Ocean gyre

An ocean gyre is a basin-scale rotating circulation driven principally by wind stress and shaped by planetary rotation and boundaries.

Sverdrup balance equates meridional advection of planetary vorticity to wind-stress curl in the weakly frictional ocean interior.

With depth transport $(-{\psi }_y,{\psi }_x)$ and a depth-mean bottom-drag closure, the wind-driven equation is $\beta {\psi }_x=S-r{\nabla }^2\psi +\nu {\nabla }^4\psi$, where $S$ is wind-stress curl divided by density and $r=\gamma /({\rho }_{0}H)$ is the drag rate. Its zonal boundary modes solve $\nu {\lambda }^3-r\lambda -\beta =0$. The Munk boundary layer and Stommel boundary layer are different limits of this combined equation.

If $r^3\gg {\beta }^2\nu$, the broad western Stommel boundary layer has thickness $r/\beta$, but small viscosity supplies no-slip skins of thickness $(\nu /r{)}^{1/2}$ at both east and west walls. The western solution therefore contains two decaying exponentials, while the east has the rapidly decaying skin. Setting viscosity identically to zero removes the modes needed to satisfy all four lateral no-slip conditions.

For a uniform normalized upper-layer wind stress curl $W$ and no lower-layer forcing, a uniform local Sverdrup balance isThe upper layer flows meridionally and the lower layer rests. Although the relative vorticity vanishes, the potential vorticity has the zonal gradients ${\bar{q}}_{1,x}=-FV$, ${\bar{q}}_{2,x}=FV$. These are essential for the perturbation stability and do not vanish merely because the background velocity is uniform.

Let $K^2=k^2+l^2$, $A=K^2+F$, $\mathcal{B}=K^2(K^2+2F)$ and use $e^{i(kx+ly-\sigma t)}$. The uniform upper-layer flow $V$ over a resting lower layer has the dispersion relationIts discriminant is $4F^2{\beta }^2{k}^2+\mathcal{B}{K}^2(K^2-2F)l^2{V}^2$. A negative value gives exponential baroclinic instability. This model includes advection across the interface-induced potential-vorticity gradients in a meridional two-layer current.

For purely meridional wavevectors, $k=0$, $l\ne 0$, the growth rate of a uniform meridional upper-layer flow $V$ over a resting lower layer isGrowth requires $V\ne 0$ and $0<l^2<2F$. The perturbation has no northward velocity, so planetary vorticity advection does not explicitly stabilize it. At fixed normalized wind stress curl, $V=W/\beta$ means stronger $|\beta |$ nevertheless reduces the shear and growth rate.

For ${\bar{\psi }}_1=Vx$ and ${\bar{\psi }}_2=0$, the two-layer quasi-geostrophic potential vorticity is ${\bar{q}}_1=f_0+\beta y-FVx$, ${\bar{q}}_2=f_0+\beta y+FVx$. A perturbation therefore feels both the northward gradient $\beta$ and opposite zonal gradients $\mp FV$. Dropping the zonal gradients removes the instability mechanism from the linearization.

The unbounded constant-depth steady transport response to a unit point wind-curl source solves $r{\nabla }_h^2\psi +\beta {\psi }_x=H\delta (x)\delta (y)$ for $r,\beta >0$. An exponential substitution reduces it to a modified Helmholtz equation, yielding the expression shown in terms of the Modified Bessel function of the second kind. Its finite-level streamfunction contours are nearly circular near the source and stretched into a western wake far away. The wake width is of order $\sqrt{(r/\beta )|x|}$.

An ocean transport streamfunction defines depth-integrated horizontal transport. For constant depth $H$, it is $H$ times the velocity streamfunction. Its difference between two boundaries gives signed transport across a connecting line.

The ocean transport streamfunction convention $Hu=-{\psi }_y$, $Hv={\psi }_x$ gives relative vorticity $\zeta =H^{-1}{\nabla }^2\psi -H^{-2}\nabla H\cdot \nabla \psi$. This identity is exact for variable positive depth and does not assume that depth variations are small.

For an island in a constant-depth ocean with Sverdrup balance, choose a counterclockwise contour running from the island's southern tip east to the basin wall, north along that wall, west to the northern tip, and south along the island's western coast. If that contour avoids dissipative boundary layers, the ocean transport streamfunction on the island isHere $\mathbf{W}$ is the wind acceleration appearing in the depth-averaged momentum equation. The basin-boundary streamfunction is zero. This integral fixes the otherwise undetermined constant on the island without requiring a friction law, as in Godfrey's original island-circulation model.

When depth varies but the Coriolis parameter is constant, forcing can balance advection of $f/H$ rather than advection of planetary vorticity. This topographic potential-vorticity gradient plays the role normally played by the beta effect.

With $f=f_0+\beta y$ and parabolic depth, the topographic potential-vorticity pseudovelocity is $(-\beta /H,-f{H}_x/H^2)$. For positive $f,\beta$, it points west and turns south on the western slope, north on the eastern slope. Its characteristics are the parabolas $y=Qax(L_x-x)/\beta -f_0/\beta$, where $Q=f/H$. Wind-driven ocean gyres are steered by these depth gradients; drag and boundary data determine the actual ocean transport streamfunction. The zero-depth shoreline is a singular limit that requires a separate physical regularization.

In the steady small-Rossby number transport equation, this coefficient field advects the ocean transport streamfunction along contours of background shallow-water potential vorticity $f/H$. It is not a physical velocity: when $\psi$ has units of volume transport, its units are inverse length squared per time. For uniform depth on a beta plane it points westward, $\tilde{\mathbf{u}}=(-\beta /H,0)$. With no forcing or drag, $\psi$ is constant on each connected characteristic; locally $\psi =\Phi (f/H)$ at regular points.

In a slowly varying inviscid layer, parcels tend to follow contours of $f/H$. If depth increases in the direction of motion, conservation of potential vorticity requires a poleward displacement or a compensating change of relative vorticity. Ekman pumping permits streamlines to cross those contours.

A western boundary current is a narrow, intense return flow that closes a broad wind-driven interior circulation. The beta effect selects the western side for this frictional boundary layer.

The Stommel boundary layer closes a Sverdrup interior using linear drag. Balancing beta-effect advection against drag gives widthfor the normalization in which quasi-geostrophic drag is $-\gamma {\nabla }^2\psi$.

A Munk boundary layer closes a Sverdrup balance interior using lateral viscosity. At a zonal wall, balancing $\beta {\psi }_x$ with $\nu {\nabla }^4\psi$ gives the width $(\nu /\beta {)}^{1/3}$.