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Avalanches, Breathers and Flow Reversal in a Continuous Lorenz-96 Model R. Blender, J. Wouters, and V. Lucarini Meteorologisches Institut, KlimaCampus, Universita¨t Hamburg, Hamburg, Germany (Dated: January 25, 2013) For the discrete model suggested by Lorenz in 1996 a one-dimensional long wave approximation with nonlinear excitation and diffusion is derived. The model is energy conserving but non- Hamiltonian. In a low order truncation weak external forcing of the zonal mean flow induces avalanche-like breather solutions which cause reversal of the mean flow by a wave-mean flow interaction. The mechanism is an outburst-rechargeprocess similar to avalanches in a sand pile model. PACSnumbers: 47.20.-k47.10.Df47.35.Bb64.60.Ht 3 1 In1996Lorenzsuggestedanonlinearchaoticmodelfor The conservative terms of the Lorenz-96 equations 0 an unspecified observable with next and second nearest (1) are obtained for Q = X . The equations are non- 2 i neighbor couplings on grid points along a latitude circle Hamiltonian [11] since the Jacobi identity n [1]. Due to its scalability the model is a versatile tool in a ∂J ∂J ∂J J statistical mechanics [2–5] and meteorology [6–8]. The Jiℓ jk + Jjℓ ki + Jkℓ ij =0 (5) nonlinear terms have a quadratic conservation law and ∂Xℓ ∂Xℓ ∂Xℓ 4 ℓ ℓ ℓ satisfyLiouville’sTheorem. Forstrongforcingthemodel X X X 2 shows intermittency [9]. is not satisfied. ] The Lorenz-96equations for the variableXi area sur- A continuous approximation is derived for a smooth O rogate for nonlinear advection in a periodic domain dependencyofXi onthespatialcoordinatex=ihinthe A limith→0. ThevariableXi is replacedbyacontinuous d function u(x,t) which is interpreted as velocity in the n. dtXi =Xi−1[Xi+1−Xi−2]−γXi+Fi (1) following. We use the infinitesimal shift operators i nl γ characterizes linear friction (γ = 1 in [1]) and F is a ∞ (±h∂ )k [ forcing. L± = x (6) InthisLetteracontinuouslongwaveapproximationof k=0 k! 1 X the Lorenz-96 model is derived. A surprising finding is v that the nonlinear terms in the Taylor expansion are as- to write the bracket (3) as 1 sociated with generic dynamic operators. Furthermore, 0 δA δB 8 the dynamics in a truncated version reveals avalanches, {A,B}= J∞ (7) δu δu 5 breather-likeexcitations andflow reversals,whichmimic Z 1. various physical processes in complex systems in a sim- with 0 plistic way. 3 Lorenz [10] has analysed the linear stability of the J∞ =(L−u)◦L+−L−◦(L−u) (8) 1 mean m of X in (1) and found that long waves with i : wavenumbersk <2π/3areunstable forapositive mean where(L−u)is amultiplicationoperator. Thebracketis iv m. anti-symmetric since the adjoint is L∗+ =L−. X For γ =F =0 the equations (1) are conservativewith By taking n-th order truncations of the operators L±, r the conservation law, H = 1/2 X2, denoted as en- we can find a hierarchy of truncated anti-symmetric op- a X i i ergyinthe following. The dynamicsinthestatespaceof erators P the X is non-divergent thus satisfying Liouville’s Theo- rem, i ∂X˙ /∂X =0. Jnm =(L−,nu)◦L+,m−L−,m◦(L−,nu) (9) i i i The dynamicsofanobservablefunctionQ(X)isgiven where P by n (±h∂ )k x Qt ={Q,HX} (2) L±,n = k! (10) k=0 X with an anti-symmetric bracket The total energy for the velocity u(x,t) is {A,B}=∂ AJ ∂ B =−{B,A} (3) i ij j 1 H= u2dx (11) and the antisymmetric matrix 2 Z Jij =Xi−1δj,i+1−Xj−1δi,j+1 (4) To each of these truncated operators corresponds a continuous Lorenz-96 model EnergyH isconservedduetotheanti-symmetryofthe X bracket. u ={u,H} (12) t nm 2 where the indices indicate the operator J (as in (1) Thus, mean flows with U >0 (U <0) are unstable (sta- nm periodic boundary conditions are assumed). ble) as in the discrete system (1) analysed in [10]. Theexpansionofthenonlineartermsin(1)uptoorder The equations (20, 22) represent a coupling between O(h2) yields for the rescaled coordinate x′ = −x/3 (the perturbations and the mean flow. A forcing drives the prime is dropped below) meanflowtowardspositivevalueswhichallowthegrowth ofperturbations. Whentheperturbationsaresufficiently u =−uu − 1 u2 + 1uu +f (13) intense they reduce the flow to negativevalues causing a t x 3 x 2 xx decay of their intensities. (cid:18) (cid:19) The nonlinear energy cycle represented by the ex- withanadvectionandfurther nonlinearterms whichare change between zonal flow and wave energy in (20) and due to the noncentered definition of the interaction in (22)isanalysedinaspectralmodelforthe unstablelong (1). wavesby Fourier expansionin a periodic domain [−π,π] Thenonlineartermsareassociatedwithantisymmetric evolution operators N u= a sin(nx)+b cos(nx) (23) 1 n n O(h):J1 =−3(u∂x+∂xu) (14) nX=0 1 Here we restrict to the low order system N =2. O(h2):J2 =− (ux∂x+∂xux) (15) 6 1 1 Thus the evolution equation (13) can be written as b˙0 =−12 a21+b21 − 3 a22+b22 +f (24) (cid:0) 1 (cid:1) 1 (cid:0) (cid:1) δH a˙1 =b0b1+ b0a1+ (a1a2+b1b2) ut =J , J =J1+J2 (16) 6 2 δu 1 + (a1b2−b1a2) (25) Note that the O(h3) expansion in (12) is represented by 4 1 1 a third operator J3 = −(1/18)(uxx∂x+∂xuxx); here we b˙1 =−b0a1+ b0b1− (a1a2+b1b2) restrict to the O(h2) expansion (13). 6 4 The evolution equation (13) has a conservation law 1 + (a1b2−b1a2) (26) 2 1 2 1 ∂t 2u2 =∂xφ, (17) a˙2 =2b0b2+ 3b0a2+ 2 b21+a1b1−a21 (27) (cid:18) (cid:19) b˙2 =−2b0a2−a1b1+ 14(cid:0)b21−a21 + 32b0(cid:1)b2 (28) 1 1 φ=− u3− u2u (18) (cid:0) (cid:1) 3 6 x The mean flow is U = b0 which is subject to a constant forcing f in the numerical experiments (24). The with the conserved current φ which leads to the conser- truncated system conserves energy vation of total energy (11). Further conservation laws could not be found. In particular momentum given as Htot =H0+H1+H2 (29) the mean flow 1 1 1 U =hui= udx (19) H0 = 2b20, H1 = 4 a21+b21 , H2 = 4 a22+b22 Z (30) (cid:0) (cid:1) (cid:0) (cid:1) is not constant. The Liouville Theorem is not satisfied In the following we consider a constant and positive forcingf (notethatthesystemisnotdissipative). Inthe 2 ∂a˙ ∂b˙ 5 n n presenceofperturbations v to the meanflow, u=U+v, + = b0 (31) ∂a ∂b 3 the mean flow energy H¯ =U2/2 changes according to nX=0 n n! Theexpansionandcontractionofthestatespacevolume ∂ U H¯ =− hv2i+Uf (20) is controlled by the sign of the mean flow. ∂t 6 x Numerical experiments reveal a flow reversal mecha- The perturbation energy nism and vanishing long term means of mean flow and wave number amplitudes, hence the Liouville Theorem 1 E′ = hv2i (21) (31) is satisfied in the mean. 2 (i) Weak forcing with f = 0.1 in the N = 1 truncation reveals periodic flow reversals (Fig. 1). The system grows for positive U starts with randomly chosen amplitudes. A mean flow ∂ U increases gradually to positive values where it becomes ∂tE′ = 6hvx2i (22) unstable due to the excited waves, denoted as breathers 3 9 and the perturbation energy is a) N = 1 b1 6 ude 3 ab10 18a2 mplit 0 H1 = cosh2(at) (34) A-3 -6 where a is related to the total energy H = 18a2. H1 100 200 300 400 500 attains its maximum during flow reversals when U = 0. 9 These approximations are compared to the forced simu- b1 de 6 a1 lation in Fig. 1b,c centered at a single flow reversal. plitu 03 b0 Inthepresenceofforcingf andforasmallwaveenergy m-3 H1themeanflowb0growslinearlyintime,b0(t)≈ft,up A -6 b) to a value b0,max. This defines an interarrival time scale 106 108 110 112 114 116 118 of flow reversals, τ = 2b0,max/f. In this range the wave energy evolves rapidly according to H1(t)∼exp(ft2/6). 20 15 20 ergy10 Htot y15 a) N = 1 HHt1ot En 5 c) HH10 Energ150 H0 0 106 108 110 112 114 116 118 0 200 210 220 230 240 250 4.5 mplitude 100 N = 2 bab220 Energy13..50 b) N = 2 HHHHt210ot A -10 d) 0 2000 2050 2100 2150 2200 200 300 400 500 600 Time 60 e) H2 FIG. 2. Intermediate forcing f =1: Energy distributions for y H0 (a) N =1, intervals≈13, (b) N =2. g40 Htot er n E20 The described flow reversal mechanism is retained for 0 viscous dissipation represented by a linear damping of 200 300 400 500 600 the wave amplitudes a1 and b1. Time For the N = 2 truncation with all modes b0,a1,b1,a2 FIG. 1. Weak forcing f = 0.1: (a) Amplitudes b0,a1,b1 for and b2 flow reversalsoccur ona time scale roughlytwice N =1,intervals≈110, (b)amplitudesduringaflowreversal as for N = 1 (Fig. 1d). Due to the weak forcing the with Eq. (33) for b0 (dashed), (c) energies H0,H1 and Eq. energy cascades to mode 2 with negligible amplitudes (34) for H1 (dashed), (d) amplitudes b0,a2,b2 for N = 2, a1,b1 and energy H1 (Fig. 1d, e). Neglecting the modes (a1,b1 vanish), and (e) energies (H1 vanishes). 1, the energy cycle for interactions among b0,a2 and b2 is 4 4 inthefollowing. Thesebreathersdrivearapidflowrever- ∂tH0 =− b0H2, ∂tH2 = b0H2 (35) 3 3 saltowardsa negativeflowwhichinitiates theircollapse. The process is energy conserving on short time scales. Thiscorrespondstoarescalingofthe H0−H1 cycle(32) The total energy increases (decreases) when the mean by t˜= 2t for time and ˜b0 = 2b0 etc. for the amplitudes, flow is positive (negative). hence the energies quadruple. For N = 1 with the amplitudes b0,a1,b1 the energy (ii) For intermediate forcing with f =1 the time scale cycle is for f =0 (compare (20, 22)) between flow reversals decreases by an order of magnitude in the N = 1 truncation (see Fig. 2a). Thus the intervalsτ approachthedurationofindividualbreathers. 1 1 ∂tH0 =− b0H1, ∂tH1 = b0H1 (32) For the complete set of modes in N = 2 (Fig. 2b) the 3 3 system is weakly nonlinear with a mixing of frequencies, ω/2,ω,3ω/2 and 2ω, where ω = 2π/τ is defined by the which is controlled by the mean flow. The solution for interarrival times of the flow reversals [12]. The lowest the mean flow is for b0(0)=0 frequency determines the amplitude modulation. (iii) For strong forcing, f = 10, the flow reversals in b0 =−6atanh(at) (33) theN =1truncationareregular(Fig. 3a)withintervals 4 75 statewheremeanflowandwaveenergyinteract. Thein- a) N = 1 Htot y50 H1 tervalsbetweentheflowreversalsareapproximatelypro- g er H0 portional to the inverse of the forcing intensity, ∼1/f. En25 (ii)TheQuasi-BiennialOscillation(QBO,[15]),aflow 0 reversal in the tropical stratosphere driven by two dif- 200 210 220 230 240 250 ferent types of upward propagating gravity waves. A 150 b) N = 2 Htot common aspect is that the driving of the mean flow by y100 H2 wavesoccurs only for a particular sign of the mean flow. Energ 50 HH10 Aicaltlhlyouthghe stihmeuQlaBtOionisincopnrseidseenret-ddtaoybweeeaxthpelarinanedddcylinmaamte- 0 models necessitates careful sub-scale parameterizations 0 50 100 150 200 or high resolution models [16]. The present model is Time clearly an oversimplification but can be considered as a FIG. 3. Strong forcing f = 10: energy distributions for (a) toy model for this phenomenon. N =1, intervals ≈3, (b) N =2. (iii)Roguewaves(alsotermedfreakormonsterwaves) atthe oceansurface are simulatedmainly by the nonlinear Schroedinger equation (e.g. [17, 18]); a Lagrangian analysis has been published recently [19]. The breather decreased by an order of magnitude relative to f = 1. solutions found in the present model show characteris- Thedominantpartofenergyisaccumulatedinwaves. In tics like the rapid evolution and the high intensity in an theN =2truncationthedynamicsbecomesintermittent almost quiescent medium. as in the regime behaviour detected by Lorenz [9] in the Due to the flow reversals the total energy of the non- discrete equations (1). The events loose their identities dissipative system remains finite for a constant forcing. and the systen becomes strongly nonlinear. The long term mean of the mean flow vanishes and the InSummary,acontinuousdynamicalequationderived LiouvilleTheorem(31)issatisfiedinthemean. Theflow from the Lorenz-96 model is able to mimic severaltypes reversals are insensitive to viscous dissipation. ofcomplexprocessesobservedingeophysics,geophysical fluid dynamics, and solid state physics: ACKNOWLEDGMENTS (i) Avalanche processes excited by continuous driving as in the sand pile model of Bak et al. [13]; see also the recent observationof quasi-periodic events in crystal JW and VL acknowledge support from the Euro- plasticitysubjecttoexternalstress[14]. Acommonchar- peanResearchCouncilundertheEuropeanCommunitys acteristicpropertyistheweaknessoftheexternalforcing Seventh Framework Programme (FP7/2007-2013)/ERC which is necessary to cause avalanches. In the present GrantagreementNo. 257106. We like to thank the clus- model the flow is driven by a constant forcig towards a ter of excellence clisap at the University of Hamburg. [1] E.N.Lorenz,Proc.SeminaronPredictability,ECMWF, [13] P. Bak, C. Tang, and K. Wiesenfeld, Reading, Berkshire 1, 1 (1996). Physical Review Letters 59, 381 (1987). [2] R. Abramov and A. J. Majda, Nonlinearity 20, 2793 [14] S.Papanikolaou,D.M.Dimiduk,W.Choi,J.P.Sethna, (2007). M. D. Uchic, C. F. Woodward, and S. Zapperi, Nature [3] S. Hallerberg, D. Pazo, J. M. Lopez, and M. A. Rodr- 490, 517 (2012). guez, Phys.Rev. E81, 066204 (2010). 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