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18.1: G.1 The Navier-Stokes equation for nearly-uniform density

  • Page ID
    18116
  • Let us look again at Equation 6.3.37, this time in the form

    \[\rho \frac{D \vec{u}}{D t}=-\rho g \hat{e}^{(z)}-\vec{\nabla} p+\mu \nabla^{2} \vec{u}+\mu \vec{\nabla}(\vec{\nabla} \cdot \vec{u}).\label{eqn:1}\]

    Here we are working in gravity-aligned coordinates, so that the gravity vector \(\vec{g} = -g\hat{e}^{(z)}\).1 The left-hand side is nonlinear in two respects. First, the material derivative contains a quadratic combination of unknown fields, \([\vec{u}\cdot \vec{\nabla}]\vec{u}\). In addition, the factor \(\rho\) multiplying the material derivative adds another layer of nonlinearity. In geophysical fluids, density variations are often small enough that this additional nonlinearity can be removed.

    To begin, we write the density \(\rho\) as the sum of a uniform “background” value \(\rho_0\) and a fluctuating part \(\rho^\prime\):

    \[\rho=\rho_{0}+\rho^{\prime}.\label{eqn:2}\]

    In addition, we decompose the pressure as

    \[p=p_{0}+p^{*},\]

    where the “background” part is in hydrostatic balance with the background density:

    \[\vec{\nabla} p_{0}=-\rho_{0} g \hat{e}^{(z)},\]

    so that

    \[\vec{\nabla} p=-\rho_{0} g \hat{e}^{(z)}+\vec{\nabla} p^{*}.\label{eqn:3}\]

    With the substitution of Equation \(\ref{eqn:2}\) and Equations \(\ref{eqn:3}\), \(\ref{eqn:1}\) becomes

    \[\left(\rho_{0}+\rho^{\prime}\right) \frac{D \vec{u}}{D t}=-\rho^{\prime} g \hat{e}^{(z)}-\vec{\nabla} p^{*}+\mu \nabla^{2} \vec{u}+\mu \vec{\nabla}(\vec{\nabla} \cdot \vec{u}),\label{eqn:4}\]

    Now assume that \(|\rho^\prime| \ll \rho_0\), and neglect \(\rho^\prime\) in favor of \(\rho_0\) on the left-hand side. Dividing through by the constant \(\rho_0\), we then have

    \[\frac{D \vec{u}}{D t}=b \hat{e}^{(z)}-\vec{\nabla} \frac{p^{*}}{\rho_{0}}+v \nabla^{2} \vec{u}+v \vec{\nabla}(\vec{\nabla} \cdot \vec{u}).\label{eqn:5}\]

    Here,

    \[b=-\frac{\rho^{\prime}}{\rho_{0}} g\]

    is the buoyancy and

    \[v=\frac{\mu}{\rho_{0}}\]

    is the kinematic viscosity. By replacing \(\rho\) with the constant \(\rho_0\) everywhere except in the buoyancy, we simplify the solution of the momentum equation considerably. Equation \(\ref{eqn:5}\) is called the Boussinesq approximation of the momentum equation.

    Note that we have yet to define the background density \(\rho_0\). The only guidance we have for this choice is that the accuracy of the approximation depends on fluctuations about the background density being small. It therefore makes sense to define \(\rho_0\) so as to minimize those fluctuations, e.g., by using the volume average.

    1This restriction is not necessary; it just simplifies the discussion.