Mu Mu F.F.Zhou,H.L.Wang and X.G.Wu
description
Transcript of Mu Mu F.F.Zhou,H.L.Wang and X.G.Wu
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)[email protected]
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