grmhd::AnalyticData::InitialMagneticFields::Poloidal Class Reference

Poloidal magnetic field for GRMHD initial data. More...

#include <Poloidal.hpp>

Classes
struct	Center
struct	CutoffPressure
struct	MaxDistanceFromCenter
struct	PressureExponent
struct	VectorPotentialAmplitude

Public Types
using	options = tmpl::list< PressureExponent, CutoffPressure, VectorPotentialAmplitude, Center, MaxDistanceFromCenter >

Public Member Functions
	Poloidal (const Poloidal &)=default
Poloidal &	operator= (const Poloidal &)=default
	Poloidal (Poloidal &&)=default
Poloidal &	operator= (Poloidal &&)=default
	Poloidal (size_t pressure_exponent, double cutoff_pressure, double vector_potential_amplitude, std::array< double, 3 > center, double max_distance_from_center)
auto	get_clone () const -> std::unique_ptr< InitialMagneticField > override
void	pup (PUP::er &p) override
void	variables (gsl::not_null< tnsr::I< DataVector, 3 > * > result, const tnsr::I< DataVector, 3 > &coords, const Scalar< DataVector > &pressure, const Scalar< DataVector > &sqrt_det_spatial_metric, const tnsr::i< DataVector, 3 > &deriv_pressure) const override
	Retrieve magnetic fields at (x) More...
void	variables (gsl::not_null< tnsr::I< double, 3 > * > result, const tnsr::I< double, 3 > &coords, const Scalar< double > &pressure, const Scalar< double > &sqrt_det_spatial_metric, const tnsr::i< double, 3 > &deriv_pressure) const override
	Retrieve magnetic fields at (x) More...
bool	is_equal (const InitialMagneticField &rhs) const override
virtual auto	get_clone () const -> std::unique_ptr< InitialMagneticField >=0
virtual void	variables (gsl::not_null< tnsr::I< DataVector, 3 > * > result, const tnsr::I< DataVector, 3 > &coords, const Scalar< DataVector > &pressure, const Scalar< DataVector > &sqrt_det_spatial_metric, const tnsr::i< DataVector, 3 > &deriv_pressure) const =0
virtual void	variables (gsl::not_null< tnsr::I< double, 3 > * > result, const tnsr::I< double, 3 > &coords, const Scalar< double > &pressure, const Scalar< double > &sqrt_det_spatial_metric, const tnsr::i< double, 3 > &deriv_pressure) const =0
virtual bool	is_equal (const InitialMagneticField &rhs) const =0

Static Public Attributes
static constexpr Options::String	help = {"Poloidal initial magnetic field"}

Friends
bool	operator== (const Poloidal &lhs, const Poloidal &rhs)
bool	operator!= (const Poloidal &lhs, const Poloidal &rhs)

Detailed Description

Poloidal magnetic field for GRMHD initial data.

The vector potential has the form

\begin{align*} A_{\phi} = A_b \varpi^2 \max(p-p_{\mathrm{cut}}, 0)^{n_s} , \end{align*}

where \(A_b\) controls the amplitude of the magnetic field, \(\varpi^2=x^2+y^2=r^2-z^2\) is the cylindrical radius, \(n_s\) controls the degree of differentiability, and \(p_{\mathrm{cut}}\) controls the pressure cutoff below which the magnetic field is zero.

In Cartesian coordinates the vector potential is:

\begin{align*} A_x & = -\frac{y}{\varpi^2}A_\phi = -y A_b\max(p-p_{\mathrm{cut}}, 0)^{n_s}, \\ A_y & = \frac{x}{\varpi^2}A_\phi = x A_b\max(p-p_{\mathrm{cut}}, 0)^{n_s}, \\ A_z & = 0, \end{align*}

On the region where the field is non-zero, the magnetic field is given by:

\begin{align*} B^x & = -\frac{A_b n_s}{\sqrt{\gamma}} (p-p_{\mathrm{cut}})^{n_s-1} x \partial_z p \\ B^x & = -\frac{A_b n_s}{\sqrt{\gamma}} (p-p_{\mathrm{cut}})^{n_s-1} y \partial_z p \\ B^z & = \frac{A_b}{\sqrt{\gamma}}\left[ 2(p-p_{\mathrm{cut}})^{n_s} + n_s (p-p_{\mathrm{cut}})^{n_s-1} (x \partial_x p + y \partial_y p) \right] \end{align*}

Taking the small- \(r\) limit gives the magnetic field at the origin:

\begin{align*} B^x&=0, \\ B^y&=0, \\ B^z&=\frac{A_b}{\sqrt{\gamma}} 2(p-p_{\mathrm{cut}})^{n_s}. \end{align*}

Note that the coordinates are relative to the Center passed in, so the field can be centered about any arbitrary point. The field is also zero outside of MaxDistanceFromCenter, so that compact support can be imposed if necessary.

Warning

This assumes the magnetic field is initialized, both in size and value, before being passed into the variables function. This is so that multiple magnetic fields can be superposed. Each magnetic field configuration does a += to make this possible.

Member Function Documentation

get_clone()

auto grmhd::AnalyticData::InitialMagneticFields::Poloidal::get_clone ( ) const -> std::unique_ptr< InitialMagneticField >

overridevirtual

Implements grmhd::AnalyticData::InitialMagneticFields::InitialMagneticField.

is_equal()

bool grmhd::AnalyticData::InitialMagneticFields::Poloidal::is_equal ( const InitialMagneticField &  rhs ) const

overridevirtual

Implements grmhd::AnalyticData::InitialMagneticFields::InitialMagneticField.

variables() [1/2]

void grmhd::AnalyticData::InitialMagneticFields::Poloidal::variables ( gsl::not_null< tnsr::I< DataVector, 3 > * >  result, const tnsr::I< DataVector, 3 > &  coords, const Scalar< DataVector > &  pressure, const Scalar< DataVector > &  sqrt_det_spatial_metric, const tnsr::i< DataVector, 3 > &  deriv_pressure  ) const

overridevirtual

Retrieve magnetic fields at (x)

Implements grmhd::AnalyticData::InitialMagneticFields::InitialMagneticField.

variables() [2/2]

void grmhd::AnalyticData::InitialMagneticFields::Poloidal::variables ( gsl::not_null< tnsr::I< double, 3 > * >  result, const tnsr::I< double, 3 > &  coords, const Scalar< double > &  pressure, const Scalar< double > &  sqrt_det_spatial_metric, const tnsr::i< double, 3 > &  deriv_pressure  ) const

overridevirtual

Retrieve magnetic fields at (x)

Implements grmhd::AnalyticData::InitialMagneticFields::InitialMagneticField.

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

• src/PointwiseFunctions/AnalyticData/GrMhd/InitialMagneticFields/Poloidal.hpp