Planetary Rossby Wave#
This experiment is designed to show how a geostrophic monopole evolves on a \(\beta\)-plane.
Configuration#
Equations#
The equations solved are the linear shallow water equations, given by $$ u_t - fv = -g \eta_x $$ $$ v_t + fu = -g \eta_y $$ $$ \eta_t + (Hu)_x + (Hv)_y = 0 $$
where \(\vec{u} = u \hat{x} + v \hat{y}\) is the barotropic velocity, \(g\) is the acceleration of gravity, \(H\) is a uniform resting fluid depth, and \(\eta\) is the deviation of the fluid free surface relative to the resting fluid. In this model, the \(x\) direction is similar to longitude and \(y\) is similar to latitude.
A \(\beta\)-plane, in geophysical fluid dynamics, is an approximation that accounts for first order variability in the (vertical component of the) coriolis parameter with latitude,
The background variation in the planetary vorticity supports Rossby waves, which propagate "westward" with higher potential vorticity to the right of phase propagation.
Domain Discretization#
In this problem, the domain is a square with \((x,y) \in [-500km, 500km]^2\). The model domain is divided into \(10\times 10\) elements of uniform size. Within each element, the solution is approximated as a Lagrange interpolating polynomial of degree 7, using the Legendre-Gauss quadrature points as interpolating knots. To exchange momentum and mass fluxes between neighboring elements, we use a local upwind (Lax-Friedrich's) Riemann solver.
Initial Condition#
The initial condition is defined by setting the free surface height to a Gaussian, centered at the origin, with a half width of 10 km and a height of 1 cm. $$ \eta(t=0) = 0.01e^{ -( (x^2 + y^2 )/(2.0*10.0^{10}) )} $$
The initial velocity field is calculated by using the pressure gradient force and using geostrophic balance; in SELF, this is handled by the LinearShallowWater % DiagnoseGeostrophicVelocity type bound procedure after setting the initial free surface height.
Boundary Conditions#
Radiation boundary conditions are applied by setting the external state to a motionless fluid with no free surface height variation ( \(u=v=0, \eta = 0\)). The model is integrated forward in time using Williamson's \(3^{rd}\) order low storage Runge-Kutta, with a time step of \(\Delta t = 0.5 s\).
Physical Parameters#
The remaining parameters for the problem are as follows
- \(g = 10 m s^{-2}\)
- \(f_0 = 10^{-4} s^{-1}\)
- \(\beta = 10^{-11} m^{-1} s^{-1}\)
- \(H = 1000 m\)