Home Work Three

1. Show that the vertical velocity within a shallow-water system is given by
   
   $$w=\frac{z-\eta_B}{h}\frac{D h}{D t} +\frac{D \eta_b}{D t}.$$
   
   
   Interpret this result, showing that it gives sensible answers at the top and bottomof the fluid layer
2. In an adiabatic shallow water fluid in a rotating reference frame show that the potential vorticity conservation law is
   
   $$\frac{D}{D t}\frac{\xi+f}{\eta-\eta_B}=0,$$
   
   
   where $\eta$ is the height of the free surface and $\eta_B$ is the height of the bottom topography, both referenced to the same flat surface.

3. Consider rotating two layer shallow water system with flat bottom $\eta_B=0$. Find the linear dispersion relation for waves in this system.
4. We have used in class the property
   
   $$\nabla\times {\bf f}\times {\bf u} = ({\bf u}\cdot\nabla){\bf f}+ {\bf f}(\nabla\cdot {\bf u}).$$
   
   
   Is this correct? If yes, prove it, if not, suggest an alternative.
5. extra credit In the shallow water equations show that, if the flow is approximately geostrophically balanced, the energy at large scales is predominantly potential energy andthat energy at small scales is predominantly kinetic energy. Define precisely what ``large scale'' and ``small scale'' mean in this context.