Lorenz attractor with and without forcing shows that there are upper and lower limits of such length of time for averaging, beyond which the system remains ...
Indian JOlII'D4I of Radio & Space Physics VoL 2&, Decembe:r 1999, pp. 271-276
Feasibility study of extended range atmospheric prediction through time average Lorenz attractor Pradip K Pal & Shivani Shah Meteorology and Oceanography Group, Space Applications Centre, Ahmedabad 380053 Received 20 September 1999
Though the theoretical limit of atmospheric predictability is only up to 1-2 weeks, temporal and spatial averages may be predictable up to certain extent for extended range. It is not known what should be the length of time for averaging to have certain predictability of time average values. The time average behaviour of Lorenz attractor with and without forcing shows that there are upper and lower limits of such length of time for averaging, beyond which the system remains chaotic.
1 Introduction Theoretical limit of deterministic predictability of the atmospheric system has been established by Lorenzi through a simple nonlinear dissipative system. Although this limit is up to 15 days, practically, it is possible to predict the instantaneous weather only up to 7-10 days in midlatitudes and 3-5 days in tropics . Therefore, there is a need for prediction beyond thi s limit. Efforts 2 .3 have been made to study the feasibility of prediction beyond the limit of determini stic predictability . Of course, there is no question of predicting instantaneous values beyond thi s limit, but spati al and temporal averages may be predicted up to certain extent for a longer time. The atmospheric system has different regimes where it may lie for a certain length of time. The long time average will depend more on boundary forcings like sea-surface temperature and soil moisture. Simple Lorenz system also has an attractor with two distinct regimes. Probability of a state, being in either of the 4 attractor's branch , is same . Pal 5 has shown that, by introduci"ng fo rcing in the Lorenz system, probability of the state lying in one of the two branches is more than that lying in the other branch . Even after certain forcing being included, integration from certain close initial condition may reac h to different state in the same branc h of the attractor after a long time integration . To obtain a mean state, time average will further reduce the error. Feasibility of long range prediction of seasonal average rainfall over whole India was shown by Indi a Meteorological Department through good skill in prediction by statistical method 6 .
The possibility of time average prediction by dynamical methods has been studied by a few 2 scientists .3 It is not clear what should be the time period for averaging (say for 15 or 30 days) to have a predictability in the extended range. Is there any limit on the time period for averaging beyond which even the average behaviour becomes chaotic? In this paper, we intend to study how the actual shape of the Lorenz attractor changes, if the time average state is plotted against time.
2 Simulation results and discussion The forced Lorenz system of equations described 5 by Pal is:
dx
-dt = -ax+ay+cFx dy dt Y -dz = xy-bz +cFZ dt The three constants a, band r determine the behaviour of the system and c and F represent some sort of forcing . In case of F = 0, the system becomes the original Lorenz sy~tem. The original system has 7 been thoroughly studied by Sparrow . The attractor of the' Lorenz system looks like a butterfly, the projection of which on x-z plane is shown in Fig 1. This clearly shows the two separate regimes of the systempne in the positive side and the other in the negative side of x. For the same system without any forcing, interesting changes are seen if sliding average is taken . With sliding average, of 20
- = - xz + rx - y + c F
272
INDIAN] RADIO &. SPACE PHYS, DECEMBER 1999
z
x
Fig. l---Lorenz attractor on x-z plane
30
10
j 1
o
~~-L~~~-L~~~-L-L~~~-L~
- '6
- I.
- 12
-10
-II
-e
-4
-2
0
2
,
e
I
,
~o
X
ig. 2---Loienz attractor wi h 20 steps avera ng
I
I..L ..
12
14
"-.J~ 1C
1"
PAL & SHAH : AlMOSPHERIC PREDICTION TIIROUGH TIME AVERAGE LORENZ ATTRACTOR
32
~
30
28
28
2_
z 22
20
18
18
,. 12 -10
-8
-8
-2
~
e
2
0
8
10
X
Fig. 3-Lorenz attractor with 100 steps averaging
30
:IV
28
27
28
2S
2'
23
22
21
20
19
18
17
-8
~
~
-2
2
0
•
e
X
Fig. 4---Lorenz attractor with 150 steps averaging
~
10
273
274
[NDIAN J RADIO & SPACE PHYS, DECEMBER 1999
z
x
Fig. 5--Lorenz attracior with 250 steps averaging
50
45
35
30
Z
25
20
x
Fig. &--Forced attractor with 20 steps averaging
PAL & SHAH : ATMOSPHERIC PREDICTION THROUGH TlME AVERAG E LORENZ ATTRACTOR
30
28
26
z
24
22
20
Ie
-2
10
x Fig. 7--Forced attractor with 100 steps averaging
30
211
28
27
26
25
z 24
23
~
22
21
1
j
./0
'v -4
-3
-2
-1
5
x
Fig. X-Forced al(raClor with 150 steps averagi ng
275
276
INDIAN J RADIO & SPACE PHYS, DECEMBER 1999
steps (Fig. 2) the attractor retains its original shape, but if the slidin g average is taken for 100 steps (Fig. 3) the shape of the attractor changes. For 100 step average, the two extreme regimes have been distinctly separated and the spread of each regime has shrinked considerably. Interesting to note that an un stabl e tran sition regime is getting developed in between two regimes. Th is interm~diate regime gets more stabilised with the average of 150 steps (Fig. 4). Now there are three regimes where the state can lie. The third regime .again gets split with the average of 250 steps (Fig. 5) and more transition regimes develop. This procedure of splitting continues as the number o( averaging steps increases further. It was shown by PaIs that in case of introduction of force (F"* 0) in the system, the forced attractor shifts towards one side. In the forced system, average of 20 steps (Fig. 6) does not change the shape of forced attractor, but average of 100 steps (Fig. 7) shrinks the size of the attractor considerably. If the number of time steps for averaging is increased to 150, the attractor (Fig. 8) again gets split into two branches. It means that though under the influence of forcing the system becomes predictable up to certain extent, with the increase of number of time steps for averaging even the temporal average becomes chaotic beyond certain limit.
3 Conclusions For the atmospheric system there may be a situation such that time average for certain time
period may still be chaotic, but for a particu lar rang of time averages it may lie in one regime and the transit io n regi me may be unstab le and unpredictabl e. It seems th at time average up ~() certain time period may be predi ctable with some skill , bu t beyond certain peri od it may not be predictable. There seems to be an upper and lower limit of the length of time fo r averaging to make it feasible for atmospheric predi cti on. Thi s study does not suggest any particul ar limit for suc h time averaging. Numerical experiments have to be conducted with full atmospheric model to 2 establish such limits. Shukla has shown such feasibility for monthly averages, but he does not conc lude th at fortnightly average will not be predictable or sixty-day average also will be predictable. For atmospheric system the length of time averaging may also depend on the area of spatial averagillg.
References I 2
3
LorenzEN . JAtmosSci (USA ). 20 (1 963 ) 130. Shukla J, cited in Problems and prospects in long and medium range weather f orecasting, edited by D M Burridge and E Kallen (Springer-Verl ag, New York ), 1984, pp. 155206. Palmer T N, Bull Am Meteorol Soc (USA), 74 (1993) 49.
4 5
Palmer T N. Proc Indian Natl. Sci. Acad, 60 (1994) 57. Pal P K. Indian J Radio & Space Phys, 25( 1996) 175.
6
Gowariker V, Thapli yaJ V, Sarker R P, Mandai G S & Sikka DR. Mausam (India ), 40 ( 1989) 11 5. Sparrow C, The Lorenz equation: Bifurcation, Chaos and Strange Altractors (Springer- Verlag, New York), 1982.
7
