Internal Waves in a Continuously Stratified Fluid
    For simplicity, assume
  • the wave motion is effectively inviscid because the velocity gradients are small and the Reynolds number is large
  • the wave amplitudes are small enough so that the nonlinear advection terms can be neglected
  • the frequency of wave motion is much larger than the Coriolis frequency so it does not affect the wave motion

The Boussinesq set then simplifies to the equations below, \( \rho_0 \) is a constant reference density \[ \underbrace{ \frac{D\rho}{Dt} = 0 }_{\text{constancy of fluid particle density}} \tag{I} \] \[ \underbrace{ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 }_{\text{incompressibility}} \tag{II} \] \[ \frac{\partial u}{\partial t} = -\frac{1}{\rho_0} \frac{\partial p'}{\partial x} \tag{III} \] \[ \frac{\partial v}{\partial t} = -\frac{1}{\rho_0} \frac{\partial p'}{\partial y} \tag{IV} \] \[ \frac{\partial w}{\partial t} = -\frac{1}{\rho_0} \frac{\partial p'}{\partial z} - \frac{\rho g}{\rho_0} \tag{V} \]

Five equations contain five unknowns (\( u, v, w, p, \rho \))

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