6.4 Eddies & Gyres

Step 1: Start from the Shallow Water Equations

Consider a single layer of incompressible fluid on a rotating planet. The momentum and continuity equations are:

$$\frac{Du}{Dt} - fv = -g\frac{\partial\eta}{\partial x}, \quad \frac{Dv}{Dt} + fu = -g\frac{\partial\eta}{\partial y}$$

$$\frac{\partial h}{\partial t} + \frac{\partial(hu)}{\partial x} + \frac{\partial(hv)}{\partial y} = 0$$

where $h$ is the layer thickness and $\eta$ is the free surface displacement.

Step 2: Derive the Vorticity Equation

Take $\partial/\partial x$ of the $v$-equation minus $\partial/\partial y$ of the $u$-equation. Define relative vorticity $\zeta = \partial v/\partial x - \partial u/\partial y$:

$$\frac{D}{Dt}(f + \zeta) = -(f + \zeta)\left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}\right)$$

The right-hand side represents vortex stretching/compression due to horizontal divergence.

Step 3: Use Continuity to Eliminate Divergence

From the continuity equation, the horizontal divergence is related to the rate of change of layer thickness:

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = -\frac{1}{h}\frac{Dh}{Dt}$$

Substituting into the vorticity equation:

$$\frac{D}{Dt}(f + \zeta) = \frac{(f + \zeta)}{h}\frac{Dh}{Dt}$$

Step 4: Obtain PV Conservation

Dividing both sides by $h$ and recognizing the material derivative of a ratio:

$$\frac{D}{Dt}\left(\frac{f + \zeta}{h}\right) = 0$$

This is Ertel's theorem for shallow water: the potential vorticity $q = (f + \zeta)/h$ is conserved following a fluid parcel in the absence of friction and diabatic effects. This is one of the most powerful constraints in geophysical fluid dynamics.

Step 5: Physical Consequences

If a parcel moves poleward ($f$ increases), it must either spin down ($\zeta$ decreases) or the layer must thicken ($h$ increases) to conserve $q$. For a barotropic ocean of depth $H$, if $\zeta \ll f$:

$$\frac{f}{H} = \text{const} \implies \text{flow follows lines of } f/H$$

Over a flat bottom, this confines flow to latitude circles. Topographic features ($H$ variations) can steer currents meridionally, acting as effective $\beta$-planes.

Step 1: Linearize the QG Vorticity Equation

Start from the quasi-geostrophic potential vorticity equation with no mean flow. Linearize about a state of rest by setting $\psi = \psi'$ (small perturbation):

$$\frac{\partial}{\partial t}\left(\nabla^2\psi' + \frac{f_0^2}{N^2}\frac{\partial^2\psi'}{\partial z^2}\right) + \beta\frac{\partial\psi'}{\partial x} = 0$$

Step 2: Assume Wave Solutions

Seek plane wave solutions of the form $\psi' = \hat{\psi}\,e^{i(kx + ly - \omega t)}\cos(mz)$, where $k$ and $l$ are horizontal wavenumbers and $m$ is the vertical wavenumber. Substituting:

$$-i\omega\left(-k^2 - l^2 - \frac{f_0^2 m^2}{N^2}\right)\hat{\psi} + ik\beta\hat{\psi} = 0$$

Step 3: Dispersion Relation

Solving for $\omega$ and defining the Rossby deformation radius $L_d = NH/(n\pi f_0)$ for vertical mode $n$:

$$\omega = \frac{-\beta k}{k^2 + l^2 + L_d^{-2}}$$

The phase speed in the $x$-direction is $c_x = \omega/k = -\beta/(k^2 + l^2 + L_d^{-2})$, which is always negative (westward). Long waves ($k^2 + l^2 \ll L_d^{-2}$) travel at the maximum speed $c_{max} = -\beta L_d^2$.

Step 4: Group Velocity

The group velocity determines energy propagation. Taking $\partial\omega/\partial k$:

$$c_{gx} = \frac{\partial\omega}{\partial k} = \frac{\beta(k^2 - l^2 - L_d^{-2})}{(k^2 + l^2 + L_d^{-2})^2}$$

For short waves ($k^2 > l^2 + L_d^{-2}$), the group velocity is eastward -- energy propagates opposite to phase. This has implications for how the ocean adjusts to wind changes: long Rossby waves carry the signal westward.

Step 1: QG Equations with a Mean Shear

Consider a mean zonal flow with constant vertical shear $\bar{U}(z) = \Lambda z$ (the Eady basic state). The linearized QG potential vorticity equation becomes:

$$\left(\frac{\partial}{\partial t} + \bar{U}\frac{\partial}{\partial x}\right)\left(\nabla^2\psi' + \frac{f_0^2}{N^2}\frac{\partial^2\psi'}{\partial z^2}\right) = 0$$

Note: the interior PV gradient vanishes in the Eady model (no $\beta$ term), and all the dynamics come from the boundary conditions.

Step 2: Boundary Conditions (Temperature)

At the upper ($z = H$) and lower ($z = 0$) rigid boundaries, the buoyancy equation provides:

$$\left(\frac{\partial}{\partial t} + \bar{U}\frac{\partial}{\partial x}\right)\frac{\partial\psi'}{\partial z} + \Lambda\frac{\partial\psi'}{\partial x} = 0 \quad \text{at } z = 0,\,H$$

Step 3: Normal Mode Analysis

Assume solutions $\psi' = \hat{\psi}(z)\,e^{ik(x - ct)}$. In the interior, the PV equation gives $\hat{\psi}(z) = A\cosh(\mu z) + B\sinh(\mu z)$ where $\mu = kN/(f_0)$. Applying the boundary conditions yields an eigenvalue problem for the phase speed $c$:

$$c = \frac{\Lambda H}{2} \pm \frac{\Lambda H}{2}\sqrt{\left(\frac{\tanh(\mu H/2)}{\mu H/2} - 1\right)\left(1 - \frac{\mu H/2}{\tanh(\mu H/2)}\right)}$$

Step 4: Instability Criterion

The expression under the square root is negative (giving complex $c$ and hence growing modes) when $\mu H < \mu_c H \approx 2.399$, i.e., when:

$$kL_d < \frac{2.399}{\pi} \approx 0.76 \quad \Longrightarrow \quad \lambda > 2\pi L_d / 0.76 \approx 8.3\,L_d$$

Disturbances with wavelengths longer than about $4L_d$ are baroclinically unstable. The Rossby deformation radius $L_d = NH/f_0$ thus sets the characteristic eddy scale.

Step 5: Maximum Growth Rate

The maximum growth rate occurs at $\mu H \approx 1.61$ and is:

$$\sigma_{max} = 0.31\,\frac{f_0}{N}\,\Lambda = 0.31\,\frac{f_0}{N}\frac{\partial\bar{U}}{\partial z} = 0.31\,\frac{|\nabla_H b|}{N}$$

For typical mid-latitude ocean conditions ($\Lambda \sim 10^{-3}$ s$^{-1}$, $N/f_0 \sim 20$), the e-folding time is $1/\sigma_{max} \sim 1$–2 weeks, explaining why mesoscale eddies develop on timescales of weeks to months.