Differentiate the $y$-momentum equation with respect to $x$, subtract the $y$ derivative of the $x$-momentum equation, and use incompressibility. With vertical perturbation vorticity
$$
\zeta'=v'_x-u'_y,
$$
one obtains
$$
\operatorname{Ro}D\zeta'
-\operatorname{Ro}w'_z-\operatorname{Ro}^2w'_y=0.
$$
Therefore
$$
\boxed{
\left(\frac\partial{\partial t}+z\frac\partial{\partial x}\right)
\left(\frac{\partial v'}{\partial x}
-\frac{\partial u'}{\partial y}\right)
=\frac{\partial w'}{\partial z}
+\operatorname{Ro}\frac{\partial w'}{\partial y}}.
$$


Solution

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

 Solution

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