korc_spatial_distribution Module

@param f Value of function. @param r Guess value of radial position of the particles. @param a Minor radius of the toroidal distribution $r_0$ @param ko Decay rate of radial distribution, see $f(r)$ of \ref korc_spatial_distribution.exponential_torus. @param P Deviate of a random uniform distribution in the interval $[0,1]$. @brief Subroutine that generates a exponentially decaying radial distribution of particles in a circular cross-section torus of major and minor radi $R_0$ and $r_0$, respectively. @details We generate this exponentially decaying radial distribution $f(r)$ following the same approach as in \ref korc_spatial_distribution.disk, but this time, the radial distribution is given by:

$$f(r) = \left\{ \begin{array}{ll} \frac{k_0^2}{4\pi^2 R_0}\frac{\exp{\left( -k_0 r \right)}}{1 - \exp{\left( -k_0r_0\right)}\left[ 1 + k_0 r_0 \right]} &\ r<r_0 \\ 0 & r>r_0 \end{array} \right.$$

The radial position of the particles $r$ is obtained using the Inverse Trasnform Sampling method, finding $r$ numerically through the Newton-Raphson method. First, we calculate the particles' radial distribution in a disk centered at $(R,Z) = (0,0)$. Then, we transfor to a new set of coordinates where the disk is centered at $(R,Z) = (R_0,Z_0)$. Finally, we generate the toroidal distribution by givin each particle a toroidal angle $\zeta$ which follows a uniform distribution in the interval $[0,2\pi]$.

@param[in] params Core KORC simulation parameters. @param[in,out] spp An instance of the derived type SPECIES containing all the parameters and simulation variables of the different species in the simulation. @param fl Variable used in the Newton-Raphson method for finding the radial position of each particle. @param fr Variable used in the Newton-Raphson method for finding the radial position of each particle. @param fm Variable used in the Newton-Raphson method for finding the radial position of each particle. @param rl Variable used in the Newton-Raphson method for finding the radial position of each particle. @param rr Variable used in the Newton-Raphson method for finding the radial position of each particle. @param rm Variable used in the Newton-Raphson method for finding the radial position of each particle. @param relerr Tolerance used to determine when to stop iterating the Newton-Raphson method for finding $r$. @param r Radial position of the particles $r$. @param theta Uniform deviates in the range $[0,2\pi]$ representing the uniform poloidal angle $\theta$ distribution of the particles. @param zeta Uniform deviates in the range $[0,2\pi]$ representing the uniform toroidal angle $\zeta$ distribution of the particles. @param pp Particle iterator. @brief Subroutine that generates an exponentially decaying radial distribution in an elliptic torus as the initial spatial condition of a given particle species in the simulation. @details As a first step, we generate an exponentially decaying radial distribution in a circular cross-section torus as in \ref korc_spatial_distribution.exponential_torus. Then we transform this spatial distribution to a one in an torus with an elliptic cross section, this following the same approach as in \ref korc_spatial_distribution.elliptic_torus.

@param[in] params Core KORC simulation parameters. @param[in,out] spp An instance of the derived type SPECIES containing all the parameters and simulation variables of the different species in the simulation. @param fl Variable used in the Newton-Raphson method for finding the radial position of each particle. @param fr Variable used in the Newton-Raphson method for finding the radial position of each particle. @param fm Variable used in the Newton-Raphson method for finding the radial position of each particle. @param rl Variable used in the Newton-Raphson method for finding the radial position of each particle. @param rr Variable used in the Newton-Raphson method for finding the radial position of each particle. @param rm Variable used in the Newton-Raphson method for finding the radial position of each particle. @param relerr Tolerance used to determine when to stop iterating the Newton-Raphson method for finding $r$. @param rotation_angle This is the angle $\Theta$ in \ref korc_spatial_distribution.elliptic_torus. @param r Radial position of the particles $r$. @param theta Uniform deviates in the range $[0,2\pi]$ representing the uniform poloidal angle $\theta$ distribution of the particles. @param zeta Uniform deviates in the range $[0,2\pi]$ representing the uniform toroidal angle $\zeta$ distribution of the particles. @param X Auxiliary vector used in the coordinate transformations. @param Y Auxiliary vector used in the coordinate transformations. @param X1 Auxiliary vector used in the coordinate transformations. @param Y1 Auxiliary vector used in the coordinate transformations. @param pp Particle iterator. @brief Subroutine that generates a Gaussian radial distribution in an elliptic torus as the initial spatial condition of a given particle species in the simulation. @details As a first step, we generate an Gaussian radial distribution in a circular cross-section torus as in \ref korc_spatial_distribution.gaussian_torus. Then we transform this spatial distribution to a one in an torus with an elliptic cross section, this following the same approach as in \ref korc_spatial_distribution.elliptic_torus.

@param[in] params Core KORC simulation parameters. @param[in,out] spp An instance of the derived type SPECIES containing all the parameters and simulation variables of the different species in the simulation. @param rotation_angle This is the angle $\Theta$ in \ref korc_spatial_distribution.elliptic_torus. @param r Radial position of the particles $r$. @param theta Uniform deviates in the range $[0,2\pi]$ representing the uniform poloidal angle $\theta$ distribution of the particles. @param zeta Uniform deviates in the range $[0,2\pi]$ representing the uniform toroidal angle $\zeta$ distribution of the particles. @param X Auxiliary vector used in the coordinate transformations. @param Y Auxiliary vector used in the coordinate transformations. @param X1 Auxiliary vector used in the coordinate transformations. @param Y1 Auxiliary vector used in the coordinate transformations. @param sigma Standard deviation $\sigma$ of the radial distribution function. @param pp Particle iterator. @brief Subroutine that generates a Gaussian radial distribution of particles in a circular cross-section torus of major and minor radi $R_0$ and $r_0$, respectively. @details We generate this exponentially decaying radial distribution $f(r)$ following the same approach as in \ref korc_spatial_distribution.disk, but this time, the radial distribution is given by:

$$f(r) = \left\{ \begin{array}{ll} \frac{1}{4\pi^2 \sigma^2 R_0}\frac{\exp{\left( -\frac{r^2}{2\sigma^2} \right)}}{1 - \exp{\left( -\frac{r_0^2}{2\sigma^2} \right)}} &\ r<r_0 \\ 0 & r>r_0 \end{array} \right.$$

The radial position of the particles $r$ is obtained using the Inverse Trasnform Sampling method, finding $r$ numerically through the Newton-Raphson method. First, we calculate the particles' radial distribution in a disk centered at $(R,Z) = (0,0)$. Then, we transfor to a new set of coordinates where the disk is centered at $(R,Z) = (R_0,Z_0)$. Finally, we generate the toroidal distribution by givin each particle a toroidal angle $\zeta$ which follows a uniform distribution in the interval $[0,2\pi]$.

@param[in] params Core KORC simulation parameters. @param[in,out] spp An instance of the derived type SPECIES containing all the parameters and simulation variables of the different species in the simulation. @param fl Variable used in the Newton-Raphson method for finding the radial position of each particle. @param fr Variable used in the Newton-Raphson method for finding the radial position of each particle. @param fm Variable used in the Newton-Raphson method for finding the radial position of each particle. @param rl Variable used in the Newton-Raphson method for finding the radial position of each particle. @param rr Variable used in the Newton-Raphson method for finding the radial position of each particle. @param rm Variable used in the Newton-Raphson method for finding the radial position of each particle. @param relerr Tolerance used to determine when to stop iterating the Newton-Raphson method for finding $r$. @param r Radial position of the particles $r$. @param theta Uniform deviates in the range $[0,2\pi]$ representing the uniform poloidal angle $\theta$ distribution of the particles. @param zeta Uniform deviates in the range $[0,2\pi]$ representing the uniform toroidal angle $\zeta$ distribution of the particles. @param pp Particle iterator.

Contents

Variables

minmax_buffer_size

Functions

fzero PSI_ROT indicator random_norm Spong_2D

Subroutines

uniform disk torus elliptic_torus exponential_torus exponential_elliptic_torus gaussian_elliptic_torus MH_gaussian_elliptic_torus gaussian_torus Spong_3D MH_psi intitial_spatial_distribution

Variables

Type	Visibility	Attributes		Name		Initial
real(kind=rp),	private,	parameter	::	minmax_buffer_size	=	10.0_rp

Functions

private function fzero(r, a, ko, P) result(f)

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=rp),	intent(in)			::	r
real(kind=rp),	intent(in)			::	a
real(kind=rp),	intent(in)			::	ko
real(kind=rp),	intent(in)			::	P

Return Value real(kind=rp)

private function PSI_ROT(R, R0, sigR, Z, Z0, sigZ, theta)

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=rp),	intent(in)			::	R	R-coordinate of MH sampled location
real(kind=rp),	intent(in)			::	R0	R-coordinate of center of 2D Gaussian
real(kind=rp),	intent(in)			::	sigR	Variance of first dimension of 2D Gaussian
real(kind=rp),	intent(in)			::	Z	Z-coordinate of MH sampled location
real(kind=rp),	intent(in)			::	Z0	Z-coordinate of center of 2D Gaussian
real(kind=rp),	intent(in)			::	sigZ	Variance of second dimension of 2D Gaussian
real(kind=rp),	intent(in)			::	theta	Angle of counter-clockwise rotation (in radians), of 2D Gaussian distribution relative to R,Z

Return Value real(kind=rp)

Argument of exponential comprising 2D Gaussian distribution

private function indicator(psi, psi_max)

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=rp),	intent(in)			::	psi
real(kind=rp),	intent(in)			::	psi_max

Return Value real(kind=rp)

public function random_norm(mean, sigma)

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=rp),	intent(in)			::	mean
real(kind=rp),	intent(in)			::	sigma

Return Value real(kind=rp)

private function Spong_2D(R0, b, w, dlam, R, Z, T)

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=rp),	intent(in)			::	R0
real(kind=rp),	intent(in)			::	b
real(kind=rp),	intent(in)			::	w
real(kind=rp),	intent(in)			::	dlam
real(kind=rp),	intent(in)			::	R
real(kind=rp),	intent(in)			::	Z
real(kind=rp),	intent(in)			::	T

Return Value real(kind=rp)

Subroutines

private subroutine uniform(spp)

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Arguments

Type	Intent	Optional	Attributes		Name
type(SPECIES),	intent(inout)			::	spp	An instance of the derived type SPECIES containing all the parameters and simulation variables of the different species in the simulation.

private subroutine disk(params, spp)

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Arguments

Type	Intent	Optional	Attributes		Name
type(KORC_PARAMS),	intent(in)			::	params	Core KORC simulation parameters.
type(SPECIES),	intent(inout)			::	spp	An instance of the derived type SPECIES containing all the parameters and simulation variables of the different species in the simulation.

private subroutine torus(params, spp)

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Arguments

Type	Intent	Optional	Attributes		Name
type(KORC_PARAMS),	intent(in)			::	params	Core KORC simulation parameters.
type(SPECIES),	intent(inout)			::	spp	An instance of the derived type SPECIES containing all the parameters and simulation variables of the different species in the simulation.

private subroutine elliptic_torus(params, spp)

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Arguments

Type	Intent	Optional	Attributes		Name
type(KORC_PARAMS),	intent(in)			::	params	Core KORC simulation parameters.
type(SPECIES),	intent(inout)			::	spp	An instance of the derived type SPECIES containing all the parameters and simulation variables of the different species in the simulation.

private subroutine exponential_torus(params, spp)

Arguments

Type	Intent	Optional	Attributes		Name
type(KORC_PARAMS),	intent(in)			::	params
type(SPECIES),	intent(inout)			::	spp

private subroutine exponential_elliptic_torus(params, spp)

Arguments

Type	Intent	Optional	Attributes		Name
type(KORC_PARAMS),	intent(in)			::	params
type(SPECIES),	intent(inout)			::	spp

private subroutine gaussian_elliptic_torus(params, spp)

Arguments

Type	Intent	Optional	Attributes		Name
type(KORC_PARAMS),	intent(in)			::	params
type(SPECIES),	intent(inout)			::	spp

private subroutine MH_gaussian_elliptic_torus(params, spp)

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Arguments

Type	Intent	Optional	Attributes		Name
type(KORC_PARAMS),	intent(in)			::	params	Core KORC simulation parameters.
type(SPECIES),	intent(inout)			::	spp	An instance of the derived type SPECIES containing all the parameters and simulation variables of the different species in the simulation.

private subroutine gaussian_torus(params, spp)

Arguments

Type	Intent	Optional	Attributes		Name
type(KORC_PARAMS),	intent(in)			::	params
type(SPECIES),	intent(inout)			::	spp

private subroutine Spong_3D(params, spp)

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Arguments

Type	Intent	Optional	Attributes		Name
type(KORC_PARAMS),	intent(in)			::	params	Core KORC simulation parameters.
type(SPECIES),	intent(inout)			::	spp	An instance of the derived type SPECIES containing all the parameters and simulation variables of the different species in the simulation. Read more…

private subroutine MH_psi(params, spp, F)

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Arguments

Type	Intent	Optional	Attributes		Name
type(KORC_PARAMS),	intent(inout)			::	params	Core KORC simulation parameters.
type(SPECIES),	intent(inout)			::	spp	An instance of the derived type SPECIES containing all the parameters and simulation variables of the different species in the simulation.
type(FIELDS),	intent(in)			::	F	An instance of the KORC derived type FIELDS.

public subroutine intitial_spatial_distribution(params, spp, P, F)

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Arguments

Type	Intent	Optional	Attributes		Name
type(KORC_PARAMS),	intent(inout)			::	params	Core KORC simulation parameters.
type(SPECIES),	intent(inout),		DIMENSION(:), ALLOCATABLE	::	spp	An instance of the derived type SPECIES containing all the parameters and simulation variables of the different species in the simulation.
type(PROFILES),	intent(in)			::	P	An instance of the KORC derived type PROFILES.
type(FIELDS),	intent(in)			::	F	An instance of the KORC derived type FIELDS.