improving prediction skill of imperfect turbulent models through statistical response...

Improving Prediction Skill of Imperfect TurbulentModels through Statistical Response and Information

Theory

Di Qi, and Andrew J. Majda

Courant Institute of Mathematical Sciences

Fall 2016 Advanced Topics in Applied Math

Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of Imperfect Models Dec. 8, 2016 1 / 61

Challenges for turbulent dynamical systems

Uncertainty quantification (UQ) deals with the probabilistic characterization of all thepossible evolutions of a dynamical system given an initial set of possible states as well asthe random forcing or parameters.

Turbulent dynamical systems are characterized by a large dimensional phasespace and high degrees of internal instability.

Instabilities through energy-conserving nonlinear interactions result in astatistical steady state that is usually non-Gaussian.

Accurate quantification for the statistical variability to general externalperturbations is important in climate change sciences.

Major Task of this talk:Investigate a concise systematic framework for measuring and optimizingconsistency and sensitivity of imperfect dynamical models.


General framework for statistical modelingThe system setup will be a finite-dimensional system of, u ∈ RN , with lineardynamics and an energy preserving quadratic part

du

dt= L [u] = (L + D) u + B (u,u) + F (t) + σk (t) Ẇk (t;ω) , (1)

L∗ = −L; D ≤ 0; u · B (u,u) ≡ 0.NEW STRATEGIES FOR REDUCED-ORDER MODELS FOR PREDICTING COMPLEX TURBULENT DYNAMICAL SYSTEMS 5

Modeling Stages Math. & Computational Tools

Model SeclectionErgodic Theory

Statistical Measures in Equilibrium

Model CalibrationEmpirical Information Theory

Linear Response TheoryTotal Statistical Energy Equations

Model PredictionAccurate and Efficient Schemes

Numerical Stability Analysis

Rigorous Math Theories

Computational Methods

Figure 1.1. The general strategy for the development of reduced-order statistical models. Threesequential stages are required to carry out the reduced-order statistical model, and rigorous math-ematical theories are combined with numerical analysis to calibrate model errors and improve theimperfect model prediction skill.

The model calibration procedure is usually carried out in a training phase before the prediction, so that theoptimal imperfect model parameters can be achieved through a careful calibration about the true higher-orderstatistics. The ideal way is to find a unified systematic strategy where various external perturbations can be predictedfrom the same set of optimal parameters through this training phase. To achieve this, various statistical theoriesand numerical strategies need to be consulted. Most importantly, we need to consider the linear response theoryto calibrate the model responses in mean and variances; and use empirical information theory to get a balancedmeasure for the error in the leading order moments. In the final model prediction stage, the optimal imperfectmodel parameters are applied for the forecast of various model responses to perturbations. In the constructionabout numerical models, numerical issues also need taking into account to make sure numerical stability andaccuracy. Especially, proper schemes with accuracy order consistent with the reduced model approximation errorshould be proposed to ensure optimal performance.

2. Statistical Theory Toolkits for Improving Model Prediction Skill

In this section we introduce the general theoretical toolkits that are useful for capturing the key statisticalfeatures in turbulent systems like (1.1) and improving imperfect model prediction skill. Despite the complexmodel statistical responses in each component as the turbulent dynamical system gets perturbed, there exists asimple and exact statistical energy conservation principle for the total statistical energy of the system describingthe overall (inhomogeneous) statistical structure in the system through a simple scalar dynamical equation. Thetheory is briefly described in Section 2.1. Then the construction about the imperfect reduced-order models concernsabout the consistency in equilibrium (climate fidelity) and the responses to perturbations (model sensitivity).Equilibrium fidelity should be guaranteed in the first place so that the reduced-order model will converge to thetrue equilibrium statistics. To further calibrate the detailed model sensitivity to perturbations in each statisticalcomponent, the linear response theory can offer useful quantities to measure for quantifying the crucial statistics inthe model structure. Combining with the relative entropy under empirical information theory, a general information-theoretical framework can be proposed to tune the imperfect model parameters in a training phase, thus optimalmodel parameters can be used for model prediction in various dynamical regimes. We will describe the basic ideasfirst in this section.

2.1. A statistical energy conservation principle. Despite the fact that the exact equations for the statisticalmean (1.4) and the covariance fluctuations (1.5) are not closed equations, there is suitable statistical symmetry sothat the energy of the mean plus the trace of the covariance matrix satisfies an energy conservation principle evenwith general deterministic and random forcing. Here we briefly introduce the theory developed in [13] about a totalstatistical energy dynamics for the abstract system (1.1).


Outline

1 Turbulent systems with quadratic nonlinearitiesGeneral setups for turbulent systems with quadratic nonlinearities

2 Statistical dynamical models and closure methodsThe Gaussian closure method for statistical predictionSingle point statistics and information barrierImperfect model calibration and empirical information theoryForecast skill of closure methods for forced responses in L-96 model

3 Strategies for reduced order models in low dimensional subspaceLow order models in reduced subspaceForecast skill of reduced models for forced responses

4 Predicting statistical response in two-layer baroclinic turbulenceTwo-layer baroclinic turbulence in ocean and atmosphere regimesLow-dimensional reduced-order models for the two-layer systemImperfect model prediction skill with 2× 102 resolved modes for originalsystem with 2× 2562 dimension


Outline






General setup of turbulent systems with quadraticnonlinearities

The system setup will be a finite-dimensional system of, u ∈ RN , with lineardynamics and an energy preserving quadratic part

du

dt= L [u] = (L + D) u + B (u,u) + F (t) + σk (t) Ẇk (t;ω) . (2)

L being a skew-symmetric linear operator L∗ = −L, representing the β-effect ofEarth’s curvature, topography etc.

D being a negative definite symmetric operator D∗ = D, representing dissipativeprocesses such as surface drag, radiative damping, viscosity etc.

B (u, u) being the quadratic operator which conserves the energy by itself so that itsatisfies B (u, u) · u = 0.F (t) + σk (t) Ẇk (t;ω) being the effects of external forcing, i.e. solar forcing, whichcan be split into a mean component F (t) and a stochastic component with whitenoise characteristics.


Exact statistical moment equationsStatistical mean and covariance dynamics, u = ū + Zi vi , Rij =

〈Zi Z∗j

〉,

d ū

dt= (L + D) ū + B (ū, ū) + Rij B

(vi , vj

)+ F (t) ,

dR

dt= Lv R + RL

∗v + QF + Qσ .

the linear dynamics operator Lv expressing energy transfers between the mean field and thestochastic modes (B), as well as energy dissipation (D), and non-normal dynamics (L)

{Lv}ij =[(L + D) vj + B

(ū, vj

)+ B

(vj , ū

)]· vi .

the positive definite operator Qσ expressing energy transfer due to external stochasticforcing

{Qσ}ij = v∗i σ∗k σk vj .

the third-order moments expressing the energy flux between different modes due tonon-linear terms

QF =〈ZmZnZj

〉B (vm, vn) · vi + 〈ZmZnZi 〉B (vm, vn) · vj .

note that energy is still conserved in this nonlinear interaction part

Tr [QF ] = 2 〈ZmZnZi 〉B (vm, vn) · vi= 2 〈B (Zmvm,Znvn) · Zi vi 〉 = 2

〈B(u′, u′

)· u′〉

= 0.


Outline






Statistical Accurate Closure ModelsThe true statistical model

d ū

dt= (L + D) ū + B (ū, ū) + Rij B (vi , vj ) + F (t) , ū ∈ RN ,

dR

dt= Lv (ū) R + RL

∗v (ū) + QF + Qσ, R ∈ RN×N .

QF ,ij = 〈ZmZnZj〉B (vm, vn) · vi + 〈ZmZnZi 〉B (vm, vn) · vj , trQF = 0.

Basic Idea:The hierarchical moment equations will never be closed by adding higherorder momentsEquations for the stochastic coefficients

dZidt

= Zj [(L + D) vj + B (ū, vj ) + B (vj , ū)] · vi+ [B (u′,u′)− Rjk B (vj , vk )] · vi + Ẇkσk · vi .

Replace nonlinear interaction term by linear damping and Gaussian noise

Statistical closure models

B (u′,u′) · vi =∑

Bij Zi Zj −→ −dM,i (R) Zi + σM,i Ẇi .Di Qi, and Andrew J. Majda (CIMS) Improving Prediction Skill of Imperfect Models Dec. 8, 2016 9 / 61

Statistical Accurate Closure ModelsThe reduced-order approximation ūM ∈ RM , M � N

d ūMdt

= (L + D) ūM + B (ūM , ūM ) + RM,ij B (vi , vj ) + F,

dRMdt

= Lv RM + RM L∗v + Q

MF + Qσ,

Basic Idea:The hierarchical moment equations will never be closed by adding higherorder momentsEquations for the stochastic coefficients

dZidt

= Zj [(L + D) vj + B (ū, vj ) + B (vj , ū)] · vi+ [B (u′,u′)− Rjk B (vj , vk )] · vi + Ẇkσk · vi .

Replace nonlinear interaction term by linear damping and Gaussian noise

Statistical closure models

B (u′,u′) · vi =∑

Bij Zi Zj −→ −dM,i (R) Zi + σM,i Ẇi .


Quasilinear Gaussian (QG) closure

The simplest closure scheme for the moment problem is to completely neglect thethird-order moments, i.e. set QF = 0.

d ū

dt= (L + D) ū + B (ū, ū) + Rij B (vi , vj ) + F (t) , (3)

dR

dt= Lv R + RL

∗v +��QF + Qσ. (4)

persistent instabilities that would cause uncontrollable growth of the unstablemodes.

for L-96 system the QG closure scheme avoids blow-up.Consider homogeneous statistical solutions of QG closure for the L-96 system: the meanū (t) is a time varying constant; the covariance multiplier is diagonal in Fourier spaceR = rjδij ; the linear operator is diagonal Fourier multiplier Lv = diag

(lj),

drj

dt= 2Relj (ū) rj , lj (ū) = (exp (2πij/J)− exp (−4πij/J)) ū − 1

dū

dt= −ū + F +

20∑j=0

rjB(vj , vj

)· v0 + B

(vj , vj

)∗ · v02

.


The statistical steady state solution requires rj,∞ = 0 unless Relj (ū) = 0, that is,

maxj

Relj (ū) = 0

⇒ ū(

cos2πj

J− cos 4πj

J

)− 1 = 0

⇒ ūcr =1

cos 2πjcrJ − cos4πjcr

J

≈ 0.8944, jcr = 8

⇒ rj 6= 0, j = 8; rj = 0, ∀j 6= 8.

Correspondingly

ūcr = F + r8B (v8, v8) · v0 + B (v8, v8)∗ · v0

2

= F + r8

(cos

4πjcrJ− cos 2πjcr

J

)< F .


Number of positive eigenvalues of Lv (ū) for L-96 system with respect to themagnitude of the mean field ū. No matter how large the external forcing F is, theQG closure method always converges to the unique critical value ūcr , satisfyingmarginal linear stability.

0 0.5 1 1.5 2 2.5 3 3.5 40

1

2

3

4

5

6

7

8

9

10

Num

ber o

f pos

itive

eig

enva

lues

F=6 F=8 F=16F=5 F=32

Equilibrium point for QG models

Figure 2: Number of positive eigenvalues of Lv (ū) for L-96 with respect to the magnitude of themean field ū. The red dashed lines indicate exact equilibrium points for di§erent value of the forcingparameter F. The green solid lines indicate equilibrium points for the DO UQ scheme for N = 10.

linearized dynamics - described by Lv (ū) - which will magnify the initial uncertainty. In Figure 2we present the number of unstable wavenumbers, i.e. the number of eigenvalue pairs with positivereal part for the linearized matrix Lv (ū) , with respect to the value of the steady state mean field(note that spatial homogeneity implies spatially constant mean field). In the same plot we showthe with dashed lines the steady state value of the mean field for specific values of the forcingparameter F. Based on the presence of persistent positive eigenvalues in the steady-state we have(for su¢ciently large F ) the following energy cycle (Figure 3):

1. Energy from the external excitation F leads to the growth of the mean field energy 12 ū.ū(equation (2)).

2. The important magnitude of ū leads to the activation of positive eigenvalues of Lv (ū) (seeFigure 2) that essentially absorb energy from the mean field and transform it to variance forthe stochastic modes that are associated with this process.

3. The nonlinear conservative term B(u0,u0) absorbs part of this energy, transferring it to thestable stochastic modes. It acts as dissipative mechanism for the unstable modes (balancingtheir positive eigenvalues) and external noise for the stable modes bringing all of them into astatistical equilibrium.

6


Energy flow in the quadratic systemsStatistical mean and covariance dynamics

d ū

dt= (L + D) ū + B (ū, ū) + Rij B

(vi , vj

)+ F (t) ,

dR

dt= Lv R + RL

∗v + QF + Qσ .

In imperfect closure models, replace

QF =〈ZmZnZj

〉B (vm, vn) · vi + c.c.

−→ QMF = QMF− + Q

MF + = −2dM,k (R) rk + σ

2M,k (R)

Figure 3: Energy flow in the L-96 system. Energy flows from the mean field to the linearly unstablemodes and then redistributed through nonlinear, conservative terms to the stable modes. Redarrows denotes dissipation, while the dashed box indicates terms that conserve energy.

4. The stable modes receive energy from the unstable ones through the nonlinear conservativeterms. A portion of this energy is dissipated and the rest is subsequently returned to the meanthrough the modes with negative eigenvalues. All modes including the mean flow dissipateenergy through the negative definite part of the linearized dynamics.

This cycle of energy in the L-96 model is representative of any general model that contains i)unstable linearized modes whose stability depends on the mean field energy level (i.e. that theyabsorb energy from the mean field), ii) stable modes, and iii) nonlinear conservative terms thattransfer energy between the modes and through this transfer the system is reaching an equilibrium.This structure is ubiquitous in turbulent systems in the atmosphere and ocean with forcing anddissipation [26, 22, 1, 2] as well as in fluid flows with lower dimensional attractors [23]. However,there are also examples of idealized truncated geophysical flows without dissipation and forcingwith a Gaussian statistical equilibrium where the linear operator at the climate mean state is stablewhile the system has many positive Lyapunov exponents [19].

7

Figure 4: Energy flow in the MQG UQ scheme. Energy flows from the mean field to the linearlyunstable modes and then redistributed through empirical, conservative fluxes to the stable modes.Red arrows denotes dissipation, while the dashed box indicates terms that conserve energy.

we substitute the nonlinear conservative mechanism by a conservative pair of positive and negativeenergy fluxes having the form of additional damping for the unstable modes and additive noisefor the stable modes (Figure 4). Note that this additional damping is chosen so that the unstableeigenvalues of the original linearized dynamics are guaranteed to have zero real part in the statisticalsteady state. In that sense this is the minimal amount of additional damping and noise required toachieve marginal stability (non-positive eigenvalues) of the correct steady state statistics. Thus, wehave a minimally changed model compared to the original equation that admits the correct steadystate statistics as a stable equilibrium stage. In the next subsections we will see that for numericalreasons it is required to add a small amount of additional damping (and noise) so that the correctstatistical steady state is not just marginally stable but it is associated with eigenvalues havingpurely negative real part. Moreover, in the transient phase of the dynamics the intensity of thenonlinear energy fluxes should depend on the energy level of the system and to this end we willapply a scaling factor to the additional damping and noise (that represent the nonlinear energyfluxes) which will take into account this dependence.Note that all of the required fluxesQ−F1, Q

+F1 are evaluated explicitly from available information

involving the linear operator, Lv (ū1), and the covariance matrix, R1 in a statistical steady state.In addition, since the nonlinear flux model is kept separate from the unmodified linear dynamics, itexpresses an inherent property of the system, a direct link between second and third-order statisticalmoments in the same spirit that Karman-Howarth equation [10] does for isotropic turbulence.

11


L-96 system with homogeneous assumptionThe homogeneous L-96 system

duj

dt= uj−1

(uj+1 − uj−2

)− duj + F , j = 0, 1, ..., J − 1,

satisfies 〈ui1 ui2 · · · uin

〉=〈ui1+l ui2+l · · · uin+l

〉, ∀l .

Fourier basis naturally becomes eigenfunctions for the L-96 operator, the moment equations aresimplified

Moment equations for homogeneous L-96

dū (t)

dt= −d (t) ū (t) +

1

J

J/2∑k=−J/2+1

rk (t) Γk + F (t) ,

drk (t)

dt= 2 [−Γk ū (t)− d (t)] rk (t) + QF ,kk , k = 0, 1, ..., J/2.

Note that Γk = cos4πk

J− cos 2πk

J, and the nonlinear flux QF becomes diagonal

QF ,kk′ = 2

(1

J

)1/2∑m

Re{〈ZmZ−m−k Zk〉

(e−2πı

2m+kJ − e2πı

m+2kJ

)}δkk′ ,

with energy conservation trQF = 0.


Models with consistent equilibrium single point statistics

QMF ,kk = QMF−,kk + Q

MF +,kk = −2dM,k (R) rk + σ

2M,k (R) .

Gaussian closure 1 (GC1-1pt): let

dM,k (R) = dM ≡ const., σ2M,k (R) = σ2M ≡ const.,

QGC1F = − (dM R + RdM ) + σ2M I ; (5)Gaussian closure 2 (GC2-1pt): let

dM,k (R) = �MJ

2

(trR)1/2

(trR∞)3/2≡ �M d̄ , σ2M,k (R) = �M

(trR)3/2

(trR∞)3/2

QGC2F = −�M(d̄R + Rd̄

)+ �M

(trR)3/2

(trR∞)3/2

I . (6)

Note that GC1-1pt includes parameters(dM , σ

2M

)and the nonlinear energy

trQGC1F = −2dMtrRM + Jσ2M may not be conserved, while GC2-1pt has one parameter�M and nonlinear energy conservation is enforced by construction trQ

GC2F = 0.

Single point statistics consistency can be fulfilled through tuning the control parameters.But these models are calibrated by ignoring spatial correlations and a natural informationbarrier is present which cannot be overcome by these imperfect models.


Equation for statistical energy

The statistical mean and variance at each individual grid point focus on the single point meanand variance of the system (and ignore the cross-correlation between different grids)

ū1pt =1

J

J−1∑j=0

uj = ū, R1pt =1

J

J/2∑k=−J/2+1

rk =1

JtrR.

By multiplying ū on both sides of the mean equation, and summing up all the modes in thevariance equation

dū2

dt= −2dū2 + 2ūF + 2

∑k

Γk rk ū.

dtrR

dt= 2

(−∑

k

Γk rk ū

)− 2dtrR + trQF .

It is convenient to define the statistical energy as

E (t) =1

2ū2 +

1

2trR.

Thus the corresponding dynamical equation for the energy E can be easily derived as

dE

dt= −2d (t) E + F · ū +

1

2trQF = −2E + F · ū,

with symmetry of nonlinear energy conservation assumed.


Realizability of 1-point statistics

In a similar way, we can calculate EM from imperfect model. With the same set ofassumptions, the responses between perfect and imperfect model satisfy

d

dt(δE − δEM ) = −2d (δE − δEM ) + F · (δū − δūM ) .

Proposition

Consider a system with homogenous statistical solution and an imperfect closuremodel with flux QMF satisfying symmetry of nonlinear energy conservation.Statistical equilibrium fidelity for one point statistics of the variancetrRM,∞ = trR∞ is satisfied if the same mean state ūM,∞ = ū∞ is achieved atequilibrium. Furthermore, with equilibrium consistency, we can successfully predictthe response for the one point statistics using the imperfect model‖trRδ,M − trRδ‖ = O

(δ2)

if we can predict the mean response successfully

‖ūδ,M − ūδ‖ = O(δ2).


Information barrier in 1-point statistics

the information distance between the truth and imperfect model prediction has the form

P(π, πM1pt

)= [S (πG )− S (π)] + P

πG , J−1∏j=0

πG1pt,j

+ J−1∑j=0

P(πG1pt,j , π

M1pt,j

).

The first part on the right hand side of is the inherent information barrier in Gaussianapproximation. Using spectral random fields

uG = ū +

J/2∑k=−J/2+1

R1/2k e

ikxj Ŵk , uG1pt = ū +

J/2∑k=−J/2+1

(∑J−1j=0 Rj

J

)1/2e ikxj Ŵk .

PropositionThe information barrier between the Gaussian random field and the uncorrelated one pointstatistics random field is given by

P

πG , J−1∏j=0

πG1pt

= J log det

(∑J−1j=0 Rj/J

)(∏J−1

j=0 det Rj

)1/J ∼ R−11pt (σmax − σmin)2 .


Hierarchy of closure models with consistent equilibrium foreach mode

QMF ,kk = QMF−,kk + Q

MF +,kk = −2dM,k (R) rk + σ

2M,k (R) .

Gaussian closure 1 (GC1): let

dM,k (R) = dM,k , σ2M,k (R) = σ

2M , dM,k = [−Γk ū∞ − d] + σ

2M/2rk,∞; (7)

Gaussian closure 2 (GC2): let

dM,k (R) = �1,k d̄ , σ2M,k (R) = �M

(trR)3/2

(trR∞)3/2

,

�1,k =2 [−Γk ū∞ − d] rk,∞ + �M

Jrk,∞/trR∞, d̄ =

J

2

(trR)1/2

(trR∞)3/2

. (8)

Above dM,k or �1,k is chosen so that the system has the same equilibrium mean ū∞ andvariance rk,∞ as the true model, therefore ensuring equilibrium consistency by finding thesteady state solutions through simple algebraic manipulations.Modified Gaussian closure (MQG): let

dM (R) =f (R)

f (R∞)N∞, σ

2M (R) = −

trQMQGF−

trQMQGF +,∞

[(Γk ū∞ + d) rk,∞δI+ + qs

],

with

N∞,kk = [Γk ū∞ + d] δI− −1

2qs r−1k,∞.

Above I− represents the unstable modes with Γk ū∞ + d > 0 while I+ is the stable oneswith Γk ū∞ + d ≤ 0.


Model calibration blending statistical response andinformation theory

Accurate modeling about the model sensitivity to various external perturbationsrequires the imperfect reduced-order models to correctly reflect the true system’s“memory” about its previous states.

the linear response operator can characterize the model sensitivity involvingthe nonlinear effects in the system regardless of the specific forms of theexternal perturbations.

empirical information theory can be used as the distance between these twooperators to calculate the unbiased and invariant measure for modeldistributions.


Empirical information theoryEmpirical information theory: A natural way to measure the lack of informationin one probability density from the imperfect model, πM , compared with the trueprobability density, π, is through the relative entropy, given by

P(π, πM

)=

∫π log

π

πM.

This functional on probability densities has two attractive features as a metric:

P (p, q) ≥ 0 with equality if and only if p = q;P (p, q) is invariant under general nonlinear changes of variables.

Model optimization: Consider a class of imperfect models M. The bestimperfect model for the coarse-grained variable u is the M∗ ∈M so that

P(π, πM

∗)

= minM∈M

P(π, πM

). (9)

Also, actual improvements in a given imperfect model with distribution πM (u)resulting in a new πMpost (u) should result in improved information for the perfect

model, so that P(π, πMpost

)≤ P

(π, πM

).


Empirical information theory

The actual model density πL (u) only reflects the best unbiased estimate of theperfect model given the L measurements ĒL.

P(π, πML′

)= P (π, πL) + P

(πL, π

ML′)

= [S (πL)− S (π)] + P(πL, π

ML′), for L′ ≤ L.

The entropy difference, S (πL)− S (π), precisely measures an intrinsic error fromthe L measurements of the perfect system.

Practical setup for calibration of contemporary models involves only mean andcovariance. If the density functions π, πM involve only the first two moments,

P(π, πM

)=

1

2(ū− ūM )T R−1 (ū− ūM ) +

1

2

(tr(RR−1M

)− J − log det

(RR−1M

)).

First term is the signal, reflecting the model error in the mean but weighted by theinverse of the covariance R−1, while the second term, the dispersion, involves onlythe model error covariance ratio, RR−1M .


Theories for improving imperfect model prediction skill

Linear response theory: The linear response of a system to small perturbationscan be predicted by observing appropriate statistics of the system in equilibriumwithout the need of applying any perturbations.

Assume the perfect model with equilibrium measure πeq, and external perturbation

ut = f (u) , fδ = f + δf ′ (t) , δf ′ = δw (u) f (t) .

For any functional A (u)

EδA (u) = Eeq (A) + δE ′A + O(δ2). (10)

Eeq (A) =

∫A (u)πeq (u) du, δE

′A =

∫ t

0

RA (t − s) δf ′ (s) ds, (11)

where RA (t) is called the linear response operator.

Linear response operator RA (t)The linear operator RA (t) can be calculated via the correlations in the unperturbed climate

RA (t) = 〈A (u (t + s)) B (u (s))〉 ,

with the special nonlinear functional perturbed by the change δw (u) f (t)

B (u) ≡LFPπeq (u)

πeq= −

divu (wπeq)

πeq.

The problem in calculating the leading order response above is that the linear response operatorRA (t) is expensive to calculate for general systems.

kicked response1:

π |t=0= πeq (u− δu) = πeq − δu · ∇πeq + O(δ2).

For δ small enough, the linear response operator RA (t) can be calculated bysolving the unperturbed system with a perturbed initial distribution

δRA (t) ≡ δu · RA =∫

A (u) δπ′ + O(δ2).

1Majda, Abramov, and Grote, Information theory and stochastics for multiscale nonlinearsystems vol. 25.


Link between statistical equilibrium fidelity and forecastingskill

Given the optimal model for the unperturbed climate πeq, how can we assess the error in theclimate change prediction P

(πδ, π

Mδ

)based on the unperturbed climate?

Under assumptions with diagonal covariance matrices R = diag (Rk ) and equilibrium modelfidelity P

(πG , π

MG

)= 0, the relative entropy between perturbed model density πMδ and the true

perturbed density πδ with small perturbation δ can be expanded as

P(πδ, π

Mδ

)= S

(πG ,δ

)− S (πδ)

+1

2

∑k

(δūk − δūM,k

)R−1k

(δūk − δūM,k

)+

1

4

∑k

R−2k(δRk − δRM,k

)2+ O

(δ3).

Here in the first line S(πG ,δ

)− S (πδ) is the intrinsic error from Gaussian approximation of the

system. Rk is the equilibrium variance in k-th component, and δūk and δRk are the linearresponse operators for the mean and variance in k-th component. (Majda & Gershogorin, PNAS,2011)


Optimization in the training phase

To summarize, consider a class of imperfect models, M. The optimal modelM∗ ∈M that ensures best information consistent responses is characterized withthe smallest additional information in the linear response operator RA, such that

∥∥∥P(

pδ, pM∗

δ

)∥∥∥L1([0,T ])

= minM∈M

∥∥P(pδ, p

Mδ

)∥∥L1([0,T ])

,

pMδ can be achieved through a kicked response procedure in the trainingphase compared with the actual observed data pδ in nature;

the information distance between perturbed responses P(pδ, p

Mδ

)can be

calculated through the expansion formula;

the information distance P(pδ (t) , p

Mδ (t)

)is measured at each time instant,

so the entire error is averaged under the L1-norm inside a time window [0,T ].


Lorenz 96 system

The Lorenz 96 model was introduced to mimick the large-scale behavior of themid-latitude atmosphere around a circle of constant latitude. The model is adiscrete periodic system described by the equations

dujdt

= uj−1 (uj+1 − uj−2)− duj + F , j = 0, 1, ..., J − 1. (12)

with J = 40 the number of grids and Fi the deterministic forcing. The quadraticpart conserves energy, that is, B (u, u) · u = 0. The L-96 model offers morecontrollable, simplified scenarios but still mimicking the complex features of thevastly more complex true turbulent system..

FLinear Analysis Mean Statistics

kbegin kend Reω ū E (ū) Ep

5 5 11 0.08 1.6344 53.4276 110.0132

6 4 12 0.0998 2.0127 81.0260 160.5063

8 4 12 0.1004 2.3418 109.6825 265.0064

16 4 12 0.0876 3.0869 190.5900 797.2113


F = 5

space

10 20 30 40

tim

e

0

5

10

15

20

-3

-2

-1

0

1

2

3

4

5

6

7

wavenumber

0 5 10 15 20P

erc

enta

ge o

f E

nerg

y %

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4F = 5

Re(uk)

-60 -40 -20 0 20 40 60

dis

trib

utio

n function

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

pdfs for Re(uk)

mode 7mode 8

time

0 5 10 15 20 25 30

corr

ela

tion function

-0.5

0

0.5

1Autocorrelation function for F = 5

F = 8

space

10 20 30 40

tim

e

0

5

10

15

20

-6

-4

-2

0

2

4

6

8

10

12

wavenumber

0 5 10 15 20

Perc

enta

ge o

f E

nerg

y %

0.02

0.04

0.06

0.08

0.1

0.12F = 8

Re(uk)

-150 -100 -50 0 50 100 150

dis

trib

ution function

0

0.005

0.01

0.015

0.02

pdfs for Re(uk)

mode 8mode 7

time

0 5 10 15 20

corr

ela

tion function

-0.2

0

0.2

0.4

0.6

0.8

1Autocorrelation function for F = 8

Figure: Numerical solutions of L-96 model in space-time through MC simulations for weaklychaotic (F = 5), and strongly chaotic (F = 8) regime.


Tuning imperfect model in the training phase

50 100 150 200

σM

2

0.1

0.2

0.3

0.4

0.5

P(δπ

,δπ

M)

tuning model, GC1

50 100 150 200

ǫM

0.1

0.2

0.3

0.4

0.5

P(δπ

,δπ

M)

tuning model, GC2

0 0.2 0.4 0.6 0.8

qsM

0.1

0.12

0.14

0.16

0.18

0.2

P(δπ

,δπ

M)

tuning model, MQG

(a) Information distance in response operator

0 0.5 1 1.5 2 2.5 3

time

-0.5

0

0.5

1

Ru

Linear response for the mean (optimal parameter)

truthGC1GC2MQG

0 0.5 1 1.5 2 2.5 3

time

-50

0

50

100

150

200

Rσ

2

Linear response for total variance (optimal parameter)

(b) Linear response operators for the mean and total variance

Figure: Training imperfect models in the training phase with full dimensional models by tuningmodel parameters. Time-averaged information distances in linear response operators for the

three closure models GC1, GC2, and MQG as the model parameter changes are shown in the

first row (the point where the information error is minimized is marked with a circle).


Prediction skill of closure methods in L-96 system

0 5 10 15 20 25 302.3

2.4

2.5State of the Mean

truthGC1GC2MQG

0 5 10 15 20 25 30

time

500

600

700Total variance

0 5 10 15 20 25 302.3

2.4


truthGC1GC2MQG

0 5 10 15 20 25 30

time

500

600

700Total variance

Model with non-optimal parameterModel with optimal parameter

Figure: Upward ramp-type forcing: imperfect model predictions in mean state and totalvariance for the closure methods GC1 (dotted-dashed blue), GC2 (dashed green), andMQG (solid red).



0 5 10 15 20 25 302.2

2.25

2.3


truthGC1GC2MQG

0 5 10 15 20 25 30

time

400

450

500

550Total variance

0 5 10 15 20 25 302.2

2.25

2.3


truthGC1GC2MQG

0 5 10 15 20 25 30

time

400

450

500

550Total variance


Figure: Downward ramp-type forcing: imperfect model predictions in mean state andtotal variance for the closure methods GC1 (dotted-dashed blue), GC2 (dashed green),and MQG (solid red).



0 5 10 15 202

2.2

2.4


truthGC1GC2MQG

0 5 10 15 20

time

400

500

600

700Total variance

0 5 10 15 202

2.2

2.4


truthGC1GC2MQG

0 5 10 15 20

time

400

500

600

700Total variance


Figure: Periodic forcing: imperfect model predictions in mean state and total variance forthe closure methods GC1 (dotted-dashed blue), GC2 (dashed green), and MQG (solidred).



0 5 10 15 20

2.2

2.4


truthGC1GC2MQG

0 5 10 15 20

time

400

500

600

Total variance

0 5 10 15 20

2.2

2.4


truthGC1GC2MQG

0 5 10 15 20

time

400

500

600

Total variance


Figure: Random forcing: imperfect model predictions in mean state and total variance forthe closure methods GC1 (dotted-dashed blue), GC2 (dashed green), and MQG (solidred).


Information errors of closure methods in L-96 system

0 10 20 308

8.2

8.4

8.6

8.8Forcing 1: upward ramp-type forcing

0 10 20 300

0.1

0.2

0.3

0.4total information error, GC1

optimal

non-optimal

0 10 20 300

0.02

0.04

0.06


optimal

non-optimal

0 10 20 300

0.005

0.01

0.015

0.02total information error, MQG

optimal

non-optimal

(a) climate change forcing

0 5 10 15 207

7.5

8

8.5

9Forcing 2: periodic forcing

0 5 10 15 200

0.1

0.2


optimal non-optimal

0 5 10 15 200

0.02

0.04


optimal non-optimal

0 5 10 15 200

0.02

0.04


optimal non-optimal

(b) periodic forcing

time

0 5 10 15 206

7

8

9Forcing 3: random forcing

time

0 5 10 15 200

0.2

0.4


optimal

non-optimal

time

0 5 10 15 200

0.02

0.04

0.06


optimal

non-optimal

time

0 5 10 15 200

0.02

0.04


optimal

non-optimal

(c) random forcing

Outline






Reduced-order models in low dimensional subspace – Meanevolution

Use s orthonormal eigenvectors of full covariance matrix R∞ to be {vi}si=1 (e.g.EOF modes). Form projection matrix P = [v1, · · · , vs ], and the reducedcovariance

Rs = P∗RP.

To compensate for the influence of quadratic terms only partially modeled due totruncation, introduce additional forcing G∞

d ū

dt= (L + D) ū + B (ū, ū) +

s∑

i,j=1

Rs,ij B (vi , vj ) + F + G∞.


Reduced-order models in low dimensional subspace – Meanevolution

d ū

dt= (L + D) ū + B (ū, ū) +

s∑

i,j=1

Rs,ij B (vi , vj ) + F + G∞.

Determine G∞ through statistical steady-state information for covariance andmean. The equilibrium equation is:

G∞ = − (L + D) ū∞ − B (ū∞, ū∞)−s∑

i,j=1

Rs∞,ij B (vi , vj )− F,

where Rs∞ = P∗R∞P, thereby guaranteeing that ū∞ is the steady state of the

truncated equation.


Reduced-order models in low dimensional subspace –Covariance evolution

The exact equation for the reduced covariance is

dRsdt

= Lv ,sRs + RsL∗v ,s + QF ,s .

QF ,s contains nonlinear dynamics between modes as well as linear dynamicsmissing from truncation. Note in general we no longer have conservationproperty, Tr [QF ,s ] 6= 0.Still we can use steady state information

QF ,s∞ = −Lv ,sRs∞ − Rs∞L∗v ,s .

As done in MQG, split QF ,s∞ into positive/negative definite parts as anoise/damping pair Q+F ,s∞, Q

−F ,s∞.


Low order models in reduced subspaceSolve the statistics in a low-order subspace spanned by leading EOFs, vk , k = 0, 1, · · · , s,{

dūdt

= −d (t) ū (t) + 1J

∑|k|≤s rk (t) Γk + F (t) + G ,

drkdt

= 2 [−Γk ū (t)− d (t)] rk (t) + QMF ,kk (trR) , k = 0, 1, ..., s.

First approach:

I Get the unresolved variances in the mean dynamics from the equilibrium statistics

G =trRs

trRs,∞G∞, G∞ = dū∞ −

1

J

∑|k|≤s

rk,∞Γk + F ;

I Scale the nonlinear flux QMF ,kk (trRs ) by only resolved modes trRs =∑|k|≤s rk .

Further improvements:

I Get unresolved variances from linear first order correction from FDT

rk,un ∼ rk,∞ + δr ′k = rk,∞ +∫ t

0Rrk (t − s) δF

′ (s) ds;

I Get the total variance from the scalar energy equation

dE

dt= −2d (t) E + F (t) · ū, trR = 2E (t)− ū2.


Tuning parameters of reduced models in a training regimeCOMPARISON OF PREDICTION SKILL OF REDUCED-ORDER GAUSSIAN CLOSURE MODELS 7

0 50 100 150 2000

0.1

0.2

0.3

0.4GC1 F = 8, Information distance with reduced models

corrected model

original model

σ2

0 50 100 150 2000

0.05

0.1

0.15GC1, corrected model

signal errordispersion error

0 50 100 150 2000

0.1

0.2

0.3

0.4GC1, original model


(a) GC1, optimal �21 = 91,�22 = 61

0 50 100 1500.02

0.04

0.06

0.08

0.1

0.12GC2 F = 8, Information distance with reduced models

corrected model

original model

ϵ

0 50 100 1500.01

0.02

0.03

0.04

0.05GC2, corrected model


0 50 100 1500.02

0.04

0.06

0.08

0.1

0.12GC2, original model


(b) GC2, optimal ✏1 =200, ✏2 = 20

0 0.2 0.4 0.6 0.8 10.02

0.04

0.06

0.08

0.1

0.12MQG F = 8, Information distance with reduced models

corrected model

original model

ds0 0.2 0.4 0.6 0.8 1

0

0.01

0.02

0.03

0.04

0.05

0.06ROMQG, corrected model


0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12ROMQG, original model


(c) ROMQG, optimalds1 = 0.455, ds2 = 0.145

Figure 2.1. Tuning parameters in the training regime for GC1, GC2, and ROMQG. The originalmodel results are shown in red while the corrected model results are in blue. It can be seen thatwith the original model there is always large error in the mean (signal error) compared with therelatively small error from the variances in the resolved modes (dispersion error). The correctedmethods can effectively reduce the signal error. And the fitting process is almost to find the pointwhere the error from mean decreases to the minimum amount. The point where the optimal valueis achieved is marked by circles.

in Figure 3.2-3.8 are compared. It can be seen that by adding the corrections from the global energy E, the errorin the variances are effectively reduced while large errors appear in the predictions for the mean. And if the linearresponse corrections for the mean dynamics are also considered, the overall improvements for both the mean andvariances are achieved with no further increasement for more computational costs. Specifically, we can summarizethe performance of these methods and the improvements through the corrections in the following several points:

• The effects of the correction from the global energy equation for total variance can be shown by comparingthe dispersion error between GC1 and GC2. In GC1, we only use constant damping and noise at eachtime instant, while in GC2 the variance equations can be corrected by the scale factor using total variancerather than that in the 1-dimensional subspace. Comparing the dispersion errors between original modeland the corrected model, it can be seen that the energy correction can effectively improve the performanceof the prediction for variances in GC2 results (and also for the ROMQG results when energy correction isapplied), while GC1 has little improvement for the variance since no correction method can be applied;

• The effects of the correction from linear response theory for mean dynamics can be seen by checking thesignal error part for all the three methods. When linear response corrections for the unresolved variances areadded to the corrected models, improvements can be seen for all three methods compared with the originalmodels with only steady state statistics applied. Especially for the case with also energy corrections, theaccuracy for the mean can in turn improve the accuracy in the global energy dynamics and further improvethe prediction for the variances. This can also be seen by comparing GC1 and GC2 results;

Figure: Tuning parameters in the training regime for reduced-order GC1, GC2, and ROMQGwith s = 3. The corrected methods can effectively reduce the signal error.


Check model sensitivity to perturbations

0 5 10 15 20 25 308

8.1

8.2

8.3

8.4

8.5

8.6

8.7

8.8

8.9

9upward ramp−type forcing

time

F

(a) upward ramp-type forcing

0 5 10 15 20 25 307.2

7.3

7.4

7.5

7.6

7.7

7.8

7.9

8downward ramp−type forcing

time

F

(b) downward ramp-type forcing

0 2 4 6 8 10 12 14 16 18 207.2

7.4

7.6

7.8

8

8.2

8.4

8.6

8.8

9periodic forcing

time

F

(c) periodic forcing

0 2 4 6 8 10 12 14 16 18 206.5

7

7.5

8

8.5

9random forcing

time

F

(d) random forcing


Upward Ramp-type Forcing

0 5 10 15 20 25 3062

64

66

68

70

72GC2, Variance in principal direction, F = 8

0 5 10 15 20 25 302.34

2.36

2.38

2.4

2.42

2.44

2.46GC2, State of the Mean, F =8

truth

original GC2

corrected GC2

0 5 10 15 20 25 3062

64

66

68

70


0 5 10 15 20 25 302.3

2.35

2.4

2.45


truth

original GC1

corrected GC1

time0 5 10 15 20 25 30

62

64

66

68

70

72ROMQG, Variance in principal direction, F = 8

time0 5 10 15 20 25 30

2.3

2.35

2.4

2.45

2.5ROMQG, State of the Mean, F =8

truth

original ROMQG

corrected ROMQG

time0 5 10 15 20 25 30

0

0.005

0.01

0.015

0.02

0.025total information distance, original method

GC2

GC1

ROMQG

time0 5 10 15 20 25 30

0

0.005

0.01

0.015

0.02

0.025total information distance, corrected method

Information Error with Upward Ramp-type Forcing— Signal part & Dispersion part

0 5 10 15 20 25 30

×10-3

0

0.2

0.4

0.6

0.8

1GC2, signal error

original GC2

corrected GC2

0 5 10 15 20 25 30

×10-3

0

0.5

1

1.5

2

2.5

3

3.5GC2, dispersion error

0 5 10 15 20 25 30

×10-3

0

1

2

3

4

5

6

7GC1, signal error

original GC1

corrected GC1

0 5 10 15 20 25 300

0.005

0.01

0.015

0.02GC1, dispersion error

time0 5 10 15 20 25 30

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014ROMQG, dispersion error

time0 5 10 15 20 25 30

×10-3

0

2

4

6

8ROMQG, signal error

original MQG

corrected MQG

Figure: Comparison of the information errors in signal and dispersion part separatelybetween the closure methods GC1, GC2, and MQG.

Downward Ramp-type Forcing

0 5 10 15 20 25 3054

56

58

60

62


0 5 10 15 20 25 302.2

2.25

2.3


truth

original GC2

corrected GC2

time0 5 10 15 20 25 30

54

56

58

60

62


time0 5 10 15 20 25 30

2.15

2.2

2.25

2.3


truth

original GC1

corrected GC1

0 5 10 15 20 25 3054

56

58

60

62


0 5 10 15 20 25 302.15

2.2

2.25

2.3

2.35ROMQG, State of the Mean F =8

truth

original ROMQG

corrected ROMQG

time0 5 10 15 20 25 30

0

0.005

0.01

0.015

0.02

0.025


GC2

GC1

ROMQG

time0 5 10 15 20 25 30

0

0.005

0.01

0.015

0.02

0.025


Periodic Forcing

0 2 4 6 8 10 12 14 16 18 2050

55

60

65

70


0 2 4 6 8 10 12 14 16 18 202.1

2.2

2.3

2.4

2.5


time0 2 4 6 8 10 12 14 16 18 20

50

55

60

65

70


time0 2 4 6 8 10 12 14 16 18 20

2.1

2.2

2.3

2.4

2.5


0 2 4 6 8 10 12 14 16 18 2050

55

60

65

70


0 2 4 6 8 10 12 14 16 18 202.1

2.2

2.3

2.4

2.5


time0 2 4 6 8 10 12 14 16 18 20

0

0.005

0.01

0.015

0.02


time0 2 4 6 8 10 12 14 16 18 20

0

0.005

0.01

0.015

0.02


Random Forcing

0 2 4 6 8 10 12 14 16 18 2050

55

60

65

70


0 2 4 6 8 10 12 14 16 18 202.1

2.2

2.3

2.4


time0 2 4 6 8 10 12 14 16 18 20

50

55

60

65

70


time0 2 4 6 8 10 12 14 16 18 20

2.1

2.2

2.3

2.4

2.5


0 2 4 6 8 10 12 14 16 18 2050

55

60

65

70


0 2 4 6 8 10 12 14 16 18 202.1

2.2

2.3

2.4

2.5


time0 2 4 6 8 10 12 14 16 18 20

0

0.01

0.02

0.03


time0 2 4 6 8 10 12 14 16 18 20

0

0.01

0.02

0.03


Check model sensitivity to perturbations in L-96 system

time0 5 10 15 20 25 30

0

0.005

0.01

0.015

0.02


GC2

GC1

ROMQG

time0 5 10 15 20 25 30

0

0.005

0.01

0.015

0.02


(g) climate change forcing

time0 2 4 6 8 10 12 14 16 18 20

0

0.005

0.01

0.015

0.02


time0 2 4 6 8 10 12 14 16 18 20

0

0.005

0.01

0.015

0.02


(h) periodic forcing

time0 2 4 6 8 10 12 14 16 18 20

0

0.01

0.02

0.03


time0 2 4 6 8 10 12 14 16 18 20

0

0.01

0.02

0.03


(i) random forcing

Outline






The two-layer flow with forcing and dissipation

The two-layer quasi-geostrophic model with baroclinic instability is one simplebut fully nonlinear fluid model capable in capturing the essential physics in oceanand atmosphere science.

Two-layer model

∂qψ

∂t+ J

(ψ, qψ

)+ J (τ, qτ ) + β

∂ψ

∂x+ U

∂∆τ

∂x= −

κ

2∆ (ψ − τ)− ν∆s qψ + Fψ ,

∂qτ

∂t+ J (ψ, qτ ) + J

(τ, qψ

)+ β

∂τ

∂x+ U

∂

∂x

(∆ψ + k2dψ

)= −

κ

2∆ (τ − ψ)− ν∆s qτ + Fτ .

Barotropic and baroclinic modes:

qψ = ∆ψ, ψ =1

2(ψ1 + ψ2) ,

qτ = ∆τ − k2dτ, τ =1

2(ψ1 − ψ2) .

Normalized energy-consistent modes:

pψ,k =qψ,k|k| = − |k|ψk,

pτ,k =qτ,k√|k|2 + k2d

= −√|k|2 + k2dτk.


Flow in high-latitude homogeneous regimes

regime N β kd U κ (kmin, kmax) σmax (kx , ky )max

ocean, high lat. 256 10 10 1 9 (2.25, 14.61) 0.411 (4, 0)

atmosphere, high lat. 256 1 4 0.2 0.2 (1.58, 6.78) 0.099 (2, 0)4 Reduced-order models with homogeneous mean flow 14

linear growth rate, ocean regime

-15 -10 -5 0 5 10 15zonal wavenumber

-15

-10

-5

0

5

10

15

me

rid

ion

al w

ave

nu

mb

er

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4nonlinear flux, sum of eigenvalues

-10 -5 0 5 10zonal wavenumber

-10

-5

0

5

10

me

rid

ion

al w

ave

nu

mb

er

-1.5

-1

-0.5

0

0.5

0 5 10 15-8

-6

-4

-2

0

2linear growth rate

0 5 10 15wavenumber

-8

-4

0

4

8nonlinear flux eigenvalues

(a) high-latitude ocean regime

linear growth rate, atmosphere regime

-6 -4 -2 0 2 4 6zonal wavenumber

-6

-4

-2

0

2

4

6

me

rid

ion

al w

ave

nu

mb

er

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09nonlinear flux, sum of eigenvalues

-4 -2 0 2 4zonal wavenumber

-4

-3

-2

-1

0

1

2

3

4

me

rid

ion

al w

ave

nu

mb

er

-0.1

-0.05

0

0.05

0.1

0 1 2 3 4 5 6-0.5

0

0.5linear growth rate

0 1 2 3 4 5 6wavenumber

-0.2

0

0.2nonlinear flux eigenvalues

(b) high-latitude atmosphere regime

Fig. 4.4: Stability from linear analysis and nonlinear flux in ocean (upper) and atmosphere (lower) regime using param-eters in Table 1. The growth rate from linear analysis including Ekman damping effect, and the eigenvaluesof the nonlinear flux trQF,k in each wavenumber combining barotropic and baroclinic mode are displayed inthe two-dimensional spectral domain. The last column shows the radial averaged growth rate and nonlinearflux eigenvalues in positive and negative components.

and validates the feasibility of using quasi-Gaussian approximation in calculating the linear response operators as a2⇥2 blocked system.

4.2 Testing reduced-order models in homogeneous regime

In the previous section we displayed the unperturbed statistical structures of the two-layer QG system in high-latituderegime with important nonlinear non-Gaussian features. The major task now is to test the reduced-order model skillsin predicting statistical responses to both stochastic and deterministic forcing perturbations as prescribed in (4.1) and(4.2) using only low-order closures. Only the large-scale modes |k| < 10 are calculated here, which cover the regime ofmost energetic directions. And to investigate the model sensitivity in each component, the perturbations in barotropicmode and baroclinic mode are applied individually in the tests. Three statistical quantities are of special importance

in characterizing the two-layer system, that is, the barotropic energy,��py,k

��2, baroclinic energy��pt,k

��2, and the heatflux ikxy⇤k tk. Due to the homogeneous statistics as we have shown before, the mean states become zero and thus wecan focus on the second-order variances in this situation. Therefore we will majorly check the reduced-order method’sability in capturing the responses in these key quantities. Like the Algorithm summarized in Section 3.3, the modelingprocess are decomposed into a training phase for finding optimal model parameters and a prediction phase for gettingresponses to various perturbations.

4.2.1 Training phase with linear response operator

Equilibrium consistency for the reduced-order methods

In testing the reduced-order models, we need to first guarantee climate consistency with the true unperturbed equilib-rium in statistical steady state. In the construction of low-order correction in Section 3.2, higher-order statistics fromequilibrium are combined with additional damping and noise corrections. It needs to be emphasized that neither theadditional damping and noise (3.7) nor the equilibrium high-order correction (3.6) is stable on its own even with the


Flow in low/mid-latitude regimes with zonal jets

5 Reduced-order models with inhomogeneous jet flow 25

0 20 40 60 80 100 120 1400

50

100

150mean in relative vorticity

barotropicbaroclinic

100 101 102

100

variance in relative vorticity


100 101 102

100

statistical energy

barotropicbaroclinicpotential

k-1.6

k-3.6

(a) ocean regime

0 20 40 60 80 100 120 1400

0.5

1

1.5

2mean in relative vorticity


100 101 102

100

variance in relative vorticity


100 101 10210-10

100

statistical energy

barotropicbaroclinicpotential

k-3.5

k-1.5

(b) atmosphere regime

Fig. 5.4: Time-averaged statistics (in radial average) in mean and second-order moments in low/mid-latitude regime.The first row compares the statistical mean states. The following two rows show the variances, and statisticalenergy, in barotropic and baroclinic modes, as well as the potential energy.

0 5 10 15 20 25 30-1

0

1mode (6,3)

0 5 10 15 20 25 30-1

0

1mode (6,2)

0 5 10 15 20 25 30-1

0

1mode (6,1)

-10 -5 0 5 100

0.2

0.4mode (6,3)

-10 -5 0 5 100

0.2

0.4mode (6,2)

-10 -5 0 5 100

0.2

0.4mode (6,1)

(a) ocean

0 10 20 30 40 50-1

0

1mode (1,0)

0 50 100 150 200-1

0

1mode (0,1)

0 50 100 150 200-1

0

1mode (0,2)

-3 -2 -1 0 1 2 30

0.5

1mode (1,0)

-3 -2 -1 0 1 2 30

0.5

1

1.5mode (0,1)

-3 -2 -1 0 1 2 30

1

2mode (0,2)

(b) atmosphere

Fig. 5.5: Autocorrelation functions and the probability distribution functions in low/mid-latitude ocean and atmo-sphere regime. The first three most energetic baroclinic modes are displayed. In the autocorrelations, thesolid lines show the real part while the dashed lines are the imaginary part of the functions. In the pdfs, thecorresponding Gaussian distributions with the same variance are also plotted in dashed black lines.


Exact statistical moment equations

The rescaled set of equations of (1) can be summarized in the abstract form

dpkdt

= Bk (pk, pk) + (Lk −Dk) pk + Fk, pk = (pψ,k, pτ,k)T ,∑

k

pk · Bk (pk, pk) ≡ 0,

where the normalized state variable pk =(pψ,k, pτ,k

)Tis in barotropic and baroclinic mode, the

linear operator is decomposed into non-symmetric part Lk involving β-effect and shear flow Uand dissipation part Dk, together with the forcing Fk combining deterministic component andstochastic component.

Lk =

ikxβ|k|2 − ikx U√1+(kd/|k|)2−ikx U 1−(kd/|k|)

2√

1+(kd/|k|)2ikxβ

|k|2+k2d

, Dk = κ2

−1 1√1+(kd/|k|)21√

1+(kd/|k|)2− 1

1+(kd/|k|)2

,

Fk =

fψ,k|k| + σψ,kẆψ,k|k|fτ,k√|k|2+k2

d

+στ,kẆτ,k√|k|2+k2

d

.



The rescaled set of equations of (1) can be summarized in the abstract form

dpkdt

= Bk (pk, pk) + (Lk −Dk) pk + Fk, pk = (pψ,k, pτ,k)T ,∑

k

pk · Bk (pk, pk) ≡ 0,

where the normalized state variable pk =(pψ,k, pτ,k

)Tis in barotropic and baroclinic mode, the

linear operator is decomposed into non-symmetric part Lk involving β-effect and shear flow Uand dissipation part Dk, together with the forcing Fk combining deterministic component andstochastic component.

Bk (pk,pk) =

[Bψ,kBτ,k

]

=

∑m+n=k

m⊥·n|k|

(|n||m|pψ,mpψ,n +

√|n|2+k2d|m|2+k2d

pτ,mpτ,n

)

∑m+n=k

m⊥·n√|k|2+k2d

(√|n|2+k2d|m| pψ,mpτ,n +

|n|√|m|2+k2d

pτ,mpψ,n

)

.



Statistical energy in each spectral mode

Rk = pk∗pk =

[|pψ,k|2 p∗ψ,kpτ,k

pψ,kp∗τ,k |pτ,k|

2

], p∗1,kp2,k = p̄

∗1,kp̄2,k + p

′∗1,kp′2,k.

Rk combines the variability in both mean and variance. The true statisticaldynamical equations form a 2× 2 system about Rk ∈ C2×2�

�dRkdt = (Lk −Dk) Rk + QF ,k + Qσ,k + c.c., |k| ≤ N,

QF ,k = pk∗Bk (pk,pk) =

[p∗ψ,kBψ,k p

∗ψ,kBτ,k

p∗τ,kBψ,k p∗τ,kBτ,k

],∑

k

trQF ,k ≡ 0.


Statistical energy conservation principle

The total statistical energy dynamical equation2 concerns the evolution of thetotal variability in mean and variance in response to external perturbations

E =1

2

∑1≤|k|≤N

|k|2 |ψk|2 +(|k|2 + k2d

)|τk|2 =

1

2

∑1≤|k|≤N

|pψ,k|2 + |pτ,k|2.

The exact dynamics for the statistical energy can be derived as

dE

dt+ Hf = −κE +

κ

2F − νH + Qσ.

Hf is the meridional heat flux due to baroclinic instability, F is the additional damping effectsdue to the non-symmetry in Ekman drag only applied on the bottom layer

Hf = k2d U

∫ψxτ = k

2d U∑

ikxψ∗k τk, F =∑

k2d |τk|2 + 2 |k|2 Reψ∗k τk.

2Majda, Statistical energy conservation principle for inhomogeneous turbulent dynamicalsystems, PNAS, 2016


Reduced-Order Statistical Energy ClosureThe true statistical model

dRkdt

= (Lk −Dk) Rk + QF ,k + Qσ,k + c.c., |k| ≤ N.

Qψ,k =∑

m+n=k

m⊥ · n|k|

|n||m|

p∗ψ,kpψ,mpψ,n +

√√√√ |n|2 + k2d|m|2 + k2d

p∗ψ,kpτ,mpτ,n

,Qτ,k =

∑m+n=k

m⊥ · n√|k|2 + k2d

√|n|2 + k2d|m|

p∗τ,kpψ,mpτ,n +|n|√|m|2 + k2d

p∗τ,kpτ,mpψ,n

.A preferred approach for the nonlinear flux QM,k combining both the detailedmodel energy mechanism and control over model sensitivity is proposed��

�

QM,k = Q−M,k+Q+M,k = f1 (E ) [− (NM,k,eq + dM ) RM,k]+f2 (E )

[Q+F ,k,eq + σ

2M,k

].

Higher-order corrections from equilibrium statistics:

QF ,k,eq = Q−F ,k,eq + Q

+F ,k,eq = − (Lk −Dk) Rk,eq + c.c., NM,k,eq =

1

2Q−F ,k,eqR

−1k,eq.

Additional damping and noise to model nonlinear flux:

QaddM,k = −dM RM,k + σ2M,k, dM =[

dM,ψ iωM−iωM dM,τ

].

Statistical energy-consistent scaling to improve model sensitivity:

f1 (E) =

(E

Eeq

)1/2, f2 (E) =

(E

Eeq

)3/2.


Reduced-Order Statistical Energy ClosureThe reduced-order approximation

dRM,kdt

= (Lk −Dk) RM,k + QM,k + Qσ,k + c.c., |k| ≤ M � N.

A preferred approach for the nonlinear flux QM,k combining both the detailedmodel energy mechanism and control over model sensitivity is proposed��

�


[Q+F ,k,eq + σ

2M,k

].




1

2Q−F ,k,eqR

−1k,eq.




].


f1 (E) =

(E

Eeq

)1/2, f2 (E) =

(E

Eeq

)3/2.


Reduced-Order Statistical Energy Closure

A preferred approach for the nonlinear flux QM,k combining both the detailedmodel energy mechanism and control over model sensitivity is proposed��

�


[Q+F ,k,eq + σ

2M,k

].




1

2Q−F ,k,eqR

−1k,eq.




].


f1 (E) =

(E

Eeq

)1/2, f2 (E) =

(E

Eeq

)3/2.


Climate fidelity for equilibrium

Equilibrium fidelity refers to the convergence to the same final unperturbedstatistical equilibrium Req in the reduced-order models RM in each resolvedcomponent.

Specifically, it requires that the model nonlinear flux correction term QMconverges to the truth, QM → QF ,eq, when no external perturbation is added

dRM,eqdt

= 0 = Lv (ūeq) RM,eq + RM,eqL∗v (ūeq) + Q

MF ,eq + Qσ → RM,eq = Req.

the first component(

NM,eq,Q+F ,eq

)comes from the true equilibrium

statistics.

climate consistency requires the second component correction makes nocontribution in the unperturbed case

ΣM =1

2dM Req, f1 (Eeq) = 1, f2 (Eeq) = 1.


Set-up for the numerical problem

The true statistics is calculated by a pseudo-spectra code with 128 spectralmodes zonally and meridionally, corresponding to 256× 256× 2 grid points intotal.

In the reduced-order methods, only the large-scale modes |k| ≤ 10 areresolved, which is about 0.15% of the full model resolution.

External forcing in stochastic and deterministic component:

The amplitude of the stochastic forcing σk Ẇk is introduced according to theequilibrium energy so that

σ2ψ,k = δσ20 |qψ,k|2eq, σ2τ,k = δσ20 |qτ,k|

2eq.

The deterministic forcing is introduced through a perturbation in thebackground shear Uδ = U + δU

δfψ,k = δUikx(− |k|2

)τk, δfτ,k = δUikx

(− |k|2 + k2d

)ψk.


Tuning parameters in the training phase4 Reduced-order models with homogeneous mean flow 18

tuning parameters with energy correction

0.50.50.5

0.5

0 0.05 0.1 0.15 0.2

dψ

0

0.05

0.1

0.15

0.2

dτ

0

0.5

1

1.5

(a) tuning with energy scaling

tuning parameters without energy correction

2.5

2.5

2.5

0 0.05 0.1 0.15 0.2

dψ

0

0.05

0.1

0.15

0.2

dτ

1.5

2

2.5

3

3.5

(b) tuning without energy scaling

0 1 2 3 4 5 6 7 8 9

wavenumber

0

0.2

0.4

0.6

0.8

1information error

w/ energy correctionw/o energy correction

0 1 2 3 4 5 6 7 8 9

wavenumber

0

2

4

6

8baroclinic energy

unperturbed spectrumperturbed responsew/ energy correctionw/o energy correction

(c) prediction and info. errors

Fig. 4.10: Tuning imperfect model parameters in the training phase. The information errors with varying model pa-rameters, dM =

�dy ,dt

�, are plotted for stochastic barotropic perturbation case. The errors using total

energy as scalar factor from the statistical equation and method without the scaling factor are compared.The prediction skill and information error with and without using the total energy correction are comparedin the last row for a typical test case of perturbing the barotropic mode.

corrections from low-order moments (that is, mean and variance) are used.The high-latitude ocean regime responses in barotropic and baroclinic energy and heat flux in large-scale wavenum-

bers to stochastic perturbations are first shown in Figure 4.11. The perturbation amplitude is chosen as ds20 = 0.5 of theequilibrium energy in the stochastic forcing (4.1) so that the response is large and nonlinear. We compare the responsesin perturbing only the barotropic mode and baroclinic mode. The most energetic and most sensitive scales take place atwavenumbers |k| = 4,5,6. Both barotropic and baroclinic perturbations can lead to large changes in a wide spectrumin both barotropic and baroclinic component due to the strong coupling between the modes. In the reduced-order meth-ods, only the first large-scale modes |k| < 10 are resolved, while the responses in these dominant modes are all capturedwith accuracy in both perturbation cases though the complicated higher-order interactions with small-scale modes arenot computed explicitly. Further the time-series with the total statistical energy from the equation (3.11) are compared.The dashed black lines mark the level of energy in unperturbed and perturbed case. In this regime, the total statisticalenergy can also be recovered exactly with little error. This in turn explains the high skill of the reduced-order modelsin predicting this regime. Instead, if we only consider the energy in the resolved subspace shown by blue lines, a largegap can be observed compared with the total energy. Figure 4.12 shows the results in the high-latitude atmosphereregime. Alternating blocked and unblocked structures appear in this regime and generate quite complicated statisticalfeatures. The leading mode |k| = 1 contains most of the energy and becomes highly sensitive to perturbations. Thereduced-order method keeps the skill in capturing the responses in the most sensitive directions in this difficult regime.Also it is observed that the baroclinic perturbation case becomes a little less accurate in both spectra and total statisticalenergy. This might be due to the stronger nonlinear energy interactions from baroclinic to barotropic mode.

A further test requires to check the model’s robustness in predicting perturbations with different amplitudes. Figure4.13 displays the prediction results with changing stochastic forcing amplitude ds20 in the barotropic modes. Thereduced-order model maintains the skill in predicting responses with various forcing strength, and the nonlinear trendsin the total resolved barotropic and baroclinic energy as well as the heat flux are captured compared with the linearprediction in the FDT shown by dashed lines.

Figure: Tuning imperfect model parameters in the training phase. The information errorswith varying model parameters, dM = (dψ, dτ ), are plotted for stochastic barotropicperturbation case.


High-latitude: Stochastic perturbation

The model is perturbed with stochastic forcing with variance amplitudeδσ20 = 0.5;

Either the barotropic or the baroclinic modes in large scales |k| ≤ 10 areperturbed.

4 Reduced-order models with homogeneous mean flow 19

0 1 2 3 4 5 6 7 8 90

5

10barotropic energy

unperturbedperturbed truthred. model

0 1 2 3 4 5 6 7 8 90

5

10baroclinic energy

0 1 2 3 4 5 6 7 8 9

wavenumber

-0.2

-0.1

0heat flux

0 1 2 3 4 5 6 7 8 90

5

10barotropic energy


0 1 2 3 4 5 6 7 8 90

5

10baroclinic energy

0 1 2 3 4 5 6 7 8 9

wavenumber

-0.2

-0.1

0heat flux

0 5 10 15 20 25 30

time

30

35

40

45time-series of total energy, barotropic perturbation

resolved

total

truth

(a) barotropic perturbation

0 5 10 15 20 25 30

time

30

35

40

45time-series of total energy, baroclinic perturbation

resolved

total

truth

(b) baroclinic perturbation

Fig. 4.11: Reduced-order model predictions to stochastic perturbations with amplitude ds20 = 0.5 in barotropic (left)and baroclinic (right) mode in high-latitude ocean regime. The spectra for the resolved modes 1 |k| < 10are compared. Black lines with circles show the perturbed model responses in the normalized variables,��py,k

��2 (barotropic energy),��pt,k

��2 (baroclinic energy), and ikxy⇤k tk (heat flux). The dashed black lines arethe unperturbed statistics. And the reduced order model predictions are in red lines. The last row shows themodel prediction of the energy equation in red lines and the energy in the resolved subspace shown in bluelines. For comparison, the unperturbed and perturbed total energy from the true system is shown in dashedblack lines.

(a) ocean regime

4 Reduced-order models with homogeneous mean flow 20

0 1 2 3 4 5 6 7 8 90

5

10

barotropic energy


0 1 2 3 4 5 6 7 8 90

2

4

baroclinic energy

0 1 2 3 4 5 6 7 8 9

wavenumber

-0.2

-0.1

0heat flux

0 1 2 3 4 5 6 7 8 90

5

10barotropic energy


0 1 2 3 4 5 6 7 8 90

2

4baroclinic energy

0 1 2 3 4 5 6 7 8 9

wavenumber

-0.2

-0.1

0heat flux

0 50 100 150 200

time

2

4

6

8

time-series of total energy, barotropic perturbation

resolved

total

truth

(a) barotropic perturbation

0 50 100 150 200

time

3

4

5

6

7time-series of total energy, baroclinic perturbation

resolved

total

truth

(b) baroclinic perturbation

Fig. 4.12: Reduced-order model predictions to stochastic perturbations with amplitude ds20 = 0.5 in barotropic (left)and baroclinic (right) mode in high-latitude atmosphere regime. The spectra for the resolved modes 1 |k| < 10 are compared. The last row shows the model prediction of the energy equation in red lines and theenergy in the resolved subspace shown in blue lines. For comparison, the unperturbed and perturbed totalenergy from the true system is shown in dashed black lines.

0 1 2 3 4 5 6 7 8 90

2

4

6

8barotropic energy

0 1 2 3 4 5 6 7 8 90

2

4

6

8baroclinic energy

0 1 2 3 4 5 6 7 8 9

wavenumber

-0.2

-0.15

-0.1

-0.05

0heat flux

(a) response in spectra

0 0.2 0.4 0.6 0.8

25

30

35

response in barotropic energy

0 0.2 0.4 0.6 0.836

38

40

42

44

46

response in baroclinic energy

0 0.2 0.4 0.6 0.8

δσ2

-0.9

-0.8

-0.7

-0.6response in heat flux

(b) total responses

Fig. 4.13: Imperfect model predictions to responses with changing perturbation amplitude ds20 in the high-latitudeocean regime (with barotropic perturbation). In the first part on the left we the predicted spectra with threedifferent perturbation amplitude, ds20 = 0.1,0.5,0.8, are shown. On the right the responses in total energyand heat flux with changing amplitudes ds20 2 [0,0.8] are plotted. For clarification in display, we only plotreduced model predictions by red markers and the truth is in black lines.

(b) atmosphere regime


High-latitude: Mean shear flow perturbation

The model is perturbed by changing the background zonal flow strength U;

The entire spectral is perturbed due to the mean flow advection in eachspectral mode.

5 Reduced-order models with inhomogeneous jet flow 21

0 1 2 3 4 5 6 7 8 90

5

10barotropic energy

0 1 2 3 4 5 6 7 8 90

5

10baroclinic energy

0 1 2 3 4 5 6 7 8 9

wavenumber

-0.2

-0.1

0heat flux

(a) U = 1.05

0 1 2 3 4 5 6 7 8 90

2

4

6barotropic energy

0 1 2 3 4 5 6 7 8 90

5

10baroclinic energy

0 1 2 3 4 5 6 7 8 9

wavenumber

-0.2

-0.1

0heat flux

(b) U = 0.95

Fig. 4.14: Reduced-order model predictions to mean shear flow perturbation dU = ±0.05 (that is, 5% of the originalvalue U0) in the ocean regime. The spectra for the resolved modes 1 |k| < 10 are compared. Black lineswith circles show the perturbed model responses in the normalized barotropic energy, baroclinic energy, andheat flux. The dashed black lines are the unperturbed statistics. And the reduced order model predictionsare in red lines.

Model responses to the perturbed mean shear dU

In checking the model responses to deterministic forcing, we introduce the forcing perturbation by changing the back-ground jet strength U as in (4.2). The same perturbation is tested in [24] for a more complicated reduced-order modifiedquasi-Gaussian closure (RoMQG), and we test the same perturbation form here under our systematic reduced-ordermodeling framework. Note that the deterministic perturbation in (4.2) forms a more difficult test case compared withthe stochastic forcing (4.1) because the forcing is applied along all wavenumbers with stronger mean-fluctuation in-teractions involved. On the other hand, for the reduced order methods, only the perturbations at the limited resolvedmodes are quantified. This gives the inherent difficulty for applying the reduced order models to this kind of perturba-tions since we have no knowledge of the unresolved modes where large amount of energy is contained. Therefore thestatistical energy equation (3.11) plays a crucial role.

The results with mean flow perturbations dU = ±0.05 in the ocean regime and perturbations dU = 0.02,�0.01 inthe atmosphere regime are shown in Figure 4.14 and 4.15 separately. The perturbation accounts for about 5%-10% ofthe original shear strength U , and the correspon

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