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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.

On a sphere of radius $R$ rotating at angular speed $\Omega$, the Coriolis parameter is $f=2\Omega \sin \varphi$. Near latitude ${\varphi }_0$, write $y=R(\varphi -{\varphi }_0)$, use locally eastward and northward Cartesian coordinates, and expandThe beta plane retains the first northward variation of planetary vorticity while neglecting higher latitude dependence and metric curvature. A midlatitude local calculation assumes $L/R\ll 1$ and $|\beta |L/|f_0|\ll 1$; near the equator the distinct equatorial beta plane has $f_0=0$, so the midlatitude low-frequency reduction used below does not apply.

For a homogeneous shallow layer, the hydrostatic approximation gives $p=p_{\mathrm{atm}}+\rho g(\eta -z)$ and hence $p_x/\rho =g{\eta }_x$, $p_y/\rho =g{\eta }_y$, independently of depth. The approximation follows from the small aspect ratio $H/L$ and neglect of vertical acceleration. Differentiate the linear horizontal momentum equations in $z$. Their depth derivatives obeyStarting from rest gives $u_z=v_z=0$ initially, and the unique solution remains zero. This establishes depth independence of hydrostatic shallow-water flow; an arbitrary pre-existing shear would not be removed merely by taking the hydrostatic approximation. Integrating the continuity equation between the rigid bottom and the moving surface gives ${\eta }_t+H(u_x+v_y)=0$ at linear order. With the depth-integrated shallow-water transports $U=Hu$, $V=Hv$, one obtains

It is useful to make the coefficient approximation in the height reduction explicit. Put ${\mathcal{F}}_y={\partial }_t^2+f(y{)}^2$. Differentiating the two momentum equations in time and eliminating the other transport givesSince ${\partial }_y({\mathcal{F}}_{y}V)={\mathcal{F}}_y{V}_y+2f\beta V$, taking their horizontal divergence and using the continuity equation givesApplying ${\mathcal{F}}_y$ proves the exact variable-Coriolis shallow-water height equationAt the reference latitude, or after the usual local freezing of undifferentiated $f$ factors to $f_0$, this gives the height relation written with ${\mathcal{F}}_0={\partial }_t^2+f_0^2$. With $f=f_0+\beta y$ over a finite region, that constant-coefficient version is a local approximation, not an exact identity. Keeping ${\mathcal{F}}_y$ as above avoids silently commuting a variable Coriolis parameter through a spatial derivative.

For the slow Rossby wave branch, take $|\omega |/|f_0|\ll 1$, approximate ${\mathcal{F}}_0$ by $f_0^2$, and discard the two time-derivative terms on the right compared with $-f_0^2{\eta }_x$. This is the regular midlatitude long-time ordering, with nondegenerate zonal variation. If $a=f_0/c$, the result isThe printed low-frequency equation has the opposite right-hand sign. The minus sign follows directly from the preceding eliminated equation and is also the sign needed for the printed isofrequency-circle centre. An independent check uses geostrophic balance: $\psi =g\eta /f_0$, $(u,v)=(-{\psi }_y,{\psi }_x)$, and the shallow-water quasi-geostrophic potential vorticity is ${\nabla }_h^2\psi -a^2\psi +\beta y$. Linear potential-vorticity conservation gives $({\nabla }_h^2\psi -a^2\psi {)}_t+\beta {\psi }_x=0$.

For $\eta =A{e}^{i(kx+ly-\omega t)}$, the shallow-water Rossby-wave dispersion relation isFor fixed $l$ and $\beta >0$, it is odd in $k$, zero at $k=0$, negative for $k>0$, and tends to zero from below as $k\to +\infty$. Its minimum occurs at $k=\sqrt{l^2+a^2}$ with ${\omega }_{\min }=-\beta /[2\sqrt{l^2+a^2}]$. Thus both long and short waves have small frequency. The zonal phase velocity is westward, $\omega /k<0$, whereas the zonal group velocity isIt changes sign at the frequency minimum.

At fixed nonzero frequency, completing the square yields the Rossby-wave isofrequency circleThe radius is real only if $|\omega |\le |\beta |/(2|a|)$. For $\beta >0$, the branch with $k>0$ has $\omega <0$ and a circle centred on the positive $k$-axis. If instead the printed plus-sign wave equation is taken literally with this same Fourier convention, its dispersion is $\omega =+\beta k/(k^2+l^2+a^2)$ and its circle is centred at $(+\beta /(2\omega ),0)$. These two conventions cannot be mixed.

For reflection of a Rossby wave at a meridional wall, the stationary wall preserves frequency, and its translation invariance in $y$ preserves the tangential wavenumber. Thus ${\omega }_r={\omega }_i$, $l_r=l_i$. Both $k$ values solveTheir sum and product giveAn incident wave in $x>0$ travels toward the wall in group velocity, not necessarily in phase velocity. For $\beta >0$, $0<k_i<\sqrt{l_i^2+a^2}$ has $c_{gx,i}<0$, while its partner $k_r>\sqrt{l_i^2+a^2}$ has $c_{gx,r}>0$. The double-root case has zero normal group velocity and does not describe a wave packet incident on the wall.

At leading quasi-geostrophic approximation, the impermeability condition is $u=-g{\eta }_y/f_0=0$. For $l_i\ne 0$, the boundary condition givesThe reflected height therefore has equal amplitude and a phase change of $\pi$. Equality here concerns the height or quasi-geostrophic streamfunction amplitudes in the reduced model. At the degenerate $l_i=0$, geostrophic no-normal-flow is automatically satisfied and alone does not determine their ratio; the usual homogeneous wall-streamfunction condition supplies $A_r=-A_i$ if imposed. Retaining the small ageostrophic transport changes the boundary condition towhich tends to $-1$ in the regular low-frequency ordering but is not generally of unit modulus. In particular, $l_i=0$ gives $A_r/A_i=-k_i/k_r$ in that more complete boundary relation. Thus the printed equal-amplitude assertion requires the nondegenerate leading quasi-geostrophic approximation, or an explicit homogeneous wall condition; it is not a general exact shallow-water reflection law.

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.

The potential-vorticity gradients in a meridional two-layer current are important here: besides ${\bar{q}}_{n,y}=\beta$, one has ${\bar{q}}_{1,x}=-FV$ and ${\bar{q}}_{2,x}=FV$. They must be retained when linearizing, even though the basic relative vorticity vanishes.

Let $K^2=k^2+l^2$ and $A=K^2+F$. For a normal mode the two-layer quasi-geostrophic potential vorticity amplitudes areBecause the basic velocities are northward $(V,0)$ in the two layers, their advection frequencies are $\sigma -lV$ and $\sigma$. The uniform forcing has no perturbation, $W^{\prime }=0$. The linear equations are thereforeIn particular, a perturbation's zonal velocity advects the basic interface-induced zonal potential-vorticity gradient. Substitution of the plane wave givesA nonzero disturbance exists exactly when the determinant vanishes. The requested relation for linear stability of a meridional two-layer current isEquivalently, with $\mathcal{B}=K^2(K^2+2F)$,Its discriminant isFor $K>0$, exponential baroclinic instability occurs when this discriminant is negative; otherwise the two frequencies are real. This also displays the stabilizing contribution of the planetary vorticity gradient when $k\ne 0$. As a check, $V=0$ gives the uncoupled barotropic mode and baroclinic mode frequencies $-\beta k/K^2$ and $-\beta k/(K^2+2F)$.

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.

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.

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.