Barotropic Instability – EXERCIZE
Go to http://www.o3d.org/ROMS-Tutorial/tutorials.html, and complete tutorials section 5. Also read
chapter 10 of Cushman.
In this homework we will continue to study the evolution of shallow water waves when the mean flow is
not at rest but posses a meridional relative vorticity gradients. Similar to the previous homework where
the planetary vorticity is the restoring force sustaining the evolution of Rossby waves you will find that
the relative vorticity can also sustain the evolution of waves. However these waves can lead to
instabilities of the mean vorticity and velocity field when certain conditions are met (see chapter 10).
The amplitude of the initial perturbation grow and we refer to this as a barotropic instability. To study
this problem numerically:
1) Begin with the same application in roms-examples/shallow2D. In MATLAB build an analytical grid
(roms-examples/shallow2D/input/shallow2d-grid.nc) with dimensions 3,000x10,000 km as shown
in the figure below, with dx=dy=50km, constant depth of 200m and constant Coriolis parameter for a
reference latitude of 30N. If you have trouble building such a grid you can look in the file:
/neo/GFD_Class/gfd_root/roms-examples/shallow2D/matlib/sw_CreateGrid.m
2) Create a relative vorticiy gradient as the initial condition using a shear flow in the meridional direction
(see chapter 10; the vorticity field should look similar to the one plotted below).
Choose L=4*dx and U=4 m/s. Apply a small wave perturbation in the zonal velocity initial condition with
amplitude 10% of the mean zonal velocity. Choose the wavenumber of the perturbation to be the one
associated with the fastest growing unstable mode (see chapter 10).
Run the model for at least 180 days. If you have configured things correctly you should see the initial
perturbation grow in the relative vorticity field (see plot below).
3) What happens after a long time to the vorticies field ? Is this what you expect from 2D turbulence?
Make some plots of the early/mid/final stages of its evolution.
4) Perform the simulation again but with a different wavenumber perturbation in the initial condition.
Describe how the evolution of the instability differs in space and time. Show some plots of comparisons.
5) Now study the evolution of the barotropic instability when you have a sloping topography in the xdirection (dh/dx=const). Show some plots of comparisons.
6) Now study the evolution of the barotropic instability when you have a beta effect in the y-direction
(df/dy=const). Show some plots of comparisons.