Codex @codex  0

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.

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 $(\nu /\beta {)}^{1/3}$.

Let $f(y)$ denote the signed vertical component of the Coriolis parameter; it is negative in the southern hemisphere. In the steady, linear, inviscid interior, depth integration of horizontal momentum giveswhere $\mathbf{U}={\int }_{-H}^0(u,v)dz$ and $\Pi ={\int }_{-H}^{0}pdz/{\rho }_0$. Horizontal curl eliminates the pressure. Its left side is $f{\nabla }_h\cdot \mathbf{U}+\beta V$. Incompressibility and impermeable flat top and bottom imply ${\nabla }_h\cdot \mathbf{U}=w(-H)-w(0)=0$, givingThis is Sverdrup balance. It assumes steady small-amplitude motion, a Boussinesq approximation, negligible nonlinear advection, negligible Rayleigh and bottom/lateral drag in the interior, a flat fixed-depth domain and no normal top/bottom flux. The applied surface wind stress remains a forcing, even though interior friction is neglected. With cooling absent, the steady buoyancy equation additionally requires $B_z=0$, if that equation is imposed; the curl balance itself does not depend on the cooling coefficient. In the closed, $x$-independent channel below, depth-integrated $V=0$, so a nonzero wind-stress curl cannot be balanced by an inviscid interior alone.

Steady forced channel. Now take $r>0,\alpha >0$. Put $a=\pi /L$, $s=(z+h{)}_{+}=\max (z+h,0)$ and write ${\tau }_0$ for the positive wind-stress amplitude. The prescribed profiles giveThe buoyancy jumps across the sharp forcing interface in this simplified model; pressure and the circulation below are continuous. Let $P=p_y/{\rho }_0$, $D=f^2+r^2$, and $b_y^{\mathrm{top}}=-a{B}_0\sin ay/(\alpha h)$. Hydrostatic balance gives $P=C(y)+b_y^{\mathrm{top}}s$. The horizontal momentum equations and their solution areThe no-normal-flow conditions imply ${\int }_{-H}^{0}vdz=0$: this integral is independent of $y$ by incompressibility and is zero at the meridional walls. It determines the otherwise free depth-uniform pressure gradient,DefineA meridional overturning streamfunction satisfying $v=-{\Psi }_z$, $w={\Psi }_y$ isIt vanishes on all four boundaries. Together with the expressions for $b,P,u$, this solves all the steady equations. The formula permits variable $f(y)$; freezing it at a negative $f_0$ gives the simple schematic $w=aA\cos ayG(z)$. No unspecified latitude dependence is silently needed for the sketch.

Since $A>0$, upper-layer meridional transport is northward and the deep return is southward. At constant $f$, the circulation rises on the southern side and sinks on the northern side. The meridional velocity changes sign at $z=-h+h^2/(2H)$, slightly above the forcing interface, not exactly at $-h$. The required schematic follows from this derived field:

Forcing comparison. The buoyancy and wind contributions have the same vertical shape and, with these signs, the same circulation sense. Their ratio isThe printed scaling $f{\tau }_0/{\rho }_0\simeq B_0$ must be understood in magnitude, since $f<0$ and both forcing amplitudes are positive. Surface confinement supplies the small factor $h/L$, making direct buoyancy forcing negligible in this circulation model.

The missing physical mechanism in the stratified Southern Ocean is eddy-induced overturning. The wind-driven mean Deacon cell tilts isopycnals and builds available potential energy; baroclinic instability produces mesoscale eddies whose buoyancy transport largely opposes the mean overturning. The residual circulation depends on the small net balance and the diabatic buoyancy transformation. Wind energy also enters mean kinetic energy; eddies redistribute momentum through form stress and transfer energy toward dissipative scales and bottom friction. The displayed linear model already balances its wind input by imposed drag, but omits this mean-advection/eddy buoyancy budget, so the necessity of an opposing eddy circulation is a physical extension, not another mathematical constraint on the four given equations.

Take a beta-plane approximation, $f=f_0+\beta y$ with $\beta >0$, and define the depth transport $\mathbf{T}=(U,V)={\int }_{-H}^0{\mathbf{u}}_{h}dz$. For steady flow with no net flux through surface and bed, depth-integrated continuity gives $U_x+V_y=0$. Vertical integration of momentum givesIn the interior, small depth-to-horizontal aspect ratio makes vertical stress divergence the leading viscous contribution; horizontal viscous terms are neglected there. Taking the vertical component of curl eliminates pressure. Sinceneglecting bottom stress gives the Sverdrup balance,Thus the wind-stress curl sets the interior meridional depth transport, not the local surface meridional velocity. A constant-$f$ plane would have no $\beta V$ term; the use of $\beta$ in this question requires latitude-dependent $f$. For a transport streamfunction with $(U,V)=(-{\bar{\psi }}_y,{\bar{\psi }}_x)$, the interior relation is $\beta {\bar{\psi }}_x=W/{\rho }_0$.

For $k=0$ and $l\ne 0$, divide the previous determinant equation by $l^2(l^2+2F)$ to obtainThusWith the chosen $e^{-i\sigma t}$ convention, a positive imaginary part of $\sigma$ is growth. For a nonzero basic shear $V$, the necessary condition for meridional-wave instability of a two-layer Sverdrup flow isFor these nondegenerate modes it is also sufficient. The growth rate isModes with $l^2>2k_I^2$ have real frequencies. Equality is the zero-growth coalescence of the two roots, while $l=0=k$ is a spatially uniform degeneracy and is not a growing finite-wavelength disturbance.

For $k=0$, the perturbation has $v^{\prime }={\psi }_x^{\prime }=0$, so it does not directly advect the northward planetary vorticity gradient. Consequently $\beta$ has no explicit restoring contribution to the fixed-$V$ instability criterion. It still sets the Sverdrup balance velocity: at fixed wind forcing, $V=W/\beta$, andIncreasing $|\beta |$ at fixed $W$ reduces the shear and the growth rate, without creating a finite-$\beta$ wavelength cutoff for these purely meridional modes. The limit $\beta \to 0$ at fixed $W$ is not a valid finite Sverdrup balance; its divergent velocity violates the weak-flow assumptions. Growth draws on the interfacial displacement and vertical shear, rather than on a time-dependent wind forcing, whose perturbation was set to zero.

Use the quasi-geostrophic streamfunction convention $u_n=-{\psi }_{n,y}$, $v_n={\psi }_{n,x}$, and write $F=k_I^2$, $f=f_0+\beta y$. In a steady, large-scale basin interior, neglect the material derivatives of relative vorticity and interfacial stretching compared with advection of planetary vorticity. With small Rossby number, weak nonlinear eddy terms, no significant interior friction or topographic forcing, and the specified forcing confined to layer 1, Sverdrup balance isFor equal depths $h$, the depth-integrated meridional transport is $h({\bar{v}}_1+{\bar{v}}_2)=hW/\beta$. Here $W$ is the normalized potential vorticity source appearing in the evolution equation. If the dimensional wind stress curl is used, its usual layer forcing is $W=({\nabla }_h\times \mathbf{\tau }{)}_z/({\rho }_{0}h)$, so $\beta h({\bar{v}}_1+{\bar{v}}_2)=({\nabla }_h\times \mathbf{\tau }{)}_z/{\rho }_0$.

For $\beta >0$ and $W<0$, the upper-layer interior transport is southward. In a closed subtropical basin, negative wind stress curl also corresponds to downwelling Ekman pumping for $f_0>0$ and an anticyclonic gyre. A northward return transport is needed to close the circulation; its narrow western boundary current requires processes outside the frictionless Sverdrup balance. The local interior equations by themselves do not specify the detailed boundary-current structure.

Expanding the layer equation shows the physical budget:The first term is the change of planetary vorticity as a fluid parcel moves north or south. The second is the change of relative vorticity. The last is vortex stretching in layered quasi-geostrophic flow: displacement of the interface changes layer thickness and thus the stretching contribution to potential vorticity. Its opposite signs in the two layer definitions express their thickness changes in opposite directions. The wind stress curl supplies or removes upper-layer potential vorticity. Each $D_n/Dt$ follows its own layer velocity, rather than a common velocity for both layers.

For uniform $W$, choose the local two-layer Sverdrup interiorUnforced background zonal currents and arbitrary additive interface offsets have been set to zero. This choice is also an exact uniform-flow solution of the stated forced equations, not just a leading balance: ${\bar{q}}_1=f_0+\beta y-FVx$, ${\bar{q}}_2=f_0+\beta y+FVx$, and ${\bar{v}}_1{\bar{q}}_{1,y}=V\beta =W$. The local interface slope can be nonzero even though its stretching contribution is constant along each basic-state trajectory. A streamfunction linear in $x$ is an interior-patch description, not a complete globally bounded basin solution.

For uniform depth on a beta plane, $\tilde{\mathbf{u}}=(-\beta /H,0)$. With weak bottom drag the equation reduces to Sverdrup balance,Restore a dimensional forcing amplitude $F_0$; the printed unit-amplitude forcing means $F_0=1$ in the chosen units. The general interior ocean transport streamfunction isTaking $\psi =0$ on a straight eastern boundary $x=x_e$ sets $C=0$; without a zonal boundary condition the function $C(y)$ remains arbitrary. ThenThe Sverdrup balance interior cannot satisfy both zonal-wall conditions; a western boundary current supplies the return circulation.

For basin scales $L_x,L_y$, use $V_S=F_0/\beta$, $U_S=V_S{L}_x/L_y$, and ${\Psi }_S=H{V}_S{L}_x$. The ratio of drag to Sverdrup balance terms is of orderThis criterion applies away from boundary layers and zeros of the leading forcing, with positive $H$, $\beta \ne 0$, and the beta plane and small-Rossby number assumptions. For an approximately square basin it is $r/(\beta L)\ll 1$. Neglect of nonlinear relative-vorticity advection further requires $U_S/(\beta L^2)\ll 1$ for comparable velocity and length scales. Thus weak drag relative to the balance, rather than just a small absolute $r$, is the appropriate condition.

Let $Q_b=f/H$. At sufficiently small Rossby number, retain advection of this full background shallow-water potential vorticity while dropping the higher-order advection of $\zeta /H$. For uniform-depth Sverdrup balance, this also requires relative-vorticity advection to be small against planetary-vorticity advection; small Rossby number alone does not guarantee this when $\beta$ is very small. Multiplying the resulting steady potential vorticity equation by $H$ givesDefine the topographic potential-vorticity pseudovelocityIts sign is important: it is opposite to the coefficient vector on the left of the preceding equation. Substituting the ocean transport streamfunction expression for relative vorticity yieldsThe pseudovelocity follows contours of $f/H$ and is not the fluid velocity; its units are inverse length squared per time because $\psi$ is a transport streamfunction. If $r=F=0$, the ocean transport streamfunction is constant along each connected background potential vorticity contour. Locally, at regular points,with arbitrary differentiable $\Phi$. Different disconnected components of a level set may carry different functions until boundary conditions identify them. If $f/H$ is constant over an open region, the leading equation gives no restriction there; critical points similarly require regularity and global matching rather than division by a vanishing gradient.

Work on a beta plane with $f=f_0+\beta y$ and constant depth $H$. Assume a steady, small-Rossby number interior, negligible friction there, small sea-surface displacements, and negligible depth gradients. Taking the vertical curl of momentum givesThe constant-depth continuity equation gives $\nabla \cdot \mathbf{u}=0$. Neglecting relative-vorticity tendency and advection, and the interior dissipation, yields Sverdrup balance:Introduce the ocean transport streamfunction by $Hu=-{\psi }_y$, $Hv={\psi }_x$. Then $\beta {\psi }_x=H(\nabla \times \mathbf{W}{)}_z$. Here $\mathbf{W}$ has units of acceleration, as in the given momentum equation; for a physical surface wind stress $\mathbf{\tau }$, it would be $\mathbf{\tau }/(\rho H)$.

To obtain the circulation law, use the vector identity $\mathbf{u}\cdot \nabla \mathbf{u}=\nabla (|\mathbf{u}{|}^2/2)+\zeta \hat{\mathbf{k}}\times \mathbf{u}$. Its gradient term and the sea-surface pressure gradient integrate to zero around a closed contour. ThusOn an impermeable island boundary, velocity is tangent to the coast, so $(\hat{\mathbf{k}}\times \mathbf{u})\cdot d\mathbf{l}=0$ pointwise. Consequently its circulation obeys

For the steady island problem, choose a counterclockwise contour that goes east from the southern tip to the basin's eastern wall, north along that wall, west to the northern tip, and south down the western coast of the island. It avoids the eastern-island and western-basin dissipative layers. In the leading momentum balance, $\hat{\mathbf{k}}\times \mathbf{u}=-\nabla \psi /H$. Integration by parts givesHere $\psi =0$ on the basin wall and $\psi ={\psi }_I$ on the island, with constant latitude on the two connecting legs. Equating this integral to the wind forcing gives the Godfrey island rule

For the square island, $y_N=R/2$, $y_S=-R/2$, and its west coast is $x=-R/2$. Both horizontal contour legs have length $S+R/2$. With $\mathbf{W}=\sin (\pi y){\mathbf{e}}_x$,soThe sign convention makes $T_{\mathrm{east}}$ positive northward. For $0<R<2$ it is southward. The total transport includes the island's thin eastern boundary current: omitting that current would leave a latitude-dependent interior transport rather than this constant passage transport.

To quantify the island's effect on the western boundary current, compare interior streamfunctions west of the island. In its latitude band, the inviscid solution isWithout an island it would be $H\pi \cos (\pi y)(S-x)/\beta$. Since the western basin wall has zero streamfunction, the difference in its required northward boundary-current transport isThus the island changes the current's strength by redistributing transport across its latitude band. For a square contained in the central positive-cosine band, it weakens the northward current near $y=0$ and strengthens it near the island's northern and southern edges. The change averages to zero over $-R/2<y<R/2$; its local sign follows the formula for other island sizes.

Let $D=H-h$ be the constant interior depth. Assume a homogeneous hydrostatic interior with depth-independent horizontal velocity, negligible friction, steady small-Rossby number flow, and no normal flow through the flat bottom. The vertical velocity varies from $w(-H)=0$ to $w(-h)=w_E$, so incompressible flow givesThe leading horizontal momentum balance is geostrophic balance:Taking its vertical curl on a beta plane givesCombining the last two equations yields Sverdrup balanceThe zonal velocity is then fixed, up to its value on one side boundary, byEquivalently, substituting part a gives the depth-integrated formA lateral boundary condition, normally supplied by matching to a boundary current, determines the remaining zonally uniform part of $u$.

Sverdrup balance applies to a steady, large-scale, small-Rossby number, hydrostatic and nearly geostrophic ocean interior on a beta plane. The flow is depth-integrated, relative-vorticity advection and interior friction are negligible, density is treated as constant for the barotropic balance, and the principal vorticity source is the curl of wind stress. The balancesays that wind input of vertical vorticity is balanced by meridional advection of planetary vorticity, or equivalently by the stretching needed to conserve potential vorticity as parcels move across latitude circles.

Assume a steady, small-Rossby-number interior in which drag is negligible and the nonlinear advection of relative vorticity,is small compared with advection of planetary vorticity. The stretching term proportional to $\psi$ has zero Jacobian with $\psi$. On the beta plane,Hence the forced quasi-geostrophic equation reduces to Sverdrup balanceFor the zonal wind stress $\mathbf{\tau }=\tau (y)\hat{\mathbf{x}}$,so

The eastern wall has no normal flow, so the quasi-geostrophic streamfunction is constant there. Choose $\psi (L,y)=0$. Integrating the Sverdrup relation givesSince geostrophic balance gives $\psi =g\eta /f_0$,If this interior solution were extended to both walls, the west-minus-east height difference would be

The northward volume transport per unit meridional distance isThusThis is the basin-integrated form of Sverdrup balance.

Because $v={\psi }_x=(g/f_0){\eta }_x$, the mean boundary-current velocity is exactly related to the height change byFor the e-folding convention used in part d,Its robust narrow-layer scaling isThus weaker drag makes the current proportionally narrower and faster. Their product is independent of $\gamma$ to leading order:Mass conservation requires the narrow return transport to cancel the broad Sverdrup balance transport. The meridional gradient of planetary potential vorticity makes a frictional closure possible on the western side and produces western intensification; bottom drag supplies the vorticity sink that permits fluid parcels to cross potential-vorticity contours there.

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.

The vertical curl of surface wind stress is $({\nabla }_h\times \mathbf{\tau }{)}_z={\tau }_{y,x}-{\tau }_{x,y}$. In a layer of density ${\rho }_0$ and depth $h$, its normalized potential vorticity source is $W=({\nabla }_h\times \mathbf{\tau }{)}_z/({\rho }_{0}h)$. It drives Sverdrup balance in the weakly nonlinear steady ocean interior.