Transcript ppt file

Viewing Weather and Climate Extremes in a Probabilistic Framework
Prashant. D. Sardeshmukh @ noaa.gov
Climate Diagnostics Center, CIRES, University of Colorado
and Physical Sciences Division/ESRL/NOAA
AMS Annual Meeting January 2009 Phoenix
Outline
1.
To understand the statistics of extreme anomalies and how they are altered in various situations, we need to
understand the dynamics behind the associated probability density functions (PDFs).
2.
We need to understand the PDF changes associated not only with shifts of the mean, but also with changes of
Variance, and perhaps also with changes of higher moments such Skewness and Kurtosis.
3.
If the PDFs are Gaussian, then one need be concerned only with changes of the mean and variance. Many largescale and time-averaged variables in the climate system have roughly Gaussian PDFs. Their statistics and
evolution are often surprisingly well captured even by low-dimensional linear models perturbed by Gaussian
stochastic noise whose statistics do not depend on the system state (additive noise). Such models can be highly
competitive with GCMs. However, they cannot represent non-Gaussian PDFs.
4.
Recently, it has been shown that if one allows the stochastic noise in such linear models to have both stateindependent (additive) and linearly state-dependent (linear multiplicative) parts, then one can generate nonGaussian PDFs with very realistic higher moments and also Power-Law tails. Such extended linear models
thus provide a dynamical basis for understanding how even non-Gaussian PDFs can change in various
situations.
Even relatively minor changes of mean and variance can
have large implications for the probabilities of extreme values
Note that in the original PDF, the probabilities of both extreme positive and extreme negative
values (defined here as having magnitudes of more than 1 sigma) are 16 %
A change of
variance has a
large effect on
the probability
of extreme
values
In some cases it
can even
completely
offset the effect
of a shifted
mean
From Sardeshmukh,
Compo and Penland
(J. Climate 2000)
See also Katz and Brown (Climatic Change, 1992)
Estimating shifts of the 500 mb height PDFs in winter (JFM) due to
anomalous SSTs during the 1987 El Nino and 1989 La Nina events
Based on Observational
El Nino and La Nina
Composites
The SST-forced height
Anomalies are contoured
Every 10 meters.
Positive values are
indicated by red and
negative values
by blue coloring.
Note! The lower La Nina
panel has the sign
flipped ! This makes for
an easier comparison with
the upper El Nino panel.
From Sardeshmukh,
Compo and Penland
(J.Climate 2000)
Based on 180 winter
simulations using the NCEP
MRF9 atmospheric GCM
For the 1987
El Nino
For the 1989
La Nina
(sign flipped)
Expected ratio of Subseasonal 500 mb height variance (2 to 90 day periods)
in La Nina and El Nino winters
Based on observations in
11 La Nina and 11 El Nino
winters in 1948-2000
This is an estimate of what
to expect in general
Based on 60 simulations
each of the 1989 La Nina
and 1987 El Nino winters
using the NCEP MRF9
atmospheric GCM
This is an estimate of what
to expect in specific
instances
From Compo, Sardeshmukh, and Penland
(J. Climate 2001)
Estimating expected changes of variance of 500 mb heights in
winter (JFM) due to anomalous SSTs during an El Nino event
Changes expected in
general from observations
of 11 El Nino and 11
“Neutral” winters
Changes expected in a
specific instance such as the
1987 El Nino from NCEP
GCM simulations
The basic result
here is that during
El Nino . . .
Storm-tracks
are shifted
south over the
north Pacific and
north America
And
Weekly variability
associated with
blocking activity
is decreased over
the north Pacific
From Compo, Sardeshmukh,
and Penland (J. Climate 2001)
The predictability of “storm tracks” is important for the
predictability of seasonal mean precipitation
Changes of both the
seasonal mean vertical
velocity and of its
variability on synoptic
time scales
determine the changes
of seasonal mean
precipitation.
The figure shows the SST-forced
signal to noise ratios of the three
quantities in 60-member
ensembles of NCEP atmospheric
GCM runs with El Nino
(JFM 1987) SST forcing.
C.I. = 0.2
Knowledge of the first two statistical moments is sufficient to determine
the probabilities of extreme values if the PDFs are Gaussian, which is true if
the system dynamics are effectively Linear and Stochastically Forced
dx
dt
x
 A x  fext  B 
=

=
fext(t) =
A(t) =
B(t) =
N-component anomaly state vector
M-component gaussian noise vector
N-component external forcing vector
N x N matrix
N x M matrix
Supporting Evidence for the Linear Stochastically Forced (LSF) Approximation
- Linearity of coupled GCM responses to radiative forcings
- Linearity of atmospheric GCM responses to tropical SST forcing
- Linear dynamics of observed seasonal tropical SST anomalies
- Competitiveness of linear seasonal forecast models with global coupled models
- Linear dynamics of observed weekly-averaged circulation anomalies
- Competitiveness of Week 2 and Week 3 linear forecast models with NWP models
- Ability to represent observed second-order synoptic-eddy statistics
Observed and Simulated Spectra of Tropical SST Variability
Spectra of the projection of tropical SST
anomaly fields on the 1st EOF of observed
monthly SST variability in 1950-1999.
Observations (Purple)
IPCC AR4 coupled GCMs
(20th-century (20c3m) runs)
(thin black, yellow, blue, and green)
A linear inverse model (LIM) constructed
from 1-week lag covariances of weeklyaveraged tropical data in 1982-2005
(Thick Blue)
Gray Shading :
95% confidence interval from the LIM,
based on 100 model runs with different
realizations of the stochastic forcing.
From Newman, Sardeshmukh and Penland (J. Climate 2009)
Seasonal Predictions of Ocean Temperatures in the Eastern Tropical Pacific :
Comparison of linear empirical and nonlinear GCM forecast skill
(Saha et al, J. Climate 2006)
Simple linear
empirical
models are
apparently
just as good
at predicting
ENSO
as
“state of
the art”
coupled
GCMs
Decay of lag-covariances of weekly anomalies is consistent with linear dynamics
Is
C( )  e M C(0)
? Yes
M is first estimated using the
observed C(  5 days) and C(0)
in this equation, and then used
to "predict" C(  21 days)
The components of the anomaly
state vector x include the 7-day
running mean PCs of 250 and 750 mb
streamfunction, SLP, tropical diabatic
heating and stratospheric height anomalies.
From Newman and Sardeshmukh
(J. Climate 2008)
dx
dt
An attractive feature of
the LSF Approximation
 A x  fext  B 
Equations for the first two moments
(Applicable to both Marginal and Conditional Moments)
<x > = ensemble mean anomaly
C = covariance of departures from ensemble mean
d
 x   A  x   fext
dt
d
C
 A C  C AT  B BT
dt
If A(t), B(t) , and fext(t) are constant, then
 x    A 1 fext
First two Marginal moments
dC
dt
First two Conditional moments
Ensemble mean forecast
Ensemble spread
 0 = A C  C AT  B BT
x̂ '(t)  < x '(t) | x '(0) 
 e At x '(0)
Ĉ(t)  < ( x̂ ' x ') ( x̂ ' x ')T > = C  e At Ce A
If x is Gaussian, then these moment equations COMPLETELY
characterize system variability and predictability
T
t
But . . daily atmospheric circulation statistics are not Gaussian . .
Observed Skew S and (excess) Kurtosis K of daily 300 mb Vorticity (DJF)
From Sardeshmukh and Sura (J. Climate 2008)
Daily Sea Surface Temperature statistics are also not Gaussian . . .
Observed Skew S and (excess) Kurtosis K of daily SSTs (DJF)
Skew
Kurtosis
From Sura and Sardeshmukh ( J. Climate 2008 )
Modified LSF Dynamics
in which the amplitude of the stochastic noise depends linearly on the system state
Model 1 :
Model 2 :
Model 3 :


 For simplicity consider a scalar  here

dx
 Ax  fext  B  (Ex)

dt
 A(t), B(t), E(t) are matrices; g(t), f (t),  are vectors
ext

dx
1
 Ax  fext  B  (Ex  g)  Eg 
dt
2

dx
 Ax  fext  B
dt
Moment Equations :
d
1
 x   M  x   fext
where M  (A  E 2 )
dt
2
d
C
 M C  C M T  B BT  E { C   x  x  T } E T  g g T
dt
A simple view of how additive and linear multiplicative noise can
generate skewed PDFs even in a deterministically linear system
Additive noise only
Gaussian
No skew
Additive and uncorrelated Additive and correlated
Multiplicative noise
Multiplicative noise
Symmetric non-Gaussian Asymmetric non-Gaussian
A 1-D system with Correlated Additive and Multiplicative (“CAM”) noise
Stochastic Differential Equation :
dx
1
 Ax  (Ex  g)  B  Eg
dt
2
Fokker-Planck Equation :
Mxp 
x
Moments :

1 d
[( E 2 x 2  2Egx  g 2  B2 ) p ]
2 dx
0

 n  1
 n  1 2 
n1
2
2
n 2


 xn    
2Eg

x


(g

B
)

x

/
M


 E 
 
 2  
2


A simple relationship between Skew and Kurtosis :
Remembering that Skew S 
 x3 

3
and Kurtosis K 
 x4 

4
 3 , we have
 2
 M  (1 / 2)E 2

3  M  E2
K 
S

3

1
 M  (3 / 2)E 2

2  M  (3 / 2)E 2 



3 2
S
2
Observed Skew S and (excess) Kurtosis K of daily 300 mb Vorticity (DJF)
Note the quadratic relationship between K and S : K > 3/2 S2
Observed Skew S and (excess) Kurtosis K of daily SSTs (DJF)
Skew
Note the quadratic relationship
between K and S : K > 3/2 S2
From Sura and Sardeshmukh (J. Climate2008 )
Kurtosis
A linear 1-D system with non-Gaussian statistics, forced by “CAM” noise
dx
1
 Ax  b1  (Ex  g)2  Eg
SDE
dt
2
1
1
 2g
 Ex  g  
2
2  
 (Ex  g)  b 
p(x) 
exp  
arctan 
PDF
 

N

b
b


Such a system satisfies K > (3 / 2)S 2 and its PDF has power-law tails
M  A  0.5 E 2
  E2 / M
Both  0
Observed and Simulated pdfs in the North Pacific
(On a log-log plot, and with the negative half folded over into the positive half)
Observed
(NCEP Reanalysis)
500 mb
Height
300 mb
Vorticity
Simulated by a simple dry adiabatic
GCM with fixed forcing
Observed and Simulated pdfs in the North Pacific
(On a log-log plot, and with the negative half folded over into the positive half)
Observed
(NCEP Reanalysis)
500 mb
Height
300 mb
Vorticity
Simulated by a simple dry adiabatic
GCM with fixed forcing
The most general linear 1-D system with non-Gaussian statistics, forced by “Radical” noise
dx
 Ax 
dt

[(Em x  gm )2  cm x] m 
m

2

fext
SDE
m
1
 2
 E2 x    
1

2 2
2  
 E x  2  x  G 
p(x) 
exp   fext   arctan 
 
N 



 

PDF
Such a system satisfies K  (3 / 2)S 2 and its PDF also has power-law tails


    Em gm 
cm 

2
E 2   Em2 , G 2   gm2
m
m
  E2G2   2
and
Note that

x  
 x   xmax 
fext
, and
M
xmax  
fext  
M  E2
 fext  
M  E2
So a change in the forcing fext does not result in a simple shift of the PDF !
M  A  0.5 E 2
  E2 / M
Both  0
Summary
1.
To understand the statistics of extreme anomalies and how they are altered in various situations, we need to
understand the dynamics behind the associated probability density functions (PDFs).
2.
We need to understand the PDF changes associated not only with shifts of the mean, but also with changes of
Variance, and perhaps also with changes of higher moments such Skewness and Kurtosis.
3.
If the PDFs are Gaussian, then one need be concerned only with changes of the mean and variance. Many largescale and time-averaged variables in the climate system have roughly Gaussian PDFs. Their statistics and
evolution are often surprisingly well captured even by low-dimensional linear models perturbed by Gaussian
stochastic noise, whose statistics do not depend on the system state (additive noise). Such models can be highly
competitive with GCMs. However, they cannot represent non-Gaussian PDFs.
4.
Recently, it has been shown that if one allows the stochastic noise in such linear models to have both stateindependent (additive) and linearly state-dependent (linear multiplicative) parts, then one can generate nonGaussian PDFs with very realistic higher moments and also Power-Law tails. Such extended linear models
thus provide a dynamical basis for understanding how even non-Gaussian PDFs can change in various
situations.