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Ilker Fer

Lecture Notes for GEOF110 Chapter 9 until mass transport= 2 hours until bottom Ekman spiral = 2 hours vorticity = 2-3 hours. Ilker Fer. Guiding for blackboard presentation, following Pond & Pickard, Introductory Dynamical Oceanography. Currents with friction.

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Ilker Fer

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  1. Lecture Notes for GEOF110Chapter 9until mass transport= 2 hoursuntil bottom Ekman spiral = 2 hoursvorticity = 2-3 hours Ilker Fer Guiding for blackboard presentation, following Pond & Pickard, Introductory Dynamical Oceanography

  2. Currents with friction • Gross surface circulation in the ocean is CW in NH and CCW in SH. This is due to wind + thermohaline forcing. Wind dominates (typically) in the upper 1000 m. • History: • Nansen. Qualitative theory. Currents in NH transport water 20-40deg to the right of the wind. • Ekman. Quantitative theory of wind driven transport at the sea surface. • Sverdrup. Theory of wind driven circulation in the Eastern Pacific. • Stommel. Theory for westward intensification of wind-driven circulation (Western boundary currents). GEOF110 Guidelines / 6

  3. Ft Ice velocity WIND FC Fb Nansen’s arguments • Why icebergs in the Arctic drifted to the right of the wind, looking downwind? • Important forces are • Tangential wind stress (Ft) • Drag at the bottom of the iceberg, due to friction (Fb) • Coriolis force (FC) • Argued • Drag oppose the ice velocity • Coriolis force acts normal to ice velocity • Forces balance in steady state GEOF110 Guidelines / 6

  4. P P C F F C Horisontal bevegelseslikningene med friksjon (ser bort fra 2cosw) PRESSURE G. + CORiOLIS + FRICTION = 0 Vector diagram closed: Forces in equilibrium. Note: Friction terms are an important part of the balance near the surface or bottom (boundary layers). Vertical Ekman number Ez = Az/fH2 is about 1 for Az = 0.1 m/s2, f = 10-4 1/s and H = 30 m. GEOF110 Guidelines / 6

  5. Ekman’s Solution Two forces cause motion: pressure gradient due to density distribution and wind friction. Because equations are linear, we apply principle of superposition (i.e. sum of two solutions is a solution). Split the velocity into two parts: Geostrophic (ug, vg) + Friction (uE, vE) First, look at friction alone. No geostrophic velocity and no horizontal pressure gradients. En eventuell geostrofisk strøm kan adderes p.g.a. linære likninger. GEOF110 Guidelines / 6

  6. Assumptions: • - No boundaries • - infinitely deep (hence no bottom friction) • constant Az • constant f (f-plane approx.) • steady wind blowing for a long time • homogenous water and level sea surface (hence no geostrophic flow) Ekmans Likninger CORIOLIS + FRICTION = 0 GEOF110 Guidelines / 6

  7. Since uE and vE vary only with depth use d/dz For constant f and Az, this is a set of coupled 2nd order ordinary differential equations. This can be solved e.g. by inserting vE in (1), yielding 4th order equation for uE. It is simpler to introduce complex velocity vector: Multiply (2) by i and add (1) and (2): GEOF110 Guidelines / 6

  8. Generell løsning (for f>0): • Need TWO boundary conditions to find the constants A and B. Grenseflatebetingelser: • Uendelig dyp: WE 0 når z  - • Dynamic boundary condition. Stresset i overflaten = vindstresset • For simplicity let wind blow along y-axis (i.e., wind stress in the y-direction). Complex wind stress vector is • Reynolds stress at the water surface Using, GEOF110 Guidelines / 6

  9. Husk: f>0 Grenseflatebetingelse 1: Uendelig dyp: WE 0 når z  -, altså B = 0 Grenseflatebetingelse 2: Stresset i overflaten = vindstresset GEOF110 Guidelines / 6

  10. Hastigheten ved overflaten: Retningen varierer med dypet. Kan strømmen bli motsatt rettet overflate-strømmen? Anta motsatt rettet i z = -DE, da må Det planetariske grensesjikt friksjonsdypet. GEOF110 Guidelines / 6

  11. + for NH - for SH Ekman spiralen GEOF110 Guidelines / 6

  12. At the surface, z = 0:surface current flows 45 to the right (left) of the wind direction in the NH (SH). In NH Below the surface, current speed decays exponentially with depth, while the direction changes CW (CCW) in NH (SH): The direction of the flow becomes opposite to the surface flow at z = -DE, i.e., where the speed has fallen to exp(-)=0.04 of the surface speed. i.e., directed opposite to surface current and reduced by 0.04 in magnitude. GEOF110 Guidelines / 6

  13. GEOF110 Guidelines / 6

  14. To estimate values for Ekman constants, we make use of the observations: Obs. 1: Empirical relation for wind stress: a: air density (about 1.3 kg/m3) CD: the drag coefficient (about 1.4x10-3) U10: wind speed in m/s measured 10 m above sea level discuss uncertainty Obs. 2: Empirical relation between surface current and wind speed: GEOF110 Guidelines / 6

  15. Some notes on assumptions behind Ekman’s solution: • f-plane is OK • steady-state and constant eddy viscosity are not likely • salient features of the theory is likely correct but details will differ in nature • since the Ekman current vanishes below DE, ocean need not be infinitely deep for the solution to be valid. It suffices if DE << H (H is total depth). An upper estimate for DE =  (2Az/f)1/2 is about 140 m using Az = 0.1 m2/s and f = 10-4 1/s. This is much less than mean ocean depth of 4 km. • In shallow areas where H<DE, wind influence will be felt through the entire water column. Ekman spiral will be modified by the bottom. Surface current will be deflected less than 45deg to the right of wind dir. GEOF110 Guidelines / 6

  16. Massetransport [per unit width] Transporten i Ekmanlaget p.g.a. ren vind-drevet strøm: Since wind driven motion ceases below DE, it will suffice to integrate down to say z = -2DE where speed will be about 0.002 of that at surface. GEOF110 Guidelines / 6

  17. Kan også se dette direkte fra bevegelseslikningene. Ekmans Likninger : GEOF110 Guidelines / 6

  18. Upwelling ved kyst Upwelling at a coast (NH). Wind direction is out of the paper. - Overflatehellingen gir en geostr. strøm (barotrop). - p.g.a. større tetthet inne ved kysten  baroklin kompensasjon Example: off the coast of California, Peru and Chile. Typical width of the upwelling zone is order 100 km. Upwelling speed is order 5-10 m/day. Water is upwelled from not deeper than 200-300 m. Brings up colder, nutrient rich water  large fisheries. GEOF110 Guidelines / 6

  19. Downwelling ved kyst Downwelling at a coast (NH). Wind direction is into the paper. If wind blows with coast to the right (NH), on-shore Ekman transport will pile water up against the coast  pressure gradient  geostrophic offshore transport below Ekman layer. Near the coast water must sink to maintain the circulation. Oxygen, CO2 rich surface waters are transported deeper and further off shore. GEOF110 Guidelines / 6

  20. Upwelling/Downwelling i åpne hav GEOF110 Guidelines / 6

  21. GEOF110 Guidelines / 6

  22. Schematic wind fields and up/downwelling zones in the Atlantic. GEOF110 Guidelines / 6

  23. p+Dp y x p Bunnfriksjon In current flow over sea bottom, friction will lead to Ekman spiral pattern, but with direction of rotation reversed relative to the wind-driven near surface spiral. Ekman equations apply but the boundary conditions are different. Anta at vi også har en geostrofisk strøm. Vi hadde Altså Ekmans likninger gjelder. Løsning: GEOF110 Guidelines / 6

  24. z ug u=ug+uE x Total velocity u=ug+uE • What are the boundary conditions? • Away from the bottom boundary velocity is geostrophic velocity • z, uug, vvg and uE0; vE0. • 2) At the bottom velocity goes to zero (u = v = 0 for z = 0) cos(-x)=cosx sin(-x)=-sin(x) GEOF110 Guidelines / 6

  25. Note: This solution is an example of the boundary layer approach (Ch. 9.14.) Bottom frictional effects on a geostrophic current (NH) GEOF110 Guidelines / 6

  26. Why veer to the left? Away from bottom two forces balance: PG (pressure gradient) and C (Coriolis). PG is constant with depth (barotropic). As bottom is approached friction becomes important. 3 Forces at play: PG, C and F. F slows the flow, hence C decreases  PG not balanced. Current veers to the left such that forces balance. PG2= PG1 Near boundary: Away from boundary: Forces do not balance PG1 F ug+uE<ug C2 < C1 ug Rotate to left, such that forces balance PG C1 Forces Balance ug+uE F C GEOF110 Guidelines / 6

  27. MyE 45o ug MxE Ageostrophic mass transport: GEOF110 Guidelines / 6

  28. y x Vorticity (Virvling) Relative Vorticity Characteristic of the kinematics of the flow. Tendency for fluid parcels to rotate. Directly related to velocity shear. Positive when CCW y x relative vorticity in the horizontal plane (the vertical component): GEOF110 Guidelines / 6

  29. Planetary Vorticity For a rotating solid object vorticity = 2x angular velocity At latitude , a parcel on Earth surface has angular velocity sin. Planetary vorticity = f = 2sin units: radians/s GEOF110 Guidelines / 6

  30. Absolute Vorticity Obtain from horizontal eq. of motion, ignoring friction and  is constant. GEOF110 Guidelines / 6

  31. Conservation of absolute vorticity (conservation of angular momentum) GEOF110 Guidelines / 6

  32. z overflat z=(x,y,t) x u(x,y,t) D= +H  = konst. z=-H(x,y,t) bunn eller termoklin eller pyknoklin Integrate the continuity equation from bottom to surface: Potential Vorticity GEOF110 Guidelines / 6

  33. Conservation of potential vorticity • D = konst. • (i) f = konst.  sonal bevegelse   = konst. • (ii) Meridional bevegelse mot nord polen  f øker   avtar (i.e., “-”, CW rotation) • (iii) Meridional bevegelse mot syd polen  f avtar   øker (i.e., “+”, CCW rotation) • D øker og (f+)>0 • (i) f = konst.  sonalt   = øker • (ii) mot nord  f øker   ? • (iii) mot sør  f avtar  øker (i.e., “+”, CCW rotation) For storskala, indre bevegelser  << f  f/D = konstant i.e., hvis D øker  f må øke  Topographic steering GEOF110 Guidelines / 6

  34. As the vertical fluid column moves from left to right, vertical stretching reduces the moment of inertia of the column, causing it to spin faster. Topographic steering: Barotropic flow over a sub-sea ridge is turned equatorward to conserve potential vorticity. From Dietrich, et al. (1980). GEOF110 Guidelines / 6

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