Elementary models for turbulent diffusion with complex physical features: eddy diffusivity, spectrum, and intermittent probability density functions

with complex physical features: eddy

diffusivity, spectrum, and intermittent

probability density functions

By Andrew J. Majda † and Boris Gershgorin Department of Mathematics, and Center for Atmosphere-Ocean Science, Courant Institute of Mathematical Sciences, New York University, New York, New York

This paper motivates, develops, and reviews elementary models for turbulent trac-ers with a background mean gradient which despite their simplicity, have complex statistical features mimicking crucial aspects of laboratory experiments and atmo-spheric observations. These statistical features include exact formulas for tracer eddy diffusivity which is nonlocal in space and time, exact formulas and simple numerics for the tracer variance spectrum in a statistical steady state, and the transition to intermittent scalar probability density functions with fat exponential tails as certain variances of the advecting mean velocity are increased while sat-isfying important physical constraints. The recent use of such simple models with complex statistics as unambiguous test models for central contemporary issues in both climate change science and the real time filtering of turbulent tracers from sparse noisy observations is highlighted throughout the paper.

Keywords: turbulent diffusion; eddy diffusivity; intermittency; exactly solvable model; white noise limit

1. Introduction

One of the important paradigm models for the behavior of turbulent systems (Avel-laneda and Majda, 1994; Majda and Kramer, 1999) involves a passive tracer T (~x, t) which is advected by a velocity field ~v(~x, t) with dynamics given by

∂T

∂t + ~v · ~∇T = κ∆T, (1.1)

where κ > 0 is molecular diffusion and the velocity field ~v is incompressible, div~v = 0. For simplicity of exposition, we assume here that ~x = (x, y) is two-dimensional. When ~v(~x, t) is a turbulent velocity field, the statistical properties of solutions of (1.1) such as their large scale effective diffusivity, energy spectrum, and probability density function (PDF) are all important in applications. These range from, for example, the spread of pollutants or hazardous plumes in environmen-tal science to the behavior of anthropogenic and natural tracers in climate change † Corresponding author address: Andrew J. Majda, Department of Mathematics, and Center for Atmospheric Ocean Sciences, Courant Institute, New York University, 251 Mercer st., New York, NY 10012.

science (Frierson, 2006, 2008; Neelin et al., 2010), to detailed mixing properties in engineering problems such as non-premixed turbulent combustion (Pope, 1976; Peters, 2000; Bourlioux and Majda, 2000). For turbulent random velocity fields, the passive tracer models in (1.1) also serve as simpler prototype test problems for closure theories for the Navier Stokes equations since (1.1) is a linear equation but is statistically nonlinear (Kraichnan, 1968, 1987; Avellaneda and Majda, 1992a,b; Majda, 1993a, 1994; Kraichnan, 1994; Kraichnan et al., 1995; Smith and Woodruff, 1998; Majda and Kramer, 1999). Avellaneda and Majda emphasized exactly solv-able and rigorous mathematical simplified models where the velocity field for (1.1) has the special form of a random shear flow with a mean sweep

~

v(~x, t) = (U (t), v(x, t)), (1.2)

(Avellaneda and Majda, 1990, 1992c, 1994; Majda, 1993a; Majda and Kramer, 1999), which despite their simplicity, capture key features of renormalization for various inertial range statistics for turbulent diffusion.

This research expository paper involves recent and ongoing developments in utilizing the simplified models in (1.1), (1.2) as the simplest prototype models which nevertheless capture qualitatively correct complex physical features that arise in laboratory experiments (Gollub et al., 1991; Lane et al., 1993; Jayesh and Warhaft, 1991, 1992), climate change science (Neelin et al., 2010; Majda and Gershgorin, 2010) and the practical need to recover the properties of a turbulent tracer as well as the associated velocity statistics through real-time filtering from sparse noisy partial observations (Majda et al., 2010; Gershgorin and Majda, 2010c). We review and expand upon recent work with the simplest mathematical models (Bourlioux and Majda, 2002; Bourlioux et al., 2006; Gershgorin and Majda, 2010b,c; Majda and Gershgorin, 2010) which capture the observed phenomena such as

A) Transitions between Gaussian and fat tailed highly intermittent PDFs for the tracer in laboratory experiments as the Peclet number varies with a mean background gradient for the tracer

(Gollub et al., 1991; Jayesh and Warhaft, 1991, 1992)

B) The nature of the sustained turbulent spectrum for scalar variance with a background gradient for the tracer (Sreenivasan, 1996) C) Fat tail PDFs for anthropogenic and natural tracers with highly

intermittent exponential tails in observations of the present climate (Neelin et al., 2010)

D) Eddy diffusivity approximations for tracers in climate change science (Frierson, 2006, 2008; Majda and Gershgorin, 2010)

(1.3)

Another important issue is the ability to recover these statistical features from sparsely observed partial noisy observations and simplified model problems pro-vide unambiguous test problems for these compex features (Gershgorin and Majda, 2008, 2010a,b,c; Majda et al., 2010). An important effect responsible for these new phenomena is the existence of a background mean gradient for the tracer

so that with (1.1) and (1.2), T0 _satisfies ∂T0 ∂t + U (t) ∂T0 ∂x + v(x, t) ∂T0 ∂y = κ∆T 0_{− αv(x, t). (1.5)}

Note that the random velocity v(x, t) in (1.5) drives the fluctuations in the tracer through the background mean gradient. In the models discussed here, the large scale sweeping flow U (t) has the form

U (t) = ¯U (t) + U0(t), (1.6)

where ¯U (t) is a deterministic mean sweep and U0(t) represents random fluctuations of the mean. The turbulent velocity field v(x, t) satisfies the stochastic PDE (readily solved by Fourier series, see section 2)

∂v(x, t) ∂t + P  ∂ ∂x, U (t)  v(x, t) = Fv(x, t). (1.7)

In (1.7), P is a pseudo-differential operator that combines both dispersive wave-like and dissipative effects on v with potential dependence on the cross-sweep U (t) and Fv is a forcing with both deterministic and random components. Assuming that the tracer fluctuations T0_{(x, t) only depend on the x variable alone and dropping} the prime results in the simplified version of (1.5) given by

∂T ∂t + U (t) ∂T ∂x = −αv(x, t) + κ ∂2_T ∂x2 − dTT. (1.8)

The term with dT > 0 is an explicit damping factor added to (1.8) besides molecular diffusion in order to damp the zero mode and arises naturally from the full multi-dimensional model in (1.5) after partial Fourier transform in y at non-zero Fourier modes (Avellaneda and Majda, 1990, 1992b; Majda and Kramer, 1999).

There are remarkably different regimes in the simplified models in (1.6)-(1.8). For example, Bourlioux and Majda (2002) considered the simplest model for (1.5), (1.6), (1.7) where U (t) = ¯U (t) with ¯U (t), an explicit time periodic function with isolated zeroes while v(x, t) is a deterministic or random spatially periodic function without dispersive properties; they identified a transparent intermittency mecha-nism where stream lines of the velocity field are blocked for U (t) 6= 0 with modest turbulent diffusion and unblocked with enhanced turbulent transport in the vicinity of the zeroes of U (t); this results in intermittency in the time averaged PDFs with the features in 1.3A) despite the fact that the PDFs are Gaussian for each (x, t). On the other hand, applications to atmospheric science require a non-negative zonal east-west mean jet, and random fluctuations consistent with this behavior so that

¯

U (t) > 0, ¯

U2− V ar(U0(t)) > 0, (1.9)

with V ar(U0(t)), the variance of U0(t) in (1.6) so that the zonal jet almost always stays positive. These principal requirememnts in (1.9) for the models in (1.6), (1.7), (1.8) still allow for highly intermittent non-Gaussian PDFs in the tracer model (Gershgorin and Majda, 2010c) as observed in 1.3C) (Neelin et al., 2010) in a

regime very different from that of Bourlioux and Majda (2002). One goal of the present paper is to understand the source of intermittency in this new regime.

Finally, we end this introduction with a brief discussion of the content of the remainder of this paper. We begin section 2 with a motivating example from climate science involving zonal jets and β-plane Rossby waves in the velocity field which naturally leads to the master models in (1.6), (1.7), (1.8). We show how to develop closed exact formulas for the mean and variance of the master model in section 2; in section 3 we develop interesting simplifications involving uncorrelated velocity statistics (Gershgorin and Majda, 2010c) and various white noise limit models and establish connections with other models which have been developed earlier (Ger-shgorin and Majda, 2008, 2010a,b) by the authors in different contexts. In section 4, we interpret exact equations for the mean statistics as non-local eddy diffusiv-ity models for the passive tracer; surprisingly the master model with correlated velocity fields has both nonlocal space-time eddy diffusivity and a nonlocal effect of the mean transverse velocity in (1.8) in the exact closed equation. Statistics of the turbulent tracer spectrum are discussed in detail in section 5 while section 6 is devoted to a systematic study of scalar intermittency in the PDFs as discussed in the previous paragraph. Both closed form analytical results and simple numeri-cal experiments are utilized throughout this paper. Section 7 is a brief concluding discussion. Comments regarding the use of such models for climate change science (Majda and Gershgorin, 2010; Gershgorin and Majda, 2010b) and real time filtering or data assimilation (Gershgorin and Majda, 2008, 2010b,c) are made throughout the paper.

2. Elementary models for turbulent tracers: physical

motivation and exact statistics for the mean and variance

Here we first provide some elementary physical motivation for the master models in (1.6), (1.7), (1.8) utilizing special exact solutions of the β-plane quasi-geostrophic equations from climate science and then show how to develop closed formulas for the mean and variance statistics.

(a) Physical motivation of the master model

The equations of β-plane quasigeostrophic flow (Pedlosky, 1990; Majda and Wang, 2006) involve a stream function

Ψ = −U (t)y + ψ(~x, t), (2.1) with velocity ~v =  −∂Ψ ∂y, ∂Ψ ∂x T , (2.2)

and potential vorticity

Q = F U (t)y + ∆ψ − F y + βy, (2.3)

linked by the conservation of potential vorticity ∂Q

The factor F = L−2_R is the inverse square of the Rossby radius, LR, and β is the differential effect of planetary rotation at a given latitude. Consider special exact solutions of (2.1), (2.2), (2.3) consisting of a zonal jet U and one-dimensional Rossby waves with the form compatible with (1.2) given by

Ψ = −U (t)y + ψ(x, t),

F (~x, t) = F yFU(U, t) + D(∆)ψ +Fq(~x, t). (2.5) The parameter D(∆)ψ represents dissipative mechanisms such as Ekmann friction. Substituting (2.5) into (2.4) yields the exact dynamics for the zonal jet flow U (t),

dU dt =FU(U, t), (2.6) and for q = ψxx− F ψ dq dt + U (t) ∂q ∂x + (F U (t) + β) ∂ψ ∂x = D(∆)ψ +Fq(x, t). (2.7) Physically, the special exact solutions in (2.6), (2.7) describe a mean zonal jet, U (t), at a fixed latitude away from the tropics and dispersive Rossby waves which both feel the β-effect and the large scale zonal jet U (t); clearly, consistent boundary conditions at a fixed latitude require q(x, t) to be 2π-periodic which is utilized as the unit length. The velocity v is recovered from the two identities

v = ψx, q = ψxx− F ψ, (2.8)

through the nonlocal pseudo-differential operator

v = R ∂ ∂x  q, R = ∂ 2 ∂x2 − F −1 _∂ ∂x. (2.9)

The symbol of R _∂x∂
at a given spatial wave number is R = − ik

k2_+F. To get the

equation for v, we apply R _∂x∂
to (2.7) and use (2.9) to obtain the dynamics ∂v ∂t + U (t) ∂v ∂x + (F U (t) + β)R  ∂ ∂x  v = −dvv + ν ∂2_v ∂x2 + fv(x, t) + σvW˙v(t).(2.10) Special choices of the forcing Fq result in (2.10) so that fv(x, t) is deterministic forcing and σv(x) ˙Wv(t) denotes spatially correlated white noise forcing (Gardiner, 1997) which is readily represented below by Fourier series (Majda et al., 2010). The natural dissipative mechanisms for v are a combination of Ekmann damping −dvv and a small scale frictional viscosity ν∂_∂x2v2 (Majda and Wang, 2006). The model for

the tracer, T , involves fluctuations with a background north-south gradient αy as in (1.4) which results in the simplified tracer equation in (1.8).

(b) Velocity field in the master model

For the zonal jet U (t) in the above model in (2.5), (2.10) as well as the general master model, we assume that the forcingFU(U, t) in (2.6) has the special form of

deterministic forcing fU(t), damping −γU and white noise forcing σUW (t) which˙ results in the dynamics

dU (t)

dt = −γUU (t) + fU(t) + σU ˙

WU(t). (2.11)

The evolution of the shear flow is given in general by the following pseudo-differential equation ∂v(x, t) ∂t + P  ∂ ∂x, U (t)  v(x, t) = Fv(x, t), (2.12)

where P is a pseudo-differential operator that combines both wave-like and dissipa-tive components of the dynamics that can also depend on the cross-sweep and Fv is a forcing term that has both deterministic and random components. We specify the pseudo-differential operator P by its symbol in Fourier space Pk= −γvk+ iωvk

and rewrite (2.12) through Fourier series dvk(t)

dt = (−γvk+ iωvk)vk(t) + fvk(t) + σvkW˙vk(t), (2.13)

where γvkis dissipation and ωvkis dispersion relation. In general, both of these

func-tionals can depend on the cross-sweep, U (t), and here we assume linear dependence on U (t) as in the above example so that γvk does not depend on U (t)

ωvk= akU (t) + bk, (2.14)

for real coefficients ak, bk. A special case of the transverse shear equation in the master model has already been motivated in (2.10) where

P ∂ ∂x, U (t)  = U (t)∂ ∂x +  F U (t) + βR ∂ ∂x  + dv− ν ∂2 ∂x2. (2.15) Here, the damping depends on spatial Fourier wave number and the dispersion relation depends on both spatial Fourier wave number and the cross-sweep U (t). As we will find out below, this model is a very rich example of a model with eddy diffusivity which is nonlocal in time and space.

The choice of a particular form of the dissipation, γvk, and the dispersion, ωvk,

depends on the situation for which the model is applied. We consider the following three situations:

1. Non-dispersive waves with selective damping: γvk = dv+ νk

2_{, ω}

vk = −ck,

where ν is the flow viscosity (Gershgorin and Majda, 2010c), 2. Uncorrelated Rossby waves: γvk= ν(k

2_{+F ), ω} vk=

βk

k2_+F, by directly plugging

in the dispersion relation for the Rossby waves, where ν denotes the large scale selective damping diffusivity, say eddy diffusivity. Here, F = L−2_R and LR is the Rossby deformation radius, β is the tangent approximation to the local Coriolis forcing. This version of the master model was introduced and utilized recently as a test model in quantifying uncertainty in climate change science (Majda and Gershgorin, 2010) together with the tracer equation in (1.8),

3. Correlated Rossby waves (a generalization of case 2 developed in (2.10) above): γvk = dv+ νk 2_{, (2.16)} ωvk = akU (t) + bk, (2.17) ak = k  F k2 − 1  , (2.18) bk = βk k2_{+ F}. (2.19)

(c) Statistics of the velocity field The solution for the cross-sweep becomes

U (t) = ¯U (t) + U0(t), (2.20) where ¯ U (t) = Z t t0 GU(s, t)fU(s)ds, (2.21) U0(t) = GU(t0, t)U0+ UW(t), (2.22) UW(t) = σU Z t t0 GU(s, t)dWU(s), (2.23)

and we use the shortcut notation for the initial condition, U0= U (t0), and for the Green’s function of the cross-sweep

GU(s, t) = e−γU(t−s). (2.24)

Then, the statistics of the Gaussian cross-sweep become hU (t)i = GU(t0, t) hU0i + Z t t0 GU(s, t)fU(s)ds, (2.25) V ar(U (t)) = G2_U(t0, t)V ar(U0) + σ2 U 2γU 1 − G2_U(t0, t)
. (2.26) We use Fourier series to compute explicit solutions of the master model (Majda et al., 2010; Gershgorin and Majda, 2010c) and then average them using identi-ties for Gaussian random fields (Gershgorin and Majda, 2008, 2010a,b; Majda and Gershgorin, 2010). Here the details are omitted since they are very similar to those carried out elsewhere on similar models by the authors. The main technique is to solve the master model explicitly path-wise and use formulas such as the following equality for any complex Gaussian z and any real Gaussian x

hzeix_{i =}_{hzi + iCov(z, x)}_eihxi−1

2V ar(x). (2.27)

The solution for each Fourier mode of the shear flow in the general case with time dependent dispersion, ωvk, has the form

vk(t) = Gvk(t0, t)vk,0+ Z t t0 Gvk(s, t)fvk(s)ds + σvk Z t t0 Gvk(s, t)dWvk(s), (2.28)

where vk,0= vk(t0) and the Green’s function for the shear flow is defined as Gvk(s, t) = e −γ_vk(t−s)+iJk(s,t)_{, (2.29)} Jk(s, t) = Z t s ωvk(s 0_)ds0_{. (2.30)}

For the case of correlated flow, we compute further Jk(s, t) = Z t s ωvk(s 0_)ds0_{= a} kL(s, t) + bk(t − s), (2.31) L(s, t) = Z t s U (s0)ds0= LD(s, t) + LW(s, t) + b0(s, t)U0, (2.32) LD(s, t) = Z t s Z s0 t0 GU(r0, s0)fU(r0)dr0ds0, (2.33) LW(s, t) = Z t s UW(s0)ds0= σU Z t s Z s0 t0 GU(s00, s0)dWU(s00)ds0, (2.34) b0(s, t) = − 1 γU (GU(t0, t) − GU(t0, s)). (2.35) Next, we find the mean of vk

hvk(t)i = hGvk(t0, t)vk,0i +

Z t

t0

hGvk(s, t)ifvk(s)ds. (2.36)

Note that the Green’s function of the shear flow, Gvk(s, t) defined in (2.29) has a

form of an exponential of a Gaussian random variable, iJk(s, t), from (2.30), with a deterministic factor, e−γvk(t−s)_{. We use the properties mentioned above in (2.27)}

of the characteristic function of a Gaussian random field (Gershgorin and Majda, 2008, 2010a,b,c) to find

hGvk(t0, t)vk,0i = (hvk,0i + iakb0(t0, t)Cov(vk,0, U0)) hGvk(t0, t)i , (2.37)

hGvk(s, t)i = e

−γ_vk(t−s)+ihJk(s,t)i−12V ar(Jk(s,t))_{, (2.38)}

hJk(s, t)i = akhL(s, t)i + bk(t − s), (2.39)

V ar(Jk(s, t)) = a2kV ar(L(s, t)), (2.40)

hL(s, t)i = LD(s, t) + b0(s, t) hU0i , (2.41)

V ar(L(s, t)) = b20(s, t)V ar(U0) + V ar(LW(s, t)), (2.42) V ar(LW(s, t)) = σ2 U 2γ3 U (−2 + 2γU|t − s| + 2e−γU|t−s|+ 2GU(2t0, s + t) −G2 U(t0, t) − G2U(t0, s)), (2.43)

(d ) Passive tracer in a mean gradient and tracer statistics in the master model For completeness, we repeat the equation for the dynamics of a passive tracer with a mean gradient in the y-direction from (1.8) which is given by

∂T ∂t + U (t) ∂T ∂x = κ ∂2 ∂x2T − dTT − αv. (2.44)

In Fourier space, this equation becomes dTk dt = (−γTk+ iωTk)Tk− αvk, (2.45) where γTk = dT + κk 2_{, (2.46)} ωTk = −U (t)k. (2.47)

It is worth noting that the form of the Fourier dynamics of the tracer is very similar to the form of the Fourier dynamics of the shear flow given by Eq. (2.13) with an important difference in the forcing and dispersion terms. The forcing for the tracer is given by the shear flow through the mean gradient.

To develop the tracer statistics, we note that the solution for each Fourier mode of the tracer is given by

Tk(t) = GTk(t0, t)Tk,0− α

Z t

t0

GTk(s, t)vk(s)ds, (2.48)

where Tk,0= Tk(t0) and the Green’s function for the tracer is given by GTk(s, t) = e −γ_Tk(t−s)−ikL(s,t)_{, (2.49)} L(s, t) = Z t s U (s0)ds0. (2.50)

Note that the shear flow vk(s) is governed by the dynamics discussed above in (2.13). The mean of the tracer is given by

hTk(t)i = hGTk(t0, t)Tk,0i − α Z t t0 hGTk(s, t)Gvk(t0, s)vk,0i ds −α Z t t0 Z s t0 hGTk(s, t)Gvk(r, s)i fvk(r)drds, (2.51)

where hGTk(s, t)Gvk(r, s)i is a characteristic function of a Gaussian and can be

computed analytically in a similar fashion to hGvki from (2.38)

hGTk(s, t)Gvk(r, s)i = e

−γ_vk(r−s)−γ_Tk(t−s)+i(hJk(s,t)i−khL(s,t)i)−12V ar(Jk(s,t)−kL(s,t))_(2.52).

In the statistically steady state, the mean of the tracer becomes hTk(t)i = −α Z t −∞ Z s −∞ hGTk(s, t)Gvk(r, s)i fvk(r)drds. (2.53)

And the covariance is given by Cov(Tj(t), Tk(t)) = α2 Z t −∞ Z t −∞  GTj(s, t)G ∗ Tk(s 0_{, t)v} j(s)v∗k(s0) − GTj(s, t)vj(s) hGTk(s 0_{, t)v} k(s0)i ∗ dsds0, (2.54)

where GTj(s, t)G ∗ Tk(s 0_{, t)v} j(s)v∗k(s 0_{) =} Z s −∞ Z s0 −∞ Gvj(r, s)G ∗ vk(r 0_{, s}0_)G Tj(s, t)G ∗ Tk(s 0_{, t) f} vj(r)f ∗ vk(r 0_)drdr0 +σ2_v kδ j k Z min(s,s0) −∞ hGTk(s, t)G ∗ Tk(s 0_{, t)G} vk(r, s)G ∗ vk(r, s 0_{)idr (2.55)}

3. Special regimes of the master model

While we have presented closed exact formulas for the first and second order statis-tics of the master model, it is difficult to process these analytic formulas in general (however, see section 4 below for the mean statistics and eddy diffusivity). Here, we develop instructive regimes of the master model which lead to both analytic and numerical tractability.

(a) Uncorrelated velocity field

One important special case of the master model has an uncorrelated velocity field with the shear flow given by independent complex OU processes with varying frequency ωvk. In physical space, the shear flow is given by

∂v(x, t) ∂t + P  ∂ ∂x  v(x, t) = Fv(x, t), (3.1)

where the pseudo-differential operator P is independent of the cross-sweep, U (t). This model with a mean jet U (t), tracer T (x, t), and v(x, t) given in (3.1) is simpler to solve analytically and yet is still very rich with physical phenomena such as turbulent spectrum and intermittent fat-tail PDFs of the tracer. This model was used by Majda and Gershgorin (2010) as a simplified climate model for testing information theory in a climate change context. In particular, we discussed the role of coarse-graining, and the importance of signal vs dispersion in the total lack of information due to model error. There are many other interesting questions that could be addressed using this simplified climate model: what is the role of the turbulent spectrum in describing the uncertainty of the model with errors, how well can one estimate the most sensitive climate change directions in a model error context? One of the particularly striking features of this model is the one inherited from the master model, the analytical expression for eddy diffusivity. Here, the eddy diffusivity is non-local in time, which poses a very practical question: how good is a local in time approximation to the eddy diffusivity? These issues are being addressed in a forthcoming article by the authors.

Another practical example where the tracer model with the uncorrelated veloc-ity field was used is in real-time data assimilation (Gershgorin and Majda, 2010c). There, we used the exactly solvable structure of the model to construct a Nonlinear Extended Kalman Filter (Gershgorin and Majda, 2008, 2010a; Gershgorin et al., 2010a,b; Harlim and Majda, 2010) and then discussed the role of sparse and par-tial observations in filtering. We studied how well the filter recovers the turbulent spectrum for the velocity field and for the tracer and the intermittent PDF with fat

tails for the tracer. We studied the role of the dispersion relation in recovering the true signal with extremely sparse observations. An interesting question here is how well the true signal can be filtered with an imperfect model with model error due to eddy diffusivity approximation. Here, the fact that the eddy diffusivity is non-local in time poses a real challenge in using the local in time approximation. Another issue to study is how well the Stochastic Parameterization Extended Kalman Filter (Gershgorin et al., 2010a,b; Harlim and Majda, 2010) can recover the true signal by estimating the parameters such as eddy diffusivity or the mean gradient “on the fly”.

The test model for the tracer with uncorrelated velocity field is given by the following equations for the Fourier modes

dU (t) dt = −γUU (t) + fU(t) + σU ˙ WU(t), (3.2) dvk(t) dt = (−γvk+ iωvk)vk(t) + fvk(t) + σvkW˙vk(t), (3.3) dTk(t) dt = (−γTk+ iωTk)Tk(t) − αvk(t), (3.4)

where for the example of uncorrelated Rossby waves ωvk = βk k2_{+ F} s , (3.5) ωTk = −kU (t), (3.6) γvk = ν(k 2_{+ F} s), (3.7) γTk = dT + κk 2_{. (3.8)}

Note that other forms of the dispersion relation for the waves can be studied since the formulas are general (Gershgorin and Majda, 2010c).

In order to find the first and second order statistics of this model, we can either compute them independently using the above model equations and the same tech-nology that was used for finding statistics of the master model, or we can consider a special case of the statistics of the master model when ak ≡ 0 in (2.17). We use the latter way and find

hTk(t)i = hGTk(t0, t)Tk,0i − α Z t t0 Gvk(t0, s) hGTk(s, t)vk,0i ds −α Z t t0 hGTk(s, t)i ¯Vk(t0, s)ds, (3.9) where ¯ Vk(t0, s) = Z s t0 Gvk(r, s)fvk(r)dr. (3.10)

Note that in this case of uncorrelated velocity field, the Green’s function for the shear flow is deterministic because it does not depend on the Gaussian cross-sweep, U (t). In the statistically steady state, the mean tracer can be obtained by setting ak= 0 in (2.53)

hTk(t)i = −α Z t

−∞

where ¯ Vk(s) = Z s −∞ Gvk(r, s)fvk(r)dr. (3.12)

The covariance in the statistically steady state can be obtained by setting ak = 0 in (2.54) Cov(Tj(t), Tk(t)) = α2 Z t −∞ Z t −∞  GTj(s, t)G ∗ Tk(s 0_{, t) hv} j(s)vk∗(s0)i −GTj(s, t) hvj(s)iG ∗ Tk(s 0_{, t) hv} k(s0)i ∗ dsds0,(3.13) where hvj(s)v∗k(s0)i = Z s −∞ Z s0 −∞ Gvj(r, s)G ∗ vk(r 0_{, s}0_)f vj(r)f ∗ vk(r 0_)drdr0 +σ2_v kδ j k Z min(s,s0) −∞ Gvk(r, s)G ∗ vk(r, s 0_{)dr. (3.14)}

The double integral in the last expression is easy to compute analytically for special forms of fvk.

The equation for the spectrum of the tracer in the case of time independent forcing for U (t) and no forcing for vk becomes

Cov(Tk(t), Tl(t)) = α2 σ2_k 2γk δl_k Z t −∞ Z t −∞ e−(γ+κk2)(2t−s−r)e−ik ¯U (r−s)e−γk|s−r|+iωk(s−r)

×e−k22(V ar(J (s,t))+V ar(J (r,t))−2Cov(J (s,t),J (r,t)))_{dsdr, (3.15)}

where for s > r Cov(J (s, t), J (r, t)) = V ar(J (s, t)) − σ 2 U 2γ3 U  e−γU(t−s)_{− e}−γU(t−r)_{− 1 + e}−γU(s−r)_.

From Eq. (3.15), it is obvious that the variance of Tk is proportional to the variance of vk for each k. However, the proportionality factor is a function of k that has to be studied separately. Below, we perform this study numerically to find that this proportionality coefficient is a power law with two different powers for small wave numbers and large wave numbers. These powers are in general functions of the parameters of the velocity field.

It is also interesting to study the role of time periodic forcing for the velocity field. In this case the spectrum of the tracer becomes time periodic although the spectrum of the waves vk is constant in time (Gershgorin and Majda, 2010b).

(b) White noise limits of the master model

We consider two separate interesting white noise limits of the velocity field in the master model and their effect on the dynamics of the tracer T . It is well-known that interesting analytical simplification for tracer statistics occurs in this regime (Kraichnan, 1968, 1994; Majda, 1993a; Majda and Kramer, 1999).

(i) White noise limit for the shear flow

Here, we consider a special limiting case of the master model, when the waves vk(t) from (2.13) decorrelate very fast and can be effectively considered as white noise. To define the white noise limit, we decompose the waves into two parts

vk(t) = ¯vk|U(t) + v0k|U(t), (3.16) where ¯vk|U(t) is a conditional mean of the shear flow for given realizations of the cross-sweep, i.e., the average over the noise of the waves given by ˙Wvk; on the other

hand, v_k|U0 (t) denotes fluctuations of the shear around the conditional mean. From (2.13) we find the dynamics of the conditional mean ¯vk|U(t)

d¯vk|U(t)

dt = (−γvk+ iωvk)¯vk|U(t) + fvk(t), (3.17)

and of the fluctuations around this mean dv_k|U0 (t)

dt = (−γvk+ iωvk)v

0

k|U(t) + σvkW˙vk(t). (3.18)

It is very important to emphasize that for a general master model, ¯vk|U(t) is a random variable because it explicitly depends on the Gaussian cross-sweep U (t) through the dispersion relation ωvk given in (2.17); and it is only in the special case

of uncorrelated velocity field discussed in the previous section that ¯vk|U(t) becomes deterministic and equal to the mean of the waves, ¯vk|U(t) = ¯vk(t). First, we define the white noise limit for the fluctuations v_k|U0 (t) from (3.18). To achieve this, we consider the limit γvk → +∞ which ensures vanishing decorrelation time of the

waves. In the statistically steady state with respect to the noise of the waves, ˙Wvk,

the autocorrelation function of v0

k(t) becomes Corrv0 k|U(τ ) = hv 0 k|U(t + τ )v0k|U(t)∗i = σ2 vk 2γvk e−γvkτ_eiJk(t,t+τ )_{. (3.19)}

Therefore, if we keep the following ratio fixed ηk =

σvk

γvk

= const, (3.20)

then the absolute value of the autocorrelation function formally approaches a delta-function |Corrv0 k|U(τ )| → η2 k 2 δ0(τ ), (3.21)

or, equivalently, the fluctuations v_k0(t) approach the white noise

v_k|U0 (t) → ηkW˙vk(t). (3.22)

Secondly, we proceed with the white noise limit for ¯vk|U(t) from (3.17). Suppose that the forcing fvk(t) grows as the dissipation γvk grows in the white noise limit

where ¯fk(t) is independent of γvk. Then, for the value of ¯vk|U(t) given by (2.28)

without the last term ¯

vk|U(t) = Gvk(t0, t)¯vk|U(t0) +

Z t

t0

Gvk(s, t)fvk(s)ds, (3.24)

we find, using (3.23) in (3.24), the white noise limit as γvk→ +∞

¯

vk|U(t) → ¯fvk(t). (3.25)

Note that in the white noise limit, ¯vk|U(t) becomes deterministic, which means that it is equal to its mean value. Therefore, formally we can achieve the white noise limit in the master model by first substituting

fk(t) → γvkf¯vk(t), (3.26)

and then using the limiting approach to the delta-function of the following terms

γvkGvk(s, t) → δt(s), (3.27)

σvkGvk(s, t) → ηkδt(s). (3.28)

Then, in this white noise limit the master model is described by the equations dU (t) dt = −γUU (t) + fU(t) + σU ˙ WU, dTk dt = (−γTk+ iωTk)Tk− α ¯fvk(t) + ηkW˙vk  ,

Note that this model is the same as the slow-fast test model introduced earlier by the authors (Gershgorin and Majda, 2008, 2010a,b) where U (t) is a“slow” independent variable and Tk(t) is a “fast” dependent variable. However, here we do not compare the time scales of both variables. This system has an exact statistical solution as well. The solution for the tracer in the white noise limit is given by

Tk(t) = GTk(t0, t)Tk,0− α Z t t0 GTk(s, t) ¯fvk(s)ds − αηk Z t t0 GTk(s, t)dWvk(s). (3.29)

The mean of the tracer becomes

hTk(t)i = hGTk(t0, t)Tk,0i − α

Z t

t0

hGTk(s, t)i ¯fvk(s)ds. (3.30)

In the statistically steady state, the mean simplifies to hTk(t)i = −α

Z t

−∞

hGTk(s, t)i ¯fvk(s)ds, (3.31)

where we took the limit t0→ −∞. Note that exactly the same expression is obtained by taking the white noise limit directly in Eq. (2.53) via Eqs. (3.27) and (3.28)

hTk(t)i = −α Z t −∞ Z s −∞ hGvk(r, s)GTk(s, t)i γvkf¯vk(r)drds → −α Z t −∞ Z s −∞ δs(r) hGTk(s, t)i ¯fvk(r)drds = −α Z t −∞ hGTk(s, t)i ¯fvk(s)ds (3.32)

Next, we find the covariance in the white noise limit in the statistically steady state Cov(Tj, Tk) = α2 Z t −∞ Z t −∞  hGTj(s, t)G ∗ Tk(s 0_{, t)i − hG} Tj(s, t)ihG ∗ Tk(s 0_{, t)i} ¯_f vj(s) ¯f ∗ vk(s 0_)dsds0 + δ j k 2γTk α2η2k. (3.33)

Similarly, we can find this expression by taking formally the white noise limit in Eq. (2.54) directly. First, we find

GTj(s, t)G ∗ Tk(s 0_{, t)v} j(s)v∗k(s 0_{) =} γvkγvj Z s −∞ Z s0 −∞ Gvj(r, s)G ∗ vk(r 0_{, s}0_)G Tj(s, t)G ∗ Tk(s 0_{, t) ¯} fvj(r) ¯f ∗ vk(r 0_)drdr0 +σ2vkδ j k Z min(s,s0) −∞ Gvk(r, s)G ∗ vk(r, s 0_)G Tk(s, t)G ∗ Tk(s 0_{, t) dr} → Z s −∞ Z s0 −∞ δs(r)δs0(r0)G_T j(s, t)G ∗ Tk(s 0_{, t) ¯} fvj(r) ¯f ∗ vk(r 0_)drdr0 +η2_kδj_khGTk(s, t)G ∗ Tk(s 0_{, t)i}Z min(s,s0₎ −∞ δs(r)δs0(r)dr =GTj(s, t)G ∗ Tk(s 0_{, t) ¯} fvj(s) ¯f ∗ vk(s 0_{) + η}2 kδ j kδs(s 0_)e−2γ_Tk(t−s)_{. (3.34)}

Now, the covariance becomes Cov(Tj, Tk) = α2 Z t −∞ Z t −∞  GTj(s, t)G ∗ Tk(s 0_{, t)v} j(s)v∗k(s0) − GTj(s, t)vj(s) hGTk(s 0_{, t)v} k(s0)i ∗ dsds0 → α2 Z t −∞ Z t −∞  GTj(s, t)G ∗ Tk(s 0_{, t) − hG} Tj(s, t)ihG ∗ Tk(s 0_{, t)i} ¯_f vj(s) ¯f ∗ vk(s 0_)dsds0 +α2η2kδ j k Z t −∞ Z t −∞ e−2γTk(t−s)_δs(s0_)dsds0 = α2 Z t −∞ Z t −∞  GTj(s, t)G ∗ Tk(s 0_{, t) − hG} Tj(s, t)ihG ∗ Tk(s 0_{, t)i} ¯_f vj(s) ¯f ∗ vk(s 0_)dsds0 + δ j k 2γTk α2η_k2 (3.35)

Next, we establish a connection between the white noise limit of the tracer and the triad model with seasonal cycle used for applications in climate change science (Gershgorin and Majda, 2010b). Recall that the triad model has the form

du1 dt = −γ1u1+ f1(t) + σ1 ˙ W1 du2 dt = (−γ2+ i(ω0+ a0u1))u2+ f2(t) + σ2 ˙ W2 (3.36)

whereas the white noise limit of the master model is given by dU0(t) dt = −γUU 0_{(t) + f}0 U(t) + σUW˙U, dTk dt = (−γTk− ik(U0+ U 0_(t)))T k− α¯vk(t) + αη ˙Wk.

where we redefined the variables such that the ensemble and time average of the jet is equal to U0 and the fluctuations around this grand average are denoted as U0(t). The forcing of the fluctuation of the jet is denoted as f_U0 (t). Here, the mean shear flow −α¯vk(t) plays the role of the forcing of the tracer through the mean gradient. Suppose that f_U0 (t) and all or some of ¯vk(t) have the same period that represents the seasonal cycle. Then, we can apply the time-periodic version of the fluctuation-dissipation theorem (FDT) to this system to study the climate sensi-tivity to external parameters. In particular, we can use the results of the earlier work on FDT for the triad system to study how the mean and the variance of both the jet and the tracer respond to the changes in the mean forcing and dissipation. Note that the exact statistical solution provides the ideal response of the system to the changes in external parameters. As we learned in the earlier study (Gershgorin and Majda, 2010b), the quasi-Gaussian approximation to FDT provides an effective algorithm for computing the corresponding response operators. This set up allows to address the following kinds of questions in a very simple model:

• How will the mean or variance of the tracer averaged over a certain season (or month) change in response to the changes in the mean of the flow (which acts like forcing here)?

• How will the mean or variance of the tracer averaged over a certain season (or month) change in response to the changes in the mean gradient (which acts like the amplitude of the forcing here)?

• How will the mean or variance of the tracer averaged over a certain season (or month) change in response to the changes in the molecular (or eddy) diffusion?

As established in (Gershgorin and Majda, 2010b), it is interesting to study the case of resonant forcing. Here, resonance happens when

kU0= ωfk, (3.37)

where ωfk is the frequency of ¯vk(t). As we learned from the triad test model, in the

case of the resonance, the pdf of the tracer becomes strongly non-Gaussian with two peaks. We have found then that in the resonant regime, the variance response to the changes in the external forcing varies in time and takes large values whereas in the Gaussian model this response would have been zero. Moreover, the quasi-Gaussian approximation proved to be very effective in recovering the ideal variance response to the changes in the external forcing. We note that the coupling parameter here is given by the wave number k. Therefore, it is expected that the Fourier modes with higher wave numbers are more non-Gaussian.

(ii) White noise limit for the cross-sweep

Now we proceed further and consider a white noise limit for the cross-sweep U (t) dU (t) dt = −γU (t) + fU(t) + σU ˙ WU (3.38) when γU → +∞, (3.39) σU → +∞, σU γU = ηU = const. fU(t) γU is independent of γU

As we have shown in the previous section, the solution of the OU process in this limit has the following form

U (t) → ¯U (t) + ηUW˙U(t), (3.40)

with ¯U (t) given through the normalized forcing fU(t)/γU. This leads us to the following Stratonovich SDE for the Fourier modes of the tracer

dTk(t)

dt = (−γTk− ik ¯U (t))Tk− α ¯fvk(t) − ikηUTk(t) ◦ ˙WU(t) − αηkW˙vk. (3.41)

This is a linear SDE with multiplicative and additive noise. The fact that we in-terpret the multiplicative noise in the Stratonovich form in the white noise limit is of crucial importance. The way we take the white noise limit assumes nonva-nishing correlation between the noise (U (t) before the limit) and the tracer, i.e., hU (t)T (t)i 6= 0 before the limit and hT (t) ◦ ˙WUi 6= 0 after the limit is taken. As usual in physics and engineering, the white noise limit of colored noise leads to the Stratonovich integral (Gardiner, 1997). We apply the white noise limit for U (t) in the formulas for the statistically steady state mean and covariance of the tracer given by Eqs. (3.31) and (3.33) after the white noise limit in the shear flow was applied. We need to find

hGTk(s, t)i = D e−γTk(t−s)−ikRstU (s 0_)ds0E → ¯GTk(s, t) D e−ik(WU(t)−WU(s)) E = ¯GTk(s, t)e −η2 2(t−s),(3.42)

where the deterministic part of the Green’s function for the tracer is given by ¯ GTk(s, t) = e −γ_Tk(t−s)−ikRt sU (s¯ 0_)ds0 (3.43) Now, the mean of the tracer becomes a white noise limit of (3.31)

hTk(t)i = −α Z t −∞ e− η2_U 2 k 2_(t−s) ¯ GTk(s, t) ¯fvk(s)ds. (3.44)

Here we note the correction to the diffusivity (the eddy diffusivity) η2U

2 that comes from the diffusion-induced advection term that appears after rewriting the SDE (3.41) in the Ito form (Gardiner, 1997)

dTk(t) dt = (−γTk− ik ¯U (t) − k2 2 η 2 U)Tk− α ¯fvk(t) − ikηUTk(t) ˙WU(t) − αηkW˙vk.(3.45)

Here, the eddy diffusivity is local in space and time and is equal to κe= η2 U 2 ∼ σ2 U 2γ2 U . (3.46)

Next, the covariance, is a white noise limit of (3.33)

Cov(Tj(t), Tk(t)) = α2 Z t −∞ Z t −∞ ¯ GTj(s, t) ¯G ∗ Tk(s 0_{, t) ¯f} vj(s) ¯f ∗ vk(s 0₎ ×e−η22U(j 2_(t−s)+k2_(t−s0₎₎ (eη2Ukj(t−max(s,s 0₎₎ − 1)dsds0 + δ j k 2γTk α2η_k2. (3.47)

Moreover, the white noise limit in U (t) of the second order statistics of Tk(t) can be found for a general case of the shear flow, not just in the case of the white noise limit of vk(t). The mean is given by the same equation (3.44). The covariance has the form Cov(Tj(t), Tk(t)) = α2 Z t −∞ Z t −∞ ¯ GTj(s, t) ¯G ∗ Tk(s 0_{, t)e}−η2U 2 (j 2_(t−s)+k2_(t−s0₎₎ ×hvj(s)vk∗(s0)ie η2 Ukj(t−max(s,s 0₎₎ − hvj(s)ihv∗k(s0)i  dsds0. (3.48)

Note that Eqs. (3.44), (3.48) can be used as the mean and covariance with model error in the form of an eddy diffusivity approximation to the original master model with uncorrelated velocity fields. A related approximation to (3.46), (3.48) has been utilized recently to illustrate the role of model error in quantifying uncertainty in climate change science (Majda and Gershgorin, 2010).

4. Eddy diffusivity

Here, we use the closed form expression for the mean statistics of the shear velocity v and passive tracer T developed in section 2 to study the actual form of eddy diffusivity in the master model both for v and for T . Even in this context, the eddy diffusivity is nonlocal in space and time. To motivate the issue of eddy diffusivity, we take the governing equation for the mean of the shear flow by averaging (2.13)

dhvk(t)i

dt = −γvkhvk(t)i + ihωvk(t)vk(t)i + fvk(t). (4.1)

Here, ωvk(t) is Gaussian, ωvk(t) = akU (t) + bk, according to its definition in (2.17)

We use the decomposition of both ωvk(t) and vk(t) into their respective

determin-istic means and random fluctuations around those means ωvk(t) = hωvk(t)i + ω 0 vk(t), vk(t) = hvk(t)i + vk0(t), and find dhvk(t)i

dt = (−γvkhvk(t)i + ihωvk(t)i)hvk(t)i + ihω

0 vk(t)v

0

k(t)i + fvk(t). (4.2)

The underlined term in (4.2) is exactly the eddy diffusivity for the shear flow that we discuss below using the closed form solutions developed in section 2.

In a very similar fashion, we obtain an eddy diffusivity form for the tracer. We average equation in (2.45) to find the dynamics of the mean of the tracer

dhTk(t)i

dt = −γTkhTk(t)i + ihωTk(t)Tk(t)i − αhvk(t)i. (4.3)

where ωTk is given by (2.47). We decompose ωTk(t) and Tk(t) into the deterministic

means and the random fluctuations around those means ωTk(t) = hωTk(t)i + ω 0 Tk(t), Tk(t) = hTk(t)i + Tk0(t), and find hTk(t)i

dt = (−γTkhTk(t)i + ihωTk(t)i)hTk(t)i + ihω

0 Tk(t)T

0

k(t)i− αhvk(t)i. (4.4) Here, the underlined term represents the eddy diffusivity for the tracer in the master model that is also discussed below.

(a) Eddy diffusivity for the horizontal shear flow in the master model We use the formula for the mean of the shear flow, Eq. (2.36), to find eddy diffusivity approximation of the shear flow

hvk(t)i = hGvk(t0, t)ihvk,0i +

Z t

t0

hGvk(s, t)i fvk(s)ds, (4.5)

where we disregarded the initial correlation between the cross-sweep and the shear flow for simplicity. The Green’s function for the shear flow is given by (2.29)

Gvk(s, t) = e −γ_vk(t−s)+iJk(s,t)_{, (4.6)} Jk(s, t) = Z t s ωvk(s 0_)ds0_{, (4.7)}

For the special case of atmospheric waves in the QG model, we have for example Jk(s, t) = k  F k2 − 1  Z t s U (s0)ds0+ βk k2_{+ F}. (4.8)

We find the time derivative of the mean of vk dhvk(t)i dt = (−γvk+ i hωvk(t)i)hvk(t)i + fvk(t) −k2_κv e(k, t0, t)hGvk(t0, t)ihvk,0i −k2Z t t0 κv_e(k, s, t)hGvk(s, t)ifvk(s)ds, (4.9)

where the eddy diffusivity in the general case becomes κv_e(k, s, t) = 1

2k2 ∂

∂tV ar(Jk(s, t)). (4.10)

Note that we factored out the k2 _{term which corresponds to the second derivative} in the physical space to compare the results with ordinary diffusion. It is convenient to introduce an eddy diffusivity functional here for the shear flow

Kv e[hvk,0i, fvk] = κ v e(k, t0, t)hGvk(t0, t)ihvk,0i + Z t t0 κve(k, s, t)hGvk(s, t)ifvk(s)ds,(4.11)

so that the differential equation for the eddy diffusivity of the shear flow becomes dhvk(t)i

dt = (−γvk+ i hωvk(t)i)hvk(t)i + fvk(t) − k

2

Kve[hvk,0i, fvk]. (4.12)

We compare this equation with the formula in (4.2) to find ihω0vk(t)v

0

k(t)i = −k2Kev[hvk,0i, fvk]. (4.13)

In physical space, the equation for the mean of the shear flow becomes dhv(x, t)i dt = P  ∂ ∂x, hU (t)i  hv(x, t)i + ∂ 2 ∂x2K˜ v e  ∂ ∂x, t0, t  , (4.14)

where ˜K is given by its Fourier symbol in (4.11). For the QG model, we find

κv_e(k, s, t) = 1 2  F k2 − 1 2 _∂ ∂tV ar Z t s U (s0)ds0  . (4.15)

The time derivative of the variance ofR_stU (s0)ds0 is given by ∂ ∂tV ar Z t s U (s0)ds0  = 2 γU  GU(t0, s) − GU(t0, t)  GU(s, t)V ar(U0) + σ 2 U γ2 U  1 − GU(s, t) − GU(2t0, s + t) + G2U(t0, t)2  , (4.16)

where we used (2.42) and GU is defined in (2.24). Next, we study the role of each component of the eddy diffusivity.

(i) Unforced waves

First, we consider the case of unforced waves, i.e., fvk(t) ≡ 0. Then, we have

hvk(t)i = hGvk(t0, t)i hvk,0i , (4.17)

and the differential equation for the mean becomes dhvk(t)i

dt = (−γvk+ i hωvk(t)i)hvk(t)i − k

2_κv

e(k, t0, t)hvk(t)i (4.18) with κv

e given explicitly by (4.10). For the QG model, the corresponding equation in physical space becomes

dhv(x, t)i dt = P  ∂ ∂x, hU (t)i  hv(x, t)i + ∂ 2 ∂x2κ˜ v e  ∂ ∂x, t0, t  hv(x, t)i , (4.19) where ˜κv

eis the pseudo-differential time dependent eddy diffusivity operator with its Fourier symbol defined in (4.15), (4.16). It is convenient to separate eddy diffusivity κv

e(k, s, t) into the product of non-local temporal and non-local spatial parts κv_e(k, s, t) = κv_e,tm(s, t) κv_e,sp(k). (4.20) Then, we find for the QG model

κv_e,tm(s, t) = 1 2 ∂ ∂tV ar Z t s U (s0)ds0  , (4.21) κv_e,sp(k) =  F k2 − 1 2 . (4.22)

In physical space, we find

˜ κv_e,sp ∂ ∂x  =  F ∂ −2 ∂x−2 − 1 2 . (4.23)

We note that for large spatial wave numbers with wavelengths within a Rossby radius, |k| > L−1_R , we have spatial localization at small scales, and the eddy diffusivity becomes a local operator

κv_e,sp(k) ≈ 1. (4.24)

At wavelengths larger than LRgenuine spatially nonlocal effects persist. Moreover, if the cross-sweep decorrelates on a short time scale, i.e., γU is large, we have temporal localization with the temporal part of the eddy diffusivity equal to

κv_e,tm(s, t) = σ 2 U 2γ2 U . (4.25)

Note that this is exactly the white-noise limit in the cross-sweep for the eddy diffusivity from section 3b). In general, this example represents an interesting test case with eddy diffusivity which is non-local in time and space.

(ii) Waves in the statistically steady state

Next, we turn to the case of the statistically steady state with nonzero forcing for the waves, fvk6= 0, which induces a non-zero statistically steady mean, ¯vk(t). Here

the initial conditions for hvki are irrelevant but inhomogeneous forcing dominates. Next, we show a nonlocal memory of the forcing. We take the mean of vk(t) from (2.28) and consider the limit t0→ −∞ to find the mean in the statistically steady state

hvk(t)i = Z t

−∞

hGvk(s, t)i fvk(s)ds. (4.26)

Now, the differential equation for the mean of vk becomes dhvk(t)i dt = (−γvk+ i hωvki)hvk(t)i + fvk(t) −k2_κv e,sp(k) Z t −∞ κv_e,tm(s, t)hGvk(s, t)ifvk(s)ds. (4.27)

The integral in the last expression is a functional of the forcing of the shear flow Hk[fvk](t) =

Z t

−∞

κve,tm(s, t)hGvk(s, t)ifvk(s)ds. (4.28)

It is important to emphasize, that Hk[fvk] is not equal to the mean wave hvki but

instead it is obtained using convolution of the forcing with the same Green’s function Gvk as for vk but also multiplied by the temporal part of the eddy diffusivity,

κv

e,tm(s, t), in (4.28), i.e., Hk[fvk](t) carries the history of the evolution of fvk(t)

and not just its value at a given moment. Then, the differential equation for the mean becomes

dhvk(t)i

dt = (−γvk+ i hωvki)hvk(t)i + fvk(t) − k

2_κv

e,sp(k)Hk[fvk](t). (4.29)

In physical space, this equation becomes dhv(x, t)i dt = P  ∂ ∂x, hU (t)i  hv(x, t)i + ∂ 2 ∂x2κ˜ v e,sp  ∂ ∂x  ˜ Hk(x, t). (4.30) Note that, in the statistically steady state, the temporal part of the eddy diffu-sivity becomes κv_e,tm= σ 2 U 2γ_U2  1 − e−γU(t−s)  (4.31) This part of eddy diffusivity introduces temporal memory through (4.28) into the equation in (4.30) and makes it non-local in time, and again the term κv_{e,sp ∂x}∂
makes the contribution of the effect of mean forcing non-local in space as well.

As we discussed above, in the general case, when we have both the initial con-dition and the forcing contributions, the derivative for the mean of the shear flow is given by (4.11). Here, the eddy diffusivity affects both parts of the solution, the one with initial condition, and the one with the forcing and we have just discussed these individual contributions.

(b) Eddy diffusivity for the tracer in the master model

Here, we obtain and analyze the exact expression for the eddy diffusivity for the tracer that was motivated in (4.4). We use the formula for the mean of the tracer given by Eq. (2.51) that we repeat here for convenience

hTk(t)i = hGTk(t0, t)ihTk,0i − α Z t t0 hGTk(s, t)Gvk(t0, s)ihvk,0ids −α Z t t0 Z s t0 hGTk(s, t)Gvk(r, s)ifvk(r)drds. (4.32)

We find the effective equation for this mean of the tracer by differentiating (4.32) dhTk(t)i

dt = (−γTk+ ihωTki)hTk(t)i − αhvk(t)i

−k2_κT e(k, t0, t0, t)hGTk(t0, t)ihTk,0i +k2α Z t t0 κT_e(k, t0, s, t)hGTk(s, t)Gvk(t0, s)ids  hvk,0i +k2α Z t t0 Z s t0 κT_e(k, r, s, t)hGTk(s, t)Gvk(r, s)ifvk(r)drds, (4.33) where κT_e(k, r, s, t) = 1 2 ∂ ∂tV ar(L(s, t)) −  F k2 − 1  ∂ ∂tCov(L(r, s), L(s, t)), (4.34) and L(s, t) is defined in (2.32). Note that here we assumed for simplicity that all the initial conditions are independent random variables. It is convenient to introduce a notation for the eddy diffusivity functional for the tracer

KT e[hvk,0i, hTk,0i, fvk] = κ T e(k, t0, t0, t)hGTk(t0, t)ihTk,0i −α Z t t0 κT_e(k, t0, s, t)hGTk(s, t)Gvk(t0, s)ids  hvk,0i −α Z t t0 Z s t0 κT_e(k, r, s, t)hGTk(s, t)Gvk(r, s)ifvk(r)drds, (4.35) highlighting the three separate contributions. Then, the equation for the mean of the tracer takes a simple form

dhTk(t)i

dt = (−γTk+ ihωTki)hTk(t)i − αhvk(t)i − k

2_KT

e[hvk,0i, hTk,0i, fvk]. (4.36)

By comparing this equation with the formula in (4.35) we find

ihω0_Tk(t)T_k0(t)i = K_eT[hvk,0i, hTk,0i, fvk], (4.37)

which is the closed form of the eddy diffusivity for each spatial wave number of the tracer. To understand the role of eddy diffusivity term by term, we consider different physical situations.

(i) Zero mean gradient

Suppose that the mean gradient is zero, α = 0, then the eddy diffusivity becomes local in space and still stays nonlocal in time. The equation for the mean of the tracer in physical space becomes

dhT (x, t)i dt + hU (t)i ∂hT (x, t)i ∂x = (κ + κ T e(t0, t)) ∂2_{hT (x, t)i} ∂x2 − dThT (x, t)i, (4.38) where κTe(t0, t) = 1 2 ∂ ∂tV ar Z t t0 U (s)ds  . (4.39)

With this simple special case, we demonstrate how the eddy diffusivity brings mem-ory into the mean of the tracer because the RHS of the equation in (4.38) depends on some earlier time t0. Note that in this case the dynamics is damped and in the statistically steady state regime, the mean tracer vanishes.

(ii) Unforced waves and zero initial condition for the tracer

Here we assume that only the initial conditions of the shear waves contribute to the mean tracer but the mean gradient is nonzero, α 6= 0. This situation is possible when the waves are unforced and the tracer has vanishing initial mean. Mathematically, this means that only the second term in the RHS of (4.32) is non-vanishing which leads to the following differential equation for the mean of the tracer

dhTk(t)i

dt = (−γTk+ ihωTki)hTk(t)i − αhvk(t)i

+k2α Z t t0 κTe(k, t0, s, t)hGTk(s, t)Gvk(t0, s)ids  hvk,0i, (4.40) where κT_e is given in (4.34). The last term here corresponds to the second derivative in physical space of the mean tracer weighted with an eddy diffusivity kernel. This eddy diffusivity is non-local in space and time.

(iii) Statistically steady state

Here we consider the limit of the eddy diffusivity in the statistically steady state regime so the initial conditions are irrelevant but the mean gradient is non-zero, α 6= 0. We use (4.32) to find the mean of the tracer in the statistically steady state by taking the limit t0→ −∞

hTk(t)i = −α Z t −∞ Z s −∞ hGTk(s, t)Gvk(r, s)ifvk(r)drds. (4.41)

Now, we find the differential equation for the mean of the tracer which is a special case of (4.36)

dhTk(t)i

dt = (−γTk+ ihωTki)hTk(t)i − αhvk(t)i − k

2

where the eddy diffusivity functional for the tracer has the following form KT e[fvk](t) = −α Z t −∞ Z s −∞ κT_e(k, r, s, t)hGTk(s, t)Gvk(r, s)ifvk(r)drds, (4.43)

and the eddy diffusivity kernel in the statistically steady state is given by κTe(k, r, s, t) = σ2 U 2γ2 U   1 − e−γU(t−s)₋F k2− 1  e−γU(t−s)_{− e}−γU(t−r)  .(4.44) Note that KT

e[fvk](t) is a nonlocal linear functional of the history of the mean shear

forcing, fvk(t). In physical space, we have the following equation

dhT (x, t)i dt + ¯U (t) ∂hT (x, t)i ∂x = κ ∂2_{hT (x, t)i} ∂x2 − dThT (x, t)i + ∂2_K˜T e[fv](x, t) ∂x2 − αhv(x, t)i.(4.45) The functional ˜KT

e[fv](x, t) has memory in both space and time so that surprisingly even the contribution from the mean forcing of the shear induces memory effects. However, for high wave numbers, this equation becomes local in space and still not local in time. Moreover, in the white noise limit for the cross-sweep given in (3.39), the eddy-diffusivity kernel in (4.44) becomes constant for all wave numbers k and all values of the parameters r, s, and t

κT_e = σ 2 U 2γ2 U . (4.46)

By comparing (4.43) with (4.41) we find that with the constant kernel κT e in the white noise limit of the cross-sweep, the spatial and temporal memories disappear from the eddy-diffusivity functional and it becomes local in space and time

KT e[fvk](t) = σ_U2 2γ2 U hTk(t)i . (4.47)

Substituting (4.47) into (4.43) and taking inverse Fourier transform in space, we find the following local in time and space equation for the mean of the tracer in physical space dhT (x, t)i dt + ¯U (t) ∂hT (x, t)i ∂x = (κ + κ T e) ∂2hT (x, t)i ∂x2 − dThT (x, t)i − αhv(x, t)i,(4.48) with a constant eddy diffusivity, κT

e, given by (4.46). Note that here, unlike in the case with the eddy diffusivity of the shear flow, the eddy diffusivity κT

e becomes local in space in the white noise limit of U (t) even for small wave numbers k. In the case of the eddy diffusivity of the shear flow, the eddy-diffusion becomes local in space only for high wave numbers regardless of the temporal scales of the velocity field due to the different role of the sweep of the jet, U (t), at large scales.

(iv) Eddy diffusivity for the tracer in the model with uncorrelated velocity field When the shear velocity field and the jet are uncorrelated, we obtain a differen-tial equation for the mean hTk(t)i from (4.33) by noting that the Green’s function

for the velocity field in (2.29) becomes deterministic and the eddy diffusivity func-tional from (4.35) becomes

KT e[hvk,0i, hTk,0i, fvk] = κ T e(t0, t)hGTk(t0, t)ihTk,0i − α Z t t0 κTe(s, t)hGTk(s, t)iGvk(t0, s)ds  hvk,0i − α Z t t0 κT_e(s, t)hGTk(s, t)i Z s t0 Gvk(r, s)fvk(r)dr  ds. (4.49) Thus the eddy diffusivity kernel is local in space and nonlocal in time

κTe(s, t) = 1 2

∂

∂tV ar(L(s, t)), (4.50)

where L(s, t) is given by (2.32). An extremely interesting question is how well can this non-local in time diffusion be approximated by a standard local in time diffusivity? To answer this question, we approximate the first term in the integrand by some constant

κT_e(s, t) ≈ keddy, (4.51)

then (4.49) becomes ∂hTk(t)i

∂t − ihωTkihTk(t)i ≈ −(κ + κeddy)k

2

hTk(t)i − γTkhTk(t)i − αhvk(t)i. (4.52)

Here, this constant, κeddy, exactly represents the eddy diffusivity that enhances the diffusivity of the system due to smaller scale nonlinear interactions. The model error due to eddy diffusivity approximation can be quantified by comparing the exact mean and its approximation given by the solution of a linear ODE (4.52). The important practical issue of model error due to eddy diffusivity approximation in the filtering context can be addressed unambiguously using this test case. This has been done using information theory to quantify such model errors in the context of climate change science by the authors (Majda and Gershgorin, 2010).

(v) Eddy diffusivity for the tracer in the white noise limit of the shear in the master model

Here, we study eddy diffusivity of the white noise limit of the master model. We differentiate the exact mean of the tracer given by Eq. (3.30)

dhTk(t)i

dt = (−γTk+ ihωTk(t)i)hTk(t)i − α ¯fvk(t) − k

2_κT e(t0, t)hGTk(t0, t)ihTk,0i +k2α Z t t0 κT_e(s, t)hGTk(s, t)i ¯fvk(s)ds,

where the initial condition for the tracer is assumed to be uncorrelated with the initial condition of the cross-sweep and the eddy diffusivity kernel becomes

κTe(s, t) = 1 2

∂

Note that here, the eddy diffusivity is local in space and non-local in time. Accord-ing to the argument presented earlier for the master model, this eddy diffusivity becomes almost local in time if the dissipation of the cross-sweep becomes strong, which “erases memory” in the eddy diffusivity kernel.

5. The variance spectrum of the tracer

A bulk statistical quantity of great interest in the turbulent fluctuations of a tracer T is the tracer variance spectrum in a statistically steady state (Kraichnan, 1968; Sreenivasan, 1996; Majda and Kramer, 1999; Smith et al., 2002). In sections 2 and 3, we wrote down explicit closed formulas for the variance spectrum of the tracer in a statistically steady state with a background mean tracer gradient for the master model with both correlated and uncorrelated mean jet and shear waves. The question we address here is the following one:

Given an energy spectrum for the shear waves, Ek = V ar(vk),

what is the corresponding variance spectrum for the tracer, (5.1) V ar(Tk), in a statistically steady state with a mean tracer gradient? While in principle this only requires processing through asymptotics and/or nu-merics of multi-dimensional quadrature formulas as developed in section 2 in the present models, this is a very cumbersome but interesting procedure which we leave for the future. Instead we answer the question in (5.1) completely in a straightfor-ward fashion in the white noise limit for the shear waves developed in section 3b) (Kraichnan, 1968; Majda, 1993a; Majda and Kramer, 1999) and then check these spectral predictions in a family of instructive numerical experiments.

(a) Spectrum of the tracer in the white noise limit of the shear waves in the master equation

Recall from section 3b) that in the white noise limit of the shear waves in the master model, the spectrum of the tracer with unforced waves is given by the equation in (3.33) with ¯fvk= 0,

V ar(Tk) = α2η_k2 2γTk

. (5.2)

Also recall that ηk is a fixed ratio of σvk and γvk as they both go to infinity in the

white noise limit, ηk = σ_vk

γ_vk. Now formally in the white noise limit starting with the initial steady state velocity spectrum in (5.2), as shown in section 3b), we have η_k2=2Ek

γ_vk and the limiting velocity field converges to

vk(t) = ηkW˙k(t), (5.3)

yielding the steeper white noise limiting velocity spectrum

V ar(vk) = η2_k 2 = 2Ek γvk . (5.4)

Thus with (5.2) and (5.4) in the white noise limit, the tracer variance spectrum is given by the exact formula

V ar(Tk) = α2(dT+ κk2)−1V ar(vk). (5.5) In the present models with a mean gradient , the scalar spectrum is always steeper in the white noise limit. For the regime of wave numbers with κk2_{1, the scalar} dissipation regime, the limiting tracer spectrum is steeper than the limiting velocity spectrum by (κk2)−1. On the other hand, for a large inertial range of wave numbers with small molecular diffusivity, so that κk2  1, for a substantial range of wave numbers, the tracer variance spectrum in the white noise limit is proportional to the velocity spectrum with constant, _dα2

T.

(b) Numerical examples of tracer variance spectrum with an inertial range and the white noise limit

Here we report on a series of numerical experiments with the model with uncor-related velocity shear and mean jet, U (t). For the random velocity shear flow, we use the dynamics of uncorrelated β-plane Rossby waves with constants suitable for the atmosphere as discussed below (2.15) in section 2; this model amounts to setting U (t) ≡ 0 in the formula for P ∂

∂x, U (t)
in (2.15) and this model has been utilized elsewhere by the authors recenty in turbulent regimes for both climate change sci-ence (Majda and Gershgorin, 2010) and for a filtering test model (Gershgorin and Majda, 2010c). The tracer variance statistics in the statistical steady state are com-puted through long-time averaging of an individual trajectory in standard fashion since the system is ergodic and mixing.

To mimic the white noise limit of the shear velocity field discussed in section 3b), we first perform an initial experiment with a prescribed energy spectrum for the shear waves, here chosen to be the Kolmogorov spectrum, V ar(vk) = Ek= |k|−5/3, |k| ≥ 1. With this initial choice of the variance parameter σvk and the damping

parameter γvk, we perform a series of experiments where we integrate the tracer

variance statistics to a statistically steady state replacing σvk by rσvk,

γvk by rγvk,

as r → ∞ (5.6)

in a fashion consistent with the white noise limit discussed above in section 5a) since ηk =

σ_vk

γ_vk is held constant with γvk → +∞. All numerical experiments are

calculated with a large inertial range for the tracer so that there are 1000 spatial wave numbers, 1 ≤ |k| ≤ 1000 in the tracer dynamics with small tracer diffusivity κ = 10−8, uniform damping dT = 0.1 and mean background gradient α = 10. All experiments use the OU equations for the mean jet from (2.11) with parameters γU = 0.04, σU = 0.4, fU = 0.09 so that the mean jet ¯U = γU−1fU = 2.25 with jet variance σU2

2γU = 2; thus the physical requirement in (1.9) is satisfied. The β-plane

Rossby dispersion relation ωvk=

βk

k2_+F is utilized with β = 8.91 and F = 16 while

the values γvk = dv+ νk

2_{are used for the Rossby wave dissipation with the inertial} range parameters dv= 0.032 for Ekmann friction and ν = 10−8for viscosity; these

values together with imposing Ek = |k|−5/3 for the velocity spectrum in the initial simulation determine σvk via σvk=

√ 2Ek

√

dv+ νk2.

In Fig. 1, we show the tracer spectrum that emerged from four simulations with the above parameters with r = 1, r = 50, r = 103_{, r = 10}4_{, respectively.} As a general trend, in accordance with the white noise limit, the tracer variance spectrum systematically increases as r increases for each fixed spatial wave number. The tracer variance spectra show a roughly k−3 spectrum for the first 100 wave numbers for r = 1, 50 and a steeper slope for higher wave numbers. The spectral plot for r = 103 _{shows a definite roll-over of the spectrum for the large scale wave} numbers 1 ≤ |k| ≤ 10 to the less steep power law k−5/3 predicted by the white noise limit in (5.5); as expected from the white noise limit, this roll-over behavior is more pronounced at large wave numbers for r = 104. As evident from the large scatter at small wave numbers, it takes a very long time for the tracer statistics to equilibrate at very high wave numbers.

6. Strongly intermittent PDFs with fat exponential tails

As discussed in the introdution in the paragraph surrounding (1.9), the uncorre-lated mean flow and shear wave models have at least two very different regimes with highly intermittent PDFs for the tracer T with a mean gradient.

Regime A) The first regime discovered (Bourlioux and Majda, 2002) involves deterministic time periodic mean flow, ¯U (t) = AUsin(ωUt), with no random fluc-tuations, U0(t) ≡ 0, and deterministic or random waves without dispersion in the shear statistics. The time periodic PDFs of the tracer T in a statistically steady state are Gaussian for every fixed x, t but the time periodic averaged PDFs admit transitions from Gaussian to highly intermittent non-Gaussian PDFs with fat tails as the Peclet number increases due to intermittent unblocked streamlines at the zeroes of ¯U (t) with enhanced transport.

Regime B) The regime discovered recently (Majda and Gershgorin, 2010; Ger-shgorin and Majda, 2010c) with highly intermittent exponential tails in the tracer with mean gradient in the uncorrelated velocity model; this velocity model has a random mean jet U (t) with uncorrelated dispersive Rossby waves for the ran-dom shear model with atmospheric parameters for the Rossby waves with the jet U (t) = ¯U (t) + U0(t) constrained to satisfy physical requirements: the mean jet ¯U is non-negative, ¯U (t) > 0, and the standard deviation of the jet fluctuations also yield a positive jet, i.e., ¯U2_{(t) − V ar(U}0_{(t)) > 0. Here the PDFs for the tracer in} the statistical steady state are intermittent for each (x,t) in contrast to Regime A) The results in this regime in the simplified model mimic actual observations of the tracers in the atmosphere with a mean gradient with highly intermittent exponential tails in the PDFs (Neelin et al., 2010).

The goal here is to uncover the source of intermittency in the tracer PDF in the regime in the interesting recent scenario in B) and to contrast these results with the seemingly unrelated intermittency Regime A).

with dispersive Rossby waves for the shear together with simple numerical experi-ments to demonstrate these results.

(a) Stronger tracer PDF intermittency with increasing mean jet fluctuations in the simplified atmospheric model

First, we consider numerical simulations of the statistical steady state for the passive tracer with a fixed mean gradient α = 2, with dT = 0.1, and κ = 0.001; as in section 5 the β-plane Rossby dispersion relation ωvk =

βk

k2_+F is utilized for the

random shear waves with the atmospheric values β = 8.91 and F = 16 with the dissipation values, γvk = dv+ νk

2_{, with d}

v = 0.6 and ν = 0.1 and the turbulent energy spectrum V ar(vk) =    1 2, for |k| ≤ 5 1 2  k 5 −θv , for |k| > 5. (6.1)

where θv = −5/3 was used in the simulation. We fix the forcing fU(t) = 2 and the dissipation γU = 0.1 in the OU process from (2.11) for the jet so that the statistically steady mean jet becomes ¯U = 20 > 0. We systematically increase the variance of the random forcing driving the fluctuations of the jet σU from 1 to 8. Note that even for the largest value of σU = 8, we have ¯U2− V ar(U0) = 60 > 0 so the physical requirements for Regime B) are satisfied.

The PDFs for the tracer in the physical space as well as the PDFs for the large scale Fourier modes of the tracer are given in Figs. 2, 3, 4, 5 for the four respective values σU = 1, 2, 4, and 8. The PDFs for the tracer are clearly Gaussian for σU = 0 and are essentially Gaussian for σU = 1; weakly intermittent non-Gaussian tails emerge for σU = 2 while stronger fat exponential tails with sub-Gaussian inner core occur for σU = 4 and these effects are even stronger for σU = 8. Thus, increasing the mean jet fluctuations through σU serves as the transition parameter to highly intermittent scalar PDFs while satisfying the physical constraints.

In Fig. 1 from section 5, we showed the transitions in the tracer variance spec-trum for large inertial range simulations in the white noise limit with mean jet parameters satisfying the physical requirements for Regime B) but with varying r = 1, 50, 103_{, 10}4_{. In Fig. 6, we show the corresponding scalar PDFs. As expected,} the case with r = 1 is highly intermittent while the tracer PDF for r = 50 is less intermittent; increasing r substantially to r = 103, 104 to mimic the white noise limit makes the tracer PDF essentially Gaussian. The PDFs of the tracer in the white-noise limit are expected to be Gaussian and these simulations confirm this trend.

(b) The role of zeros in the cross-sweep for PDF intermittency

Here, we study the scenario of intermittency described in Regime A) (Bourlioux and Majda, 2002). In this regime, the cross-sweep is purely deterministic and time periodic. It was shown that if the cross-sweep has zeros, then the transport of the tracer increases significantly at the moment of zero cross-sweep and this process leads to intermittency with fat exponential tails for the time averaged pdfs. Note that here, at any fixed time the tracer is Gaussian, however, the variance of the