Quasi-geostrophic approximation (QG approximation)

The quasi-geostrophic approximation describes slowly evolving, nearly geostrophic flow with small Rossby number. Its dynamics are governed by advection and forcing of potential vorticity.

For equal-depth layers coupled by an interface,The Laplacian terms are relative vorticity; the streamfunction differences encode opposite layer-thickness changes. Each material derivative follows that layer's own velocity. The sum has a barotropic mode without interface displacement, while the difference contains a baroclinic mode.

For inviscid unforced two-layer QG dynamics, $H_1{F}_1=H_2{F}_2=f_0^2/g\prime$ makes the coupling symmetric in the layer-depth inner product. Multiplication by $H_i{\psi }_i$ gives the positive energy $E=\frac{1}{2}[{\sum }_i{H}_i|\nabla {\psi }_i{|}^2+(f_0^2/g\prime )({\psi }_1-{\psi }_2{)}^2]$. It is conserved after integration when the lateral energy flux vanishes. The interface term is available potential energy; the gradient terms are kinetic energy.

Depth-weighted decomposition gives reduced depth $H_r=H_1{H}_2/(H_1+H_2)$ and internal radius $R_d^2=g\prime H_r/f_0^2$. On scale $l$, the ratio of interface available potential energy to baroclinic kinetic energy is of order $(l/R_d{)}^2$. Comparable depths recover $l^2{f}_0^2/(g\prime H_i)$; for strongly unequal depths the thinner layer sets the scale.

The stretching contribution to upper-layer potential vorticity is $-F({\psi }_1-{\psi }_2)$, with the opposite sign below. An interface displacement thickens one layer and thins the other. In a parcel's potential-vorticity equation, its material change balances changes of relative and planetary vorticity and any forcing. The two layer derivatives cannot in general be replaced by one common advection operator.

Define $J(a,b)=a_x{b}_y-a_y{b}_x$ and $\zeta ={\nabla }_h^2\psi$. Combining quasi-geostrophic potential vorticity dynamics with buoyancy evolution eliminates the pressure tendency and givesThe beta term remains when the nonlinear terms are omitted. For a steady weak response to heating about rest, $w=R/N^2$ solves this equation using $\beta {\psi }_x=f_0{R}_z/N^2$ and suitable no-through-flow and decay conditions.

The quasi-geostrophic streamfunction defines horizontal velocity by $u=-{\psi }_y$ and $v={\psi }_x$. For a one-layer free-surface model, geostrophic balance gives $\psi =g\eta /f_0$.

A geostrophic flow is a horizontal flow in leading geostrophic balance. Its velocity follows pressure contours rather than pointing directly down the pressure gradient.

Ageostrophic flow is the part of velocity beyond leading geostrophic flow. Its small horizontal divergence supplies vertical motion and the stretching term in the quasi-geostrophic potential-vorticity equation.

For constant buoyancy frequency $N$ on a beta plane, three-dimensional quasi-geostrophic potential vorticity can be written

The hydrostatic convention $b=f_0{\psi }_z$ may refer to full buoyancy or to its anomaly relative to $N^{2}z$. Subtracting the uniform reference profile changes $\psi$ by $-N^2{z}^2/(2f_0)$ and changes $q=f_0+{\nabla }_h^2\psi +(f_0^2/N^2){\psi }_{zz}$ by the uniform constant $-f_0$. Horizontal velocity and potential-vorticity gradients are unchanged, so the advection and stability dynamics are identical. Stating the convention avoids mistaking this reference constant for a dynamical gradient.

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.

A quasi-geostrophic vertical mode separates horizontal and vertical dependence of the streamfunction. With rigid horizontal boundaries and constant stratification, the vertical eigenfunctions are $\cos (n\pi z/H)$.

A barotropic mode has horizontal velocity independent of depth. It is the zeroth quasi-geostrophic vertical mode and usually has the largest gravity-wave speed and Rossby deformation radius.

A baroclinic mode has nontrivial vertical shear and at least one change of vertical structure through the fluid depth. Higher quasi-geostrophic vertical modes have smaller gravity-wave speeds and Rossby deformation radii.

The quasi-geostrophic potential-vorticity equation states that three-dimensional quasi-geostrophic potential vorticity is materially conserved in inviscid, adiabatic flow. On a beta plane with constant buoyancy frequency,

With basic zonal velocity $U(z)$, basic horizontal buoyancy gradient $b_y=-f_0{U}_z$, and $b^{\prime }=f_0{\psi }_z^{\prime }$, adiabatic buoyancy evolution at a flat impermeable wall givesA normal mode thus obeys $(U-c){\hat{\psi }}^{\prime }-U_z\hat{\psi }=0$ at the wall. This dynamical condition, rather than vanishing streamfunction, sets the frequency of an Eady edge wave.

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

In the weak-forcing steady response about rest, diabatically forced quasi-geostrophic potential vorticity gives $\beta {\psi }_x=(f_0/N^2)R_z$. For $\beta \ne 0$, an eastern zero-flow condition fixes the horizontal circulation through $\psi =-(f_0/\beta N^2){\int }_x^{\infty }{R}_{z}dX$, up to a horizontally uniform pressure gauge. The thermodynamic equation gives $w=R/N^2$. A western zonal wake can remain even though the heating and vertical velocity vanish there. This is a formal steady interior solution: specified initial boundary buoyancy can prevent it from being the attained steady state. For example, under a rigid lid where the heating vanishes, initially uniform boundary buoyancy remains uniform at linear order.

In a log-pressure coordinate, quasi-geostrophic potential vorticity uses density-weighted vertical stretching. For stratification $N^2=RS/H$ and geostrophic flow ${\mathbf{u}}_g=(-{\psi }_y,{\psi }_x,0)$, the inviscid equation is $D_{g}q/Dt=0$. Combining horizontal vorticity balance with the thermal equation eliminates ageostrophic flow and gives this conservation law.

For scales $U,L,D$, thermal balance estimates $w\sim f_0{U}^2/(N^{2}D)$. Density-weighted continuity gives horizontal ageostrophic flow of order $Lw/\min (D,H)$. It is small relative to $U$ only ifThis condition supplements small Rossby number and small relative variation $\beta L/|f_0|$ of the Coriolis parameter.

An isopycnal displacement is the vertical displacement $\xi$ of a constant-density surface. In linear stratified quasi-geostrophic flow,

A quasi-geostrophic mountain wave is a stationary Rossby wave forced when a geostrophic basic flow crosses topography. Its vertical wavenumber is real only when the intrinsic Rossby-wave phase speed permits vertical propagation.

Thermal damping makes the vertical wavenumber of a stationary topographic Rossby wave complex. A vertically propagating wave then decays while oscillating, whereas an inviscidly evanescent disturbance acquires both decay and phase tilt; both can exert wave drag.