Ekman spiral

As used in meteorology, an idealized mathematical description of the wind distribution in the atmospheric boundary layer, within which the earth's surface has an appreciable effect on the air motion.

The model is simplified by assuming that within this layer the eddy viscosity and density are constant, the motion is horizontal and steady, the isobars are straight and parallel, and the geostrophic wind is constant with height. The x direction is taken along the pressure gradient; the resulting approximate equations for the component wind speeds U and V in the x and y directions, respectively, at any level z are

where G is the geostrophic wind speed, β = z(f/2K_M)^½, f is the Coriolis parameter, and K_M is the eddy viscosity. The lowest level H where U = 0, so that the true wind and the geostrophic wind have the same direction, is called the geostrophic wind level (or gradient wind level). It is given by

where α₀ is the angle between the surface wind and the surface isobars. At this height the magnitude of the true wind will exceed that of the geostrophic wind by a small amount, depending on the value of β. The Ekman spiral is an equiangular spiral having the geostrophic wind as its limit point. Below the geostrophic wind level the wind blows across the isobars toward low pressure, at an angle that is a maximum at the surface and does not exceed 45°. The deviation of the wind vector from the geostrophic wind vector diminishes upward at an exponential rate. The theory of this spiral was developed by Ekman in 1902 for motion in the upper layers of the ocean under the influence of a steady wind. It was applied to the atmosphere by Åkerblom in 1908.

As originally applied by Ekman to ocean currents, a graphic representation of the way in which the theoretical wind-driven currents in the surface layers of the sea vary with depth.
In an ocean that is assumed to be homogeneous, infinitely deep, unbounded, and having a constant eddy viscosity, over which a uniform steady wind blows, Ekman has computed that the current induced in the surface layers by the wind will have the following characteristics: 1) At the very surface the water will move at an angle of 45° cum sole from the wind direction; 2) in successively deeper layers the movement will be deflected farther and farther cum sole from the wind direction, and the speed will decrease; and 3) a hodograph of the velocity vectors would form a spiral descending into the water and decreasing in amplitude exponentially with depth. The depth at which the vector first points 180° from the wind vector is called the depth of frictional influence (or depth of frictional resistance). At this depth the speed is e^-π times that at the surface. The layer from the surface to the depth of frictional influence is called the layer of frictional influence. If the velocity vectors from the surface to the depth of frictional influence are integrated, the resultant vertically integrated motion is 90° cum sole from the wind direction.