Chapter 9 - Response of the
Upper Ocean to Winds

 Chapter 9 Contents (9.1) Inertial Motion (9.2) Ekman Layer at the Sea Surface (9.3) Ekman Mass Transports (9.4) Application of Ekman Theory (9.5) Important Concepts

If you have had a chance to travel around the United States, you may have noticed that the climate of the east coast differs considerably from that on the west coast. Why? Why is the climate of Charleston, South Carolina so different from that of San Diego, although both are near 32°N, and both are on or near the ocean? Charleston has 125-150 cm of rain a year, San Diego has 25-50 cm, Charleston has hot summers, San Diego has cool summers. Or why is the climate of San Francisco so different from that of Norfolk, Virginia?

If we look closely at the characteristics of the atmosphere along the two coasts near 32°N, we find more differences that may explain the climate. For example, when the wind blows onshore toward San Diego, it brings a cool, moist, marine, boundary layer a few hundred meters thick capped by much warmer, dry air. On the east coast, when the wind blows onshore, it brings a warm, moist, marine, boundary layer that is much thicker. Convection, which produces rain, is much easier on the east coast than on the west coast. Why then is the atmospheric boundary layer over the water so different on the two coasts? The answer can be found in part by studying the ocean's response to local winds, the subject of this chapter.

chapter contents

9.1 Inertial Motion

To begin our study of currents near the sea surface, let's consider first a very simple solution to the equations of motion, the response of the ocean to an impulse that sets the water in motion. For example, the impulse can be a strong wind blowing for a few hours. The water then moves under the influence of coriolis force and gravity. No other forces act on the water.

Such motion is said to be inertial. The mass of water continues to move due to its inertia. If the water were in space, it would move in a straight line according to Newton's second law. But on a rotating Earth, the motion is much different.

From (7.12) the equations of motion for a frictionless ocean are:

 (9.1a) (9.1b) (9.1c)

where p is pressure, Ω = 2 π/(sidereal day) = 7.292 × 10-5 rad/s is the rotation of the Earth in fixed coordinates, and φ is latitude. We have also used Fi = 0 because the fluid is frictionless.

Let's now look for simple solutions to these equations. To do this we must simplify the momentum equations. First, if only gravity and coriolis force act on the water, there must be no horizontal pressure gradient:

Furthermore, we can assume that the flow is horizontal, and (9.1) becomes:

 (9.2a) (9.2b)

where:

 (9.3)

is the Coriolis Parameter and Ω = 7.292 × 10-5/s is the rotation rate of Earth.

Equations (9.2) are two coupled, first-order, linear, differential equations which can be solved with standard techniques. If we solve the second equation for u, and insert it into the first equation we obtain:

Rearranging the equation puts it into a standard form we should recognize, the equation for the harmonic oscillator:

 (9.4)

which has the solution (9.5). This current is called an inertial current or inertial oscillation:

 (9.5)

Note that (9.5) are the parametric equations for a circle with diameter Di = 2V/f and period Ti = (2π)/f = Tsd /(2 sin φ) where Tsd is a sidereal day.

Ti is the inertial period, and it is one half the time required for the rotation of a local plane on Earth's surface (Table 9.1). The direction of rotation is anti-cyclonic: clockwise in the northern hemisphere, counterclockwise in the southern. Inertial currents are the free motion of parcels of water on a rotating plane.

Table 9.1 Inertial Oscillations
 Latitude (j) Ti (hr) D (km) for V = 20 cm/s 90° 11.97 2.7 35° 20.87 4.8 10° 68.93 15.8

Inertial currents are the most common currents in the ocean (Figure 9.1). Webster (1968) reviewed many published reports of inertial currents and found that currents have been observed at all depths in the ocean and at all latitudes. The motions are transient and decay in a few days. Oscillations at different depths or at different nearby sites are usually incoherent.

 Figure 9.1 Inertial currents in the North Pacific in October 1987 (days 275?300) measured by holey-sock drifting buoy drogued at a depth of 15m. Positions were observed 10?12 times per day by the Argos system on NOAA polar-orbiting weather satellites and interpolated to positions every three hours. The largest currents were generated by a storm on day 277. Note these are not individual eddies. The entire surface is rotating. A drogue placed anywhere in the region would have the same circular motion. From van Meurs (1998).

Inertial currents are caused by rapid changes of wind at the sea surface, with rapid changes of strong winds producing the largest oscillations. Although we have derived the equations for the oscillation assuming frictionless flow, friction cannot be completely neglected. With time, the oscillations decay into other surface currents. (See, for example, Apel, 1987: ?6.3 for more information.)

chapter contents