Transcript An inhomogeneous statistical dynamical turbulence closure and its application to

An inhomogeneous statistical dynamical
turbulence closure and its application to
problems in atmospheric dynamics
Terry O’Kane & Jorgen Frederiksen
Centre for Australian Weather & Climate
Research
CSIRO Marine & Atmospheric Research
Hobart Australia
Motivation for a tractable inhomogeneous closure
Subgrid-scale parameterization
Current generation coupled climate models in particular typically have non
eddy resolving oceans with eddy forces in the momentum equation
typically formulated as a diffusive process based on the assumption that
eddies should damp the flow toward a state of relative rest. However
statistical mechanics tells us that eddies can drive mean flows!
Ensemble prediction/Data assimilation: non-Gaussian terms, memory, regime
transitions
The skill of numerical weather forecasts is determined by the instability
properties of the atmospheric flow, analysis errors, and model deficiencies.
In ensemble prediction /data assimilation there is the severe restriction that
only very small ensembles can be employed. The role of non-Gaussian
effects in predicting regime transitions is poorly understood as is the effect
of sampling error on small scale error growth. GCM’s have slight
deviations from Gaussian statistics but capture mid-latitude regime
transitions well .
•Atmospheric spectra nearly 2-D
• large scale Rossby waves
• large scale flow instabilities
• Inhomogeneous large scales
• small scale turbulent eddies
• homogeneous small scales
•complex (emergence/coherent
structures/instabilities)
Atmospheric Regime transitions
Northern Hemisphere blocking, Gulf of Alaska 6 Nov 1979
Transition from strong zonal to “wavy” flow and the emergence of a
coherent high low blocking dipolar structure. Rapidly growing large
scale flow instabilities and a loss of predictability i.e. rapid growth of
the error field.
Obstacles to an accurate tractable inhomogeneous
non-Markovian statistical closure
•
•
Generalize two-point two-time homogeneous
closure theory to general 2-D flow over topography.
Tractable representations of the two- and threepoint cumulants. Generalize special case of Kraichnan (1964): Boussinesq
convection: diagonalizing closure for a mean horizontally averaged temperature field with
zero fluctuations to general 2-D flow over topography.
•
•
•
Incorporate large scale Rossby waves (β-plane).
Long integrations of time-history integrals:
atmospheric regime transitions require timehistory information over many days
Vertex renormalization problem→ Regularization
Ensemble averaged DNS code for comparison
2-D barotropic vorticity on a generalized β-plane
Small scales
Large scales
Invariants

  J (  Uy,   h  y  k02Uy) ˆ 2  f 0
t
U 1

t
S

S h x dS   (U  U )
1 2 11
2
E U 
(


)
dS

2
2SS
1
 2 11
2
Q  (k0U  ) 
(


h
)
dS

2
k0
2SS
 k   k  ˆk
Mean and transient evolution equations


 k    (k  p  q) K (k , p, q)   p  q  C p ,  q (t , t )
t
p q

   (k  p  q) A(k , p, q)   p h q
p
q

 ˆ
 k    (k  p  q) K (k , p, q)   p ˆq  ˆ p  q  ˆ pˆq  C p ,  q (t , t )
t
p q
   (k  p  q) A(k , p, q)ˆ p hq
p
k   0
q
Ck (t , t )  ˆk (t )ˆk (t )
C p ,  q (t , t )  ˆ p (t )ˆ q (t )
We have written spectral BVE with differential rotation describing small scales and large scales using
the same compact form as for f-plane through specification and extension of the interaction
coefficients from  to 0. Mathematically elegant and avoids massive re-writing of codes.

The two-time cumulant equation

2
I
H
  0 k k Ck (t , t ' )  N k (t , t ' )  N k (t , t ' )
 t

N kI (t , t ' )    k  p  q A k , p, q C p ,  q (t , t ' )h q

p
q


 

   k  p  q K k , p, q

p

q
   p (t ) C q ,  k (t , t ' )  C p ,  k (t , t ' )   q (t )

 


N kH (t , t ' )    k  p  q K k , p, q ˆ q (t )ˆ p (t )ˆ k (t ' )
p
q
Renormalized closure theory
The Closure Problem:

t
~ 
  
~   
t
   
~   
t
   
~   
t
Functional Forms

(t , t ' )C , C

,, h 
Ck ,  l (t , t ' )  CkQDIA
,  l (t , t ' ) C k , C l , Cl  k , Rk ,  ,  k ,  , hk
Rk ,  l (t , t ' )  RkQDIA
, l
k
l
, Cl  k , Rk ,  ,  k
ˆk (t )ˆl (t )ˆ(l  k ) (t )  ˆk (t )ˆl (t )ˆ(l  k ) (t )
DIA
k
C , C
k
l
Response functions
Rk (t , t ' )  Rk , k (t , t ' )
ˆ (t )


k
ˆ (t , t ' ) 
Rk ,l (t , t ' )  R
k ,l
fˆl 0 (t ' )
, Cl  k , Rk , 
Isotropic turbulence → DIA Closure
Nonlinear noise
t'
ˆl (t )ˆl  k  (t )ˆk (t ' )  2  ds K k ,l , l  k Cl (t , s)C(l  k ) (t , s) Rk (t ' , s )
t0
t
 2  ds K  l , l  k , k Rl (t , s)C(l  k ) (t , s )Ck (t ' , s )
t0
t
 2  ds K l  k ,l , k R(l  k ) (t , s )Cl (t , s )Ck (t ' , s )
}
Nonlinear damping
t0
Kraichnan, J. Fluid Mech.,1959
We also require modifying the DIA to include vertex correction and to allow
for non-Gaussian initial conditions.
T.J. O’Kane & J.S. Frederiksen, J. Fluid Mech. (2004)
Two-time two-point cumulant equation with updates

t
Ck ,  l (t , t ' )   dsRk (t , s )Cl ( s, t ' ) A(k ,l , l  k )h( k l )  2 K (k ,l , l  k )  ( k l ) ( s )
t0
t'

  dsRl (t ' , s )Ck (t , s ) A(l , k , l  k )h( k l )  2 K (l , k , l  k )  ( k l ) ( s )


t0
~
 Rk (t ,t 0 ) Rl (t ' , t0 ) K k( 2, ) l (t0,t0 )
~ ( 2)
K
Here k , l (t0,t0 ) is the contribution to the off-diagonal covariance matrix at
initial time t0 and in reduced notation
~ 2 
~ 2 
QDIA
K  p ,  k T , T   C p ,  k T , T   K  p ,  k t0 , t0 R p T , t0 R k T , t0 
In a similar manner we derive the two-time two-point response function
t

Rk ,l (t , t ' )   dsRk (t , s) Rl ( s, t ' ) A(k ,l , l  k )h( k l )  2 K (k ,l , l  k )  ( k l ) ( s)
t'

DIA Closure + 3-point cumulant update
ˆl (t )ˆl  k  (t )ˆk (t ' )
t'
 2  ds K k ,l , l  k Cl (t , s )C( l  k ) (t , s ) Rk (t ' , s )
t0
t
 2  ds K  l , l  k , k Rl (t , s )C( l  k ) (t , s )Ck (t ' , s )
t0
t
 2  ds K l  k ,l , k R( l  k ) (t , s )Cl (t , s )Ck (t ' , s )
t0
~
 Rl (t , t0 ) R( l  k ) (t , t0 ) Rk (t ' , t0 ) K ( 3l ,)( l  k ), k (t0 , t0 , t0 )
QDIA
~ 3 
ˆ
ˆ
ˆ
K q , p ,  k T , T , T    q (T ),   p (T ),  k (T )
~
 K 3q ,  p ,  k t0 , t0 , t0 Rq T , t0 R p T , t0 Rk T , t0 
Here
~
K (3l ,)(l k ),k (t0 , t0 , t0 ) allows for non-Gaussian initial conditions.
Thus the QDIA equations including off-diagonal and non-Gaussian initial conditions and Eqs. for
the single time cumulants and response functions may be used to periodically truncate time history
integrals to obtain a computationally efficient closure
Generalization of operator formalism of U. Deker Phys. Rev. A, 1979; H. Rose, Physica D, 1985
Kraichnan 1959, Wyld 1961, Herring 1965, McComb 1974 (homogeneous & isotropic hydrodynamic)
Lee 1965 (homogeneous magneto-hydrodynamic)
Feynman (1951); Schwinger (1951) (quantum electrodynamics)
Crocce & Scoccimarro 2006 (homogeneous cosmological)
Frederiksen & O’Kane 2004, 2005, 2008 (inhomogeneous hydrodynamic)
Diagnostics Herring et al 1974
(t )   k 2 N k t , t 
Palinstrophy production
 (t )  ˆk 2Ck t , t 
Enstrophy dissipation
k
k
Eˆ (t )  1 / 2 Ck t , t  / k 2
Transient energy
k
Fˆ (t )  1 / 2 Ck t , t 
Transient enstrophy
Pˆ (t )  1 / 2 Ck t , t k 2
Transient palinstrophy
k
k
1
3
RL (t )  Eˆ /(ˆ )
Large-scale Reynolds number
1
2
S K (t )  2 /( Pˆ Fˆ )
Skewness
Statistical closures at Finite Resolution
• For infinite resolution and moderate Reynolds
numbers the DIA (isotropic turbulence) under predicts
the inertial range kinetic energy. Other variants
(McComb LET, Herring SCFT ) based on differing
applications of the FDT
i.e. Ck(t,t’) (t-t’) = Rk(t,t’) Ck(t,t) (time ordering).
• For finite resolution (>C30) and moderate RL (>200)
numbers all homogeneous two-point non-Markovian
closures underestimate the evolved small-scale KE
and dramatically underestimate the skewness.
Decaying 2D isotropic turbulence
Frederiksen and Davies 2000, GAFD
Comparison of DNS with DIA,
LET and SCFT closures
Initial spectrum
2
Ck (0,0)  1.8  10 1 k 5 exp(  k ),
3
ˆ   0  0.0025,
Large-scale
Reynolds number
RL (0)  307, RLDNS (0.8)  280,
1  k  48
In case with large amplitude small scale topography (fixed random
phase) where QDIA skewness is underestimated by 25% spectra
are in close agreement. Conclusion is that topography acts to
localize transfers.
In case with small amplitude small scale topography QDIA
skewness is underestimated by >50% . Regularization is required
for spectra (and skewness) to be in close agreement.
A regularized approach to vertex renormalization
• The regularization procedure consists of zeroing the
interaction coefficients K(k,p,q) if p < k/α1 or q < k/α1 and
A(k,p,q) if p < k/α2 or q < k/α2 in the two-time cumulant
and response function equations of the QDIA equations i.e
K (k , p, q)  ( p  k / 1 )(q  k / 1 ) K (k , p, q)
A(k , p, q)  ( p  k /  2 )(q  k /  2 ) A(k , p, q)
where Θ is the heavyside step function.
• The interaction coefficients are unchanged in the single
time cumulant equations.
Evolved kinetic energy spectra
CUQDIA transients
Mean CUQDIA
RCUQDIA transients
DNS Transients
Mean: 1000 DNS, RCUQDIA, 100 DNS
J.S. Frederiksen & T.J. O’Kane, J. Fluid Mech. (2005)
Day 10 DNS (a) and QDIA (b) mean eddy streamfunction when 7.5m/s eastward U impinges on
conical mountain (c)
1800 DNS
Height
=2500m
QDIA
Correlation
= 0.99
Kinetic energy spectra
Eastward 7.5 m/s flow plus turbulence over topography
Initial mean
Day 10 mean
DNS and QDIA
indistinguishable
Initial transients
Day 10 transients
DNS and QDIA
indistinguishable
DNS sampling problem for strong transients
Day 0 Transients
Mean …..
QDIA
mean .....
Day 2
800 DNS
mean ----
Day 10
1800 DNS
mean -------
Role of transients in evolution of mean
fields: Consider the contribution to the
mean vorticity tendency from the transient
eddies
Then the streamfunction tendency is
anticorrelated with the streamfunction
(positive nonlinear damping) and transient
perturbation structures that develop on a
given mean state act to weaken those basic
state eddies.
Branstator & Frederiksen (2003) showed in a
study of observed climatological mean
300hPa atmospheric streamfunction, that
streamfunction tendencies are anticorrelated
with the climatological zonally asymmetric
mean in each month of the year.
 
-  -2 J ˆ , ˆ   J 
Jk
QDIA
1
 2
k

if

1
k2
  k  p  q K k , p, q C
p
QDIA
 p,q
(t , t )
q
t
 t


ds

(
t
,
s
)

(
s
)

h
ds

(
t
,
s
)
 

k
k
k
k
 t0

t0

1 
~ ( 2)
QDIA





k

p

q
K
k
,
p
,
q
C
R
(
t
,
t
)
R
(
t
,
t
)
K
(
t
t
)


 p,q p
0
q
0
 p ,  q 0, 0 
k2  p q

 k (t , s)  k ( s)  hk  k (t , s )
~
and K ( 2p),  q (t0,t0 ) is small
Ensemble Prediction
•
Statistics of error prediction (pdf) ↔ statistical theory
of turbulence (higher moments).
• Epstein(1969) (3rd and higher moment discard): (statistical
dynamical prognostic equations based on a third- and higher-order cumulant-discard hypothesis
to enable the direct forecasting of mean and variance information)
•
Fleming (1971,1979; MC, QN & EDQN): (Monte-Carlo and
non-realizable closures)
•
•
Leith (1971, 1974 TFM): (error growth from isotropic initial conditions)
Herring etal(1973; DIA): (decaying 3D isotropic turbulence)
Generating initial perturbations
•
In EP independent initial disturbances are generated as fast growing disturbances with
structures and growth rates typical of the analysis errors.
•
Random isotropic initial perturbations grow more slowly and lead to underestimated error
variances.
•
Generate independently perturbed initial conditions such that the covariance of the ensemble
perturbations ≈ initial analysis error covariance at the time of the forecast.
•
Breeding method (Toth and Kalnay 1993,1997; Wei & Toth 2006).
•
Chaos is not random but generated by physical instabilities. Breeding is a simple method to
find the growth and shape of these instabilities
•
Analysis cycle acts as a nonlinear perturbation model on the evolution of the real atmosphere
(nudging) resulting in error growth associated with the evolving atmospheric state to develop
within the analysis cycle and dominate forecast error growth.
•
All random perturbations will assume the structure of the LLV given time thereby reducing the
spread to 1. BV’s are local, finite-time generalization of Lyapunov vectors
•
What are the relative contributions of non-Gaussian terms (three-point) and Inhomogeneity
(two-point cumulant terms) to the growth of the transient field? How much memory do we
need.
T.J. O’Kane & J.S. Frederiksen (2008), J. Atmos. Sci., 65, pp426--447
Comparison of bred vector ensemble averaged DNS
and RQDIA zonally asymmetric streamfunction in 5
day breeding / 5 day forecast experiment during
block formation and maturation.
•We are interested in evolution of transient error fields on
trajectories similar to the atmospheric 500-hPa field
between 26 October and 8 November 1979.
•For the mean fields to follow this trajectory we specify
time evolving source terms S(t)=κ(ζobs-ζ) where the obs are
the linearly interpolated daily observed 500-hPa
streamfunction fields at 1200 UTC.
•We generate a truth trajectory beginning on 26 October
1979 by running the barotropic model with a relaxation
term interpolating down to the required time step and use
an e-folding relaxation time of 2 days.
•The source term is calculated at each time step of the
unperturbed truth simulation, stored, and then applied to
both perturbed ensemble DNS runs and the mean field
equation of the closure. This ensures that the truth field
closely follows the observed trajectory, but that the
perturbations are not additionally damped.
Palinstrophy production measure
P M (t )  I K (t )  S K (t )
Inhomogeneity
t
t


K
2
I (t ) 
  k   dsPk (t , s ) R k (t , s )   ds k (t , s )C k (t , s )
1
t0

t0
2 k
1
2
Ck t , t k   Ck t , t 

2 k
 k

2
Skewness
t
t

0
S (t ) 
  k   ds S k (t , s )  Fk (t , s ) R k (t , s )   ds k (t , s )C k (t , s )
1
t0

t0
2 k
1
2
Ck t , t k   Ck t , t 

2 k
 k

K
2
Note: Small scale measure
2


Ensemble prediction and error growth studies
Isotropic 26/10/79
Breeding 1/11/79
Day 2-5: rescaling of error
variances
Day1: growth of Ck,-l
and organization of
error structures
Forecast 5/11/79
Day 5-7: Error fields
with LLV structures
amplify rapidly
Day 7 onwards reduced
growth as errors saturate
•PM slope indicates growth rate.
•Drop in PM corresponds to
growth of instability vectors at
the large scales.
•When error KE growing PM
largely determined by dynamics
of large scale flow instabilities.
•As errors saturate non-Gaussian
terms become of increasing
importance as is the case for
increasing resolution.
•For decaying homogeneous
turbulence PM = SK and saturates
at a nearly constant value.
Integral contributions to error growth.
How important are memory effects?
•QDIA: direct interactions only, inhomogeneous + non-Gaussian initial forecast perturbations
•RQDIA: direct + indirect interactions, inhomogeneous + non-Gaussian initial forecast
perturbations
•ZQDIA as for RQDIA but with homogeneous initial forecast perturbations, neglect information
from off-diagonal covariances and non-Gaussian terms at time of forecast
•CD & QN variants
ˆk (t )ˆl (t )ˆl k (t ' )  0
,
 ˆ
 k (t )ˆl (t )ˆl  k (t ' )  0
t
respectively.
Spread / Model error;
EDQNM Stochastic Backscatter Forcing Incorporated.
Fb (k ; t , t )  fˆk t  fˆk* t 
Deterministic Filters: KF, EKF & GEKF
•
•
•
•
•
•
Statistically optimal analysis methods for nonlinear systems involve, theoretically,
an infinite number of equations for the moments. This is formally identical to the
closure problem in turbulence theory.
Kalman filter theory it is also assumed that the background or forecast vorticity
field satisfies a linear equation. The Kalman filter method simply discard
moments of third order and higher.
The extended Kalman filter (EKF) is formulated entirely in terms of covariances
and completely neglects third and higher order moments. It is a method of
successive linearizations about the evolving nonlinear trajectory; it is a tangent
linear approximation for calculating Cf(x,t;y,t).
In order to tackle strongly nonlinear systems where bifurcations may occur Miller
et al. 1994 developed the generalized EKF. The GEKF uses a moment expansion
method in terms of Taylor series to estimate the contributions of the higher order
moments to the Kalman gain.
The generalized EKF extends the EKF methodology to strongly nonlinear systems
by including contributions from the third and fourth order moments to the
calculations of the error variance Cf(x,t;y,t). This enables the tracking of regime
transitions by increasing the estimated Kalman gain which is typically
underestimated in EKF comparisons.
The inclusion of approximations to the third and fourth moments into the EKF
was found to be sufficient to track the reference solution and to capture the
bifurcation behavior of chaotic trajectories in both the three-component Lorenz
and double well models; however, it was found to be impractical for higher
dimensional models.
Ensemble Methods
•
EnKF propagates an ensemble of model states with a fully nonlinear model allowing the full
matrix error covariance matrix to be calculated in principle. Obs are perturbed in order that
the Kalman filter update equation be satisfied.
Ensemble Kalman Filter
ˆia  (1  KH )ˆi f  Kdˆi
C f  ˆi f ˆi fT
Ensemble Square Root Kalman Filter
Don’t perturb observations but still satisfy KF update equation
~ ˆf
a
ˆ
  (1  KH )
~
K  C f H T (( HC f H T  D ) 1 )T ( HC f H T  D  D ) 1
equivalent
ˆ a   1  KH ˆ f
Statistical/deterministic versus stochastic filters
•
•
•
•
1)
2)
3)
4)
Truth trajectory is calculated by running the barotropic vorticity equation over the desired period
with a relaxation 1 day e-folding time. The source term is calculated at each timestep of the
unperturbed simulation, stored and then applied to both the perturbed ensemble DNS and mean field
equation of the closure for data assimilation and prediction.
We choose the observational error to have an rms of 1x106m2s-1 and define the nondimensional
variance Dk(t,t)=1.826E-06k, which results in an almost flat kinetic energy spectrum (Anderson
2001).
Throughout we assume the model error variance Qk(t,t) to be zero, circularly truncated wavenumber
space k=16 (C16) resolution (which has 797 degrees of freedom).
In the calculations that follow we have chosen to examine 3 & 4 (1 & 2 are current work)
Full Kalman filter equations in NxN, MxN, NxM, MxM matrices in grid point space.
The Kalman filter Eqs. in KxK spectral space with Hk(t) the diagonal elements of H.
The spectral Kalman filter Eqs and a diagonal Kalman gain Kk and a diagonal observation
operator Hk
If Hk(t) are nonzero then we also have spectral equations in terms of E=<eeT>=H-1E0H-1 where e=H01d and K (t)=K (t)H (t)
k
k
k
a EnKF
ˆ

 I  KH ˆ f  Kdˆ
I  KH ˆ f
a SDKF
C
 I  KH C f
a EnSF
ˆ


SDKF vs EnKF

 ka   kf  K k d k  H k  kf

ˆka  ˆkf  K k dˆk  H kˆkf


Cka (t , t )  (1  K k H k)Ckf (t , t )
Cka,  l (t , t )  ˆka (t )ˆal (t )  (1  K k H k)Ckf, l (t , t )(1  K l H l)  K k Dk , l (t , t ) K l
ˆka (t )ˆal (t )ˆla k (t )  (1  K k H k)(1  K l H l)(1  K l  k H l  k) ˆkf (t )ˆfl (t )ˆl f k (t )
 K k K l K l  k dˆk (t )dˆl (t )dˆl  k (t )
If d̂ k Gaussian then the 3-point observational error term vanishes, however for the
EnKF this is not true if sampling errors are present. We will examine cases where we
have sufficient realizations for Ck,-l(t,t’) to be considered well resolved but higher order
terms are not.
Choose Ck,-l(t0,t0) to be isotropic and evolving to Ck,-l(t,t’)
SDKF has been shown to closely match the
performance of the EnSF (3600 realizations) but
without issues related to sampling error.
EnKF badly lags both SDKF and EnSF in
comparison of the growth of error variances.
5 day assimilation, 5 day forecast
30 day assimilation – 4 blocks: Systematic differences
between quasi-diagonal EnSF and control over 30 days
using 12 hourly data assimilation


S  (t )  abs   kf (t )  fk (t )   kt (t )  t k (t ) 
 kS

D (k , t ) 
dˆ o k (t )dˆ o  k (t ) / k 4

kS
500hPa mean eddy streamfunction (km2s-1) and variance (km4s-2) 6th Nov 1979
Subgrid-scale parameterizations





b)
d)
a)
c)
2
  0 k k Ck (t , t ' )   ds S k t , s   Pk t , s  R k t ' , s    ds  k t , s    k t , s  C k t ' , s 
 t

t0
t0
t'

 
t


2
  0 k k   k    k  p  q [ K k , p, q   p (t )   q (t ) 
 t

p q
  k  p  q Ak , p, q  
p
q
t
p
(t ) h q ]   ds k t , s   k ( s)
e)
t0
g)
t
 hk  ds k t , s 
f)
t0
a) The nonlinear damping which gives rise to an subgrid eddy viscosity
b) The nonlinear noise leading to a subgrid stochastic backscatter
c) Eddy topographic-eddy mean field dissipation terms
d) Eddy topographic-eddy mean field noise terms
e) Gives rise to the residual Jacobian
f) Gives rise to the eddy topographic force
g) Gives rise to eddy mean field interactions
Maximum entropy / equilibrium SM arguments have been employed with some
success but how do these compare to nonequilibrium SD?
CT reduced to CR in circularly truncated wavenumber space

R  p, q | p  C

,q  C 
T  p, q | p  CT , q  CT
R
R
Reduce resolution from T to S, then one or both of
these inequalities hold
S T R
 p, q  S
C R  p  CT , C R  q  CT
 r (k )k 2   0 (k )k 2   ks
Renormalized mean viscosity
ˆr (k )k 2   0 (k )k 2   ks   ks
Renormalised eddy drain viscosity
fˆ r k  fˆ 0 k  f Ss k  f Ps k
Renormalized random force
t
f  S k  hk  ds ks (t , s)
Eddy topographic force
t0
f
r
k
 f
0
k
 f  s k  jk
Renormalized mean force
Holloway (1992), J. Phys. Oceanogr.
Maltrud & Holloway (2008), Ocean
modeling
 T (k )k 2  ˆT (k )k 2
 T (k )k 2  k eq   0 (k )k 2   kS   k eq
f  S k   0 (k )k 2  k
f
r
 f
k
0
k
eq
at equilibrium
fˆ  f
always
SS
k
 f
PS
k
valid : if  k
 fˆ  0
s.s.
fˆ r k  fˆ 0 k  f Ss k  f Ps k
s.s.
f
f
eq
 k
hh  ds (t , s)   k
t0
ˆr (k )k 2   0 (k )k 2   ks   ks
S
t
k
 hk  ds ks (t , s)
in general
t0
0
k
Horizontal Laplacian eddy viscosity to relax
model toward sign definite flow along slopes has
been very successful.
t
 r (k )k 2   0 (k )k 2   ks
 f S k
jk  0
r
k
J.S. Frederiksen, J. Atmos. Sci., (1999)
T.J. O’Kane & J.S. Frederiksen (2008), Physica Scripta
J.S. Frederiksen & T.J. O’Kane (2008), Entropy
then
t
 ds (t , s)
to
However if  T (k )k 2   0 (k )k 2  ks
Then there is no stochastic backscatter
to balance eddy drain viscosity
r
k
 f
f k S
0
k
 f  s k  jk
small scale
jk  0 : generally

 
  p h q

 
  p  q
p , q S
p , q S
Correlated with topography
Correlated with Rossby waves
QDIA Canonical Equilibrium
ETF
ETF
ESF
MV
TOP
Overhead 8
f
r
k
f
r
k
 1x106
 dsS t , s   P t , s R t ' , s 
t
S
S
k
k
k
t0

t
  ds 
S
k
t , s    k t , s Ck (t , s)
S
t0
ˆr (k )k 2
t

  ds 
t0
S
k
t , s    k t , s 
S
t
 t

   ds S k t , s   k s     ds S k t , s   k s  
t0
 t0

t
 t

 ds S k t , s  ds S k t , s 


t
 t

0
 0

*
*
Dissipative January Flow
ESF
SRJ
U
MSF
SETF
V
Beta-plane, January, C48 truncated to C24, Steady state, Forced-dissipative
(kx , ky) space
t'
 dsS S k (t , s) Rk (t ' , s)
t0
t'
 dsS S k (t , s) R k (t ' , s)
t0
t
  ds S k (t , s) R k (t ' , s)
t0
t
  ds S k (t , s) Rk (t ' , s )
t0
t'
  dsP S k (t , s ) R k (t ' , s)
t0
t
  ds S k (t , s) Rk (t ' , s)
t0
t'
 dsP S k (t , s) R k (t ' , s)
t0
t
  ds S k (t , s) R k (t ' , s)
t0
Overheads 4 7
Subgrid eddy-mean field & eddy-topographic terms
beta-plane
t
  ds
S
k
t , s   k (s)
t
  ds
S
k
t , s   k ( s)
hk  ds
t0
t0
 k  p  q[K k , p, q 
 p ,q S

t
t
S
k
hk  ds
t , s 
t0
t0
p


(t )  q (t )  A k , p, q   p (t ) hq ]

k
t , s 
Residual streamfunction Jacobian decomposition
  k  p  q Ak , p, q  
( p , q )S
p
(t ) hq
 k  p  q[K k , p, q 
 p ,q S
p
(t )  q (t ) ]
The End
References
• J.S. Frederiksen (1999) Subgrid-scale parameterizations of eddy-topographic
force, eddy viscosity, and stochastic backscatter for flow over topography, J.
Atmos. Sci., 56, pp1481--1494
• T.J. O’Kane & J.S. Frederiksen (2004) The QDIA and regularized QDIA
closures for inhomogeneous turbulence over topography, J. Fluid Mech., 504,
pp133--165
• J.S. Frederiksen & T.J. O’Kane (2005) Inhomogeneous closure and statistical
mechanics for Rossby wave turbulence over topography, J. Fluid Mech., 539,
pp137--165
• T.J. O’Kane & J.S. Frederiksen (2008) A comparison of statistical dynamical
and ensemble prediction methods during blocking, J. Atmos. Sci., 65, pp426-447
• T.J. O’Kane & J.S. Frederiksen (2008) Statistical dynamical subgrid-scale
parameterizations for geophysical flows, Physica Scripta,T132 014033
• J.S. Frederiksen & J.S. Frederiksen (2008) Entropy, closures & subgrid
modelling, Entropy, 10, pp635-683; DOI: 10.3390/e10040635
• T.J. O’Kane & J.S. Frederiksen (2008) Comparison of statistical dynamical,
square root and ensemble Kalman filters, Entropy, 10, pp684-721; DOI:
10.3390/e10040684
Forced-dissipative equilibrium: C48 reduced to C24
(kx,ky) space
t
  ds k (t , s) R k (t ' , s)
S
t0
t'
 dsS S k (t , s) Rk (t ' , s)
t0
t
  ds
t0
S
k
t , s   k (s)
t
  ds S k (t , s) Rk (t ' , s)
t0
t
  ds S k t , s   k ( s)
t0
t'
  dsP S k (t , s ) Rk (t ' , s)
t0
t
hk  ds S k t , s 
t0
t
hk  ds S k t , s 
t0
Spread / Model error;
EDQNM Stochastic Backscatter Forcing Incorporated.
Fb (k ; t , t )  fˆk t  fˆk* t 
Data assimilation: Kalman Filter
•
The Kalman filter theory implicitly assumes that both the observations and priori (forecast)
distributions, based on the background state, are Gaussianly distributed. The posteriori distribution
for the analysis is then derived based on the products of these two Gaussians and yields simply the
Kalman filter equations. In the standard Kalman filter theory it is also assumed that the background
or forecast vorticity field satisfies a linear equation. The Kalman filter method simply discard
moments of third order and higher.
 a   f  K d o  H
f

    ˆ
d  d  dˆ
K  C f H T ( HC f H T  D) 1
a  
f

 K do  H 
C a  ( I  KH )C f
C f  ˆi f , ˆi fT
f
