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Chapter 11

Chapter 11. Wind Driven Ocean Circulation Physical oceanography Instructor: Dr. Cheng-Chien Liu Department of Earth Sciences National Cheng Kung University Last updated: 7 December 2003. Introduction. Winds drive the circulation?

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Chapter 11

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  1. Chapter 11 Wind Driven Ocean Circulation Physical oceanography Instructor: Dr. Cheng-Chien Liu Department of Earth Sciences National Cheng Kung University Last updated: 7December 2003

  2. Introduction • Winds drive the circulation? • Spanish navigators in the 16th century  Strong northward currents along the Florida coast are unrelated to the wind? • Strong currents are found offshore of east coasts but not offshore of west coasts? • Answers  three great papers • Harald Sverdrup (1947) • Henry Stommel (1948) • Walter Munk (1950)

  3. Sverdrup's Theory of the Oceanic Circulation • Assumptions • Stationary flow • Lateral friction and molecular viscosity are small • Turbulence near the sea surface can be described using an eddy viscosity • Baroclinic flow • Depth of no motion

  4. Sverdrup's Theory of the Oceanic Circulation (cont.) • Momentum equation • x-component • y-component • Integrated from the surface (0) to a depth –D • The mass transports in the wind-driven layer • Boundary conditions • Sea surface • Depth -D

  5. Sverdrup's Theory of the Oceanic Circulation (cont.) 0 • Momentum equation (cont.) • x-component • y-component • Continuity equation • Integrated from the surface to a depth –D • Boundary conditions

  6. Sverdrup's Theory of the Oceanic Circulation (cont.) 0 • Sverdrup’s solution • BC: Mx = 0 at x = 0 • Dx: the distance from the eastern boundary of the ocean basin • Brackets indicate zonal averages of the wind stress • Fig 11.1

  7. Sverdrup's Theory of the Oceanic Circulation (cont.) • Test of Sverdrup’s theory • The eastern tropical Pacific • Known wind  Mx, My • Hydrographic data  Mx, My • Comparison: Fig 11.2 • Accurate calculation of transports • Successfully predict wind-driven currents going upwind

  8. Sverdrup's Theory of the Oceanic Circulation (cont.) • Comments on Sverdrup's Solutions • Sverdrup assumed • The internal flow in the ocean is geostrophic  Chapter 10 • There is a uniform depth of no motion  know little about the depth • Ekman's transport is correct  Chapter 9 • Limited to the east side of the oceans • Because Mx grows with x  Neglecting friction which would eventually balance the wind-driven flow • Only one BC can be satisfied • Mx = 0 at x = 0 • Give no information on the vertical distribution • Limited data to test the theory • Wunsch’s comments

  9. Stream Lines, Path Lines, and the Stream Function • the stream lines • The instantaneous curves that are everywhere tangent to the direction of the vectors • Unsteady flow  the pattern of stream lines = fn(t) • The path line • The trajectory of a fluid particle, the path followed by a Lagrangean drifter • Steady flow  the path line = the stream line • They are different for an unsteady flow

  10. Stream Lines, Path Lines, and the Stream Function (cont.) • Stream function y • Definition • Use • Simplify the description for two-dimensional, incompressible flows • Visualizing the flow • For steady flow: lines of constant y // streamlines // path lines • The volume rate of flow • Between any two stream lines of a steady flow = dy • Between any two stream lines of a steady flow = y1 - y2 • Fig 11.4

  11. Stream Lines, Path Lines, and the Stream Function (cont.) • Stream function y (cont.) • The sea surface is a stream function • The mass-transport stream function

  12. Stommel's Theory of Western Boundary Currents • Momentum equation • Same as Sverdrup’s work • Boundary conditions • Sea surface • Depth –D • Simple bottom stress proportional to velocity • F and R are constants • Stommel’s solution • Steady-state solutions for flow in a rectangular basin 0 yb, 0 x l of constant depth D filled with water of constant r • No rotation: Fig 11.5 left • Symmetric flow pattern with no western boundary current • f = f(j): Fig 11.5 right • Crowding of stream lines in the west  Gulf Stream

  13. Munk's Solution • Momentum equation • x-component • y-component • lateral eddy friction with constant AH = Ax = Ay • Used the mass-transport stream function y • The biharmonic operator • A fourth-order partial differential equation (PDE) • 4 BCs • flow at a boundary is parallel to a boundary • No slip at the boundary • Assuming the flow was in a rectangular basin extending from x = 0 to x = r, and from y = -s to y = +s • The wind stress was zonal and in the form

  14. Munk's Solution (cont.) • Munk's Solution • Fig 11.6 • ~ Sverdrup’s solution in the eastern part • Show a strong western boundary current in the west • Using AH = 5 × 103m2/s gives a boundary current roughly 225km wide with a shape similar to the flow observed in the Gulf Stream and the Kuroshio • My fn(AH) • Munk’s calculation • Gulf Stream: 36Sv, Kuroshio: 39Sv • About one half of the measured values of the flow  very good agreement considering T was not well known

  15. Observed Circulation in the Atlantic • North Atlantic Circulation • Fig 11.7: time-averaged circulation • As we expect from Sverdrup's theory • A broad, basin-wide, mid latitude gyre • In the west, a western boundary current, the Gulf Stream, completes the gyre • In the north a sub-polar gyre includes the Labrador current • low latitudes: an equatorial current system and countercurrent • The strong cross equatorial flow in the west which flows along the northeast coast of Brazil toward the Caribbean • Fig 11.8: far north Atlantic • Complex • Based on hydrographic observations

  16. Observed Circulation in the Atlantic (cont.) • North Atlantic Circulation (cont.) • Fig 11.9: tracks of a 110 buoys • Spaghetti tracks • Turbulent, especially in the Gulf Stream • The turbulent eddies seem to have a diameter of a few degrees • In the air, the large eddies are called storms with diameters of 10° - 20° • Thus oceanic "storms" are much smaller than atmospheric storms • Average in 2° × 2° boxes • The flow is so variable, that the average is not stable. • The kinetic energy of the eddies is 8 to 37 times larger than the kinetic energy of the mean flow • Thus the oceanic turbulence is very different than laboratory turbulence • In the lab, the mean flow is typically much faster than the eddies

  17. Observed Circulation in the Atlantic (cont.) • Gulf Stream Recirculation Region • Observations don’t match theory • Increase from 26Sv in the Florida Strait (between Florida and Cuba) to 55Sv offshore of Cape Hatteras • Increases from 30Sv in the Florida Strait to 150Sv near 40°N • Niiler’s summary • Not from the Antilles Current • The flow seems to come from the Gulf Stream itself • two subtropical gyres • The Gulf Stream recirculation region: a small gyre directly south of the Stream centered on 65°W • The broad, wind-driven gyre near the surface that extends all the way to Europe

  18. Observed Circulation in the Atlantic (cont.) • Gulf Stream Recirculation Region (cont.) • Negative viscosity • Eddies in the Stream convert the potential energy to kinetic energy through baroclinic instability • Figure 11.10: • Gulf Stream meanders lead to the formation of a spinning eddy, a ring

  19. Important concepts • The theory for wind-driven, geostrophic currents was first outlined in a series of papers by Sverdrup, Stommel, and Munk between 1947 and 1951. • They showed that realistic currents can be calculated only if the Coriolis parameter varies with latitude. • Sverdrup showed that the curl of the wind stress drives a northward mass transport, and that this can be used to calculate currents in the ocean away from western boundary currents. • Stommel showed that western boundary currents are required for flow to circulate around an ocean basin when the Coriolis parameter varies with latitude.

  20. Important concepts (cont.) • Munk showed how to combine the two solutions to calculate the wind-driven geostrophic circulation in an ocean basin. In all cases, the current is driven by the curl of the wind stress. • The observed circulation in the ocean is very turbulent. many years of observations may need to be averaged together to obtain a stable map of the mean flow. • The Gulf Stream is a region of baroclinic instability in which turbulence accelerates the stream. This creates a Gulf Stream recirculation. Transports in the recirculation region are much larger than transports calculated from the Sverdrup-Munk theory

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