Let $D=\partial_t+z\partial_x$. Retaining terms linear in the primed fields gives
$$
\boxed{
\operatorname{Ro}(Du'+\operatorname{Ro}w')-v'=-p'_x},
$$
$$
\boxed{\operatorname{Ro}Dv'+u'=-p'_y},
$$
$$
\boxed{\operatorname{Ro}^2Dw'=-p'_z+B\theta'},
$$
$$
\boxed{D\theta'+w'-\frac1Bv'=0},
\qquad
\boxed{u'_x+v'_y+\operatorname{Ro}w'_z=0}.
$$
The terms $\operatorname{Ro}^2w'$ and $-v'/B$ respectively arise from perturbation advection of the velocity and temperature gradients in the basic state.


Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 331 / 3 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-331/3/c/solution