Principle of Oceanography Gholamreza Mashayekhinia 2013_2014 Response of the Upper Ocean to Winds Contents • Inertial Motion • Ekman Layer at the Sea Surface • Ekman Mass Transport • Application of Ekman Theory • Langmuir Circulation • Important Concepts Inertial 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. In physics, the Coriolis effect is a deflection of moving objects when they are viewed in a rotating reference frame. This low pressure system over Iceland spins counter-clockwise due to balance between the Coriolis force and the pressure gradient force. 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. The equations of motion for a frictionless ocean are: 푑푢 1 휕푝 = − + 2Ω푣 푠푖푛휑 (1a) 푑푡 휌 휕푥 푑푣 1 휕푝 = − − 2Ω푢 푠푖푛휑 (1b) 푑푡 휌 휕푦 푑휔 1 휕푝 = − + 2Ω푢 푐표푠휑 − 푔 (1c) 푑푡 휌 휕푥 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: 휕푝 휕푝 = = 0 휕푥 휕푦 Furthermore, we can assume that the flow is horizontal, and (1) becomes: 푑푢 = 2Ω푣 푠푖푛휑 = 푓 (2a) 푑푡 푣 푑푣 = −2Ω푢 푠푖푛휑 = −푓 (2b) 푑푡 푢 Where: 푓 = 2Ω 푠푖푛휑 (3) is the Coriolis Parameter and Ω = 7.292 × 10-5/s is the rotation rate of Earth. Equations (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: 푑푢 1 푑2푣 = − = 푓 푑푡 푓 푑푡2 푣 Rearranging the equation puts it into a standard form we should recognize, the equation for the harmonic oscillator: 2 푑 푣 2 + 푓 푣 = 0 (4) 푑푡2 which has the solution (5). This current is called an inertial current or inertial oscillation: 푢 = 푉 sin 푓푡 푣 = 푉 cos 푓푡 (5) 푉2 = 푢2 + 푣2 Note: That (5) are the parametric equations for a circle with diameter Di = 2V/f and period Ti = (2ð)/f = Tsd /(2 sin ö) 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 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 (1) Inertial Oscillations 퐿푎푡푖푡푢푑푒 푻풊(풉풓) 퐷(푘푚) (흋) 풇풐풓 푽 = ퟐퟎ 풄풎/풔 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 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 1 Figure (1) Inertial currents in the north Pacific in October 1987 (days 275-300) measured by holey-sock drifting Buoy drogue at a depth of 15m. Position 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. Ekman Layer at the Sea Surface • Steady winds blowing on the sea surface produce a thin, horizontal boundary layer, the Ekman layer. • By thin, a layer that is at most a few-hundred meters thick, which is thin compared with the depth of the water in the deep ocean. A similar boundary layer exists at the bottom of the ocean, the bottom Ekman layer, and at the bottom of the atmosphere just above the sea surface. The Ekman layer is named after Professor Walfrid Ekman, who worked out its dynamics for his doctoral thesis. • Ekman's work was the first of a remarkable series of studies conducted during the first half of the twentieth century that led to an understanding of how winds drive the ocean's circulation Table (2) Contributions to the Theory of the Wind-Driven Circulation Qualitative theory, currents transport water at an Fridtjof Nansen (1898) angle to the wind. Quantitative theory for wind-driven transport at Vagn Walfrid Ekman (1902) the sea surface. Theory for wind-driven circulation in the eastern Harald Sverdrup (1947) Pacific. Theory for westward intensification of wind- Henry Stommel (1948) driven circulation (western boundary currents). Quantitative theory for main features of the Walter Munk (1950) wind-driven circulation Kirk Bryan (1963) Numerical models of the oceanic circulation. Bert Semtner and Global, eddy-resolving, realistic model of the (1988) Robert Chervin ocean's circulation. Nansen's Qualitative Arguments Fridtjof Nansen noticed that wind tended to blow ice at an angle of 20°–40° to the right of the wind in the Arctic, by which he meant that the track of the iceberg was to the right of the wind looking downwind (See Figure 2). He later worked out the balance of forces that must exist when wind tried to push icebergs downwind on a rotating Earth. Figure (2) The balance of forces acting on an iceberg in a wind on a rotating Earth. Nansen argued that three forces must be important: • Wind Stress, W; • Friction F (otherwise the iceberg would move as fast as the wind); • Coriolis Force, C. Nansen argued further that the forces must have the following attributes: • Drag must be opposite the direction of the ice's velocity; • Coriolis force must be perpendicular to the velocity; • The forces must balance for steady flow. 푊 + 퐹 + 퐶 = 0 Ekman's Solution Nansen asked Vilhelm BjerkBjer to let one of Bjerknes students make a theoretical study of the influence of Earth's rotation on wind-driven currents. Walfrid Ekman was chosen, and he presented the results in his thesis at Uppsala. Ekman later expanded the study to include the influence of continents and differences of density of water (Ekman, 1905). Ekman assumed a steady, homogeneous, horizontal flow with friction on a rotating Earth. Thus horizontal and temporal derivatives are zero: 휕 휕 휕 = = = 0 (6) 휕푡 휕푥 휕푦 For flow on a rotating earth, this leaves a balance between frictional and Coriolis forces. Ekman further assumed a constant vertical eddy viscosity of the form of flow in a boundary layer. 휕푢 휕푢 2Ω푣푠푖푛휑 + 푣 − 푢′휔′ = 0 휕푧 휕푧 휕푢 휕푣 푇푥푧 = 휌휔퐴푧 , 푇푦푧 = 휌휔퐴푧 (7) 휕푧 휕푧 where Txz, Tyz are the components of the wind stress in the x, y directions, and ρw is the density of sea water. With these assumptions, and using (7) in 2 휕푇푥푧 휕 휕푢 휕 푢 = (퐴푧 ) ≈ 퐴푧 2 휕푧 휕푧 휕푧 휕푧 the x and y components of the momentum equation have the simple form: 휕2푢 푓 + 퐴 = 0 (8a) 푣 푧 휕푧2 휕2푣 −푓 + 퐴 = 0 (8b) 푢 푧 휕푧2 where f is the Coriolis parameter. It is easy to verify that the equations (9) have solutions: 휋 푢 = 푉 exp 푎푧 sin − 푎푧 (9푎) 0 4 휋 푣 = 푉 exp 푎푧 c표푠 − 푎푧 (9푎) 0 4 To evaluate the constants, let's assume that the wind blows toward the north. There is nothing special about this choice of wind direction. We just need to pick a direction, and north is convenient. When the wind is blowing to the north (T = Tyz ). The constants are 푇 푓 푉0 = 푎푛푑 푎 = (10) 2 2퐴 휌 휔푓퐴푧 푧 and V0 is the velocity of the current at the sea surface. Now let's look at the form of the solutions. At the sea surface z = 0, exp(z = 0) = 1, and 휋 푢 0 = 푉 cos (11푎) 0 4 휋 푣 0 = 푉 sin (11푏) 0 4 The current has a speed of V0 to the northeast. In general, the surface current is 45° to the right of the wind when looking downwind in the northern hemisphere. The current is 45° to the left of the wind in the southern hemisphere. Below the surface, the velocity decays exponentially with depth (Figure 3): 1 2 2 [푢 푧 + 푣 푧 ]2 = 푉0 exp 푎푧 (12) Figure 3. Ekman current generated by a 10m/s wind at 35°N. Values for Ekman's Constants • To proceed further, we need values for any two of the free parameters: the velocity at the surface, V0; the coefficient of eddy viscosity, Az; or the wind stress T.