Maximum Entropy Production Principle: Where is the positive feedback?

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1 Maximum Entropy Production Principle: Where is the positive feedback? Remi Tailleux Department of Meteorology, U. of Reading HHH - 30 November 2009

2 Outline The context: Horizontal convection Formulation of the MEP principle Importance of stirring: Zeldovich s result Basic ideas on turbulent diapycnal mixing Power input due to surface buoyancy fluxes The positive feedback

3 Horizontal Convection buoyancy-driven circulation forced by stabilizing and destabilizing buoyancy fluxes applied at the same level (as in the oceans) Physical ingredients High-latitude cooling-> dense plumes Turbulent molecular diffusion to avoid the oceans to fill up with cold water

4 Stabilizing heating Destabilizing Cooling

5 Entropy balance Equation dσ dt = V ρ Q T dv + V ρε T dv ρ Q = (κρc p T ) S κρc p T nds T = V ρκc p T 2 T 2 dv + V ρε T dv Production Dissipation by molecular diffusion Viscous dissipation

6 MEPP assumption Maximization of Entropy production: V κρc p T nds T = [ Qin T in + Q out T out ] = ( Tin T out T out ) Qin T in Equivalent to maximizing Entropy destruction V κρc p T 2 T 2 dv + V ρε T dv ρκc p T 2 0 V T 2 dv

7 Puzzling points What is the relevant optimization problem? Is the MEPP a property of the Navier- Stokes equations, or a new principle of Nature? The controversy about statistical versus thermodynamic entropy

8 Approach Reaching a steady state requires: A positive feedback to make irreversible entropy production grow A negative feedback to limit the rate of irreversible entropy production What are these? Can we identify both a positive and negative feedback?

9 Boussinesq HC, linear eos Dv Dt +2Ω v + 1 P = bz + ν 2 v ρ 0 v =0 Db Dt = κ 2 b Dimensionless parameters R h = gα TH3 νκ Horizontal Rayleigh number P r = ν κ Prandtl number

10 Numerical Examples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

11 Importance of stirring: Zeldovich (1937) result Φ = V κ T 2 dv V κ T c 2 dv > 1 T and Tc solutions of the following problems, with same boundary conditions κ 2 T c =0 κ 2 T = v T Stirring required for MEP principle!

12 Stirring versus Mixing STIRRING: Adiabatic lifting up of dense parcels and pushing down of light parcels Stirring requires power! Does not affect pdf of entropy MIXING: Destruction of tracer variance by molecular diffusion Affects pdf of entropy

13 How Stirring enhances Mixing

14 Bounds on horizontal convection Bounds on epsilon Bounds on chi

15 Available Poten-al Energy (APE) and APE density Φ a (x, t) = N 2 ζ 2 2 ζ(x,y,z,t) APE(t) = V Φ a (x,t)dm 15

16 Energetics Issues Buoyany forcing Buoyany forcing D(KE) KE C(APE,KE) APE D(APE) IE o G(APE) IE exergy GPEr

17 Mechanical Energy (APE+KE) balance: G(AP E) =D(KE)+D(AP E) Production by buoyancy Viscous dissipation Diffusive dissipation

18 APE production by cooling

19 Diffusive APE dissipation

20 The nature of G(APE) G(AP E) = S αgz r C p Q surf ds αg C p [h cooling h heating ] Q cooling hheating 0 hcooling = penetration depth of dense plumes Measure of thermocline depth Key Result: The power input due to buoyancy forcing depends on the total amount of cooling and on the thermocline structure!

21 The Answer V ρε T dv D(KE) Cooling APE KE D(APE) Thermocline depth??? ρ 0 κc p T 2 0 V T 2 dv

22 Postulate MEPP seems to require the maximization of the so-called mixing efficiency Γ, often thought to be close to Γ=0.2 Γ = D(AP E) D(KE)

23 Numerical Results (200 x 200) G(KE) G(APE) D(APE) Γ Ψ I II III IV I = Thermally direct buoyancy driven circula-on II = Direct Mechanical Forcing III = Weak Indirect Forcing IV = Strong Indirect Forcing 23

24 Dependency on Ra (Pr=10) Buoyancy driven horizontal convec-on R a = gα TH3 νκ R a = in the Oceans Ra Γ ? Numerical Resolu-on: 150 x

25 Summary MEPP requires maximizing stirring and dissipation Maximizing stirring requires maximizing power input and minimizing dissipation Conflict Way out: Minimizing viscous dissipation (maximize stirring) and maximizing diffusive dissipation (maximize G(APE))

26 Summary (cont d) This is equivalent to maximizing the socalled mixing efficiency It seems possible for horizontal convection to satisfy MEPP Is that useful? (i.e., to make predictions of overturning strength, to predict behavior as function of Rh, and so on...)

The effect of mechanical stirring on horizontal convection

Tellus, () Printed 2 October 29 (Tellus LATEX style file v2.2) The effect of mechanical stirring on horizontal convection By R TAILLEUX 1 and L ROULEAU 1 1 Department of Meteorology, University of Reading,