CONSISTENT AND LUMPED MASS MATRICES IN DYNAMICS AND THEIR IMPACT ON FINITE ELEMENT ANALYSIS RESULTS
Gaussian statistics are consistent with linear dynamics.
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![Page 1: Gaussian statistics are consistent with linear dynamics.](https://reader035.fdocuments.net/reader035/viewer/2022062422/56812f08550346895d94a50a/html5/thumbnails/1.jpg)
Reconciling Non-Gaussian Climate Statistics with Linear Dynamics
Prashant. Sardeshmukh and Philip Sura
Climate Diagnostics Center/CIRES/CU, and Physical Sciences Division/ESRL/NOAA
Maurice Blackmon Symposium 29 October 2007
Gaussian statistics are consistent with linear dynamics.
Are non-Gaussian statistics necessarily inconsistent with linear dynamics ?
In particular, are skewed pdfs, implying different behavior of positive and negative anomalies, inconsistent with linear dynamics ?
Thanks also to : Barsugli, Compo, Newman, Penland , and Shin
For additional information, see: http://www.cdc.noaa.gov/people/prashant.d.sardeshmukh/publications.html
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dx
dt A x fext B
Supporting EvidenceSupporting Evidence
- 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
The Linear Stochastically Forced (LSF) Approximation
x = N-component anomaly state vector = M-component gaussian noise vector fext(t) = N-component external forcing vector A(t) = N x N matrix B(t) = N x M matrix
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Decay of lag-covariances of weekly anomalies is consistent with linear dynamics
Is C( ) eM C(0) ?
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
(2007)
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dx
dt A x fext B
d
dt x A x fext
d
dt C A C C AT B BT
Equations for the first two momentsEquations for the first two moments
(Applicable to both Marginal and Conditional Moments)
<x > = ensemble mean anomaly
C = covariance of departures from ensemble mean x A 1 fext
dC
dt 0 = A C C AT B BT
x '(t) < x '(t) | x '(0) eAt x '(0)
C(t) < (x ' x ') (x ' x ')T > = C eAtCeAT t
If A(t), B(t) , and fIf A(t), B(t) , and fextext(t)(t) are constant, then are constant, then
First two Marginal moments
First two Conditional moments Ensemble mean forecast Ensemble spread
If x is Gaussian, then these moment equations COMPLETELY characterize system variability and predictability
An attractive feature of the LSF Approximation
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But . . . atmospheric circulation statistics are not Gaussian . . .
Observed Skew S and (excess) Kurtosis K of daily 300 mb Vorticity (DJF)
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Model 1 : dx
dtAx fext B
Model 2 : dx
dtAx fext B (Ex)
For simplicity consider a scalar here
A(t), B(t), E(t) are matrices; fext (t), are vectors
Moment Equations :
d
dt x M x fext where M (A
1
2E2 )
d
dt C M C C MT B BT E C ET
Modified LSF Dynamics
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Departures from Gaussianity
Sura et al (2005) showed that the skew of observed 7-day running streamfunction is similar to
that of the dominant barotropic Rossby eigenmode of the observed DJF mean flow, if the mode
is stochastically damped, and also forced with a fixed forcing. The model is of the form
dx
dtAx fext (Ex) x M 1 fext M (A 0.5E2 )
This is encouraging, but there is a difficulty. The model only produces skew if fext is non-zero, but in that
case, x cannot have zero mean. Such a model cannot therefore explain the skew of centered anomalies.
Joint PDF of real and imaginary parts of least damped barotropic Mode with stochastic damping
Joint PDF of first two EOFsof 750 mb streamfunctionDJF 1950-2002
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Model 1 : dx
dtAx fext B
Model 2 : dx
dtAx fext B (Ex)
Model 3 : dx
dtAx fext B (Ex g)
1
2Eg
For simplicity consider a scalar here
A(t), B(t), E(t) are matrices; g(t), fext (t), are vectors
Moment Equations :
d
dt x M x fext where M (A
1
2E2 )
d
dt C M C C MT B BT E C ET g gT
Modified LSF Dynamics
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A simple rationale for Correlated Additive and Multiplicative (CAM) noise
(xy)' x y x y x y x y
x y (x x ) y x y
In a quadratically nonlinear system with “slow” and “fast” components x and y, the anomalous nonlinear tendency has terms of the form :
This is the CAM noise term And this is its mean
“Noise-Induced Drift”
Note that it is the STOCHASTICITY of y’ that enables the mean drift to be parameterized in terms of the noise amplitude parameters
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Moments of a 1-D system with CONSTANT coefficients
dx
dtAx fext B (Ex g)
1
2Eg
xn
n 1
2
M n 1
2
E2
2Eg xn 1 (g2 B2 ) xn 2
Equations for high-order moments involve lower-order moments !
x 0 x2 B2 g2
2M E2
Mxp 1
2
d
dx [ (g2 B2 E2x2 2Egx) p ]
Fokker-Planck Equation
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A simple relationship between Kurtosis and Skew in 1-d systems
x3 2Eg
M E2 x2
x4
3
2
M 3
2
E2
2Eg x3 (g2 B2 ) x2
Remembering that Skew S x3 3
and Kurtosis K x4 4
3
we have
K 3
2
M E2
M (3 / 2)E2
S2 3
M (1 / 2)E2
M (3 / 2)E2 1
3
2 S2
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Note the quadratic relationship between K and S : K > 3/2 S2
Observed Skew S and (excess) Kurtosis K of daily 300 mb Vorticity (DJF)
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Understanding the patterns of Skewness and Kurtosis
Are diabatic or adiabatic stochastic transients more important ?
To clarify this, we examined the circulation statistics in a 1200winter simulation generated with a T42 5-level dry adiabatic GCM (“PUMA”) with the observed time-mean diabatic forcing specified asa fixed forcing.
There is thus NO transient diabatic forcing in these runs.
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1-point anomaly correlations of synoptic (2 to 6 day period) variations
with respect to base points in the Pacific and Atlantic sectors
Simulated Observed
•
• •
•
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Observed (NCEP, Top) and Simulated (PUMA, Bottom) S and K of 300 mb Vorticity
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Observed (Left) and Simulated (Right) pdfs in the North Pacific (On a log-log plot, and with the negative half folded over into the positive half)
500 mb Height
300 mbVorticity
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Observed (Left) and Simulated (Right) pdfs in the North Pacific (On a log-log plot, and with the negative half folded over into the positive half)
500 mb Height
300 mbVorticity
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Summary
1. Strong evidence for “coarse-grained” linear dynamics is provided by ( a ) the observed decay of correlations with lag( b ) the success of linear forecast models, and( c ) the approximately linear system response to external forcing.
2. The simplest dynamical model with the above characteristics is a linear model perturbed by additive Gaussian stochastic noise. Such a model, however, cannot generate
non-Gaussian statistics. 3. A linear model with a mix of multiplicative and additive noises can generate non-Gaussian
statistics; but not odd moments (such as skew) without external forcing; and therefore are not viable models of anomalies with zero mean.
4. Linear models with correlated multiplicative and additive noises can generate both oddand even moments, and can also explain the remarkable quadratic K-S relationship between Kurtosis and Skew, as well as Power-Law tails of the pdfs.
5. There is strong evidence that the skew and kurtosis of upper troposheric vorticity are associated with fast nonlinear local adiabatic dynamics that can be approximated as stochastic dynamics with correlated additive and multiplicative noises..
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Summary Visual
Additive noise onlyGaussianNo skew
Additive and uncorrelatedMultiplicative noise
Symmetric non-Gaussian
Additive and correlatedMultiplicative noise
Asymmetric non-Gaussian
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dXi
dtLij X j Nijk X j Xk Fi
d Xi
dt[Lij (Nijk Nikj )Xk ] X j Nijk ( X j Xk X j Xk ) Fi
Let X x
and X
x
d xi
dt [Lij (Nijm Nimj )m ] x j Linear terms (= Aij x j )
[(Nijm Nimj ) x j {Lim (Nijm Nimj )x j }] m Correlated additive and multiplicative noise
(Nijm Nimj ) x j m Mean noise-induced drift
Nimn ( m n m n ) Other additive noise ( = Bikk )
Nijk ( x j xk x j xk ) Hard nonlinearity
fi External forcing
Neglecting the hard nonlinearity, and using the FPE to derive the noise-induced drift, we obtain
d xi
dt ; Aij x j + (Eijm x j Lim Eijmx j ) m
1
2Eijm (L jm E jkmxk ) + Bikk fi
; Aij x j + (Eijm x j Gim ) m 1
2EijmG jm + Bikk fi
where Eijm = (Nijm Nimj ), and Gim = Lim (Nijm Nimj )x j = Lim Eijmx j
Rationalizing linear anomaly dynamics
with correlated additive and linear multiplicative stochastic noise
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Scatter plots of Fifth Moments versus Skew in the PUMA run
300 mb Vorticity 500 mb Heights
The 1-d model predicts 5 x5 5
10s 3S3 for S 0
10s 3S3 for S 0 !!