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A buoyancy perturbation is the departure $b^{\prime }$ from a prescribed basic buoyancy field. In a fluid with constant buoyancy frequency $N$, vertical velocity $w^{\prime }$ changes it according to $({\partial }_t+\mathbf{U}\cdot \nabla )b^{\prime }=-N^2{w}^{\prime }$.

A uniform-density perfect-gas sphere supported by decreasing pressure has buoyancy frequency squared $N^2=-2{\omega }_d^2{r}^2/[\gamma (R^2-r^2)]$ in the interior. Its specific entropy decreases outward and adiabatic displacements are buoyantly unstable. For the nonradial polynomial family, the product of the two normalized squared mode frequencies is $-l(l+1)$, displaying one growing branch even when radial compression modes are stable.

A stable stratification gives a restoring buoyancy force and positive squared buoyancy frequency for an adiabatically displaced parcel. An adverse specific entropy gradient can instead drive convection; other forces such as rotation can modify the full stability criterion.

For $S<0$ and constant positive buoyancy frequency, the critical height is $U_0/|S|$. A stationary upward ray approaches it with vertical group velocity proportional to $U^2$, requiring infinite ideal ray travel time. Large gradient Richardson number can preserve the short-wavelength WKB approximation, but parcel-displacement amplitude grows and eventually invalidates linearity.

A critical level is a height at which the intrinsic frequency of an internal gravity wave vanishes. For finite positive buoyancy frequency, its local vertical wavenumber diverges as $\hat{\omega }\to 0$. The inviscid Taylor–Goldstein equation is singular there; viscosity, diffusion or nonlinear wave breaking can regularize the resulting small scales.

A smooth zero-interior-potential vorticity mode hasDecay for $z\to \infty$ gives $\hat{\psi }=C{e}^{-mz}$. Its penetration depth $1/m$ increases with horizontal wavelength and decreases with buoyancy frequency. Singular neutral interior sheets are a distinct continuous-spectrum class.

For a Boussinesq approximation, constant buoyancy frequency $N$, and buoyancy heating rate $R$, the quasi-geostrophic potential-vorticity equation becomes $D_{g}q/Dt=(f_0/N^2)R_z$, where $q={\nabla }_h^2\psi +(f_0^2/N^2){\psi }_{zz}+\beta y$. Heating changes the vertical gradient of buoyancy and therefore produces a potential-vorticity tendency proportional to its height derivative.

For a semi-infinite fluid above a rigid horizontal boundary, constant buoyancy frequency $N$, and thermal wind $U=\Lambda z$, a zero-interior-potential vorticity perturbation has vertical dependence $e^{-\mu z}$, where $\mu =NK/|f_0|$ and $K=(k^2+l^2{)}^{1/2}$. Material conservation of boundary buoyancy gives $c{\varphi }^{\prime }(0)+\Lambda \varphi (0)=0$. Thus its phase velocity relative to the boundary flow is $c=\Lambda /\mu$. The wave's time dependence resides in its boundary condition; the interior field follows by potential-vorticity inversion. MIT's Eady-edge-wave notes develop this boundary-wave interpretation.

For constant Coriolis parameter $f_0$ and buoyancy frequency $N$, thermal-wind balance and transformed mean-flow equations givewhere ${\chi }^{\ast }$ is the residual-circulation streamfunction and $\mathbf{F}$ is the Eliassen–Palm flux.

For constant shear $U=\Lambda z$ between impermeable horizontal walls at $0,D$, with constant buoyancy frequency and no interior potential-vorticity gradient, define $m=NK/|f_0|$. The boundary conditions yieldThe bracket is negative for $0<mD<2.399\dots$, giving baroclinic instability for nonzero zonal wavenumber. For large separation the boundary waves decouple; the lower-wave speed tends to $\Lambda /m$, matching the dispersion relation of a semi-infinite Eady edge wave.

For a stratified parallel shear flow, the gradient Richardson number compares the squared buoyancy frequency with squared vertical shear. The Miles–Howard theorem excludes exponentially growing inviscid normal modes when this ratio is at least $1/4$ everywhere. Where $U^{\prime }=0$, the equivalent criterion is written directly as $N^2\ge (U^{\prime }{)}^2/4$.

The Hazel model is a smooth stratified shear flow whose base velocity has a hyperbolic-tangent profile and whose squared buoyancy frequency has a hyperbolic-secant-squared profile. The equal-width nondimensional example is $U=\tanh z$ and $N^2=J{\mathrm{sech}}^{2}z$. For $J\ge 0$, its gradient Richardson number is $J{\cosh }^{2}z$, with minimum $J$.

At a step in buoyancy frequency without a jump in background mass density, vertical velocity and fluid pressure are continuous. For a fixed horizontal wavenumber and angular frequency, these are continuity of $W$ and $W^{\prime }$. They determine reflection and transmission between propagating internal gravity waves and evanescent waves.

The linearized Boussinesq equations describe small velocity, pressure, and buoyancy perturbations about a motionless, stably stratified reference state. With kinematic viscosity $\nu$, mass diffusivity $\kappa$, and buoyancy frequency $N$, they are