1 Equations of Motion Buoyancy Ekman and Inertial Motion September 17

2 Recall:

3 Or: where: x-momentum y-momentum z-momentum continuity

4 Figure 8.4 in Stewart The buoyancy force acting on the displaced parcel is: ‘ Buoyancy:

5 The acceleration of the displaced parcel is:

6 Stability Equation Stability is defined such that: E > 0 stable E = 0 neutral stability E < 0 unstable Influence of stability is expressed by a stability frequency N (also known as Brunt-Vaisala frequency):

7 Figure 8.6 in Stewart: Observed stratification frequency in the Pacific. Left: Stability of the deep thermocline east of the Kuroshio. Right: Stability of a shallow thermocline typical of the tropics. Note the change of scales.

8 Dynamic Stability and Richardson's Number If velocity changes with depth in a stable, stratified flow, then the flow may become unstable if the change in velocity with depth, the current shear, is large enough. The simplest example is wind blowing over the ocean. In this case, stability is very large across the sea surface. We might say it is infinite because there is a step discontinuity in , and N is infinite. Yet, wind blowing on the ocean creates waves, and if the wind is strong enough, the surface becomes unstable and the waves break. If velocity changes with depth in a stable, stratified flow, then the flow may become unstable if the change in velocity with depth, the current shear, is large enough. The simplest example is wind blowing over the ocean. In this case, stability is very large across the sea surface. We might say it is infinite because there is a step discontinuity in , and N 2 is infinite. Yet, wind blowing on the ocean creates waves, and if the wind is strong enough, the surface becomes unstable and the waves break. This is an example of dynamic instability in which a stable fluid is made unstable by velocity shear. Another example of dynamic instability, the Kelvin-Helmholtz instability, occurs when the density contrast in a sheared flow is much less than at the sea surface, such as in the thermocline or at the top of a stable, atmospheric boundary layer This is an example of dynamic instability in which a stable fluid is made unstable by velocity shear. Another example of dynamic instability, the Kelvin-Helmholtz instability, occurs when the density contrast in a sheared flow is much less than at the sea surface, such as in the thermocline or at the top of a stable, atmospheric boundary layer

9 Figure 8.7 in Stewart: Billow clouds showing a Kelvin-Helmholtz instability at the top of a stable atmospheric boundary layer. Some billows can become large enough that more dense air overlies less dense air, and then the billows collapse into turbulence. Photography copyright Brooks Martner, NOAA Environmental Technology Laboratory.

10 Richardson Number The relative importance of static stability and dynamic instability is expressed by the Richardson Number: The relative importance of static stability and dynamic instability is expressed by the Richardson Number: R i > 0.25 Stable Ri < 0.25 Velocity shear enhances turbulence

11 Ekman Flow

12 Again:

13 Define c (current speed) as: u and v change, but c stays constant: Coriolis force does no work!

14 Flow is in a circle: Inertial or Centripetal force = Coriolis force where Inertial radius If c ~ 0.1 m/s f ~ then r ~ 1 km

15 Inertial Period is given by T where: If f ~ then T f ~ 6.28x10 4 sec ~ 17.4 hrs

16 r fc c c c c c 2 /r

17 Latitude () T i (hr)D (km) for V = 20 cm/s 90° ° ° Table 9.1 in Stewart Inertial Oscillations Note: V is equivalent to c from previous slides, D is equal to the diameter or twice the radius, r

18 Figure 9.1 in Stewart Inertial currents in the North Pacific in October 1987

19 Ekman flow Fridtjof Nansen noticed that wind tended to blow ice at an angle of 20°-40° to the right of the wind in the Artic Fridtjof Nansen noticed that wind tended to blow ice at an angle of 20°-40° to the right of the wind in the Artic Nansen hired Ekman (Bjerknes graduate student) to study the influence of the Earth’s rotation on wind- driven currents Nansen hired Ekman (Bjerknes graduate student) to study the influence of the Earth’s rotation on wind- driven currents Ekman presented the results in his thesis and later expanded the study to include the influence of continents and differences of density of water (Ekman, 1905) Ekman presented the results in his thesis and later expanded the study to include the influence of continents and differences of density of water (Ekman, 1905)

20 Figure 9.2 in Stewart Balances of forces acting on an iceberg on a rotating earth

21 So again…

22 Can ignore all terms except Coriolis and vertical eddy viscocity

23 Balance in the surface boundary layer is between vertical friction (as expressed by the eddy viscocity) and Coriolis – all other terms are neglected

24 friction c.f. 45° At the surface (z=0)

25 If we assume the wind is blowing in the x-direction only, we can show:

26 Wind Stress Frictional force acting on the surface skin Frictional force acting on the surface skin ρ a : density of air C D : Drag coefficient – depends on atmospheric conditions, may depend on wind speed itself W: wind speed - usually measured at “standard anemometer height” ~ 10m above the sea surface

27 Ekman Depth (thickness of Ekman layer) For Mid-latitudes: A v = 10 ρ = 10 3 f = Plug these into the D equation and: meters

28 Figure 9.3 in Stewart

29 U 10 (m/s) Latitude 15°45° 5 40m30m 1090m50m 20180m110m Typical Ekman Depths Table 9.3 in Stewart

30 At z=0 y x v u 45° Remember, we have assumed that wind stress is in the x-direction only

31 At z=-D: e -    y,v x,u v u -π/4-π-π

32 Ekman Number The depth of the Ekman layer is closely related to the depth at which frictional force is equal to the Coriolis force in the momentum equation The depth of the Ekman layer is closely related to the depth at which frictional force is equal to the Coriolis force in the momentum equation The ratio of the forces is known as Ekman depth The ratio of the forces is known as Ekman depth Solving for d:

33 Ekman Transport In general, net transport in the Ekman Layer is 90° to the right of the wind stress in Northern Hemisphere

34 Ekman Pumping but w(0) = 0

35 By definition, the Ekman velocities approach zero at the base of the Ekman layer, and the vertical velocity at the base of the layer w E (-d) due to divergence of the Ekman flow must be zero. Therefore: vector mass transport due to Ekman flow horizontal divergence operator

36 If we use the Ekman mass transports in we can relate Ekman pumping to the wind stress. wind stress

37 MEME Ek pile up of water wEwE wEwE Hi P Lo P anticyclonic