Fluid disk orbiting a Kerr black hole. More...

#include <FishboneMoncriefDisk.hpp>

Classes
struct	BhDimlessSpin
	The dimensionless black hole spin, \(\chi = a/M\). More...
struct	BhMass
	The mass of the black hole, \(M\). More...
struct	InnerEdgeRadius
	The radial coordinate of the inner edge of the disk, in units of \(M\). More...
struct	IntermediateVariables
struct	MaxPressureRadius
	The radial coordinate of the maximum pressure, in units of \(M\). More...
struct	Noise
	The magnitude of noise added to pressure/energy of the Disk to drive MRI. More...
struct	PolytropicConstant
	The polytropic constant of the fluid. More...
struct	PolytropicExponent
	The polytropic exponent of the fluid. More...

Public Types
using	equation_of_state_type = EquationsOfState::PolytropicFluid<true>
using	options
template<typename DataType>
using	tags
Public Types inherited from RelativisticEuler::AnalyticSolution< 3 >
using	tags

Public Member Functions
	FishboneMoncriefDisk (const FishboneMoncriefDisk &)=default
FishboneMoncriefDisk &	operator= (const FishboneMoncriefDisk &)=default
	FishboneMoncriefDisk (FishboneMoncriefDisk &&)=default
FishboneMoncriefDisk &	operator= (FishboneMoncriefDisk &&)=default
	FishboneMoncriefDisk (double bh_mass, double bh_dimless_spin, double inner_edge_radius, double max_pressure_radius, double polytropic_constant, double polytropic_exponent, double noise)
auto	get_clone () const -> std::unique_ptr< evolution::initial_data::InitialData > override
template<typename DataType>
DataType	sigma (const DataType &r_sqrd, const DataType &sin_theta_sqrd) const
template<typename DataType>
DataType	inv_ucase_a (const DataType &r_sqrd, const DataType &sin_theta_sqrd, const DataType &delta) const
template<typename DataType>
DataType	four_velocity_t_sqrd (const DataType &r_sqrd, const DataType &sin_theta_sqrd) const
template<typename DataType>
DataType	angular_velocity (const DataType &r_sqrd, const DataType &sin_theta_sqrd) const
template<typename DataType>
DataType	potential (const DataType &r_sqrd, const DataType &sin_theta_sqrd) const
void	pup (PUP::er &p) override
const EquationsOfState::PolytropicFluid< true > &	equation_of_state () const

template<typename DataType, typename... Tags>
tuples::TaggedTuple< Tags... >	variables (const tnsr::I< DataType, 3 > &x, const double, tmpl::list< Tags... >) const
	The variables in Cartesian Spherical-Kerr-Schild coordinates at (x, t)
template<typename DataType, typename Tag>
tuples::TaggedTuple< Tag >	variables (const tnsr::I< DataType, 3 > &x, const double, tmpl::list< Tag >) const
	The variables in Cartesian Spherical-Kerr-Schild coordinates at (x, t)
Public Member Functions inherited from hydro::TemperatureInitialization< FishboneMoncriefDisk >
auto	variables (const tnsr::I< DataType, Dim > &x, 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 RelativisticEuler::AnalyticSolution< 3 >
static constexpr size_t	volume_dim

Protected Member Functions
template<typename DataType>
auto	variables (const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::RestMassDensity< DataType > >, gsl::not_null< IntermediateVariables< DataType > * > vars) const -> tuples::TaggedTuple< hydro::Tags::RestMassDensity< DataType > >
template<typename DataType>
auto	variables (const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::ElectronFraction< DataType > >, gsl::not_null< IntermediateVariables< DataType > * > vars) const -> tuples::TaggedTuple< hydro::Tags::ElectronFraction< DataType > >
template<typename DataType>
auto	variables (const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::SpecificEnthalpy< DataType > >, gsl::not_null< IntermediateVariables< DataType > * > vars) const -> tuples::TaggedTuple< hydro::Tags::SpecificEnthalpy< DataType > >
template<typename DataType>
auto	variables (const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::Pressure< DataType > >, gsl::not_null< IntermediateVariables< DataType > * > vars) const -> tuples::TaggedTuple< hydro::Tags::Pressure< DataType > >
template<typename DataType>
auto	variables (const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::Temperature< DataType > >, gsl::not_null< IntermediateVariables< DataType > * > vars) const -> tuples::TaggedTuple< hydro::Tags::Temperature< DataType > >
template<typename DataType>
auto	variables (const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::SpecificInternalEnergy< DataType > >, gsl::not_null< IntermediateVariables< DataType > * > vars) const -> tuples::TaggedTuple< hydro::Tags::SpecificInternalEnergy< DataType > >
template<typename DataType>
auto	variables (const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::SpatialVelocity< DataType, 3 > >, gsl::not_null< IntermediateVariables< DataType > * > vars) const -> tuples::TaggedTuple< hydro::Tags::SpatialVelocity< DataType, 3 > >
template<typename DataType>
auto	variables (const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::LorentzFactor< DataType > >, gsl::not_null< IntermediateVariables< DataType > * > vars) const -> tuples::TaggedTuple< hydro::Tags::LorentzFactor< DataType > >
template<typename DataType>
auto	variables (const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::MagneticField< DataType, 3 > >, gsl::not_null< IntermediateVariables< DataType > * > vars) const -> tuples::TaggedTuple< hydro::Tags::MagneticField< DataType, 3 > >
template<typename DataType>
auto	variables (const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::DivergenceCleaningField< DataType > >, gsl::not_null< IntermediateVariables< DataType > * > vars) const -> tuples::TaggedTuple< hydro::Tags::DivergenceCleaningField< DataType > >
template<typename DataType, typename Tag, Requires< not tmpl::list_contains_v< tmpl::push_back< hydro::grmhd_tags< DataType >, hydro::Tags::SpecificEnthalpy< DataType >, hydro::Tags::SpatialVelocity< DataType, 3 >, hydro::Tags::LorentzFactor< DataType > >, Tag > > = nullptr>
tuples::TaggedTuple< Tag >	variables (const tnsr::I< DataType, 3 > &x, tmpl::list< Tag >, gsl::not_null< IntermediateVariables< DataType > * > vars) const
template<typename DataType, typename Func>
void	variables_impl (gsl::not_null< IntermediateVariables< DataType > * > vars, Func f) const

Protected Attributes
double	bh_mass_ = std::numeric_limits<double>::signaling_NaN()
double	bh_spin_a_ = std::numeric_limits<double>::signaling_NaN()
double	inner_edge_radius_ = std::numeric_limits<double>::signaling_NaN()
double	max_pressure_radius_ = std::numeric_limits<double>::signaling_NaN()
double	polytropic_constant_ = std::numeric_limits<double>::signaling_NaN()
double	polytropic_exponent_ = std::numeric_limits<double>::signaling_NaN()
double	angular_momentum_ = std::numeric_limits<double>::signaling_NaN()
double	rho_max_ = std::numeric_limits<double>::signaling_NaN()
double	noise_ = std::numeric_limits<double>::signaling_NaN()
EquationsOfState::PolytropicFluid< true >	equation_of_state_ {}
gr::Solutions::SphericalKerrSchild	background_spacetime_ {}
double	background_density_ = 1.e-7
double	background_temperature_ = 2.e-5
double	background_pressure_ = std::numeric_limits<double>::signaling_NaN()
double	background_specific_internal_energy_

Friends
class	grmhd::AnalyticData::MagnetizedFmDisk
bool	operator== (const FishboneMoncriefDisk &lhs, const FishboneMoncriefDisk &rhs)

Detailed Description

Fluid disk orbiting a Kerr black hole.

The Fishbone-Moncrief solution to the 3D relativistic Euler system [74], representing the isentropic flow of a thick fluid disk orbiting a Kerr black hole. In Boyer-Lindquist coordinates \((t, r, \theta, \phi)\), the flow is assumed to be purely toroidal,

\begin{align*}u^\mu = (u^t, 0, 0, u^\phi), \end{align*}

where \(u^\mu\) is the 4-velocity. Then, all the fluid quantities are assumed to share the same symmetries as those of the background spacetime, namely they are stationary (independent of \(t\)), and axially symmetric (independent of \(\phi\)).

Self-gravity is neglected, so that the fluid variables are determined as functions of the metric. Following the treatment by Kozlowski et al. [123] (but using signature +2) the solution is expressed in terms of the quantities

\begin{align*}\Omega &= \dfrac{u^\phi}{u^t},\\ W &= W_\text{in} - \int_{p_\text{in}}^p\frac{dp}{e + p}, \end{align*}

where \(\Omega\) is the angular velocity, \(p\) is the fluid pressure, \(e\) is the energy density, and \(W\) is an auxiliary quantity interpreted in the Newtonian limit as the total (gravitational + centrifugal) potential. \(W_\text{in}\) and \(p_\text{in}\) are the potential and the pressure at the radius of the inner edge, i.e. the closest edge to the black hole. Here we assume \(p_\text{in} = 0.\) The solution to the Euler equation is then

\begin{align*}(u^t)^2 &= \frac{A}{2\Delta\Sigma}\left(1 + \sqrt{1 + \frac{4l^2\Delta \Sigma^2}{A^2\sin^2\theta}}\right)\\ \Omega &= \frac{\Sigma}{A (u^t)^2}\frac{l}{\sin^2\theta} + \frac{2Mra}{A}\\ u^\phi &= \Omega u^t\\ W &= l\Omega - \ln u^t, \end{align*}

where

\begin{align*}\Sigma = r^2 + a^2\cos^2\theta\qquad \Delta = r^2 - 2Mr + a^2\qquad A = (r^2 + a^2)^2 - \Delta a^2 \sin^2\theta \end{align*}

and \(l = u_\phi u^t\) is the so-called angular momentum per unit intertial mass, which is a parameter defining an individual disk. In deriving the solution, an integration constant has been chosen so that \( W\longrightarrow 0\) as \(r\longrightarrow \infty\), in accordance with the Newtonian limit. Note that, from its definition, equipotential contours coincide with isobaric contours. Physically, the matter can fill each of the closed surfaces \(W = \text{const}\), giving rise to an orbiting thick disk. For \(W > 0\), all equipotentials are open, whereas for \(W < 0\), some of them will be closed. Should a disk exist, the pressure reaches a maximum value on the equator at a coordinate radius \(r_\text{max}\) that is related to the angular momentum per unit inertial mass via

\begin{align*}l = \dfrac{M^{1/2}(r_\text{max}^{3/2} + aM^{1/2})(a^2 - 2aM^{1/2} r_\text{max}^{1/2} + r_\text{max}^2)}{2aM^{1/2}r_\text{max}^{3/2} + (r_\text{max} - 3M)r_\text{max}^2}. \end{align*}

Once \(W\) is determined, an equation of state is required in order to obtain the thermodynamic variables. If the flow is isentropic, the specific enthalpy can readily be obtained from the first and second laws of thermodynamics: one has

\begin{align*}\frac{dp}{e + p} = \frac{dh}{h} \end{align*}

so that

\begin{align*}h = h_\text{in}\exp(W_\text{in} - W), \end{align*}

and the pressure can be obtained from a thermodynamic relation of the form \(h = h(p)\). Here we assume a polytropic relation

\begin{align*}p = K\rho^\gamma. \end{align*}

Following [173], the rest mass density is normalized to be 1. at maximum.

Once all the variables are known in Boyer-Lindquist (or Kerr) coordinates, it is straightforward to write them in Cartesian Kerr-Schild coordinates. The coordinate transformation in gr::KerrSchildCoords helps read the Jacobian matrix, which, applied to the azimuthal flow of the disk, gives

\begin{align*}u_\text{KS}^\mu = u^t(1, -y\Omega, x\Omega, 0), \end{align*}

where \(u^t\) and \(\Omega\) are now understood as functions of the Kerr-Schild coordinates. Finally, the spatial velocity can be readily obtained from its definition,

\begin{align*}\alpha v^i = \frac{u^i}{u^t} + \beta^i, \end{align*}

where \(\alpha\) and \(\beta^i\) are the lapse and the shift, respectively.

Note: Kozlowski et al. [123] denote \(l_* = u_\phi u^t\) in order to distinguish this quantity from their own definition \(l = - u_\phi/u_t\).; When using Kerr-Schild coordinates, the horizon that is at constant \(r\) is not spherical, but instead spheroidal. This could make application of boundary condition and computing various fluxes across the horizon more complicated than they need to be. Thus, we use Spherical Kerr-Schild coordinates, see gr::Solutions::SphericalKerrSchild, in which constant \(r\) is spherical. Because we compute variables in Kerr-Schild coordinates, there is a necessary extra step of transforming them back to Spherical Kerr-Schild coordinates.

Member Typedef Documentation

◆ options

using RelativisticEuler::Solutions::FishboneMoncriefDisk::options

Initial value:

 
      tmpl::list<BhMass, BhDimlessSpin, InnerEdgeRadius, MaxPressureRadius,
                 PolytropicConstant, PolytropicExponent, Noise>

◆ tags

template<typename DataType>

using RelativisticEuler::Solutions::FishboneMoncriefDisk::tags

Initial value:

 
      tmpl::append<hydro::grmhd_tags<DataType>,
                   typename gr::Solutions::SphericalKerrSchild::tags<DataType>>

Member Function Documentation

◆ get_clone()

auto RelativisticEuler::Solutions::FishboneMoncriefDisk::get_clone ( ) const -> std::unique_ptr< evolution::initial_data::InitialData >

overridevirtual

Implements evolution::initial_data::InitialData.

Member Data Documentation

◆ background_specific_internal_energy_

double RelativisticEuler::Solutions::FishboneMoncriefDisk::background_specific_internal_energy_

protected

Initial value:

=

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

std::numeric_limits::signaling_NaN

T signaling_NaN(T... args)

◆ help

Options::String RelativisticEuler::Solutions::FishboneMoncriefDisk::help

staticconstexpr

Initial value:

= {

"Fluid disk orbiting a Kerr black hole."}

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

src/PointwiseFunctions/AnalyticSolutions/RelativisticEuler/FishboneMoncriefDisk.hpp

Classes

Public Types

Public Member Functions

Static Public Attributes

Protected Member Functions

Protected Attributes

Friends

Detailed Description

Member Typedef Documentation

◆ options

◆ tags

Member Function Documentation

◆ get_clone()

Member Data Documentation

◆ background_specific_internal_energy_

◆ help