A comparison of statistical dynamical and A comparison of statistical dynamical and ensemble prediction methods during the ensemble prediction methods during the

formation of large-scale coherent structures in formation of large-scale coherent structures in the atmospherethe atmosphere

Terence J. O’Kane Terence J. O’Kane Collaborator: Collaborator: Jorgen S. FrederiksenJorgen S. Frederiksen

Bureau of Meteorology Research CentreBureau of Meteorology Research Centre

CSIRO Marine & Atmospheric ResearchCSIRO Marine & Atmospheric Research

Melbourne AustraliaMelbourne Australia

• MotivationMotivation

• Inhomogeneous statistical closure theoryInhomogeneous statistical closure theory

• Flow regimesFlow regimes

• Ensemble predictionEnsemble prediction

• Random and bred initial forecast Random and bred initial forecast perturbationsperturbations

• Atmospheric blocking transitionsAtmospheric blocking transitions

Global Atmospheric spectraGlobal Atmospheric spectra

CSIRO MK3 Global Climate Model CSIRO MK3 Global Climate Model T63 JanuaryT63 January

Atmospheric turbulenceAtmospheric turbulence•Atmospheric spectra nearly 2-DAtmospheric spectra nearly 2-D

• large scale Rossby waveslarge scale Rossby waves

• large scale flow instabilitieslarge scale flow instabilities

• Inhomogeneous large scalesInhomogeneous large scales

• small scale turbulent eddiessmall scale turbulent eddies

• homogeneous small scaleshomogeneous small scales

• Quasi 2-D at the large scales Quasi 2-D at the large scales

•complex (emergence/coherent structures/instabilities)complex (emergence/coherent structures/instabilities)

Ensemble weather predictionEnsemble weather predictionNWP NWP → → ensemble forecasting. Vast computational cost ensemble forecasting. Vast computational cost →→ very small ensembles (<100) very small ensembles (<100) Insufficient to accurately resolve the forecast error covariances. Hence a variety of deficiencies Insufficient to accurately resolve the forecast error covariances. Hence a variety of deficiencies including spurious long range correlations and grossly underestimated error variances requiring including spurious long range correlations and grossly underestimated error variances requiring heuristic approximation methods such as covariance localization and inflation. Recent studies heuristic approximation methods such as covariance localization and inflation. Recent studies (Denholm-Price 2003) have suggested that ensemble NWP schemes have little capacity to (Denholm-Price 2003) have suggested that ensemble NWP schemes have little capacity to produce anything beyond Gaussian statistics. Higher order cumulants have been shown to be produce anything beyond Gaussian statistics. Higher order cumulants have been shown to be necessary to track regime transitions in low dimensional (Miller et al 1994) & atmospheric data necessary to track regime transitions in low dimensional (Miller et al 1994) & atmospheric data assimilation (O’Kane & Frederiksen 2006) studies and are of no less importance in the accurate assimilation (O’Kane & Frederiksen 2006) studies and are of no less importance in the accurate determination of the predictability of atmospheric flows. determination of the predictability of atmospheric flows.

~200km resolution

Obstacles to an accurate tractable inhomogeneous Obstacles to an accurate tractable inhomogeneous non-Markovian statistical closurenon-Markovian statistical closure

1)1) Generalize two-point two-time homogeneous Generalize two-point two-time homogeneous closure theory to general 2-D flow over topography.closure theory to general 2-D flow over topography.

2)2) Tractable representations of the two- and three-Tractable representations of the two- and three-point cumulants. point cumulants. Generalize special case of Kraichnan (1964): Boussinesq Generalize special case of Kraichnan (1964): Boussinesq convection: diagonalizing closure for a mean horizontally averaged temperature field with convection: diagonalizing closure for a mean horizontally averaged temperature field with zero fluctuations to general 2-D flow over topography. zero fluctuations to general 2-D flow over topography.

3)3) Incorporate large scale Rossby waves (Incorporate large scale Rossby waves (ββ-plane).-plane).4)4) Long integrations of time-history integrals (non-Long integrations of time-history integrals (non-

Markovian integro-differential equations requiring Markovian integro-differential equations requiring large storage on super computers)large storage on super computers)

5)5) Vertex renormalization problem→ RegularizationVertex renormalization problem→ Regularization6)6) Ensemble averaged DNS code for comparisonEnsemble averaged DNS code for comparison

2-D barotropic vorticity on a generalized 2-D barotropic vorticity on a generalized ββ-plane-plane

SUUdS

xh

St

U)(

1

S

dSS

UE 22 )(1

2

1

2

1

S

dShSk

UkQ 22

00 )(

1

2

1)(

2

1

0220 ˆ),( fUykyhUyJ

t

Small scalesSmall scales

Large scalesLarge scales

InvariantsInvariants

Mean and transient evolution equationsMean and transient evolution equations

)(ˆ)(ˆ),(, ttttC qpqp

0k

p qqp

p qqpqpk

kkk

hqpkAqpk

ttCqpkKqpkt

),,()(

),(),,()(

ˆ

,

p qqp

qpqpqpqpp q

k

hqpkAqpk

ttCqpkKqpkt

ˆ),,()(

),(ˆˆˆˆ),,()(ˆ,

)(ˆ)(ˆ),( ttttC kkk

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 equationThe two-time cumulant equation

)'(ˆ)(ˆ)(ˆ,,)',(

)()',()',()(

,,

)',(,,)',(

)',()',()',(

,,

,

20

tttqpkKqpkttN

tttCttCt

qpkKqpk

httCqpkAqpkttN

ttNttNttCkkt

kpqp q

Hk

qkpkqp

p q

p qqqp

Ik

Hk

Ikk

Functional FormsFunctional Forms

kk

DIA

kllkkllk

kkkkQDIA

lklk

kkkkQDIA

lklk

RCtttttt

hRCttRttR

hRCttCttC

,)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ

,,,)',()',(

,,,)',()',(

)()(

,,

,,

)',()',( , ttRttR kkk

)'(ˆ)(ˆ)(ˆ,,)',( tttqpkKqpkttCt kpq

p qk

The closure problem (homogeneous isotropic turbulence)The closure problem (homogeneous isotropic turbulence)

)'(ˆ)(ˆ

)',(ˆ)',(0,,

tf

tttRttR

l

klklk

Response functionsResponse functions

3-point 3-point →→ DIA Closure (+ correction?) DIA Closure (+ correction?)

),'(),(),(,,2

),'(),(),(,,2

),'(),(),(,,2)'(ˆ)(ˆ)(ˆ

)(

)(

)(

'

0

0

0

stCstCstRklklKds

stCstCstRkkllKds

stRstCstCkllkKdsttt

klkl

t

t

kkll

t

t

kkll

t

t

kkll

Kraichnan, J. Fluid Mech.,1959

} Nonlinear damping

Nonlinear noise

2 point terms 2 point terms → → Renormalized perturbation theoryRenormalized perturbation theory

)(ˆ)(ˆ)(ˆ 10 ttt kkk

)(ˆ),()(ˆ),()(ˆ

)(ˆ)(ˆ)(ˆ

00,0

00

0,

0

00

00002

0

0

sfstRdstttRt

ttttftkkt

kkk

t

t

kkkk

kkk

Solution in terms of Greens function

)(ˆˆˆ,,,,

2

1

ˆˆˆˆ,,

ˆ,,)(ˆ

01

000

0000

0120

tttpqkAqpkAqpk

qpkKqpk

hqpkAqpktkkt

kp q

pqqp

p qqpqp

p qqpk

To order

To zeroth order

(1)

(2)

(3)

(4)

Similarly expand the two-time two-point cumulant

p qpqqp

p qqpqp

p qqp

kk

t

t

k

kkkkk

sssspqkAqpkAqpk

ssssqpkKqpk

hsqpkAqpk

stRdst

ttttRt

QDIA

QDIA

)(ˆ)()(ˆ)(,,,,2

1

)(ˆ)(ˆ)(ˆ)(ˆ,,

)(ˆ,,

),()(ˆ

)(ˆ)(ˆ),()(ˆ

00

0000

0

0,

1

10

10

0,

1

0

)(

)(

)'(ˆ)(ˆ)'(ˆ)(ˆ)'(ˆ)(ˆ)',( 100100

, ttttttttCQDIAQDIA

lklklkQDIA

lk

2 point terms 2 point terms → → Renormalized perturbation theoryRenormalized perturbation theory

Thus

(5)

(6)

(7)

Where formal soln to (4) is

Renormalized perturbation theory cont.Renormalized perturbation theory cont.

lklkkkkkkkkk CCCCRR ,1

,,0,,

0, ,,,1

Assume initially multivariate Gaussian

To Zeroth Order Diagonal dominance

Renormalize

),(,,)',()',(

)'(ˆ)(ˆ)'(ˆ)(ˆ)',(

00,00

001

,1

,

10011,

)(

)(

ttCttRttRttCttC

ttttttC

lklklklk

lklklk

QDIA

QDIAQDIAQDIA

)'(ˆ)(ˆ)',()'(ˆ)(ˆ 000,

00 ttttCtt kklklklk

)(ˆ)(ˆ)(ˆ)(ˆ0000 tttt kklklk

So to first order

Substituting in solution Eq. (6) gives

Sufficient conditions for diagonal dominance are that h & <> be sufficiently small.

At canonical equilibrium the off-diagonal elements of the equal time covariance vanish

regardless of the size of h & <> .

),(),(),(

)(),,(2),,(),(),(

)(),,(2),,(),(),(),(

'

),(),'(),()',(

),(),'(),(

)(),,(2),,(),(),'(

)(),,(2),,()',(),()',(

00,00

)()(

)()(,

,,

,

)()(

'

)()(,

0

0

ttCtTRtTR

spkpkKhpkpkAsTCsTRds

skpkpKhpkkpAsTCsTRdsTTC

Ttt

TTCTtRTtRttC

TTCTtRTtR

sklklKhklklAstCstRds

skllkKhkllkAtsCstRdsttC

kplk

pkpkpk

T

t

pkpkkp

T

t

kp

lklklkQDIA

lklk

lklkkl

t

T

lklklk

t

T

lk

Thus for Ttt '

And for

Using a reduced notation and a re-ordering of the wavevectors

Two-time two-point cumulant equation with updatesTwo-time two-point cumulant equation with updates

)(~

0,0)2(

, ttK lk

)(

~),'(),(

)(),,(2),,(),(),'(

)(),,(2),,()',(),()',(

0,0)2(

,00

)()(

'

)()(,

0

0

ttKttRttR

sklklKhklklAstCstRds

skllkKhkllkAtsCstRdsttC

lklk

lklkkl

t

t

lklklk

t

t

lk

)(),,(2),,()',(),()',( )()(

'

, skllkKhkllkAtsRstdsRttR lklk

t

t

lklk

In a similar manner we derive the two-time two-point response functionIn a similar manner we derive the two-time two-point response function

Here is the contribution to the off-diagonal covariance matrix at initial time t0 and in reduced notation

0000,2

,,2 ,,,

~,,

~tTRtTRttKTTCTTK kpkp

QDIAkpkp

DIA Closure + 3-point cumulant updateDIA Closure + 3-point cumulant update

),,(~

000)3(

),(, tttK kkll

),,(~

),'(),(),(

),'(),(),(,,2

),'(),(),(,,2

),'(),(),(,,2)'(ˆ)(ˆ)(ˆ

000)3(

),(,00)(0

)(

)(

)(

'

0

0

0

tttKttRttRttR

stCstCstRklklKds

stCstCstRkkllKds

stRstCstCkllkKdsttt

kkllkkll

klkl

t

t

kkll

t

t

kkll

t

t

kkll

Generalization of H. Rose, Physica D, 1985

Here 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

000000,,3

,,3

,,,,,~

)(ˆ),(ˆ),(ˆ,,~

tTRtTRtTRtttK

TTTTTTK

kpqkpq

QDIA

kpqkpq

Two-point two-time closures at Finite ResolutionTwo-point two-time closures at Finite Resolution

• For infinite resolution and moderate Reynolds numbers the DIA 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) (Frederiksen & Davies 2004, McComb & Kiyani 2006)

• For finite resolution (>C48) and moderate RL (>200) numbers all homogeneous two-point non-Markovian closures underestimate the evolved small-scale KE and dramatically underestimate the skewness.

• This is due to the fact that while we renormalized the propagators the vertices are bare i.e. interaction coefficients and some information about the higher order information is absent.

• What to do in a practical sense? How does the inhomogeneous closure perform given small scale topography may act to localize transfers?

• Large scale Reynolds number in terms of transient energy and enstrophy dissipation Herring et al 1974 JFM

3

1

2

2

,ˆˆ

/,21

kk

kk

L

ttCk

kttC

tR

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

where Θ is the heavyside step function.

• The interaction coefficients are unchanged in the single time cumulant equations.

• It was found that α1= α2=4 was a universal best choice for the strong turbulence weak mean field flow regime on an f-plane.

• And further found that α1= α2=4 was again the universal best choice where both strong mean fields and Rossby waves where dominating the large scale dynamics.

• Thus with the choice α1= α2=4 the regularized QDIA in cumulant update form offers a one parameter two-time renormalized non-Markovian closure for inhomogeneous turbulence that compares very closely to ensemble averaged DNS regardless of the relative strengths of mean flow, topography, eddies and where large-scale waves are present.

),,()/()/(

),,()/()/(

22

11

qpkAkqkp

qpkKkqkp

Flow RegimesFlow Regimes• For homogeneous isotropic turbulence skewness is commonly used as a For homogeneous isotropic turbulence skewness is commonly used as a

sensitive measure of the growth of non-Gaussian terms.sensitive measure of the growth of non-Gaussian terms.

• What happens in the Inhomogeneous case?What happens in the Inhomogeneous case?

• Does skewness grow at comparable rates when topography and mean Does skewness grow at comparable rates when topography and mean fields are present?fields are present?

• What are the relative contributions of non-Gaussian terms (three-point) What are the relative contributions of non-Gaussian terms (three-point) and Inhomogeneity (two-point off-diagonal covariances) to the growth of and Inhomogeneity (two-point off-diagonal covariances) to the growth of the transient field?the transient field?

• What proportion of error growth is due to each of these components in What proportion of error growth is due to each of these components in data assimilation / ensemble prediction regime?data assimilation / ensemble prediction regime?

• Palinstrophy production measure Palinstrophy production measure

k k kkkk

HIM ttCkttCttNktP2

1

22 ),(2

1),(

2

1/),(2)(

Inhomogeneity - 2 pointInhomogeneity - 2 point

),(),(),(

~

,,)(,,,,

),(),(),(),(

,,21

2)(

0000)2(,

2

2

1

2 0 0

ttRttRttK

hqpkAtqkpAqpkAqpk

stCstdsstRstdsPk

ttCkttC

tI

kpqp

p qqq

t

t

t

t

kkkkk

kk

kk

K

),(),(),(,,~

,,

),(),(),(),(),(

,,2

1

2)(

000000000)3(

,,

02

2

1

2 0 0

ttRttRttRtttKqpkKqpk

stCstdsstRstFstSdsk

ttCkttC

tS

kpqp q

kpq

t

t

t

t

kkkkkk

kk

kk

K

Skewness - 3 pointSkewness - 3 point

Note: Small scale measureNote: Small scale measure

)()()( tStItP KKM

Flow regimesFlow regimes

RL ~ 4000 at C92

RL ~ a few 100’s C64

RL ~ small C48

Ensemble PredictionEnsemble Prediction• 2-D Navier-Stokes equation could be used to describe systems whose 2-D Navier-Stokes equation could be used to describe systems whose

many scales of motion can be simultaneously excited and that such systems many scales of motion can be simultaneously excited and that such systems have a finite range of predictability.have a finite range of predictability.

• Statistics of error prediction (pdf) ↔ statistical theory of turbulence (higher Statistics of error prediction (pdf) ↔ statistical theory of turbulence (higher moments). moments).

• Forecast depends critically on random initial errors, model errors and Forecast depends critically on random initial errors, model errors and observational error.observational error.

• Ensemble / probabilistic predictions give specific information on the nature Ensemble / probabilistic predictions give specific information on the nature and extent of the uncertainty of the forecast.and extent of the uncertainty of the forecast.

• Our closure methodology extends 1) the work of Epstein(1969) (3and Our closure methodology extends 1) the work of Epstein(1969) (3and higher moment discard) and Fleming (1971; MC, QN & EDQN) and Leith higher moment discard) and Fleming (1971; MC, QN & EDQN) and Leith (1971, 1974 TFM) and 2) extends the homogeneous statistical closure (1971, 1974 TFM) and 2) extends the homogeneous statistical closure methods used by Herring etal(1973; DIA) & Fleming (1979) to non-methods used by Herring etal(1973; DIA) & Fleming (1979) to non-Gaussian and strongly inhomogeneous flows.Gaussian and strongly inhomogeneous flows.

• All the above studies based on random isotropic initial conditions.All the above studies based on random isotropic initial conditions.

Generating initial perturbationsGenerating initial perturbations• In EP independent initial disturbances are generated as fast growing disturbances In EP independent initial disturbances are generated as fast growing disturbances

with structures and growth rates typical of the analysis errors. Random isotropic with structures and growth rates typical of the analysis errors. Random isotropic initial perturbations grow more slowly and lead to underestimated error variances.initial perturbations grow more slowly and lead to underestimated error variances.

• Generate independently perturbed initial conditions such that the covariance of the Generate independently perturbed initial conditions such that the covariance of the ensemble perturbations ≈ initial analysis error covariance at the time of the forecast.ensemble perturbations ≈ initial analysis error covariance at the time of the forecast.

• Breeding method (Toth & Kalnay, Mon. Wea. Rev., 1997) forecast perturbations Breeding method (Toth & Kalnay, Mon. Wea. Rev., 1997) forecast perturbations are transformed into analysis perturbations in order to sare transformed into analysis perturbations in order to simulate the effect of obs by imulate the effect of obs by rescaling nonlinear perturbations (Toth and Kalnay 1993,1997). Sample subspace rescaling nonlinear perturbations (Toth and Kalnay 1993,1997). Sample subspace of most rapidly growing analysis errorsof most rapidly growing analysis errors

• Extension of linear concept of Lyapunov Vectors into nonlinear environmentExtension of linear concept of Lyapunov Vectors into nonlinear environment• Fastest growing nonlinear perturbations (Toth et al 2004, Wei & Toth 2006)Fastest growing nonlinear perturbations (Toth et al 2004, Wei & Toth 2006)

Bred PerturbationsBred Perturbations

)',()'()()',(

),()()(),(

)(ˆ)()(ˆ

/),(

/),()(

2

1

4

400

ttCtgtgttC

ttCtgtgttC

ttgt

kttC

kttCtg

fk

ak

fk

ak

fk

ak

kk

kk

•Analysis cycle acts as a nonlinear perturbation Analysis cycle acts as a nonlinear perturbation model on the evolution of the real atmosphere model on the evolution of the real atmosphere (nudging) resulting in error growth associated with (nudging) resulting in error growth associated with the evolving atmospheric state to develop within the evolving atmospheric state to develop within the analysis cycle and dominate forecast error the analysis cycle and dominate forecast error growth. growth. •Other approaches include mixed initial and Other approaches include mixed initial and evolved singular vectors, Ensemble square root evolved singular vectors, Ensemble square root filters (i.e. non-perturbed observations) etc.filters (i.e. non-perturbed observations) etc.•Bred vectors are superpositions of the leading local Bred vectors are superpositions of the leading local time dependent Lyapunov vectors (LLV’s).time dependent Lyapunov vectors (LLV’s).•All random perturbations will assume the structure All random perturbations will assume the structure of the LLV given time thereby reducing the spread of the LLV given time thereby reducing the spread to to 11

Atmospheric Regime transitionsAtmospheric Regime transitionsNorthern Hemisphere blocking, Gulf of Alaska 6 Nov 1979Northern Hemisphere blocking, Gulf of Alaska 6 Nov 1979

Transition from strong zonal to “wavy” flow and the emergence of a Transition from strong zonal to “wavy” flow and the emergence of a coherent high low blocking dipolar structure. Rapidly growing large coherent high low blocking dipolar structure. Rapidly growing large scale flow instabilities and a loss of predictability i.e. rapid growth of scale flow instabilities and a loss of predictability i.e. rapid growth of the error field.the error field.

Comparison of DNS and Comparison of DNS and RQDIA zonally asymmetric RQDIA zonally asymmetric

streamfunction in 5 day streamfunction in 5 day breeding / 5 day forecast breeding / 5 day forecast experiment during block experiment during block

formation and maturation.formation and maturation.

Ensemble prediction and error growth studiesEnsemble prediction and error growth studies

Breeding 1/11/79

Forecast 5/11/79

Isotropic 26/10/79

Day1: growth of Ck,-l

and organization of error structures

Day 2-5: rescaling of error variances 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.Integral contributions to error growth.How important are memory effects?How important are memory effects?

0)'(ˆ)(ˆ)(ˆ ttt kllk

•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 , respectively. 0)'(ˆ)(ˆ)(ˆ

tttt kllk

Maintaining spread;Maintaining spread;EDQNM Stochastic Backscatter Forcing Incorporated.EDQNM Stochastic Backscatter Forcing Incorporated.

tftfttkF kkb*ˆˆ),;(

ConclusionsConclusions

• The November 1979 Gulf of Alaska block is typical of a large scale coherent The November 1979 Gulf of Alaska block is typical of a large scale coherent structure in the atmosphere and is associated with markedly increased flow structure in the atmosphere and is associated with markedly increased flow instability and a corresponding loss of predictability. instability and a corresponding loss of predictability.

• Comparison of the DNS, QDIA closure and variants enabled quantification of Comparison of the DNS, QDIA closure and variants enabled quantification of the respective contributions of off-diagonal and non-Gaussian terms to error the respective contributions of off-diagonal and non-Gaussian terms to error growth.growth.

• For ensemble prediction instantaneous error growth is largely due to the For ensemble prediction instantaneous error growth is largely due to the inhomogeneity and not the non-Gaussian terms which at any given time are inhomogeneity and not the non-Gaussian terms which at any given time are small. However the cumulative effect of the non-Gaussian terms determines the small. However the cumulative effect of the non-Gaussian terms determines the correct amplitude of the evolved kinetic energy variances. correct amplitude of the evolved kinetic energy variances.

• Instantaneous non-Gaussian terms only become important after the transients Instantaneous non-Gaussian terms only become important after the transients saturate the mean and the flow enters a regime where turbulence dominates. saturate the mean and the flow enters a regime where turbulence dominates.

• Other applications include statistical dynamical data assimilation methods andOther applications include statistical dynamical data assimilation methods and subgrid-scale parameterizations.subgrid-scale parameterizations.• Future work focussing on developing baroclinic QDIA and averaging over the Future work focussing on developing baroclinic QDIA and averaging over the

large wavenumberslarge wavenumbers

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 (2007) A comparison of statistical dynamical and ensemble prediction methods during the formation of large-scale coherent structures in the atmosphere, J. Atmos. Sci. In Press

• T.J. O’Kane & J.S. Frederiksen (2007) Comparison of statistical dynamical, square root and ensemble Kalman filters, Submitted Tellus

• T.J. O’Kane & J.S. Frederiksen (2007) The structure and functional form of the subgrid scales for inhomogeneous barotropic flows in the atmosphere using the QDIA, To appear Physics Scripta