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

Substitute the stated scales and divide the momentum equation by $2\Omega V$. The ratio of inertial to Coriolis acceleration is the Rossby numberwhile the dimensionless buoyancy coefficient isThusThe dimensionless $B$ is the Burgers number for this rotating stratified flow.

Forboth material derivatives vanish because the fields are independent of $x$. Since $\hat{\mathbf{z}}\times \mathbf{U}=z\hat{\mathbf{y}}$, steady momentum balance requiresThese derivatives are compatible and integrate toThe cross-stream temperature gradient and vertical stratification are in thermal-wind balance with the vertical shear.

Let $D={\partial }_t+z{\partial }_x$. Retaining terms linear in the primed fields givesThe terms ${\mathrm{Ro}}^2{w}^{\prime }$ and $-v^{\prime }/B$ respectively arise from perturbation advection of the velocity and temperature gradients in the basic state.

Differentiate the $y$-momentum equation with respect to $x$, subtract the $y$ derivative of the $x$-momentum equation, and use incompressibility. With vertical perturbation vorticityone obtainsTherefore

At leading order as $\mathrm{Ro}\to 0$, horizontal geostrophic balance and vertical hydrostatic balance giveThe temperature equation givesMeanwhile ${\zeta }^{\prime }=p_{xx}^{\prime }+p_{yy}^{\prime }$. The leading vertical-vorticity equation is $D{\zeta }^{\prime }=w_z^{\prime }$. Since derivatives in $z$ do not commute with $D$,The $p_{xz}^{\prime }$ terms therefore cancel, leaving the conserved three-dimensional quasi-geostrophic potential vorticityNo normal flow at $y=0,1$ means $v^{\prime }=p_x^{\prime }=0$. At $z=0,1$, $w^{\prime }=0$, so

InsertThe side-wall condition is automatic, and the potential-vorticity equation givesDefineThenThe top and bottom conditions becomeSetting the determinant of these two homogeneous equations for $A,C$ to zero and simplifying gives

For every $\lambda >0$, $\tanh \lambda <\lambda$, soThe radicand in part f is therefore negative precisely when its second factor is negative. The two wave speeds are then complex conjugates, one with positive imaginary part and exponential growth. Hence