Define the <transformed Eulerian mean> circulation by
$$
\boxed{
\overline v_a^*=\overline v_a-\frac1{N^2}
\frac{\partial\overline{v'\sigma'}}{\partial z},
\qquad
\overline w_a^*=\overline w_a+\frac1{N^2}
\frac{\partial\overline{v'\sigma'}}{\partial y}.}
$$
The added terms form a nondivergent eddy-induced circulation, so $\partial_y\overline v_a^*+\partial_z\overline w_a^*=0$. The buoyancy equation becomes
$$
\overline\sigma_t+N^2\overline w_a^*=0.
$$
Substituting $\overline v_a=\overline v_a^*+N^{-2}\partial_z\overline{v'\sigma'}$ into the momentum equation gives
$$
\overline u_t-\beta y\overline v_a^*
=\frac{\partial F^{(y)}}{\partial y}
+\frac{\partial F^{(z)}}{\partial z},
$$
where the components of the <Eliassen–Palm flux> are
$$
\boxed{
F^{(y)}=-\overline{u'v'},
\qquad
F^{(z)}=-\overline{u'w'}
+\frac{\beta y}{N^2}\overline{v'\sigma'}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 333 / 4 / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-333/4/i/solution