Postgraduate Seminar: Duncan Sutherland -- Chebyshev spectral methods for quasigeostrophic shallow-water flow

Quasigeostrophic shallow-water flow on the beta plane typically involves turbulent
motion and the formation of jets, which can be modelled by considering inviscid flow in
a channel.  I will present a review of the use of Chebyshev spectral methods for the
solution of the stream-function vorticity formulation of flow in a channel, periodic in
the streamwise direction and with Dirichlet boundary conditions on the channel wall.
The vorticity equation is handled with standard pseudo-spectral techniques, while
Poisson’s equation for the streamfunction is solved by transforming in both x and y and
deriving a recursion relation for the Chebyshev coefficients in y.  The recursion
relation can be expressed as a quasipentadiagonal matrix equation, which may be solved
in O(N) operations.  Example flows used to test convergence and accuracy of the scheme
will be discussed.