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Rotating Nonlinear Shallow Waters

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

$\displaystyle h_t + \nabla \cdot (h \vec u)$ $\textstyle =$ $\displaystyle 0,$ (19)
$\displaystyle \vec{u}_t + (\vec{u} \cdot \nabla) \vec{u} + \nabla h + \vec{u}^{\perp}$ $\textstyle =$ $\displaystyle 0\, .$ (20)

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

\begin{displaymath}
\frac{D}{Dt} \left(\frac{1+v_x-u_y}{h}\right) = 0 \,
\end{displaymath} (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

\begin{displaymath}q=\frac{1+v_x-u_y}{h}=q_0,\end{displaymath}

where $q_0$ is an arbitrary potential vorticity, so this is the hypothesis to make for the analogue of irrotational flows:
\begin{displaymath}
q_0 h = 1 + v_x - u_y \, .
\end{displaymath} (22)

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

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

to rewrite ([*]) as

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

Taking the divergence and the two-dimensional curl $(\nabla^{\perp} \cdot)$ of the above equations, we obtain the following pair:
$\displaystyle \phi_t + \frac{1}{2} \vert\nabla\phi+\nabla^{{\perp}}\psi\vert^2 ...
...la \cdot
\left[\Delta\psi (\nabla^{{\perp}}\phi - \nabla\psi) \right]
+ h -\psi$ $\textstyle =$ $\displaystyle 0,$  
$\displaystyle \psi_t +
\Delta^{-1} \nabla^{{\perp}} \cdot
\left[\Delta\psi (\nabla^{{\perp}}\phi - \nabla\psi) \right]+\phi$ $\textstyle =$ $\displaystyle 0\, .$  

By noticing that

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

we can rewrite these equations, together with ([*]) in the form
$\displaystyle h_t + \nabla \cdot (h\, (\nabla \phi + \nabla^{{\perp}} \psi))$ $\textstyle =$ $\displaystyle 0,$  
$\displaystyle \phi_t + \frac{1}{2} \vert\nabla\phi+\nabla^{{\perp}}\psi\vert^2 ...
...abla \cdot
\left[(1+\Delta\psi) (\nabla^{{\perp}}\phi - \nabla\psi) \right]
+ h$ $\textstyle =$ $\displaystyle 0,$  
$\displaystyle \psi_t +
\Delta^{-1} \nabla^{{\perp}} \cdot
\left[(1+\Delta\psi) (\nabla^{{\perp}}\phi - \nabla\psi) \right]$ $\textstyle =$ $\displaystyle 0\, .$  

The constraint ([*]) on the potential vorticity takes the form

\begin{displaymath}
1 + \Delta \psi = q_0 h \, ,
\end{displaymath} (23)

under which the equations above reduce to the pair
$\displaystyle h_t + \nabla \cdot (h\, (\nabla \phi + \nabla^{{\perp}}
\Delta^{-1}(q_0 h-1)))$ $\textstyle =$ $\displaystyle 0$  
$\displaystyle \phi_t + \frac{1}{2}
(\nabla\phi+\nabla^{{\perp}}\Delta^{-1}(q_0 h-1))^2 +$      
$\displaystyle q_0 \Delta^{-1} \nabla \cdot \left[h\, (\nabla^{{\perp}}\phi -
\nabla\Delta^{-1}(q_0 h-1)) \right] + h$ $\textstyle =$ $\displaystyle 0\, .$  

These equations are Hamiltonian, with conjugate variables $\phi$ and $h$, and Hamiltonian
\begin{displaymath}
{\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} \, ,
\end{displaymath} (24)

representing again the sum of kinetic and potential energies.


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Next: Nonlinear Internal Waves in Up: Hamiltonian formalism for long Previous: Linear Shallow Waters in
Dr Yuri V Lvov 2007-01-17