The Omega equation is a time independent partial differential which governs the vertical velocity of water. Normally, equations governing fluids are time independent, but a particular approximation, known as the quasigeostrophic approximation, allows one to reduce the complicated Navier-Stokes equations into this simpler form.
This approximation has been hisotrically used in the open ocean, but recently, Libe Washburn and Chris Gotschalk discovered the presence of phytoplankton well below the depth which sunlight allows them to live. Using data that they gathered in the Santa Barbara channel, one can solve the Omega equation for the vertical velocity of the water column as a possible explination for the surprising depth of the phytoplankton.
Simply clone the repo and run the Matlab scipt omega_eqn_zero_neumann.m.
The *.mat files are the necessary data in order to run the code. The omega equation is a non-constant coefficient second order elliptic pde, so we solve it iteratively by GMRES. The domain is rectangular prism, with constant step size in each axis direction.
The two matlab files correspond to different boundary conditions:
omega_eqn_zero_dirichlet_fft_z.m corresponds to zero Dirichlet boundary conditions on all faces of the box. This is not physically resonable, and is included for testing purposes, and for comparing against the academic literature.
omega_eqn_zero_neumann.m is a mixed type Neumann-Dirichlet boundary condition. On the sides of the box we use a natural boundary condition. This amounts to choosing a Neumann boundary condition with flux coming from the Q vector. On the bottom of the box we use a zero Neumann boundary condition, while on the top we use a zero Dirichlet boundary conditions
Omega Equation: https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/96JC01144
GMRES: https://en.wikipedia.org/wiki/Generalized_minimal_residual_method