydro::transport_velocity	(	gsl::not_null< tnsr::I< DataType, Dim, Fr > * >	result,
		const tnsr::I< DataType, Dim, Fr > &	spatial_velocity,
		const Scalar< DataType > &	lapse,
		const tnsr::I< DataType, Dim, Fr > &	shift
	)

Function computing the transport velocity.

Computes the transport velocity, using $v_t^i=\alpha v^i-{\beta }^i$, with $v^i$ being the spatial velocity, $\alpha$ the lapse, and ${\beta }^i$ the shift.