Gravitational Potential Energy and Oceanic Circulation: Understanding the Driving Forces

1. Gravitational potential energy and Oceanic circulationRui Xin HuangWoods Hole Oceanographic Institution

2. Can surface thermal forcing drive the oceanic general circulation ? NO ! • 1) The “Sandstrom theorem” • A closed steady circulation can only be maintained in the ocean if the heat sources is situated at a lower level than the cold source. This “theorem” is inaccurate! • 2) Three types of thermal forcing • 1) Type 1: Heating source is below the cooling source • There is strong circulation (Rayleigh-Barnard) • 2) Type 2: Heating source is above the cooling source • There is a very weak circulation only • 3) Type 3: Heating/cooling at the sea surface (or bottom) source • There is weak circulation, this is also called horizontal convection.

3. “Sandstrom Theory”

4. Energy conversion in a compressible fluid, A modified version of Paparella and Young (2002) This is the upper limit

5. Rate of internal to mechanical energy conversion in the world oceans Energy conversion rate Total conversion rate from internal to mechanical energy is less than In comparision, total mechanical energy from wind (64TW), tides (3.5TW), this is totally negligible !

6. Balance of mechanical energy in the world oceans (in units of TW)

7. Tidal dissipation in the world oceans(Munk & Wunsch, 1998)(Modified)

8. M2 Tidal dissipation rate over the past 500ma Changes of M2 tide dissipation over the past 600 millions years (after Kagan & Sudermann, 1996).

9. Deep Mixing during the Last Glaciations Maximum (LGM) • 1) Sea level was more than 100 m lower than present day. • 2) Tidal dissipation in the shallow seas was much reduced. As a result, tidal flow in the deep oceans was much fast. • 3) Global tidal dissipation during LGM was 50% higher than present day (Egbet, Ray and Bills, JGR, 2003). • 4) The meridional temperature in the atmosphere was larger, so wind was much stronger; thus, wind energy input into the ocean was much stronger. • It would be a great project to simulate the oceanic circulation and climate during LGM with a new parameterization of diapycnal mixing that changes with climate.

11. Two types of upwelling • 1) Upwelling in the deep ocean induced by diapycnal mixing (roughly in the vertical direction) through internal wave breaking, which is supported by mechanical energy from tidal dissipation and wind stress. • 2) Upwelling driven by wind-driven Ekman divergence, especially the westerly in ACC and the easterley in the equatorial regime. The energy required is the wind energy input to the surface geostrophic current, which may be the most important source of mechanical energy supporting the global thermohaline circulation.

12. Two types of diapycnal mixing • To maintain stratification in the deep ocean diapycnal mixing is needed, and this requires mechanical energy. • During the northward movement in the Southern Ocean, water within the Ekman layer is warmed by surface heating. Mechanical energy required for the uphill (the surface elevation) movement is provided by the wind energy input to the geostrophic flow, so it does not require additional mechanical energy. • However, during this northward movement heating in the whole mixed layer leads to increase of GPE, this energy is provided by wind stirring. • Horizontal mixing in the mixed layer may require additional mechanical energy, which can be supported from wind energy input through the Ekman layer or surface waves.

13. 2-Cell structure of meridional overturning circulation in the Southern Ocean Upwelling driven by surface wind, but mixing in the surface layer requires no additional mechanical energy Deep mixing requires mechanical energy

14. Horizontal geostrophic current and Ekman flux Ekman upwelling Energy input to the geostrophic current is the same as the work for push the Ekman flux uphill It is also equal to the work supporting the Ekman upwelling

15. Balance of mechanical energy inthe modern oceanic circulation (with ACC on)

16. Scaling of the meridional overturning: The mass balance is The thermocline depth equation (Gnanadesikan, 1999) This is reduced to a simple equation

17. Introduce a non-dimensional depth Assuming a bulk mixing rate Under the current climate, NADW is primarily controlled by wind stress over ACC; Eddy Term is very important, and diapycnal mixing plays a minor role only.

18. A z-coordinate model underthe inviscid limit(Toggweiler and Samuel, 1998) With wind forcing, a model with no vertical mixing has a non-zero overturning rate and poleward heat flux

19. The “unbelivable” balance of mechanical energy for the oceans • (70%?) Wind energy input into the geostrophic current -- this is also the energy supporting the northward uphill flow in the Ekman layer. Diapycnal mixing in the mixing layer requires no additional mechanical energy for supporting. • (15%?) Mixing in the mixed layer, including loss of GPE during diurnal/annual cycle of mixed layer, especially convection due to cooling. • (<15%?) Diapycnal mixing in the abyssal ocean is required, but it is a small part of the total budget.

20. Why wind is important for thermohaline circulation and climate variability? • Wind energy input (60+3+1 TW) is much larger than the tidal dissipation in the open ocean (less than 1TW) • Wind stress dominates upper ocean circulation which is the primary focus for many application, including climate, environment, fishery…. • Wind energy input varies over a very broad time scale, interannual, decadal to centennial, tidal dissipation vary on millennial time scale.

21. GPE balance in a model ocean • A. A model based on the pressure coordinates: The model conserves mass and GPE (gravitational potential energy) • B. Vertical (Diapycnal) mixing is assumed constant. C. Heating/cooling from above or from below, with a linear reference temperature profile, T=0-25C. D. A basin, 5 km deep. E. A linear equation of state A nonlinear equation of state

22. How to diagnose source/sink of GPE is the bottom pressure, is the reference bottom pressure Density is diagnostic, potential temperature is prognostic GPE balance for a steady state: ADV+VM+HM+SF+CA=0 ADV

23. Model Ocean with different thermal forcing and equation of state

24. Distribution of temperature and circulation

25. Difference in the circulation

26. Difference in SSH/Bottom pressure

27. Balance of GPE for four cases

28. The generalized Reynolds transport theorem • The Leibnitz’s theorem: • For a three dimensional space, the generalized Leibnitz’ theorem is

29. Basic conservation equations:

30. Basic conservation equations:

31. At steady state, there is no net vertical mass transport and no net change in GPE, so the net conversion rate should be zero. Does this mean the balance of GPE is dynamically unimportant? NO! Generation of GPE, its transportation and dissipation may be the most important physical processes what regulates the oceanic general circulation. • The conversion from GPE -> KE: • The perturbation term can be interpreted as GPE source due to turbulence and internal waves in the oceanic interior with stable stratification and sink of GPE due convective adjustment. • Using the horizontal-mean density and vertical velocity, then the perturbation term can be interpreted as GPE conversion to kinetic energy.

32. Meridional distribution of sources/sinks of GPE

33. Conversion to Kinetic Energy and the Meridional transport of GPE Souce Vertical Velocity deviation Density deviation Poleward transport

34. Source and poleward flux of GPE due to vertical mixing and surface forcing

35. Vetical distribution of sources of GPESum=2.6GW Souce due to mixing Sink due to Convective adjustment

36. Seasonal cycle of the basin-integrated sources/sinks of GPE for Case A GPE source due to surface forcing becomes positive Source due to vertical mixing Net flux of GPE Sink due to Convective Adjustment

37. GPE source and sinkdue to thermal forcing on the bottom

38. Conclusions and Speculations • For the cases of thermal forcing from above, cooling appears as a big sink of GPE. Due to the convective adjustment the effective centre of cooling is not on the sea surface; instead it is half way of the mixed layer depth. • The basic balance of GPE is: source due to diapycnal mixing, and sink due to convective adjustment. • Cabbeling effect reduces the source/sink of GPE associated with vertical mixing and convective adjustment. • For the case of thermal forcing from below: Under the assumption of a constant mixing rate and a non-linear equation of state, there can be a large amount of mechanical energy generated. However, this energy may be offset by the loss of GPE due to cabbeling.

39. GPE & SGPE Gravitational potential energy (GPE) is defined as Value of GPE depends on the choice of reference level. Using the mean depth of the ocean, 3750 m, as the reference level, the total amount of GPE in the world oceans is estimated as 2.1E25J and the density of GPE for the world oceans is 1.4E7J/m^3. Stratified GPE (another index for the stratification) GPE of a water column can be separated into two parts: Note SGPE is negative, and it represents the strength of the stratification.

40. Available potential energy in the world oceans • APE in an incompressible ocean can be calculated by a simple sorting program (Huang, 1998, JPO). • APE in a compressible ocean includes two components: AGPE and AIE (available internal energy). The vital point is to find a reference state of minimal gravitational potential energy. • This reference state can be found through an iterative program.

41. Define the AGPE for a two-layer ocean

42. Using z=-H as the reference level, the total GPE in a water column • The AGPE of the system is For the warm water pool in the Pacific ocean, H=100m, L=10000km, and , the meridional width of the pool is 200km, so the total amount of AGPE is . The density of AGPE is If all this energy is converted to kinetic energy, it is enough to accelerate a barotropic current from a speed of zero to 0.5 m/s.

43. Available Gravitational Potential Energy in the ocean Two definitions: 1) The meso-scale (QG) APE: Oort et al. (1987) is the horizontal mean density is the horizontal mean stratification 2) The exact definition:

44. Balance of GPE (upper) and AGPE (lower) in a model ocean (flux in mW/m^3, APE in J/m^3)

45. Contribution to AGPE through adiabatic adjustment, in units of E+11 J/m2

46. Contribution to AGPE through adiabatic adjustment, in units of E+13 J/m

47. A simple case demonstrating the difference between the exact definition and the QG definition

50. Table 3. APE dependence on the pressure effect, in units of J/m3.