Geophysics Physical oceanography Proposition

Ekman spiral

Builds on Ekman layer Coriolis force Viscosity

Statement

Consider sea water of constant density $\rho$ filling the half-space $z \le 0$ , in a frame rotating with Coriolis parameter $f > 0$ (the northern hemisphere). Let $\nu > 0$ be a constant vertical eddy viscosity. Let a constant wind stress of magnitude $\tau > 0$ act on the surface $z = 0$ in the $x$ -direction. Assume the horizontal velocity $(u(z), v(z))$ is steady and horizontally uniform, that there is no horizontal pressure gradient, and that it solves $-f v = \nu u'', \qquad f u = \nu v'', \qquad \rho \nu u'(0) = \tau, \quad v'(0) = 0, \qquad (u, v) \to 0 \text{ as } z \to -\infty.$ Set $\delta = \sqrt{2 \nu / f}$ and $V_0 = \tau / (\rho \sqrt{f \nu})$ . Then the unique solution is $u = V_0 \, e^{z/\delta} \cos\!\Big( \frac{z}{\delta} - \frac{\pi}{4} \Big), \qquad v = V_0 \, e^{z/\delta} \sin\!\Big( \frac{z}{\delta} - \frac{\pi}{4} \Big).$ In particular:

the surface current has speed $V_0$ and points $45^\circ$ to the right of the wind;
with increasing depth the current turns clockwise and its speed decays like $e^{z/\delta}$ ;
the total transport is $\int_{-\infty}^{0} (u, v) \, dz = \big( 0, -\tau / (\rho f) \big)$ , at right angles to the right of the wind.

Remarks

Set $w = u + i v$ . The two equations combine into $w'' = (i f / \nu) w = \big( (1 + i)/\delta \big)^2 w$ , whose only solutions that vanish as $z \to -\infty$ are $w = W e^{(1 + i) z / \delta}$ . The stress condition $\rho \nu w'(0) = \tau$ gives $W = V_0 e^{-i \pi / 4}$ , which is the formula above. Integrating $w$ over $z \le 0$ gives $W \delta / (1 + i) = -i \tau / (\rho f)$ , which is clause 3.

At the depth $z = -\pi \delta$ the current points opposite to the surface current and is weaker by the factor $e^{-\pi} \approx 0.04$ . This depth is often taken as the thickness of the Ekman layer. In the southern hemisphere, $f < 0$ , and the same computation gives a spiral turning to the left. The hypothesis $f \ne 0$ is needed: without rotation the equations reduce to $u'' = v'' = 0$ , and no solution decays at depth.

Going deeper

For a student

The spiral was computed by Vagn Walfrid Ekman from a balance between the Coriolis force and vertical friction. Real ocean spirals are flatter than the model predicts, because the eddy viscosity is not constant and the wind is not steady. The robust prediction is clause 3: the Ekman transport $\tau / (\rho f)$ does not depend on $\nu$ . Its convergence and divergence pump water down or up out of the Ekman layer, and this pumping drives the wind-driven circulation of the ocean interior and coastal upwelling.

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Cross product concept
Derivative concept
Ekman layer concept
Euclidean space object
Euclidean vector concept
Identity matrix object
Matrix definition
Ordinary differential equation concept
Vector space definition
Viscosity concept
Velocity concept
Classical mechanics concept
Square matrix definition
Transpose definition
Linear map definition
Momentum concept
Inertial frame of reference concept
Multilinear map definition
Orthogonal matrix definition
Newton's laws of motion statement
Alternating multilinear map definition
Orthogonal group object
Determinant definition
Special orthogonal group object
Angular velocity concept
Rotating reference frame concept
Fictitious force concept
Coriolis force concept
Ekman spiral statement

Tag	`AHBC400C4X`
URI	`m2://AHBC400C4X`
Kind	statement, proposition
Writing	W0 stub
MSC	86A05, 76U05
Status	This copy of the site has no build data. Run `m2 status AHBC400C4X`.
About	Ekman layer
Uses	Ekman layer, Coriolis force, Viscosity
Changed	3 November 2026