pg_utils.pg_model.expand_stream_force_orth

Expansion configuration file - Expansion for the streamfunction-force formulation with self-orthogonal forcing basis.

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

\[ \begin{align}\begin{aligned}\Psi^{nm}(s) = s^{|m|} H^3 J_n^{(\frac{3}{2}, |m|)}(2s^2 - 1)\\F^{nm}(s) = s^{|m|} J_n^{(0, |m|+\frac{1}{2})}(2s^2 - 1)\end{aligned}\end{align} \]

This has the orthogonality:

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

However, under this configuration, a compact streamfunction spectrum does not translate to a compact forcing spectrum. This is due to the fact that the forcing basis is not directly linked with the streamfunction basis.

Module Attributes

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