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CoriolisPressure Gradient x z CURRENTS WITH FRICTION Nansen’s qualitative argument on effects of friction CoriolisPressure Gradient x y CoriolisPressure.

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Presentation on theme: "CoriolisPressure Gradient x z CURRENTS WITH FRICTION Nansen’s qualitative argument on effects of friction CoriolisPressure Gradient x y CoriolisPressure."— Presentation transcript:

1 CoriolisPressure Gradient x z CURRENTS WITH FRICTION Nansen’s qualitative argument on effects of friction CoriolisPressure Gradient x y CoriolisPressure Gradient x y CoriolisPressure Gradient x y ?

2 EKMAN SOLUTION (1905) Assumptions:homogeneous fluid no horizontal pressure gradients infinitely deep and wide ocean no horizontal friction constant eddy viscosity steady wind northward

3 Boundary conditions: EKMAN DEPTH or depth of frictional influence Wind-induced current speed

4 DEDE V 0 is 45° to the right of the wind (in the northern hemisphere) V 0 decreases exponentially with depth as it turns clockwise (NH) At depth z = - D E the flow speed falls to e - π = 0.04 that at the surface and in opposite direction

5 Alternatively, Von Kármán constant = 0.4 Charnock constant = 0.0185 sea surface roughness

6 From observations outside 10º by Ekman:

7 Indicates that surface currents are ~ 1% of the wind speed at the poles ~2.5% of the wind speed at 45 0 ~11% of the wind speed at 20 0 Combining and V 0 / W Empirically, it is seen that V 0 / W oscillates between 1 and 5%

8 Ekman Transport

9 Ekman Equations Rearranging: Integrating over two Ekman layers, from depth z = 2 D E to z = 0: Ekman Transport

10 Ekman Transport x y x y Ekman transport is inversely proportional to f Water is replaced from the side – but what happens at the coast? m 2 /s

11 Equatorward winds on ocean eastern boundaries Poleward wind on ocean western boundaries Poleward winds on ocean eastern boundaries Equatorward wind on ocean western boundaries

12 Consequence of Upwelling

13 Bottom Friction and Shallow Water Effects Assumptions: z Bottom at z = 0 uvz u, v = 0 at z = 0 (no flow at the bottom) uu g vzD B u = u g, v = 0 at the top of the bottom Ekman layer ( z = D B ) u g u g is geostrophic flow

14 ugug BOTTOM EKMAN LAYER ugug D B = 15 m N 45° PLAN VIEW PERSPECTIVE VIEW North East Distance from bottom (m)

15 DEDE Flow at interior? Flow at bottom? DBDB -x-x z Overlap of bottom and surface Ekman layers Importance of shelf break depth Problems with Ekman theory – constant A z, constant wind, linear flow, steady state, infinite ocean, no pressure gradients

16 SVERDRUP SOLUTION Assumed gradients in the wind field -- in contrast to Ekman’s spatially uniform wind x  ConvergenceDivergence x y Differentiating Ekman equations with respect to y and x, to look at gradients in wind field: Ekman Transport

17 Adding the two yields: 0 Sverdrup’s Equation x  ConvergenceDivergence x y

18 Sverdrup’s Equation QxQx DBDB -x-x z DEDE

19 0 using Meridional transport of water given by the curl of the wind How about zonal (in ‘ x ’) transport?

20 Trades Doldrums EQUATOR x y 20° 10° Integrating from Eastern Boundary ( x =0) to the west ‘- x ’

21 Trades Doldrums EQUATOR x y 20° 10° + + - Divergence Convergence SOUTH EQUATORIAL CURRENT NORTH EQUATORIAL CURRENT NORTH EQUATORIAL COUNTERCURRENT

22 NORTH EQUATORIAL CURRENT SOUTH EQUATORIAL CURRENT NORTH EQUATORIAL COUNTERCURRENT Streamlines of mass transport from mean wind stress (Reid, 1948)

23 Conservation of mass is forced by including north-south currents confined to a thin, horizontal boundary layer. From Tomczak and Godfrey (1994).

24 Q x x Limited to the east side of the oceans because Q x grows with x. Neglects friction which would eventually balance the wind-driven flow Solutions may be used for describing the global system of surface currents. Conservation of mass is forced at the western boundaries by including north-south currents confined to a thin, horizontal boundary layer Only one boundary condition can be satisfied, no flow through the eastern boundary. More complete descriptions of the flow require more complete equations. Solutions give no information on the vertical distribution of the current. SVERDRUP SOLUTION What happens on the western part of the oceans?

25 STOMMEL SOLUTION (1948) Currents are fast and narrow on the western part of ocean basins slow and broad on the eastern part of basins using a streamfunction definition: Solution with:

26 STOMMEL SOLUTION (1948)

27 45°N 15°N Westerlies Easterlies f = 0 f = const Addition of linearized bottom friction allows a closed circulation in the basinstreamlines sea surface height f Change of f with latitude is the main responsible for western intensification of ocean currents

28 Western Intensification can also be understood with vorticity arguments Vorticity – tendency for portions of fluid to rotate We will consider: RELATIVE, PLANETARY, ABSOLUTE, AND POTENTIAL E N u u v v v v u u - +

29 Relative vorticity: E, x N, y u u v v v v u u - + f. Planetary vorticity: Equals f. A stationary object on the earth has planetary vorticity that varies with latitude

30 Absolute vorticity: Planetary plus relative vorticities Changes in absolute vorticity are useful to help understand tendencies for fluids to rotate To describe changes in absolute vorticity, take: Changes of absolute vorticity in time are related to divergences

31 Plan View Side View Convergence = Gain of Absolute Vorticity = Column Stretching

32 Plan View Side View Divergence = Loss of Absolute Vorticity = Column Squashing _

33 D Now consider a layer of thickness D, whose equation of continuity is: Changes of layer thickness are given by divergences/convergences D Convergences = increase in D D Divergences = decrease in D Combining with: CONSERVATION OF POTENTIAL VORTICITY

34 Consider: Df D constant and f changing Df D changing and f constant Potential Vorticity Conservation Western Intensification Use concept of Potential Vorticity Conservation to explain Western Intensification 45°N 15°N Westerlies Easterlies -

35

36 MUNK’S SOLUTION (1950) Extended domain from Stommel’s Added horizontal friction More realistic wind Integrating, using a streamfunction definition: and cross-differentiating: Sverdrup’s solution friction biharmonic operator:

37 MUNK’S SOLUTION (1950) subtropical gyre NEC NECC SEC 4 th order d.e. with b.c.: (from Munk, 1950)

38 L = 6000 km J = 3000 km Beta = 2x10 -11 m -1 s -1 r = 2x10 -6 s -1 (from George Veronis) /J


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