User Reference¶

This page documents the eight classes, and its members, corresponding to the eight analytical solutions provided in the access python package.

Assess is a python package that implements analytical solutions to the Stokes equations in cylindrical and spherical domains.

class assess.CylindricalStokesSolutionSmoothFreeSlip(n, k, Rp=2.22, Rm=1.22, nu=1.0, g=1.0)¶

Bases: assess.cylindrical.CylindricalStokesSolutionSmooth

Analytical Solution in cylindrical domain with smooth r^k forcing and free-slip boundary conditions

Parameters:	n – wave number k – polynomial order of forcing Rp – outer radius Rm – inner radius nu – viscosity g – forcing strength

delta_rho(r, phi)¶

Perturbation density \(\rho'\) in forcing term: \(g\rho'\hat r\)

Parameters:	r – radius phi – angle with x-axis

delta_rho_cartesian(X)¶

Perturbation density \(\rho'\) in forcing term: \(g\rho'\hat r\)

Parameters:	X – 2D Cartesian coordinate

dpsi_rdr(r)¶

Radial derivative of radial part of streamfunction

Parameters:	r – radius

dpsi_rdr2(r)¶

Second radial derivative of radial part of streamfunction

Parameters:	r – radius

p(r, phi)¶

Pressure solution

Parameters:	r – radius phi – angle with x-axis

pressure_cartesian(X)¶

Return pressure solution at Cartesian location.

Parameters:	X – 2D Cartesian location

psi_r(r)¶

Radial part of streamfunction

Parameters:	r – radius

radial_stress(r, phi)¶

Return radial component of stress.

Parameters:	r – radius phi – angle with x-axis.

radial_stress_cartesian(X)¶

Return radial component of stress at Cartesian location.

Parameters:	X – 2D Cartesian location

tau_rphi(r, phi)¶

Return shear stress \(\tau_{r\varphi}\).

Parameters:	r – radius phi – angle with x-axis.

tau_rr(r, phi)¶

Return radial component of deviatoric stress.

Parameters:	r – radius phi – angle with x-axis.

u_phi(r, phi)¶

Return tangential component of velocity.

Parameters:	r – radius phi – angle with x-axis.

u_r(r, phi)¶

Return radial component of velocity.

Parameters:	r – radius phi – angle with x-axis.

velocity_cartesian(X)¶

Return Cartesian velocity at Cartesian location.

Parameters:	X – 2D Cartesian location

class assess.CylindricalStokesSolutionSmoothZeroSlip(n, k, Rp=2.22, Rm=1.22, nu=1.0, g=1.0)¶

Bases: assess.cylindrical.CylindricalStokesSolutionSmooth

Analytical Solution in cylindrical domain with smooth r^k forcing and zero-slip boundary conditions

Parameters:	n – wave number k – polynomial order of forcing Rp – outer radius Rm – inner radius nu – viscosity g – forcing strength

delta_rho(r, phi)¶

Perturbation density \(\rho'\) in forcing term: \(g\rho'\hat r\)

Parameters:	r – radius phi – angle with x-axis

delta_rho_cartesian(X)¶

Perturbation density \(\rho'\) in forcing term: \(g\rho'\hat r\)

Parameters:	X – 2D Cartesian coordinate

dpsi_rdr(r)¶

Radial derivative of radial part of streamfunction

Parameters:	r – radius

dpsi_rdr2(r)¶

Second radial derivative of radial part of streamfunction

Parameters:	r – radius

p(r, phi)¶

Pressure solution

Parameters:	r – radius phi – angle with x-axis

pressure_cartesian(X)¶

Return pressure solution at Cartesian location.

Parameters:	X – 2D Cartesian location

psi_r(r)¶

Radial part of streamfunction

Parameters:	r – radius

radial_stress(r, phi)¶

Return radial component of stress.

Parameters:	r – radius phi – angle with x-axis.

radial_stress_cartesian(X)¶

Return radial component of stress at Cartesian location.

Parameters:	X – 2D Cartesian location

tau_rphi(r, phi)¶

Return shear stress \(\tau_{r\varphi}\).

Parameters:	r – radius phi – angle with x-axis.

tau_rr(r, phi)¶

Return radial component of deviatoric stress.

Parameters:	r – radius phi – angle with x-axis.

u_phi(r, phi)¶

Return tangential component of velocity.

Parameters:	r – radius phi – angle with x-axis.

u_r(r, phi)¶

Return radial component of velocity.

Parameters:	r – radius phi – angle with x-axis.

velocity_cartesian(X)¶

Return Cartesian velocity at Cartesian location.

Parameters:	X – 2D Cartesian location

class assess.CylindricalStokesSolutionDeltaFreeSlip(n, sign, Rp=2.22, Rm=1.22, rp=1.72, nu=1.0, g=1.0)¶

Bases: assess.cylindrical.CylindricalStokesSolutionDelta

Analytical Solution in cylindrical domain with delta(r-r’) forcing and free-slip boundary conditions

Parameters:	n – wave number sign – +1 for upper half solution r’<r<Rp -1 for lower half solution Rm<r<r’ Rp – outer radius Rm – inner radius nu – viscosity g – forcing strength

dpsi_rdr(r)¶

Radial derivative of radial part of streamfunction

Parameters:	r – radius

dpsi_rdr2(r)¶

Second radial derivative of radial part of streamfunction

Parameters:	r – radius

p(r, phi)¶

Pressure solution

Parameters:	r – radius phi – angle with x-axis

pressure_cartesian(X)¶

Return pressure solution at Cartesian location.

Parameters:	X – 2D Cartesian location

psi_r(r)¶

Radial part of streamfunction

Parameters:	r – radius

radial_stress(r, phi)¶

Return radial component of stress.

Parameters:	r – radius phi – angle with x-axis.

radial_stress_cartesian(X)¶

Return radial component of stress at Cartesian location.

Parameters:	X – 2D Cartesian location

tau_rphi(r, phi)¶

Return shear stress \(\tau_{r\varphi}\).

Parameters:	r – radius phi – angle with x-axis.

tau_rr(r, phi)¶

Return radial component of deviatoric stress.

Parameters:	r – radius phi – angle with x-axis.

u_phi(r, phi)¶

Return tangential component of velocity.

Parameters:	r – radius phi – angle with x-axis.

u_r(r, phi)¶

Return radial component of velocity.

Parameters:	r – radius phi – angle with x-axis.

velocity_cartesian(X)¶

Return Cartesian velocity at Cartesian location.

Parameters:	X – 2D Cartesian location

class assess.CylindricalStokesSolutionDeltaZeroSlip(n, sign, Rp=2.22, Rm=1.22, rp=1.72, nu=1.0, g=1.0)¶

Bases: assess.cylindrical.CylindricalStokesSolutionDelta

Analytical Solution in cylindrical domain with delta(r-r’) forcing and zero-slip boundary conditions

Parameters:	n – wave number sign – +1 for upper half solution r’<r<Rp -1 for lower half solution Rm<r<r’ Rp – outer radius Rm – inner radius nu – viscosity g – forcing strength

dpsi_rdr(r)¶

Radial derivative of radial part of streamfunction

Parameters:	r – radius

dpsi_rdr2(r)¶

Second radial derivative of radial part of streamfunction

Parameters:	r – radius

p(r, phi)¶

Pressure solution

Parameters:	r – radius phi – angle with x-axis

pressure_cartesian(X)¶

Return pressure solution at Cartesian location.

Parameters:	X – 2D Cartesian location

psi_r(r)¶

Radial part of streamfunction

Parameters:	r – radius

radial_stress(r, phi)¶

Return radial component of stress.

Parameters:	r – radius phi – angle with x-axis.

radial_stress_cartesian(X)¶

Return radial component of stress at Cartesian location.

Parameters:	X – 2D Cartesian location

tau_rphi(r, phi)¶

Return shear stress \(\tau_{r\varphi}\).

Parameters:	r – radius phi – angle with x-axis.

tau_rr(r, phi)¶

Return radial component of deviatoric stress.

Parameters:	r – radius phi – angle with x-axis.

u_phi(r, phi)¶

Return tangential component of velocity.

Parameters:	r – radius phi – angle with x-axis.

u_r(r, phi)¶

Return radial component of velocity.

Parameters:	r – radius phi – angle with x-axis.

velocity_cartesian(X)¶

Return Cartesian velocity at Cartesian location.

Parameters:	X – 2D Cartesian location

class assess.SphericalStokesSolutionSmoothFreeSlip(l, m, k, Rp=2.22, Rm=1.22, nu=1.0, g=1.0)¶

Bases: assess.spherical.SphericalStokesSolutionSmooth

Analytical Solution in cylindrical domain with smooth r^k forcing and free-slip boundary conditions

Parameters:	l – degree of the harmonic m – order of the harmonic k – polynomial order of forcing Rp – outer radius Rm – inner radius nu – viscosity g – forcing strength

Pl(r)¶

Radial part of poloidal function

Parameters:	r – radius

dPldr(r)¶

Radial derivative of radial part of poloidal function

Parameters:	r – radius

dPldr2(r)¶

Second radial derivative of radial part of poloidal function

Parameters:	r – radius

delta_rho(r, theta, phi)¶

Perturbation density \(\rho'\) in forcing term: \(g\rho'\hat r\)

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

delta_rho_cartesian(X)¶

Perturbation density \(\rho'\) in forcing term: \(g\rho'\hat r\)

Parameters:	X – 3D Cartesian coordinate

p(r, theta, phi)¶

Pressure solution

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

pressure_cartesian(X)¶

Return pressure solution at Cartesian location.

Parameters:	X – 3D Cartesian location

radial_stress(r, theta, phi)¶

Return radial component of stress.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

radial_stress_cartesian(X)¶

Return radial component of stress at Cartesian location.

Parameters:	X – 3D Cartesian location

tau_rphi(r, theta, phi)¶

Return longitudinal component of shear stress.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

tau_rr(r, theta, phi)¶

Return radial component of deviatoric stress.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

tau_rtheta(r, theta, phi)¶

Return colatitudinal component of shear stress.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

u_phi(r, theta, phi)¶

Return longitudinal (eastward) component of velocity.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

u_r(r, theta, phi)¶

Return radial component of velocity.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

u_theta(r, theta, phi)¶

Return colatitudinal (southward) component of velocity.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

velocity_cartesian(X)¶

Return Cartesian velocity at Cartesian location.

Parameters:	X – 3D Cartesian location

class assess.SphericalStokesSolutionSmoothZeroSlip(l, m, k, Rp=2.22, Rm=1.22, nu=1.0, g=1.0)¶

Bases: assess.spherical.SphericalStokesSolutionSmooth

Analytical Solution in cylindrical domain with smooth r^k forcing and zero-slip boundary conditions

Parameters:	l – degree of the harmonic m – order of the harmonic k – polynomial order of forcing Rp – outer radius Rm – inner radius nu – viscosity g – forcing strength

Pl(r)¶

Radial part of poloidal function

Parameters:	r – radius

dPldr(r)¶

Radial derivative of radial part of poloidal function

Parameters:	r – radius

dPldr2(r)¶

Second radial derivative of radial part of poloidal function

Parameters:	r – radius

delta_rho(r, theta, phi)¶

Perturbation density \(\rho'\) in forcing term: \(g\rho'\hat r\)

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

delta_rho_cartesian(X)¶

Perturbation density \(\rho'\) in forcing term: \(g\rho'\hat r\)

Parameters:	X – 3D Cartesian coordinate

p(r, theta, phi)¶

Pressure solution

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

pressure_cartesian(X)¶

Return pressure solution at Cartesian location.

Parameters:	X – 3D Cartesian location

radial_stress(r, theta, phi)¶

Return radial component of stress.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

radial_stress_cartesian(X)¶

Return radial component of stress at Cartesian location.

Parameters:	X – 3D Cartesian location

tau_rphi(r, theta, phi)¶

Return longitudinal component of shear stress.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

tau_rr(r, theta, phi)¶

Return radial component of deviatoric stress.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

tau_rtheta(r, theta, phi)¶

Return colatitudinal component of shear stress.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

u_phi(r, theta, phi)¶

Return longitudinal (eastward) component of velocity.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

u_r(r, theta, phi)¶

Return radial component of velocity.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

u_theta(r, theta, phi)¶

Return colatitudinal (southward) component of velocity.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

velocity_cartesian(X)¶

Return Cartesian velocity at Cartesian location.

Parameters:	X – 3D Cartesian location

class assess.SphericalStokesSolutionDeltaFreeSlip(l, m, sign, Rp=2.22, Rm=1.22, rp=1.72, nu=1.0, g=1.0)¶

Bases: assess.spherical.SphericalStokesSolutionDelta

Analytical Solution in cylindrical domain with delta(r-r’) forcing and free-slip boundary conditions

Parameters:	l – degree of the harmonic m – order of the harmonic sign – +1 for upper half solution r’<r<Rp -1 for lower half solution Rm<r<r’ Rp – outer radius Rm – inner radius nu – viscosity g – forcing strength

Pl(r)¶

Radial part of poloidal function

Parameters:	r – radius

dPldr(r)¶

Radial derivative of radial part of poloidal function

Parameters:	r – radius

dPldr2(r)¶

Second radial derivative of radial part of poloidal function

Parameters:	r – radius

p(r, theta, phi)¶

Pressure solution

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

pressure_cartesian(X)¶

Return pressure solution at Cartesian location.

Parameters:	X – 3D Cartesian location

radial_stress(r, theta, phi)¶

Return radial component of stress.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

radial_stress_cartesian(X)¶

Return radial component of stress at Cartesian location.

Parameters:	X – 3D Cartesian location

tau_rphi(r, theta, phi)¶

Return longitudinal component of shear stress.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

tau_rr(r, theta, phi)¶

Return radial component of deviatoric stress.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

tau_rtheta(r, theta, phi)¶

Return colatitudinal component of shear stress.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

u_phi(r, theta, phi)¶

Return longitudinal (eastward) component of velocity.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

u_r(r, theta, phi)¶

Return radial component of velocity.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

u_theta(r, theta, phi)¶

Return colatitudinal (southward) component of velocity.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

velocity_cartesian(X)¶

Return Cartesian velocity at Cartesian location.

Parameters:	X – 3D Cartesian location

class assess.SphericalStokesSolutionDeltaZeroSlip(l, m, sign, Rp=2.22, Rm=1.22, rp=1.72, nu=1.0, g=1.0)¶

Bases: assess.spherical.SphericalStokesSolutionDelta

Analytical Solution in cylindrical domain with delta(r-r’) forcing and zero-slip boundary conditions

Parameters:	l – degree of the harmonic m – order of the harmonic sign – +1 for upper half solution r’<r<Rp -1 for lower half solution Rm<r<r’ Rp – outer radius Rm – inner radius nu – viscosity g – forcing strength

Pl(r)¶

Radial part of poloidal function

Parameters:	r – radius

dPldr(r)¶

Radial derivative of radial part of poloidal function

Parameters:	r – radius

dPldr2(r)¶

Second radial derivative of radial part of poloidal function

Parameters:	r – radius

p(r, theta, phi)¶

Pressure solution

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

pressure_cartesian(X)¶

Return pressure solution at Cartesian location.

Parameters:	X – 3D Cartesian location

radial_stress(r, theta, phi)¶

Return radial component of stress.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

radial_stress_cartesian(X)¶

Return radial component of stress at Cartesian location.

Parameters:	X – 3D Cartesian location

tau_rphi(r, theta, phi)¶

Return longitudinal component of shear stress.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

tau_rr(r, theta, phi)¶

Return radial component of deviatoric stress.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

tau_rtheta(r, theta, phi)¶

Return colatitudinal component of shear stress.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

u_phi(r, theta, phi)¶

Return longitudinal (eastward) component of velocity.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

u_r(r, theta, phi)¶

Return radial component of velocity.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

u_theta(r, theta, phi)¶

Return colatitudinal (southward) component of velocity.

Parameters:	r – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

velocity_cartesian(X)¶

Return Cartesian velocity at Cartesian location.

Parameters:	X – 3D Cartesian location

assess.Y(l, m, theta, phi)¶

Real-valued spherical harmonic function \(Y_{lm}(\theta, \varphi)\)

This is based on the following definition:

\[Y_{lm}(\theta, \varphi) = \sqrt{\frac{(2l+1)}{4\pi}\frac{(l-m)!}{(l+m)!}}P_l^m(\cos(\theta))\cos(m\varphi)\]

which is equal to the real part of scipy.special.sph_harm.

Parameters:	l – degree of the harmonic m – order of the harmonic theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

assess.dYdphi(l, m, theta, phi)¶

Colatitudinal derivative of spherical harmonic function \(Y_{lm}(\theta, \varphi)\)

Parameters:	l – degree of the harmonic m – order of the harmonic theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

assess.dYdtheta(l, m, theta, phi)¶

Longitudinal derivative of spherical harmonic function \(Y_{lm}(\theta, \varphi)\)

Parameters:	l – degree of the harmonic m – order of the harmonic theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]

assess.to_spherical(X)¶

Convert Cartesian 3D coordinates X to spherical r, theta, phi

Returns r, theta, phi:
Parameters:	X – Cartesian 3D coordinates
	radius, colatitude, longitude

assess.from_spherical(r, theta, phi)¶

Convert spherical r, theta, phi to 3D Cartesian coordinates.

Parameters:	r, – radius theta – co-latitude in [0, pi] phi – longitude in [0, 2*pi]
Returns X:	Cartesian 3D coordinates