Linear Shallow Waters in a Rotating Environment

In a rotating environment, the linearized shallow-water equations are

$$\eta_t + \nabla \cdot \vec{u} = 0\, ,$$

$$\vec{u}_t + \nabla \eta + \vec{u}^{\perp} = 0\, .$$

Here

$$\vec{u} = \left( \begin{array}{r} u \\ v \end{array} \right) \,$$

is the velocity field, and

$$\vec{u}^{\perp} = \left( \begin{array}{r} -v \\ u \end{array} \right) \, .$$

The Coriolis parameter $f$ has been absorbed in the nondimensionalization of time, so it is effectively equal to one.

These equations do not preserve vorticity, so irrotationality cannot be assumed. However, they preserve the potential vorticity

$$q = v_x - u_y - \eta \, .$$ (11)

The assumption corresponding to irrotationality in the non-rotating case is therefore that of zero potential vorticity, i.e. $q=0$. We can in fact generalize this hypothesis, and consider an arbitrary, though constant, potential vorticity. We shall employ such generalization when we consider internal waves in a rotating environment. In order to exploit the irrotationality assumption, it is convenient to decompose the flow into a potential and a divergence-free part:

$$\vec{u} = \nabla \phi + \nabla^{\perp} \psi \, ,$$ (12)

where

$$\nabla^{\perp} =\left( \begin{array}{r} -\partial_y \\ \partial_x \end{array} \right) \, .$$ (13)

In terms of $\phi$ and $\psi$, the equations take the form

$$\eta_t + \Delta \phi = 0\, ,$$

$$\phi_t + \eta - \psi = 0\, ,$$

$$\psi_t + \phi = 0\, .$$ (14)

The condition of zero potential vorticity reads

$$q = v_x - u_y - \eta = \Delta \psi - \eta = 0 \, ,$$ (15)

so the system above reduces to

$$\eta_t + \Delta \phi = 0$$ (16)

$$\phi_t + \eta - \Delta^{-1} \eta = 0\, .$$ (17)

This system is Hamiltonian, with canonical variables $\phi$ and $\eta$, and Hamiltonian

$${\cal H}= \frac{1}{2} \int \left( \left\vert\nabla\phi + \nabla \Delta^{-1} \eta \right\vert^2 + \eta^2 \right) d x \, .$$ (18)

Again, the Hamiltonian agrees with the total energy of the system.