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Mu Mu F.F.Zhou,H.L.Wang and X.G.Wu

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Applications of Conditional

Nonlinear Optimal Perturbations

Mu Mu F.F.Zhou,H.L.Wang and X.G.Wu

State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics (LASG),

Institute of Atmospheric Physics (IAP),

Chinese Academy of Sciences (CAS)

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

- 1. 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

: (nonlinear) propagator of (1)

Let

be the solutions to (1)

1. Conditional Nonlinear Optimal Perturbation

(1)

Conditional Nonlinear Optimal Perturbation(CNOP)

Constraint condition

- 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

[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

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 difference between CNOP and LSV represented by two facts:

- 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 difference between CNOP and LSV represented by two facts:

- 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 methods may differ much. Which one is better is still in discussion (Majumdar et al.2006).
- Conditional nonlinear optimal perturbation (CNOP), which is a natural extension of linear singular vector (SV) into the nonlinear regime, is in the advantage of considering nonlinearity(Mu et al, 2003; Mu and Zhang,2006).

Applications of CNOP to Adaptive Observations differ much. Which one is better is still in discussion

- Rainstorms

- Tropical cyclones

Rainstorms differ much. Which one is better is still in discussion

- 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 differ much. Which one is better is still in discussion
- SPG2(Spectral projected gradient,
- Birgin etal,2001)

Characters: box or ball constraints

linearity convergence

high dimensions

The constraint in this study is

The optimization time interval is 24 hours.

- Experimental design differ much. Which one is better is still in discussion

- Model: 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 differ much. Which one is better is still in discussion ：

where,

The integration extends the full horizontal

domain D and the vertical direction .

Nonlinear evolutions differ much. Which one is better is still in discussion

a (CNOP)

b(CNOP)

Case A

Figure1.

The temperature (shaded, unit:K) and wind (vector, unit: m/s) componentsof CNOP(a,b), FSV (c,d) and loc CNOP (e,f)

on level

at 0000 UTC 4 July (a,c,e) and their nonlinear evolutions

at 0000 UTC 5 July (b,d,f).

c (FSV)

d (FSV)

f (loc CNOP)

e (loc CNOP)

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 2. Case A

The evolution of the total dry energy on targeting area during the optimization time interval. CNOP (solid), local CNOP(dashed), FSV (dot) and -FSV (dashdotted). The TE showed is divided by the initial.

Nonlinear evolutions K), zonal and meridional wind (unit: m/s) on level

b (CNOP)

a (CNOP)

c (FSV)

d (FSV)

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)

Table 2. Same as table 1, but for case B K), zonal and meridional wind (unit: m/s) on level

time

type

0000 UTC 5 Aug, 1996

0000 UTC 6 Aug, 1996

Figure 4

Same as Fig.2,

but for case B

- Sensitivity experiments K), zonal and meridional wind (unit: m/s) on level

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

- 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 K), zonal and meridional wind (unit: m/s) on level
- SPG2(Spectral projected gradient,
- Birgin etal,2001)

The constraints are

for case C,

and

for case D.

The optimization time intervals for these two cases

are still 24 hours.

- Experimental design K), zonal and meridional wind (unit: m/s) on level

Model: 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

- Metrics K), zonal and meridional wind (unit: m/s) on level

dynamic energy

total dry energy

where,

The integration extends the full horizontal

domain D and the vertical direction .

Simulation of case C (Mindulle) K), zonal and meridional wind (unit: m/s) on level

a

a: model domain

b: target area

Figure 6.

Simulation track from MM5 (red)

and the observation track (blue) from CMA

b

- Simulation of case D (Matsa) K), zonal and meridional wind (unit: m/s) on level

a

a: model domain

b: target area

a

Figure 7.

Simulation track from MM5 (red)

and the observation track (blue) from CMA

b

b

Results K), zonal and meridional wind (unit: m/s) on level

CNOP

Mindulle

FSV

dynamic energy, 24-h

at 0000 UTC 28 Jun

FSV

CNOP

at 0000 UTC 29 Jun

Nonlinear evolutions

Mindulle K), zonal and meridional wind (unit: m/s) on level

CNOP

dry energy, 24-h

FSV

at 0000 UTC 28 Jun

CNOP

FSV

at 0000 UTC 29 Jun

Nonlinear evolutions

Case C (Mindulle) K), zonal and meridional wind (unit: m/s) on level

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

Matsa K), zonal and meridional wind (unit: m/s) on level

CNOP

dynamic energy, 24-h

FSV

at 0000 UTC 5 Aug

CNOP

FSV

at 0000 UTC 6 Aug

Nonlinear evolutions

Matsa K), zonal and meridional wind (unit: m/s) on level

FSV

CNOP

dry energy, 24-h

at 0000 UTC 5 Aug

CNOP

FSV

at 0000 UTC 6 Aug

Nonlinear evolutions

Case D (Matsa) K), zonal and meridional wind (unit: m/s) on level

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

- Sensitivity experiments K), zonal and meridional wind (unit: m/s) on level

Define:

Where is the projection operator, is a constant

less than one.

Benefits obtained from the reductions of CNOP or FSV

are evaluated by:

- Benefits obtained from the reductions of CNOP or FSV K), zonal and meridional wind (unit: m/s) on level

Case C Mindulle

Case D Matsa

- Conclusions K), zonal and meridional wind (unit: m/s) on level

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

- 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 (THC) to finite amplitude initial perturbations and decadal variability

The day after tomorrow? (THC) to finite amplitude initial perturbations and decadal variability

Floods & Impacts to New York (THC) to finite amplitude initial perturbations and decadal variability

Stommel box Model (THC) to finite amplitude initial perturbations and decadal variability

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 (THC) to finite amplitude initial perturbations and decadal variability

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 (THC) to finite amplitude initial perturbations and decadal variability

(2a)

(2b)

: the diffusion coefficient

Steady state (THC) to finite amplitude initial perturbations and decadal variability

Thermally-driven, TH

Salinity-driven, SA

Perturbation

Norm

The effect of WDOG on the existence of multi-equilibrium (THC) to finite amplitude initial perturbations and decadal variability

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.

when ,we have (THC) to finite amplitude initial perturbations and decadal variability

,

Fig.1 shows ,hence

(numerical result)

nonlinear stability analysis (THC) to finite amplitude initial perturbations and decadal variability

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.

Fig.2 a, b (THC) to finite amplitude initial perturbations and decadal variability

WDOG stabilizes the TH-state

Fig.2 c, d

WDOG destabilizes the SA-state

Understanding nonlinear stable regime (THC) to finite amplitude initial perturbations and decadal variability

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

Figure 3. (THC) to finite amplitude initial perturbations and decadal variability 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.

Why WDOG stabilizes (destabilizes) TH-state (SA-state) ? (THC) to finite amplitude initial perturbations and decadal variability

Recall (numerical)

We can prove that theoretically

Conclusion (THC) to finite amplitude initial perturbations and decadal variability

There exists a physical mechanism,

WDOG stabilizes (destabilizes)

TH-state (SA-state).

Thanks！ (THC) to finite amplitude initial perturbations and decadal variability

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