(\overline{v}_{a}^{*}, \overline{w}_{a}^{*}) = (\overline{χ}_{a, z}^{*}, - \overline{χ}_{a, y}^{*}) .

This satisfies residual mass conservation identically. Let

G = \nabla \cdot \overline{F}, N^{2} = - \frac{g}{ρ _{0}} \frac{d ρ _{s}}{d z} .

\overline{u}_{t} - f_{0} \overline{χ}_{a, z}^{*} = G .

f_{0} \overline{u}_{z} = \frac{g}{ρ _{0}} \overline{ρ}_{y} .

After a time derivative, the transformed density equation gives

f_{0} \overline{u}_{t z} = \frac{g}{ρ _{0}} \overline{ρ}_{t y} = - N^{2} \overline{χ}_{a, yy}^{*} .

On the other hand, the

z

\overline{u}_{t z} - f_{0} \overline{χ}_{a, zz}^{*} = G_{z} .

Eliminating

\overline{u}_{t z}

f_{0}^{2} \overline{χ}_{a, zz}^{*} + N^{2} \overline{χ}_{a, yy}^{*} = - f_{0} (\nabla \cdot \overline{F})_{z} .

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