Implements the source terms in the Hall terms that are added to Ideal MHD to produce the Hall MHD equations arising from a cylindrical geometry. More...

#include <cyl_source.h>

Inheritance diagram for wxm::apps::hallmhd::CylSource:

Detailed Description

Implements the source terms in the Hall terms that are added to Ideal MHD to produce the Hall MHD equations arising from a cylindrical geometry.

See also: Latex Documentation: MHD Cylindrical Source Terms File; Warpy constructor: warpy.apps.mhd.hallmhd.hallmhd.cyl_source

For Hall MHD, the Ohm's Law terms are:

\begin{align*} \boldsymbol{E}_\mathrm{Hall} =& \frac{1}{n_{\mathrm{e}}}\left[ \boldsymbol{j}\times\boldsymbol{B} - \left(\frac{\delta_{p}}{L}\right)\nabla p_{\mathrm{e}} \right] \mathrm{,} \end{align*}

or, in component form:

\begin{align*} \boldsymbol{E}_{\mathrm{Hall}} =& \frac{1}{n_{\mathrm{e}}} \left[\left(j_{r}B_{\theta} - j_{\theta}B_{r}\right) - \left(\frac{\delta_{p}}{L}\right)\left(\nabla p_{e}\right)_{z}\right] *\boldsymbol{\hat{z}} \nonumber \\ +& \frac{1}{n_{\mathrm{e}}} \left[\left(j_{\theta}B_{z} - j_{z}B_{\theta}\right) - \left(\frac{\delta_{p}}{L}\right)\left(\nabla p_{e}\right)_{r}\right] *\boldsymbol{\hat{r}} \nonumber \\ +& \frac{1}{n_{\mathrm{e}}} \left[\left(j_{z}B_{r} - j_{r}B_{z}\right) - \left(\frac{\delta_{p}}{L}\right)\left(\nabla p_{e}\right)_{\theta}\right] \boldsymbol{\hat{\theta}} \mathrm{.} \end{align*}

The components of \(\boldsymbol{j}\) are given by

\begin{align*} \boldsymbol{j} =& \begin{pmatrix} j_{r}\\ j_{\theta}\\ j_{z} \end{pmatrix} = \left(\frac{\delta_{p}}{L}\right) \begin{pmatrix} \left(\nabla\times\boldsymbol{B}\right)_{r}\\ \left(\nabla\times\boldsymbol{B}\right)_{\theta}\\ \left(\nabla\times\boldsymbol{B}\right)_{z} \end{pmatrix} = \left(\frac{\delta_{p}}{L}\right) \begin{pmatrix} -\frac{\partial B_{\theta}}{\partial z}\\ \frac{\partial B_{r}}{\partial z} - \frac{\partial B_{z}}{\partial r}\\ \frac{\partial B_{\theta}}{\partial r} + \frac{B_{\theta}}{r} \end{pmatrix} \mathrm{.} \end{align*}

This leads to the Hall Terms in the 2D Axisymmetric MHD equations resulting in

\begin{align*} \frac{\partial}{\partial t} \begin{pmatrix} \rho \\ \rho v_{z} \\ \rho v_{r} \\ \rho v_{\theta} \\ e_{t} \\ B_{z} \\ B_{r} \\ B_{\theta} \end{pmatrix} +& \frac{\partial}{\partial z} \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ E_{\mathrm{Hall,}r}B_{\theta} - E_{\mathrm{Hall,}\theta}B_{r} \\ 0 \\ -E_{\mathrm{Hall,}\theta}\\ +E_{\mathrm{Hall,}r} \end{pmatrix} + \frac{\partial}{\partial r} \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ E_{\mathrm{Hall,}\theta}B_{z} - E_{\mathrm{Hall,}z}B_{\theta} \\ +E_{\mathrm{Hall,}\theta} \\ 0 \\ -E_{\mathrm{Hall,}z} \\ \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ -\frac{E_{\mathrm{Hall,}\theta}B_{z} - E_{\mathrm{Hall,}z}B_{\theta}}{r} \\ -\frac{E_{\mathrm{Hall,}\theta}}{r} \\ 0 \\ 0 \end{pmatrix} \mathrm{,} \end{align*}

where the source terms on the RHS are what are implemented here.

To avoid \(\frac{1}{r}\) issues, this should be used with Gaussian Quadrature nodes.

Public Member Functions
	CylSource ()

	~CylSource ()

void	setup (const WxCryptSet &wxc) override

const std::vector< int > &	getInputVariableIndexes (int flag) const override

const std::vector< int > &	getAuxiliaryVariableIndexes (int flag) const override

const std::vector< int > &	getOutputVariableIndexes (int flag) const override

real	source (const real q, const real aux, const elementGeometry_t pEG, real source) const override

Public Member Functions inherited from WmApplication
	WmApplication ()

virtual	~WmApplication ()=default

virtual void	setup (const WxCryptSet &wxc)

virtual const std::vector< int > &	getAuxiliaryVariableIndexes (int flag=WMAPPLICATIONFLAG_NONE) const

virtual const std::vector< int > &	getCrossVariableIndexes (int flag=WMAPPLICATIONFLAG_NONE) const

const std::vector< std::string > &	getBoundaryNames () const

bool	isOnBoundary (const std::string &boundaryName) const

bool	has (int flag) const

virtual real	numerical_flux (const real q_l, const real q_r, const real aux_l, const real aux_r, const solverVariables_t pFV, real numericalFlux) const

virtual real	internal_flux (const real q, const real aux, const solverVariables_t *pSV, std::vector< std::vector< real > > &internalFlux) const

virtual real	source (const real q, const real aux, const elementGeometry_t pEG, real source) const

virtual void	bc_q (const real q_in, const real aux_in, const real aux_out, const solverVariables_t pFV, real *q_out) const
	Boundary Condition Application which sets the boundary condition on ghost nodes.

virtual real	bcNumericalFlux (const real q_l, const real q_r, const real aux_l, const real aux_r, const solverVariables_t pFV, real numericalFlux) const

virtual void	conserved_to_primitive (const real q, const real aux, real *w) const

virtual void	primitive_to_conserved (const real w, const real aux, real *q) const

virtual void	evaluate_function (const real q, const real aux, const solverVariables_t pSV, real result) const

virtual void	bc_q_kinetic (const real q_in, const real aux_in, const solverVariables_t pFV, real q_out) const

std::shared_ptr< std::string >	app_name ()

virtual const std::vector< int > &	getInputVariableIndexes (int flag=0) const

virtual const std::vector< int > &	getOutputVariableIndexes (int flag=0) const

Protected Attributes
real	_gas_gamma

real	_skin_depth_norm

real	_Ai

real	_Zi

real	_Ze

real	_Ae

real	_theta

int	_radial_index

int	_ir
	r index

int	_ith
	theta index

int	_iz
	z index

real	_min_density_floor

std::vector< int >	_input_variable_indices

std::vector< int >	_aux_variable_indices

Protected Attributes inherited from WmApplication
std::string	_app_name

WxLogStream	_debStrm

std::vector< int >	_allowedFlags

std::vector< std::string >	_onBoundaries

Additional Inherited Members
Public Types inherited from WmApplication
typedef Application_Variable	WmApplication_Variable

typedef std::map< std::string, std::string >	replaceMap_t

Static Public Attributes inherited from wxm::app_base
static std::vector< int >	empty

Protected Member Functions inherited from wxm::app_base
virtual	~app_base ()=default

Constructor & Destructor Documentation

◆ CylSource()

wxm::apps::hallmhd::CylSource::CylSource ( )

◆ ~CylSource()

wxm::apps::hallmhd::CylSource::~CylSource ( )

Member Function Documentation

◆ getAuxiliaryVariableIndexes()

const std::vector< int > & wxm::apps::hallmhd::CylSource::getAuxiliaryVariableIndexes ( int flag ) const

inlineoverridevirtual

Reimplemented from WmApplication.

◆ getInputVariableIndexes()

const std::vector< int > & wxm::apps::hallmhd::CylSource::getInputVariableIndexes ( int flag ) const

inlineoverridevirtual

Reimplemented from wxm::app_base.

◆ getOutputVariableIndexes()

const std::vector< int > & wxm::apps::hallmhd::CylSource::getOutputVariableIndexes ( int flag ) const

inlineoverridevirtual

Reimplemented from wxm::app_base.

◆ setup()

void wxm::apps::hallmhd::CylSource::setup ( const WxCryptSet & wxc )

overridevirtual

Reimplemented from WmApplication.

◆ source()

real wxm::apps::hallmhd::CylSource::source	(	const real *	q,
		const real *	aux,
		const elementGeometry_t *	pEG,
		real *	source
	)		const

overridevirtual

Reimplemented from WmApplication.

Member Data Documentation

◆ _Ae

real wxm::apps::hallmhd::CylSource::_Ae

protected

◆ _Ai

real wxm::apps::hallmhd::CylSource::_Ai

protected

◆ _aux_variable_indices

std::vector<int> wxm::apps::hallmhd::CylSource::_aux_variable_indices

protected

◆ _gas_gamma

real wxm::apps::hallmhd::CylSource::_gas_gamma

protected

◆ _input_variable_indices

std::vector<int> wxm::apps::hallmhd::CylSource::_input_variable_indices

protected

◆ _ir

int wxm::apps::hallmhd::CylSource::_ir

protected

r index

◆ _ith

int wxm::apps::hallmhd::CylSource::_ith

protected

theta index

◆ _iz

int wxm::apps::hallmhd::CylSource::_iz

protected

z index

◆ _min_density_floor

real wxm::apps::hallmhd::CylSource::_min_density_floor

protected

◆ _radial_index

int wxm::apps::hallmhd::CylSource::_radial_index

protected

◆ _skin_depth_norm

real wxm::apps::hallmhd::CylSource::_skin_depth_norm

protected

◆ _theta

real wxm::apps::hallmhd::CylSource::_theta

protected

◆ _Ze

real wxm::apps::hallmhd::CylSource::_Ze

protected

◆ _Zi

real wxm::apps::hallmhd::CylSource::_Zi

protected

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

src/apps/mhd/hallmhd/cyl_source.h

Detailed Description

Public Member Functions

Protected Attributes

Additional Inherited Members

Constructor & Destructor Documentation

◆ CylSource()

◆ ~CylSource()

Member Function Documentation

◆ getAuxiliaryVariableIndexes()

◆ getInputVariableIndexes()

◆ getOutputVariableIndexes()

◆ setup()

◆ source()

Member Data Documentation

◆ _Ae

◆ _Ai

◆ _aux_variable_indices

◆ _gas_gamma

◆ _input_variable_indices

◆ _ir

◆ _ith

◆ _iz

◆ _min_density_floor

◆ _radial_index

◆ _skin_depth_norm

◆ _theta

◆ _Ze

◆ _Zi