Mu Mu F.F.Zhou,H.L.Wang and X.G.WuState Key Laboratory of Numerical Modeling for Atmospheric Scie

nces and Geophysical Fluid Dynamics (LASG), Institute of Atmospheric Physics (IAP),

Chinese Academy of Sciences (CAS)mumu@lasg.iap.ac.cn

http://web.lasg.ac.cn/staff/mumu/

Some New Progresses in the Applications of Conditional Nonlinear Optimal Perturbations

OutlineOutline1. Concept of conditional nonlinear optimal perturbation (CNOP) and the difference between CNOP and LSV

2.Adaptive observations (MM5 model)

3.The sensitivity of ocean’s thermohaline circulation (THC) to the finite amplitude initial perturbations

1. Conditional Nonlinear Optimal Perturbation1. Conditional Nonlinear Optimal Perturbation

00|

0),,(

ww

txwFt

w

t

)(),( 0wMTxw T

TM : (nonlinear) propagator of (1)

00000 |)),(),(( ,| uUtxutxUUU tt

),(),()( ),,()( 000 txutxUuUMtxUUM TT

),(),( ),,( txutxUtxU Let be the solutions to (1)

(1)

)(max)( 0||||

00

uJuJu

0u

Conditional Nonlinear Optimal Perturbation(CNOP)

||||0u

Constraint condition

||)()(||)( 0000 UMuUMuJ TT

1. The initial error which has largest effect on the uncertainty at prediction time.

2. The initial anomaly mode which will evolve into certain climate event most probably (ENSO)

3. The most unstable (or sensitive ) initial mode of nonlinear model with the given finite time period

Physical meaning of CNOP Physical meaning of CNOP

||||

||)(M||)(

0

00 w

wwJ T

*0w is LSV if and only if ,

00|

0),,(|)(

ww

wtxww

F

t

w

t

Uw

)(M),( 0wTxw T

TM : (linear) propagator of (1)

),(max)( 0*0

0

wJwJw

where

(2)

[1] Mu Mu, Duan Wansuo, Wang Bin, 2003, Nonlinear Processes in Geophysics, 10, 493-501.

[2] Duan Wansuo, Mu Mu, Wang Bin, 2004,. JGR Atmosphere, 109, D23105, doi:10.1029/2004JD004756.

[3] Mu Mu, Sun Liang, D.A. Henk, 2004, J. Phys. Oceanogr., 34, 2305-2315.

[4] Sun Liang, Mu Mu, Sun Dejun, Yin Xieyuan, 2005, JGR-Oceans, 110, C07025,doi: 10.1029/2005JC002897.

[5] Mu Mu and Zhiyue Zhang,2006,J.Atmos.Sci..

[6] Mu Mu ,Hui Xu and Wansuo Dun(2007),GRL

[7] Mu Mu ,Wansuo Duan and Bin Wang (2007),JGR

[8] Mu Mu and Wang Bo,2007, Nonlinear Processes in Geophysics[9]Olivier Riviere et al,2008,JAS

ReferenceReference

When nonlinearity is of importance , there exist distinct difference between CNOP and LSV represented by two facts:

a. The initial patterns are different

Note: LSV stands for the optimal growing direction , but CNOP the “pattern”

b. Linear and nonlinear evolutions of CNOP and LSV are different.

Mu Mu and Zhiyue Zhang,2006.J.Atmos.Sci.

2. Adaptive Observation

• FASTEX (Snyder 1996)

• NORPEX (Langland et al.1999)

• WSR (Szunyogh et al. 2000,2002)

• DOTSTAR (Wu et al.2005)

• NATReC (Petersen et al. 2006 )

• THORPEX (in process)

Methods used in Adaptive Observations

• SV (Palmer et al.1998)

• Adjoint Sensitivity (Ancell and Mass 2006)

• ET (Bishop and Toth 1999)

• EKF (Hamill and Snyder 2002)

• ETKF (Bishop et al. 2001)

• Quasi-inverse Linear Method (Pu et al.1997)

• ADSSV (Wu et al. 2007)

• The sensitive areas identified by different me

thods may differ much. Which one is better i

s still in discussion (Majumdar et al.2006).

• Conditional nonlinear optimal perturbation (C

NOP), which is a natural extension of linear

singular vector (SV) into the nonlinear regim

e, is in the advantage of considering nonline

arity (Mu et al, 2003; Mu and Zhang,2006).

Applications of CNOP to

Adaptive Observations

• Rainstorms

• Tropical cyclones

Rainstorms

• Case A:

Rainfall during 0000 UTC 4 July~ 0000 UTC 5 July, 2003 on the Jianghuai drainage basin in China

• Case B:

Rainfall during 0000 UTC 5 Aug~ 0000 UTC 6 Aug, 1996 on the Huabei plain in China

• optimization algorithm

SPG2(Spectral projected gradient,

Birgin etal,2001)

Characters: box or ball constraints

linearity convergence

high dimensions

The constraint in this study is 860.37 J/kg

The optimization time interval is 24 hours.

• Experimental designModel: MM5 and its Adjoint

Grid number: 51*61*10 Grid distance: 120km

Top level: 100hPa

Physical parameterizations:

dry-convective adjustment

grid-resolved large scale precipitation

high resolution PBL scheme

Anthes-Kuo cumulus parameterization scheme

Data: NCEP analysis

ECMWF reanalysis

routine observations

• Total dry energy is chosen as a metric：2

12 2 2 2

0

1[ ( ) ]p s

a rDr r

cR T d ds

D T p

pu v T

where,1 11005.7 J kg Kpc

1 1287.04J kg KaR

270KrT

1000hparp

The integration extends the full horizontal

domain D and the vertical direction .

Figure1.

The temperature (sha

ded, unit:K) and wind

(vector, unit: m/s) com

ponents of CNOP(a,b),

FSV (c,d) and loc CN

OP (e,f)

on level

at 0000 UTC 4 July (a,

c,e) and their nonlinea

r evolutions

at 0000 UTC 5 July (b,

d,f).

0.45

Case A

c (FSV)

b(CNOP)a (CNOP)

d (FSV)

e (loc CNOP) f (loc CNOP)

Nonlinear evolutions

Figure 2. Case AThe evolution of the total dry ener

gy on targeting area during the o

ptimization time interval. CNOP (s

olid), local CNOP(dashed), FSV (d

ot) and -FSV (dashdotted). The TE

showed is divided by the initial.

0.45 Table 1.Case A: The maxima (minima) of temperature (unit: K), zonal

and meridional wind (unit: m/s) on level

time type

0000 UTC 4 July, 2003

0000 UTC 5 July, 2003

Figure 3. Same as Fig.1(a,b,c,d), but for case B at 0000 UTC 5 Aug, 1996 (a,c) and at 0000 UTC 6 Aug, 1996 (b,d)

a (CNOP) b (CNOP)

c (FSV) d (FSV)

Nonlinear evolutions

Table 2. Same as table 1, but for case B

Figure 4

Same as Fig.2,

but for case B

0000 UTC 5 Aug, 1996

0000 UTC 6 Aug, 1996

time type

• Sensitivity experiments

Case A Case B

Figure 5. the variations of the cost function due to the reductions

of CNOP (solid) or FSV (dashed) during the optimization time

interval for case A and case B.

Tropical Cyclones

• Case C: Mindulle, North-West Pacific Tropical cyclones

0000 UTC 28 Jun ~ 0000 UTC 29 Jun, 2004

• Case D: Matsa, North-West Pacific Tropical cyclones

0000 UTC 5 Aug ~ 0000 UTC 6 Aug , 2005

• optimization algorithm

SPG2(Spectral projected gradient,

Birgin etal,2001)

The constraints are 729 J/kg for case C,

and 900 J/kg for case D.

The optimization time intervals for these two cases

are still 24 hours.

• Experimental designModel: MM5 and its Adjoint

Grid number: 41*51*11(case C), 55*55*11(case D)

Grid distance: 60km

Top level: 100hPa

Physical parameterizations:

dry-convective adjustment

grid-resolved large scale precipitation

high resolution PBL scheme

Anthes-Kuo cumulus parameterization scheme

Data: NCEP reanalysis

12 2 2

0( )

Dd ds u v

212 2 2 2

0[ ( ) ]p s

a rDr r

cR T d ds

T p

pu v T

1 11005.7 J kg Kpc 1 1287.04J kg KaR

270KrT

1000hparp

• Metrics

total dry energy

dynamic energy

where,

The integration extends the full horizontal

domain D and the vertical direction .

•Simulation of case C (Mindulle)

a

b

a: model domainb: target area

Figure 6.

Simulation

track from

MM5 (red)

and the

observation

track (blue)

from CMA

a

b

a: model domainb: target area

Figure 7.

Simulation

track from MM5

(red)

and the

observation

track (blue)

from CMA

•Simulation of case D (Matsa)

a

b

729

ResultsMindulle

dynamic energy, 24-h

Nonlinear evolutions

0.7

CNOP

at 0000 UTC 28 Jun

at 0000 UTC 29 Jun

FSV

CNOP FSV

Mindulle

dry energy, 24-h0.7 729

at 0000 UTC 28 Jun

at 0000 UTC 29 Jun

Nonlinear evolutions

CNOP FSV

CNOP FSV

Case C (Mindulle)

The evolutions of the dynamic energies (KE) and total dry energies (T

E) of CNOP (blue) and FSV (red) on targeting area during the optimizat

ion time interval. Unit: J/kg

24h nonl i near devel opment (TE)

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

0 3 6 9 12 15 18 21 24

t i me(h)

TE(J

/kg)

CNOPFSV

24h nonl i near devel opment (KE)

0

10000

20000

30000

40000

50000

60000

70000

0 3 6 9 12 15 18 21 24

t i me(h)

KE(J

/kg)

CNOPFSV

Matsadynamic energy, 24-h

0.7 900

at 0000 UTC 5 Aug

at 0000 UTC 6 Aug

Nonlinear evolutions

CNOP

CNOP

FSV

FSV

Matsadry energy, 24-h

0.7 900

at 0000 UTC 5 Aug

at 0000 UTC 6 Aug

Nonlinear evolutions

CNOP

CNOP

FSV

FSV

Case D (Matsa)

The evolutions of the dynamic energies (KE) and total dry energies (TE)

of CNOP (blue) and FSV (red) on targeting area during the optimization

time interval. Unit: J/kg

24h nonl i near devel opment (TE)

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

50000

0 3 6 9 12 15 18 21 24

t i me(h)

TE(J

/kg)

CNOPFSV

24h nonl i near devel opment (KE)

0

5000

10000

15000

20000

25000

30000

0 3 6 9 12 15 18 21 24

t i me(h)

KE(J

/kg)

CNOPFSV

• Sensitivity experiments

2

1( )J 0 0 0 0δX P (X + δX ) - P (X )M MDefine:

2

2 ( )J c0 0 0 0δX P (X + δX ) - P (X )M M

Where is the projection operator, is a constant

less than one.

cP

1 2

1

( ) ( )

( )

J J

J

0 0

0

δX δX

δX

Benefits obtained from the reductions of CNOP or FSV

are evaluated by:

• Benefits obtained from the reductions of CNOP or FSV

KE TE

CNOP FSV CNOP FSV

0.25 91.6% 47.8% 84.8% 25.1%

0.50 62.2% 27.3% 53.8% 7.5%

0.75 26.6% 15.3% 24.1% -5.3%

Case C Mindulle

KE TE

CNOP FSV CNOP FSV

0.25 91.3% 63.3% 86.4% 46.5%

0.50 69.5% 38.3% 69.9% 26.3%

0.75 42.3% 17.8% 49.4% 15.3%

Case D Matsa

c

c

• Conclusions

The pattern of CNOP may differ from that of FSV,

and its nonlinear evolutions are larger than those of

FSV, as well as the loc CNOP and –FSV.

The forecasts are more sensitive to the CNOP kind

errors than the FSV kind. It is indicated that reduction

of the CNOP kind errors benefits more than reduction

of the FSV kind errors.

Discussions• The determination of the sensitivity area

according to CNOP

• Comparisons with other methods

• Choice of the constraints

• Optimization algorithm: L-BFGS, no constraint

• Evaluations of the effectiveness of adaptive

observation

• Feasibility and the time limitation

3.The sensitivity of ocean’s thermohaline circulation (THC) to finite amplitude initial perturbations and decadal variability

Mu Mu, Sun Liang, D.A. Henk, 2004, J. Phys. Oceanogr., 34, 2305-2315

Sun Liang, Mu Mu, Sun Dejun, Yin Xieyuan, 2005, JGR-Oceans, 110,C07025,doi:10.1029/2005JC002897.

Wu Xiaogang,Mu Mu, 2008,J.P.O. in review

Sensitivity and stability study of THC

The day after tomorrow?

Floods & Impacts to New York

Stommel box Model

Strength of the thermal forcing

Strength of the freshwater forcing

Ratio of the relaxation time of T and S to surface forcing

One disadvantage of S-model

The ignoring the effect of wind-stress

To consider the impact of small- and meso-scale motions of wind-driven ocean gyres (WDOG) of THC, Longworth et al (2005,J.of Climate) introduce a diffusion term to represent the effect of WDOG.

Longworth’s model

(2a)

(2b)

: the diffusion coefficient

1 41dT

T T Sdt

2 3 4

dSS T S

dt

4

Thermally-driven, TH

Salinity-driven, SA

Steady state

Perturbation

Norm

The effect of WDOG on the existence of multi-equilibrium

Figure 1. The bifurcation diagram of box models for ， as a plot of versus . The curves from left to right : 0.0, 0.01, 0.05, 0.09 and 0.17. Circles in the figure represent the bifurcation points, which separate the linearly stable equilibrium TH-states and unstable ones. Besides, negative corresponds to the linearly stable SA-state.

1 3.0 3 0.6

2 4

when ,we have

,

Fig.1 shows ,hence

(numerical result)

4 0

1 2 0T T T 1 2 0S S S

2 1

1 2 0T S

T S

nonlinear stability analysis

Figure 2. The evolution of (a) (c) cost function J and (b) (d) overturning function versus t computed with CNOPs superposed on the equilibrium state as initial conditions for , . (a) (b) : the TH-state with , and (c) (d) : SA-state with . Solid (dashed) curve is for L (S) model.

1 3.0 3 0.6 2 1.84

2 1.83

TH state TH state

SA state SA state

Fig.2 a, b WDOG stabilizes the TH-state

Fig.2 c, d

WDOG destabilizes the SA-state

The smallest magnitude of a finite perturbation which induces a transition from TH state to SA state and vise versa.

Understanding nonlinear stable regimeUnderstanding nonlinear stable regime

Figure 3. The critical value versus control parameter for , in the case of (a) TH-state and (b) SA-state. Solid (dashed) curve corresponds to L (S) model.

c1 3.0 3 0.6 2

Why WDOG stabilizes (destabilizes) TH-state (SA-state) ?

Recall (numerical)

We can prove that theoretically

T S

2 1

2 1

0S T

Conclusion

There exists a physical mechanism,

WDOG stabilizes (destabilizes)

TH-state (SA-state).

WDOG S T