Artificial diffusivity/viscosity in Eulerian models. Models in intrinsic coordinate?

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1 Energetics of mixing Artificial diffusivity/viscosity in Eulerian models Models in intrinsic coordinate? Rui Xin Huang Woods Hole Oceanographic Institution Woods Hole, USA

2 Ocean circulation is maintained by external mechanical energy Vertical mixing is regulated by mechanical energy There is strong lateral mixing in the ocean, but it is treated almost arbitrarily Physical constraint for lateral mixing? Our goal: 1) New model with no artificial diffusion 2) Parameterizing lateral mixing with real physics 3) Using GPE as a criterion to evaluate lateral mixing in models

3 Do eddy-resolving models really resolve eddies? No!! Lateral diffusivity in high-resolution models is set to zero(or very small). People believe that such models really resolve eddyeddy interaction, so eddies calculated are real eddies Observations indicated lateral diffusion in the ocean is finite, not zero; thus, models are not honest There is strong artificial diffusion associated with the horizontal advection in the Eulerian models.

4 Mechanical energy balance

5 Main source/sink of mechanical energy Tidal dissipation in open ocean: TW0.18TW Wind to surface geostrophic current 1TW Convective adjustment: 0.24TW Bottom drag: TW (?) Assuming a constant vertical diffusivity GPE changes (TW):Stirring:1.32, Cabbeling:-0.36, Net:0.96 Energy available:0.2tw constant vertical diffusivity 2 should set to. A 0.2 cm / s v 2 Av 1 cm / s Upper limit for lateral advection/diffusion: tw? Anything larger than this limit is no good!

6 Numerical diffusion due to horizontal advection in Eulerian Coordinates The tracer equation Eulerian Coordinates t u Diffusion Lagrangian Coordinates J Diffusion J is the Jacobian There is no diffusion associated with advection in Lagrangian coordinates. Advection stretches water parcel s volume only.

7 S 2 us A S t In x-y coordinates Finite difference us 0.5 u S 4 S5 ( S5 S6) S2 S5 ( S5 S8) / y 0.5 u S S S S / y In X-Y coordinates US 0.5 U S1 S5 ( S5 S9) / Y U S1 S9 Y u S1 S9 y The equivalent diffusivity (the worst case) 0.5 / 0.5 / U 2 2x A S S S S S S S S S S S x A F R, F Ux, R S S S S S S S S S S 4S

8 Equivalent diffusivity can be negative Normalize salinity S S S S i ' i S S S 2 8 S Advection in x-y coordinates increases salinity in box 5 S1 S Advection in X-Y coordinates reduces salinity in box 5 S S S S 4S Advection in x-y coor. Overpowers that in X-Y coor., leads to increase in S5 equivalent negative diffusivity

9 Universal Factor Use the normalized tracer variable Dropping the primes S S S S S S S S S S R, D S2 S4 S6 S8 4S5 2D Thus, the equivalent diffusivity obtained is independent to flow field and the amplitude of tracer variability A F R F ux is a function of u and grid size only i ' i

10 The R factor from a Monte-Carlo experiment 101 runs (5,000,000 casts each run) Probability of negative equivalent diffusion

11 Universal Factor For the general case: A F R, F Ux, R S S S S S S S S S S 4S sin o R 1 A R F F ux It is a function of mean velocity and grid size only

12 Numerical diffusion associated with advection in Eulerian coordinates is intrinsic for this coordinates Artificial diffusion due to advection overpowers the traditional diffusion F. Bryan (1987) This problem exists even if model resolution goes to 1 mm Current climate models L ( Obuko) Small /zero diffusivity in highresolution runs, (1/6 o in ECCO2?)

13 What happened for x = 10km? Modelers set A h =0 there is no explicit diffusion and modelers claimed that models resolve eddies. Really?? Observations indicated that there should be lateral Laplacian diffusion with A h =10 m 2 /s. Models cannot truthfully simulate turbulence in the ocean. Models do have self-generate lateral diffusion, but this is numerical diffusion, which is out of our control, and it is not the same as what observed in the oceans.

14 Separating lateral eddy diffusion as 2 steps: stirring & subscale turbulent mixing 14

15 Using GPE change as a criterion to evaluate lateral advection/diffusion Advection Diffusion Change mean density at a grid Cabbeling leads to increase of density Push the water column up/down Change GPE of the system

16 Effect of lateral advection/diffusion t 2 Cu A C A C C Cw, C (, S) h h h h z z v z Lateral advection/diffusion spread the fluid and thus change temperature, salinity and density fields. Similarly, lateral advection/dissipation of momentum can also affect the velocity field. It is now realized: eddies grow/decay mostly through lateral processes: instability, eddy-eddy interaction, eddy decaying through lateral dissipation and breaking down. Accurate eddy simulation critically depends on better representing dynamical processes in quasi-horizontal surfaces: advection/dissipation(diffusion).

17 GPE source/sink inferred from SODA SODA data: , 0.5 o x0.5 o, 40 levels vertically Multi-year mean (S,T,u,v) with grid resolution of 1 o x1 o and 0.5 o x0.5 o. Z model, assuming p(db)=1.035z (m); Assuming horizontal velocity is the same before projection to different coordinates. Isopycnal model: use mean stratification between two grids to define the mean isopycnal slope, and (S,T,u,v) obtained by linear interpolation. Sigma model: use the topography from SODA to construct an equivalent sigma coordinates; (S,T,u,v) obtained by linear interpolation.

18 GPE source/sink (Stirring, TW) Coordinates Terms Source Sink Net Lateral diffusion Isopycnal Lateral advection Z sigma Lateral diffusion Lateral advection Lateral diffusion Lateral advection

19 GPE source/sink (Cabbeling, TW) Negative value indicates negative diffusivity Coordinates Terms Source Sink Net Lateral diffusion Isopycnal Lateral advection Z sigma Lateral diffusion Lateral advection Lateral diffusion Lateral advection

20 GPE due to diffusion (A H =1000m 2 /s) Huge source/sink in middle/deep ocean in sigma model is artificial

21 GPE source/sink Isopycnal model: Temperature front in oceanic interior Sigma model: Temp. difference at the sea floor (due to sea floor depth difference).

22 GPE source/sink Large source/sink in isopycnal model is due to fronts in the upper ocean Huge source/sink in sigma model is due to strong bottom temperature difference

23 GPE due to advection This is a problem for Eulerian models Small in deep ocean due to slow velocity Huge source/sink in sigma model --- potentially a big problem

24 Sigma mixing is mostly artificial A real section from WOA09 A section obtained by horizontal averaging A section in sigma coordinates 24

25 Over steep topography, artificial diffusion due to lateral advection/mixing can induce large errors, in particular in sigma coordinates Artificial lateral advection brings & mixes water from top/valley 不是迎着困难上而是绕着困难走 Not much vertical movements Mostly horizontal movements

26 Sigma-coordinate allows cross-slope exchange, which is artificial outside of the bottom boundary layer H H Sigma coordinate gives rise to uphill/downhill exchange of tracer Z and isopycnal coordinates have no uphill/downhill exchange of tracer

27 Increase model resolution cannot reduce the artificial GPE source/sink (Source/sink due to stirring, TW) Coordinates sigma Terms Grid Source Sink Net Lateral diffusion Lateral Advection

28 GPE source/sink (Cabbeling, TW) Coordinates sigma Terms Grid Source Sink Net Lateral diffusion Lateral Advection

29 Artificial viscosity in momentum equation u 2 uu Amh, u t Local velocity is a small perturbation to the mean velocity of large scale u U u and u U X Y U isthe slowly varying function ', ',, Linearization leads to u ' t 2 mh, 2 Uu ' A u ' There is artificial dissipation in the momentum equation, similar to the tracer equation, plus a factor of 2. There can be negative viscosity, but it may not be the same as the negative viscosity inferred from observations in quasi 2D turbulence.

30 The challenges High resolution <=0.1 o : the lateral diffusion is set to zero mix tensor rotation & GM90 scheme are outdated. This does not mean zero lateral diffusion. In Eulerian coordinate, horizontal advectionlarge artificial diffusion Ocean circulation is regulated by mechanical energy Use mechanical energy as a constraint for lateral mixing parameterization; eliminating artificial diffusion in models. New light: Lagrangian-type coordinates: Intrinsic coordinates Unified Coordinates.

31 The unified coordinates Hui Transformation (1999) dt d dx Ud Ad Ld dy Vd Bd Md This new coordinates combines the advantage of Euler and Lagrangian coordinates and avoid their problems. The coordinates is self-generated, it fits the boundary perfectly.

32 Free fall of a rectangular plate (Hui, 2007) Vorticity field