Breeding:New)Insightsinto Predic3ng)Regime)Transi3on)in ... · Forecast evolution Initial random...

Breeding: New Insights into Predic3ng Regime Transi3on in

the Lorenz 63 Model

Ying Zhang, Kayo Ide, and Eugenia Kalnay Dec. 19, 2011

UMD-‐PSU DA Workshop, Dec. 19, 2011

Outline •  IntroducGon •  Basic Experiment •  CharacterisGcs of Bred Vectors •  PredicGng Regime Changes Based on z •  Summary


The Lorenz (1963) Model

bzxydtdz

xzyrxdtdy

xyadtdx

−=

−−=

−= )(

1=a 3/8=b 28=rwhere , , and

A simple system exhibits chaotic dynamics, in particular, regime transitions cold regimes warm regimes. UMD-‐PSU DA Workshop, Dec. 19, 2011

The solution integrated 4001 time steps, dt = 0.01

Forecast evolution

Initial random perturbation Perturbed forecast xf

Bred Vectors dx (difference between two nonlinear forecasts)

Unperturbed control forecast xc Rescaling intervals (8 steps)

time

Bred Vectors are rescaled and added to control forecast.

σdx0,

dx

xc

xf

Tw

Breeding •  Breeding is an effecGve method to produce perturbaGons for

ensemble forecasGng (Toth and Kalnay, 1993).

•  Two important quanGGes in breeding:

-‐-‐ Bred vectors (BVs): dx= xf -‐ xc

-‐-‐ Growth rate:

•  Three parameters: -‐-‐ The size of perturbaGon: σ ;

-‐-‐ The direcGon of iniGal perturbaGon: dx0= (dx0, dy0, dz0); hence,

indicates iniGal perturbaGon;

-‐-‐ The rescaling interval: Tw = n*dt, where dt is the Gme step;

σ*0

0

dxdx


)ln(*1σ

dxTw

(Evans et al., 2004)

•  Evans et al. (2004) found the regime changes in the Lorenz 63 model were PREDICTABLE and followed the two rules:

1. If σ ≥ σcr (≈ 1.67) indicated by * in Fig.4 , the current regime will end aCer it completes the current orbit.

2. The length of the new regime is proporIonal to the number of * as shown in Fig. 5.


PredicGon Rules by Evans et al.(2004)

Fig. 4, Evans et al., 2004

Colored with growth rate

x

Fig. 5, Evans et al., 2004

Number of * in old regime

Num

ber o

f cyc

les

in n

ew re

gim

e

ObjecGves •  To examine the characterisGcs of bred vectors through sensiGvity experiments of the direcGon of iniGal perturbaGon, dx0 = (dx0, dy0, dz0)

•  To see if we can find another way to predict regime changes


Basic Experiments

•  MoGvaGon -‐-‐ Introduce bred vector and verify the applicaGon of breeding. •  Parameters in breeding -‐-‐ The size of perturbaGon, σ =1 -‐-‐ The direcGon of iniGal perturbaGon, dx0=[1; 1; 1] -‐-‐ The rescaling interval, Tw=8*dt, where dt=0.01 is Gme step


Lorenz Agractor and Bred Vectors

UMD-‐PSU DA Workshop, Dec. 19, 2011 bred vector

direction

size

Growth Rate:

PredicGng Regime Changes QuanGfy the rules of Evans et al. (2004)


High growth rate is significantly correlated to next regime length.

Lower growth rate is not significantly correlated to next regime length.

•  MoGvaGon -‐-‐ How many bred vectors that the system has and how they affect

the predicGon of regime changes

•  Experiments Design: -‐-‐ fix σ =1 and Tw = 8*dt -‐-‐ change dx0


Characteristics of Bred Vectors

Exp 01: dx0=[1; 1; 1] (basic experiment) Exp 02: dx0=[1; 1; 3] Exp 03: dx0=[1; 1; -1] Exp 04: dx0=[1; 1; -3] Exp 05: dx0=[3; 1; 1] Exp 06: dx0=[-1; 1; 1] Exp 07: dx0=[-3; 1; 1] Exp 08: dx0=[1; 3; 1] Exp 09: dx0=[1; -1; 1] Exp 10: dx0=[1; -3; 1] Exp 11: dx0=[3; 3; 1] Exp 12: dx0=[-1; -1; 1] Exp 13: dx0=[-3; -3; 1] Exp 14: dx0=[-1; -1; -1]

Changing in dz0

Changing in dx0

Changing in dy0

Changing in dx0 & dy0

Changing in dx0, dy0 & dz0


14 experiments are done under the same initial condition x0=[-6.11, -9.99, 15.70]

Sensitivity Experiments of dx0

Growth Rates in the 14 Exps differ from each other at the very beginning. They converged into two lines afterward. So separate them into two groups.


Growth Rate of the 14 Exps

Group 1

Group 2


Separate the 14 Exps into two Groups Composite BVs on the attractor

Movie of Bred Vectors on the Agractor in the two groups


Group 1

Colored with growth rate of Group 2


Growth Rate

Composite Bred Vectors of the Two Groups

Angle between BVs in the two groups

Group 2

Dimensionality of Bred Vectors (Patil et al., 2001)

Three perturbations: dx0=[1, 1, 1], [-1, 1, 1], [1, 1, -1] σ = 1 Colored with composite growth rate of the three members.

l  Bred vector has dimensionality between 1 and 2. l  It approaches to 1 dimension with high growth rate (red stars). l  It has much more than 1 dimension with less growth rate (other color stars).


Initial condition: x0=[-6.11, -9.99, 15.70] Growth Rate

17

Group1 Group2 Growth Rate

Impacts of Bred Vectors on PredicGng Regime Changes

l  Warm à Cold: For same numbers of red stars in warm regime, the next cold regime is shorter for Group1 than Group2.

l  Cold à Warm: For same numbers of red stars in cold regime, the next warm regime is longer for Group1 than Group2.

Conclusion 1

•  Growth rates of the 14 experiments diverge into two groups ajer they mature. This means the model has two bred vectors.

•  For moderate or low growth rate, the system has two bred vectors (dimensionality =2). For high growth rate that can be used for predicGon, the two bred vectors become nearly idenGcal (dimensionality à1).

•  When separaGng the transiGons from warm to cold and from cold to warm, differences exist in the two groups. This may relate to the direcGon of bred vectors with high growth rate.


MoGvaGon: •  The results of the basic experiments: -‐-‐ regime changes are related to high growth rate indicated by red stars (*) -‐-‐ red stars (*) always occur at low values of z

PredicGng Regime Changes Based on z


l  z > zcr (≈ 13.96), stay in current regime l  z ≤ zcr (≈ 13.96), regime will change; smaller z is, longer next regime lasts

PredicGon Rules Based on z


These ranges of z for each step do not overlap.

value of z in lower attractor VS. number of orbits in next regime Step Function


We also can predict how long current regime will last after lowest z occurs

l  Larger local minimum z, longer lasting of current regime l  Smaller local minimum z, shorter lasting of current regime

Integrate 19000 time steps

How long current regime will last?

•  Beside doing breeding, we can more accurately predict regime changes in the Lorenz model with respect to the value of z.

•  When z > zcr (≈ 13.96), it will stay in the current regime.

When z ≤ zcr (≈ 13.96), the current regime will end soon and transit to the new regime.

•  The smaller z is, the sooner the current regime ends and the longer the next regime lasts.

Conclusion 2


Final Summary

•  The Lorenz 63 system has two bred vectors. The two bred vectors approaches to idenGcal during high-‐growth periods. The regime transiGons are associated with the direcGon of bred vectors having high-‐growth rates.

•  The value of z can also be used to predict regime changes. -‐-‐ When z > zcr (≈ 13.96), it will stay in the current regime.

-‐-‐ When z ≤ zcr (≈ 13.96), smaller z is, sooner the current will end and longer the new regime lasts.


References •  Evans, E., N. Bhap, J. Kinney, L. Pann, M. Pena, S.-‐C. Yang, E. Kalnay, and J.

Hansen, 2004: RISE undergraduates find the regime changes in Lorenz’s model are predictable. Bull. Amer. Meteor. Soc., 520-‐524.

•  PaGl, D. J., B. R. Hunt, E. Kalnay, J. A. York, and E. Og, 2001: Local low dimensionality of atmospheric dynamics. Phys. Rev. LeS., 86: 5878-‐5881.

•  Toth, Zoltan, and E. Kalnay, 1993: Ensemble forecasGng at NMC: the generaGon of perturbaGons. Bull. Amer. Meteor. Soc., 74: 2317-‐2330.



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