Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2022 / Section B / 4

The buoyancy frequency, also called the Brunt–Väisälä frequency, is the angular frequency of a small vertical displacement restored by buoyancy in a stably stratified fluid. Take $z$ upward, gravitational acceleration $-g\hat{\mathbf{z}}$, and a background in hydrostatic equilibrium,Displace a small gas element upward by $\xi$, sufficiently slowly for its pressure to match its surroundings but without heat exchange. For this adiabatic process, $p/{\rho }^{\gamma }$ is constant along the element's motion, so its mass density changes by $\rho {\gamma }^{-1}(d\ln p/dz)\xi$. The surrounding mass density changes by $(d\rho /dz)\xi$. The element's excess mass density is consequentlyIts acceleration is $\ddot{\xi }=-g\Delta \rho /\rho =-N^2\xi$. ThusFor a gas of fixed composition obeying the ideal gas law, $\rho \propto p/T$. The adiabatic buoyancy frequency of an ideal gas is thereforewhere $c_p=\gamma \mathcal{R}/(\gamma -1)$ is the specific heat capacity at constant pressure, and $\mathcal{R}=p/(\rho T)$ is the specific gas constant. Here $N^2$ has units ${\mathrm{s}}^{-2}$; $N$ is an angular frequency, not a cyclic frequency. Positive $N^2$ gives simple harmonic motion, negative $N^2$ gives convective instability, and $N^2=0$ is marginal. The temperature form assumes fixed composition; the preceding density form is more general.

Write the background mass density as $\bar{\rho }(z)={\rho }_0+{\rho }_0^{\prime }z$ and its hydrostatic pressure as $\bar{p}(z)$, whereApply an Eulerian perturbation of a fluid variable to the background fields:Thus $\pi$ is the Eulerian pressure perturbation, in pascals; it is not the background hydrostatic pressure.

For incompressible flow, mass conservation becomes $D{\rho }_{\mathrm{tot}}/Dt=0$. Linearizing about rest givesThe Euler equations are ${\rho }_{\mathrm{tot}}D\mathbf{u}/Dt=-\nabla p_{\mathrm{tot}}-{\rho }_{\mathrm{tot}}g\hat{\mathbf{z}}$. The background pressure gradient and weight cancel. Dropping products of perturbations givesUnder $|{\rho }_0^{\prime }z|\ll {\rho }_0$, replace $\bar{\rho }$ by ${\rho }_0$ in inertia but retain the background mass density gradient in the displacement-induced buoyancy. This is the Boussinesq approximation. Its linear Boussinesq equations here areThe absolute value makes explicit that the small-density-variation condition concerns magnitude, including the usual negative ${\rho }_0^{\prime }$ of stable density stratification.

Use plane wave perturbations proportional to $e^{i(k_{x}x+k_{z}z-\omega t)}$, with amplitudes $\hat{u},\hat{w},\hat{\pi },\hat{\varrho }$. The linear equations becomeTo eliminate pressure without dividing by $k_x$, multiply the vertical momentum equation by $k_x$ and subtract $k_z$ times the horizontal momentum equation:Use incompressible flow and the equation for mass density to obtain ${\omega }^2{\rho }_0(k_x^2+k_z^2)\hat{w}=-g{\rho }_0^{\prime }{k}_x^2\hat{w}$. Hence, for a nontrivial internal gravity wave,For stable density stratification, ${\rho }_0^{\prime }<0$ and $N=\sqrt{-g{\rho }_0^{\prime }/{\rho }_0}$ is the buoyancy frequency. Then $0\le {\omega }^2\le N^2$ and the normal modes oscillate rather than grow. For ${\rho }_0^{\prime }>0$, a mode with $k_x\ne 0$ has ${\omega }^2<0$ and grows exponentially: denser fluid lies above lighter fluid. Uniform density, ${\rho }_0^{\prime }=0$, is marginal for this restoring mechanism.

The case $k_x=0$, $k_z\ne 0$, has zero frequency in the displayed relation. Incompressible flow then requires $\hat{w}=0$, so it is a degenerate non-propagating case rather than a finite-frequency vertical oscillation. These equations concern the incompressible model of part (b); its $N^2$ does not contain the adiabatic compression correction needed in part (a).

For $N^2>0$, write $K=\sqrt{k_x^2+k_z^2}$ and choose either signed branch $\omega =sN{k}_x/K$, with $s=\pm 1$. These two branches cover the roots of the squared dispersion relation. The group velocity of an internal gravity wave isIf instead the positive-frequency root is chosen for both signs of $k_x$, set $s=+1$ and multiply the displayed components by $\mathrm{sgn}(k_x)$, away from the cusp $k_x=0$.

Taking the dot product with the wave vector givesThus the group velocity, which describes the travel of a narrow wave packet and its energy, is perpendicular to the wave vector, and hence perpendicular to the phase velocity ${\mathbf{c}}_p=\omega \mathbf{k}/K^2$. This is the internal-wave phase and group velocity geometry. It also follows from phase-group orthogonality for degree-zero dispersion: the frequency depends only on the direction of $\mathbf{k}$, not its magnitude. For $k_z=0$, the group velocity is zero, consistent with the stationary extremum $|\omega |=N$.