Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 333 / 3

For an inviscid Boussinesq approximation fluid in a nonrotating frame, write buoyancy as $\sigma$ and kinematic pressure as $p$. The governing equations areThey require density variations to be small compared with a constant reference density, while retaining those variations in buoyancy; the flow scale must be small compared with the background density scale height. Ideal flow additionally neglects viscosity and scalar diffusion.

Linearize aboutwhere $N$ is the buoyancy frequency, and defineFor two-dimensional disturbances, the linear equations areThe condition $N^2>0$ expresses stable density stratification.

Differentiate horizontal momentum with respect to $z$ and vertical momentum with respect to $x$. When the two equations are subtracted, the terms proportional to ${\overline{u}}_z(u_x^{\prime }+w_z^{\prime })$ vanish by incompressible flow. The pressure derivatives also cancel, leaving

Differentiate this equation with respect to $x$. Since continuity givesone obtainsApply $D_t$ and use the buoyancy equation $D_t{\sigma }^{\prime }=-N^2{w}^{\prime }$. Multiplication by $-1$ then gives

For the normal modethe linear material derivative becomesSubstitution into the equation from part b and division by $-k^2(\overline{u}-c{)}^2$ gives the Taylor–Goldstein equationwhereand the Scorer parameter is

For a ridge-fixed disturbance, $c=0$. If the Scorer parameter varies on a height scale much longer than the vertical wavelength, the local WKB approximation giveswhere $m^2=l^2-k^2>0$. The disturbance is vertically oscillatory there and can carry wave activity upward. Where $l^2<k^2$, $m$ is imaginary and the solution is vertically evanescent.

A rigid ridge supplies the lower boundary condition and an upper radiation or decay condition selects the physical solution. If $l^2$ decreases through $k^2$, the crossing is a turning level: the wave is reflected or decays above it and is trapped beneath it. Increasing $\overline{u}$ normally reduces $N^2/{\overline{u}}^2$, while decreasing $N^2$ reduces it directly; subject to the curvature term $-{\overline{u}}_{zz}/\overline{u}$, either change therefore promotes vertical trapping of an atmospheric gravity wave.

Multiplyby $\hat{w}$ and integrate from the rigid bottom to an upper endpoint $z$ at which $\hat{w}{\hat{w}}_z=0$. The bottom term also vanishes because $\hat{w}(0)=0$. Integration by parts givesTherefore the horizontal wavenumber has the Rayleigh quotientFor a trapped wave the upper endpoint may be taken to infinity because the eigenfunction decays.

LetBecause the Rayleigh quotient is stationary with respect to first-order changes of its eigenfunction, only the explicit dependence of $l^2$ on $c=\omega /k$ contributes when the quotient is differentiated. ThuswhereIt follows thatSince $\omega =kc$, the horizontal group velocity is

For a stationary ridge wave, $c=0$. Assume $\overline{u}>0$, so downstream is the positive $x$ direction. The stated inequality isThe denominator in the group-velocity formula is then positive, andThe atmospheric internal gravity waves generated by the ridge consequently carry their wave packet downstream.