Geophysical fluid dynamics (GFD)

Geophysical fluid dynamics studies fluid motion on rotating, stratified planets, including oceans, atmospheres, cores, and planetary interiors.

A flow along latitude circles, conventionally positive eastward. Its speed Doppler shifts the intrinsic frequency of a disturbance. A stationary Rossby wave in a uniform current requires the eastward zonal flow to balance its westward intrinsic phase velocity.

An atmospheric vertical coordinate using pressure instead of geometric height. Under hydrostatic balance, pressure decreases upward and geopotential obeys ${\partial }_p\Phi =-1/\rho$. With the ideal gas law, ${\partial }_{\ln p}\Phi =-RT$. Horizontal gradients taken at fixed $p$ lie on an isobaric surface. A log-pressure coordinate is a useful related coordinate with the opposite upward sign.

A surface of constant pressure. Its intersection with a fixed-height horizontal plane is an atmospheric isobar. An isobaric surface is generally not a geopotential surface: the horizontal geopotential variation on it determines the geostrophic wind.

In the tropical free troposphere, small Coriolis parameter and rapid internal gravity wave adjustment keep horizontal temperature gradients weak. An idealized model therefore suppresses horizontal temperature perturbations at leading order and balances diabatic heating by vertical advection of the reference stratification. This is not a claim that surface temperature or moisture is spatially uniform, nor is it ordinary midlatitude geostrophic balance.

The ratio of viscous force to Coriolis force in a rotating fluid at length scale $L$. This convention uses rotation frequency $2\Omega$; conventions omitting the factor two are also common. For small $E$, bulk viscosity is weak but no-slip boundaries can form Ekman layers of thickness of order $L\sqrt{E}$, with relatively larger boundary shear stress.

A zonal-mean meridional overturning circulation associated with Southern Ocean wind forcing. In a southern-hemisphere channel with eastward wind stress, the mean circulation has northward surface transport and a southward deep return. Its Eulerian strength need not equal the residual water-mass transport because eddy-induced overturning can oppose it.

The effective meridional circulation associated with eddy buoyancy transport. In a stratified Southern Ocean, mesoscale eddies tend to flatten the isopycnals tilted by the wind-driven mean circulation, so their contribution can largely oppose the mean Deacon circulation. The residual combines mean and eddy contributions with the diabatic water-mass budget.

A streamfunction for an incompressible zonal-mean flow in the meridional-vertical plane. Its derivatives give meridional and vertical velocity, and differences in its values measure volume transport per zonal span. Constant values on closed impermeable boundaries encode zero net normal flow.

A horizontal scale comparing gravitational or buoyancy restoring forces with rotation. For a shallow-water gravity mode with propagation speed $c$, the deformation radius is $c/|f|$. The barotropic deformation radius uses $c=\sqrt{gH}$; internal layered modes use a speed set by reduced gravity and layer depths.

For an equatorial beta plane and a shallow water wave speed $c$, the natural meridional trapping scale is $R_{\mathrm{eq}}=\sqrt{c/\beta }$. A decaying equatorial Kelvin wave has $\eta \propto e^{-y^2/(2R_{\mathrm{eq}}^2)}$. An envelope written $e^{-y^2/L^2}$ instead uses $L=\sqrt{2}{R}_{\mathrm{eq}}$; stating the exponent convention prevents a factor-of-two error in the meridional Cauchy momentum equation.

The two-layer quasi-geostrophic potential vorticity coupling defines $R_d^{-2}=F_1+F_2$. With $F_i=f_0^2/(g\prime H_i)$ this gives the stated radius. The baroclinic energy ratio and deformation scale depends on $(l/R_d{)}^2$, so the thinner layer matters when the depths are strongly unequal.

The Burger number compares the square of a relevant deformation radius with the square of the horizontal flow scale: $\mathrm{Bu}=R_d^2/l^2$. In a two-layer geostrophic model $R_d^2=g\prime H_1{H}_2/[f_0^2(H_1+H_2)]$. Its inverse measures the scale ratio of baroclinic available potential energy to kinetic energy. Keeping it of order unity retains both interface and relative-vorticity contributions in QG scaling.

A general circulation model evolves rotating fluid dynamics together with heating/cooling and suitable transport parameterizations. Atmospheric applications can couple radiative transfer, clouds and chemistry to resolve longitudinal and latitudinal structure. For an exoplanet atmosphere, it can predict winds and an exoplanet thermal phase curve. Unresolved turbulence and cloud microphysics still require approximate closures.

An Eulerian mean flow is obtained by averaging the velocity field at fixed spatial positions, for example over longitude or over an ensemble. It differs from a parcel-following average. In the transformed Eulerian mean, its meridional circulation is corrected by an eddy-induced contribution to form the residual mean circulation.

Thermal-wind balance relates vertical shear of geostrophic flow to horizontal density or temperature gradients. With constant Coriolis parameter $f_0$ and reference density ${\rho }_0$,for the sign conventions used here.

For a steady density field sampled by two ships at angles $\theta$ and zero, write $\mathbf{y}=C\left(\begin{array}{cc}\sin \theta & -\cos \theta \\ 0& -1\end{array}\right)(u,v{)}^{\mathsf{T}}$. Independent measurement errors of variance ${\sigma }_y^2$ giveNearly parallel tracks poorly constrain $u$ and induce positive component covariance for $0<\theta <\pi /2$. Orthogonal tracks remove this covariance and give equal component precision under the stated independent-noise model.

Combining geostrophic balance, $fu=-{\Phi }_y$, $fv={\Phi }_x$, with hydrostatic balance, ${\Phi }_p=-RT/p$, and commuting the horizontal and vertical derivatives proves the displayed equations. Horizontal derivatives hold pressure fixed. This relates vertical shear to horizontal temperature gradients away from the equator; the derivation requires nonzero Coriolis parameter and the validity of geostrophic balance.

For $b=A(y)+B(z)$, constant nonzero $f_0$, and zero basic velocity at $z=0$, thermal-wind balance givesWith full buoyancy represented by $f_0{\bar{\psi }}_z$, the constant-coefficient three-dimensional quasi-geostrophic potential vorticity is $\bar{q}=f_0+z{A}^{\prime \prime }/f_0+f_0{B}^{\prime }/N^2$. The reference-buoyancy convention in quasi-geostrophic potential vorticity changes only the uniform reference contribution.

A log-pressure coordinate uses atmospheric pressure to define an effective height. Its density weight is proportional to $e^{-z/H}$, and its continuity equation has vertical divergence $e^{z/H}{\partial }_z(e^{-z/H}w)$. Here $H$ is the chosen pressure scale height; the coordinate is not generally identical to geometric altitude.

The Rossby number compares inertial acceleration with Coriolis acceleration. For speed $U$, horizontal length $L$, and Coriolis parameter $f$, it is $\mathrm{Ro}=U/(|f|L)$. Small Rossby number supports geostrophic balance and the quasi-geostrophic approximation.

For a convective star, the activity convention compares its rotation period with a convective turnover time. It differs by fixed factors from the fluid Rossby number estimated from speed, length and angular velocity. Small ${\mathrm{Ro}}_{\ast }$ denotes strong rotational influence on convection. Magnetic activity usually rises as this ratio decreases in the unsaturated regime, but levels off in a saturated rapid-rotation regime.

The Burgers number compares stratification with rotational effects. In shallow-water and stratified flows it is commonly the squared ratio of deformation radius to horizontal length scale.

The Boussinesq approximation treats density as constant except where a small density variation multiplies gravity to produce buoyancy. The velocity remains incompressible.

In the constant-property convection convention, the strong Boussinesq approximation supplements small relative mass density changes with constant dynamic viscosity, thermal diffusivity, specific heat capacity and thermal expansion coefficient throughout the modeled temperature interval. A uniform reference density replaces density in inertia and mass conservation, while a linear density anomaly is retained in gravitational buoyancy. The temperature obeys an advection-diffusion equation with constant coefficients. The term “strong” is not uniform across all fluid-mechanics literature: some stratified-flow treatments instead emphasize the use of a globally constant reference density. The constant-property convection usage is explained in courses.physics.ucsd.edu/2021/Spring/physics218c/Spiegel%20Convection.pdf .

Uniformly stratified, nonrotating, two-dimensional ideal Boussinesq equations in coordinates $(x,z)$ are equivalent to unstratified rotating equations independent of $Y$, with $(X,Y,Z)=(z,y,-x)$, rotation $2\mathbf{\Omega }=N{\mathbf{e}}_Z$, velocity $(U,V,W)=(w,\sigma /N,-u)$, and the same reduced pressure. The transverse rotating velocity represents buoyancy perturbation, while the rotation-axis velocity represents the stratified horizontal velocity. Matching prescribed forces may have components in the invariant plane, but no transverse component. Physical free surface conditions are not preserved: the physical vertical direction and its normal velocity are different under the mapping.

A weak horizontal force $f^{\prime }(x)e^{imz}$, with $m>0$ and $\int f^{\prime }(x)dx=\epsilon$, admits many steady pressures differing by an arbitrary function of height. Switching the force on in initially motionless, unbounded, uniformly stratified fluid selects $p_{\infty }=[f(x)-\epsilon /2]e^{imz}$, up to a spatially constant pressure gauge. For horizontal wavenumber $q$, the transient frequency is $\Omega (q)=Nq/\sqrt{q^2+m^2}$. Its long-wave group velocity is $N/m$, and oppositely propagating outgoing fronts leave pressure limits $-\epsilon /2$ and $+\epsilon /2$. The limit is at fixed $x$ after the fronts have passed; dispersive tails need not vanish at any finite time.

For a weak horizontal forcing proportional to $e^{imz}$ in an ideal uniformly stratified nonrotating fluid, the low-frequency hydrostatic approximation gives the displayed one-dimensional wave equation for the streamfunction amplitude. Two oppositely propagating characteristic integrals determine the response. A source confined to $|x|\le a$ and switched on at time zero produces no response beyond $|x|\le ct+a$ in this reduced equation. The bound on mixed forcing derivative $|f_{xt}|\ll |mN{f}_{\max }|$ controls the neglected vertical acceleration in characteristic magnitude.

With mechanically and thermally equivalent upper and lower boundaries, the Boussinesq approximation permits reflection through the layer midplane together with reversal of the temperature perturbation and vertical velocity. In a reduced convection temperature equation this induces temperature inversion. A quadratic term which changes differently from the linear terms breaks this symmetry; a cubic term can preserve it. The Boussinesq approximation alone does not impose symmetry on unequal boundary conditions.

The inviscid Boussinesq equations for velocity $\mathbf{u}$, buoyancy $b$, and reference density ${\rho }_0$ aretogether with an advection or advection–diffusion equation for $b$.

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

An inertial wave is a rotating-fluid wave restored by the Coriolis force. Its frequency depends on the angle between its wavevector and the rotation axis.

For fixed vertical wavenumber $k_z$, the inertial-wave dispersion relation is ${\omega }^2=4{\Omega }^2{k}_z^2/(k_x^2+k_z^2)$. Its horizontal group velocity is the derivative displayed above, opposite in sign to $\omega k_x$. In a half-channel $x>0$, an outgoing propagating mode therefore requires $\omega k_x<0$. With time factor $e^{-i\omega t}$ and $q=|k_z|\sqrt{4{\Omega }^2/{\omega }^2-1}$, choose $e^{-i\mathrm{sgn}(\omega )qx}$. Above the inertial-frequency cutoff, the horizontal wavenumber is imaginary and the appropriate radiation condition instead selects the exponentially decaying solution. Energy transport, rather than the direction of phase travel alone, determines the outgoing sign.

Linear incompressible motion about rest in a frame rotating with $\mathbf{\Omega }=\Omega \mathbf{k}$ obeys ${\mathbf{u}}_t+2\mathbf{\Omega }\times \mathbf{u}=-\nabla P/\rho$. If $\zeta =v_x-u_y$ is its vertical vorticity, taking curl gives ${\zeta }_t=2\Omega w_z$, while eliminating pressure using continuity gives ${\partial }_t{\nabla }^{2}w+2\Omega {\zeta }_z=0$. Differentiating in time yields the displayed scalar equation. All three velocity components must be retained, even when the fields depend on only two spatial coordinates. The Coriolis acceleration generally generates a transverse velocity that is absent in a strictly two-component description.

For $\omega =\sigma 2\Omega k_{\parallel }/k$, differentiating the dispersion relation gives the displayed group velocity. It is perpendicular to the wavevector. If $|k_{\parallel }|\ll k_{\perp }$, energy propagates almost along the rotation axis while the phase velocity is almost horizontal. In an orthonormal basis ${\mathbf{e}}_1,{\mathbf{e}}_2$ normal to $\mathbf{k}$, with ${\mathbf{e}}_2=\hat{\mathbf{k}}\times {\mathbf{e}}_1$, the complex velocity is $A({\mathbf{e}}_1-\sigma i{\mathbf{e}}_2)$ and its real oscillation is circular. In the anisotropic limit the vertical and transverse-horizontal velocity amplitudes are comparable, while the horizontal component parallel to ${\mathbf{k}}_{\perp }$ is smaller by $|k_{\parallel }|/k_{\perp }$.

Insert a transverse plane wave $\mathbf{u}={\mathbf{u}}_0{e}^{i(\mathbf{k}\cdot \mathbf{r}-\omega t)}$ into the inertial-wave vorticity equation, with $\mathbf{k}\cdot {\mathbf{u}}_0=0$. It gives $\omega \mathbf{k}\times {\mathbf{u}}_0=2i(\mathbf{\Omega }\cdot \mathbf{k}){\mathbf{u}}_0$. Applying $\mathbf{k}\times$ again and using transversality gives the title dispersion relation. If $\theta$ is the angle to the rotation axis, $|\omega |=2|\Omega \cos \theta |$. At fixed direction the angular frequency does not depend on wavelength. Parallel wavevectors give the maximum $2|\Omega |$; perpendicular ones have zero frequency and belong to the nonoscillating balanced limit.

Linearize the Euler equation in a uniformly rotating frame about rest and impose incompressible flow, $\nabla \cdot \mathbf{u}=0$. Taking the curl eliminates the reduced pressure. For constant $\mathbf{\Omega }$, $\nabla \times (\mathbf{\Omega }\times \mathbf{u})=-(\mathbf{\Omega }\cdot \nabla )\mathbf{u}$, giving the displayed vorticity equation. Taking another curl and time derivative yields ${\partial }_t^2{\nabla }^2\mathbf{u}+4(\mathbf{\Omega }\cdot \nabla {)}^2\mathbf{u}=0$. These equations describe the restoring influence of background rotation on small disturbances without buoyancy or viscosity.

In a uniform-density incompressible flow inside a smooth rigid sphere rotating at angular speed $\Omega$, a normal mode with phase $e^{i\sigma t+im\varphi }$ has intrinsic frequency $s=\sigma +m\Omega$. For $s\ne 0,\pm 2\Omega$, eliminating velocity from the linearized Euler equations gives the displayed pressure equation, with $W=p^{\prime }/\rho$. Impermeability gives $R{W}_R+(2m\Omega /s)W+(1-4{\Omega }^2/s^2)z{W}_z=0$ on the sphere. There is no condition $W=0$: the rigid wall supplies variable normal pressure. These inertial waves are restored by Coriolis acceleration, and their pressure equation is hyperbolic for $|s|<2|\Omega |$.

For an axisymmetric normal mode in a rigid rotating sphere, substitute $W=z(A{R}^2+B{z}^2+C)$ into the pressure equation and its impermeability condition. With $\beta =1-4{\Omega }^2/{\sigma }^2$, the interior equation gives $2A+3\beta B=0$. The boundary polynomial gives $3\beta B=(2+\beta )A$ and $\beta C=-(2+\beta )A{r}_0^2$. A nontrivial oscillatory solution has $\beta =-4$, hence the displayed frequency and $B=A/6$, $C=-A{r}_0^2/2$. It satisfies both equations identically.

A rotating fluid is described in a rotating reference frame, where the Coriolis acceleration and centrifugal acceleration supplement the physical forces.

Under critical balance in a rapidly rotating fluid, define ${\mathbf{u}}_{\perp }^{(0)}=\hat{\mathbf{z}}\times {\nabla }_{\perp }\Phi$ and $D_{\perp }={\partial }_t+\{\Phi ,\cdot \}$. Leading horizontal incompressibility gives zero divergence, but at the next order ${\nabla }_{\perp }\cdot {\mathbf{u}}_{\perp }^{(1)}=-{\partial }_{z}w$. Its Coriolis acceleration contribution survives in the vertical vorticity equation, producing the displayed coupled system. The vertical momentum equation uses the leading pressure $p_0=2\Omega \Phi$. A flow that stays purely horizontal is independent of $z$ in this reduced system; the assertion is not an exact theorem for arbitrary fast solutions of the full rotating Euler equations.

For homogeneous, incompressible, inviscid flow with constant rotation vector and negligible inertia, taking the curl of $2\mathbf{\Omega }\times \mathbf{u}=-\nabla p$ gives $(\mathbf{\Omega }\cdot \nabla )\mathbf{u}=0$. Thus a geostrophic flow is independent of distance along the rotation axis. Body forces modify the relation by their curl; it applies unchanged in unforced exterior regions. A nontrivial exterior column cannot both obey this leading balance and decay along the rotation axis.

In nearly columnar flow between rigid boundaries, a geostrophic contour is a contour of column height divided by the Coriolis parameter. If relative vorticity is negligible, conservation of shallow-water potential vorticity constrains leading geostrophic flow to these contours. Motion across them stretches or squashes columns, generating relative vorticity; for constant $f$, the contours are simply contours of $H$.

For a layer rotating at angular speed $\Omega$, of depth $d$ and kinematic viscosity $\nu$, the Taylor number compares the Coriolis force to viscous damping through the squared ratio $(2\Omega d^2/\nu {)}^2$. Definitions in other geometries may include different geometric factors; this is the plane-layer convention.

The traditional approximation retains only the locally vertical component of planetary rotation in the Coriolis force. It gives the horizontal acceleration $f\hat{\mathbf{z}}\times {\mathbf{u}}_h$ and neglects terms involving the horizontal component of the rotation vector and vertical velocity.

The Coriolis parameter is $f=2\Omega \sin \phi$ for planetary rotation rate $\Omega$ and latitude $\phi$. It controls the vertical component of the Coriolis acceleration acting on horizontal flow.

An f-plane approximates the Coriolis parameter by a constant $f_0$ over a limited horizontal region.

A beta plane retains the leading meridional variation $f=f_0+\beta y$ of the Coriolis parameter.

An inertial oscillation is the horizontal circular motion of an unforced parcel in a rotating fluid. On an f-plane, $\dot{u}-fv=0$ and $\dot{v}+fu=0$, so the speed is constant and the angular frequency is $|f|$.

On a beta plane, the turning rate of an inertial oscillation is larger on the poleward side of its orbit than on the equatorward side. For initial speed $V$ and $\beta V/f_0^2\ll 1$, this asymmetry produces westward mean drift of magnitude $\beta V^2/(2f_0^2)$.

Potential vorticity combines vorticity with vortex stretching and density stratification. It also applies to nonrotating stratified flow through Ertel potential vorticity. In the inviscid shallow water equations,where $\zeta$ is relative vorticity and $h$ is layer thickness.

Potential-vorticity inversion recovers a balanced velocity field from potential vorticity and specified boundary and global data. In two-dimensional vortex dynamics this is a Poisson equation; shallow-water quasi-geostrophic inversion and stratified quasi-geostrophic inversion use different elliptic boundary value problems. Advance the materially conserved scalar, then invert again to recover the evolving flow. Boundary buoyancy can induce flow with zero interior anomaly. The balance relation and boundary conditions are essential: unrestricted gravity waves are additional degrees of freedom. McIntyre's potential-vorticity account explains the general invertibility principle.

In two-dimensional vortex dynamics, solve the Poisson equation for the streamfunction with the prescribed boundary conditions. On the plane its fundamental solution is $(2\pi {)}^{-1}\log |\mathbf{x}|$, so differentiating the displayed inverse gives the planar Biot-Savart kernel. The harmonic function ${\psi }_h$ represents boundary and global-flow data. An additive constant changes no velocity; circulation around holes and a uniform flow are additional degrees of freedom if their boundary or far-field values are not prescribed. The same inverse acts independently on each level in nonrotating layerwise-two-dimensional vortex dynamics.

The shallow-water quasi-geostrophic potential vorticity relation is a modified Helmholtz equation for the quasi-geostrophic streamfunction. Its inverse with specified boundary conditions recovers horizontal geostrophic flow and surface elevation $\eta =f_0\psi /g$. The Rossby deformation radius $L_D=\sqrt{gH}/|f_0|$ is the range of the balanced response: for a horizontal Fourier component, $\hat{\psi }=-\hat{q}/(|{\mathbf{k}}_h{|}^2+L_D^{-2})$, after subtracting background potential vorticity.

The three-dimensional quasi-geostrophic potential vorticity relation defines an anisotropic elliptic boundary value problem. Interior potential vorticity and boundary buoyancy perturbation $b=f_0{\psi }_z$, together with lateral and global conditions, determine the quasi-geostrophic streamfunction up to its allowed constant. If two solutions have the same data, their difference has zero elliptic source and homogeneous boundary data. Integration by parts gives zero integral of $|{\nabla }_h\psi {|}^2+(f_0^2/N^2)|{\psi }_z{|}^2$, proving uniqueness modulo a constant in a connected domain when the boundary terms vanish. Eady edge waves show that zero interior source alone is insufficient.

The Prandtl ratio compares horizontal and vertical scales in rotating density stratification. In the three-dimensional quasi-geostrophic potential vorticity operator, horizontal curvature scales as $\psi /L^2$ and vertical stretching as $(f_0^2/N^2)\psi /H^2$. Comparable terms therefore require $H/L\sim |f_0|/N$. For constant $N$, the stretched coordinate $Z=(N/|f_0|)z$ makes the operator isotropic. This aspect ratio concerns rotational and stratification effects, not the viscous-to-thermal diffusivity Prandtl number.

For constant nonzero Coriolis parameter and positive constant buoyancy frequency, the anisotropic quasi-geostrophic potential vorticity operator becomes the ordinary three-dimensional Laplacian in the displayed coordinates. Its decaying whole-space inverse is $\psi (\mathbf{X})=-\int [Q({\mathbf{X}}^{\prime })-f]/(4\pi |\mathbf{X}-{\mathbf{X}}^{\prime }|)d^3{X}^{\prime }$. Boundary or far-field data are needed to exclude arbitrary harmonic additions.

A uniform positive quasi-geostrophic potential vorticity anomaly in a sphere of stretched radius $R$ has the displayed regular, decaying streamfunction. Both it and its radial derivative match at the sphere. In its equatorial plane the horizontal relative vorticity is $2Q_0/3$ inside and $-Q_0{R}^3/(3r^3)$ outside. The exterior potential-vorticity anomaly remains zero because its vertical stretching contribution compensates that relative vorticity. A stretched sphere corresponds to a flattened ellipsoid in physical coordinates when $N\gg |f|$.

The spatial gradient of potential vorticity controls the anomaly created when a parcel is displaced. The linear potential-vorticity equation contains ${\mathbf{u}}^{\prime }\cdot \nabla \bar{q}$. A gradient of the Coriolis parameter contributes planetary vorticity, but gradients of relative vorticity and layer stretching can be equally important.

A potential vorticity budget has the formThe material derivative follows the fluid motion and $S_q$ collects forcing or dissipation. With $S_q=0$ it becomes potential-vorticity conservation. For a quasi-geostrophic streamfunction, horizontal advection is $J(\psi ,q)={\psi }_x{q}_y-{\psi }_y{q}_x$.

In unforced adiabatic inviscid motion, potential vorticity is conserved following a parcel:The linearization about a basic state is $({\partial }_t+\bar{\mathbf{u}}\cdot \nabla )q^{\prime }+{\mathbf{u}}^{\prime }\cdot \nabla \bar{q}=0$. This separates advection of a disturbance from its generation by displacement across a basic potential-vorticity gradient.

If quasi-geostrophic potential vorticity initially has gradient $\beta$ across a channel of width $W$ and is homogenized, a height-independent change of zonal velocity obeys $\delta Q=-\delta U_y$. Requiring $\delta U(0)=\delta U(W)=0$ gives uniform final potential vorticity equal to its original channel mean, $\delta U=\beta y(y-W)/2$, and channel-mean momentum change $-\beta W^2/12$. The negative change for $\beta >0$ is wave drag associated with mixing, not a loss of total momentum in an isolated unforced system.

In zonally averaged barotropic quasi-geostrophic approximation flow with zero mean meridional transport, the Taylor identity for quasi-geostrophic flux relates the meridional potential vorticity flux to the convergence of Reynolds stress. A local downgradient closure $\overline{v^{\prime }{q}^{\prime }}=-\kappa {\overline{Q}}_y$, with $\kappa \ge 0$, therefore gives westward mean acceleration where the background gradient is positive. Rossby wave breaking can stir PV into fine filaments; coarse-graining or weak diffusion produces the inferred mixing. Globally the momentum-flux convergence integrates to boundary stresses, so localized westward changes need compensating transport or momentum changes elsewhere in an isolated system.

Ertel potential vorticity is the scalar product of absolute vorticity with the gradient of a materially conserved stratifying scalar. In Boussinesq flow with buoyancy $b$,It is materially conserved by inviscid, adiabatic motion.

For dry atmospheric thermodynamics, choose potential temperature as the materially conserved scalar in Ertel's theorem. The displayed density-normalized scalar has units $\mathrm{K}{\mathrm{m}}^2\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-1}$. Its conservation in adiabatic inviscid motion depends on conservative body forces and a composition-fixed equation of state. It differs in normalization from the Boussinesq approximation convention using buoyancy without the variable density factor. Large-scale balanced weather can be organized by its distribution, with potential-vorticity inversion additionally requiring balance and boundary information.

For a composition-fixed equation of state, let $\dot{\theta }=D\theta /Dt$ and let $\mathbf{F}$ be nonconservative force per unit mass. The vorticity and scalar-gradient evolution equations give the source terms in the title. Spatially varying heating projected along absolute vorticity and the curl of friction can generate or destroy Ertel potential vorticity. Heating that changes potential temperature does not automatically change PV everywhere: its spatial gradient and orientation matter. Unresolved irreversible mixing can also alter the parcelwise conservation picture.

On potential-temperature surfaces, $q=({\zeta }_{\theta }+f)/{\sigma }_{\theta }$, where ${\sigma }_{\theta }=-(1/g)\partial p/\partial \theta$ is isentropic mass density. In adiabatic frictionless flow it is materially conserved. Diagnosing a velocity field also requires balance and boundary data.

For an adiabatic, inviscid parcel, reduced isentropic mass thickness requires reduced absolute vertical vorticity; increased mass thickness requires the reverse. A rigid-upper-boundary geometric cartoon with layer depth $H-h_m$ gives an anticyclonic tendency while crossing a mountain in the Northern Hemisphere. Actual isentropes need not behave like a rigid lid. Blocking, meridional displacement, heating and friction must be assessed before applying that cartoon to real terrain.

${\sigma }_{\theta }=-(1/g)\partial p/\partial \theta$ measures mass per horizontal area per unit potential-temperature interval under hydrostatic balance. It is positive in a statically stable atmosphere.

Heating over land often favours a deeper cyclonic thermal low, while cooling favours a shallower anticyclonic thermal high. Diabatic heating and surface friction violate the assumptions of simple parcel PV conservation; a vertical profile is needed for a quantitative statement.

Spatially varying drag has a vorticity tendency beyond simple damping of existing relative vorticity. The cross-gradient term depends on wind direction and the gradient of surface resistance. Increasing roughness on moving from ocean to land does not alone specify a universal cyclonic or anticyclonic sign. Seasonal heating also changes stratification and isentropic mass thickness, so it cannot generally be replaced by adiabatic PV conservation.

An inviscid fluid with conservative body force, a materially conserved stratifying scalar $\alpha$, and an equation of state making $\nabla \rho ,\nabla p,\nabla \alpha$ coplanar conserves its Ertel potential vorticity. The stretching of absolute vorticity per unit density cancels the deformation of the scalar gradient; the remaining baroclinic triple product vanishes. In an incompressible density-stratified ideal fluid, choosing the advected density as $\alpha$ gives that cancellation directly.

Shallow-water potential vorticity is materially conserved by inviscid, unforced shallow water equations. Linearizing about rest with depth $H_0$ gives an anomaly proportional to

For inviscid nonrotating shallow water equations, columnar horizontal flow has material vorticity equation $D\zeta /Dt=-\zeta {\nabla }_h\cdot {\mathbf{u}}_h$ and depth equation $Dh/Dt=-h{\nabla }_h\cdot {\mathbf{u}}_h$. Thus $\zeta /h$ is conserved, not $\zeta$ itself. Squashing a column to much smaller depth decreases its vertical vorticity. This permits a late draining flow to approximate an irrotational $u_{\theta }=C/r$ profile outside the drain, despite nonzero initial solid-body vorticity. At finite nonzero depth the residual vorticity need not be exactly zero. In a rotating frame, replace $\zeta$ by absolute vorticity $\zeta +f$ in the conserved ratio. Kelvin's circulation theorem remains compatible with these changes because it follows a material loop, not a fixed spatial circle.

For an upstream irrotational current on an f-plane in a rigid layer of height $h$, a circular region of height $h(1-ϵ)$ has the displayed relative vorticity if its fluid retains upstream potential vorticity $f/h$. To leading order in the height contrast, take $u=-{\Psi }_y$, $v={\Psi }_x$ and $\Psi =-Uy+\psi$. Solving the Poisson equation for $\psi$ gives $\psi =-C{r}^2/2$ inside and $\psi =C{a}^2[\log (a/r)-1/2]$ outside, where $C=fϵ/2$. The disturbance is clockwise for positive $fϵ$. Variable-depth transport and the transition across the rim introduce higher-order corrections to this leading columnar inversion.

For the circular topographic anticyclone in a positive uniform current $U$, the opposing disturbance speed is at most $Ca=fϵa/2$. Below the displayed threshold $u$ remains positive everywhere, so no streamline can close. Above it an interior stagnation point at $(0,-U/C)$ is a center, while the exterior point $(0,-C{a}^2/U)$ is a saddle. At equality the two meet at the rim. This is the onset criterion; once closed contours form, the potential vorticity of trapped fluid is not determined solely by its present upstream conditions.

For a steady, slowly varying rotating channel, negligible transverse velocity and zero shallow-water potential vorticity give $u_y=f$. Cross-channel geostrophic balance gives $g{h}_y=-fu$, producing the displayed profiles. The specific energy $E=h+u^2/(2g)$ is constant across the channel, and $E+H$ is constant along it. Zero potential vorticity is not the value in an ordinary finite-depth reservoir at rest in the rotating frame; it requires vanishing absolute vorticity or a deep-reservoir approximation.

At a hydraulic control, the specific-energy function of a zero-potential-vorticity rotating channel flow is stationary with respect to wall depth at fixed discharge and width. Writing $v=u_1+fb/2$ gives $v(v+fb/2)=g{h}_1$ and $Q_c=b{v}^3/g$, hence the displayed discharge. The fully wetted positive-flow solution requires $fb<\sqrt{2g{h}_1}$; at equality the other wall dries and the first wall velocity vanishes. Beyond that limit the assumed two-wet-wall profile must be replaced.

For a constant-depth resting layer on an f-plane, the depth-scaled linear potential vorticity anomaly isIt satisfies $q_t=0$ under the linearized shallow water equations. The physical perturbation of $(f+\zeta )/(H+\eta )$ is $q/H$. The signed scalar $q$, not its absolute value, is used in potential-vorticity inversion and balanced energy pairings.

For small Rossby number and small free-surface displacement, shallow-water potential vorticity reduces, up to a constant factor and additive constant, to relative vorticity plus background PV gradients and the stretching term $-\psi /L_D^2$.