Internal gravity wave

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

For a plane inertia-gravity wave independent of one horizontal coordinate, period-mean kinetic and available potential energy obey $⟨K⟩-⟨A⟩=2⟨K_y⟩$. The transverse rotating velocity is in quadrature with the in-plane velocity. The modified oscillator balance is $⟨K_{xz}⟩=⟨A⟩+⟨K_y⟩$; ordinary kinetic/potential equipartition occurs only when the transverse amplitude vanishes.

Nonlinear disruption of an internal gravity wave, for example when large displacement gradients overturn the stable density arrangement or strong shear becomes unstable. The resulting turbulence dissipates wave energy and mixes the stratification, limiting ideal geometric focusing.

A monochromatic internal gravity wave in uniform stratification has a spatially linear phase. In the nonrotating inviscid Boussinesq approximation, its dispersion relation is ${\omega }^2=N^2|{\mathbf{k}}_h{|}^2/(|{\mathbf{k}}_h{|}^2+m^2)$. The wave vector is normal to its phase planes, while the group velocity lies along them.

For a plane internal gravity wave with vertical displacement $\zeta$, linear density advection gives ${\rho }^{\prime }=-{\bar{\rho }}_z\zeta$. In a constant stable gradient, ${\rho }_z={\bar{\rho }}_z(1-{\zeta }_z)$, so static overturning begins when the maximum of ${\zeta }_z$ exceeds one. A monochromatic displacement has maximum $|m{A}_{\zeta }|$. The criterion predicts the failure of the small-amplitude field, rather than a valid continuation of linear theory beyond overturning.

For two incompressible inviscid layers at rest between rigid horizontal boundaries, with the heavier fluid below and no surface tension, the displayed dispersion relation follows from Laplace equation for the velocity potentials, no normal flow at the walls, equal normal velocities at the interface, and pressure continuity. The positive density contrast supplies the restoring force. In the long-wave limit the squared wave speed is $g({\rho }_2-{\rho }_1)/({\rho }_1/h_1+{\rho }_2/h_2)$.

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.

A fixed terrain obstacle forces a zero laboratory-frequency internal gravity wave through the kinematic boundary condition $w=U{h}_x$. Its intrinsic frequency is $-kU$. Propagation, radiation and the transient arrival of its permanent component are distinct questions.

For a localized terrain source in uniform flow, the stationary phase method applied to $kx+m(k)z$ gives phase $\Phi =(N/U)\sqrt{x^2+z^2}$ on the downstream branch. Permanent phase lines are circular arcs, whereas an individual monochromatic component has straight phase lines. Reflected arcs use an image source; transmitted phase is continuous across the velocity interface.

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.

For upward internal gravity waves, let $B=c_{gz}{N}^2{A}_{\zeta }^2/\hat{\omega }$ be constant. Combining $c_{gz}=(N/k)\sin \theta {\cos }^2\theta$ with the monochromatic internal-wave overturning criterion gives ${\cot }^4\theta <B{k}^3\hat{\omega }/(N^3{\sin }^3\theta )$. Near a critical level the onset balance is $\cot \theta \sim (B{k}^3\hat{\omega }/N^3{)}^{1/4}$. This is compatible with $\hat{\omega }=N\cos \theta$; it fixes the overturning point rather than redefining the dispersion relation.

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 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.

For ${\psi }^{\prime \prime }={\alpha }^2\psi$ on $-1<z<1$ with endpoint values $P=\psi (1)$ and $Q=\psi (-1)$, the interior derivatives are ${\psi }^{\prime }(1)=\alpha [\coth (2\alpha )P-\mathrm{csch}(2\alpha )Q]$ and ${\psi }^{\prime }(-1)=\alpha [\mathrm{csch}(2\alpha )P-\coth (2\alpha )Q]$. Fitting the hyperbolic-function solution proves the map. It turns matching conditions into a finite-dimensional interface system, with exponentially small cross-interface coupling at large $\alpha$.

At an isolated interface with continuous base velocity, a stable density drop $\Delta \rho /2$ and a jump $[U^{\prime }]$ of base shear, decaying disturbances $w\propto e^{-k|z-z_i|}$ obey the displayed dispersion relation. It follows by inserting their derivative jump $[w^{\prime }]=-2kw$ into the jump conditions for stratified inviscid shear flow. Both the density jump and the vorticity jump contribute to the intrinsic phase velocity $c-U_i$.

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 wave amplitudes and upward phase-line angles measured from the vertical ${\theta }_1,{\theta }_2$, define $r={\hat{\omega }}_2/{\hat{\omega }}_1$. Then the transmission coefficient isThe displacement condition is $A_i+A_r=A_t\cos {\theta }_2/(r\cos {\theta }_1)$, while pressure continuity gives $A_i-A_r=r{A}_t\sin {\theta }_2/\sin {\theta }_1$. Adding eliminates the reflected amplitude and gives the coefficient. It assumes nonzero intrinsic frequencies and propagating outgoing branches.

At a material velocity jump with continuous background mass density, vertical displacement $\eta$ and fluid pressure are continuous, while $w=ikU\eta$ need not be. For a stationary component, $p=U^2{\eta }_z$ and the signed upward-wave impedance is $Z=U^{2}m$. Reflection and vertical-displacement transmission are $R=(Z_1-Z_2)/(Z_1+Z_2)$ and $T=2Z_1/(Z_1+Z_2)$.

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.

For constant buoyancy frequency $N$ and $U=U_0+Sz$ with $S>0$, a stationary component of horizontal wavenumber $k$ turns at $z_t=(N/k-U_0)/S$ if it propagates at the source. Its vertical wavenumber vanishes there; WKB approximation fails and an Airy function transition connects reflection to an upper evanescent wave.

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.

The ocean mixed layer is a near-surface layer with relatively weak vertical variations of temperature and mass density, maintained by mixing. Beneath it, stable density stratification can isolate colder water from recent surface heating. The mixed-layer depth is not necessarily the full local water depth.

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.

Homogenizing the top depth $h$ of a constant-gradient stable stratification over horizontal area $\mathcal{A}$ creates mean density ${\rho }_t+{\rho }_0{N}^{2}h/(2g)$. Comparing gravitational potential energies before and after gives $\Delta P={\rho }_0\mathcal{A}{N}^2{h}^3/12$. The density jump at the base has reduced gravity $N^{2}h/2$. The cubic energy cost leads to a quadratic-in-depth coefficient multiplying the deepening rate.

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.

The radial buoyancy frequency describes restoration of a radially displaced parcel in a stratified astrophysical disk. With a dimensionless specific entropy $S$, its square is proportional to $-{\rho }^{-1}(dP/dr)(dS/dr)$. Negative $N_r^2$ means an adverse entropy stratification, but epicyclic motion can still stabilize the parcel. For smooth thin disk profiles, $|N_r^2|$ is usually of order $(H/r{)}^2{\Omega }^2$.

For ideal axisymmetric radial displacements in a Keplerian shearing sheet, the hydrodynamic squared oscillation frequency is $N_r^2+{\kappa }^2$, with radial epicyclic frequency $\kappa =\Omega$. Exponential instability occurs when $N_r^2+{\kappa }^2<0$. This is the radial special case of rotating-fluid convective stability, rather than the complete set of three-dimensional Solberg–Høiland criteria.

For a dry, homogeneous ideal gas in hydrostatic equilibrium, a displaced parcel undergoing an adiabatic process follows $dT/dz=-g/C_p$, where $C_p$ is specific heat capacity at constant pressure. At a common pressure its buoyancy depends on its temperature difference from the environment, giving $N^2=(g/T)(dT/dz+g/C_p)$. Negative $N^2$ means growing displacements; zero $N^2$ means neutral stability. This is the homogeneous dry limit of the Schwarzschild criterion.

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

A local ratio of stabilizing buoyancy to shear across an interface of thickness $\delta$, using reduced gravity jump $g^{\prime }$ and characteristic velocity $u$. The dimensionless value is ${\mathrm{Ri}}_i=g^{\prime }\delta /u^2$. It is distinct from a whole-layer bulk Richardson number based on the total mixed depth. Local entrainment laws use this ratio when the transition thickness and velocity characterize the turbulence-producing region.

Combine a fixed-energy mixed-layer relation $u^{2}h=A{\Omega }^2{R}^3$, jump $g^{\prime }=N^{2}h/2$, and stress power proportional to $u^3$ with an energy-transfer factor $(\Omega /N{)}^{\alpha }(\delta /R{)}^{\beta }$. Eliminating depth in favor of the interfacial Richardson number gives $E\propto (\Omega /N{)}^{\alpha -1}(\delta /R{)}^{\beta +3/2}{\mathrm{Ri}}_i^{-3/2}$. A local-only entrainment closure requires $\alpha =1$, $\beta =-3/2$ and exponent $3/2$ on the Richardson number.

A bulk Richardson number compares a characteristic buoyancy difference across a layer with the squared characteristic flow speed. It uses large-scale differences rather than the pointwise gradient Richardson number. Its normalization depends on the chosen velocity, density and length scales.

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 $U=sz$, $N>0$, $k>0$ and the rising branch $m_0<0$, the Hamiltonian is $\omega =ksz+Nk/\sqrt{k^2+m^2}$. It gives constant $\omega ,k$ and $m=m_0-kst$. The height approaches $z_c=\omega /(ks)$ as time tends to infinity. With $\theta =\arctan (|m|/k)$, the ray slope is $dx/dz=\tan \theta +ksz/(N\sin \theta {\cos }^2\theta )$. The intrinsic and absolute frequencies must not be confused.

A periodic spatial internal-wave ray toward which neighbouring ray paths converge after reflection in a confined stratified basin. A fixed point of an internal-wave ray return map is attracting when its derivative has magnitude below one. Viscous attenuation of an internal-wave beam and internal-wave breaking regularize the ideal concentration of energy. A spatial ray construction does not remove turning-level limitations of the WKB approximation.

Map one boundary intersection of an internal-wave ray to the next intersection with the same boundary after a specified reflection itinerary. A periodic ray is a fixed point. Linear contraction, $|R^{\prime }|<1$, gives an internal-wave attractor; the reversed itinerary, when invertible, has reciprocal multiplier and is locally repelling.

A level where the squared vertical wavenumber of an internal gravity wave touches zero quadratically without changing sign. The ordinary WKB approximation fails there; unlike a simple sign-changing turning point, this does not itself provide an evanescent region on one side. Finite-wavelength analysis determines how solutions connect across or meet a boundary at the level.

In uniform positive flow $U$, upward stationary internal gravity waves have laboratory group velocity $U({\cos }^2\theta ,\sin \theta \cos \theta )$. Packets emitted by a localized source at startup lie on $(x-Ut/2{)}^2+z^2=(Ut/2{)}^2$, with later emissions inside. This envelope belongs to the permanent stationary component, not every possible startup transient.

A lower ray reaches $H$ after ${\tau }_1=H/[U_1\sin {\theta }_1\cos {\theta }_1]$, at $x_H=H\cot {\theta }_1$. Conserved horizontal wavenumber gives $\cos {\theta }_2=(U_2{N}_1/U_1{N}_2)\cos {\theta }_1$. The transmitted front is $(x_H,H)+(t-{\tau }_1){\mathbf{c}}_{g2}$ on propagating branches; reflection reverses the vertical group velocity.

A broad-spectrum stationary source below height $H$ first sends an upward packet there after $2H/U$, at $x=H$. This uses the maximum vertical group velocity $U/2$ at $\theta =\pi /4$. A source lacking the corresponding wavenumber has a later actual arrival.

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.

In the prescribed along-beam scalar model $U=A(\xi )\cos \omega t$ with $A=U_0{e}^{-\lambda \xi }$, the first-order fluid displacement is $A\sin \omega t/\omega$. Its leading sampled-velocity correction is $-\lambda A^2\sin (2\omega t)/(2\omega )$, whose period average vanishes. This result concerns the supplied component model and does not determine an Eulerian mean streaming flow.

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.