Rotating Nonlinear Shallow Waters

The fully nonlinear equations for shallow waters in a rotating environment are

$$h_t + \nabla \cdot (h \vec u) = 0,$$ (19)

$$\vec{u}_t + (\vec{u} \cdot \nabla) \vec{u} + \nabla h + \vec{u}^{\perp} = 0\, .$$ (20)

The statement of conservation of potential vorticity now takes the form (Sec 12-2 in [30])

$$\frac{D}{Dt} \left(\frac{1+v_x-u_y}{h}\right) = 0 \,$$ (21)

(That is: the total vorticity of a vertical column of water divided by its height remains constant as the column moves.) The unperturbed state has

$$q=\frac{1+v_x-u_y}{h}=q_0,$$

where $q_0$ is an arbitrary potential vorticity, so this is the hypothesis to make for the analogue of irrotational flows:

$$q_0 h = 1 + v_x - u_y \, .$$ (22)

We introduce the potentials $\phi$ and $\psi$ as in (), and use the fact that

$$(\vec u \cdot \nabla )\vec u = \frac{1}{2}\nabla \vert \nabla... ...^2 + \Delta \psi \left(\nabla^\perp \phi - \nabla \psi \right),$$

to rewrite () as

$$\nabla \phi_t + \nabla^\perp \psi_t + \frac{1}{2}\nabla \vert... ... \nabla \psi \right)+\nabla h +\nabla^\perp\phi-\nabla\psi = 0.$$

Taking the divergence and the two-dimensional curl $(\nabla^{\perp} \cdot)$ of the above equations, we obtain the following pair:

$$\phi_t + \frac{1}{2} \vert\nabla\phi+\nabla^{{\perp}}\psi\vert^2 ... ...la \cdot \left[\Delta\psi (\nabla^{{\perp}}\phi - \nabla\psi) \right] + h -\psi = 0,$$

$$\psi_t + \Delta^{-1} \nabla^{{\perp}} \cdot \left[\Delta\psi (\nabla^{{\perp}}\phi - \nabla\psi) \right]+\phi = 0\, .$$

By noticing that

$$-\psi= \Delta^{-1} \nabla \cdot (\nabla^{{\perp}}\phi - \nabla\psi),$$

$$\phi= \Delta^{-1} \nabla^\perp \cdot (\nabla^{{\perp}}\phi - \nabla\psi),$$

we can rewrite these equations, together with () in the form

$$h_t + \nabla \cdot (h\, (\nabla \phi + \nabla^{{\perp}} \psi)) = 0,$$

$$\phi_t + \frac{1}{2} \vert\nabla\phi+\nabla^{{\perp}}\psi\vert^2 ... ...abla \cdot \left[(1+\Delta\psi) (\nabla^{{\perp}}\phi - \nabla\psi) \right] + h = 0,$$

$$\psi_t + \Delta^{-1} \nabla^{{\perp}} \cdot \left[(1+\Delta\psi) (\nabla^{{\perp}}\phi - \nabla\psi) \right] = 0\, .$$

The constraint () on the potential vorticity takes the form

$$1 + \Delta \psi = q_0 h \, ,$$ (23)

under which the equations above reduce to the pair

$$h_t + \nabla \cdot (h\, (\nabla \phi + \nabla^{{\perp}} \Delta^{-1}(q_0 h-1))) = 0$$

$$\phi_t + \frac{1}{2} (\nabla\phi+\nabla^{{\perp}}\Delta^{-1}(q_0 h-1))^2 +$$

$$q_0 \Delta^{-1} \nabla \cdot \left[h\, (\nabla^{{\perp}}\phi - \nabla\Delta^{-1}(q_0 h-1)) \right] + h = 0\, .$$

These equations are Hamiltonian, with conjugate variables $\phi$ and $h$, and Hamiltonian

$${\cal H}=\frac{1}{2} \int\left( h\, \left\vert\nabla\phi+\... ...Delta^{-1}(q_0 h-1)\right\vert^2 + h^2 \right) d {\bf r} \, ,$$ (24)

representing again the sum of kinetic and potential energies.