WallGo.grid3Scales.Grid3Scales

class Grid3Scales(M, N, tailLengthInside, tailLengthOutside, wallThickness, momentumFalloffT, ratioPointsWall=0.5, smoothing=0.1, wallCenter=0, spacing='Spectral')

Bases: Grid

Redefinition of the Grid class to take into account the different scales present in the z direction. More specifically, the \(z\) mapping function should scale as

\[\begin{split}z(\chi \to -1) &= \lambda_- \log(1+\chi),\\ z(\chi \to 1) &= -\lambda_+ \log(1-\chi),\end{split}\]

where \(\lambda_-\) and \(\lambda_+\) are the lengths of the solution’s tails inside and outside the bubble, respectively. Furthermore, the mapping should be approximately linear in the region \(-r<\chi<r\), where \(r\) is roughly the ratio of points that are used to resolve the wall’s interior. The slope in that region should be \(L/r\), where \(L\) is the wall thickness, so that

\[z'(\chi) \approx \frac{L}{r}, \quad \chi \in [-r, r].\]

It is easier to find the derivative of a function that has these properties, and then integrate it. We choose here

\[z'(\chi)=\frac{f(\chi)}{1-\chi^2},\]

where \(f(\chi)\) is a smoothed step function equal to

\[\begin{split}f(\chi) \approx \begin{cases} 2\lambda_-,& \chi<-r,\\ L/r,& -r<\chi<r,\\ 2\lambda_+,& \chi>r. \end{cases}\end{split}\]

We choose \(f(\chi)\) to be a sum of functions like \(\frac{\chi-\chi_0}{\sqrt{a^2+(\chi-\chi_0)^2}}\), which allows us to find analytically the mapping function with

\[z(\chi) - z(\chi_0) =\int^\chi_{\chi_0} \frac{f(\chi')}{1-\chi'^2} d\chi'.\]

The parameter \(a\) can be adjusted to control the smoothness of the mapping function.

Parameters:

• M (int) – Number of basis functions in the \(\xi\) (and \(\chi\)) direction.

• N (int) – Number of basis functions in the \(p_z\) and \(p_\Vert\) (and \(\rho_z\) and \(\rho_\Vert\)) directions.

• tailLengthInside (float) – Decay length of the solution’s tail inside the wall. Should be larger than wallThickness*(1/2+smoothing)/ratioPointsWall. Should be expressed in physical units (the units used in EffectivePotential).

• tailLengthOutside (float) – Decay length of the solution’s tail outside the wall. Should be larger than wallThickness*(1/2+smoothing)/ratioPointsWall. Should be expressed in physical units (the units used in EffectivePotential).

• wallThickness (float) – Thickness of the wall. Should be expressed in physical units (the units used in EffectivePotential).

• momentumFalloffT (float) – Temperature scale determining transform in momentum directions. Should be close to the plasma temperature.

• ratioPointsWall (float, optional) – Ratio of grid points inside the wall. The remaining points are distributed equally between the 2 tails. The default is 0.5.

• smoothing (float, optional) – Controls the smoothness of the mapping function. Its first derivative becomes discontinuous at \(\chi=\pm r\) when smoothness is 0. Should be smaller than 1, otherwise the function would not be linear at \(\chi=0\) anymore. As explained above, the decay length is controlled by adding 2 smoothed step functions. ‘smoothing’ is the value of these functions at the origin, in units of \(L/r\). The default is 0.1.

• wallCenter (float, optional) – Position of the wall’s center (in the z coordinates). Default is 0.

• spacing ({'Spectral', 'Uniform'}) – Choose ‘Spectral’ for the Gauss-Lobatto collocation points, as required for WallGo’s spectral representation, or ‘Uniform’ for a uniform grid. Default is ‘Spectral’.

Return type:

None.

__init__(M, N, tailLengthInside, tailLengthOutside, wallThickness, momentumFalloffT, ratioPointsWall=0.5, smoothing=0.1, wallCenter=0, spacing='Spectral')

Parameters:

• M (int) – Number of basis functions in the \(\xi\) (and \(\chi\)) direction.

• N (int) – Number of basis functions in the \(p_z\) and \(p_\Vert\) (and \(\rho_z\) and \(\rho_\Vert\)) directions.

• tailLengthInside (float) – Decay length of the solution’s tail inside the wall. Should be larger than wallThickness*(1/2+smoothing)/ratioPointsWall. Should be expressed in physical units (the units used in EffectivePotential).

• tailLengthOutside (float) – Decay length of the solution’s tail outside the wall. Should be larger than wallThickness*(1/2+smoothing)/ratioPointsWall. Should be expressed in physical units (the units used in EffectivePotential).

• wallThickness (float) – Thickness of the wall. Should be expressed in physical units (the units used in EffectivePotential).

• momentumFalloffT (float) – Temperature scale determining transform in momentum directions. Should be close to the plasma temperature.

• ratioPointsWall (float, optional) – Ratio of grid points inside the wall. The remaining points are distributed equally between the 2 tails. The default is 0.5.

• smoothing (float, optional) – Controls the smoothness of the mapping function. Its first derivative becomes discontinuous at \(\chi=\pm r\) when smoothness is 0. Should be smaller than 1, otherwise the function would not be linear at \(\chi=0\) anymore. As explained above, the decay length is controlled by adding 2 smoothed step functions. ‘smoothing’ is the value of these functions at the origin, in units of \(L/r\). The default is 0.1.

• wallCenter (float, optional) – Position of the wall’s center (in the z coordinates). Default is 0.

• spacing ({'Spectral', 'Uniform'}) – Choose ‘Spectral’ for the Gauss-Lobatto collocation points, as required for WallGo’s spectral representation, or ‘Uniform’ for a uniform grid. Default is ‘Spectral’.

Return type:

None.

Methods

__init__(M, N, tailLengthInside, ...[, ...])
changeMomentumFalloffScale(newScale)	Change the momentum falloff scale.
changePositionFalloffScale(tailLengthInside, ...)	Change the position falloff scale.
compactificationDerivatives(zCompact, ...)	Derivative of transforms coordinates to [-1, 1] interval
compactify(z, pz, pp)	Transforms coordinates to [-1, 1] interval
decompactify(zCompact, pzCompact, ppCompact)	Transforms coordinates from [-1, 1] interval (inverse of compactify).
getCompactCoordinates([endpoints, direction])	Return compact coordinates of grid.
getCompactificationDerivatives([endpoints])	Return derivatives of compactified coordinates of grid, with respect to uncompactified derivatives.
getCoordinates([endpoints])	Return coordinates of grid, not compactified.

changeMomentumFalloffScale(newScale)

Change the momentum falloff scale.

Parameters:

newScale (float) – New momentum falloff scale.

Return type:

None

changePositionFalloffScale(tailLengthInside, tailLengthOutside, wallThickness, wallCenter)

Change the position falloff scale.

Parameters:

• newScale (float) – New position falloff scale.

• tailLengthInside (float)

• tailLengthOutside (float)

• wallThickness (float)

• wallCenter (float)

Return type:

None

compactificationDerivatives(zCompact, pzCompact, ppCompact)

Derivative of transforms coordinates to [-1, 1] interval

Parameters:

• zCompact (ndarray)

• pzCompact (ndarray)

• ppCompact (ndarray)

Return type:

tuple[ndarray, …]

compactify(z, pz, pp)

Transforms coordinates to [-1, 1] interval

Parameters:

• z (array-like) – Physical z (or xi) coordinate.

• pz (array-like) – Physical pz coordinate.

• pp (array-like) – Physical p_par coordinate.

Returns:

• z_compact (array-like) – Compact z coordinate (chi).

• pz_compact (array-like) – Compact pz coordinate (rho_z).

• pp_compact (array-like) – Compact p_par coordinate (rho_par).

Return type:

tuple[ndarray, …]

decompactify(zCompact, pzCompact, ppCompact)

Transforms coordinates from [-1, 1] interval (inverse of compactify).

Parameters:

• zCompact (ndarray)

• pzCompact (ndarray)

• ppCompact (ndarray)

Return type:

tuple[ndarray, …]

getCompactCoordinates(endpoints=False, direction=None)

Return compact coordinates of grid.

Parameters:

• endpoints (Bool, optional) – If True, include endpoints of grid. Default is False.

• direction (string or None, optional) – Specifies which coordinates to return. Can either be ‘z’, ‘pz’, ‘pp’ or None. If None, returns a tuple containing the 3 directions. Default is None.

Returns:

• chiValues (array_like) – Grid of the \(\chi\) direction.

• rzValues (array_like) – Grid of the \(\rho_z\) direction.

• rpValues (array_like) – Grid of the \(\rho_\Vert\) direction.

Return type:

tuple[ndarray, …] | ndarray

getCompactificationDerivatives(endpoints=False)

Return derivatives of compactified coordinates of grid, with respect to uncompactified derivatives.

Parameters:

endpoints (Bool, optional) – If True, include endpoints of grid. Default is False.

Returns:

• dchiValues (array_like) – Grid of the \(\partial_\xi\chi\) direction.

• drzValues (array_like) – Grid of the \(\partial_{p_z}\rho_z\) direction.

• drpValues (array_like) – Grid of the \(\partial_{p_\Vert}\rho_\Vert\) direction.

Return type:

tuple[ndarray, …]

getCoordinates(endpoints=False)

Return coordinates of grid, not compactified.

Parameters:

endpoints (Bool, optional) – If True, include endpoints of grid. Default is False.

Returns:

• xiValues (array_like) – Grid of the \(\xi\) direction.

• pzValues (array_like) – Grid of the \(p_z\) direction.

• ppValues (array_like) – Grid of the \(p_\Vert\) direction.

Return type:

tuple[ndarray, …]