Quasi-geostrophic potential-vorticity equation (QGPV equation)

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,

For a small steady lower-boundary height $b(x,y)$, impermeability gives $w={\mathbf{u}}_g\cdot \nabla b$ at the reference plane. Substitution in the buoyancy equation $D_g(f_0{\psi }_z)+N^{2}w=0$ yields the displayed boundary condition. For nonzero Fourier modes in a uniform crossing current, this fixes ${\psi }_z(0)=-N^{2}b/f_0$. An upper radiation condition or decay condition is still needed to select the physical response.

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.

For constant buoyancy frequency, density forcing $r_0{e}^{-\mu z}\cos kx\cos {\omega }_{0}t$ gives two frequencies $\omega =\pm {\omega }_0$. The quasi-geostrophic streamfunction satisfies ${\lambda }_{\omega }^2=(N^2/f_0^2)(k^2+\beta k/\omega )$ and $C_{\omega }=ig\mu r_0/(2{\rho }_0{f}_0\omega )$. With zero streamfunction at the bottom, a positive ${\lambda }_{\omega }^2$ selects the decaying solution $C_{\omega }(e^{-\mu z}-e^{-{\lambda }_{\omega }z})/({\mu }^2-{\lambda }_{\omega }^2)$. For $\beta ,k>0$, the westward branch propagates vertically when ${\omega }_0<\beta /k$; a radiation condition selects positive vertical group velocity, not merely bounded amplitude. Coincident exponential exponents are handled by the finite limit $-C_{\omega }z{e}^{-\mu z}/(2\mu )$.

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.