\overline{v}_{a}^{*} = \overline{v}_{a} - \frac{1}{N ^{2}} \frac{\partial v ^{'} σ ^{'}}{\partial z}, \overline{w}_{a}^{*} = \overline{w}_{a} + \frac{1}{N ^{2}} \frac{\partial v ^{'} σ ^{'}}{\partial y} .

The added terms form a nondivergent eddy-induced circulation, so

\partial_{y} \overline{v}_{a}^{*} + \partial_{z} \overline{w}_{a}^{*} = 0

\overline{σ}_{t} + N^{2} \overline{w}_{a}^{*} = 0.

Substituting

\overline{v}_{a} = \overline{v}_{a}^{*} + N^{- 2} \partial_{z} \overline{v^{'} σ^{'}}

into the momentum equation gives

\overline{u}_{t} - β y \overline{v}_{a}^{*} = \frac{\partial F ^{(y)}}{\partial y} + \frac{\partial F ^{(z)}}{\partial z},

where the components of the Eliassen–Palm flux are

F^{(y)} = - \overline{u^{'} v^{'}}, F^{(z)} = - \overline{u^{'} w^{'}} + \frac{β y}{N ^{2}} \overline{v^{'} σ^{'}} .

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