Fluctuation-Dissipation and Linear ResponseTheory

Instructor: Nan Chen

Department of MathematicsUniversity of Wisconsin-Madison

Graduate Course, MATH 801, Spring 2020MWF 8:50AM – 9:40AM, B239 Van Vleck Hall

Main reference

Majda, Andrew, Rafail V. Abramov, and Marcus J. Grote. Information theory and stochastics formultiscale nonlinear systems. Vol. 25. American Mathematical Soc., 2005.

Majda, Andrew, and Nan Chen. "Model error, information barriers, state estimation and predictionin complex multiscale systems." Entropy 20.9 (2018): 644.

Leith, C. E. Climate response and fluctuation dissipation. Journal of the Atmospheric Sciences32.10 (1975): 2022-2026..

1 / 38

A simple example.

Computing the mean response to the change of forcing in linear models.Consider a general linear system with noise

dudt

= Lu + F + σW. (1)

I L is a linear operator whose eigenvalues all have negative real part, whichguarantees the existence of a Gaussian statistical steady state of u.

I F is an external forcing,

I W is stochastic white noise.

Now we impose a forcing perturbation δF to the original system in (1)

duδ

dt= Luδ + F + δF + σW. (2)

What is the mean response 〈uδ − u〉 due to δF when t →∞?

2 / 38

A simple example.

Computing the mean response to the change of forcing in linear models.Consider a general linear system with noise

dudt

= Lu + F + σW. (1)

I L is a linear operator whose eigenvalues all have negative real part, whichguarantees the existence of a Gaussian statistical steady state of u.

I F is an external forcing,

I W is stochastic white noise.

Now we impose a forcing perturbation δF to the original system in (1)

duδ

dt= Luδ + F + δF + σW. (2)

What is the mean response 〈uδ − u〉 due to δF when t →∞?

2 / 38

dudt

= Lu + F + σW. (1)

duδ

dt= Luδ + F + δF + σW. (2)

Since both (1) and (2) are linear models, the mean values 〈u〉 and 〈uδ〉 at the statisticalsteady state can be written explicitly,

〈u〉 = L−1F, and 〈uδ〉 = L−1 (δF + F) .

Therefore, the mean response of u to the forcing perturbation δF is given by

〈δu〉 = 〈uδ − u〉 = L−1δF. (3)

In practice, model error is usually inevitable. A suitable imperfect model is expected togenerate at least the same mean response as in the perfect model in (3) in addition tothe model fidelity.

3 / 38

A slightly more complicated example.

A slow-fast system and reduced model.

An important practical issue for complex dynamical systems is how to account for theindirect influence of the unresolved variables uII on the response of the resolvedvariables uI beyond bare truncation formulae.

Therefore, developing reduced stochastic models for the variables uI with high skill forthe low-frequency response is a central issue.

Below, we will show that the stochastic mode reduction techniques are able to producea reduced stochastic model for the low-frequency variables uI. Despite its simplicity,such a reduced stochastic model has exactly the same mean response operator asthat in the complete stochastic system!

4 / 38

Consider a linear multiscale stochastic model for variables u = (uI,uII)T given by

duI

dt= L11uI + L12uII + FI,

duII

dt= L21uI −

Γ

εL22uII + FII +

σ

ε1/2W,

(4)

which can also be written in a compact form

dudt

= Lεu + σεW + F, with Lε =

(L11 L12

L21 − Γε

). (5)

The parameter ε > 0 in (4) can be large or small. Here we require that Lε haseigenvalues with a negative real part for all ε and in particular

(L11u,u) < 0, (Γu,u) > 0. (6)

for u 6= 0. These requirements guarantee that Lε is invertible and the climate meanstate is given by 〈u〉 = (Lε)−1〈F〉. This together with (5) and (6) implies in particularthat the change in the first components of the climate mean state, δ〈uI〉, in response toa change in forcing, δF1, is given exactly by

δ〈uI〉 = (Lε)−111 δFI,

(Lε)−111 = (L11 + εL12Γ−1L21)−1.

(7)

5 / 38

Stochastic mode reduction techniques systematically produce a reduced stochasticmodel for the variables uI alone, which is a valid model in the limit ε→ 0; such modelsoften have significant skill for moderate variables of ε. Here we focus on their skill inproducing infinite time-mean response in (7) of the full dynamics from (4) independentof ε.

duI

dt= L11uI + L12uII + FI,

duII

dt= L21uI −

Γ

εL22uII + FII +

σ

ε1/2W,

(4)

First, the local equations in (4) can be rewritten exactly as an equivalent equation withmemory in time for the uI variable alone given by

duI

dt= L11uI+FI+L12

∫ t

0e−(Γ/ε)(t−s)

[L21uI(s)+FII(s)

]ds+L12ε

−1/2∫ t

0e−(Γ/ε)(t−s)σdW(s).

(8)For simplicity in exposition, zero initial data are assumed for uII. The second and thirdterms in (8) simplify in the limit ε→ 0 and yield reduced simplified local stochasticdynamics for uI alone given by

d uI

dt=(L11 + εL12Γ−1L21

)uI + ε1/2L12(−Γ)−1σW + FI + εL12(−Γ)−1FII. (9)

6 / 38

d uI

dt=(L11 + εL12Γ−1L21

)uI + ε1/2L12(−Γ)−1σW + FI + εL12(−Γ)−1FII. (9)

This is an explicit example of stochastic mode reduction where the variables uII havebeen eliminated and there is a reduced local stochastic equation for uI alone withexplicit corrections that reflect the interaction with the unresolved variables.

Here, we address the skill of the approximation in (9) in recovering the exact meanclimate response in (7) independent of ε. Reasoning as discussed earlier in generalbelow (6), the response of the climate mean in (9) to a change in forcing is givenexactly by

δ〈uI〉 =(L11 + εL12Γ−1L21

)−1δFI. (10)

Remarkably, the mean response operator in (10) coincides exactly with the projectedmean climate response operator in (7) for the complete stochastic system in (4) for anyvalue of ε > 0! This general result points to the surprisingly high skill of the responsefor the reduced stochastic model in calculating the mean climate response.

7 / 38

Different from the linear models above, direct calculations of the response in generalnonlinear models becomes a great challenge.

Nevertheless, fluctuation-dissipation theorem (FDT) provides an efficient and practicalway for computing the response in nonlinear systems.

8 / 38

The general framework of the fluctuation-dissipation theorem(FDT)

Consider a general nonlinear dynamical system with noise

dudt

= F(u) + σ(u)W, (11)

where u ∈ RN is the state variables, σ is an N × K noise matrix and W ∈ RK is aK -dimensional white noise. The evolution of the PDF p(u) associated with u is drivenby the so-called Fokker-Planck equation,

∂p∂t

= −divu[F(u)p] +12

divu∇u(Σp) ≡ LFPp, (12)

where Σ = σσT and p|t=0 = p0(u). Let peq(u) be the smooth equilibrium PDF thatsatisfies LFPpeq = 0. The statistics of some function A(u) are determined by

〈A(u)〉 =

∫A(u)peq(u)du. (13)

9 / 38

Now consider the dynamical in (11) by a small external forcing perturbation δF(u, t).The perturbed system reads

dudt

= F(u) + δF(u, t) + σ(u)W, (14)

We further assume a very natural explicit time-separable structure for δF(u, t), namely

δF(u, t) = δw(u)f (t). (15)

Then the Fokker-Planck equation associated with the perturbed system (14) is given by

∂pδ

∂t= LFPpδ + δLext pδ,

where δLext pδ = Lext p·δf (t), Lext p = −∂

∂ui

(wi (u)p

), 1 ≤ i ≤ N.

(16)

Similar to (13), for the perturbed system (16) the expected value of the nonlinearfunctional A(u) is given by

〈A(u)〉δ =

∫A(u)pδ(u)du. (17)

The goal here is to calculate the change in the expected value

δ〈A(u)〉 = 〈A(u)〉δ − 〈A(u)〉. (18)

10 / 38

To this end, let’s take the difference between (12) and (16),

∂

∂tδp = LFPδp + δLext peq + δLextδp, (19)

where δp = pδ − peq is the small perturbation in the PDF. Ignoring the higher orderterm δLextδp assuming δ is small, (19) reduces to

∂

∂tδp = LFPδp + δLext peq ,

δp|t=0 = 0.(20)

Since LFP is a linear operator, with the semigroup notation, exp[tLFP ], for this solutionoperator, the solution of (20) is written concisely as

δp =

∫ T

0exp

[(t − t ′)LFP

](δLext (t ′)peq

)dt ′. (21)

11 / 38

Now combining (21) with (18) and (16), we arrive at the linear response formula

δ〈A(u)〉(t) =

∫RN

A(u)δp(u, t)du =

∫ t

0R(t − t ′) · δF(t ′)dt ′, (22)

where the vector linear response operator is given by

R(t) =

∫RN

A(u)(

exp[tLFP ][Lext peq ])(u)du. (23)

This general calculation is the first step in the FDT. However, for nonlinear systems withmany degrees of freedom, direct use of the formula in (23) is completely impracticalbecause the exponential exp[tLFP ], cannot be calculated directly.

12 / 38

FDT states that if δ is small enough, then the leading-order correction to the statisticsin (13) becomes

δ〈A(u)〉(t) =

∫ t

0R(t − s)δf (s)ds, (24)

where R(t) is the linear response operator, which is calculated through correlationfunctions in the unperturbed climate:

R(t) = 〈A[u(t)]B[u(0)]〉, B(u) = −divu(wpeq)

peq. (25)

Clearly, calculating the correlation functions in (25) via FDT is much cheaper andpractical than directly computing the linear response operator (23).

13 / 38

Some comments.Recall the perturbed system

dudt

= F(u) + δF(u, t) + σ(u)W, (14)

We further assume a very natural explicit time-separable structure for δF(u, t), namely

δF(u, t) = δw(u)f (t). (15)

I if w has no dependence on u, then δF(t) naturally represents the forcingperturbation.

I If w(u) is a linear function of u, then δF(u, t) represents the perturbation indissipation.

I if the functional A(u) in (24) is given by A(u) = u, then the response computed isfor the statistical mean.

I Likewise, A(u) = (u− u)2 is used for computing the response in the variance.

Notably, FDT (24)–(25) does not require any linearization of the underlyingdynamics in (11). Therefore, it captures the nonlinear features in the underlyingturbulent systems.

14 / 38

Approximate FDT methods.

FDT:

δ〈A(u)〉(t) =

∫ t

0R(t − s)δf (s)ds, (24)

R(t) = 〈A[u(t)]B[u(0)]〉, B(u) = −divu(wpeq)

peq. (25)

One major issue in applying FDT directly in the form of (25) is that the equilibriummeasure peq(u) is not known exactly. Therefore, different approximate methods havebeen proposed to compute the linear response operator.

15 / 38

Quasi-Gaussian (qG) FDT.Among all the approximate methods, the quasi-Gaussian (qG) approximation is one ofthe most effective approach. It uses the approximate equilibrium measure

pGeq = CN exp

[−

12

(u− u)∗R−1(u− u)

], (26)

where the mean u and covariance matrix R match those in the equilibrium peq . Onethen calculates

BG(u) = −divu(wpG

eq)

pGeq

(27)

and replace B(u) by BG(u) in the qG FDT. The correlation in (25) with thisapproximation is calculated by integrating the original system in (11) over a longtrajectory or an ensemble of trajectories covering the attractor for shorter timesassuming mixing and ergodicity for (25).

For the special case of changes in external forcing w(u)i = ei , i ≤ i ≤ N, the responseoperator for the qG FDT is given by the matrix

RG(t) = 〈A(u(t))C−1(u− u)(0)〉. (28)

16 / 38

Kicked FDT.One strategy to approximate the linear response operator which avoids directevaluation of πeq through the FDT formula is through the kicked response of anunperturbed system to a perturbation δu of the initial state from the equilibriummeasure, that is,

π |t=0= πeq (u− δu) = πeq − δu · ∇πeq + O(δ2). (29)

One important advantage of adopting this kicked response strategy is that higher orderstatistics due to nonlinear dynamics will not be ignored (compared with other linearizedstrategy using only Gaussian statistics). Then the kicked response theory gives thefollowing fact for calculating the linear response operator.

Fact: For δ small enough, the linear response operator R (t) can be calculated bysolving the unperturbed system (11) with a perturbed initial distribution in (29).Therefore, the linear response operator can be achieved through

R (t) =

∫A (u) δπ + O

(δ2). (30)

Here δπ is the resulting leading order expansion of the transient density function fromunperturbed dynamics using initial value perturbation.

17 / 38

Information barrier for linear reduced models in capturing theresponse in the second order statistics

I We use a simple 2D example to systematically illustrate the procedure of theFDT as introduced above.

I We also aim at showing the information barrier for linear reduced models incapturing the response beyond the first-order statistics.

18 / 38

The perfect model here is the SPEKF type of non-Gaussian model, except that forsimplicity we adopt a constant forcing fu in the equation of u,

dudt

= −γu + fu + σuWu ,

dγdt

= −dγ(γ − γ) + σγWγ .

(31)

The following parameters are used in (31) in order to generate non-Gaussian statisticsof u,

σu = 0.5, dγ = 1.3, σγ = 1, γ = 1, fu = 1. (32)

19 / 38

250 300 350 400 450 500 550 600 650 700 750t

0

2

4

6

8

10(a) Sample trajectory of u

0 5 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8(b) PDF of u

TruthGaussian fit

250 300 350 400 450 500 550 600 650 700 750t

-2

-1

0

1

2

3

4(c) Sample trajectory of

-2 0 2 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7(d) PDF of

0 5 10

10 -4

10 -2

10 0Log scale

Figure: Panels (a)–(b): Sample trajectories of u and γ in the SPEKF type of non-Gaussian model (31). Panels(c)–(d): The corresponding PDFs. The subpanel within panel (b) shows the PDF in logarithm scale, with the redcurves representing the Gaussian fit. The parameters associated with these figures are given in (32).

20 / 38

Now we add a forcing perturbation δfu(t) to the model in (31),

dudt

= −γu + fu + δfu(t) + σuWu ,

dγdt

= −dγ(γ − γ) + σγWγ .

(33)

The function δfu(t) is a ramp-type perturbation with the following form

δfu(t) = A0tanh(a(t − tc)) + tanh(atc)

1 + tanh(atc), (34)

withA0 = 0.1, a = 1, tc = 2. (35)

21 / 38

-10 -5 0 5 101.2

1.3

1.4

1.5

1.6(a) u(t)

-10 -5 0 5 100.6

0.7

0.8

0.9

1(b) Var(u(t))

-10 -5 0 5 10t

1

1.02

1.04

1.06

1.08

1.1

(c) fu and f

u + f

u(t)

Response to forcing perturbation Equilibrium of the unperturbed model New equilibriumRelaxation towards equilibrium

Forcing perturbationfu

fu + f

u(t)

Figure: Time evolution of the mean 〈u〉 and variance Var(u(t)) of u (panels (a) and (b)) and thecorresponding forcing fu(t) (panel (c)). The forcing fu(t) is perturbed at time t = 0 with δf (t) given in (34). Themean and variance of u have corresponding responses and eventually arrive at a new equilibrium.

22 / 38

Note that these responses are computed by using the analytical formulas of the timeevolutions of the statistics, which are accurate. They are known as the idealizedresponses.

23 / 38

In most realistic scenarios, the true dynamics is unknown or it is too expensive to runthe full perfect model. Therefore, simplified or reduced models are widely used incomputing the responses.

One type of the simple models that are widely adopted is the linear model,

duM = −dMu uM + f M

u + σMu W . (36)

Note that adopting such a linear model to compute the responses shares the samephilosophy as one of the adhoc-FDT procedures, where linear regression approximatestochastic model is used for the variables of interest before applying FDT.

The three parameters in (36) are calibrated by matching the equilibrium mean,equilibrium variance and decorrelation time with those of u in the perfect model (31),where

〈u〉eq =f Mu

dMu, Var(u)eq =

(σMu )2

2dMu, τcorr =

1dM

u. (37)

With such calibrations, the linear model (36) automatically fit the unperturbed meanand variance at t = 0. Now we add the same forcing perturbation to the linear model,

duM = −dMu uM + f M

u + δfu(t) + σMu W . (38)

24 / 38

Since the statistics in the linear model is Gaussian, the qG FDT formulas in (26)–(28)become rigorous with no approximation. In computing the responses to the forcingperturbation δfu(t) in the mean and variance of u, the functional A(u(t)) is set to be

Response in the mean : A(u(t)) = uM ,

Response in the variance : A(u(t)) = (uM − uM )2,(39)

respectively.

As comparison, we also show the responses using the qG FDT based on the perfectmodel (31). Since the forcing perturbation is only on the direction of u, w(u) in (27) isgiven by w(u) = [1, 0]T .

25 / 38

0 2 4 6 8 10 12t

1

1.1

1.2

1.3

1.4

1.5

1.6(a) u(t)

Idealized responsePerfect model with qG ApproxLinear model

0 2 4 6 8 10 12t

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1(b) Var(u(t))

Figure: Responses to the mean 〈u〉 and variance Var(u(t)) of u with the forcing perturbation δf (t) given in(34). The perturbation starts at time t = 0, which is consistent with that in Figure 2.

26 / 38

Using information theory for finding the most sensitive changedirections

I An important question in climate change is how to find the most sensitivedirections for climate change given the present climate.

I Consider a family of parameters λ ∈ Rp with πλ the PDF of the climate as afunction of λ. Here λ = 0 corresponds to the unperturbed state or the presentclimate π.

I In light of the information theoretic framework, the most sensitive perturbedclimate is the one with the largest uncertainty related to the unperturbed one,

P(πλ∗ , π) = maxλ∈Rp

P(πλ, π).

I For small values of λ, we have

P(πλ, π) = λ · I(π)λ + O(|λ|3), (40)

where λ · I(π)λ is the quadratic form in λ given by the Fisher information

λ · I(π)λ =

∫(λ · ∇λπ)2

π, (41)

and the elements of the matrix of this quadratic form are given by

Ikj (π) =

∫ ∂π∂λk

∂π∂λj

π. (42)

27 / 38

Derivations.Consider the two PDFs πθ(u) and πθ′ (u) with θ′ = θ + δθ and δθ is a smallincrement. Applying the Talyor’s expansion to πθ and lnπθ yields

πθ′ = πθ +∇θπθδθ +12δθT∇2

θπθδθ + O(δθ3)

lnπθ′ = lnπθ +1πθ∇θπθδθ +

12δθT

(−

1π2θ

(∇θ)2 +1πθ∇2

θπθ

)δθ + O(δθ3).

(43)

28 / 38

With (43) in hand, now we compute the relative entropy in (40).P(πθ′ , πθ)

=

∫πθ′ ln

πθ′

πθ

=

∫πθ′ lnπθ′ −

∫πθ′ lnπθ

=

∫ (πθ +∇θπθδθ +

1

2δθ

T∇2θπθδθ

)(lnπθ +

1

πθ

∇θπθδθ +1

2δθ

T(−

1

π2θ

(∇θ)2 +1

πθ

∇2θπθ

)δθ

)

−∫ (

πθ +∇θπθδθ +1

2δθ

T∇2θπθδθ

)lnπθ + O(δθ3)

=

∫πθ lnπθ +∇θπθδθ +

1

2πθδθ

T(−

1

π2θ

(∇θ)2 +1

πθ

∇2θπθ

)δθ +∇θπθ lnπθδθ + δθ

T 1

πθ

(∇θπθ)2δθ

+1

2lnπθδθ

T∇2θπθδθ − πθ lnπθ −∇θπθ lnπθδθ −

1

2lnπθδθ

T∇2θπθδθ + O(δθ3)

= ∇θ

∫πθδθ +

1

2δθ

T(∫ 1

πθ

(∇θπθ)2)δθ +

1

2δθ

T(∇2

θ

∫πθ

)δθ + O(δθ3)

=1

2δθ

T ·(∫ 1

πθ

(∇θπθ)2)· δθ + O(δθ3)

=1

2

(∫(δθ · ∇θπθ)2

πθ

)+ O(δθ3),

(44)

where we have made use of the fact that∫πθ ≡ 1 and therefore ∇θ

∫πθ = 0. some

regularity assumptions are also required such that the integral and gradient operatorcan be interchanged. Clearly, the final result is the Fisher information. Note that δθhere is λ in (41).

29 / 38

Example 1. An one-dimensional linear model,

dudt

= −au + f + σW ,

the equilibrium PDF of which is Gaussian and is given by N (u,C),

π(u) = NC exp

(−

(u − u)2

2C

),

with

u =fa, C =

σ2

2a.

The two dimensional parameters λ = (f , a)T ∈ R2 for external forcing and dissipationare the natural parameters which are varied in this model. Therefore, thecorresponding I(λ) is a 2× 2 matrix with entries Iij , i, j = 1, 2.

30 / 38

It is straightforward to compute the first-order derivatives of π with respect to f and a,

∂π

∂f=

u − uaC

π,

∂π

∂a=

σ2

4a2Cπ −

f (u − u)

a2Cπ −

σ2(u − u)2

4a2C2π.

(45)

In light of (42) and (45), the four elements of I have the following explicit expressions,

I11 =

∫ (∂π∂f

)2

πdu =

∫(u − u)2

C2

π

a2du =

1Ca2

,

I12 = I21 =

∫ ∂π∂f

∂π∂a

πdu =

∫u − u

aC

(σ2

4a2C−

f (u − u)

a2C−σ2(u − u)2

4a2C2

)πdu = −

fa3C

,

I22 =

∫ (∂π∂a

)2

π=

∫ (σ2

4a2C−

f (u − u)

a2Cπ −

σ2(u − u)2

4a2C2

)2

πdu

=

∫ ( σ2

4a2C

)2

+

(f (u − u)

a2C

)2+

(σ2(u − u)2

4a2C2

)2

− 2σ2

4a2Cσ2(u − u)2

4a2C2

πdu

= −f 2

Ca4+

σ4

8C2a4.

(46)

31 / 38

Now let’s implement numerical experiments. The following two group of parameters areused,

(a) : a = 1, f = 1, σ = 1,

(b) : a = 1, f = 1, σ = 3.(47)

Since I is a 2× 2 matrix, there are only two eigenmodes. The eigenvector wassociated with the larger eigenvalue corresponds to the most sensitive direction withrespect to the perturbation (δf , δa)T .By plugging the model parameters (47) into the I matrix in (46), we find the mostsensitive direction in both the cases:

(a) : e∗π =

(−0.66180.7497

), (b) : e∗π =

(−0.35540.9347

). (48)

32 / 38

To gain more intuition on the results of these most sensitive directions, we make use ofthe simple structure of the model to solve this problem in an alternative way. In fact,given small perturbations (δf , δa)T to (f , a)T , the corresponding perturbed mean andvariance can be written down explicitly

uδ =f + δfa + δa

, Cδ =σ2

2(a + δa). (49)

Since both the unperturbed and perturbed PDFs are Gaussian, we can easily makeuse of the explicit formula of the relative entropy to compute P(π, πδ) and find the mostsensitive direction in the two-dimensional parameter space,

Signal =12

(fa−

f + δfa + δa

)2(σ2

2a

)−1

=12

(fa + fδa− fa− aδf )2

a2(a + δa)2

2aσ2

=(fδa− aδf )2

aσ2+ o

(δa3)

+ o(δa2δf

)+ o

(δaδf 2

)Dispersion = −

12

ln(

a + δaa

)+

12

(a + δa

a− 1)

= −12

ln(

1 +δaa

)+

12δaa

= −12

(δaa−

12

(δaa

)2+ o

(δaa

)3)

+12δaa

=14

(δaa

)2+ o

(δaa

)3.

(50)

Note that the dispersion part depends only on the perturbation in the dissipation δasince f has no effect on the variance. In addition, it is clear that δa and δf should haveopposite signs in order to maximize the relative entropy in the signal part. 33 / 38

34 / 38

Example 2. Consider the cubic nonlinear model,

dudt

= (f + au + bu2 − cu3) + (A− Bu)WC + σWA. (51)

Assume A = B = 0. The equilibrium PDF is given by the following explicit formula

π(u) = N0 exp(

2σ2

(fu +

a2

u2 +b3

u3 −c4

u4))

. (52)

Again look at the perturbation in the two-dimensional parameter space λ = (f , a)T ,which representing the changes in forcing and damping. We aim at solving theeigenvectors of the 2× 2 matrix I(λ). To explicitly write down the elements in I(λ), wedefine

Hk =

∫ukψ(u)du, k ≥ 0 with ψ(u) = exp

(2σ2

(fu +

a2

u2 +b3

u3 −c4

u4))

,

(53)Straightforward calculations show that

I11 =

∫( ∂π∂f )2

πdu =

4σ4H4

0(H0H2 − H2

1 ),

I12 = I21 =

∫ ∂π∂f

∂π∂a

πdu =

2σ4H4

0(H0H3 − H1H2),

I22 =

∫( ∂π∂a )2

πdu =

1σ4H4

0(H0H2 − H2

2 ).

(54)

35 / 38

Now we focus on the case studies in the following three regimes,

Regime I : f = 1.8, a = 0, b = −5.4, c = 4, σ =√

0.5,

Regime II : f = −0.005, a = −0.018, b = 0.006, c = 0.003, σ = 0.226,

Regime III : f = −1.44, a = −0.55, b = −0.073, c = 0.003, σ = 0.253.(55)

The PDF in Regime I is unimodal with skewness and an one-side fat tail. Interestingly,the time series in Regime I shows a distinct regimes of behavior. Regimes II and IIIcorrespond to PC-1 and NAO for the low frequency data as discussed above, whereRegime II has a slight skewed PDF with sub-Gaussian tails while Regime III is nearlyGaussian.The most sensitive direction of the parameter perturbation in the two-dimensionalspace (δf , δa)T is given by respectively

Regime I : e∗π = (0.9545, 0.2981)T ,

Regime II : e∗π = (0.9685, 0.2488)T ,

Regime III : e∗π = (−0.0760, 0.9971)T .

(56)

36 / 38

37 / 38

I Note that both the simple examples contain the perfect knowledge of the presentclimate given by the unperturbed equilibrium PDFs.

I However, it is often quite difficult in practice to know the exact expression ofthese PDFs or it is computationally unaffordable to compute the gradient in highdimensions.

I One common practical strategy is to adopt some approximated PDFs based on afew measurements such as the mean and covariance.

I It is also common to use imperfect or reduced models from a practical point ofview, where Fluctuation-Dissipation Theorem can also be incorporated tocalculate the gradient of the present climate.

38 / 38