Some New Progresses in the
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)
[email protected]
http://web.lasg.ac.cn/staff/mumu/
Outline
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
1. Conditional Nonlinear Optimal Perturbation
 w
  F ( w, x, t )  0
 t
 w |t  0  w0
(1)
w( x, T )  M T ( w0 )
M T : (nonlinear) propagator of (1)
Let U ( x, t ), U ( x, t )  u( x, t ) be the solutions to (1)
U |t 0  U0 , (U ( x, t )  u( x, t )) |t 0  U0  u0
M T (U0 )  U ( x, t ), M T (U0  u0 )  U ( x, t )  u( x, t )
 


J (u0 ) || M T (U0  u0 )  M T (U0 ) ||


J (u0 )  max J (u0 )
||u0 ||

u0

|| u0 || 
Conditional Nonlinear Optimal
Perturbation(CNOP)
Constraint condition
Physical meaning of CNOP
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
  (w) F

|wU ( w, x, t )w  0

w
 t
w |t 0  w0
w( x, T )  MT (w0 )
M T : (linear) propagator of (1)
w
* is LSV if and only if ,
0
J (w )  max J (w0 ),
*
0
where
w0
|| M T (w0 ) ||
J (w0 ) 
|| w0 ||
(2)
Reference
[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
• 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
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
• 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 design
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：
2
2
1
cp 2
ps
1
2
2
   [u  v  T  RaTr ( ) ]d ds
D D 0
Tr
pr
where,
Tr  270K
pr  1000hpa
c p  1005.7J  kg 1  K 1
Ra  287.04J  kg 1  K 1
The integration extends the full horizontal
domain D and the vertical direction  .
Nonlinear evolutions
a (CNOP)
b(CNOP)
Case A
Figure1.
The temperature
(shaded, unit:K) and
c (FSV)
d (FSV)
wind (vector, unit: m/s)
components of
CNOP(a,b), FSV (c,d)
and loc CNOP (e,f)
on level   0.45
at 0000 UTC 4 July
e (loc CNOP)
f (loc CNOP)
(a,c,e) and their
nonlinear evolutions
at 0000 UTC 5 July
(b,d,f).
Table 1.Case A: The maxima (minima) of temperature (unit: K), zonal
and meridional wind (unit: m/s) on level   0.45
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
a (CNOP)
c (FSV)
b (CNOP)
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
time
type
0000 UTC 5
Aug, 1996
0000 UTC 6
Aug, 1996
Figure 4
Same as Fig.2,
but for case B
• 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
and
  729J/kg for case C,
  900J/kg for case D.
The optimization time intervals for these two cases
are still 24 hours.
• Experimental design
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
dynamic energy
2


1
D 0
(u2  v 2 )d ds
total dry energy
2
cp
2
ps
   [u  v  T  RaTr ( ) ]d ds
D 0
Tr
pr
where,
1
2
2
Tr  270K
2
c p  1005.7J  kg 1  K 1
1
1
R

287.04J

kg

K
pr  1000hpa
a
The integration extends the full horizontal
domain D and the vertical direction
.
•Simulation of case C (Mindulle)
a
a: model domain
b: target area
b
Figure 6.
Simulation
track from
MM5 (red)
and the
observation
track (blue)
from CMA
•Simulation of case D (Matsa)
a
a: model domain
b: target area
a
bb
Figure 7.
Simulation
track from MM5
(red)
and the
observation
track (blue)
from CMA
Results
CNOP
Mindulle
FSV
dynamic energy, 24-h
  0.7   729
at 0000 UTC 28 Jun
FSV
CNOP
at 0000 UTC 29 Jun
Nonlinear evolutions
Mindulle
CNOP
dry energy, 24-h
  0.7   729
FSV
at 0000 UTC 28 Jun
CNOP
FSV
at 0000 UTC 29 Jun
Nonlinear evolutions
Case C (Mindulle)
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 nonlinear development (TE)
24h nonlinear development (KE)
90000
70000
80000
60000
70000
60000
40000
CNOP
FSV
30000
TE(J/kg)
KE(J/kg)
50000
50000
CNOP
FSV
40000
30000
20000
20000
10000
10000
0
0
0
3
6
9
12
time(h)
15
18
21
24
0
3
6
9
12
time(h)
15
18
21
24
Matsa
CNOP
dynamic energy, 24-h
  0.7   900
FSV
at 0000 UTC 5 Aug
CNOP
FSV
at 0000 UTC 6 Aug
Nonlinear evolutions
Matsa
CNOP
dry energy, 24-h
FSV
  0.7   900
at 0000 UTC 5 Aug
CNOP
FSV
at 0000 UTC 6 Aug
Nonlinear evolutions
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 nonlinear development (KE)
24h nonlinear development (TE)
30000
50000
45000
25000
40000
35000
CNOP
FSV
15000
10000
TE(J/kg)
KE(J/kg)
20000
30000
CNOP
FSV
25000
20000
15000
10000
5000
5000
0
0
0
3
6
9
12
time(h)
15
18
21
24
0
3
6
9
12
time(h)
15
18
21
24
• Sensitivity experiments
Define:
J1 (δX0 )  PM (X0 + δX0 ) - PM (X0 )
2
J 2 (δX0 )  PM (X0 + cδX0 ) - PM (X0 )
2
Where P is the projection operator,
less than one.
c
is a constant
Benefits obtained from the reductions of CNOP or FSV
are evaluated by:
J1 (δX0 )  J 2 (δX0 )
J1 (δX0 )
• Benefits obtained from the reductions of CNOP or FSV
Case C Mindulle
c
0.25
0.50
0.75
KE
CNOP
FSV
TE
CNOP
FSV
91.6%
62.2%
26.6%
84.8%
53.8%
24.1%
47.8%
27.3%
15.3%
25.1%
7.5%
-5.3%
Case D Matsa
c
0.25
0.50
0.75
KE
CNOP
FSV
TE
CNOP
FSV
91.3%
69.5%
42.3%
86.4%
69.9%
49.4%
63.3%
38.3%
17.8%
46.5%
26.3%
15.3%
• 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
dT
 1  T 1   4  T  S
dt
dS
 2  S 3  4  T  S
dt
 4: the diffusion coefficient


(2a)
(2b)
Steady state
Thermally-driven, TH
Salinity-driven, SA
Perturbation
Norm
The effect of WDOG on the
existence of multi-equilibrium
Figure 1. The bifurcation diagram of box models for 1  3.0，3  0.6 as a plot
4
0.09 and 0.17.
 of versus  2 . The curves from left to right : 0.0, 0.01, 0.05,
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 4  0 ,we have
T  T1  T2  0 , S  S1  S 2  0
Fig.1 shows  2  1 ,hence
T  S  1   2  0
T   S
(numerical result)
nonlinear stability analysis
TH  state
TH  state
SA  state
SA  state
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1  3.0, 3  0.6. (a) (b) : the TH-state with 2  1.84, and
(c) (d) : SA-state with 2  1.83. Solid (dashed) curve is for
L (S) model.
Fig.2 a, b
WDOG stabilizes the TH-state
Fig.2 c, d
WDOG destabilizes the SA-state
Understanding nonlinear stable regime
The smallest magnitude of a finite
perturbation which induces a
transition from TH state to SA state
and vise versa.
Figure 3. The critical value  c versus control
parameter  2 for 1  3.0 , 3  0.6 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) ?
Recall T  S (numerical)
We can prove that theoretically
 2  1   S  T   0
  2  1
Conclusion
There exists a physical mechanism,
WDOG  S  T

WDOG stabilizes (destabilizes)
TH-state (SA-state).