Avalanches, Breathers and Flow Reversal in a Continuous Lorenz-96 Model

0.25 MB·English

by  Richard Blender

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 approxima-
tion 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 inter-
action. 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 con-
tinuous 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 con-
stant 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 trunca-
tion 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 magni-
tude 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 nonlin-
ear 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). [15] M.P.Baldwin,L.J.Gray,T.J.Dunkerton,K.Hamilton,
[4] V.Lucarini, J. Stat. Phys.146, 774 (2012). P. H. Hayne, W. J. Randel, J. R. Holton, M. J. Alexan-
[5] V.LucariniandS.Sarno,Nonlin.ProcessesGeophys18, der, I. Hirota, T. Horinouchi, D. B. A. Jones, J. S. Kin-
728 (2011). nersley, C. Marquardt, K. Sato, and M. Takahasi, Rev.
[6] J. T. Ambadan and Y. Tang, J. Atmos. Sci. 66, 261 Geophys. 39, 179 (2001).
(2009). [16] Y. Kawatani, K. Sato, T. J. Dunkerton, S. Watanabe,
[7] S. Khare and L. A. Smith, Mon. Wea. Rev. 139, 2080 S. Miyahara, and M. Takahashi, J. Atmos. Sci. 67, 963
(2011). (2010).
[8] J. W. Messner and G. J. Mayr, Mon. Wea. Rev. 139, [17] M. Onorato, A. R. Osborne, and M. Serio,
1960 (2011). Phys. Rev.Lett. 96, 014503 (2006).
[9] E. Lorenz, J. Atmos.Sci 63, 2056 (2006). [18] A. Calini and C. M. Schober, Nonlinearity 25, R99
[10] E. Lorenz, J. Atmos.Sci. 62, 1574 (2005). (2012).
[11] A.SergiandM.Ferrario,Phys.Rev.E64,056125(2001). [19] A. Abrashkin and A. Soloviev,
[12] V. Lucarini and K. Fraedrich, Phys. Rev. E 80, 026313 Phys. Rev.Lett. 110, 014501 (2013).
(2009).