dg::BathRZ Struct Reference

\(f(R,Z) = A B \sum_\vec{k} \sqrt{E_k} \alpha_k \cos{\left(k \kappa_k + \theta_k \right)} \) More...

Public Member Functions
	BathRZ (unsigned N_kR, unsigned N_kZ, double R_min, double Z_min, double gamma, double L_E, double amp)
	Functor returning a random field in the RZ-plane or in the first RZ-plane.
double	operator() (double R, double Z) const
	Return the value of the Bath.
double	operator() (double R, double Z, double) const
	Return the value of the Bath.

Detailed Description

\(f(R,Z) = A B \sum_\vec{k} \sqrt{E_k} \alpha_k \cos{\left(k \kappa_k + \theta_k \right)} \)

A random bath in the R-Z plane with

\[ B := \sqrt{\frac{2}{N_{k_R} N_{k_Z}}} \\ k:=\sqrt{k_R^2 + k_Z^2} \\ k_R:=2 \pi \left( i -N_{k_R}/2\right)/N_{k_R} \\ k_Z:=2 \pi \left( j -N_{k_Z}/2\right)/N_{k_Z} \\ k_0:=2 \pi L_E / N_k\\ N_k := \sqrt{N_{k_R}^2 + N_{k_Z}^2} \\ E_k:=\left(4 k k_0/(k+k_0)^2\right)^{\gamma} \\ \alpha_k := \sqrt{\mathcal{N}_1^2 + \mathcal{N}_2^2} \\ \theta_k := \arctan{\left(\mathcal{N}_2/\mathcal{N}_1\right)} \\ \kappa_k(R,Z) := (R-R_{min}) \mathcal{U}_1 + (Z-Z_{min}) \mathcal{U}_2 \\ \]

where \(\mathcal{N}_{1,2}\) are random normal distributed real numbers with a mean of \(\mu = 0\) and a standard deviation of \(\sigma=1 \), \(\mathcal{U}_{1,2}\) are random uniformly distributed real numbers \(\in \left[0, 2 \pi \right) \) and \( A \) is the amplitude.

Constructor & Destructor Documentation

BathRZ()

dg::BathRZ::BathRZ ( unsigned N_kR, unsigned N_kZ, double R_min, double Z_min, double gamma, double L_E, double amp )

inline

Functor returning a random field in the RZ-plane or in the first RZ-plane.

Parameters

N_kR	Number of Fourier modes in R direction
N_kZ	Number of Fourier modes in Z direction
R_min	Minimal R (in units of rho_s)
Z_min	Minimal Z (in units of rho_s)
gamma	exponent in the energy function \(E_k\) (typically around 30)
L_E	is the typical eddysize (typically around 5)
amp	Amplitude

Member Function Documentation

operator()() [1/2]

double dg::BathRZ::operator() ( double R, double Z ) const

inline

Return the value of the Bath.

\[f(R,Z) = A B \sum_\vec{k} \sqrt{E_k} \alpha_k \cos{\left(k \kappa_k + \theta_k \right)} \]

with

\[ \mathcal{N} := \sqrt{\frac{2}{N_{k_R} N_{k_Z}}} \\ k:=\sqrt{k_R^2 + k_Z^2} \\ k_R:=2 \pi \left( i -N_{k_R}/2\right)/N_{k_R} \\ k_Z:=2 \pi \left( j -N_{k_Z}/2\right)/N_{k_Z} \\ k_0:=2 \pi L_E / N_k\\ N_k := \sqrt{N_{k_R}^2 + N_{k_Z}^2} \\ E_k:=\left(4 k k_0/(k+k_0)^2\right)^{\gamma} \\ \alpha_k := \sqrt{\mathcal{N}_1^2 + \mathcal{N}_2^2} \\ \theta_k := \arctan{\left(\mathcal{N}_2/\mathcal{N}_1\right)} \\ \kappa_k(R,Z) := (R-R_{min}) \mathcal{U}_1 + (Z-Z_{min}) \mathcal{U}_2 \\ \]

where \(\mathcal{N}_{1,2}\) are random normal distributed real numbers with a mean of \(\mu = 0\) and a standard deviation of \(\sigma=1 \), \(\mathcal{U}_{1,2}\) are random uniformly distributed real numbers \(\in \left[0, 2 \pi \right) \) and \( A \) is the amplitude

Parameters

R	R - coordinate
Z	Z - coordinate

Returns

the above function value

operator()() [2/2]

double dg::BathRZ::operator() ( double R, double Z, double  ) const

inline

Return the value of the Bath.

\[f(R,Z) = A B \sum_\vec{k} \sqrt{E_k} \alpha_k \cos{\left(k \kappa_k + \theta_k \right)} \]

with

\[ \mathcal{N} := \sqrt{\frac{2}{N_{k_R} N_{k_Z}}} \\ k:=\sqrt{k_R^2 + k_Z^2} \\ k_R:=2 \pi \left( i -N_{k_R}/2\right)/N_{k_R} \\ k_Z:=2 \pi \left( j -N_{k_Z}/2\right)/N_{k_Z} \\ k_0:=2 \pi L_E / N_k\\ N_k := \sqrt{N_{k_R}^2 + N_{k_Z}^2} \\ E_k:=\left(4 k k_0/(k+k_0)^2\right)^{\gamma} \\ \alpha_k := \sqrt{\mathcal{N}_1^2 + \mathcal{N}_2^2} \\ \theta_k := \arctan{\left(\mathcal{N}_2/\mathcal{N}_1\right)} \\ \kappa_k(R,Z) := (R-R_{min}) \mathcal{U}_1 + (Z-Z_{min}) \mathcal{U}_2 \\ \]

where \(\mathcal{N}_{1,2}\) are random normal distributed real numbers with a mean of \(\mu = 0\) and a standard deviation of \(\sigma=1 \), \(\mathcal{U}_{1,2}\) are random uniformly distributed real numbers \(\in \left[0, 2 \pi \right) \) and \( A \) is the amplitude

Parameters

R	R - coordinate
Z	Z - coordinate

Returns

the above function value

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

• functors.h