In class, we discussed the role of nonlinearity as measured
by the nonlinearity parameter [U/(Beta L^2)]. For small values of the nonlinearity parameter,
linear Rossby-wave dynamics dominate, whereas for large values of the nonlinearity
parameter, vortex dynamics are important. As we discussed in class,
there are also exact solutions to the full nonlinear equations in the form
of monopolar and dipolar vortices. Below you will find numerical solutions
for the barotropic vorticity equation on the sphere for an initially localized
disturbance.
For large nonlinearity parameter (delta ~ 4.55), the flow rapidly adjusts to a steadily propagating dipole---a modon.
Strong barotropic dipole on the sphere (modon).
For delta ~ 1, the localized initial condition starts out like a dipole, but because Rossby wave processes are of comparable importance, the dipole weakens steadily due to the loss of energy by Rossby-wave radiation. Ultimately, the dipole separates and weakens rapidly by wave radiation.
Moderate barotropic dipole on the sphere.
Finally, for delta ~ 0.1, the localized disturbance behaves like a bundle of uncoupled Rossby waves. These waves radiate away as if they did not know the others were present, and the solution is essentially linear.
Weak barotropic dipole on the sphere.
We proceed now to the more general case of unsteady flow. Nonlinear cascades are illustrated below by a numerical simulation for barotropic flow on the sphere subject to random initial conditions. Notice how the small-scale noise organizes into vortices (small L, and large nonlinearity parameter). These vortices then merge and build to large scales (upscale energy cascade), which reduces the nonlinearity parameter. Gradually, the flow becomes more linear and large scale as a single steady jet develops over the equator.
Barotropic turbulence on the sphere.