a stochastic perturbation theory for non-autonomous systems · 2014-01-18 · journal of...

JOURNAL OF MATHEMATICAL PHYSICS 54, 123303 (2013)

A stochastic perturbation theory for non-autonomoussystems

W. Moon1,a) and J. S. Wettlaufer1,2,b)

1Yale University, New Haven, Connecticut 06520-8109, USA2Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom

(Received 5 July 2013; accepted 2 December 2013; published online 27 December 2013)

We develop a perturbation theory for a class of first order nonlinear non-autonomousstochastic ordinary differential equations that arise in climate physics. The perturba-tive procedure produces moments in terms of integral delay equations, whose orderby order decay is characterized in a Floquet-like sense. Both additive and multi-plicative sources of noise are discussed and the question of how the nature of thenoise influences the results is addressed theoretically and numerically. By invoking theMartingale property, we rationalize the transformation of the underlying Stratonovichform of the model to an Ito form, independent of whether the noise is additive ormultiplicative. The generality of the analysis is demonstrated by developing it bothfor a Brownian particle moving in a periodically forced quartic potential, which actsas a simple model of stochastic resonance, as well as for our more complex climatephysics model. The validity of the approach is shown by comparison with numericalsolutions. The particular climate dynamics problem upon which we focus involves alow-order model for the evolution of Arctic sea ice under the influence of increasinggreenhouse gas forcing �F0. The deterministic model, developed by Eisenman andWettlaufer [“Nonlinear threshold behavior during the loss of Arctic sea ice,” Proc.Natl. Acad. Sci. U.S.A. 106(1), 28–32 (2009)] exhibits several transitions as �F0

increases and the stochastic analysis is used to understand the manner in which noiseinfluences these transitions and the stability of the system. C© 2013 AIP PublishingLLC. [http://dx.doi.org/10.1063/1.4848776]

I. INTRODUCTION

A. Physical motivation

Arctic sea ice is coupled to low latitudes through atmospheric and oceanographic transport.Radiative forcing is mediated by the atmospheric dynamics and the spatial structure of hydrometeorsand aerosols. The stratification and stability of the water column control the ocean-to-ice heattransport. Hence, the Arctic “feels” the rest of the globe through the fluids that straddle the icecover, and it is the ice cover that provides the climatological diagnostic pulse of the Arctic. In turn,changes in Arctic sea ice can impact global climate, either inducing shifts in atmospheric processesor modifying the oceanic circulation that affects the transport of heat from low to high latitudes.

A mathematical description that couples Arctic sea ice to climate can predict the temporalevolution of its thickness as a function of the atmospheric and ocean heat fluxes using a numericaltreatment of the heat equation for the ice and solving the associated two-moving boundary (upperand lower surfaces) problem.2 A simplification of such an approach can be developed by buildinga so-called energy-balance model, which can be autonomous or non-autonomous (see Refs. 3

a)Present address: Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics,Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom. Electronic mail:[email protected].

b)Electronic mail: [email protected]

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123303-2 W. Moon and J. S. Wettlaufer J. Math. Phys. 54, 123303 (2013)

and 4 and references therein). Constant or periodic forcing obtained from observed or estimatedclimatology can drive the evolution of the ice cover, the conduction of heat through which is simplifiedsystematically.1, 5–8 Solutions to these latter-deterministic-models are useful for understanding thenature of the transitions in the state of the ice cover under scenarios in which the greenhouse gasforcing, here represented by a radiative flux �F0, increases. Depending on �F0, one can find threestable fixed points: a perennial-ice state, a seasonal-ice state and an ice-free state, and transitionsbetween. For example, one finds that upon increase in �F0, the seasonal state is lost to the ice-freestate through a saddle-node bifurcation.

Analysis of data reveals that the time evolution of the ice coverage and its albedo during thesatellite era is described by a multi-fractal exhibiting multiple intrinsic time scales.9, 10 Hence, weunderstand that such deterministic models, while useful in revealing many aspects of the system,need to be extended to include the influence of stochastic forcing. We save for a paper in thegeophysical literature questions regarding the origin and realism of various sources of noise. Forour purposes here, it is sufficient to understand that there are a range of physical origins of noise,both multiplicative and additive, and hence we proceed by constructing a stochastic theory from thedeterministic energy-balance model of Ref. 1 (hereafter referred to as EW09). We will focus ourattention on a noise source that originates in sea ice albedo,11 and we assume that this noise is aWiener process whose magnitude is ultimately to be determined from observations. Due to the factthat we understand the deterministic solutions rather well, when the amplitude of the noise is smalla stochastic perturbation theory is possible.

B. Relevance to stochastic resonance

Clearly, the condition of small noise amplitude is a principal assumption in a wide range ofstochastic modelling problems and underlies the behavior of many physical and biological systems.A crucial aspect of a stochastic perturbation theory concerns how to treat the combined influence ofperiodicity associated with external sources and noise forcing. For example, in one of the problemswe examine here, Arctic sea ice is seasonally forced by the solar shortwave radiance but fluctuationsassociated with weather systems provide high frequency noise sources. In all cases we must assessthe stability of the system over the period of the external forcing and then construct a stochasticsolution for weak background noise.

A common and general example exhibiting this behavior is the over-damped motion of aBrownian particle in a double-well potential with a periodic external forcing. This system has beenused to explain essential characteristics of “stochastic resonance,” which is a mechanism to explainthe enhancement of a weak periodic signal by background noise,12 and can be described by thefollowing stochastic differential equation:

dx

dt= − ∂

∂xV + Acos(�t + φ) + σξ, (1)

where the double-well potential is V (x) = −2x2 + 14 x4, the periodic forcing with a general phase

φ is Acos(�t + φ), and the zero mean Gaussian white noise is given by σξ , where the amplitude isσ . A principal goal is to find the transition probability of a particle from one fixed point to another,which is known as Kramer’s problem. In the absence of periodic forcing the transition probabilitydriven by the weak background noise is negligibly small, due to the local maximum in the potentialwell that separates the two stable fixed points. However, the essence of stochastic resonance is thatthe periodic forcing can resonate with the background noise, which facilitates the hopping of aparticle from one minimum to the other. Thus, in a specific problem governed by an equation that ismore complex than Eq. (1) constructing a steady state stochastic solution around a single fixed pointis an essential starting point for examining the transition probability. Moreover, combining this withthe assumption that magnitude of the noise σ is small allows one in principle to apply a perturbativetheory. Indeed, the general approach taken here can provide new perspectives and methodologies inthe study of stochastic resonance, which is commonly invoked in climatic transitions. In this paper,while we focus mainly on a different physical system, we use stochastic resonance as a simple modelto demonstrate our stochastic perturbation theory.


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Our paper is structured as follows. The general stochastic perturbation theory is developed inSec. II after which we describe a wide range of tests of our solutions in Sec. III. This section has anumber of parts. First, in Sec. III A, we examine how the approach plays out for the canonical modelof stochastic resonance described by Eq. (1). Second, in Sec. III B, we summarize the deterministictheory that is the jumping off point for our stochastic analysis of the Arctic sea ice dynamics, whichis described in Sec. III C. Third, this theory is examined in Sec. III D both by comparison withnumerical solutions and by contrasting the difference between additive and multiplicative noiseeffects. We conclude in Sec. IV.

II. STOCHASTIC PERTURBATION THEORY

We write our stochastic model for the state variable x in a Langevin-Stratonovich form as

dx = a(x, t)dt + σb(x, t) ◦ dW, (2)

where the Wiener process, W (t), is related to the noise in Eq. (66) as ξ = dW (t)/dt , which is asmooth function of time and hence not “purely white.” Note that one may approximate this continuousnoise by a sequence of stochastic processes, {ξ (t)m}∞m=1, and although physically relevant problemsare intuitively treated with a finite decorrelation time, it is more convenient mathematically toinvoke the Martingale property and approximate W (t) as “pure white” noise. Hence, W (t) will haveindependent increments such that 〈�Wi�W j 〉 = δi j , where δij = 1 (δij = 0) when i = j (i �= j).

In this manner, the Langevin-Stratonovich form in Eq. (2) can be converted to the Langevin-Itoform as

dx =[

a(x, t) + σ 2

2b(x, t)

∂

∂xb(x, t)

]dt + σb(x, t)dW, (3)

where the ◦ distinguishes the Stratonovich interpretation of the noise dW from the Ito interpretation.We comment here that Eq. (3) is unstable with respect to changing the noise term to a sequencesuch as {ξ (t)m}∞m=1, and this is often the motivation in studies seeking to test sensitivity to finitedecorrelation times to work within the Stratonovich framework.13

Motivated by the situation in which we know the deterministic solution xS, we write x = xS

+ η with |xS| � |η|, where

dxS

dt= a(xS, t).

Therefore, η satisfies approximately

dη = [c(t)η + d(t)η2 + σ 2 N (t)g(t) + . . .]

dt + σ[N (t) + g(t)η + h(t)η2 + . . .

]dW, (4)

with

c(t) = ∂a

∂x|x=xS , (5)

d(t) = 1

2

∂2a

∂x2|x=xS , (6)

N (t) = b(xS, t), (7)

g(t) = ∂b

∂x|x=xS , and (8)

h(t) = 1

2

∂2b

∂x2|x=xS , (9)

where σ is the amplitude of the noise, which is sufficiently small so that our perturbation ansatzinvolves a convergent power series written as

η = η0 + ση1 + σ 2η2 + . . . (10)


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Given this we can write

d[η0 + ση1 + σ 2η2 + . . .

] = [c(t)(η0 + ση1 + σ 2η2 + . . .)

+ d(t)(η0 + ση1 + σ 2η2 + . . .)2

+ 1

2σ 2(N (t) + g(t)η0 . . .)(g(t) + 2h(t)η0 + . . .)

]+ σ N (t)dW + σg(t)(η0 + ση1 + σ 2η2 + . . .)dW

+ σh(t)(η0 + ση1 + σ 2η2 + . . .)2dW, (11)

which can be examined at each order systematically;

O(1) : dη0 = [c(t)η0 + d(t)η20]dt,

O(σ ) : dη1 = [c(t) + 2d(t)η0]η1dt + [N (t) + g(t)η0 + h(t)η20]dW,

O(σ 2) : dη2 ={

[c(t) + 2d(t)η0]η2 + d(t)η21 + 1

2N (t)g(t)

}dt + [g(t)η1 + 2h(t)η0η1]dW. (12)

The solution at order O(1) is

η0(t) = η0(0)exp[∫ t

0 c(s) ds]

1 − η0(0)∫ t

0 d(r )exp[∫ r

0 c(s) ds] dr. (13)

We are interested in the dominant behavior of the solutions with time at each order. As we will seein the climate physics problem we study in Sec. III, when the radiative forcing �F0 is less than22.2, the steady state solutions are stable because

∫ T0 a(s) ds is always negative and hence η0 decays

to 0 as t increases. Thus, the solution at O(1) eventually becomes negligible. At O(σ ), if we letc(t) = c(t) + 2d(t)η0, observe that the solution depends upon

∫ t0 c(s) ds but, as seen in the O(1)

solution, η0 decays to 0 and thus the contribution 2d(t)η0 for∫ t

0 c(s) ds becomes negligible. For longtime stochastic solutions, the O(1) contribution can be ignored, and hence the first order at whichthe stochastic solutions are non-negligible is O(σ ).

Before proceeding to the next step, we consider the response time scales of the system. When∫ T0 a(s) ds is negative and of large magnitude, the response time scale for the stochastic system is

controlled by a(s). However, when∫ T

0 a(s) ds is small, we must consider the magnitude of 2d(t)η0.The combination of the initial condition and the nonlinearity represented by d(t) can lead to a muchlonger response time scale. Hence the initial condition, triggered by some transient forcing, becomesquite important in determining the variability of x in general and Arctic sea ice in particular overgeophysical time scales. We treat this problem numerically in separate papers but here use numericalsolutions as a test of the perturbation theory.

Considering now the long time solutions, we can simplify the stochastic equations as

O(σ ) : dη1 = c(t)η1dt + N (t)dW, and (14)

O(σ 2) : dη2 =[

c(t)η2 + d(t)η21 + 1

2N (t)g(t)

]dt + g(t)η1dW, (15)

from which we determine the moments for η perturbatively in the following sections.

A. Solution at O(σ )

From Eq. (14), we have

η1(t) = η1(0)exp

[∫ t

0c(s) ds

]+ exp

[∫ t

0c(s) ds

] ∫ t

0N (r )exp

[−∫ r

0c(s) ds

]dW, (16)


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where the function η1(t) is Gaussian and hence the first and second moments completely determineits probability density function. Thus, here we have

η1(t) = η1(0)exp

[∫ t

0c(s) ds

], (17)

which comes from the fact that∫ t

0N (r )exp

[−∫ r

0c(s) ds

]dW = 0, (18)

where from here on we use (·) and 〈( · )〉 interchangeably to represent the ensemble average of( · ). The quantity

∫ T0 c(s) ds, where T is the period of c(s), is negative and therefore η1(t) becomes

negligible for long times. At this order, the stationary average of the stochastic variable η1 is 0;however, the second moment can differ from 0. Because

σ 2 = η21(t) − η1(t)

2 ∼= η21(t), (19)

we have

η21(t) = η2

1(0)exp

[2∫ t

0c(s) ds

]+ exp

[2∫ t

0c(s) ds

] ∫ t

0N 2(r )exp

[−2∫ r

0c(s) ds

]dr. (20)

As considered before, for long times the first term on the right side becomes negligible because∫ T0 c(s) ds is negative. Thus for large t, we have

η21(t) exp

[2∫ t

0c(s) ds

] ∫ t

0N 2(r )exp

[−2∫ r

0c(s) ds

]dr, (21)

where we used the fact that for any square integrable function f

[∫ t

0f (t) dW

]2

=∫ t

0f 2(t) dt. (22)

Now we seek further simplification of the right side for long time. Consider that c(s) is a periodicfunction of period T (take this to be one year in anticipation of our application) and hence define

γ = −∫ T

0c(s) ds, (23)

where we see that γ is a positive constant. Now let t = nT + t , where 0 ≤ t < T , to give

exp

[2∫ t

0c(s) ds

]= e−2nγ exp

[2∫ t

0c(s) ds

], (24)

and hence∫ t

0N 2(r )exp

[−2∫ r

0c(s) ds

]dr

= e2γ n∫ t

0N 2(r )exp

[−2∫ r

0c(s) ds

]dr + e2γ n − 1

e2γ − 1

∫ T

0N 2(r )exp

[−2∫ r

0c(s) ds

]dr. (25)

Based on this calculation, we conclude that the following expression

η21(t) exp

[2∫ t

0c(s) ds

]{∫ t

0N 2(r )exp

[−2∫ r

0c(s) ds

]dr

+ 1 − e−2γ n

e2γ − 1

∫ T

0N 2(r )exp

[−2∫ r

0c(s) ds

]dr,

}(26)


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captures the long time behavior. Therefore, we have

limn→∞ η2

1(t) = exp

[2∫ t

0c(s) ds

][At + 1

e2γ − 1AT

], (27)

in which we define

At ≡∫ t

0N 2(r )exp

[−2∫ r

0c(s) ds

]dr, and (28)

AT ≡∫ T

0N 2(r )exp

[−2∫ r

0c(s) ds

]dr. (29)

To O(σ ), the stochastic variable η1 is Gaussian with zero mean. The second moment is given byEq. (27). Therefore, at this order noise linearly affects the value of x such that the standard deviationdepends on time and the mean of x does not deviate from the deterministic solution xS. Finally, x isGaussian with a second moment controlled mainly by η1.

B. Solution at O(σ 2)

At this order, the equation of motion is

dη2 =[

c(t)η2 + d(t)η21 + 1

2N (t)g(t)

]dt + g(t)η1dW, (30)

in which the stochastic variables, η21 and g(t)η1dW can be considered as forcing terms. Thus, we

can intuit that η2 is not a Gaussian variable and hence contains information concerning the deviationof the mean value from the deterministic solution and the skewness. Clearly the non-Gaussiancharacteristics emerge from the nonlinearity d(t)η2

1 and the multiplicative nature of the noise forcingin η1dW . As considered before, we focus on the long time behavior of η2 and so the contribution ofthe initial condition can be ignored. Therefore, we write η2 as

η2 = M1 + 1

2M2 + M3, (31)

where

M1 ≡ exp

[∫ t

0c(s) ds

] ∫ t

0d(r )η2

1exp

[−∫ r

0c(s) ds

]dr, (32)

M2 ≡ exp

[∫ t

0c(s) ds

] ∫ t

0N (r )g(r )exp

[−∫ r

0c(s) ds

]dr, and (33)

M3 ≡ exp

[∫ t

0c(s) ds

] ∫ t

0g(r )η1exp

[−∫ r

0c(s) ds

]dW. (34)

Therefore, the ensemble average is

η2 = M1 + 1

2M2 + M3. (35)

Let us first consider the mean of M1,

M1 = exp

[∫ t

0c(s) ds

] ∫ t

0d(r )η2

1exp

[−∫ r

0c(s) ds

]dr, (36)

where

η21 = exp

[2∫ t

0c(s) ds

] ∫ r

0N 2(r )exp

[−2∫ r

0c(s) ds

]dr. (37)


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After a detailed calculation (given in Appendix A), we find

limt→∞ M1(t)=exp

[∫ t

0c(s) ds

][1

(e2γ −1)(eγ − 1)AT BT + 1

eγ − 1(AB)T + 1

e2γ −1AT Bt +(AB)t

],

(38)

where γ is again written as previously and Bt and (AB)t are

γ = −∫ T

0c(s) ds, (39)

Bt ≡∫ t

0d(r )exp

[∫ r

0c(s) ds

]dr, and (40)

(AB)t ≡∫ t

0d(r )exp

[∫ r

0c(s) ds

]Ar dr, (41)

and 0 ≤ t < T represents the steady-state seasonal cycle of M1.Similarly, following the process used to find Eq. (27), we determine the long time form of M2

as

limt→∞ M2(t) = exp

[∫ t

0c(s) ds

](1

eγ − 1GT + Gt

), (42)

where

Gt ≡∫ t

0g(r )N (r )exp

[−∫ r

0c(s) ds

]dr. (43)

Finally, M3, which contains an Ito-integral, can be shown to be zero. The exact proof canbe found in Appendix B. Therefore, we conclude that limt→∞ η2(t) = limt→∞[M1(t) + 1

2 M2(t)],which is clearly not zero. At O(σ 2), the mean of η1 deviates from zero, so that stochastic meansdiffer from the deterministic solutions. The expression for η2 has two contributions. Within thefirst, M1, the presence of AT originates from the amplitude of the white noise, N(t), and Bt comesfrom the nonlinear term, d(t). The other contribution, 1

2 M2, reflects the role of multiplicative noise.Taken together—the asymmetry associated with the nonlinear term, the deterministic stability andthe multiplicative noise—these effects are responsible for the deviation of the stochastic mean fromthe deterministic solutions.

The asymmetry is captured by the skewness of the PDF. The skewness also reflects the amplitudeof the noise and the nonlinearity just discussed. Accordingly, we must calculate (η − η)3, which canbe represented approximately as

(η − η)3 (ση1 + σ 2η2 − σ 2η2)3

= σ 3(η31 + 3ση2

1(η2 − η2) + O(σ 2))3

σ 3(η31 + 3σ (η2

1η2 − η2η21)). (44)

Recall from the results from the analysis at O(σ ) that η1 is a Gaussian variable such that odd momentsare zero. Therefore, we have

(η − η)3 3σ 4(η21η2 − η2η

21), (45)


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and hence, to calculate the skewness, we focus on η21(t)η2(t), which we write

η21(t)η2(t)

= exp

[3∫ t

0c(r ) dr

]{∫ t

0N (r )exp

[−∫ r

0c(s) ds

]dW

}2

×{∫ t

0d(r )η2

1exp

[−∫ r

0c(s) ds

]dr

+∫ t

0N (r )g(r )exp

[−∫ r

0c(s) ds

]dr +

∫ t

0g(r )η1exp

[−∫ r

0c(s) ds

]dW

}

≡ S1 + 1

2S2 + S3, (46)

where

S1 ≡ exp

[3∫ t

0c(r ) dr

]{∫ t

0N (r )exp

[−∫ r

0c(s) ds

]dW

}2 ∫ t

0d(r )η2

1exp

[−∫ r

0c(s) ds

]dr

S2 ≡ exp

[3∫ t

0c(r ) dr

]{∫ t

0N (r )exp

[−∫ r

0c(s) ds

]dW

}2 ∫ t

0N (r )g(r )exp

[−∫ r

0c(s) ds

]dr

and

S3 ≡ exp

[3∫ t

0c(r ) dr

]{∫ t

0N (r )exp

[−∫ r

0c(s) ds

]dW

}2 ∫ t

0g(r )η1exp

[−∫ r

0c(s) ds

]dW.

(47)

Now we determine the appropriate ensemble mean quantities, beginning with S1, which is

S1 = 2exp

[3∫ t

0c(s) ds

] ∫ t

0d(r )exp

[∫ r

0c(s) ds

]{∫ r

0N 2(r ′)exp

[−2∫ r ′

0c(s) ds

]dr ′}2

dr

+ exp

[3∫ t

0c(s) ds

] ∫ t

0N 2(r )exp

[−2∫ r

0c(s) ds

]dr

×∫ t

0d(r )exp

[∫ r

0c(s) ds

] ∫ r

0N 2(r ′)exp

[−2∫ r ′

0c(s) ds

]dr ′dr

= S∗1 + η2

1(M1 + M3), (48)

where we define

S∗1 ≡ 2exp

[3∫ t

0c(s) ds

] ∫ t

0d(r )exp

[∫ r

0c(s) ds

]{∫ r

0N 2(r ′)exp

[−2∫ r ′

0c(s) ds

]dr ′}2

dr,

(49)

and the details of the calculation are given in Appendix C. The term S2 can be easily averaged and is

S2 = 1

2η2

1 M2. (50)

The last term, S3, can also be considered in a similar manner to the calculation of M3. The simplifi-cation via Riemann summation of the integral form is represented as

S3 = 2∫ t

0g(r )N (r )exp

[−∫ r

0c(s) ds

] ∫ r

0N 2(r ′)exp

[−2∫ r ′

0c(s) ds

]dr ′ dr. (51)


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We give the details of the calculation in Appendix D. This allows us to write

η21η2 − η2

1η2 = S∗1 + S3. (52)

Next we extract the periodicity from the integral forms and examine the limit t → ∞. First we lett = nT + t , and divide the total time integral into n + 1 sub-integrals as∫ nT +t

0(·) dr =

∫ nT +t

nT(·) dr +

n∑k=1

∫ kT

(k−1)T(·) dr. (53)

Thus, using this decomposition, we find

limt→∞

(η2

1η2 − η21η2

)

= 2exp

[3∫ t

0c(r ) dr

][1

(e2γ − 1)2A2

T Bt + 1

(e2γ − 1)2(e3γ − 1)A2

T BT

+ 2

e2γ − 1AT (AB)t + 2

(e2γ − 1)(e3γ − 1)AT (AB)T + (A2 B)t + 1

e3γ − 1(A2 B)T

+ 1

(e2γ − 1)(e3γ − 1)AT GT + 1

e2γ − 1AT Gt + 1

e3γ − 1(G A)T + (G A)t

], (54)

where At, Bt, ABt, and Gt are as defined previously in Eqs. (28), (40), (41), and (43) and the otherterms are defined as

(A2 B)t ≡ ∫ t0 d(r )exp

[∫ r0 c(s) ds

] {∫ r0 N 2(r ′)exp

[−2∫ r ′

0 c(s)]

dr ′}2

, (55)

(G A)t ≡∫ t

0g(r )N (r )exp

[−∫ r

0c(s) ds

] ∫ r

0N 2(r ′)exp

[−2∫ r ′

0c(s) ds

]dr ′ dr, (56)

the details of which can be found in Appendix E.

III. EXAMINING THE VALIDITY OF THE SOLUTIONS

A. A first test using stochastic resonance

As described in Sec. I B, we first test our theory using the canonical example of a Brownianparticle in a double-well potential with a periodic forcing and a weak background noise, which is amodel of stochastic resonance. We rewrite the model equation (1) as

dx

dt= 4x − x3 + cos(2π t) + σξ, (57)

with A = 1 and explicitly write (minus) the gradient of the potential V (x) = 14 x4 − 2x2. This form

of the model has two stable fixed points, ± 2, and one unstable fixed point 0. The periodic forcingcos(2πx) modulates V (x) with period 1.0. Clearly, for initial data near one of the two stable fixedpoints, the final deterministic steady state exercises a periodic cycle around that fixed point, as seenin Fig. 1 for the fixed point + 2. Now, given that we know the deterministic periodic cycle, we canconsider the effect of weak background noise on the solution.

To apply the perturbation method developed in Sec. II, we need only know c(t) and d(t) evaluatedon the deterministic solution xS(t) shown in Fig. 1, and these are

c(t) = 4 − 3x2S(t) and (58)

d(t) = −3xS(t). (59)

Thus, we are equipped to compare the perturbative solutions with numerically determined solutions,which we do for two values of the noise amplitude σ : 0.2 and 0.4. These choices are arbitrary so


130.132.173.175 On: Sat, 18 Jan 2014 15:22:18


0 0.2 0.4 0.6 0.8 11.85

1.9

1.95

2

2.05

2.1

time

x

FIG. 1. The stable deterministic steady state periodic cycle around 2.0 showing that the periodic forcing cos(2π t) determinesthe periodicity of the final solution.

long as their magnitude is small relative to xS(t). The numerical method, which we describe in moredetail in Sec. III D, uses a variant of so-called weak order two Runge-Kutta stochastic methods.14

We use a time step of 10− 6 and collect 106 realizations after having reached a periodic steady state,from which we determine the standard deviation, the deviation of the stochastic means from thedeterministic solutions xS(t) and the skewness. Clearly, as seen in Fig. 2, the analytical and numericalsolutions compare quite favorably. The standard deviations of x have a smaller magnitude than theamplitude of the background noise, which is due to the stabilizing effect provided by the potential Vin the vicinity of the fixed point. The other two moments are negative because in this model d(t) isalways negative.

It is hoped that the success of the approach may stimulate exploration of its implications in anysystem represented by a periodic non-autonomous stochastic dynamical system. In particular, due tothe fact that the simple model examined in this section is used in the study of stochastic resonance,our perturbative approach suggests it may be fruitful to consider the non-Gaussian effects realizedat second order (as described in Sec. II) when calculating the statistics of hopping between fixedpoints. Indeed, this perturbation method is expected to provide a more exact quantification in theprocess of stochastic resonance, but that should be left to an independent study. In what followshere, we apply this general methodology to a stochastic Arctic sea ice model.

B. Summarizing the deterministic energy balance model of Eisenman and Wettlaufer

To insure that the reader can follow this approach we summarize EW09 here. The state variableE is the energy (with units W m− 2 yr) stored in sea ice as latent heat when the ocean is ice-covered

0 0.2 0.4 0.6 0.8 10.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

time

std

0 0.2 0.4 0.6 0.8 1−10

−8

−6

−4

−2

x 10−3

time

dev−

mea

n

0 0.2 0.4 0.6 0.8 1−0.2

−0.18

−0.16

−0.14

−0.12

−0.1

−0.08

−0.06

time

skew

ness

(a) (b) (c)

FIG. 2. Comparison of the analytical solutions (dashed lines) and the numerical solutions (thick lines) for several keystatistical moments: (a) the standard deviation, (b) the deviation of the stochastic means from the deterministic solution xS(t),and (c) the skewness. The case of σ = 0.2 (σ = 0.4) is given by the blue (red) lines.


130.132.173.175 On: Sat, 18 Jan 2014 15:22:18


or in the ocean mixed layer as sensible heat when the ocean is ice-free, viz.,

E ≡⎧⎨⎩

−Li hi E < 0 [sea ice]

cml Hml Tml E ≥ 0 [ocean], (60)

where Li is the sea ice latent heat of fusion, hi its thickness, cml is the specific heat capacity of theocean mixed layer, Hml is its depth, and Tml its temperature. Ignoring salinity effects, the temperatureT(t, E) of the sea ice or ocean, determined by energy balance across the layer, is measured relativeto the freezing point Tm as

T (t, E) = −R[

FD(t)

ki Li/E − FT (t)

], (61)

where the ramp function is R(x ≥ 0) = x and R(x < 0) = 0, the thermal conductivity of the ice iski, and the radiative flux quantity FD is discussed in detail immediately following Eq. (65), and FT(t)is described presently.

The Stefan-Boltzmann equation is linearized, σ (T)4 ≈ (σ 0 + σ T�Ti), where σ is the Stefan-Boltzmann constant, �T is the deviation of the surface temperature T(t, E) from the freezingpoint, σ 0 = 316 Wm− 2 and σ T = 3.9Wm− 2K− 1 are chosen such that the equation is exact whenT = − 30 ◦C and when T = 0 ◦C (the approximate values of T during most of the winter andsummer, respectively). This allows us to express the temperature dependence of the outgoing longwave flux as F0(t) + FT(t)T(t, E), where F0(t) is σ 0 plus the specified sensible and latent heat fluxesfrom observation, and FT(t) = σ T. An atmospheric model incorporating observations of Arcticcloudiness, atmospheric transport from lower latitudes and the meridional temperature gradient isused to determine the seasonally varying values of F0(t) and FT(t)1 but here we choose representativeconstant or seasonal values as described in Table I.

An essential aspect of the transitions we discuss is the nature of the ice albedo feedback. Here,the Beer-Lambert law of exponential attenuation of radiative intensity with depth in a mediummotivates a treatment of the dependence of the surface albedo with E using a mixture formula witha characteristic ice thickness hα for the extinction of shortwave radiation as

α(E) = αml + αi

2+ αml − αi

2tanh

(E

Li hα

). (62)

This describes the fraction, 1 − α(E), of the incident shortwave radiation FS(t) absorbed by the ice,with albedo αi and the ocean, with albedo αml.

The evolution of the state of the ice (or ocean) cover is determined by the balance of radiativeand sensible heat fluxes at the upper surface, FD − FT(t)T(t, E), the upward heat flux from the oceanFB, and the fraction of ice exported from the domain v0R(−E) through a first order nonautonomousenergy balance model as

d E

dt= f (t, E), (63)

with

f (t, E) = FD − FT (t)T (t, E) + FB + v0R(−E), (64)

where

FD(t, E) ≡ [1 − α(E)] FS(t) − F0(t) + �F0. (65)

The term FD − FT(t)T(t, E) is thought of as the difference between the incoming shortwaveradiation at the surface [1 − α(E)]FS(t) and the outgoing longwave radiation (∝T4), augmentedhere by sensible and latent heat fluxes as described above and an additional amount associated withgreenhouse gas forcing �F0. Finally, we note that the ice export v0R(−E) is typically ∼10% yr− 1,but the nonlinear relationship between ice thickness and ice growth rate highlights the possibilitythat in changing climates a time dependent value may be important in determining multiple ice states(most recently see Ref. 12 and references therein).


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We render the deterministic model (Eqs. (60)–(65)) dimensionless using the scaling factor E0

= Lih0, where h0 is equal to 3.0m, which is approximately the climatological mean thickness of seaice. Thus, for example E ≡ E/E0 is the dimensionless energy, but we immediately omit the ’s.

C. Stochastic sea ice model

Hasselman first considered the global climate in direct analogy to Brownian motion in whichthe “noise” is the weather and the Brownian “particle” is the climate.15 Because the structure of thestatistical dynamical system is autonomous, a fluctuation theorem follows directly. Clearly, such isnot the case in the problem considered here.

The deterministic model successfully captures the seasonal cycle of Arctic sea ice thickness andpredicts the nature of the transitions as the greenhouse gas forcing increases. However, a realisticaspect of the fate of the ice cover is its variability, due to internal dynamics and/or external forcing.Observations demonstrate that the variability in the principal observables has white noise structureon annual to bi-annual time scales and different noise characteristics on longer and shorter timescales.9, 10 How this variability influences the trends is an essential question in the field.

As a starting point to address these and related questions we add a term, σb(E, t)ξ to thedeterministic model, where, ξ represents stochastic noise as

d E

dt= a(E, t) + σb(E, t)ξ. (66)

As is standard we begin by assuming that ξ is Gaussian white noise satisfying 〈ξ (t)ξ (t′)〉 = δ(t− t′), where 〈•〉 represents the ensemble average. We take the amplitude of the noise b(E, t) intwo general forms—additive and multiplicative—to capture different classes of physical processes.For example, the simplest version of additive noise is to consider the variability due to corpus ofhigh frequency processes affecting the ice and thus the stochastic term b(E, t)ξ , is written as σξ ,with a constant noise magnitude σ throughout the year. The next level of complexity models theseasonal variation in the noise magnitude viz., σb(t)ξ . Examples of multiplicative noise, in whichcase we have b(E, t) are numerous. As noted above, the ice export from the Arctic basin, representedby the term v0R(−E) in Eq. (63), which is typically treated as a constant. However, because theatmospheric wind field drives ice motion, we can consider large scale variability by writing v0 asv0 + σξ , where now v0 is an average value (say ∼10% yr− 1) and σξ captures the randomness ofatmospheric motions. Thus, b(E, t) = − σE, representing a multiplicative noise process, but we notethis is but one of many potential examples.

In Sec. III D, the approximate stochastic solution is analyzed, wherein we treat b(E, t) in themost general sense. We then examine specific cases, such as those mentioned above, in order to testthe approach through comparison with numerical simulations.

D. Interpretation of results: Additive and multiplicative noise

Having constructed perturbative (under small amplitude noise; σ � 1) expressions for severalmoments of the stochastic solutions for this one-dimensional periodic non-autonomous dynamicalsystem, we now test these expressions against those obtained numerically. Our knowledge of thedeterministic solutions enables us to Taylor expand about their stable steady states. For the stochasticsolution at O(σ ), the important moments are the standard deviation, mean, and skewness. Theapproximate moments principally reflect the magnitude of the noise forcing N(t, E) and the first andsecond terms in the Taylor expansion, c(t) and d(t), respectively.

The first term c(t) controls the stability of the deterministic solution, which is intimately relatedto the standard deviation of the stochastic solution to O(σ ). The second term d(t) is the degree of theasymmetry of the response to (signed) perturbations: a non-zero d(t) insures that the response timescale of a positive perturbation will be different from that of a negative perturbation. This generatesthe asymmetry in the stochastic solutions to O(σ 2).

The magnitude of the noise forcing, with N(t) representing its periodic variation, is an essentialingredient. When combined with the stability of the deterministic solutions c(t), this determines the


130.132.173.175 On: Sat, 18 Jan 2014 15:22:18


standard deviation of the stochastic solutions to O(σ ). The multiplicative noise g(t) appears at O(σ 2)and affects the deviation of the stochastic means from the deterministic solutions, as well as theskewness of the stochastic solutions.

The applicability of our theoretical approach to the transitions in the states of Arctic sea iceis limited to a range of greenhouse gas forcing �F0. For example, when �F0 is in the seasonallyvarying state as determined from the deterministic model, we cannot guarantee an O(σ ) stochasticsolution is applicable. According to our comparison with numerical simulations (discussed below),our perturbation analysis applies to perennial sea ice states and hence for �F0 up to about 18.Therefore, this perturbation analysis is particularly useful for understanding the contemporary andnear-future variability of the ice cover. However, a deeper qualitative understanding regardingstochastic solutions is quite generally applicable to a range of ice states so long as the deterministicsolutions are available.

Importantly, we can distill the stochastic behavior of Arctic sea ice from the physical interpreta-tions of c(t), d(t), N(t), and g(t). The deterministic stability of the ice, controlled by the competitionbetween the destabilizing sea ice-albedo feedback during the melt season and the stabilizing influ-ence of longwave radiative loss during winter, is captured by c(t). The second term in the Taylorexpansion, d(t), represents the asymmetric response of the ice to perturbations, which originatesin the different response time scales of the competing effects just described underlying c(t) due totheir ice thickness dependencies. Finally, as discussed previously, the terms N(t) and g(t) reflect theseasonal variation of the noise forcing and the effect of the multiplicative noise forcing, respectively.

Here, we consider two examples relevant to the variability of contemporary Arctic sea ice:constant additive noise and multiplicative noise. In the additive case g(t) = 0 thereby eliminatingmultiplicative noise. Hence, we focus on the role of the stability c(t) and the asymmetry d(t), bothcaptured in the statistics of η. We consider the noise associated with the variability of sea ice export.Here, as discussed in Sec. III C, the export vo is converted to v0 + σξ where ξ is Gaussian whitenoise. Hence, the magnitude of the noise forcing depends on the ice thickness itself, and is therebymultiplicative. This may be the simplest example of multiplicative noise forcing, providing an idealtheoretical test bed of geophysical relevance because the variability of sea ice export is believed toplay an important role in the recent decline of Arctic sea ice. The sea ice export out of Arctic iscontrolled by large scale atmospheric and oceanographic motions. Under the assumption that thephase of these motions is determined randomly, it is reasonable to consider ice export as a randomvariable determined by a Wiener process.

The validity of our perturbation theory is best tested by comparison with numerical solutions. Asmentioned in Sec. III A, the numerical scheme used here is a simple Runge-Kutta method which hassecond numerical order accuracy in the sense of convergence of moments, so-called 2nd weak order.In other words, just as in the case of deterministic Runge-Kutta methods, the stochastic expansionof the approximation is compared to the corresponding Taylor expansion. Tocino and Vigo-Aguiardetermined the conditions on the stochastic Runge-Kutta method required for weak order two tohold.14 We simulate the full model Eq. (3) without any approximations using a time step 10− 6

yr, and collect 106 realizations to obtain ensemble statistics. To reach a stable steady state, eachsimulation is run for 20 years beyond the initial data.

The first test treats constant additive white noise as σξ , where σ is the magnitude of thenoise and ξ Gaussian white noise. In Fig. 3, we compare statistical moments from the theory withthose generated numerically for four values of the greenhouse gas forcing �F0 and σ = 0.05. Thecomparison is excellent for low �F0, only deviating for large values due to the increasing magnitudeof the sea ice-albedo feedback, during the summer.

The second test, shown in Fig. 4, uses the simplest multiplicative case where the noise forcingis − σEξ and for the same external heat fluxes �F0’s as shown in Fig. 3. Clearly, the match betweenthe numerical and analytical solutions is even better than with constant additive noise, because theabsolute magnitude of E decreases as �F0 increases. Therefore, the overall magnitude of the noiseforcing remains sufficiently small that the perturbative solutions are valid.

Further tests (not shown) confirm that the perturbative method is highly accurate for the con-temporary perennial ice states. As the secular stability of sea ice decreases, a strong ice albedofeedback, increasing during summer, generates large variability, and hence our perturbative method


130.132.173.175 On: Sat, 18 Jan 2014 15:22:18


0 2 4 6 8 10 12

0.04

0.045

0.05

0.055

0.06

Month

Std

0 2 4 6 8 10 12−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5x 10

−3

Month

Mea

n

0 2 4 6 8 10 12−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Month

Ske

wne

ss

(a) (b) (c)

FIG. 3. Comparison of several key statistical moments between the analytic solutions and the numerical solutions in thecase of constant additive noise. Thick lines represent the results from the numerical simulations and dashed lines from theanalytical solutions. The four colors show the statistical moments for four different �F0, 10 (black), 12 (blue), 14 (green),and 16 (red). The moments are (a) the standard deviation, (b) the deviation of the stochastic mean from the deterministicsolution, and (c) the skewness. The value of σ is 0.05.

is no longer valid. However, clearly our perturbative method provides a quantitative understandingof the variability of the system under present conditions and a mathematical framework for the studyof future states. Now we describe in more detail the two cases treated.

1. Constant additive noise

The simplest interpretation of this stochastic model is gleaned from the constant additive noisecase. Here, N(t) = 1 and g(t), Gt and (GA)t are all zero. First consider the standard deviationappearing at the first non-trivial order, O(σ ), where the stochastic solution η1 is a Gaussian variablewith zero mean. The standard deviation is determined by the noise intensity σ and the stability ofthe deterministic solution c(t). Thus, we can expect that small amplitude noise forcing is amplified(or decays) with positive (negative) c(t). However, the interpretation of the effect of the stability,viz., c(t), is subtle because it is not an instantaneous effect, but rather it describes the accumulatedinfluence of noise up to the time of interest. In other words, it captures the “memory” of the noise(Fig. 5).

To see this we write Eq. (26) as

η21(t) =

∫ t

0N 2(r )exp

[2∫ t

rc(s) ds

]dr, (67)

which represents the accumulation of past uncertainties at a specific time t. This is analogous tothe well-known physical phenomenon of photon absorption-emission in a layer of gas such as theatmosphere. First, assume c(s) is a constant c, which simplifies Eq. (67) as

η21 =

∫ t

0N 2(r )ec(t−r ) dr, (68)

where we take c < 0 to represent photon absorption, and the number of photons emitted at positionr is N2(r).

0 2 4 6 8 10 120.018

0.02

0.022

0.024

0.026

0.028

0.03

0.032

0.034

0.036

Month

Std

0 2 4 6 8 10 12−2.2

−2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4x 10

−3

Month

Mea

n

0 2 4 6 8 10 12−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

Month

Ske

wne

ss

(a) (b) (c)

FIG. 4. Same as Figure 3 but with multiplicative noise of magnitude σE.


130.132.173.175 On: Sat, 18 Jan 2014 15:22:18


10 12 14 16 180.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

Δ F0

1/(e2γ−1)× AT

γ

FIG. 5. The memory effect, 1/(e2γ − 1) (red line) and γ (blue line) for different values of �F0. The memory effect ismainly controlled by γ and hence the two quantities depend similarly on �F0.

The number of the photons that reach the ground is net amount of emission/absorption summedfrom the top of the atmosphere (r = 0) to the ground (r = t) as given by Eq. (68). However, notethat we do not have to consider all of photons in the layer to find a reasonable approximate numberarriving at the ground. For, if the layer is sufficiently deep, distant photons are unlikely to reachthe ground due to absorption. The depth scale 1/c characterizes that over which the total number ofphotons arriving at the ground originate.

We can translate the photon emission-absorption model to our stochastic ice model results. Ateach time, the system is forced randomly with the magnitude σ . The number of photons emitted ata certain height is analogous to the magnitude of uncertainty in the state of the ice cover at a certaintime. The height of the layer can be interpreted as the time lapsed over which we wish to scrutinizethe state of the ice. The absorption rate c corresponds to the dynamical stability of the ice cover,characterized by a decay time scale as in Eq. (23).

However, the analogy becomes complicated by the fact that c(t) is periodic in time and thusbears only a slight relationship to the constant c case. During the one year period, c(t) changessign, which can amplify the standard deviation. Whereas, for long times, the standard deviation issaturated because −γ ≡ ∫ T

0 c(t) dt is negative. Thus, the “memory” governing the calculation ofthe standard deviation is determined by 1/γ , which is the sea ice response time scale. Therefore, thestandard deviation of sea ice thickness at any time in its evolution depends on its state in the past.

This overall interpretation emerges from the exact solution shown in Eq. (27). First, as discussedpreviously, 1/(e2γ − 1)AT represents the memory effect by quantifying the accumulated past uncer-tainty and its decay. Because for additive noise N(r) = 1, the memory effect is entirely controlled byc(t). Figure 5 shows the comparison between 1/(e2γ − 1)AT and γ for this constant additive noisecase, which behave in a similar manner. We see that, as the overall stability of sea ice characterizedby γ increases, the memory effect diminishes. Increased stability is associated with rapid relaxationin response to perturbations, which implies that only the recent past influences the present nature ofthe statistical fluctuations. As analyzed by us previously,6 two main processes govern the stabilityof Arctic sea ice, the sea ice-albedo feedback, and the longwave radiative stabilization due to theStefan-Boltzmann Law. When the ice is thick, the latter effect essentially dominates the responsetime scale. In Fig. 5, we see increased stability (decreased γ ) as �F0 increases from 10 to 15. Thisis due to the faster longwave radiative heat loss of thin ice during winter when in this range of�F0 the sea ice-albedo feedback is not effective. However, when �F0 increases, summer sea iceis sufficiently thin to activate the sea ice-albedo feedback, driving destabilization and extending theresponse time scale.


130.132.173.175 On: Sat, 18 Jan 2014 15:22:18


0 2 4 6 8 10 120.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Months

ST

D (

Sta

ndar

d D

evia

tion)

10.013.015.018.0

FIG. 6. The seasonal cycle of the standard deviation of Arctic sea ice thickness with different �F0.

In this case the standard deviation is the sum of the memory effect and the seasonal variation.

The seasonal variation is normally controlled by the prefactor exp[∫ t

0 c(r ) dr]and hence the seasonal

cycle of the standard deviation should follow the seasonal cycle of the stability of the ice.As the external heat flux �F0 increases, the sea ice thins and the magnitude of both the sea ice-

albedo feedback and the longwave stabilization increase. Namely, the albedo of the sea ice changesmore rapidly during summer and the growth of thinner ice is faster during winter. Therefore, weexpect that as �F0 increases, the standard deviation increases during summer but decreases duringwinter. This is clearly demonstrated in Fig. 6.

Interestingly, while our stochastic forcing does not have any memory, the memory effect isgenerated by the rectified stability of the ice as embodied in γ . The magnitude of uncertainty of thepresent depends on the accumulated uncertainty of the past. Past uncertainties propagate throughthe time domain in a manner that depends on the characteristic response time scale of the stability.Importantly, because the stability of the ice decreases under global warming modelled here as anincrease in �F0, we infer that the predictability of the future status of the system will be limited bythe memory effect. Therefore, quantitative knowledge of the statistical characteristics of the small-scale physics normally ignored in many models could be crucial for improving our predictability ina sense not typically considered, due to stochastic amplification.

At O(σ 2), we can examine the effect of the asymmetry embedded in d(t), because it generates asign-dependent response to a given perturbation. Because the noise forcing is additive, S2 is irrelevantand the deviation of the stochastic means from the deterministic solutions, as well as the skewnessare mainly generated by the sign of d(t) upon which we focus presently.

First, consider Eq. (32) which shows the deviation of the stochastic mean, and can be rewrittenas

M1 =∫ t

0d(r )η2

1exp

(∫ t

rc(s) ds

)dr, (69)

clearly displaying the memory effect via exp(∫ t

r c(s) ds)

. The integral describes the additive con-

tribution of the accumulated past asymmetry emerging with “density” c(t). Before focusing on theexact interpretation of what is integrated through the time domain, namely d(t)η2

1, we construct anargument for the deviation of the stochastic mean.

We can rewrite the original stochastic perturbation equation as

dη

dt= − ∂

∂ηV + noise forcing, (70)


130.132.173.175 On: Sat, 18 Jan 2014 15:22:18


FIG. 7. The dependence of the shape of V upon the signs of c and d. The dotted line represents V for d = 0. When c ispositive (negative), a given perturbation due to noise forcing grows (decays). The term − 1

3 dη3 amplifies the asymmetries asfollows. If d(t) is positive (negative), the stochastic solutions tend to move to the positive (negative) side.

where V ≡ − 12 c(t)η2 − 1

3 d(t)η3. This form facilitates the interpretation of the behavior of thestochastic solution in terms of the shape of the “ice-potential” V .

Figure 7 shows how the ice potential V depends on the sign of c and d. Consider a ball rolling ina V -shaped bowl. When c is positive (negative), V is convex (concave), and the absolute magnitudeof c characterizes the degree of stability (or instability). The value of d provides additional structureto the curvature of the bowl, which generates the asymmetry of the statistics of the fluctuations of theball within it. The left side of Fig. 7 represents stable situations when c is negative. When d is positive(top left), the asymmetry assures that a ball perturbed to the positive side takes a longer time to returnto the origin than one perturbed to the negative side. When c and d are both positive (top right), aball positioned in the positive side rolls more rapidly away from the origin than one on the negativeside. The other cases can be interpreted analogously generating a rich time-dependent asymmetryin the statistics of fluctuations, wherein unstable configurations are more strongly amplified.

This construct facilitates our interpretation of how the deviation of the stochastic mean isreflected in the term d(t)η2

1. At this order, the standard deviation η21 follows the previously discussed

trend of c(t) with �F0. During the summer melt season, η21 is large due to the sea ice-albedo feedback,

the potential V is convex, and a small perturbation is amplified asymmetrically as discussed in theprevious paragraph. Conversely, when sea ice is stable during the winter, the magnitude of theasymmetric response is comparatively smaller.

As in the case of the standard deviation, the memory effect is also important in determiningthe deviation of the stochastic mean, which is shown in Eq. (32). Recall too that the characteristictime scale of the memory effect is 1/γ . The essential question here is to explain how the degree ofasymmetry, d(t)η2

1, is accumulated in terms of the main physical processes underlying the evolutionof the potential V . The sea ice-albedo feedback is the main destabilizing process during summer, thethinner ice being more vulnerable to it. On the other hand, the wintertime longwave stabilization isalso more effective for thinner sea ice because thinner ice grows faster than does thicker ice. Thus,during the summer melt season the shape of V is as in Fig. 7 with both c and d positive, and duringwinter both c and d are negative. This demonstrates the seasonal dependence of the asymmetry. Aprecise analysis is possible by examining d(t) and η2

1 at the same time of the year.

Figure 8 shows the evolution of d(t) and the standard deviation η21 throughout the year. The

standard deviation is a maximum in August and a minimum in March, which is commensurate withthe evolution of c(t). The asymmetry factor d(t) shows a positive maximum in July and sharplydecreases from positive to negative values in October, finally reaching to a minimum in November.


130.132.173.175 On: Sat, 18 Jan 2014 15:22:18


0 2 4 6 8 10 12−5

−4

−3

−2

−1

0

1

2

3

4

5

6

7

8

d(t)

Month0 2 4 6 8 10 12

0.5

0.7

0.9

1.1

1.3

0 2 4 6 8 10 12−5

−4

−3

−2

−1

0

1

2

3

4

5

6

7

8

d(t)

Month0 2 4 6 8 10 12

0.5

0.7

0.9

1.1

1.3

0 2 4 6 8 10 12−5

−4

−3

−2

−1

0

1

2

3

4

5

6

7

8

d(t)

Month0 2 4 6 8 10 12

0.5

0.7

0.9

1.1

1.3

0 2 4 6 8 10 12−5

−4

−3

−2

−1

0

1

2

3

4

5

6

7

8

d(t)

Month0 2 4 6 8 10 12

0.5

0.7

0.9

1.1

1.3

(a) (b)

(c) (d)

FIG. 8. For each �F0, d(t) and η21 are shown as the blue solid line and the red dotted line, respectively. The deviation of

the stochastic mean is the product of d(t) and η21 at each time. The largest contribution comes from the period between

June and November when the standard deviation η21 and the asymmetry d(t) are maximal. The peaks shown in d(t) are

approximate representations of delta functions generated by differentiation of a discontinuity. (a) �F0 = 10, (b) �F0 = 13,(c) �F0 = 15, and (d) �F0 = 16.

Physically, this reflects two main physical processes. The sea ice-albedo feedback begins in Mayand intensifies over the summer. The longwave stabilization becomes operative in August when theice begins to cool. This is reflected in the shape of the potential V , which changes from having bothc and d positive to both negative during this period, exhibiting the positive asymmetry of the seaice-albedo feedback and the negative asymmetry of the longwave stabilization. As shown in Fig. 8,when the external heat flux �F0 increases and the ice thins, the two main processes are amplified.

Now we are equipped to interpret the deviation of the stochastic mean from the deterministicsolutions. Figure 9 shows the seasonal deviation of the stochastic means for several values of�F0, also reflecting the asymmetry generated by the sea ice-albedo feedback and the longwavestabilization. For lower �F0, sea ice is sufficiently thick that the sea ice-albedo feedback is ineffectiveat destabilizing the ice, and the sole asymmetry in the response is the longwave stabilization. InFig. 9, we see that for �F0 < 14 the value of η2 is always negative, whereas when �F0 > 14 theice-albedo feedback begins to generate a positive asymmetry in the response to stochastic forcing.

The skewness of η2 can be understood in a similar manner. Recall that in the case of constantnoise S2 = 0 and the skewness is calculated from S∗

1 . Equation (49) can be rewritten as

S∗1 = 2

∫ t

0d(r )(η2

1)2exp

[3∫ t

rc(s) ds

]dr, (71)

exhibiting the interaction between d(t) and η21 as well as the memory effect. However, Eq. (71) shows

an enhanced interaction between d(t) and η21 because d(t) is multiplied by (η2

1)2, rather than by η21.

Therefore, as �F0 increases, the positive skewness during summer will be magnified relative to thedeviation of the mean, as shown in Fig. 10.


130.132.173.175 On: Sat, 18 Jan 2014 15:22:18


0 2 4 6 8 10 12−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Month

10.013.014.016.0

FIG. 9. The deviation of the stochastic means from the deterministic solutions η2 for different values of �F0. For lower�F0, the deviation is negative during the entire year. However, at higher values of �F0, a positive deviation appears duringsummer.

The seasonal variation of the skewness is quite similar to that of the deviation from the stochasticmean, the only difference being the weighting of the asymmetries generated by sea ice-albedofeedback and longwave stabilization. Therefore, we see the amplification of the skewness as �F0

increases with the positive (negative) asymmetry amplified in the summer (winter) due to theweighting associated with (η2

1)2.In summary for the constant noise case, we are able to interpret the essential aspects of the

behavior of the system in light of the two main competing processes in the seasonal cycle of Arcticsea ice: the ice-albedo feedback and the stabilization due to longwave radiative fluxes in winter.1, 6

We thereby distilled the behavior of the stochastic solutions in terms of the interaction between thestability and the asymmetry contained in the nonlinearity of the deterministic forcing. At O(σ ), wefind that the stability of sea ice is represented by c(t), which is responsible for both the memoryeffect and the seasonal evolution of the standard deviation. At O(σ 2), the asymmetry is approximatelyrepresented by d(t) and, combined with the standard deviation, it determines the deviation of the

0 2 4 6 8 10 12−1

−0.5

0

0.5

1

1.5

Month

Ske

wne

ss

10.013.015.016.0

FIG. 10. The skewness of η2 for different values of �F0. For lower �F0, the skewness is negative during the entire year.However, for higher �F0, the skewness becomes positive during the summer.


130.132.173.175 On: Sat, 18 Jan 2014 15:22:18


stochastic means from the deterministic solutions as well as the skewness. Next we come to the roleof the multiplicative nature of the noise forcing.

2. Multiplicative noise: Sea ice export

Physically, constant additive noise can be understood as describing the sum of all of the small-scale physical processes that drive direct fluctuations of Arctic sea ice thickness. However, we canalso consider the variability of the ice thickness that originates from a specific physical process. Therecent decline in Arctic sea ice is believed to be strongly influenced by the export of sea ice drivenby large scale atmospheric motions.16 Because the time scale of synoptic atmospheric motions isapproximately 10 days, ice export driven principally by atmospheric motions is characterized bytime scales that are short relative to the overall evolution of the deterministic model. In this sense,the rate of sea ice export can be considered as a stochastic process. We note that a qualitativelyequivalent argument can be made for the role of oceanographic motions, but these are less accessibleto large-scale observational methods so the reader may consider our use of the term “atmosphericmotions” to apply similarly to those of the ocean.

We begin with consideration of the noise amplitude. In the deterministic model, sea ice exportis represented by νR( − E), the fraction of the ice cover exported annually. Hence, a coefficient of ν

= 0.1, means that 10 percent of the ice is exported due to large-scale atmospheric and oceanographicmotions annually. Therefore, rather than using a constant value, we can use a stochastic variable forν in order to capture the effect of the variability of atmospheric and oceanographic motions. If we letν = ν0 + σξ , where ν0 is the average and ξ is a Wiener process, then N(t) = − Es(t) and g(t) = − 1.

First, consider the standard deviation to O(σ ). As �F0 increases, Arctic sea ice is less stableand we may naively expect the standard deviation to increase, as was found when the noise wasconstant. However, in the case of stochastic ice export, the magnitude of the noise is proportional tothe magnitude of sea ice energy Es(t), and thus increasing the external heat flux �F0 can lead to anoverall decline of the fluctuations about the mean. This displays a fundamental difference betweenadditive and multiplicative noise.

Figure 11 shows the seasonal evolution of the standard deviation for several values of �F0. Asexpected, for a range of �F0 the overall standard deviation decreases as �F0 increases. However,as the ice cover begins to thin, the ice-albedo feedback begins to play a more important role in thesummer and the standard deviation displays a reentrant behavior increasing with �F0 in the summermonths as was the case for all months in the additive noise case.

0 2 4 6 8 10 120.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

Month

Sta

ndar

d D

evia

tion

10.0

13.0

15.0

18.0

FIG. 11. The seasonal standard deviation of η1 for several values of �F0 at O(σ ). As �F0 increases, the overall standarddeviation decreases. But, it is reentrant as sea ice-albedo feedback becomes more effective during summer, and the standarddeviation begins to grow with �F0.


130.132.173.175 On: Sat, 18 Jan 2014 15:22:18


0 2 4 6 8 10 12−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Month

10.013.015.016.0

FIG. 12. The seasonal cycle of the deviation of the stochastic mean η2 for different values of �F0 at O(σ 2). Relative tothe constant noise case, the effect of the multiplicative noise dominates that of the nonlinear asymmetry in the deterministicsolution, and the means shift toward negative values. When �F0 is small, the deviation is negative all season, but as �F0

increases, the sea ice albedo feedback becomes more effective and the deviation becomes positive in summer.

The influence of multiplicative noise becomes more interesting at O(σ 2). Here, the deviationof the stochastic mean and the skewness are both affected by N(t)g(t), as shown in Fig. 12 for theformer. Within the range of �F0 in which the deterministic solutions represent perennial ice states,the deviation is negative for the entire season. For low values of �F0 when the ice is thick andthe sea ice-albedo is not effective during summer, the deviation is negative throughout the year. As�F0 increases, the ice thins and the sea ice-albedo feedback becomes more prominent in generatinga positive asymmetric response during summer. To understand the overall seasonal cycle of thedeviation of the stochastic means, we have to compare the asymmetry contained in the deterministicsolution with the effect of the multiplicative noise.

Figure 13 compares M1 and M2, which represent the contribution of the asymmetry due to thedeterministic nonlinearity and the role of the multiplicative noise to the deviation of the stochastic

0 2 4 6 8 10 12−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Month

15.0, nonlinear15.0, multiplicative16.0, nonlinear16.0, multiplicative

FIG. 13. The comparison between M1 (blue) and M2 (red) for a �F0 of 15 and 16, where the sea ice-albedo feedback isinfluential.


130.132.173.175 On: Sat, 18 Jan 2014 15:22:18


0 2 4 6 8 10 12−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

Month

Ske

wne

ss

10.013.015.016.0

FIG. 14. The skewness for the multiplicative noise case for several values of �F0 at O(σ ). Unlike the constant noise case,the skewness is negative during the entire season as �F0 increases.

mean, respectively. The distinction between the two contributions during summer is particularlylarge when �F0 ≥ 15. The asymmetric response to a given perturbation due to the nonlinearityin the sea ice-albedo feedback contributes positively to the deviation during summer, whereas themultiplicative noise provides a negative contribution. Clearly, overall the magnitude of M2 is largerthan that of M1.

The effect of the multiplicative noise influences the skewness in a similar manner, as seen inFig. 14. As opposed to the additive noise case, in this range of �F0 the skewness is negative duringthe entire season. The decrease of the magnitude of the skewness during summer as �F0 increases iscaused by the positive asymmetry in the sea ice-albedo feedback. The contributions to the asymmetryof the deterministic nonlinearity (S∗

1 ) and the multiplicative noise (S3) are shown in Fig. 15. The

0 2 4 6 8 10 12−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Month

15.0, nonlinear15.0, multiplicative16.0, nonlinear16.0, multiplicative

FIG. 15. The comparison of S∗1 (blue) and S3 (red) for a �F0 of 15 and 16. Even though S∗

1 , which represents the contributionsof the asymmetry embodied in d(t), are positive the contributions from the multiplicative nature of the noise forcing (S3) arenegative and of larger magnitude.


130.132.173.175 On: Sat, 18 Jan 2014 15:22:18


overall magnitude of the effect of the latter is larger than that of the former, and both effects aremaximized during summer, which is connected to the stability of sea ice.

The contribution of the multiplicative noise M2 to the deviation of the stochastic mean and theskewness S3 can be rewritten as

M2 =∫ t

0g(r )N (r )exp

[∫ t

rc(s) ds

]dr, and (72)

S3 = 2∫ t

0g(r )N (r )exp

[3∫ t

rc(s) ds

](η2

1)2 dr. (73)

The integral delay forms in the above equations represent the time integration of the effect of themultiplicative noise g(t)N(t) combined with the effect of the stability, which is understood in termsof the memory effect discussed previously. In M2, the seasonal change comes from c(r) embeddedin the memory effect

∫ tr c(s) ds. Because g(t)N(t) is always negative, the contribution from the most

recent past dominates M2, which is a maximum at the end of the summer. The contribution ofthe multiplicative noise to the skewness in Eq. (73) is explained in a similar manner; however, theseasonal change is controlled by the multiplicative effect combined with the square of the standarddeviation (η2

1)2. Moreover, because the standard deviation is a maximum during summer so too isS3.

It is reasonable to consider why the contribution from the multiplicative nature of the noiseforcing associated with sea ice export is always negative. Qualitatively, we can consider two oppositeextremes. In one case more ice is generated by the stochastic forcing and hence less ice exported,whereas in the other case less ice is generated and more ice exported. Assume that the originalamount of ice is 100 and the size of a perturbation is 10. Hence, 1% less (more) of sea ice is exportedfor the first case (the second case). Therefore, there exists an asymmetry between the two cases,and using this simple analogy, we can understand why the deviation of the stochastic means and theskewness are negative.

This simple example of multiplicative noise is related to the variation of ice export due principallyto the short time scale variability of atmospheric and oceanographic motions in the Arctic basin.Namely, while the deterministic Arctic sea ice model evolves over monthly time scales, the highfrequency variability of geophysical fluid motions is of order days to weeks. According to ouranalysis, the overall impact of stochastic atmospheric and oceanographic motions generates moresea ice. Clearly, different mechanisms of multiplicative noise may have a net influence that is of theopposite sign and these should be taken on a case by case basis.

IV. CONCLUSION

A quantitative understanding of the present variability of Arctic sea ice relies on understandingthe essential small-scale physical processes in the system. Due to the ability to include leadingorder physics, a stochastic model provides a useful tool for understanding the controlling factorsin driving sea ice variability. Although we cannot pinpoint the specific physical causes of theobserved fluctuations of one state variable of the ice, the approach provides the ability to analyzelinkages in the key processes influencing the variability of Arctic sea ice. Moreover, the generalperturbative approach motivated by this class of energy balance models is applicable to a wide swathof non-autonomous ordinary differential equations.

Our perturbation theory provides a unique view of the variability of perennial sea ice statesunder the influence of small amplitude stochastic forcing, and we analyzed the cases of constantadditive noise and multiplicative noise, the latter case representing the stochastic variability of sea iceexport. The statistical properties of the solutions are interpreted readily in terms of the stability andthe asymmetry that originate in the nonlinearity of the deterministic theory as well as the magnitudeand the multiplicative nature of the noise forcing. The first order approximation of the stochasticsolution is a Gaussian variable whose mean is the same as that of the deterministic solution. Thestandard deviation is determined by the combination of the stability of the deterministic solutionand the magnitude of the noise. The role of the stability is crucial because it determines the range


130.132.173.175 On: Sat, 18 Jan 2014 15:22:18


of time, which here is referred to as the “memory effect,” that affects the stationary solution as wellas determining the seasonal cycle of the standard deviation. At the next order in the expansion, theasymmetry contained in the deterministic solution and the multiplicative nature of the noise forcingare combined with the standard deviation and the memory effect to determine the seasonal cycles ofthe deviations of the stochastic means from the deterministic solutions and the skewness.

In the constant noise case, the asymmetry originating from the deterministic solution underliesthe deviation of the stochastic means and the skewness. Physically, the asymmetry comes fromthe ice-thickness-dependent response time scales to perturbations. Thinner ice is more vulnerableto the ice-albedo feedback and because it also grows faster in winter due to longwave radiativeheat loss, these two competing effects depend on the magnitude of greenhouse gas forcing �F0.For lower �F0, only the longwave stabilization is effective so that the deviation of the stochasticmeans and the skewness are negative during the entire year. However, as �F0 increases and thesea ice-albedo feedback becomes significant during summer, a positive deviation of the stochasticmeans and skewness emerges.

We analyzed a simple case of multiplicative noise forcing by treating ice export, driven byvariability in atmospheric and oceanographic motions, as a stochastic process. The stability and theasymmetry played similar roles qualitatively as in the constant noise case, but multiplicative noisehas a novel form of skewness. Here, the overall skewness is always negative. This may be of someuse in in understanding the recent variability of Arctic sea ice driven by variability in large-scaleatmospheric motions.

The analysis developed here provides basic insight into the qualitative nature of Arctic sea icevariability in the perennial ice state, and how the system may change under increased greenhouse gasforcing. For example, using a wide range of observations on many time scales,10 one may be able toassess which quantities dominate the statistical variability and extract noise amplitudes intrinsic tothe system. We note, however, that when seasonally varying states emerge, this perturbation methodmay have to be modified due the detailed structure of the strong nonlinearity. For our understandingof the future state of Arctic sea ice, we must perform systematic detailed numerical simulationsof the seasonally varying states, which is the focus of our next papers on the topic. Having testedthe analytical results here using robust numerical methods, many of the challenges faced are welldefined. Finally, it is hoped that our general approach will be taken into other, mathematically similar,problems in statistical dynamical systems.

ACKNOWLEDGMENTS

W.M. thanks NASA for a graduate fellowship and J.S.W. thanks the John Simon GuggenheimFoundation, the Swedish Research Council, and a Royal Society Wolfson Research Merit Award forsupport.

APPENDIX A: CALCULATION OF M1

The mean of M1 is written as

M1 = exp

[∫ t

0c(s) ds

] ∫ t

0d(r )η2

1exp

[−∫ r

0c(s)

]dr, (A1)

where

η21 = exp

[2∫ t

0c(s) ds

] ∫ t

0N 2(r )exp

[−2∫ r

0c(s) ds

]dr. (A2)

After substituting this expression into Eq. (A1), we find

M1 = exp

[∫ t

0c(s) ds

] ∫ t

0d(r )exp

[∫ r

0c(s) ds

]×∫ r

0N 2(r )exp

[−2∫ r

0c(s) ds

]dr dr. (A3)


130.132.173.175 On: Sat, 18 Jan 2014 15:22:18


Now, let t = nT + t and∫ T

0 c(s) ds = −γ , where 0 ≤ t < T and γ is positive. First, we see that

exp

[∫ t

0a(s) ds

]= e−nγ exp

[∫ t

0c(s) ds

], (A4)

and hence find that

∫ t

0d(r )exp

[∫ r

0c(s) ds

] ∫ r

0N 2(r )exp

[−2∫ r

0c(s) ds

]dr dr

=∫ nT +t

nTd(r )exp

[∫ r

0c(s) ds

] ∫ r

0N 2(r )exp

[−2∫ r

0c(s) ds

]dr dr

+∫ nT

0d(r )exp

[∫ r

0c(s) ds

] ∫ r

0N 2(r )exp

[−2∫ r

0c(s) ds

]dr dr. (A5)

Next we can express

∫ kT +r

0N 2(r)exp

[−2∫ r

0c(s) ds

]dr

=k∑

l=1

∫ lT

(l−1)TN 2(r )exp

[−2∫ r

0c(s) ds

]dr + e2kγ

∫ r

0N 2(r )exp

[−2∫ r

0c(s) ds

]dr

=k∑

l=1

e2(l−1)γ AT + e2kr Ar = e2kγ − 1

e2γ − 1AT + e2kr Ar (A6)

Using this formula we can write

∫ nT +t

nTd(r )exp

[∫ r

0c(s) ds

] ∫ r

0N 2(r)exp

[−2∫ r

0c(s) ds

]dr dr

= enγ (AB)t + enγ − e−nγ

e2γ − 1AT Bt , (A7)

where

Bt ≡∫ t

0d(r )exp

[∫ r

0c(s) ds

]dr

(AB)t ≡∫ t

0d(r )exp

[∫ r

0c(s) ds

]Ar dr. (A8)

Based on the same procedure, we have

∫ nT

0d(r )exp

[∫ r

0c(s) ds

]{∫ r

0N 2(r)exp

[−2∫ r

0c(s) ds

]}dr

=n∑

k=1

∫ kT

(k−1)Td(r )exp

[∫ r

0c(s) ds

]{∫ r

0N 2(r )exp

[−2∫ r

0c(s) ds

]dr

}dr

= e(k−1)γ − e−(k−1)γ

e2γ − 1AT BT + e(k−1)γ (AB)T , (A9)


130.132.173.175 On: Sat, 18 Jan 2014 15:22:18


and therefore,

η2 = exp

(∫ t

0c(s) ds

)[1 − e−2nγ

e2γ − 1AT Bt + (AB)t+

e−nγ

(n∑

k=1

e(k−1)γ − e−(k−1)γ

e2γ − 1AT BT +

n∑k=1

e(k−1)γ (AB)T

)]. (A10)

For the stationary solution, we go over to the limit

limt→∞ M1(t)=exp

[∫ t

0c(s) ds

][1

(e2γ −1)(eγ −1)AT BT + 1

eγ −1(AB)T + 1

e2γ −1AT Bt +(AB)t

].

(A11)APPENDIX B: CALCULATION OF M3

The term, M3, is written as

∫ t

0g(r )η1exp

[−∫ r

0c(s) ds

]dW

=∫ t

0g(r )

{∫ r

0N (r )exp

[−∫ r

0c(s) ds

]dW

}dW , (B1)

where we have used

η1 =∫ t

0N (r )exp

[−∫ r

0c(s) ds

]dW. (B2)

Now, if we let f (r ) ≡ N (r )exp(− ∫ r

0 c(s) ds), then we can write

M3 =∫ t

0g(r )

(∫ r

0f (r ) dW

)dW . (B3)

The Ito-integral form can be approximated by a Riemann sum. It is important to note that weuse the left Riemann sum, which means that in S =∑if(x∗)(xi − xi − 1), we take xi − 1 for x*. Thisguarantees the Martingale property of the Riemann sum. The integral can be represented as

n−1∑i=0

g(ri )�Wi

⎛⎝ i−1∑

j=0

f (r j )�W j

⎞⎠ , (B4)

where �Wi = Wi+1 − Wi . The Wiener process, W (t), has independent increments such that�Wi�W j = δi j , where δij = 1 when i = j, otherwise δij = 0. In the sum, i and j are alwaysdifferent. Therefore,

n−1∑i=0

g(ri )�Wi

⎛⎝ i−1∑

j=0

f (r j )�W j

⎞⎠ = 0, (B5)

and we can conclude that M2 = 0.


130.132.173.175 On: Sat, 18 Jan 2014 15:22:18


APPENDIX C: CALCULATION OF S1

The calculation of S1 is based upon the Martingale property of the Ito-integral. First, we canstart from the expression for S1, which is

S1 = exp

[3∫ t

0c(r ) dr

]{∫ t

0N (r )exp

[−∫ r

0c(s) ds

]dW

}2

×∫ t

0d(r )exp

[∫ r

0c(s) ds

]{∫ r

0N (r ′)exp

[−∫ r ′

0c(s) ds

]dW

}2

= exp

[3∫ t

0c(s) ds

] ∫ t

0d(r )exp

[∫ r

0c(s) ds

]{∫ t

0N (r )exp

[−∫ r

0c(s) ds

]dW

}2

×{∫ r

0N (r ′)exp

[−∫ r ′

0c(s) ds

]dW

}2

dr

= exp

[3∫ t

0c(s) ds

] ∫ t

0d(r )exp

[∫ r

0c(s) ds

]

×{∫ r

0N (r ′)exp

[−∫ r ′

0c(s) ds

]dW +

∫ t

rN (r ′)exp

[−∫ r ′

0c(s) ds

]dW

}2

×{∫ r

0N (r ′)exp

[−∫ r ′

0c(s) ds

]dW

}2

dr. (C1)

Now, we use the Martingale property of the Ito-integral for the ensemble average of S1. First, wehave

{∫ r

0N (r ′)exp

[−∫ r ′

0c(s) ds

]dW

}3 ∫ t

rN (r ′)exp

[−∫ r ′

0c(s) ds

]dW

={∫ r

0N (r ′)exp

[−∫ r ′

0c(s) ds

]dW

}3

×∫ t

rN (r ′)exp

[−∫ r ′

0c(s) ds

]dW

= 0, (C2)

where∫ t

0 (.)dW is a Gaussian variable whose odd moments are zero. Therefore,

S1 = exp

[3∫ t

0c(s) ds

] ∫ t

0d(r )exp

[∫ r

0c(s) ds

]

=⎧⎨⎩(∫ r

0N (r ′)exp

[−∫ r ′

0c(s) ds

]dW

)4

+(∫ t

rN (r ′)exp

[−∫ r ′

0c(s) ds

]dW

)2

×(∫ r

0N (r ′)exp

[−∫ r ′

0c(s) ds

]dW

)2⎫⎬⎭ , (C3)


130.132.173.175 On: Sat, 18 Jan 2014 15:22:18


where we can use

(∫ r

0N (r ′)exp

[−∫ r ′

0c(s) ds

]dW

)4

= 3

{∫ r

0N 2(r ′)exp

[−2∫ r ′

0c(s) ds

]dr ′}2

(∫ t

rN (r ′)exp

[−∫ r ′

0c(s) ds

]dW

)2

=∫ t

rN 2(r ′)exp

[−2∫ r ′

0c(s) ds

]dr ′

(∫ r

0N (r ′)exp

[−∫ r ′

0c(s) ds

]dW

)2

=∫ r

0N 2(r ′)exp

[−2∫ r ′

0c(s) ds

]dr ′. (C4)

Therefore, finally we have

S1 = exp

[3∫ t

0c(s) ds

] ∫ t

0d(r )exp

[∫ r

0c(s) ds

]×⎧⎨⎩2

(∫ r

0N 2(r ′)exp

[−2∫ r ′

0c(s) ds

]dr ′)2

+∫ t

0N 2(r ′)exp

[−2∫ r ′

0c(s) ds

]dr ′ ×

∫ r

0N 2(r ′)exp

[−2∫ r ′

0c(s) ds

]dr ′}

dr

= 2exp

[3∫ t

0c(s) ds

] ∫ t

0d(r )exp

[∫ r

0c(s) ds

](∫ r

0N 2(r ′)exp

[−2∫ r ′

0c(s) ds

]dr ′)2

dr

+ exp

[3∫ t

0c(s) ds

] ∫ t

0N 2(r )exp

[−2∫ r

0c(s) ds

]dr

×∫ t

0d(r )exp

[∫ r

0c(s) ds

] ∫ r

0N 2(r ′)exp

[−2∫ r ′

0c(s) ds

]dr ′dr. (C5)

APPENDIX D: CALCULATION OF S3

Let �(r ) ≡ N (r )exp(− ∫ r

0 c(s) ds). Here, we can write S3 as

S3 = exp

[3∫ t

0c(s) ds

](∫ t

0�(r ) dWr

)2 ∫ t

0g(r )

∫ r

0�(r ′) dWr ′ dWr , (D1)

and use the Riemann summation to approximate the integral form, which is

(∫ t

0�(r ) dWr

)2 ∫ t

0g(r )

∫ r

0�(r ′) dWr ′ dWr

(

n−1∑i=0

�(ri )�Wi

)2 n−1∑j=0

g(r j )

(j−1∑k=0

�(rk)�Wk

)�W j (D2)

Finally, we use the fact that the Wiener process, Wt , has independent increments and thus satisfies

�Wi�W j = δi j�ti . (D3)


130.132.173.175 On: Sat, 18 Jan 2014 15:22:18


Therefore, we find that

(n−1∑i=0

�(ri )�Wi

)2 n−1∑j=0

g(r j )

(j−1∑k=0

�(rk)�Wk

)�W j

= 2n−1∑j=0

g(r j )�(r j )(�W j )2

j−1∑k=0

�2(rk)(�Wk)2

= 2n−1∑j=0

g(r j )�(r j )�t j

j−1∑k=0

�2(rk)�tk

2∫ t

0g(r )N (r )exp

[−∫ r

0c(s) ds

] ∫ r

0N 2(r ′)exp

[−2∫ r ′

0c(s) ds

]dr ′ dr. (D4)

APPENDIX E: SIMPLIFICATION OF S∗1 + S3

Let t = nT + t . In this case, the prefactor exp(

3∫ t

0 c(s) ds)

is equal to e−3nγ

× exp(

3∫ t

0 c(s) ds)

, which will be used in the last step to find the limit t → ∞ (n → ∞).

Let

I1 ≡∫ t

0d(r )exp

[∫ r

0c(s) ds

]{∫ r

0N 2(r ′)exp

[−2∫ r ′

0c(s) ds

]dr ′}2

dr, (E1)

I2 ≡∫ t

0g(r )N (r )exp

[−∫ r

0c(s) ds

] ∫ r

0N 2(r ′)exp

[−2∫ r ′

0c(s) ds

]dr ′ dr. (E2)

The time integral decomposes as∫ nT +t

0(·) dr =

∫ nT +t

nT(·)dr +

n∑k=1

∫ kT

(k−1)T(·)dr, (E3)

and thus here,

∫ kT

(k−1)Td(r )exp

[∫ r

0c(s) ds

]{∫ r

0N 2(r ′)exp

[−2∫ r ′

0c(s) ds

]dr ′}2

dr

=∫ T

0d(r )e−γ (k−1)exp

[∫ r

0c(s) ds

](e2(k−1)γ − 1

e2γ − 1AT + e2(k−1)γ Ar

)2

dr

= e3(k−1)γ − 2e(k−1)γ + e−(k−1)γ

(e2γ − 1)2A2

T BT + e3(k−1)γ − e(k−1)γ

e2γ − 1AT (AB)T + e3(k−1)γ (A2 B)T . (E4)

Therefore,

n∑k=1

∫ kT

(k−1)Td(r )exp

[∫ r

0c(s) ds

]{∫ r

0N 2(r ′)exp

[−2∫ r ′

0c(s) ds

]dr ′}2

dr

= 1

(e2γ − 1)2

[e3nγ − 1

e3γ − 1− 2enγ − 2 − e + e−(n−1)γ

eγ − 1

]A2

T BT

+ 2

e2γ − 1

[e3nγ − 1

e3γ − 1− enγ − 1

eγ − 1

]AT (AB)T + e3nγ − 1

e3γ − 1(A2 B)T . (E5)


130.132.173.175 On: Sat, 18 Jan 2014 15:22:18


In a similar manner, we also find that

∫ nT +t

nTd(r )exp

[∫ r

0c(s) ds

]{∫ r

0N 2(r ′)exp

[−2∫ r ′

0c(s) ds

]dr ′}2

dr

= e3nγ − enγ + e−nγ

(e2γ − 1)2A2

T Bt + e3nγ − enγ

e2γ − 1AT (AB)t + e3nγ (A2 B)t . (E6)

The simplification of I2 can be performed similarly. First, we find∫ kT

(k−1)Tg(r )N (r )exp

[−∫ r

0c(s) ds

] ∫ r

0N 2(r ′)exp

[−2∫ r ′

0c(s) ds

]dr ′ dr

= e3(k−1)γ − e(k−1)γ

e2γ − 1GT AT + e3(k−1)γ (G A)T , (E7)

and hence,

n∑k=1

∫ kT

(k−1)Tg(r )N (r )exp

[−∫ r

0c(s) ds

] ∫ r

0N 2(r ′)exp

[−2∫ r ′

0c(s) ds

]dr ′ dr

= 1

e2γ − 1

[e3nγ − 1

e3γ − 1− enγ − 1

eγ − 1

]GT AT + e3nγ − 1

e3γ − 1(G A)T . (E8)

We then determine that∫ nT +t

nTg(r )N (r )exp

[−∫ r

0c(s) ds

] ∫ r

0N 2(r ′)exp

[−2∫ r ′

0c(s) ds

]dr ′ dr

= e3nγ − 1

e2γ − 1AT Gt + e3nγ (G A)t . (E9)

Finally, we arrive at

limt→∞(η2

1η2 − η21η2)

= limn→∞ e−3nγ exp

[3∫ t

0c(r ) dr

](I1 + I2)

= 2exp

[3∫ t

0c(r ) dr

][1

(e2γ − 1)2A2

T Bt + 1

(e2γ − 1)2(e3γ − 1)A2

T BT

+ 2

e2γ − 1AT (AB)t + 2

(e2γ − 1)(e3γ − 1)AT (AB)T + (A2 B)t + 1

e3γ − 1(A2 B)

+ 1

(e2γ − 1)(e3γ − 1)AT GT + 1

e2γ − 1AT Gt + 1

e3γ − 1(G A)T + (G A)t

]. (E10)

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