Lyapunov Vectors and Bred Vectors in a Fast- Slow Model

Transcription

1 Lyapuov Vectors ad Bred Vectors i a Fast- Slow Model Adriee Norwood With great thaks to: Eugeia Kalay Kayo Ide Christopher Wolfe ad Yig Zhag April

2 Outlie Wolfe- Samelso (2007) algorithm for fidig all Lyapuov vectors (LVs) of a system Compariso of Lyapuov ad bred vectors (BVs) for the Lorez63 system Compariso of LVs for the Fast- Slow Coupled Model developed by Peña ad Kalay (2004)

3 A Few Notes If T is the taget liear model the the sigular value decomposiyo of T is T ηˆ Σξ ˆ Trevisa ad Paco[ (1998) showed SVs are orthogoalized LVs Let η ad ξ be the asymptoyc sigular vectors of T. Wat to use easy- to- fid SVs to fid all LVs of a system.

4 Wolfe- Samelso Algorithm (WS07): Fial SVS Let η j (t) represet fial (or le^) sigular vectors iiyalized i the distat past ad itegrated forward. Most perturbayos alig with the leadig LV (LLV) over Yme so η 1( t) c 1 ϕ1( ) φ LV. 1 t SVs are orthogoal η ( ) 2 2 t c ϕ1( t) + c ϕ 2( t) > ηj( t) c ϕ( t) j i 1 j i

5 WS07 Cot d: IiYal SVs Let ξ j (t) represet iiyal (or right) sigular vectors iiyalized i the distat future ad itegrated backwards. Most perturbayos will alig with the most rapidly decayig LV φ N (t) N is the dimesio of the system. The N ξ j( t ) d ϕi( t) i j j i

6 WS07 Cot d: CreaYg the LVs η(t) ad ξ(t) provide two bases for R N thus droppig the depedece upo Yme where <> is a ier product. Se[g (1) (2) ad takig the ier product with ξ ad η gives (2) (1) 1 > < > < j j N i i i j η ϕ η ϕ ξ ϕ ξ ϕ (4) (3) 1 k k k N i i i k j k j j k > >< < > < > >< < > < η ξ ϕ ξ η ϕ ξ η ϕ η ξ ϕ

7 WS07 Cot d: First LVs SubsYtuYg (2) ito (3) ad oyg meas the first LVs ca be obtaied by solvig D () y () 0 where > >< < N i ki j i i k 1 δ η ξ ξ η k y j k D k k i j k i i kj 1... ) ( 1 1 ) ( > < > >< < ϕ η η ξ ξ η

8 WS07 Cot d: Last LVs Similarly subsytuyg (1) ito (4) the last LVs ca be obtaied by solvig C () x () 0 where ) ( ) ( + > < + > >< < N k x N i k C k k N i i j j k ki ϕ ξ ξ η η ξ

9 Lorez63 Model dx dt dy dt dz dt σ(y-x) ρx y xz xy βz with stadard parameters σ 10 β 8/3 ad ρ 28 ad a itegrayo Yme step of.01.

10 Lyapuov Vectors ad Expoets of LV1 LE.91 Lorez63 LV2 LE 0 LV3 LE Lorez63 LEs

11 LV Growth ad Decay Rates LV1 Growth LV2 Growth LV3 Decay LV1 Growth LV2 Growth LV3 Decay 1 LV growth is give by ϕ gr ϕ d ϕ dt Decay paier of LV3 is similar to growth paier of LV2

12 Compariso of LV1 ad LV2 Cosie of Agle b/w LV1 & LV2 Cosie o X Compoet Cosie o Airactor Red stars i first plot idicate fast LV2 growth. Closer aligmet betwee LV1 ad LV2 whe LV2 grows fastest Red stars i last two plots idicate LV1 ad LV2 are approximately parallel. Blue stars idicate they are approximately ayparallel. Closer aligmet betwee LV1 ad LV2 before ad upo eterig a ew regime.

13 Bred Vectors BVs are iiyally created usig a resizig Yme period (rtp) of two 9me steps ad a perturbayo of 0.1. This provides a idea of the liear chages of perturbayo growth. They are the created usig a rtp of eight 9me steps ad a perturbayo of 1. This is to study the effect of olieariyes o perturbayo growth.

14 Vector Growth Rates LV1 Growth LV2 Growth LV3 Decay Liear BV Growth Noliear BV Growth BV gr ϕgr 1 rtp*dt l( 1 d ϕ ϕ dt BV ) δ 0 LV1 exhibits a growth paier similar to that of the BVs

15 Vector Growth Rates LV1 Growth LV2 Growth LV3 Decay Liear BV Growth Noliear BV Growth

16 Compariso of LV1 LV2 ad BVs Cosie of Agle b/w LV1 & Liear BVs Cosie of Agle b/w LV1 & Noliear BVs LV1 Cosie of Agle b/w LV2 & Liear BVs Cosie of Agle b/w LV2 & Noliear BVs LV2 NolieariYes cause some separayo of LV1 ad BVs but LV1 is syll highly correlated with BVs for the oliear case. Noliear BVs become more correlated with LV2 (alog the flow).

17 Lorez63 Coclusios LV1 growth best predicts regime chages compared to the other LVs BVs with small rtps ad perturbayos alig most closely with LV1. BVs with larger rtps ad perturbayos alig more closely with LV2. LV3 s decay paier mimics LV2 s growth paier: whe the flow accelerates perturbayos coverge faster

18 Fast- Slow Coupled Model Ocea Tropical Atmosphere Extratropical Atmosphere X Compoet of Each Subsystem Ocea is strogly coupled to the tropical atmosphere Tropical ad extratropical atmospheres are weakly coupled Ocea has a amplitude 10x larger ad a Yme scale 10x slower tha that of the fast modes.

19 Fast- Slow Coupled Model dx dt dy dt dz dt dx dt dy dt dz dt t t t e e e dx dt dy dt dz dt σ ( y ρx x xtyt e y e e e ρxt yt y e βz e x (Peña ad Kalay 2004) xe) ce( Sxt + k1) xtzt czz e z e ctzt σ ( yt xt) c( SX + k βzt + + c( SY + k + ctze τσ ( Y X ) c( xt τsxy τβz czz + ce( Syt 2 + k ) ce( Sx τρx τy τsxz + c( yt + k 2) 2 ) 2 + k e 1 ) + k1) ) + ce( Sye + k1) Subscript e s (extratropics) ad t s (tropics) represet fast modes upper cases (ocea) are slow modes τ.1 ad S 1 temporal ad space scalig factors Extratropics ad tropics are 10x faster tha ocea Ocea s amplitude is 10x that of the fast modes. k 1 10 ad k 2-11 uceterig parameters c e c t.08 weak couplig betwee the extratropics ad tropics c c z 1 strog couplig betwee the tropics ad ocea.

20 LEs of Fast- Slow Coupled Model Lorez63 LEs FSCM LEs Couplig creates a itermiglig effect o the growth of the LVs There are two almost eutral LVs correspodig to the fast ad slow modes Gree > extratropics blue > tropics red > ocea

21 Extratropical LVs LV1 Growth LV4 Growth LV8 Decay LV1 Growth LV4 Growth LV8 Decay LV1 LV4 ad LV8 are iflueced by the fastest chagig system. LV1 growth is similar to that of the growig LV of the Lorez63 model. LV8 decay is similar to that of the decayig LV of the Lorez63 model.

22 LV2 Growth Ocea LVs LV3 Growth LV7 Decay LV2 Growth LV3 Growth LV7 Decay LV2 ad LV3 have more growth durig a ocea chage LV7 decay idicates the ocea is goig back to ormal LV2 ad LV3 are syll iflueced by chages i the other systems.

23 LV5 Growth Tropical LVs LV6 Growth LV9 Decay LV5 Growth LV6 Growth LV9 Decay LV5 LV6 ad LV9 caot be used as predictors for the tropical system but their growth rates are most iflueced by chages i this subsystem.

24 LV1 ad Fast Mode BV Cosie of Agle b/w LV1 & Fast Mode BV Fast Mode BV Growth Fast Mode BV Growth Short rtp of 5 Yme steps small perturbayo of size.05. Red stars idicate rapid BV growth BV growth correspods to chages i the extratropical atmosphere ad is closely aliged with LV1

25 Ocea LVs ad Slow Mode BV Slow Mode BV Growth Slow Mode BV Growth Cosie of Agle b/w LV2 & Liear BV Cosie of Agle b/w LV2 & Liear BV Cosie of Agle b/w LV2 & Liear BV Log rtp of 50 Yme steps large perturbayo of size 100 Red stars idicate rapid BV growth BV growth correspods to chages i the ocea subsystem but it is ot closely alig with ay of the LVs associated with this system

26 Coclusios LVs may have predicyve capabiliyes for parycular subsystems i a fast- slow model. LV1 growth is related to regime chages i the fast subsystem. LV2 growth is related to regime chages i the slow subsystem. Some LVs cotai iformayo about more tha oe subsystem. LV1 is similar to BVs with short rtps ad small iiyal perturbayos There is a BV associated with the slow subsystem but there is liile correlayo betwee it ad the LVs associated with that system.

27 Future Work More thorough compariso of LVs ad BVs for the Fast- Slow Coupled Model Iclusio of SVs for both models All of the above for the quasi- geostrophic model: more like the atmosphere Compare the LVs with the LETKF perturbayos Ca LVs be useful i EKF? SuggesYos?

28 Refereces Evas E. N. Bha[ J. Kiey L. Pa M. Pea S.- C. Yag E. Kalay ad J. Hase RISE udergraduates fid the regime chages i Lorez s model are predictable. Bull. Amer. Meteor. Soc Oseledec V A mulyplicayve ergodic theorem. Lyapuov characterisyc umbers for dyamical systems. Tras. Moscow. Math. Soc Peña M. ad E. Kalay SeparaYg fast ad slow modes i coupled chaoyc systems. Noliear Processes i Geophysics 11: Trevisa A. ad F. Paco[ Periodic orbits Lyapuov vectors ad sigular vectors i the Lorez system. J. Atmos. Sci Wolfe C. L. ad R. M. Samelso A efficiet method for recoverig Lyapuov vectors from sigular vectors. Tellus A 59(3):

29 Thak you

30 Compariso of LV1 ad LV3 Cosie of Agle b/w LV1 & LV3 Cosie o X Compoet Cosie o Airactor Red stars i the first graph idicate fast growth of LV1 Red stars i last two graphs idicate LV1 ad LV3 are approximately parallel. Gree stars idicate LV1 ad LV3 are approximately perpedicular.

31 Compariso of LV2 ad LV3 Cosie of Agle b/w LV2 & LV3 Cosie o X Compoet Cosie o Airactor Red stars i the first graph idicate fast growth of LV2 Red stars i last two graphs idicate LV2 ad LV3 are approximately parallel. Blue stars idicate LV2 ad LV3 are approximately ayparallel Gree stars idicate LV1 ad LV3 are approximately perpedicular.

32 Compariso of LV2 ad BVs Cosie of Agle b/w LV2 & Liear BV Cosie o X Compoet Cosie o Airactor Cosie of Agle b/w LV2 & Noliear BVs Cosie o X Compoet Cosie o Airactor With icreased rtp BV2 is more iflueced by olieariyes like LV2.

33 Compariso of LV3 ad BVs Cosie of Agle b/w LV3 & Liear BV Cosie o X Compoet Cosie o Airactor Cosie of Agle b/w LV3 & Noliear BV Cosie o X Compoet Cosie o Airactor Oly sigificat correlayo is upo eterig ew regime

34 Lorez63 Coclusios LV1 predicts regime chages LV2 may be more sesiyve to chages caused by olieariyes. LV3 s decay paier mimics LV2 s growth paier (CONTRADICTION???) LV1 is most closely aliged with BVs with small rtps ad perturbayos. LV2 is most closely aliged with BVs with larger rtps ad perturbayos.

35 Bred Vectors The extratropical rate was computed by (rtp*dt)^- 1 l(extratr BV/del). Similarly for the tropical ad oceaic rates.