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Introduction to Observational Physical Oceanography 12.808 Class 18, 17 November, 2009

Introduction to Observational Physical Oceanography 12.808 Class 18, 17 November, 2009 1:05 to 2:25 these slides are online at www.whoi.edu/class/12808. Topic: Momentum Balances, Geostrophy and the large-scale circulation of the upper ocean. Momentum Equation: F = ma

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Introduction to Observational Physical Oceanography 12.808 Class 18, 17 November, 2009

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  1. Introduction to Observational Physical Oceanography 12.808 Class 18, 17 November, 2009 1:05 to 2:25 these slides are online at www.whoi.edu/class/12808

  2. Topic: Momentum Balances, Geostrophy and the large-scale circulation of the upper ocean Momentum Equation: F = ma Inertial/non-inertial reference frames Centrifugal Acceleration Coriolis Acceleration Hydrostatic Balance Pressure Gradient Force Summary Forces in Momentum Equations Geostrophy Friction

  3. Equations of Motion Force = mass x acceleration Force = mass x acceleration on a rotating planet Force = mass x acceleration on a rotating planet for a fluid Particle acceleration Local acceleration Advection

  4. Equations of Motion • Sum of forces acting on a parcel of fluid: • Gravity • Pressure gradient • Friction

  5. Equations of Motion • Sum of forces acting on a parcel of fluid: • Gravity • Mostly in the vertical direction • Fz = -g g = 9.81 m2/s

  6. Equations of Motion • Sum of forces acting on a parcel of fluid: • Gravity  Fz =-g • Pressure gradient – due to gradients in the pressure acting on a fluid parcel

  7. Summary Equations of Motion Coriolis Gravity Frictional Forces Pressure gradient

  8. Vertical Momentum Equation (Vertical component of Coriolis is small) Hydrostatic Balance and the hydrostatic approximation Steady motion (no acceleration) and neglecting friction:

  9. Hydrostatic Balance z η Z=0 x + Pa Atmospheric pressure displaced sea surface density variations at depth

  10. Horizontal Momentum Equations • where f = 2Ωsinθ • Coriolis Force • Increases amplitude with latitude • Proportional to the velocity (no flow – no force) • At right angles to the motion

  11. Geostrophic Balance Coriolis force balances the horizontal pressure gradient • Where does this apply: • Away from boundaries where friction is important • Over large enough spatial scales • forcing is mostly steady

  12. Mass and pressure fields in the ocean • Over most of the ocean and atmosphere, the dominant • (most common or leading ) momentum balance for large • scale winds and currents is a geostrophic balance between • the Coriolis force and the horizontal pressure gradient. • The horizontal pressure gradient is due to differences in • mass distribution (mass ‘pile up’ = high pressure) • Mass distribution results from • Sloping sea-surface height • (more water in one place than another) • ii) Horizontal density gradients • (denser water in on place than another) • Very often these two effects run in the opposite directions • and cancel at great depth.

  13. Geostrophic in a Barotropic Ocean • Barotropic – Flow is invariant with depth • (isopycnals are not sloped but vary only with depth) • Sloping sea-surface height • (more water in one place than another)

  14. Geostrophic Balance in a Barotropic Fluid sea-surface height η x1 x2 z If we ignore horizontal density variations, the pressure gradient force is tied to sea-surface height variations x

  15. In class Exercise: A satellite altimeter observes a meridional sea-surface slope of 1 m over 100 km, at mid-latitudes. Assuming geostrophy, what is the speed and direction of flow of the current if the sea-surface height is higher to the south? Ignore density variations and assume we are in the northern hemisphere. Take f = 2Ωsinθ ~ 10-4 s-1

  16. the mean sea surface of the N Atlantic -0.8 m 0.9 m Steve Jayne and Grace

  17. Labrador current L Sverdrup interior North Atlantic current Gulf Stream western boundary current H Sverdrup interior Antilles, Loop and Fla current North Eq. current Steve Jayne and Grace

  18. Geostrophic Balance in a Baroclinic Ocean • Baroclinic – flow varies with depth and the isopcynals are not flat • Sloping sea-surface height • (more water in one place than another) • ii) Horizontal density gradients • (denser water in on place than another)

  19. Geostrophic Balance in a Baroclinic Ocean Consider a fluid with sloping isopycnals but a flat sea-surface η = 0 ρ1 ρ2 z x

  20. Baroclinic Geostrophic Flow η = 0 Consider a 2 layer fluid, flat sea-surface. According to geostrophy, the flow in the top layer is due to the sea-surface slope alone: ρ1 ρ2 z x In the second layer, geostrophic balance reduces to:

  21. In Class Exercise – Baroclinic Geostrophic flow η = 0 Consider a 2 layer fluid, flat sea-surface. According to geostrophy, the flow in the top layer is due to the sea-surface slope alone: ρ1 ρ2 z x • Taking ∆z = 500m, ∆x = 100km, ∆ρ= 2 kg/m3 • What is the magnitude of the current in the lower layer, and the direction? • How does the isopcynal slope compare to that of the sea-surface in the previous example ?

  22. In reality both the sea-surface and the isopycnals are sloped but in opposite directions sea surface from Grace gradient P ~ 0 (?)

  23. Baroclinic and Barotropic Flow in the Gulf Stream Left: Relative current as a function of depth calculated from hydrographic data collected by the Endeavor cruise south of Cape Cod in August 1982. The Gulf Stream is the fast current shallower than 1000 decibars. The assumed depth of no motion is at 2000 decibars. Right: Cross section of potential density across the Gulf Stream along 63.66°W. The Gulf Stream is centered on the steeply sloping contours shallower than 1000m between 40° and 41°. http://oceanworld.tamu.edu/resources/ocng_textbook

  24. Friction • Necessary for the • Driving of ocean currents by the wind • Exchange of momentum between adjacent fluid parcels • Dissipation at the boundaries

  25. Friction At the molecular scale – momentum is transferred from one molecule to the next – in the same way as heat or salt  molecular viscosity coefficient In a fluid, however, turbulence is much more efficient at transferring momentum  Eddy Viscosity (analogous to eddy diffusivity) Ah ~ 102-105 m2/s Az ~ 10-4-10-2 m2/s Compare to molecular viscosity ν ~ 10-6 m2/s

  26. What is driving changes in this Greenland Fjord? • Fjord warmed between July and September • Observed heat flux was negative during the two surveys – i.e. driving cooling and, therefore cannot explain the July-September change

  27. Some other clues • Very stratified ocean (i.e. small influence of surface fluxes) • T/S diagram reveals that the watermasses have changed – i.e. the change cannot be explained by mixing

  28. SST 2003 Some other clues This suggests that the change is due to advection of different water masses from outside the fjord  what goes on outside the fjord? Subtropical Waters (STW) Polar Waters (PW) Sutherland & Pickart ‘08

  29. How to obtain data from remote, ice-covered regions?

  30. Temperature profiles collected by 19 Hooded Seals For a total of > 5000 dives 2003-2007 What we found is that there is an increase in the warm waters on the shelf from July to September – in agreement with what we are seeing in the fjord.

  31. How do shelf waters get into the fjord? (NB the fjord is not geostrophic – too narrow). Moored data  abrupt cooling during wind events

  32. Preview – Impact of along-shore wind events on the shelf

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