6.1 Surface Currents

Step 1: Governing Equations in the Ekman Layer

In a steady, horizontally homogeneous Ekman layer with no pressure gradient, the horizontal momentum equations reduce to a balance between the Coriolis force and vertical turbulent friction:

$$f v = A_z \frac{\partial^2 u}{\partial z^2}, \quad -f u = A_z \frac{\partial^2 v}{\partial z^2}$$

Step 2: Form a Complex Variable

Define the complex velocity $W = u + iv$. Multiply the second equation by $i$ and add to the first:

$$A_z \frac{\partial^2 W}{\partial z^2} = if W$$

This is a second-order ODE with constant coefficients. The characteristic equation is $A_z m^2 = if$, giving $m = \pm(1+i)\sqrt{f/(2A_z)}$.

Step 3: Define the Ekman Depth Scale

Introduce the Ekman depth parameter $a = \sqrt{f/(2A_z)} = \pi/D_E$, so $D_E = \pi\sqrt{2A_z/f}$. The general solution is:

$$W(z) = C_1 e^{(1+i)az} + C_2 e^{-(1+i)az}$$

Step 4: Apply Boundary Conditions

Require $W \to 0$ as $z \to \infty$ (taking $z$ positive downward), so $C_1 = 0$. At the surface ($z = 0$), the wind stress provides the boundary condition $\rho A_z\,\partial W/\partial z\big|_{z=0} = \tau$. For wind along the $y$-axis ($\tau = \tau_y$):

$$C_2 = \frac{\tau_y}{\rho A_z (1+i)a} = V_0\,e^{i\pi/4}$$

Step 5: Extract Real Components

Substituting and taking real and imaginary parts yields the Ekman spiral velocity components:

$$u(z) = V_0\,e^{-az}\cos\!\left(\frac{\pi}{4} - az\right), \quad v(z) = V_0\,e^{-az}\sin\!\left(\frac{\pi}{4} - az\right)$$

The surface current is deflected 45 degrees to the right of the wind (NH). The velocity decays exponentially and rotates clockwise with depth. At $z = D_E$, the velocity is $e^{-\pi} \approx 4\%$ of the surface value.

Step 6: Depth-Integrated Ekman Transport

Integrating the velocity over the full Ekman layer depth and using $\int_0^\infty e^{-(1+i)az}dz = 1/[(1+i)a]$:

$$\mathbf{M}_E = \int_0^\infty \rho\,\mathbf{u}\,dz = \frac{\boldsymbol{\tau} \times \hat{k}}{f}$$

The net Ekman transport is exactly perpendicular to the wind (90 degrees to the right in the NH), regardless of the eddy viscosity profile. This remarkable result is independent of $A_z$.

Step 1: Start from the Horizontal Momentum Equations

For large-scale, steady ocean flow, the Reynolds-averaged horizontal momentum equations on a rotating Earth are:

$$\frac{Du}{Dt} - fv = -\frac{1}{\rho}\frac{\partial p}{\partial x} + \text{friction}, \quad \frac{Dv}{Dt} + fu = -\frac{1}{\rho}\frac{\partial p}{\partial y} + \text{friction}$$

Step 2: Geostrophic Approximation

For Rossby number $Ro = U/(fL) \ll 1$, the acceleration and friction terms are negligible compared to the Coriolis and pressure gradient terms, yielding geostrophic balance:

$$fv = \frac{1}{\rho}\frac{\partial p}{\partial x}, \quad -fu = \frac{1}{\rho}\frac{\partial p}{\partial y}$$

The flow is along isobars (pressure contours), with high pressure to the right in the NH. Using the hydrostatic relation $p = \rho g \eta$ at the surface, geostrophic velocity can be computed from sea surface height gradients.

Step 3: Depth-Integrated Vorticity Equation

Take $\partial/\partial x$ of the $v$-equation minus $\partial/\partial y$ of the $u$-equation, then integrate vertically over the full ocean depth. Using $f = f_0 + \beta y$:

$$\beta V = \text{curl}_z\!\left(\frac{\boldsymbol{\tau}}{\rho}\right) + \text{bottom friction terms}$$

Step 4: Sverdrup Balance (Interior)

In the ocean interior, far from boundaries, bottom friction is negligible. The depth-integrated meridional transport $V$ is determined solely by the wind stress curl:

$$\beta V = \frac{1}{\rho}\text{curl}_z(\boldsymbol{\tau}) = \frac{1}{\rho}\left(\frac{\partial \tau_y}{\partial x} - \frac{\partial \tau_x}{\partial y}\right)$$

This is Sverdrup's (1947) result: the beta effect converts wind stress curl into a meridional mass transport. It is the fundamental relation linking the wind field to the interior ocean circulation.

Step 5: Sverdrup Streamfunction

Integrating from the eastern boundary $x_E$ (where $\psi = 0$) westward, and using the depth-integrated continuity equation $\partial U/\partial x + \partial V/\partial y = 0$:

$$\psi(x,y) = \frac{1}{\beta\rho}\int_x^{x_E}\text{curl}_z(\boldsymbol{\tau})\,dx'$$

The streamfunction increases westward, and mass balance requires that the return flow occurs in a narrow western boundary layer -- the Sverdrup balance alone cannot close the circulation.

Step 1: Stommel's Barotropic Vorticity Equation

Stommel (1948) considered a flat-bottomed rectangular ocean with linear bottom friction $-r\mathbf{u}$. The depth-integrated vorticity equation becomes:

$$r\nabla^2\psi + \beta\frac{\partial\psi}{\partial x} = \frac{1}{\rho H}\text{curl}_z(\boldsymbol{\tau})$$

where $\psi$ is the barotropic streamfunction and $r$ is the linear drag coefficient ($\sim 10^{-6}$ s$^{-1}$).

Step 2: Non-Dimensionalize and Identify the Boundary Layer

Non-dimensionalize with basin width $L$. The ratio $\epsilon = r/(\beta L) \ll 1$ means the friction term $r\nabla^2\psi$ is negligible in the interior (recovering Sverdrup balance). Near the western boundary, the cross-basin derivative $\partial^2\psi/\partial x^2$ becomes large, creating a boundary layer of width:

$$\delta_S = \frac{r}{\beta}$$

Step 3: Boundary Layer Solution

In the boundary layer, define the stretched coordinate $\xi = x/\delta_S$. The vorticity equation reduces to:

$$\frac{\partial^2\psi}{\partial\xi^2} + \frac{\partial\psi}{\partial\xi} = 0$$

The solution is $\psi_{BL} = C(y)(1 - e^{-\xi})$, where $C(y)$ is chosen to match the interior Sverdrup transport. The exponential decay $e^{-x/\delta_S}$ occurs only at the western boundary because the $\beta$-effect breaks east-west symmetry.

Step 4: Why Western, Not Eastern?

The physical argument: a fluid parcel moving from the subtropical to the subpolar gyre acquires negative relative vorticity (from the wind curl). Conservation of total vorticity $(f + \zeta)$ requires friction to dissipate this vorticity. At the western boundary, the intense shear in the narrow current provides the needed frictional torque. Mathematically, the $\beta\partial\psi/\partial x$ term means only the western boundary solution has the correct sign for exponential decay:

$$\text{Western: } e^{-x/\delta_S} \to 0 \text{ as } x \to \infty \quad \checkmark$$

$$\text{Eastern: } e^{+x/\delta_S} \to \infty \text{ as } x \to \infty \quad \text{(rejected)}$$

Step 5: Munk's Extension with Lateral Friction

Munk (1950) replaced Stommel's bottom friction with lateral (horizontal) eddy viscosity $A_H\nabla^4\psi$. The vorticity equation becomes:

$$A_H\nabla^4\psi + \beta\frac{\partial\psi}{\partial x} = \frac{1}{\rho H}\text{curl}_z(\boldsymbol{\tau})$$

The boundary layer is now of width $\delta_M = (A_H/\beta)^{1/3}$. The fourth-order equation admits oscillatory decay, producing a weak recirculation gyre east of the western boundary current that better matches observations.

Step 6: Velocity Amplification Factor

The full Sverdrup transport $\psi_{interior}$ must be returned in the narrow western boundary layer. The ratio of boundary layer velocity to interior velocity scales as:

$$\frac{v_{WBC}}{v_{interior}} \sim \frac{L_x}{\delta_S} \sim \frac{5000\;\text{km}}{100\;\text{km}} = 50$$

This explains why the Gulf Stream flows at 2.5 m/s while the broad interior Sverdrup flow is only a few cm/s.