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\setcounter{chapter}{7}
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\section{The Shallow-Water Approximation}
 \setcounter{equation}{0}


The  continuity equation  reads 
\[ \underbrace{\f{\p u}{\p x}}_{\f{U}{L}} +  \underbrace{ \f{\p v}{\p
y}}_{\f{U}{L}} +   \underbrace{\f{\p w}{\p z}}_{\f{W}{D}}  = 0 \]

If the horizontal length scale of motion $L$ is much greater than the sea depth $D$,  i.e., shallow water, 
\be \f{D}{L} \ll 1 \ee we  have
\be  W=\f{DU}{L}\ll U\ee Momentum conservation requires
\[ \ub{\f{\p u}{\p t}}_{\f{U}{T}} + \big(\ub{u \f{\p u}{\p x}}_{\f{U^2}{L}} + \ub{v
\f{\p u}{\p {y}}}_{\f{U^2}{L}} + \ub{w \f{\p u}{\p z}}_{\f{U^2}{L}} \big) -\ub{fv
}_{fU}= \ub{\f{1}{\rho}
\, \f{\p p_d}{\p x}}_{\f{P}{\rho L}} \]
 \[ \f{\p v}{\p t} + \lp u \f{\p v}{\p x} + v \f{\p v}{\p y} + w \f{\p v}{\p z} \rp +
fu = -\f{1}{\rho} \, \f{\p p_d}{\p y} , \]
\[ \ub{\f{\p w}{\p t}}_{\f{W}{T}} + \big(\ub{u \f{\p w}{\p x}}_{\f{UW}{L}} +
\ub{v \f{\p w}{\p {y}}}_{\f{UW}{L}} + \ub{w \f{\p w}{\p z}}_{\f{UW}{L}} \big) =
\ub{\f{1}{\rho}
\, \f{\p p_d}{\p z}}_{\f{P}{\rho D}} \] where $p_d$ stands for  the dynamic
pressure
\[ p = -\rho g z + p_d \]
 
 Examine first the horizontal momentum balance.   In order
that pressure gradient forces the rotating flow, the pressure gradient has to balance either the  Coriolis force (rotation   important), assuming that the Rossby number is no greater than $O(1)$. The scale of pressure is
\[ P = \f{\rho UL}{T}, ~~\mbox{or}~~ \rho UfL   \]  From the vertical momentum we find, if  $P =  \rho UL/T  $,
\[ \f{\f{\p w}{\p t}}{\f{1}{\rho}\f{\p_d}{\p z}}\sim \f{\f{DU}{LT}}{\f{\rho UL/T}{\rho D}}\sim \f{D^2}{L^2}
\]
 and if $P=\rho UfL $,
\[ \f{\f{\p w}{\p t}}{\f{1}{\rho}\f{\p_d}{\p z}}\sim \f{\f{DU}{LT}}{\f{  UfL}{\rho D}}\sim \f{D^2}{L^2} \f{1}{Tf}
\]
Since the time scale of interest is of a day or so, $fT=O(1)$. We conclude that the vertical pressure gradient is practially zero with error of order $D^2/L^2\ll 1$,  implying that 
\be p_d \simeq \rho g\eta \ee 
or  the total pressure is hydrostatic
\be p_{\mbox{total}} \simeq \rho g (\eta - z)
\ee Thus, \be \f{\p p_d}{\p x}   =   \rho g \f{\p \eta}{\p x} \qquad \f{\p p_d}{\p
y} = \rho g \f{\p \eta}{\p y} \ee

   
We now show    by a formal 
perturbation scheme for horizontal bottom,  that      $u,v$
are independent of $z$ to the leading order or approximation,
\be \f{\p u}{\p x}+ \f{\p v}{\p y}+\f{\p w}{\p z} =0 \ee
\be \f{\p u}{\p t} + u \f{\p
u}{\p x} + v
\f{\p u}{\p y} + w\f{\p u}{\p z}  - fv = -g\f{\p \eta}{\p x} \label{x-mom-2}\ee 
\be \f{\p v}{\p t} + u \f{\p v}{\p x} + v \f{\p v}{\p y} + w\f{\p v}{\p z}  + fu = - g
\f{\p
\eta}{\p y}\label{y-mom-2}\ee 
On the bottom \be w=0\label{bottombc}\ee
On the free surface \be \f{\p \eta}{\p t} + u\f{\p \eta}{\p x}+v\f{\p \eta}{\p y} =
w,
\quad z = \eta\label{freesurfbc}\ee
Assuming the following expansions: 
\be u = u_0  + \f{z+h}{L}\, u_1 + \f{(z+h)^2}{2L^2}\, u_2+ \cdots \ee
\be v = v_0  + \f{z+h}{L}\, v_1 + \f{(z+h)^2}{2L^2}\, v_2+ \cdots \ee
\be w =   \f{z+h}{L} \,w_1 + \f{(z+h)^2}{2L^2} \,w_2+ \cdots \ee
where $u_n,v_n,w_n$ are independent of $z$. From the x
momentum equation we get from (\ref{x-mom-2})
\begin{eqnarray}
\lefteqn{
\f{\p u_0}{\p t} + u_0\f{\p u_0}{\p x} + v_0\f{\p u_0 }{\p y} -fv_0 +g\f{\p
\eta}{\p x} }\nonumber \\
&& +\f{(z+h)}{L}
 \lp\f{\p u_1}{\p t} +
  u_1\f{\p u_0}{\p x} +u_0\f{\p u_1}{\p x}+
 v_1\f{\p u_0}{\p y} +v_0\f{\p u_1}{\p y} + \f{w_1u_1 }{L}-fv_1\rp\nonumber
\\
&& + \lp\f{z+h}{L}\rp^2 (\cdots) + \cdots = 0
\end{eqnarray}
 with a similar equation for the y momentum. Separating the zeroth
power of
$(z+h)/L$ we get 
\be \f{\p u_0}{\p t} + u_0 \f{\p
u_0}{\p x} + v_0
\f{\p u_0}{\p y} - fv_0 = -g\f{\p \eta}{\p x} \label{x-mom-3}\ee 
\be \f{\p v_0}{\p t} + u_0 \f{\p v_0}{\p x} + v_0) \f{\p v_0}{\p y} + fu_0 = - g
\f{\p
\eta}{\p y}\label{y-mom-3}\ee 
Thus $u,v$ are depth-independent to the leading order. Also from  continuity, 
\be \f{\p u_0}{\p x} + \f{\p v_0}{\p y} + \f{w_1}{L} = 0\ee
and from the free surface condition
\be 
\f{\p \eta}{\p t} + u_0 \f{\p \eta}{\p x } = \f{\eta + h}{L} w_1 \ee
Hence after eliminating $w_1$, 
\be \f{\p \eta}{\p t} + \f{\p[ (\eta +h)u_0]}{\p x} + \f{\p [(\eta +h)v_0]}{\p
y} = 0 \label{mass}\ee

In summary, for shallow seas, 
\be \f{\p u}{\p t} + u \f{\p u}{\p x} + v \f{\p u}{\p y} - fv = - g \f{\p
\eta}{\p x} \label{x-mom-1}\ee 
\be \f{\p v}{\p t} + u \f{\p v}{\p x} + v \f{\p v}{\p y} + fu = - g \f{\p
\eta}{\p y}\label{y-mom-1}\ee  
to leading order. 

If Rossby number is small, then  the convective inertia is negligible; the
momentum  equations reduce  to :
 \be \f{\p u}{\p t}   - fv = - g  \f{\p
\eta}{\p x} \label{x-mom-0}\ee 
\be \f{\p v}{\p t}   + fu = - g \f{\p
\eta}{\p y}\label{y-mom-0}\ee  
where are linear.  

Take for estimates,  $U =0.1,  1 m/s, \Omega = 2.31\times10^{-5}, L =100,  1000 km =10^5,  10^6 m$, Rossby
number $ U/2\Omega L = 0.043$.  
We leave it as an exercise  to work out the equations for $h(x,y)$ with a small
slope. 



  


\subsection{Geostrophic motion} 
For steady flow at small Rossby number, 
\[ H =\eta +h \simeq h \]
 the momentum equations reduce  to 
 \begin{eqnarray*} -fv & = & -g \f{\p \eta}{\p x} \\
 fu& = & -g \f{\p\eta}{\p y} .
\end{eqnarray*} 
Thus, Coriolis force and pressure gradient are in balance
\[ u = -\f{g}{f} \, \f{\p \eta}{\p y} \qquad v = \f{g}{f} \, \f{\p \eta}{\p x} \]
implying
\be u \f{\p \eta}{\p x} + v \f{\p \eta}{\p y} = 0 \ee
 or
\[ \vec{q} \cdot \nabla \eta = 0 . \] 

Physically along a streamlines, the free surface height  remains constant. Hence, the surface contours  are parallel to the streamlines and to isobars.   This state is called {\bf geostrophic}.

\subsection{Remark: Depth-integrated mass conservation}
The depth integrated mass  conservation (\ref{mass}) is a general an exact result which holds   for variable depth as well. 
On the seabed   $z = - h(x,y)$, vanishing of the normal veocity requires
\[ w = -u\f{\p h}{\p x} - v \f{\p h}{\p y} . \] 
On the free
surface $z = \eta (x,y,t)$, the kinematic boundary condition reads, 
\[ w = \f{\p \eta}{\p t} + u \f{\p \eta}{\p x} + v \f{\p \eta}{\p y} \qquad z=\eta 
\] Integrating the continuity equation and using Leibniz's rule
\begin{eqnarray*} 0&=&  \int^\eta_{-h} \lp \f{\p u}{\p x} + \f{\p v}{\p y} + \f{\p
w}{\p z} \rp dz \\ & = & \big[w\big]^\eta _{-h} + \f{\p}{\p x} \int^\eta_{-h} udz +
\f{\p}{\p y} \int^\eta_{-h} vdz \\ & & + \f{\p \eta}{\p x} u(\eta) + \f{\p \eta}{\p
y} v(\eta)  -\f{\p h}{\p x} u(-h) - \f{\p h}{\p y} v(-h) .
\end{eqnarray*} Define the total depth to be $H=\eta+h$
\[ 0= \lp w - u \f{\p \eta}{\p x} - v \f{\p \eta}{\p y} \rp_\eta  -\lp w + u \f{\p
h}{\p x} + v \f{\p h}{\p x} \rp_{-h}  + \f{\p Hu}{\p x}  + \f{\p(Hv)}{\p y} \] Using
the boundary conditions we get 
\be  \f{\p H}{\p t} + \f{\p Hu}{\p x} + \f{\p Hv}{\p y} = 0 .\label{mass}\ee


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