Shallow-Water Equations
The condition of incompressibility \[ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \]
\(x\)-momentum: \( \underbrace{\frac{\partial u}{\partial t}}_{\displaystyle \frac{U}{T}} + \underbrace{u \frac{\partial u}{\partial x}}_{\displaystyle \frac{U^2}{L}} + \underbrace{v \frac{\partial u}{\partial y}}_{\displaystyle \frac{U^2}{L}} + \underbrace{w \frac{\partial u}{\partial z}}_{\displaystyle \frac{UW}{D}} - \underbrace{fv}_{\displaystyle fU} = -\underbrace{\frac{1}{\rho} \frac{\partial \tilde{p}}{\partial x}}_{\displaystyle \frac{P}{\rho L}} \)
\(y\)-momentum: \( \underbrace{\frac{\partial v}{\partial t}}_{\displaystyle \frac{U}{T}} + \underbrace{u \frac{\partial v}{\partial x}}_{\displaystyle \frac{U^2}{L}} + \underbrace{v \frac{\partial v}{\partial y}}_{\displaystyle \frac{U^2}{L}} + \underbrace{w \frac{\partial v}{\partial z}}_{\displaystyle \frac{UW}{D}} + \underbrace{fu}_{\displaystyle fU} = -\underbrace{\frac{1}{\rho} \frac{\partial \tilde{p}}{\partial y}}_{\displaystyle \frac{P}{\rho L}} \)
\(z\)-momentum: \( \underbrace{\frac{\partial w}{\partial t}}_{\displaystyle \frac{W}{T}} + \underbrace{u \frac{\partial w}{\partial x}}_{\displaystyle \frac{UW}{L}} + \underbrace{v \frac{\partial w}{\partial y}}_{\displaystyle \frac{UW}{L}} + \underbrace{w \frac{\partial w}{\partial z}}_{\displaystyle \frac{W^2}{D}} = -\underbrace{\frac{1}{\rho} \frac{\partial \tilde{p}}{\partial z}}_{\displaystyle \frac{P}{\rho D}} \)

The linearized continuity and momentum equations are then:

\[ \begin{align*} &\frac{\partial u}{\partial t} + \cancel{u \frac{\partial u}{\partial x}} + \cancel{v \frac{\partial u}{\partial y}} + \cancel{w \frac{\partial u}{\partial z}} - fv = -g \frac{\partial \eta}{\partial x} \\[1.5ex] &\frac{\partial v}{\partial t} + \cancel{u \frac{\partial v}{\partial x}} + \cancel{v \frac{\partial v}{\partial y}} + \cancel{w \frac{\partial v}{\partial z}} + fu = -g \frac{\partial \eta}{\partial y} \\[1.5ex] &\overbrace{\nabla \cdot \mathbf{u} + \frac{\partial w}{\partial z} = 0}^{\text{Continuity}} \quad \underset{\text{Integrate from } z=0 \text{ to } z=\eta}{\Rightarrow} \quad \int_0^\eta \left( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right) dz = -\left. w \right|_0^\eta = \frac{\partial \eta}{\partial t} \\ &\underset{\text{Assume } u,v \text{ uniform over depth } H}{\Rightarrow} \quad \boxed{ \frac{\partial \eta}{\partial t} + H \left( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right) = 0 } \end{align*} \]

These are shallow-water equations govern the motion of a layer of fluid in which the horizontal scale is much larger than the depth of the layer. These equations are used for studying various types of gravity waves.

\( \frac{\partial p}{\partial z} = -\rho g \quad \underset{\text{integrate from } z \text{ to } \eta}{\Longrightarrow} \quad p(x,y,z,t) = \rho g \underbrace{[\eta(x,y,t) - z]}_{\text{height above point}} \)

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