Exercises

1. Consider a tidal wave whose wavelength is very much less than the radius of the Earth. This corresponds to the limit $n\gg 1$ , where $n$ is an azimuthal mode number. Suppose, however, that the ocean depth, $d$ , is allowed to vary with position. Show that, in this limit, the Laplace tidal equations are written

   $$\frac{\partial \zeta}{\partial t} = - \frac{1}{a\,\sin\theta}\left[\frac{\partial}{\partial\theta}\,(\sin\theta\,u\,d)+\frac{\partial (v\,d)}{\partial\phi}\right],$$

   $$\frac{\partial u}{\partial t} - 2\,{\mit\Omega}\,\cos\theta\,v =- \frac{g}{a}\,\frac{\partial}{\partial\theta}\,(\zeta-\skew{5}\bar{\zeta}'),$$

   $$\frac{\partial v}{\partial t} + 2\,{\mit\Omega}\,\cos\theta\,u = - \frac{g}{a\,\sin\theta}\,\frac{\partial}{\partial\phi}\,(\zeta-\skew{5}\bar{\zeta}'),$$

   
   
   where
   $$\skew{5}\bar{\zeta}'=(1+k_2-h_2)\,\skew{5}\bar{\zeta}_2.$$

2. Consider short-wavelength tidal waves in a region of the ocean that is sufficiently localized that it is a good approximation to treat $\cos\theta$ and $\sin\theta$ as constants. We can define local Cartesian coordinates, $x$ , $y$ , $z$ , such that $dx=a\,d\theta$ , $dy=a\,\sin\theta\,d\phi$ , and $dz = r-a$ . It follows that the $x$ -axis is directed southward, the $y$ -axis is directed eastward, and the $z$ -axis is directed vertically upward. Show that, when expressed in terms of this local coordinate system, the Laplace tidal equations derived in the previous exercise reduce to

   $$\frac{\partial \zeta}{\partial t} =-\frac{\partial (u\,d)}{\partial x} -\frac{\partial (v\,d)}{\partial y},$$

   $$\frac{\partial u}{\partial t} -f\,v = -g\,\frac{\partial(\zeta-\skew{5}\bar{\zeta}')}{\partial x},$$

   $$\frac{\partial v}{\partial t} +f\,u = -g\,\frac{\partial(\zeta-\skew{5}\bar{\zeta}')}{\partial y},$$

   
   
   where
   $$f= 2\,{\mit\Omega}\,\cos\theta$$
   is a constant. In fact, the previous equations are the linearized equations of motion of a body of shallow water confined to a tangent plane that touches the Earth at the angular coordinates $\theta$ , $\phi$ . This plane is known as the $f$ -plane because, as a consequence of the Earth's diurnal rotation, it rotates about ${\bf e}_z$ at the angular velocity $f$ .

3. Demonstrate that the set of equations derived in the previous exercise can be written in the coordinate-free (in the $x$ -$y$ plane) form

   $$\frac{\partial \zeta}{\partial t} = -\nabla\cdot (d\,{\bf v}),$$

   $$\frac{\partial{\bf v}}{\partial t} + f\,{\rm e}_z\times {\bf v} =-g\,\nabla(\zeta-\skew{5}\bar{\zeta}'),$$

   
   
   where ${\bf v}= u\,{\bf e}_x+v\,{\bf e}_y$ . Let $\omega = {\bf e}_z\cdot\nabla \times {\bf v}$ and ${\mit\Pi} = \nabla\cdot{\bf v}$ . Assuming that $d$ and $f$ are constants, demonstrate that the previous equations are equivalent to

   $$\frac{\partial^{\,2}\omega}{\partial t^{\,2}} + f^{\,2}\,\omega-g\,d\,\nabla^{\,2}\,\omega=-f\,g\,\nabla^{\,2}\skew{5}\bar{\zeta}',$$

   $$\frac{\partial^{\,2}{\mit\Pi}}{\partial t^{\,2}} + f^{\,2}\,{\mit... ...i}=g\,\nabla^{\,2}\left(\frac{\partial\skew{5}\bar{\zeta}'}{\partial t}\right),$$

   
   
   where $\nabla^{\,2}$ denotes a two-dimensional Laplacian (in the $x$ -$y$ plane).

4. Consider free (i.e., $\skew{5}\bar{\zeta}'=0$ ) plane-wave solutions to the $f$ -plane equations, derived in Exercise ii, of the form

   $$\zeta({\bf r},t) = \zeta_0\,{\rm e}^{\,{\rm i}\,(\sigma\,t-{\bf k}\cdot{\bf r})},$$

   $$u({\bf r},t) = u_0\,{\rm e}^{\,{\rm i}\,(\sigma\,t-{\bf k}\cdot{\bf r})},$$

   $$v({\bf r},t) = v_0\,{\rm e}^{\,{\rm i}\,(\sigma\,t-{\bf k}\cdot{\bf r})}.$$

   
   
   Here, ${\bf r}=(x,\,y)$ , and $\zeta_0$ , $u_0$ , $v_0$ are constants. Assuming that $d$ is constant, show that

   $$u_0 = g\left(\frac{\sigma\,k_x-{\rm i}\,f\,k_y}{\sigma^{\,2}-f^{\,2}}\right)\zeta_0,$$

   $$v_0 = g\left(\frac{\sigma\,k_y+{\rm i}\,f\,k_x}{\sigma^{\,2}-f^{\,2}}\right)\zeta_0,,$$

   
   
   and
   $$\sigma^{\,2} = f^{\,2}+c^{\,2}\,k^{\,2},$$
   where $c=\sqrt{d\,g}$ . This type of wave is known as a Poincaré wave.

5. Suppose that the region $y\leq 0$ corresponds to an ocean of constant depth $d$ , whereas the region $y>0$ corresponds to land. Consider free solutions to the $f$ -plane equations in the region $y<0$ . We can trivially satisfy the constraint $v(x,0,t)=0$ by searching for solutions which are such that $v=0$ for all $y\leq 0$ . Show that the most general such solution takes the form
   $$\zeta(x,y,t)= Z_1(x+c\,t,y)+Z_2(x-c\,t,y),$$
   where $c=\sqrt{d\,g}$ , and

   $$Z_1(x+c\,t,y) = Z_1(x+c\,t,0)\,{\rm e}^{+s\,y/l},$$

   $$Z_2(x-c\,t,y) = Z_2(x-c\,t,0)\,{\rm e}^{-s\,y/l}.$$

   
   
   Here, $Z_1(x,0)$ and $Z_2(x,0)$ are arbitrary functions, $l=\sqrt{d\,g}/\vert f\vert$ , and $s={\rm sgn}(f)$ . These solutions are known as Kelvin waves. Deduce that Kelvin waves propagate along coastlines, at the speed $c$ , in such a manner as to keep the coastline to the right of the direction of propagation in the Earth's northern hemisphere, and to the left in the southern hemisphere.

6. We can take into account the latitude dependence of the parameter $f$ by writing $f= f_0-\beta\,x$ , where $f_0=2\,{\mit\Omega}\,\cos\theta$ and $\beta=2\,{\mit \Omega}\,\sin\theta/a$ . Let us assume that the ocean is of constant depth, $d$ . Furthermore, let us search for an almost incompressible, free solution of the Laplace tidal equations which is such that $u\simeq \partial\psi/\partial y$ and $v\simeq -\partial\psi/\partial x$ . By eliminating $\zeta$ from the final two $f$ -plane equations, show that
   $$\nabla^{\,2}\left(\frac{\partial\psi}{\partial t}\right)+\beta\,\frac{\partial\psi}{\partial y} \simeq 0.$$
   By searching for a wavelike solution of the previous equation of the form $\psi=\psi_0\,{\rm e}^{\,{\rm i}\,(\sigma\,t-{\bf k}\cdot{\bf r})}$ , deduce that
   $$\sigma\,k^{\,2}+\beta\,k_y=0.$$
   This is the dispersion relation of a so-called Rossby wave. Demonstrate that Rossby waves always travel with a westward component of phase velocity. Finally, show that it is reasonable to neglect compression provided that $\sigma\ll \vert f_0\vert$ .