Internal wave

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

For a lower uniformly stratified half-space and an upper well-mixed half-space, the vertical velocity amplitude is continuous and its derivative jumps by $[W^{\prime }]=-k^2\Delta bW/{\omega }^2$. A wave incident upwards with vertical phase factor $e^{-imz}$ therefore has upper evanescent amplitude $T/W_i=2im/[k(1-S)+im]$, where $m=k\sqrt{N^2/{\omega }^2-1}$ and $S=\Delta bk/{\omega }^2$. Its magnitude peaks at $S=1$, with amplitude twice the incident amplitude. Lossless reflection carries all incident energy back down; the evanescent upper field carries no mean outgoing vertical energy flux.

Incompressibility makes pressure adjust throughout the surrounding fluid. Projecting the linear Boussinesq equations perpendicular to a Fourier wavevector gives $w_t=[k_h^2/(k_h^2+m^2)]b$ and $b_t=-N^{2}w$. A localized spherical disturbance contains many wavevector directions and therefore many internal gravity wave frequencies, rather than one oscillator at $N$. Its shape evolves and energy radiates into the surroundings; a rigid-sphere added mass analogy illustrates extra inertia but does not give an exact coherent frequency for a free material blob.

In a nonrotating ideal Boussinesq layer with uniform current $U>0$, constant buoyancy frequency $N$ and rigid walls, any stationary superposition of channel wave modes with $k_n^2+(n\pi /H{)}^2=N^2/U^2$ obeys the displayed relations. With $(u,w)=({\psi }_z,-{\psi }_x)$, the total streamfunction is $\Psi =Uz+{\psi }^{\prime }$ and total buoyancy is $\sigma =(N^2/U)\Psi$. Its material derivative vanishes. The perturbation vorticity is proportional to ${\psi }^{\prime }$, so its nonlinear self-advection vanishes as well. The modes are therefore exact finite-amplitude solutions, although large amplitudes may violate physical approximation assumptions or become unstable.

A slowly established upward plane internal gravity wave has $\overline{u^{\prime }{w}^{\prime }}=-m|a{|}^2/(2k)$. The mean horizontal conservation of momentum equation gives ${\overline{u}}_t=-{\partial }_z\overline{u^{\prime }{w}^{\prime }}$. Using envelope transport and an initially vanishing mean gives $\overline{u}=-m|a{|}^2/(2k{c}_{gz})$. Its energy density per unit mass is $\mathcal{E}=(k^2+m^2)|a{|}^2/(2k^2)$, so $\overline{u}=(k/\omega )\mathcal{E}$. This is the mean flow left behind the advancing wave envelope, rather than a claim that an established uniform wavetrain has nonzero local stress divergence.

For a traveling sinusoidal boundary forcing with horizontal phase speed $c_p$, the pressure and horizontal velocity polarization gives the displayed relation between upward energy input and vertical wave momentum flux. A slowly established outgoing wave envelope deposits horizontal mean momentum as its front advances. In the frame moving at $c_p$, the order-amplitude-squared change of mean kinetic energy is $-{\rho }_0{c}_p\int \overline{u}dz$. Its loss rate equals the boundary work in the laboratory frame. The comparison uses a finite perturbation of the infinite uniform background energy.

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.

The displayed displacement convention is the negative of the usual positive-frequency wave pseudomomentum $k\overline{E}/\omega$ for a rightgoing internal gravity wave. Here energy per unit reference mass density is $\overline{E}=|\hat{\sigma }{|}^2/(2N^2)$. From the buoyancy polarization of a plane internal gravity wave, horizontal averaging gives $\overline{{\zeta }_{x}p}=km|\hat{\sigma }{|}^2/(2N^2{K}^2)=c_{gz}P$, where $p$ is kinematic pressure. The sign convention changes both density and flux and leaves their propagation at the group velocity unchanged.

With weak kinematic viscosity $\nu$, zero buoyancy diffusion and locally unchanged plane internal gravity wave polarization, the displacement pseudomomentum of an internal gravity wave obeys $P_t+{\partial }_z(c_{gz}P)=-\nu K^{2}P$. For constant wavenumbers and a stationary wave envelope, $P\propto |\hat{\sigma }{|}^2$ proves the displayed amplitude decay. Decay is along the group velocity, so $c_{gz}<0$ describes a source above the receiving level. In a slowly varying shear flow, use wave action and the changing intrinsic frequency rather than treating $P/|\hat{\sigma }{|}^2$ as constant.

At a jump in buoyancy frequency with continuous background density, incident, reflected and transmitted internal gravity waves share frequency and horizontal wavenumber. Continuity of pressure and normal velocity determines their amplitudes. If amplitudes multiply unit particle-displacement vectors directed along each ray with positive horizontal component, the reflection coefficient is $[\sin {\theta }_2/\sin {\theta }_1-\cos {\theta }_2/\cos {\theta }_1]/[\sin {\theta }_2/\sin {\theta }_1+\cos {\theta }_2/\cos {\theta }_1]$. Using vertical-displacement amplitudes reverses the reflected coefficient's sign because the reflected polarization reverses its vertical component.

A reflecting bottom beneath a jump in buoyancy frequency creates a standing field from repeated transmitted and reflected internal gravity waves. For ray angles ${\theta }_1=\pi /3$ and ${\theta }_2=\pi /6$, the downward particle-displacement amplitude has magnitude $\sqrt{3}|{\eta }_i|/[1+8{\sin }^2(m_{2}H){]}^{1/2}$. It reaches $\sqrt{3}|{\eta }_i|$ when $m_{2}H$ is an integer multiple of $\pi$. The upper reflected wave has the same amplitude magnitude as the incident wave because the lossless closed cavity admits no net downward energy flux.

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.

For a plane internal gravity wave in uniform buoyancy frequency, substitute $\zeta =A(\mu t,\mu z)e^{i(kx+mz-\omega t)}$ into the displacement equation. The leading order gives ${\omega }^2=N^2{k}^2/(k^2+m^2)$; the next order gives the displayed transport equation with $c_{gz}=-\omega m/(k^2+m^2)$. A boundary amplitude $a(T)$ radiates as $A=a(T-Z/c_{gz})$, choosing the sign of $m$ so the group velocity points into the fluid. The wavelength-averaged energy is $N^2|A{|}^2/2$ and its vertical flux is $c_{gz}$ times that energy. A causal ramp therefore creates a translating front, not an instantaneous monochromatic field throughout the half-space.

A right-going, boundary-forced internal gravity wave has horizontal momentum per unit inertial density $P=(k/\omega )\overline{E}$. Its envelope-induced Eulerian mean acceleration is ${\overline{u}}_t=P_t$ to leading slow order. In the frame moving at horizontal phase velocity $c=\omega /k$, the change of mean kinetic energy relative to the initial current $-c$ is $-{\rho }_{0}c\int \overline{u}dz$ at quadratic wave order. Its time derivative is minus the boundary work $\overline{W}={\rho }_{0}d\int \overline{E}dz/dt$. The lab-frame mean kinetic energy alone is fourth order in wave amplitude; retaining the moving-frame cross term is essential.

For a nonrotating uniform-stratification fluid under the Boussinesq approximation driven at $0<\omega <N$ by a localized vertical-wall velocity, positive vertical wavenumbers radiate down and right and negative ones up and right. The real velocity separates into functions of $Z_{+}$ and $Z_{-}$. Its in-phase component is half the imposed boundary profile on each beam; the quadrature component is its Hilbert transform partner. Thus compact localization at maximum boundary speed does not imply compact localization of all components at every phase. A finite-duration source contains several frequencies and spreads energy over their different ray angles and group speeds.

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 sinusoidal small-amplitude bed, constant current $U>0$ and buoyancy frequency $N$, put $m=\sqrt{N^2/U^2-k^2}>0$. The linear vertical velocity amplitude is $W(z)=ikU{\eta }_0\sin [m(H-z)]/\sin (mH)$. The upward-wave coefficient is $A_{+}=-kU{\eta }_0{e}^{-imH}/[2\sin (mH)]$. At $mH=n\pi$, the prescribed nonzero bed motion and homogeneous rigid-lid condition are incompatible with a bounded stationary inviscid solution. With angle $\theta$ between intrinsic group velocity and vertical, $\tan \theta =m/k$, giving $kH\tan \theta =n\pi$.

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.

For a slowly switched-on plane internal gravity wave in the half-space $z>0$, the method of multiple scales gives transport of its amplitude at the vertical group velocity. For ${\omega }^2=N^2{k}^2/(k^2+m^2)$, $c_{gz}=-\omega m/(k^2+m^2)$. Initially undisturbed fluid and forcing at the bottom require $c_{gz}>0$. The outgoing radiation condition therefore selects $m<0$ when $\omega >0$, although the vertical phase velocity is downward.

Select outgoing waves as a limit of a problem with weak absorption. With time convention $e^{-i\omega t}$ use $\mathrm{Im}\omega >0$; with $e^{i\omega t}$ use $\mathrm{Im}\omega <0$. The resulting Green function and transform-contour prescriptions determine which poles represent incoming or outgoing waves. This is especially useful when real-axis singularities obscure a Wiener-Hopf factorization.

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 positive background velocity $U=U_0{e}^{Mz}$ and buoyancy frequency $N=N_0{e}^{Mz}$, a stationary nonzero-horizontal-wavenumber Taylor–Goldstein equation reduces to the displayed constant-coefficient equation. Its structures are oscillatory, affine or exponential according to the sign of $N_0^2/U_0^2-M^2-k^2$. The disturbance Laplacian is a constant times its streamfunction, but its buoyancy anomaly is $N_0^2{e}^{Mz}\varphi /U_0$. Therefore vorticity self-advection vanishes while buoyancy self-advection produces a nonzero second harmonic for $M\ne 0$. These nontrivial real linear modes are not exact finite-amplitude solutions without additional fields. For $M=0$, both polarizations are constant and these nonlinear self-interactions vanish.

For two-dimensional flow under the Boussinesq approximation with $u={\psi }_z$, $w=-{\psi }_x$, a linear mode whose Laplacian is proportional to its streamfunction has zero self-advection of vorticity. Its buoyancy anomaly is $\sigma =a(z)\varphi$, with $a=N^2/(U-c)$. Self-advection of buoyancy vanishes only when the displayed Jacobian is zero. Constant stratification and constant flow make $a$ constant and yield finite-amplitude exact wave solutions. A varying polarization coefficient generally creates a second harmonic, even when the vorticity self-interaction vanishes.

For a regular real mode between rigid boundaries, multiply the Taylor–Goldstein equation by its vertical structure and integrate to get the stationary quotient $k^2=\int ({\ell }^2{\varphi }^2-{\varphi }_z^2)dz/\int {\varphi }^{2}dz$. Along a differentiable dispersion branch, stationarity removes the variation of the eigenfunction, giving $2k=(\partial I/\partial c)dc/dk$. Since ${\partial }_c{\ell }^2=N^2/(U-c{)}^3+{\ell }^2/(U-c)$, the displayed group velocity follows from $d(kc)/dk$. Critical levels and vanishing branch derivatives require a separate limiting analysis.

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.

A vertically propagating stationary topographic internal gravity wave meets a turning level when its squared vertical wavenumber changes from positive to negative. The stationary internal-wave WKB solution then fails locally. A simple zero is resolved by an Airy function transition connecting incident and reflected waves below to a decaying evanescent wave above. For negligible wind curvature the condition is $|kU(z_t)|=N(z_t)$, distinct from the zero-intrinsic-frequency critical layer in a shear flow.

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 density gradient is the gradient of a spatial mass density field. With an upward vertical coordinate in the Boussinesq approximation, the squared buoyancy frequency is $N^2=-(g/{\rho }_{\ast }){\partial }_z{\rho }_0$. A negative vertical density gradient therefore gives stable density stratification.

A layer with a pronounced vertical change in salinity is a halocline. When relatively fresh surface water lies over saltier water, its contribution to density stratification inhibits vertical exchange. A cold Arctic halocline can separate sea ice from warmer Atlantic water; increased heat at depth does not imply an equally increased ocean heat flux at the ice unless the intervening exchange also changes.

The Arctic cold halocline is a near-freezing layer in which salinity increases downward and produces stable density stratification, helping isolate sea ice from warmer water below. It can be maintained through shelf-water input, cooling and limited winter convection of inflowing waters, followed by freshwater capping and advection. Its formation cannot be inferred from shelf area alone: water-mass volumes, salt budgets and residence times matter.

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.

Equating an inertial eddy turnover time to the inverse buoyancy frequency gives the scale separating buoyancy-affected motions from a smaller approximately isotropic cascade. It is distinct from the viscous Kolmogorov microscales. In very stable local surface-layer scaling it is proportional to the Obukhov length, up to closure constants.

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.

The ratio of stable buoyancy destruction to shear production of turbulent kinetic energy. With down-gradient transport and turbulent Prandtl number ${\mathrm{Pr}}_t$, it obeys ${\mathrm{Ri}}_f={\mathrm{Ri}}_g/{\mathrm{Pr}}_t$, rather than automatically equaling the gradient Richardson number. Unstable buoyancy production gives a negative value in this convention.

With positive stable buoyancy destruction $B$ and positive dissipation $ϵ$, this normalized sink fraction need not equal the Flux Richardson number $B/P$ away from equilibrium. If $R=(B+ϵ)/P$, then $J={\mathrm{Ri}}_f/R$. In the fixed-correlation equilibrium model for stratified turbulence, a minimum of $R=a/q+bq$ has $B=ϵ$, hence $J=1/2$. The actual flux Richardson number there is $R_{\min }/2$ and equals $1/2$ only when that minimum also satisfies the equilibrium condition $R=1$.

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.

Choose $k,m>0$, $K=(k^2+m^2{)}^{1/2}$ and the positive angular frequency $\omega =Nk/K$. Then the phase velocity is $Nk(k,m)/K^3$ and the group velocity is $N(m^2,-km)/K^3$. Their dot product vanishes and their sum is horizontal with magnitude $N/K$. Draw the group velocity from the tip of the phase velocity: the two form the perpendicular sides of a right triangle. The horizontal hypotenuse is their sum, not the horizontal phase velocity $\omega /k$.