The following are examples of solutions to the equations studied in ATOC 512 in specific settings. The data was generated with the MITgcm and the animations were made with python's matplotlib (with help of xmitgcm).
A 2-D version of the Navier-Stokes equations for incompressible fluids and no coriolis term reads as: \[u_t + uu_x + vu_y = \phi_x + \nu(u_xx + u_yy)\] \[v_t + uv_x + vv_y = \phi_y + \nu(v_xx + v_yy)\] \[u_x + v_y = 0\] However, the simplicity of these equations allows us to take their curl and get an equation for the vorticity \(\zeta = v_x - u_y\): \[\zeta_t + \mathbf{u}\cdot\nabla\zeta = \nu\nabla^2\zeta\] Here, we see that the vorticity simply obeys a 2-D advection-diffusion equation.
In the following simulation, the \(u\) and \(v\) fields were initialized with a checker board pattern so as to generate a field of vorticies. The domain has periodic boundary conditions and the simplest configuration of the NS equations was used (which in the MITgcm is more then the example above, but difference are minimal). Notice on the video that the speed of the animation changes. Even with noise inserted at the beginning, it takes some time befored the perturbation triggers motion everywhere in the domain. The streamfunction is \(\nabla^2\psi = \zeta\).
Download the video.This video shows an initial setup with only two vortices of opposite sign near each other. In the first case (left) they have the same magnitude and each advects the other, so they start moving to the right. The second case (right) has the positive vortex stronger then the negative one. A similar process is seen, but with the negative vortex slowly rotation around the positive one. Again, boundary conditions are doubly periodic and there is no coriolis term.
Download the video.This example is a modification of a tutorial experiment of the MITgcm. The domain is a x-z plane with a sloping bottom. Part of the surface just above the shallowest region is cooled down, which generates a circulation where the colder (denser) water flows down the slope. We studied three cases:
The following simulations show comparison between the first case and each of the others. Special "Orlanski" boundary conditions are used for the right side of the domain, but that doesn't really change the interesting process here. In the case with a coriolis term, the flow is deflected to the right (out of the screen) as it speeds down the slope, until geostrophic balance is achieved (not really shown in the videos as the cooling never stops). On the other hand, addition of stratification makes the flow stops in the middle of the slope, when it attains the depth of similar density.
This is the same animation as above, but also showing the velocity through the screen \(v\).
Download the video.