This page intentionally left blank Non-linear dynamics and statistical theories for basic geophysical flows Non-linear dynamics and statistical theories for basic geophysical flows ANDREW J MAJDA New York University XIAOMING WANG Florida State University cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521834414 © Cambridge University Press 2006 This publication is in copyright Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published in print format 2006 isbn-13 isbn-10 978-0-511-16813-0 eBook (EBL) 0-511-16813-6 eBook (EBL) isbn-13 isbn-10 978-0-521-83441-4 hardback 0-521-83441-4 hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate Contents Preface page xi Barotropic geophysical flows and two-dimensional fluid flows: elementary introduction 1.1 Introduction 1.2 Some special exact solutions 1.3 Conserved quantities 1.4 Barotropic geophysical flows in a channel domain – an important physical model 1.5 Variational derivatives and an optimization principle for elementary geophysical solutions 1.6 More equations for geophysical flows References The 2.1 2.2 2.3 response to large-scale forcing Introduction Non-linear stability with Kolmogorov forcing Stability of flows with generalized Kolmogorov forcing References The 3.1 3.2 3.3 3.4 3.5 3.6 3.7 selective decay principle for basic geophysical flows Introduction Selective decay states and their invariance Mathematical formulation of the selective decay principle Energy–enstrophy decay Bounds on the Dirichlet quotient, t Rigorous theory for selective decay Numerical experiments demonstrating facets of selective decay References v 1 33 44 50 52 58 59 59 62 76 79 80 80 82 84 86 88 90 95 102 vi Contents A.1 A.2 Stronger controls on t The proof of the mathematical form of the selective decay principle in the presence of the beta-plane effect 103 107 Non-linear stability of steady geophysical flows 4.1 Introduction 4.2 Stability of simple steady states 4.3 Stability for more general steady states 4.4 Non-linear stability of zonal flows on the beta-plane 4.5 Variational characterization of the steady states References 115 115 116 124 129 133 137 Topographic mean flow interaction, non-linear instability, and chaotic dynamics 5.1 Introduction 5.2 Systems with layered topography 5.3 Integrable behavior 5.4 A limit regime with chaotic solutions 5.5 Numerical experiments References Appendix Appendix 138 138 141 145 154 167 178 180 181 Introduction to information theory and empirical statistical theory 6.1 Introduction 6.2 Information theory and Shannon’s entropy 6.3 Most probable states with prior distribution 6.4 Entropy for continuous measures on the line 6.5 Maximum entropy principle for continuous fields 6.6 An application of the maximum entropy principle to geophysical flows with topography 6.7 Application of the maximum entropy principle to geophysical flows with topography and mean flow References Equilibrium statistical mechanics for systems of ordinary differential equations 7.1 Introduction 7.2 Introduction to statistical mechanics for ODEs 7.3 Statistical mechanics for the truncated Burgers–Hopf equations 7.4 The Lorenz 96 model References 183 183 184 190 194 201 204 211 218 219 219 221 229 239 255 Contents vii Statistical mechanics for the truncated quasi-geostrophic equations 8.1 Introduction 8.2 The finite-dimensional truncated quasi-geostrophic equations 8.3 The statistical predictions for the truncated systems 8.4 Numerical evidence supporting the statistical prediction 8.5 The pseudo-energy and equilibrium statistical mechanics for fluctuations about the mean 8.6 The continuum limit 8.7 The role of statistically relevant and irrelevant conserved quantities References Appendix 256 256 258 262 264 Empirical statistical theories for most probable states 9.1 Introduction 9.2 Empirical statistical theories with a few constraints 9.3 The mean field statistical theory for point vortices 9.4 Empirical statistical theories with infinitely many constraints 9.5 Non-linear stability for the most probable mean fields References 289 289 291 299 309 313 316 10 Assessing the potential applicability of equilibrium statistical theories for geophysical flows: an overview 10.1 Introduction 10.2 Basic issues regarding equilibrium statistical theories for geophysical flows 10.3 The central role of equilibrium statistical theories with a judicious prior distribution and a few external constraints 10.4 The role of forcing and dissipation 10.5 Is there a complete statistical mechanics theory for ESTMC and ESTP? References 11 Predictions and comparison of equilibrium statistical theories 11.1 Introduction 11.2 Predictions of the statistical theory with a judicious prior and a few external constraints for beta-plane channel flow 11.3 Statistical sharpness of statistical theories with few constraints 11.4 The limit of many-constraint theory (ESTMC) with small amplitude potential vorticity References 267 270 285 285 286 317 317 318 320 322 324 326 328 328 330 346 355 360 viii 12 13 14 15 Contents Equilibrium statistical theories and dynamical modeling of flows with forcing and dissipation 12.1 Introduction 12.2 Meta-stability of equilibrium statistical structures with dissipation and small-scale forcing 12.3 Crude closure for two-dimensional flows 12.4 Remarks on the mathematical justifications of crude closure References Predicting the jets and spots on Jupiter by equilibrium statistical mechanics 13.1 Introduction 13.2 The quasi-geostrophic model for interpreting observations and predictions for the weather layer of Jupiter 13.3 The ESTP with physically motivated prior distribution 13.4 Equilibrium statistical predictions for the jets and spots on Jupiter References The statistical relevance of additional conserved quantities for truncated geophysical flows 14.1 Introduction 14.2 A numerical laboratory for the role of higher-order invariants 14.3 Comparison with equilibrium statistical predictions with a judicious prior 14.4 Statistically relevant conserved quantities for the truncated Burgers–Hopf equation References A.1 Spectral truncations of quasi-geostrophic flow with additional conserved quantities A mathematical framework for quantifying predictability utilizing relative entropy 15.1 Ensemble prediction and relative entropy as a measure of predictability 15.2 Quantifying predictability for a Gaussian prior distribution 15.3 Non-Gaussian ensemble predictions in the Lorenz 96 model 15.4 Information content beyond the climatology in ensemble predictions for the truncated Burgers–Hopf model 361 361 362 385 405 410 411 411 417 419 423 426 427 427 430 438 440 442 442 452 452 459 466 472 16.6 Statistical theories with a few and many constraints 537 preserved under the barotropic quasi-geostrophic dynamics and the energy and circulation in the northern hemisphere are conserved in time Hence our statistical theory will apply to flows on the northern hemisphere This is similar to the flat geometry case where we considered the channel geometry or swimming pool geometry (see Chapters and for details) Energy–circulation theory This is the parallel of Subsection 9.2.1 for the spherical geometry Here we consider two conserved quantities, the energy EN and the circulation N in the northern hemisphere (see (16.2.2)) EN = − − SN2 with a prior distribution N =− d ds = s SN2 As usual we postulate the one-point statistics SN2 × R1 ∈ s→ ∈ s ≥0 − (16.189) R1 (16.190) s d ds = s SN2 for all s ∈ SN2 d =1 ae s∈ s q+ q− q s − s SN2 (16.191) SN2 = Prob q− ≤ q s ≤ q+ s We define the relative Shannon entropy = −− as SN2 d ds ln (16.192) We may represent the energy and circulation as well as the mean field in terms of the one-point statistics as q¯ s = E s d N = E q¯ = − − ¯ ¯ SN2 =− SN2 s d ds (16.193) We then proceed to search for the most probable states with given constraints on the energy and circulation 538 Barotropic quasi-geostrophic equations on the sphere Going through a standard Lagrangian multiplier argument procedure (see Section 9.2.1 for more details), we deduce that the most probable one-point statistics ∗ must satisfy exp exp = s s − s − 0 s s d (16.194) where and are the Lagrange multipliers for the energy and circulation, respectively, and ∗ is the most probable mean stream function As usual we introduce the partition function s = log s − exp s d (16.195) We then have q = s d = Hence the most probable mean state satisfies the following equation + sin + h = (16.196) For the special choice of prior distribution as a uniform distribution on an given interval, i.e q0+ − = + I s q q0 − q0− we recover the Langevin theory For another special choice of prior distribution as a Gaussian = c exp − 2 we then recover the previous theory with energy and enstrophy conservation See Subsection 9.2.1 for more details In the case with ground state topography, we need to consider the statistics of the high modes with odd symmetry The energy–circulation theory still applies, since they are conserved for the high modes in the northern hemisphere (see (16.89)) The predictions are very much the same except there is no geophysical effects for the predictions of the high modes We leave the detail to the interested reader The point vortex theory Here we consider equilibrium statistical theories for point vortices This is parallel to Section 9.3.1 for the flat geometry case 16.6 Statistical theories with a few and many constraints 539 We exclude geophysical effects here and hence h≡0 ≡0 (16.197) Since this implies that we have ground state topography and odd symmetry may be imposed, we will consider statistics for the high modes (modes in the second energy shell or higher) in the northern hemisphere with odd symmetry Recall that the energy and circulation for the high modes in the northern hemisphere are conserved in this case (see Section 16.2.2) and they can be as represented in terms of the one-point statistics for the high modes EN = − − ¯ ¯ = E0 SN2 N =− SN2 ¯ =− ¯ = s SN2 (16.198) d We also consider a prior distribution − d ds = s SN2 0 on the northern hemisphere d ds = s (16.199) In order to accommodate point vortices, we not enforce the one-point statistics to be probability density at each point on the northern hemisphere We instead postulate − s SN2 d ds = As usual we define the relative Shannon entropy = −− SN2 (16.200) as ln d ds (16.201) Going through a standard Lagrangian multiplier argument procedure we have that the most probable one-point statistics ∗ must satisfy s = s − exp − SN2 s − exp s 0 s (16.202) d ds where and are Lagrange multipliers for the energy and circulation respectively, and ∗ is the most probable mean stream function This implies, since q = = s d 540 Barotropic quasi-geostrophic equations on the sphere that we have the mean field equation = = − SN2 exp s − exp s − s ds s d ds (16.203) For the special choice of prior distribution = s (16.204) we have = exp − SN2 exp (16.205) ds Note that can be determined by the energy constraint and by the circulation constraint = can be determined The reader is referred to Chapter for more details 16.6.1 Infinitely many constraint statistical theory In this last subsection of this chapter we discuss the empirical statistical theory with infinitely many constraints consists of all generalize potential enstrophy for the inviscid unforced barotropic quasi-geostrophic equations on the unit sphere (16.62) The result and argument here are very similar to those for the flat geometry, which we treated in detail in Section 9.4 The reader is referred to that section for more ingredients and motivations One catch here is that we cannot apply the empirical statistics with many conserved quantities (ESTMC) to the high modes even if the topography lives on the ground energy shell, since high moments of the high modes are not necessarily conserved (see Subsection 16.2.2 for more details) This raises the question of realizability of the predictions of the ESTMC in the case of ground state topography, since the ground state modes not possess enough mixing, a required ingredient of the physical realizability of the most probable states As in the flat geometry case (see Section 9.4), we avoid postulating infinitely many constraints in the Lagrange multiplier method by using the distribution function for the initial condition 16.6 Statistical theories with a few and many constraints 541 Let Pq denote the distribution function of q Thus if q0 is the initial potential enstrophy and q its evolution, the conservation of generalized enstrophy is equivalent to, utilizing the distribution functions dPq = dPq0 ∈ supp dPq0 for all This can be reformulated, in terms of the one-point statistics vorticity q as − S2 s ds d = dPq = dPq0 s (16.206) of the potential ∈ supp dPq0 for all (16.207) Assumption: For convenience in exposition, we consider the special case that is absolutely continuous with respect to the Lebesgue measure, i.e dPq0 = dPq0 q0 d (16.208) for some integrable function q0 As usual we define the relative entropy with prior distribution Pq0 = − − S2 ln d ds = − − ¯ ¯ = E0 S2 − S2 s ds = s as (16.209) q0 We recall the constraints on the one-point statistics E q0 are d = for all s ∈ S (16.210) for all q0 ∈ supp q0 The usual Lagrange multiplier method implies that the most probable state must satisfy s − exp q0 (16.211) s = s − d exp q0 where and are Lagrange multipliers for the energy and generalized enstrophy constraint respectively, and ∗ is the stream function for the most probable mean ∗d field q∗ = Again we introduce the partition function = log exp − s d = q0 We have q = = d (16.212) 542 Barotropic quasi-geostrophic equations on the sphere and hence the mean field equation takes the form + sin + h = The Lagrange multipliers Eq = E0 (16.213) = are determined by the constraints − S2 ds = q0 for all ∈ supp q0 (16.214) The stability of the most probable state we just derived can be discussed using the stability results we derived in Subsection 16.2.3 This is very similar to the flat geometry case (see for instance Section 9.5) We formulate the result here for the sake of completeness Theorem 16.4 The most probable mean field given by (16.213) are exact steady state solutions to the inviscid unforced barotropic quasi-geostrophic equation on the sphere (16.62) These steady state solutions are non-linearly stable provided one of the following conditions holds: (A) (B) > and supp q0 is bounded; < supp q0 ⊂ −a a − a2 < This is an application of Proposition 16.2 and the interested reader is referred to Section 9.5 for details References Courant, R and Hilbert D (1962), Methods of mathematical physics Vols I and II New York-London: Interscience Publishers (a division of John Wiley & Sons) Frederiksen, J S and Sawford, B L (1980), Statistical dynamics of two-dimensional inviscid flow on a sphere J Atmospheric Sci 37(4), 717–732 Frederiksen, J S and Sawford, B L (1981), Topographic waves in nonlinear and linear spherical barotropic models J Atmospheric Sci 38, 69–86 Jones, M N (1985), Spherical Harmonics and Tensor for Classical Field Theory John Wiley & Sons Inc Lax, P D (1997), Linear Algebra New York: John Wiley and Sons Pedlosky, J (1987), Geophysical Fluid Dynamics, 2nd edn., New York: Spring-Verlag Sawford, B L and Frederiksen, J S (1983), Mountain torque and angular momentum in barotropic planetary flows: equilibrium solutions Quart J R Met Soc 109, 309–324 Appendix The purpose of this Appendix is to elaborate on the invariant dynamics on the first two energy shells This is a special case of the exact dynamics of the ground state modes and the nth energy shell discussed in Subsection 16.2.1 with n = Appendix 543 For this purpose let us recall the following tables: Eigenfunctions corresponding to − sin z w1 cos cos x wa11 − sin − 3z2 w2 cos sin × cos zx wa21 = −2 n = cos sin y wb11 Eigenfunctions corresponding to − cos sin × sin yz wb21 (16.215) = −6 n = cos2 × cos x2 − y wa22 cos2 × sin 2xy wb22 (16.216) It is easy to see, utilizing (16.6) J w2 z = J − 3z2 z = J wa21 z = −wb21 J zx z = −yz J wb21 z = wa21 J yz z = zx J wa22 z = −2wb22 J x2 − y2 z = −2 2xy J wb22 z = 2wa22 J 2xy z = x2 − y2 This implies, since x2 = x2 − y2 + − z2 , that J x2 z = −2xy J y2 z = 2xy This further implies, thanks to the rotation symmetry of the sphere and the eigenfunctions (which are homogeneous polynomials of degree and 2), J w2 x = J −3z2 x = −3 2yz = −6wb21 J wa21 x = J zx x = xy = 5wb22 J wb21 x = J yz x = y2 − z2 = 5w2 − 5wa22 J wa22 x = J −y2 x = 2yz = 2wb21 J wb22 x = J 2xy x = −2zx = −2wa21 J w2 y = J −3z2 y = 6xz = 6wa21 J wa21 y = J zx y = z2 − x2 = −0 5wa22 − 5w2 J wb21 y = J yz y = −xy = −0 5wb22 J wa22 y = J x2 y = 2xz = 2wa21 J wb22 y = J 2xy y = 2yz = 2wb21 544 Barotropic quasi-geostrophic equations on the sphere Thus for t = n an t wn + n=1 anm t wanm + bnm t wbnm n=1 m=1 we have +2 z = J J z +J =2 J =2 z + 4J n=1 +4a1 J n m=1 −manm t wbnm + mbnm t wanm z + 4a11 J x + 4b11 J y We observe J z = −a21 wb21 + b21 wa21 − 2a22 wb22 + 2b22 wa22 = b21 wa21 − a21 wb21 + 2b22 wa22 − 2a22 wb22 J x = −6a2 wb21 + 5a21 wb22 + 5b21 w2 − 5b21 wa22 + 2a22 wb21 −2b22 wa21 = 5b21 w2 − 2b22 wa21 + 2a22 − 6a2 wb21 − 5b21 wa22 +0 5a21 wb22 J y = 6a2 wa21 − 5a21 wa22 − 5a21 w2 − 5b21 wb22 +2a22 wa21 + 2b22 wb21 = −0 5a21 w2 + 6a2 + 2a22 wa21 + 2b22 wb21 −0 5a21 wa22 − 5b21 wb22 Hence the dynamics of the coefficients are given by, for the simple case of no topography h = , no damping = and no external forcing =0 da2 dt da21 dt db21 dt da22 dt db22 dt 1 = − b11 a21 + a11 b21 3 4 = b + a b − a b + 4b11 a2 + b11 a22 21 21 11 22 4 =− a21 − a1 a21 + a11 a22 − 4a11 a2 + b11 b22 3 3 1 = b + a b − a b − b a 22 22 11 21 11 21 1 a22 − a1 a22 + a11 a21 − b11 b21 =− 3 3 Appendix 545 This system of linear equations can be written in a compact form da = Aa a = a2 a21 b21 a22 b22 tr dt where the matrix A is given by ⎛ ⎞ − 13 b11 0 a11 ⎜ 4b − 43 a11 ⎟ ⎜ 11 ⎟ + a1 b11 ⎜ ⎟ 4 A = ⎜−4a11 − − a1 a b ⎟ 11 11 ⎜ ⎟ ⎠ ⎝ + a − 13 b11 − 13 a11 3 1 − 13 b11 − 23 − 43 a1 a11 (16.217) (16.218) In general the above matrix A is periodic in t with period / (since a11 b11 are periodic in t with period / ) and Floquet theory is needed to solve such system Special Case 1: a11 = a b11 = This implies a11 = a cos + 2a1 = ac t b11 = a sin t and the system reduce to ⎛ ⎞ − sin t cos t 0 ⎜ 12 sin t c sin t −4 cos t ⎟ ⎟ a⎜ ⎜ ⎟ A = ⎜−12 cos t −c cos t sin t ⎟ ⎟ 3⎜ ⎝ ⎠ − sin t − cos t 2c cos t − sin t −2c The eigenvalues of this matrix are independent of time and are given by a a + c2 i − + c2 i 3 2a 2a + c2 i − + c2 i 3 Numerical evidence indicates aperiodic motion for certain choices of parameters, √ for instance a = 1, c = Special Case 2: = (no Coriolis force), the matrix is a constant matrix ⎛ ⎞ − 13 b11 13 a11 0 4 ⎜ 4b ⎟ ⎜ 11 a1 b11 − a11 ⎟ ⎜ ⎟ A = ⎜−4a11 − 23 a1 a11 43 b11 ⎟ ⎜ ⎟ ⎝ − 13 b11 − 13 a11 ⎠ a1 1 0 a11 − b11 − a1 546 Barotropic quasi-geostrophic equations on the sphere The eigenvalues of this matrix can be calculated explicitly as 2 a + a211 + b11 i 2 a + a211 + b11 i − − 2 a + a211 + b11 i 2 a + a211 + b11 i The eigenvalues must be zero or purely imaginary which is consistent with the fact that the system conserves energy Thus the motion is periodic in time However the period can be any real number This gives us an indication of the complexity of the dynamics Appendix The purpose of this Appendix is to prove claim (16.120) from section 16.3 We need to postulate the following assumption: Assumptions: d > and P2 is uniformly bounded in time Remark: The first condition is equivalent to saying that there is real dissipation in the system The second condition is satisfied provided that is uniformly bounded in time In order to prove the smallness of F˜ we have to deal with the interaction term between different surface spherical harmonics This is not a trivial task We recall from Jones (1985, page 180), formula (46), that the Euler’s equation on the unit sphere can be written in the form in terms of the coefficients of the surface spherical harmonics dˆn m + dt nmn m n n+1 n n +1 n n +1 Knm nmnm ˆ n m ˆ nm = (16.219) where (see Jones, 1985, page 175, equation (30)) Knm nmnm i = −1 m 2n + 2n + 2n + n n +1 n n +1 n n+1 × n +n−n × n n n −1 0 n + n − n −n + n + n + n + n + n + n n n −m m m (16.220) Appendix 547 = is the vorticity† , and there are restrictions on the summation indices which must satisfy (see Jones, 1985 page 193) m =m + m m ≤ n m ≤n m ≤n n −n < n < n +n Since the non-linear term in Euler’s equation is J P2 J = m n m nm = n n+1 6n n + n n+1 m n m nm (16.221) , we must have m mm K2n ˆ n m ˆ nm Y2m n n n+1 n n +1 m mmˆ ˆ K2n n m nm Y2m n (16.222) We may then identify the contribution from different sources (I) Contribution from the interaction between the first shell and the second shell This must correspond to −4J P1 P2 (II) Contribution from the interaction between the second shell and higher modes n n+1 m mmˆ ˆ K22n 2m nm Y2m m m m n≥3 (III) Contribution from the interaction between higher modes ≥ n n+1 m m m n≥3 n ≥3 n n+1 n n +1 m mmˆ ˆ K2n n m nm Y2m n (IV) Contribution from the interaction of the ground energy shell and higher modes n ≥ is zero since J P1 Pn ∈ Wn We observe that the combination of contributions of type II and III is the total effect of F˜ To estimate the contribution from type II terms, we notice the summation restriction (16.221) in this special case is 2−n < n ≥ which leads to n=3 † Notice Jones used instead of to denote the vorticity and he didn’t write down all the indices 548 Barotropic quasi-geostrophic equations on the sphere Therefore n n+1 m mmˆ ˆ K22n 2m nm Y2m m m m n≥3 = m mm 123 K223 ˆ 2m ˆ 3m m m m ≤ P2 2 P3 Thanks to the restricted stability 16 109 ≤ exp −dt P2 (By our assumption) ≤ exp −dt (16.223) To estimate the contribution from type III terms we have, thanks to the summation restriction on the indices in this case n−n < n≥3 n ≥3 and hence n and n are equivalent and we may exchange them freely at the expenses of introducing a uniformly bounded coefficient We also notice, thanks to the explicit three-symbol formula (see Jones, 1985 pages 193–195) n n −1 ∼ n− 00 n n ∼ n− −m m m We also notice that 2n + 2n + 24 n n + n n + 1 n +n−2 n +2−n −n + n + n + n + ∼1 Combining the above three relations together with the formula for K (16.220) we deduce m mm K2n ≤ (16.224) n n where is a generic constant independent of the indices This further implies, for the contribution from type III n n+1 m m n≥3 n ≥3 n n+1 n n +1 2 m mmˆ K2n nm n ˆ nm Appendix 549 (Since n and n are equivalent and the estimate on K.) n n+1 ≤ 2ˆ nm ˆ nm m m n≥3 n ≥3 ≤ n2 n + ˆ nm m m n≥3 n ≥3 n n +1 ˆn m m m n≥3 n ≥3 (By the summation restriction) n2 n + ≤ m n≥3 ˆ nm n n +1 ˆn m m n ≥3 (Thanks to the restricted stability (16.109)) ≤ exp −2dt (16.225) Combining the estimates on contributions from type II and type III terms and their relationship with F˜ we deduce the claim (16.120) Index advective derivative, auto-correlation function, 252 enstrophy, 30, 40, 42, 61 enstrophy identity, 61 EOF basis, 457 equations for large-scale and small-scale flow interaction via topographic stress, 38, 43, 116 ergodicity, 228 ESTMC, 318, 540 ESTP, 318, 536 Euler equations, 6, 12 exact dynamics of the ground state modes and the nth energy shell, 495 barotropic beta-plane equations, 19 barotropic one-layer model, 54 barotropic quasi-geostrophic equations, 2, 59 barotropic quasi-geostrophic equations in a channel with topographic stress, 46 barotropic quasi-geostrophic equations on the unit sphere, 483 beta plane effect, 2, 17 beta plane equations with mean shear flow, 28 Casimir invariants, 443 Cauchy–Schwarz inequality, 216, 315 channel geometry, 45, 49 circulation, 48, 50 complete statistical mechanics, 256, 303, 529 conserved quantities, 33, 504 continuously stratified quasi-geostrophic equations, 53 continuum limit, 270 Coriolis force, correlation deficit, 365 correlation function, 25, 238 correlation time, 238 crude closure algorithm, 390 dilute PV theory, 332 Dirichlet quotient, 84 dispersion relation, 20 dispersive wave, 20 dissipation mechanism, 2, 18, 30 EEST, 318 Ekman drag, empirical statistical theory with infinitely many constraints, 309 energy, 22, 33, 42, 61 energy–circulation theory, 292, 537 energy–circulation impulse theory, 297 energy–enstrophy theory, 211, 525 energy spectrum, 435 ensemble prediction, 452, 453 flatness, 201 Fourier series representation of barotropic quasi-geostrophic equations, 68 Fourier series tool kit, 10 F-plane equation, 53 free-slip boundary condition, 16 Gamma distribution, 422 Gaussian distribution, 195, 452 generalized enstrophy, 40, 46, 49 geophysical fluid dynamics, geostrophic balance, Gibbs measure, 226 Great Red Spot (GRS), 411 Gronwall inequality, 65 ground state eigenmode, 12 ground state modes dynamics, 492 Hamiltonian system, 442 impulse, 47, 49 information-theoretic entropy, see Shannon entropy invariant measure, 227 isobars, Jacobi identity, 443 Jacobian determinant, kinetic energy identity, 36 Kolmogorov forcing, 17, 60, 76 550 Index Langevin theory, 332 Laplace–Beltrami operator, 488 large-scale enstrophy, 39, 43, 46, 49 latitude, layered topographic equations, 143 Legendre functions, 489 Limaye profile, 412 Liouville equation, 222, 453 Liouville property, 221 longitude, Lorenz 96 model (L96), 239 maximum entropy principle, 188, 196 maximum relative entropy principle, 192, 196, 204 mean field equation, 207, 214 mean flow, 17 mean velocity–impulse difference, 49 mesoscale, 2, meta-stability, 362 Navier–Stokes equations, no-penetration boundary condition, 16 nonlinear stability, 62, 116, 117 one and one-half layer model, 56 one layer model, 53 one-point statistics, 203, 211 Parseval identity, 11 partition function, 189, 293 periodic boundary condition, Poincaré inequality, 11 point-vortex equations, 304 point-vortex theory, 300, 538 Poisson bracket, 443 potential enstrophy, 40 potential vorticity, potential vorticity extrema, 47, 50 predictability, 452, 454, 456 predictive information content, 461, 463, 465 prior distribution, 203 pseudo-energy, 269 PVST, 318 radiative damping, relative entropy, 192, 195, 204, 452, 455 relative entropy identity, 459 relative vorticity, relaxation, 473 551 response to large-scale forcing, 67, 511 restricted stability of motion on the first two energy shells, 508 Reynolds number, 97 Rhines measure of anisotropy, 96 Rossby waves, 19, 496 selective decay principle, 81, 85, 518, 519 selective decay state, 81 Shannon entropy, 185, 195, 212, 225 shear flow, 13 signal and dispersion decomposition, 460 sine-bracket truncation, 430 skewness, 201, 421 slaving principle, 78 spectrally truncated quasi-geostrophic equations, 259 spin-up, 394 stability of simple steady states, 116, 505 statistical relevance, 427 statistical solution, 219 statistically sharp, 347 Stokes formula, 488 stratification, 53 stream function, surface spherical coordinates, 482 surface spherical harmonics, 489 swimming pool geometry, 41 swirling eddies, 13 symplectic matrix, 443 tangent plane approximation, Taylor vortices, 15 topographic stress, 37 topography, 2, 25 total energy, 43, 46, 49 truncated Burgers–Hopf equation (TBH), 230 two layer model, 54 undamped unforced L-96 model (IL96), 244 utility, 474 variational characterization, 133 variational derivative, 50 velocity, viscosity (Newtonian, eddy, hyper), whitening, 459 White Ovals, 413 zonal flow, 129 ... blank Non-linear dynamics and statistical theories for basic geophysical flows Non-linear dynamics and statistical theories for basic geophysical flows ANDREW J MAJDA New York University XIAOMING... United States of America by Cambridge University Press, New York www .cambridge. org Information on this title: www .cambridge. org/97 80521834414 © Cambridge University Press 2006 This publication is... applicability of equilibrium statistical theories for geophysical flows: an overview 10.1 Introduction 10.2 Basic issues regarding equilibrium statistical theories for geophysical flows 10.3 The central