Statistical Mechanics for the Truncated Quasi-Geostrophic Equations

Transcription

1 Statistical Mechanics for the Truncated Quasi-Geostrophic Equations Di Qi, and Andrew J. Majda Courant Institute of Mathematical Sciences Fall 6 Advanced Topics in Applied Math Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 / 36

2 and = is the region occupied by the fluid. Recall that, in the empirical theory introduced in Chapter 6, we postulated empirical one-point statistics which have no direct relationship with the dynamic equations Di Qi, and Andrew (8.). J. Majda In(CIMS) this chaptertruncated we build Quasi-Geostrophic an alternative Equations theory, involving Nov. 3, complete 6 / 36 Introduction 8. Introduction In this lecture, we set-up equilibrium statistical mechanics and apply the theory to suitable In this chapter truncations we set-up of the statistical barotropicmechanics quasi-geostrophic and applyequations the theorywithout from Chapter 7 to suitable and forcing truncations of the barotropic quasi-geostrophic equations without damping damping and forcing q dv + J q = t dt t = h x (8.) where q = + h + y = = V t y + q = + h ( ) v = V t = +

3 Statistical theory for truncated QG equations Step I: We make a finite-dimensional truncation of the barotropic quasi-geostrophic equations. The equation is projected on to a finite-dimensional subspace with a Galerkin approximation. Step II: We then observe the special properties: The truncated equations conserve the truncated energy and enstrophy. However, higher moments of the truncated vorticity are no longer conserved. The truncated equations satisfy the Liouville property. The truncated equations define an incompressible flow in phase space. Therefore the resulting flow map is measure preserving, and we can transport measures with the flow. Step III: We apply the equilibrium statistical mechanics theory for ODEs to the truncated system, i.e. find the probability measure that maximizes the Shannon entropy, subject to the constraints of fixed energy and enstrophy. Step IV: We study the limit behavior of this mean state in the limit when the dimension N. This is the continuum limit of the system. Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 3 / 36

4 Lecture 9: Statistical Mechanics for the Truncated Quasi-Geostrophic Equations The finite-dimensional truncated quasi-geostrophic equations The spectral truncated quasi-geostrophic equations Conserved quantities for the truncated system Nonlinear stability to the truncated system The Liouville property The statistical predictions for the truncated systems 3 Numerical evidence supporting the statistical prediction 4 The pseudo-energy and equilibrium statistical mechanics for the fluctuations Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 4 / 36

5 Outline The finite-dimensional truncated quasi-geostrophic equations The spectral truncated quasi-geostrophic equations Conserved quantities for the truncated system Nonlinear stability to the truncated system The Liouville property The statistical predictions for the truncated systems 3 Numerical evidence supporting the statistical prediction 4 The pseudo-energy and equilibrium statistical mechanics for the fluctuations Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 5 / 36

6 where the amplitudes ˆ k ˆ k and ĥ k satisfy the reality conditions ˆ k = ˆ k ˆ k = Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 6 / 36 Truncated variable in Fourier basis The truncated system is a Galerkin approximation of the barotropic quasi-geostrophic equations with the use of standard Fourier basis. 8. The finite-dimensional truncated quasi-geostrophic equations 59 The truncated small-scale stream function, ψ Λ, the truncated vorticity, ω Λ, and the truncated dynamics equations we proceed as follows. We first introduce the the truncated topography, h Λ, in terms of the basis Fourier series expansions of the truncated small-scale stream function, the truncated { vorticity, and the truncated topography h in terms of the basis ( ) B = exp i k x k } h k k k ˆ k t ei x k = k k ˆ k t ei x k ĥ k t ei x k (8.) ˆ k t ei x k = k k ˆ k t ei x k

7 d ˆ k i k l m Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 7 / 36 k Galerkin approximation to finite-dimensional subspace k ˆ k t ei x k = k k ˆ k t ei x k We where denote the amplitudes by P Λ the orthogonal ˆ k ˆ k and ĥ projection k satisfy the onto reality the finite-dimensional conditions ˆ k = space ˆ k ˆ k = Vˆ Λ = spac {B Λ } k, and ĥ k = ĥ with denoting complex P Λ f = conjugation here. We also assume that k the mean value h ˆf k e ik x of the topography is zero, so that. the solvability condition for the associated steady state equation is k Λ automatically satisfied. We also denote by P the orthogonal projection on to the finite-dimensional space V = span B. The truncated dynamical equations are obtained by projecting the barotropic quasi-geostrophic equations in (8.) onto V q t + x + V q x + P ( ) q = q = + h (8.3) dv dt h = (8.4) x Then, we get the finite-dimensional system of the ordinary differential equation (ODE), the truncated dynamical equations, for the Fourier coefficients with k

8 quasi-geostrophic equations in (8.) onto V The finite-dimensional truncated dynamical equations q t + x + V q x + P ( ) q = q = + h (8.3) dv dt h = (8.4) x Then, we get the finite-dimensional system of the ordinary differential equation (ODE), the truncated dynamical equations, for the Fourier coefficients with k d ˆ k dt dv t dt i k k ˆ k + ivk ˆ k + ĥ k i k l+ m= k l m l m ˆ l l ˆ m + ĥ m = (8.5) k ĥ k ˆ k k = (8.6) 8.. Conserved quantities for the truncated system According to Subsection.3.5, the quasi-geostrophic equations (8.) possess two or more robust conserved quantities. An advantage of utilizing Fourier truncation Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 8 / 36

9 Conserved quantities in original QG equations The (non-truncated) quasi-geostrophic equations (8.) possess two or more robust conserved quantities. Utilizing Fourier truncation, the linear and quadratic conserved quantities survive the truncation. Candidates for Physically Conserved Quantities: Total Kinetic Energy: E = V + ψ ; Enstrophy and Potential Enstrophy: E = ˆ ω, E = ˆ q ; Large-scale Enstrophy: Generalized Enstrophy: Q = βv (t) + (ω + h) ; ˆ Q = G (q). Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 9 / 36

10 Conserved quantities for the truncated system 6 Statistical mechanics for the truncated quasi-geostrophic equations is that the linear and quadratic conserved quantities survive the truncation. More precisely we have: Proposition 8. The truncated energy E and enstrophy are conserved in the finite-dimensionally truncated dynamics, where E = V + d x = V + k ˆ k (8.7) = V + q d x = V + k k k ˆ k + ĥ k (8.8) Proof: In order to show that the truncated total energy E is conserved in time it is equivalent to proving that the time derivative of it is identically zero for all time. For this purpose we take the time derivative of the truncated total energy E and utilize the truncated dynamic equations (8.3) and (8.4) and we repeatedly carry out integration by parts d dv Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 / 36

11 Proof k Proof: In order to show that the truncated total energy E is conserved in time We it istake equivalent the timetoderivative proving that of the thetruncated time derivative total energy of it ise Λ identically and utilizezero thefor all truncated time. Fordynamic this purpose equations we take (8.3) theand time (8.4) derivative and we of repeatedly the truncated carry out total energy integration E by parts and utilize the truncated dynamic equations (8.3) and (8.4) and we repeatedly carry out integration by parts d dt E = V dv dt t d x = V dv dt q = V = V = t d x x + x d x + V ( + P ) q h h x + V k q x d x h x d x + ( ) q The conservation of the truncated total enstrophy can be shown in a similar Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 / 36 k

12 The same argument as in Section 4. implies that the steady state solution q V is non-linearly stable for > in general, or for > when V. Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 / Non-linear stability of some exact solutions to the truncated system Nonlinear stability to the truncated system It is easy to see that the truncated system possesses exact solutions having a linear q The truncated system relation similar to those introduced in Chapter for the continuum possesses exact solutions having a linear q Λ ψ Λ relation equations 8. The finite-dimensional truncated quasi-geostrophic equations 6 q = (8.9) This The non-linear condition is stability the same of as these type of steady states can be studied using the same techniques as those in Section 4.. More precisely, + h = V = we consider a linear combination of the truncated energy and enstrophy to form a positive quadratic (8.) form for perturbations. Let q V be the perturbations, we then have, following We consider Section 4. a linear and utilizing combination (8.) of the truncated energy and enstrophy E q V + q V = E q V + q V + q V (8.) where q V = V + = V V + k ( ) + q k k k + k ˆ k ˆ k (8.)

13 The same argument as in Section 4. implies that the steady state solution q V is non-linearly stable for > in general, or for > when V. Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 / Non-linear stability of some exact solutions to the truncated system Nonlinear stability to the truncated system It is easy to see that the truncated system possesses exact solutions having a linear q The truncated system relation similar to those introduced in Chapter for the continuum possesses exact solutions having a linear q Λ ψ Λ relation equations 8. The finite-dimensional truncated quasi-geostrophic equations 6 q = (8.9) This The non-linear condition is stability the same of as these type of steady states can be studied using the same techniques as those in Section 4.. More precisely, + h = V = we consider a linear combination of the truncated energy and enstrophy to form a positive quadratic (8.) form for perturbations. Let q V be the perturbations, we then have, following We consider Section 4. a linear and utilizing combination (8.) of the truncated energy and enstrophy E q V + q V = E q V + q V + q V (8.) where q V = V + = V V + k ( ) + q k k k + k ˆ k ˆ k (8.)

14 is non-linearly stable for > in general, or for > when V. The Liouville property 8..4 The Liouville property Next we verify the Liouville property for the truncated equations (8.3) and (8.4). These equations can be written as a system of ODEs with real coefficients. Indeed, let { } S = k k M (8.3) be a defining set of modes for k satisfying k S k S S S = k (8.4) Let N = M + and define X N ) X (V Re ˆ k Im ˆ k Re ˆ km Im ˆ km X R M+ = N N (8.5) We then notice that each point X in a big space N can represent the entire state of the finite-dimensional system. With these notations, the truncated dynamic equations Di Qi, and Andrew (8.5) J. Majda and (CIMS) (8.6) can be Truncated written Quasi-Geostrophic in a more Equationscompact form Nov. 3, 6 3 / 36

15 ) X (V Re ˆ k Im ˆ k Re ˆ km Im ˆ km X R M+ = N N (8.5) We then notice that each point X in a big space N can represent the entire state of the finite-dimensional system. With these notations, the truncated dynamic equations (8.5) and (8.6) can be written in a more compact form d X dt = F X X t= = X (8.6) 6 Statistical mechanics for the truncated quasi-geostrophic equations with the vector field F satisfying the property F j X = F j X X j X j+ X N (8.7) i.e. F j does not depend on X j. This immediately implies the Liouville property. In fact, F satisfies the so-called detailed Liouville property, since it is satisfied locally in terms of the Fourier coefficients. It is easy to see why (8.7) is true. It is obvious that F is independent of X = V. Observe that F j and F j+ correspond to ˆ k in (8.5). It is obvious that j the contributions from the linear terms in (8.5) are either independent of ˆ or Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 4 / 36

16 Outline The finite-dimensional truncated quasi-geostrophic equations The spectral truncated quasi-geostrophic equations Conserved quantities for the truncated system Nonlinear stability to the truncated system The Liouville property The statistical predictions for the truncated systems 3 Numerical evidence supporting the statistical prediction 4 The pseudo-energy and equilibrium statistical mechanics for the fluctuations Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 5 / 36

17 theory introduced in Section 7.. We base the theory on the truncated energy and Gibbs enstrophy measure from (8.7) based and (8.8), on which truncated are the two energy conserved andquantities enstrophy in this truncated system. Let and be the Lagrange multipliers for the enstrophy and energy E respectively. Let = if (8.8) Thanks to the formula in (7.7), and (8.7) (8.8), the Gibbs measure is then given by = c exp V + k ˆ k + ĥ k V + k k k ˆ k (8.9) where and are determined by the average enstrophy and energy constraints. Due (θ, to α) the areresemblance determined of bythe themost energy probable and enstrophy distribution constraints; (8.9) to the thermal equilibrium ensemble, is sometimes referred to as the inverse temperature and θ is referred to as the inverse temperature; as the thermodynamic potential. α is referred to as the thermodynamic potential. Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 6 / 36

18 Gibbs measure as a probability measure 8.3 The statistical predictions for the truncated systems 63 In order to guarantee that the Gibbs measure is a probability measure we need to ensure that the coefficients of the quadratic terms are negative, i.e. k 4 + k > for all k satisfying k and > if V This implies either This implies i) or ii) or iii) > > (8.) α >, µ > ; V V >, α > >, µ > ; (8.) α <, µ < Λ, θ >. < < > (8.) Obviously, case (8.) is a spurious condition due to the truncation only and Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 7 / 36

19 Mean or state exact solution < < > (8.) The mean states are exact steady state solutions to the truncated system with linear Obviously, potential case vorticity-stream (8.) is a spurious function condition relation. due to the truncation only and hence is not physically relevant. Under the realizability condition in (8.), (8.), we may introduce V = ˆ k = ĥ k (8.3) + k and x t = ˆ ei x k k (8.4) k We then observe that V satisfies equation (8.) and hence it is non-linearly stable under the realizability condition (8.). If V, then is a solution to the first equation in (8.) and it is non-linearly stable if the realizability condition (8.) is satisfied. We may rewrite the Gibbs measure, thanks to (8.) and (8.), as Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 8 / 36

20 stable under the realizability condition (8.). If V, then is a solution to the first equation in (8.) and it is non-linearly stable if the realizability condition The Gibbs measure as a product of Gaussian measure (8.) is satisfied. We may rewrite the Gibbs measure, thanks to (8.) and (8.), as = c exp + E = c exp + E = c exp q V ( ( = c exp V V + k k k + ˆ k ˆ k (8.5) Equivalent the invariant Gibbs measure for the dynamics is a product of Gaussian measures with a non-zero mean ( ) N G α,µ X = Gα,µ j (X j ). j= Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 9 / 36

21 Coherent large-scale mean flow ( Thus, ) the invariant Gibbs measure for the dynamics is a product of Gaussian V, ψ Λ is exactly the ensemble average, mean state, of (V, ψ measures with a non-zero mean. Moreover, V is exactly Λ the ) with ensemble respect toaverage, the Gibbs ormeasure mean state, of V with respect to the Gibbs measure ) X = X X d X = (V ˆ k ˆ km (8.7) N This implies, assuming ergodicity for the Gibbs measure as described in Chapter 7, the following remarkable prediction lim T lim T T T T +T T T +T T x t dt = V t dt = V = (8.8) Hence the statistical theory predicts that the time average of the solutions to the The statistical theory predicts that the time average of the solutions converge truncated to the non-linearly equations converge stable to exact the non-linearly steady state stable solutions exact to steady the truncated state solutions to equations. the truncated equations. In other words, a specific coherent large-scale mean flow A specific will emerge coherent from large-scale the dynamics mean of (8.3), flow will (8.4) emerge with generic from the initial dynamics data after of computing (8.3), (8.4) a long with time generic average. initial data after computing a long time average. Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 / 36

22 Outline The finite-dimensional truncated quasi-geostrophic equations The spectral truncated quasi-geostrophic equations Conserved quantities for the truncated system Nonlinear stability to the truncated system The Liouville property The statistical predictions for the truncated systems 3 Numerical evidence supporting the statistical prediction 4 The pseudo-energy and equilibrium statistical mechanics for the fluctuations Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 / 36

23 Set-ups for the numerical simulations Here we report some numerical evidence supporting the convergence of the time averages of numerical solutions to the most probable mean state predicted by the equilibrium statistical 8.4 Numerical mechanics evidence theory. supporting the statistical prediction 65 As usual, the time average of a function f t over an averaging window T is defined as f T = T T +T T f t dt (8.9) as a function of T, where T is a time when averaging begins. In our case we choose T = and T goes up to 5. In order to make the the visualization easy, we choose the following layered topography h = cos x + 4 cos x (8.3) so that the most probable mean state predicted by the statistical theory is also The initial conditions are generated using a generic random initial data with energy layered, i.e. function of x only. The initial conditions are generated using a E = 7 and potential enstrophy E = ; generic The parameter random initial µ is determined data with at energy the end E = of 7 the and calculation potential numerically, enstrophy and = is. The approximately parameter is 99; determined at the end of the calculation numerically, and is approximately The numerical 99. simulation Thewas numerical performed simulation the was performed grid of Fourier on the coefficients. grid of Fourier coefficients; however, the results presented here are the stream Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 / 36

24 grid of Fourier coefficients; however, the results presented here are the stream functions The convergence and flows is(velocity measuredfields) in terms on of the the 3 relative 3 grid L in error, physical whichspace is defined after as interpolation of these results. The convergence is measured L in terms of the relative L error, which is defined f g = f g (8.3) as f for two functions f and g. L f g = f g (8.3) According to the above definition, L f T denotes the relative Lfor -error two functions for the stream f andfunction, g. and L T denotes the relative L -error According for thetovelocity above (flow). definition, L T denotes the relative L -error for the T stream L function, and L T denotes the relative T L T L -error for the velocity (flow) T L 38 T L 34 T (8.3) (8.3) Table (8.3) lists the relative errors against the averaging window size T. As we Error in time-average Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 3 / 36

25 Time average of stream function and velocity 66 Statistical mechanics for the truncated quasi-geostrophic equations Y 3 3 X (a) Y X (b) Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 4 / 36 Y Figure: time average with T = 5 Y 3

26 Time average of stream function 3 and velocity Y 3 3 X (a) 3 3 X (c) Y X (b) X (d) Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 4 / 36 Y Figure: time average with T = Y 3

27 Time average of stream function.5 and velocity Y 3 3 X (c) 3 3 X (e) Y X (d) X (f) Figure 8. (a) time average Figure: of time stream average function with with T = T 5 = 5, (b) time average of velocity with T = 5, (c) time average of stream function with T =, (d) time average of velocity with T =, (e) time average of stream function with T = 5, (f) time average of velocity with T = 5. averaging time increases. For a time average window of 5, the relative error Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 4 / 36

28 Time-averaged solution with T = 5 (left) and the most probable mean 8.5 The state pseudo-energy (right) and equilibrium statistical mechanics 67 Y 3 3 X (a) Y 3 3 X (b) Y X Y X Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 5 / 36

29 Outline The finite-dimensional truncated quasi-geostrophic equations The spectral truncated quasi-geostrophic equations Conserved quantities for the truncated system Nonlinear stability to the truncated system The Liouville property The statistical predictions for the truncated systems 3 Numerical evidence supporting the statistical prediction 4 The pseudo-energy and equilibrium statistical mechanics for the fluctuations Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 6 / 36

30 68 Statistical mechanics for the truncated quasi-geostrophic equations Fluctuations about the mean state of view is an important one for stochastic modeling. See for instance, Majda, Timofeyev, and Vanden Eijnden (). Here we show how dynamical equations for fluctuations and equilibrium statistical mechanics can be applied directly for In geophysical fluids, we can assume that the time averaged mean state, the the truncated quasi-geostrophic equations treated earlier in this chapter. For simplicity in exposition, we assume that there is no large-scale mean flow, and the climate, is known with reasonable accuracy and that the statistical fluctuations about the mean state are the quantities of interest. only geophysical effect is topography, so that the equations in (8.3) (8.4) become q t + P q = (8.33) We consider the exact steady solutions from (8.9) or (8.) q = q = + h (8.34) and consider perturbations about this mean state, i.e. = + q = q + = (8.35) Substituting (8.35) into (8.33) yields the equations We assume there is no large-scale mean flow, and the only geophysical effect is topography. t + P q + P q + P Utilizing (8.34) we have + P = (8.36) Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 7 / 36

31 and consider perturbations q about = this mean q = state, i.e. + h (8.34) Equations for perturbations and consider perturbations = about + thisq mean = q state, + i.e. = (8.35) Substituting (8.35) into = (8.33) + yields qthe = equationsq + = (8.35) Substituting (8.35) t + P into (8.33) qyields + Pthe equations q + P t + P q = + P q + P (8.36) Utilizing (8.34) + we P have = (8.36) Utilizing (8.34) we have P q = P q = P P P q = Hence (8.36) Pbecomes q the equations = P for perturbations = P Hence (8.36) becomes t + P the equations for perturbations + P = (8.37) t + P = + P = (8.37) The equations for perturbations involve = the familiar truncated non-linear terms of ordinary fluid flow The equations for perturbations involve the familiar truncated non-linear terms of ordinary fluid flow P (8.38) Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 8 / 36

32 and consider perturbations q about = this mean q = state, i.e. + h (8.34) Equations for perturbations and consider perturbations = about + thisq mean = q state, + i.e. = (8.35) Substituting (8.35) into = (8.33) + yields qthe = equationsq + = (8.35) Substituting (8.35) t + P into (8.33) qyields + Pthe equations q + P t + P q = + P q + P (8.36) Utilizing (8.34) + we P have = (8.36) Utilizing (8.34) we have P q = P q = P P P q = Hence (8.36) Pbecomes q the equations = P for perturbations = P Hence (8.36) becomes t + P the equations for perturbations + P = (8.37) t + P = + P = (8.37) The equations for perturbations involve = the familiar truncated non-linear terms of ordinary fluid flow The equations for perturbations involve the familiar truncated non-linear terms of ordinary fluid flow P (8.38) Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 8 / 36

33 Conserved quantities for (8.37) We would like to set-up a statistical theory for fluctuations directly from the dynamics in (8.37). ( The standard non-linear terms, P Λ ψ ω ), conserve both the energy and enstrophy; ( )) The linear operator, P Λ ( ψ Λ ω µ ψ, conserves neither the energy nor the enstrophy. The non-linear stability analysis suggests that there exists a single conserved quantity, Ẽ, involving fluctuations about the mean state given by the pseudo-energy Ẽ = ( ω µ ψ ω ) = E Λ + µe Λ. Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 9 / 36

34 Conservation of pseudo-energy Ẽ = = + E (8.4) which is an appropriate linear combination of energy and enstrophy. To check the conservation of the pseudo-energy directly for the dynamic equations in (8.37), We we focus focus only only on on the the contribution contribution arising arising from from the the linear linear terms, terms since in (8.39), the since non-linear the non-linear terms automatically terms in (8.38) conserve automatically both conserve energy and both enstrophy energy and separately. enstrophy separately. We calculate from (8.37) and (8.39) that dẽ dt = t = ( ) = = (8.4) which verifies the conservation of the pseudo-energy. From Section 4. and Subsection 8..3, the quadratic form defining the pseudoenergy in (8.4) is given by ( ) Ẽ = + k k (8.4) k Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 3 / 36

35 which verifies the conservation = of the pseudo-energy. (8.4) which Fromverifies Sectionthe 4.conservation and Subsection of the 8..3, pseudo-energy. the quadratic form defining the pseudoenergy Fromin Section (8.4) is 4.given and Subsection by 8..3, the quadratic form defining the pseudoenergy in (8.4) is given by ( ) Ẽ = ( ) Ẽ = + + k k k (8.4) k k k (8.4) and is positive definite only for > for arbitrary truncations. In this case, we andintroduce is positive thedefinite pseudo-energy only for variables, > for p arbitrary, definedtruncations by. In this case, k we introduce the pseudo-energy variables, ( p) k, defined by ( ) p pk = + / k = + / k (8.43) k k (8.43) k Next we sketch how to set up equilibrium statistical mechanics directly for the fluctuation Next we sketch described howby to the set dynamic up equilibrium equations statistical (8.37). mechanics directly for the fluctuation With (8.43) described we useby the the natural dynamic coordinates equations (8.37). With (8.43) we use the natural coordinates ) X = (Re p k Im p k Re p km Im p km ) (8.44) X = (Re p k Im p k Re p km Im p km (8.44) over a suitably defining set S as in (8.3) (8.4). It is a straightforward exercise Itfor is over straightforward the a suitably reader todefining check check set thatsthat the as in the Liouville (8.3) (8.4). Liouville property property It is isasatisfied straightforward satisfied directly directly exercise for for the the fluctuation for the reader dynamics to check with that the variables the Liouville in (8.43). property is satisfied directly for the Checking Liouville property Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 3 / 36

36 Equilibrium statistical Gibbs measure for the fluctuations The equilibrium statistical theory applies directly to the fluctuations with the equilibrium statistical Gibbs measure given by ( ) ( G α = C α exp αẽ = C α exp α ) p k. This Gibbs measure for fluctuations has the same structure in the pseudo-energy variables as the Gibbs measure for the truncated Burgers equations; This leads to predictions of equi-partition of pseudo-energy for fluctuations about the mean state as the truncated Burgers equation; The same result would have been obtained directly from the approach by merely diagonalizing the quadratic form in (8.5) G α,µ = c α,µ exp [ α ( µ ( V V ) ( ) ( + k k + µ ˆψk ˆψ ) )] k. Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 3 / 36

37 Numerical results confirming the equi-partition of pseudo-energy 7 Statistical mechanics for the truncated quasi-geostrophic equations pseudo-energy pseudo-energy (a) (b) Figure 8.3 Pseudo-energy spectrum. Solid line with circles numerical result, dashed line analytical prediction. (a) case: = = 99, (b) 3 3 case: = 9 5 = dynamics (8.37) with the variables in (8.43) with the same structure as (8.6) Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 33 / 36

38 A list of the topics that we have studied in the previous lectures i) Exact solutions showing interesting physics ii) Conserved quantities iii) Response to large-scale forcing iv) Large- and small-scale interaction via topographic stress. v) Non-linear stability of certain steady geophysical flows vi) Equilibrium statistics mechanics and statistical theories for large coherent structures vii) Dynamic modeling for geophysical flows viii) How information theory can be used to quantify predictability for ensemble predictions in geophysical flows Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 34 / 36

39 Modern Applied Math Paradigm Modern Applied Math Paradigm Rigorous Math Theory Qualitative or Quantitative Models Novel Numerical Algorithm Crucial Improved Understanding of Complex System 3 / 5 Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 35 / 36

40 Questions & Discussions Di Qi, and Andrew J. Majda (CIMS) Truncated Quasi-Geostrophic Equations Nov. 3, 6 36 / 36