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18.2: G.2 Alternative derivation

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  • Another way to derive Equation 18.1.7 is by rearranging Equation 18.1.6 as

    \[\rho_{0} \frac{D \vec{u}}{D t}=\rho^{\prime}\left(\vec{g}-\frac{D \vec{u}}{D t}\right)-\vec{\nabla} p^{*}+\mu \nabla^{2} \vec{u}+\mu \vec{\nabla}(\vec{\nabla} \cdot \vec{u}).\]

    Now consider the first two terms in parentheses. If we assume that all accelerations are small compared with gravity, then the second term, \(D\vec{u}/Dt\), can be discarded. We then divide through by \(\rho_0\) to obtain the Boussinesq equation Equation 18.1.7 as before. Note that the smallness of accelerations compared with gravity is the same assumption that justifies neglecting the inertial terms in the baroclinic torque (section 7.4).