Gravity wave

A gravity wave is a fluid disturbance restored by gravity or buoyancy. A surface gravity wave displaces a free surface, while an internal gravity wave displaces density surfaces within a stratified fluid. Rotation can additionally contribute to restoration, giving an inertia-gravity wave.

In a uniformly stratified fluid with buoyancy frequency $N$, a plane wave of horizontal and vertical wavenumbers $k,m$ obeys

A stratified layer between unstratified regions can trap internal gravity waves, because the exterior disturbance is an evanescent wave. Its discrete normal modes obey Robin boundary conditions representing the exterior layers. Periodic boundary forcing at a trapped normal mode frequency produces resonance in the ideal undamped model.

A moving boundary excites an internal gravity wave through its kinematic boundary condition. For wavenumber $k$ and angular frequency $\omega$, the vertical velocity field satisfies $W^{\prime \prime }+k^2(N^2/{\omega }^2-1)W=0$. The radiation condition selects outward group velocity; when $\omega >N$, the bounded response is an evanescent wave.

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.

An interfacial gravity wave is a displacement of an interface between two fluids of different mass density, restored by buoyancy when the denser fluid is below. For two deep layers in the Boussinesq approximation, density difference $\delta \rho$ and horizontal wavenumber $k$ give intrinsic squared phase velocity $g\delta \rho /(2{\rho }_{0}k)$. Background flow shifts the laboratory phase velocity by its local speed.

For two inviscid layers of equal depth $h$ between rigid lids, with lower density ${\rho }_2$ and upper density ${\rho }_1$, a horizontal mode of wavenumber $k$ satisfiesThe sign reverses when the heavier fluid is above, producing the Rayleigh-Taylor instability.

An atmospheric internal gravity wave is an internal gravity wave propagating through a stably stratified atmosphere. Flow across mountains can generate nearly stationary waves whose vertical propagation is controlled by the wind and buoyancy-frequency profiles.

The intrinsic frequency is the frequency observed in a frame moving with the basic flow. For a wave of frequency $\omega$ and horizontal wavenumber $k$ in a uniform current $U$, it is $\hat{\omega }=\omega -kU$.

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 radiation condition selects the wave solution whose energy propagates away from its source. For an internal gravity wave, the vertical phase and group velocities have opposite signs, so upward radiation fixes the sign of the vertical wavenumber.

The wave momentum flux ${\rho }_0\overline{u^{\prime }{w}^{\prime }}$ is the vertical transport of horizontal momentum by correlated velocity perturbations. Its vertical convergence exerts the mean-flow force $-{\rho }_0{\partial }_z\overline{u^{\prime }{w}^{\prime }}$.

The Taylor–Goldstein equation governs linear two-dimensional disturbances of an inviscid, vertically sheared, stably stratified parallel flow. For horizontal phase velocity $c$, base velocity $U(z)$, buoyancy frequency $N(z)$, and horizontal wavenumber $k$,

For a mode with nonreal phase velocity $c$, let $V=U-c$, choose a continuous branch of $V^a$, and put $\hat{w}=V^{a}q$. Under impermeable boundary conditions, multiplying the transformed Taylor–Goldstein equation by $V^{2a}\overline{q}$ and applying integration by parts givesChoosing $a=1/2$ gives the Miles–Howard theorem; choosing $a=1$ makes the real phase velocity of an unstable mode a weighted mean of $U$.

A smooth inviscid stratified parallel flow with gradient Richardson number at least $1/4$ everywhere has no exponentially growing two-dimensional normal modes. Put $a=1/2$ in the power-transformed Taylor–Goldstein energy identity and take its imaginary part:The integral is positive for a nonzero mode when the numerator is nonnegative, forcing $c_i=0$. This is a modal stability theorem, not a prohibition on transient growth. Maslowe's review discusses the theorem and the role of critical layers.

For an unstable Taylor–Goldstein equation mode in a finite channel, take $a=1$ in the power-transformed Taylor–Goldstein energy identity. Its imaginary part givesThus the real phase velocity lies strictly between the extremes of a nonconstant smooth shear profile. Equality would force the regular eigenfunction to vanish on an interval, hence everywhere by uniqueness for its ordinary differential equation.

At a material interface without surface tension, continuity of displacement and of the pressure evaluated on the displaced interface givesThe first follows from the kinematic boundary condition $ik(U-c)\hat{\eta }=\hat{w}$. The second uses $\hat{p}={\rho }_0[(U-c){\hat{w}}^{\prime }-U^{\prime }\hat{w}]/(ik)$ and hydrostatic displacement $[\hat{p}-g\rho \hat{\eta }]=0$. These conditions apply to density, velocity and vorticity jumps within the Boussinesq approximation.

Across a horizontal material interface with continuous background mass density but a velocity jump, an internal gravity wave preserves laboratory frequency and horizontal wavenumber. The matching conditions are continuity of displacement $w/\hat{\omega }$ and pressure, not continuity of $w$. For velocity amplitudes and upward phase-line angles ${\theta }_1,{\theta }_2$, define $r={\hat{\omega }}_2/{\hat{\omega }}_1$. Then the incident-to-transmitted amplitude ratio isThis follows by adding the displacement and pressure matching equations after eliminating the reflected amplitude. It assumes nonzero intrinsic frequencies and propagating outgoing branches.

Two equal stable density jumps of size $\Delta \rho /2$ at $z=\pm h/2$ lie in the global linear shear flow $U=\Delta Uz/h$. Decaying normal modes haveThe jump conditions for stratified inviscid shear flow give the determinant equationWriting $\alpha =kh/2$, $J=g\Delta \rho h/({\rho }_0\Delta U^2)$ and $\tilde{c}=2c/\Delta U$, this isThe two roots for ${\tilde{c}}^2$ are real; one is negative exactly when $2\alpha /(1+e^{-2\alpha })<J<2\alpha /(1-e^{-2\alpha })$. At large $\alpha$, this narrow band centres on $J=2\alpha$, where the isolated counterpropagating interfacial gravity waves have the same zero laboratory speed. This realizes counterpropagating wave instability.

The Scorer parameter is the height-dependent coefficient $l^2$ in the stationary atmospheric-wave equation ${\hat{w}}^{\prime \prime }+(l^2-k^2)\hat{w}=0$. Regions with $l^2>k^2$ support vertically oscillatory disturbances, whereas $l^2<k^2$ makes them vertically evanescent.

For a stationary internal gravity wave, the Taylor–Goldstein equation gives $m^2=N^2/U^2-U^{\prime \prime }/U-k^2$. In a propagating interval with slowly varying $m$, the WKB approximation has amplitude proportional to $m^{-1/2}$ and phase derivative $m$. The conditions $|m^{\prime }|\ll m^2$ and $|m^{\prime \prime }|\ll |m{|}^3$ fail at a regular turning level. With $U>0$, the upward branch has $m>0$ and negative intrinsic frequency; laboratory group velocity is $U(k^2,km)/(k^2+m^2)$ when the curvature term is negligible.

An atmospheric internal gravity wave is vertically trapped when its squared vertical wavenumber changes from positive to negative with altitude. A decrease of the Scorer parameter, caused for example by increasing wind speed or decreasing buoyancy frequency, creates a turning level and an evanescent upper region.

Density stratification is variation of a fluid's mass density with height. Its stable or unstable character depends on whether the buoyancy force restores or amplifies a displaced parcel.

A fluid has stable density stratification when a small vertical displacement produces a restoring buoyancy force. In a gravitational field this normally means that mass density increases downward.

A fluid has unstable density stratification when a vertical displacement amplifies itself. Denser fluid above lighter fluid can overturn by Rayleigh-Taylor instability or develop thermal convection when the density difference is thermal.

The buoyancy frequency is the natural angular frequency of small vertical oscillations in a stably stratified fluid.

A Richardson number compares gravitational stratification with inertial or shear effects. Its precise form depends on the characteristic scales or local gradients being compared.

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 $N^2\ge (U^{\prime }{)}^2/4$.

In a spherically symmetric stellar model with inward gravity magnitude $g$, the squared buoyancy frequency isPositive $N^2$ gives stable stratification. Regular central profiles have $g=O(r)$ and logarithmic pressure and density gradients of order $r$, so $N^2=O(r^2)$ near the center.

For a uniformly stratified fluid, internal-wave phase velocity is parallel to the wavevector while group velocity is perpendicular to it. Energy from a localized monochromatic source propagates along four beams forming a St Andrew's cross.

For a propagating, non-cutoff two-dimensional internal gravity wave proportional to $e^{i(kx+mz-\omega t)}$, incompressibility gives $U=-mW/k$. Thus the oscillatory velocity field is perpendicular to the wavevector and parallel to the group velocity, whereas the phase velocity is parallel to the wavevector.

For a two-dimensional internal gravity wave with phase $kx+\int m(z)dz-\omega t$, a constant-phase line at fixed time has slope $dz/dx=-k/m(z)$. Its normal spacing from the next crest is $2\pi /(k^2+m^2{)}^{1/2}$; the wavevector $(k,m)$ is normal to the line. The intrinsic energy ray is tangent to these phase lines, while mean-flow advection changes the laboratory ray direction.

At a stationary slope, an internal wave preserves frequency and tangential wavenumber. Unlike specular reflection, the reflected energy ray preserves its angle to the vertical; a supercritical slope can reverse its horizontal propagation direction.

Let an internal gravity wave energy ray make angle $\theta$ with the horizontal, so that $\sin \theta =\omega /N$. A boundary of local slope magnitude $s$ is subcritical, critical, or supercritical according as $s$ is smaller than, equal to, or larger than $\tan \theta$. At criticality the inviscid reflected wavelength tends to zero.

In subcritical internal-wave reflection, the boundary is everywhere less steep than the energy ray. The reflected ray leaves the boundary without the singular shortening associated with a critical slope.

At critical internal-wave reflection, the reflected group velocity is tangent to the boundary and the inviscid reflected wavenumber diverges. Kinematic viscosity, mass diffusivity, nonlinear steepening, and wave breaking regularize the ideal singularity.

Reflection from periodic topography couples an incident horizontal wavenumber $k$ to sidebands $k+j{k}_T$. Expanding the impermeability condition on a boundary of small amplitude $h_0$ produces the $j=\pm 1$ sidebands at order $h_0$ and the $j=0,\pm 2$ corrections at order $h_0^2$.

Internal-wave ray tracing follows the group velocity while preserving the wave frequency. In a uniformly stratified fluid, each straight ray keeps the angle ${\sin }^{-1}(\omega /N)$ to the horizontal until reflection or a change in the medium.

For an internal gravity wave ray at angle $\theta$ to the horizontal reflecting from a slope at angle $\alpha$, conservation of tangential wavenumber givesAt a subcritical internal-wave reflection, $\gamma >1$: the reflected wavelength shortens and its wavelength-averaged energy density increases by ${\gamma }^2$.

When kinematic viscosity and mass diffusivity are equal to $ϵ$, a monochromatic internal-wave beam with wavenumber $k$ and ray angle $\theta$ has leading stream-function amplitudeThe cubic dependence on $k$ makes slope-focused, short internal waves dissipate especially rapidly.

For the planar strain $\mathbf{U}=\gamma (x,0,-z)$, an internal-wave ray hasIf it initially propagates upward with $m_0<0$, its height remains positive at finite time but decays to zero exponentially as $t\to \infty$.

A steady hill pattern seen by a uniform flow of speed $U$ has intrinsic frequency magnitude $Uk$. It radiates a propagating internal wave only for $k\le N/U$; above this cutoff the vertical wavenumber is imaginary and the disturbance is evanescent.