pg_utils.pg_model.expand_stream_force_cpt

Expansion configuration file - Expansion for the streamfunction-force formulation with forcing spectrum as compact as streamfunction.

Note

This set of bases and expansions has to be used together with the reduced system of equations, where the only equations present are Psi and F_ext

Note

This is only relevant to eigenvalue problems or linearized systems. Nonlinear PG equations do not generally simplify into streamfunction - force formulation, but requires full set of PG variables or their conjugate counterparts

Using this expansion configuration, the basis functions takes the form

\[ \begin{align}\begin{aligned}\Psi^{nm}(s) = s^{|m|} H^3 J_n^{(\frac{3}{2}, |m|)}(2s^2 - 1)\\F^{nm}(s) = s^{|m|} J_n^{(\frac{3}{2}, |m|)}(2s^2 - 1)\end{aligned}\end{align} \]

Unlike :py:mod:~pg_utils.pg_model.expand_stream_force_orth``, the forcing basis in this configuration is not self-orthogonal. The orthgonality property reads

\[ \begin{align}\begin{aligned}\int_0^1 \Psi^{n'm}(s) \Psi^{nm}(s) \frac{s}{H^3} ds = N_{\Psi}^{nm} \delta_{nn'}\\\int_0^1 \Psi^{n'm}(s) F^{nm}(s) ds = N_{F}^{nm} \delta_{nn'}\end{aligned}\end{align} \]

Hence the two fields share the same set of test functions.

This configuration has the desirable property that a compact streamfunction spectrum translates to a compact forcing spectrum. This is due to the fact that the forcing basis is directly linked with the streamfunction basis.

Module Attributes

subscript_str	Fourier expansion
fourier_expand	Radial expansion
rad_expand	Test functions
test_s_expression	Inner products