## Abstract

This paper motivates, develops and reviews elementary models for turbulent tracers with a background mean gradient which, despite their simplicity, have complex statistical features mimicking crucial aspects of laboratory experiments and atmospheric observations. These statistical features include exact formulas for tracer eddy diffusivity which is non-local 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 satisfying 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.

## 1. Introduction

One of the important paradigm models for the behaviour of turbulent systems [1,2] involves a passive tracer *T*(** x**,

*t*) which is advected by a velocity field

**(**

*v***,**

*x**t*) with dynamics given by 1.1 where

*κ*>0 is molecular diffusion and the velocity field

**is incompressible, div**

*v***=0. For simplicity of exposition, we assume here that**

*v***=(**

*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 environmental science to the behaviour of anthropogenic and natural tracers in climate change science [3–5], to detailed mixing properties in engineering problems such as non-premixed turbulent combustion [6–8]. 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 [2,9–17]. Avellaneda & Majda emphasized exactly solvable 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 [1,2,13,18,19] 1.2 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 using the simplified models in (1.1) and (1.2) as the simplest prototype models which nevertheless capture qualitatively correct complex physical features that arise in laboratory experiments [20–23], climate change science [5,24] 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 [25,26]. We review and expand upon recent work with the simplest mathematical models [24,26–29], which capture the observed phenomena such as
1.3
Another important issue is the ability to recover these statistical features from sparsely observed partial noisy observations and simplified model problems provide unambiguous test problems for these complex features [25,26,29,31,32]. An important effect responsible for these new phenomena is the existence of a background mean gradient for the tracer
1.4
so that with (1.1) and (1.2), *T*′ satisfies
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
1.6
where is a deterministic mean sweep and *U*′(*t*) represents random fluctuations of the mean. The turbulent velocity field *v*(*x*,*t*) satisfies the stochastic PDE (readily solved by Fourier series, see §2)
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 *F*_{v} is a forcing with both deterministic and random components. Assuming that the tracer fluctuations *T*′(*x*,*t*) depend only on the *x* variable alone and dropping the prime results in the simplified version of (1.5) given by
1.8
The term with *d*_{T}>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 [2,12,18].

There are remarkably different regimes in the simplified models in (1.6)–(1.8). For example, Bourlioux & Majda [27] considered the simplest model for (1.5)–(1.7) where , with an explicit time periodic function with isolated zeros while *v*(*x*,*t*) is a deterministic or random spatially periodic function without dispersive properties; they identified a transparent intermittency mechanism where stream lines of the velocity field are blocked for *U*(*t*)≠0 with modest turbulent diffusion and unblocked with enhanced turbulent transport in the vicinity of the zeros of *U*(*t*); this results in intermittency in the time-averaged PDFs with the features in (1.3 (A)) 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 behaviour so that
1.9
with Var(*U*′(*t*)) the variance of *U*′(*t*) in (1.6) so that the zonal jet almost always stays positive. These principal requirements in (1.9) for the models in (1.6)–(1.8) still allow for highly intermittent non-Gaussian PDFs in the tracer model [26] as observed in (1.3 (C)) [5] in a regime very different from that of Bourlioux & Majda [27]. 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 §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)–(1.8). We show how to develop closed exact formulas for the mean and variance of the master model in §2; in §3, we develop interesting simplifications involving uncorrelated velocity statistics [26] and various white noise limit models and establish connections with other models which have been developed earlier [29,31,32] by the authors in different contexts. In §4, we interpret exact equations for the mean statistics as non-local eddy diffusivity models for the passive tracer; surprisingly the master model with correlated velocity fields has both non-local space–time eddy diffusivity and a non-local 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 §5 while §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 numerical experiments are used throughout this paper. Section 7 is a brief concluding discussion. Comments regarding the use of such models for climate change science [24,29] and real-time filtering or data assimilation [26,29,31] 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)–(1.8) using special exact solutions of the *β*-plane quasi-geostrophic (QG) 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 QG flow [33,34] involve a stream function
2.1
with velocity
2.2
and potential vorticity
2.3
linked by the conservation of potential vorticity
2.4
The factor is the inverse square of the Rossby radius, *L*_{R}, and *β* is the differential effect of planetary rotation at a given latitude. Consider special exact solutions of (2.1)–(2.3) consisting of a zonal jet *U* and one-dimensional Rossby waves with the form compatible with (1.2) given by
2.5
The parameter *D*(*Δ*)*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*),
2.6
and for *q*=*ψ*_{xx}−*Fψ*
2.7
Physically, the special exact solutions in (2.6) and (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 2*π*-periodic which is used as the unit length. The velocity *v* is recovered from the two identities
2.8
through the non-local pseudo-differential operator
2.9
The symbol of *R*(∂/∂*x*) at a given spatial wavenumber is *R*=−i*k*/(*k*^{2}+*F*). To get the equation for *v*, we apply *R*(∂/∂*x*) to (2.7) and use (2.9) to obtain the dynamics
2.10
Special choices of the forcing result in (2.10) so that *f*_{v}(*x*,*t*) is deterministic forcing and denotes spatially correlated white noise forcing [35], which is readily represented below by Fourier series [25]. The natural dissipative mechanisms for *v* are a combination of Ekmann damping −*d*_{v}*v* and a small-scale frictional viscosity *ν*(∂^{2}*v*/∂*x*^{2}) [34]. 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) and (2.10) as well as the general master model, we assume that the forcing in (2.6) has the special form of deterministic forcing *f*_{U}(*t*), damping −*γ*_{U} and white noise forcing which results in the dynamics
2.11
The evolution of the shear flow is given in general by the following pseudo-differential equation:
2.12
where *P* is a pseudo-differential operator that combines both wave-like and dissipative components of the dynamics that can also depend on the cross-sweep and *F*_{v} is a forcing term that has both deterministic and random components. We specify the pseudo-differential operator *P* by its symbol in Fourier space *P*_{k}=−*γ*_{vk}+i*ω*_{vk} and rewrite (2.12) through Fourier series
2.13
where *γ*_{vk} is dissipation and *ω*_{vk} is dispersion relation. In general, both of these functionals can depend on the cross-sweep, *U*(*t*), and here we assume linear dependence on *U*(*t*) as in the above example so that *γ*_{vk} does not depend on *U*(*t*)
2.14
for real coefficients *a*_{k}, *b*_{k}. A special case of the transverse shear equation in the master model has already been motivated in (2.10) where
2.15
Here, the damping depends on spatial Fourier wavenumber and the dispersion relation depends on both spatial Fourier wavenumber 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 non-local in time and space.

The choice of a particular form of the dissipation, *γ*_{vk}, and the dispersion, *ω*_{vk}, depends on the situation for which the model is applied. We consider the following three situations.

Non-dispersive waves with selective damping:

*γ*_{vk}=*d*_{v}+*νk*^{2},*ω*_{vk}=−*ck*, where*ν*is the flow viscosity [26].Uncorrelated Rossby waves:

*γ*_{vk}=*ν*(*k*^{2}+*F*),*ω*_{vk}=*βk*/(*k*^{2}+*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, and*L*_{R}is the Rossby deformation radius,*β*is the tangent approximation to the local Coriolis forcing. This version of the master model was introduced and used recently as a test model in quantifying uncertainty in climate change science [24] together with the tracer equation in (1.8).Correlated Rossby waves (a generalization of case 2 developed in (2.10)): 2.16 2.17 2.18 2.19

### (c) Statistics of the velocity field

The solution for the cross-sweep becomes
2.20
where
2.21
2.22
2.23
and we use the shortcut notation for the initial condition, *U*_{0}=*U*(*t*_{0}), and for Green's function of the cross-sweep
2.24
Then, the statistics of the Gaussian cross-sweep become
2.25
and
2.26

We use Fourier series to compute explicit solutions of the master model [25,26], and then average them using identities for Gaussian random fields [24,29,31,32]. 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*:
2.27
The solution for each Fourier mode of the shear flow in the general case with time-dependent dispersion, *ω*_{vk}, has the form
2.28
where *v*_{k,0}=*v*_{k}(*t*_{0}) and Green's function for the shear flow is defined as
2.29
and
2.30
For the case of correlated flow, we compute further
2.31
2.32
2.33
2.34
2.35
Next, we find the mean of *v*_{k}
2.36
Note that Green's function of the shear flow, *G*_{vk}(*s*,*t*), defined in (2.29) has a form of an exponential of a Gaussian random variable, i*J*_{k}(*s*,*t*), from (2.30), with a deterministic factor, . We use the properties mentioned above in (2.27) of the characteristic function of a Gaussian random field [26,29,31,32] to find
2.37
2.38
2.39
2.40
2.41
2.42
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
2.44
In Fourier space, this equation becomes
2.45
where
2.46
and
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 equation (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
2.48
where *T*_{k,0}=*T*_{k}(*t*_{0}) and Green's function for the tracer is given by
2.49
and
2.50
Note that the shear flow *v*_{k}(*s*) is governed by the dynamics discussed above in (2.13). The mean of the tracer is given by
2.51
where 〈*G*_{Tk}(*s*,*t*)*G*_{vk}(*r*,*s*)〉 is a characteristic function of a Gaussian and can be computed analytically in a similar fashion to 〈*G*_{vk}〉 from (2.38)
2.52
In the statistically steady state, the mean of the tracer becomes
2.53
And the covariance is given by
2.54
where
2.55

## 3. Special regimes of the master model

While we have presented closed exact formulas for the first- and second-order statistics of the master model, it is difficult to process these analytical formulas in general (however, see §4 for the mean statistics and eddy diffusivity). Here, we develop instructive regimes of the master model which lead to both analytical 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 Ornstein–Uhlenbeck (OU) processes with varying frequency *ω*_{vk}. In physical space, the shear flow is given by
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 & Gershgorin [24] 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 versus 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 velocity field was used is in real-time data assimilation [26]. There, we used the exactly solvable structure of the model to construct a nonlinear extended Kalman filter [31,32,36–38], and then discussed the role of sparse and partial 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 owing 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 parametrization extended Kalman filter [36–38] 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: 3.2 3.3 3.4 where for the example of uncorrelated Rossby waves 3.5 3.6 3.7 3.8 Note that other forms of the dispersion relation for the waves can be studied since the formulas are general [26].

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 technology 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 *a*_{k}≡0 in (2.17). We use the latter way and find
3.9
where
3.10
Note that in this case of uncorrelated velocity field, 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 *a*_{k}=0 in (2.53)
3.11
where
3.12
The covariance in the statistically steady state can be obtained by setting *a*_{k}=0 in (2.54)
3.13
where
3.14
The double integral in the last expression is easy to compute analytically for special forms of *f*_{vk}.

The equation for the spectrum of the tracer in the case of time independent forcing for *U*(*t*) and no forcing for *v*_{k} becomes
3.15
where for *s*>*r*
From equation (3.15), it is obvious that the variance of *T*_{k} is proportional to the variance of *v*_{k} 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 wavenumbers and large wavenumbers. 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 *v*_{k} is constant in time [29].

### (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 [2,9,13,15].

#### (i) White noise limit for the shear flow

Here, we consider a special limiting case of the master model, when the waves *v*_{k}(*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
3.16
where 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 ; on the other hand, *v*′_{k|U}(*t*) denotes fluctuations of the shear around the conditional mean. From (2.13), we find the dynamics of the conditional mean
3.17
and of the fluctuations around this mean
3.18
It is very important to emphasize that for a general master model, is a random variable because it explicitly depends on the Gaussian cross-sweep *U*(*t*) through the dispersion relation *ω*_{vk} given in (2.17); and it is only in the special case of uncorrelated velocity field, discussed in the previous section, that becomes deterministic and equal to the mean of the waves, . First, we define the white noise limit for the fluctuations *v*′_{k|U}(*t*) from (3.18). To achieve this, we consider the limit which ensures vanishing decorrelation time of the waves. In the statistically steady state with respect to the noise of the waves, , the autocorrelation function of *v*′_{k}(*t*) becomes
3.19
Therefore, if we keep the following ratio fixed:
3.20
then the absolute value of the autocorrelation function formally approaches a delta-function
3.21
or, equivalently, the fluctuations *v*′_{k}(*t*) approach the white noise
3.22
Second, we proceed with the white noise limit for from (3.17). Suppose that the forcing *f*_{vk}(*t*) grows as the dissipation *γ*_{vk} grows in the white noise limit
3.23
where is independent of *γ*_{vk}. Then, for the value of given by (2.28) without the last term
3.24
we find, using (3.23) in (3.24), the white noise limit as
3.25
Note that in the white noise limit, 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
3.26
and then using the limiting approach to the delta-function of the following terms:
3.27
and
3.28
Then, in this white noise limit the master model is described by the equations
and
Note that this model is the same as the slow–fast test model introduced earlier by the authors [29,31,32] where *U*(*t*) is a ‘slow’ independent variable and *T*_{k}(*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
3.29
The mean of the tracer becomes
3.30
In the statistically steady state, the mean simplifies to
3.31
where we took the limit . Note that exactly the same expression is obtained by taking the white noise limit directly in equation (2.53) via equations (3.27) and (3.28)
3.32
Next, we find the covariance in the white noise limit in the statistically steady state
3.33
Similarly, we can find this expression by taking formally the white noise limit in equation (2.54) directly. First, we find
3.34
Now, the covariance becomes
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 [29]. Recall that the triad model has the form
3.36
whereas the white noise limit of the master model is given by
and
where we redefined the variables such that the ensemble and time average of the jet is equal to *U*_{0} and the fluctuations around this grand average are denoted as *U*′(*t*). The forcing of the fluctuation of the jet is denoted as *f*′_{U}(*t*). Here, the mean shear flow plays the role of the forcing of the tracer through the mean gradient. Suppose that *f*′_{U}(*t*) and all or some of 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 sensitivity 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 [29], the quasi-Gaussian approximation to FDT provides an effective algorithm for computing the corresponding response operators. This set up allows one 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 the study of Gershgorin & Majda [29], it is interesting to study the case of resonant forcing. Here, resonance happens when
3.37
where *ω*_{fk} is the frequency of . 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 then found 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 wavenumber *k*. Therefore, it is expected that the Fourier modes with higher wavenumbers 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*)
3.38
when
3.39
As we have shown in the previous section, the solution of the OU process in this limit has the following form:
3.40
with given through the normalized forcing *f*_{U}(*t*)/*γ*_{U}. This leads us to the following Stratonovich stochastic differential equation (SDE) for the Fourier modes of the tracer:
3.41
This is a linear SDE with multiplicative and additive noise. The fact that we interpret 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 non-vanishing correlation between the noise (*U*(*t*) before the limit) and the tracer, i.e. 〈*U*(*t*)*T*(*t*)〉≠0 before the limit and after the limit is taken. As usual in physics and engineering, the white noise limit of coloured noise leads to the Stratonovich integral [35]. 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 equations (3.31) and (3.33) after the white noise limit in the shear flow was applied. We need to find
3.42
where the deterministic part of Green's function for the tracer is given by
3.43
Now, the mean of the tracer becomes a white noise limit of (3.31)
3.44
Here we note the correction to the diffusivity (the eddy diffusivity) that comes from the diffusion-induced advection term that appears after rewriting the SDE (3.41) in the Ito form [35]
3.45
Here, the eddy diffusivity is local in space and time and is equal to
3.46
Next, the covariance is a white noise limit of (3.33)
3.47
Moreover, the white noise limit in *U*(*t*) of the second-order statistics of *T*_{k}(*t*) can be found for a general case of the shear flow, not just in the case of the white noise limit of *v*_{k}(*t*). The mean is given by the same equation (3.44). The covariance has the form
3.48
Note that equations (3.44) and (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) and (3.48) has been used recently to illustrate the role of model error in quantifying uncertainty in climate change science [24].

## 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 §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 non-local 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)
4.1
Here, *ω*_{vk}(*t*) is Gaussian, *ω*_{vk}(*t*)=*a*_{k}*U*(*t*)+*b*_{k}, according to its definition in (2.17), and we encounter a moment closure problem manifested in the term 〈*ω*_{vk}(*t*)*v*_{k}(*t*)〉. We use the decomposition of both *ω*_{vk}(*t*) and *v*_{k}(*t*) into their respective deterministic means and random fluctuations around those means
and
and find
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 §2.

In a very similar fashion, we obtain an eddy diffusivity form for the tracer. We average in equation (2.45) to find the dynamics of the mean of the tracer
4.3
where *ω*_{Tk} is given by (2.47). We decompose *ω*_{Tk}(*t*) and *T*_{k}(*t*) into the deterministic means and the random fluctuations around those means
and
and find
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, equation (2.36), to find eddy diffusivity approximation of the shear flow
4.5
where we disregarded the initial correlation between the cross-sweep and the shear flow for simplicity. Green's function for the shear flow is given by (2.29)
4.6
and
4.7
For the special case of atmospheric waves in the QG model, we have for example
4.8
We find the time derivative of the mean of *v*_{k}
4.9
where the eddy diffusivity in the general case becomes
4.10
Note that we factored out the *k*^{2} term which corresponds to the second derivative in physical space to compare the results with ordinary diffusion. It is convenient to introduce an eddy diffusivity functional here for the shear flow
4.11
so that the differential equation for the eddy diffusivity of the shear flow becomes
4.12
We compare this equation with the formula in (4.2) to find
4.13
In physical space, the equation for the mean of the shear flow becomes
4.14
where is given by its Fourier symbol in (4.11). For the QG model, we find
4.15
The time derivative of the variance of is given by
4.16
where we used (2.42) and *G*_{U} 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. *f*_{vk}(*t*)≡0. Then, we have
4.17
and the differential equation for the mean becomes
4.18
with *κ*^{v}_{e} given explicitly by (4.10). For the QG model, the corresponding equation in physical space becomes
4.19
where is the pseudo-differential time-dependent eddy diffusivity operator with its Fourier symbol defined in (4.15) and (4.16). It is convenient to separate eddy diffusivity *κ*^{v}_{e}(*k*,*s*,*t*) into the product of non-local temporal and non-local spatial parts
4.20
Then, we find for the QG model
4.21
and
4.22
In physical space, we find
4.23
We note that for large spatial wavenumbers 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
4.24
At wavelengths larger than *L*_{R}, genuine spatially non-local effects persist. Moreover, if the cross-sweep decorrelates on a short time scale, i.e. *γ*_{U} is large, we have *temporal localization* with the temporal part of the eddy diffusivity equal to
4.25
Note that this is exactly the white noise limit in the cross-sweep for the eddy diffusivity from §3*b*. 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 non-zero forcing for the waves, *f*_{vk}≠0, which induces a non-zero statistically steady mean, . Here the initial conditions for 〈*v*_{k}〉 are irrelevant but inhomogeneous forcing dominates. Next, we show a non-local memory of the forcing. We take the mean of *v*_{k}(*t*) from (2.28) and consider the limit to find the mean in the statistically steady state
4.26
Now, the differential equation for the mean of *v*_{k} becomes
4.27
The integral in the last expression is a functional of the forcing of the shear flow
4.28
It is important to emphasize that is not equal to the mean wave 〈*v*_{k}〉, but instead it is obtained using convolution of the forcing with the same Green's function *G*_{vk} as for *v*_{k} but also multiplied by the temporal part of the eddy diffusivity, *κ*^{v}_{e,tm}(*s*,*t*), in (4.28), i.e. carries the history of the evolution of *f*_{vk}(*t*) and not just its value at a given moment. Then, the differential equation for the mean becomes
4.29
In physical space, this equation becomes
4.30

Note that, in the statistically steady state, the temporal part of the eddy diffusivity becomes
4.31
This part of the 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 *κ*^{v}_{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 condition 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 analyse 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 equation (2.51) that we repeat here for convenience
4.32
Note that we have already computed earlier in (2.52) an explicit formula for the average of the product of Green's function in (4.32). We find the effective equation for this mean of the tracer by differentiating (4.32)
4.33
where
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
4.35
highlighting the three separate contributions. Then, the equation for the mean of the tracer takes a simple form
4.36
By comparing this equation with the formula in (4.35), we find
4.37
which is the closed form of the eddy diffusivity for each spatial wavenumber 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 non-local in time. The equation for the mean of the tracer in physical space becomes
4.38
where
4.39
With this simple special case, we demonstrate how the eddy diffusivity brings memory into the mean of the tracer because the right-hand side of equation (4.38) depends on some earlier time *t*_{0}. 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 non-zero, *α*≠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 right-hand side of (4.32) is non-vanishing which leads to the following differential equation for the mean of the tracer:
4.40
where 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*≠0. We use (4.32) to find the mean of the tracer in the statistically steady state by taking the limit
4.41
Now, we find the differential equation for the mean of the tracer which is a special case of (4.36)
4.42
where the eddy diffusivity functional for the tracer has the following form:
4.43
and the eddy diffusivity kernel in the statistically steady state is given by
4.44
Note that is a non-local linear functional of the history of the mean shear forcing, *f*_{vk}(*t*). In physical space, we have the following equation:
4.45
The functional 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 wavenumbers, 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 wavenumbers *k* and all values of the parameters *r*, *s* and *t*
4.46
By comparing (4.43) with (4.41), we find that with the constant kernel 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
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:
4.48
with a constant eddy diffusivity, , given by (4.46). Note that here, unlike in the case with the eddy diffusivity of the shear flow, the eddy diffusivity *κ*^{T}_{e} becomes local in space in the white noise limit of *U*(*t*) even for small wavenumbers *k*. In the case of the eddy diffusivity of the shear flow, the eddy diffusion becomes local in space only for high wavenumbers regardless of the temporal scales of the velocity field owing 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 differential equation for the mean 〈*T*_{k}(*t*)〉 from (4.33) by noting that Green's function for the velocity field in (2.29) becomes deterministic and the eddy diffusivity functional from (4.35) becomes
4.49
Thus, the eddy diffusivity kernel is local in space and non-local in time
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
4.51
then (4.49) becomes
4.52
Here, this constant, *κ*_{eddy}, 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 [24].

#### (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 equation (3.30) 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 4.53 Note that here, the eddy diffusivity is local in space and non-local in time. According 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 [2,9,30,39]. In §§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:
5.1

While in principle this only requires processing through asymptotics and/or numerics of multi-dimensional quadrature formulas as developed in §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 straightforward fashion in the white noise limit for the shear waves developed in §3*b* [2,9,13] 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 §3*b* 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 equation (3.33) with :
5.2
Also recall that *η*_{k} is a fixed ratio of *σ*_{vk} and *γ*_{vk} as they both go to infinity in the white noise limit, *η*_{k}=*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 §3*b*, we have and the limiting velocity field converges to
5.3
yielding the steeper white noise limiting velocity spectrum
5.4
Thus with (5.2) and (5.4) in the white noise limit, the tracer variance spectrum is given by the exact formula
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 wavenumbers with *κk*^{2}≫1, the scalar dissipation regime, the limiting tracer spectrum is steeper than the limiting velocity spectrum by (*κk*^{2})^{−1}. On the other hand, for a large inertial range of wavenumbers with small molecular diffusivity, so that *κk*^{2}≪1, for a substantial range of wavenumbers, the tracer variance spectrum in the white noise limit is proportional to the velocity spectrum with constant *α*^{2}/*d*_{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 uncorrelated 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 §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 used elsewhere by the authors recently in turbulent regimes for both climate change science [24] and for a filtering test model [26]. The tracer variance statistics in the statistical steady state are computed 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 §3*b*, we first perform an initial experiment with a prescribed energy spectrum for the shear waves, here chosen to be the Kolmogorov spectrum, Var(*v*_{k})=*E*_{k}=|*k*|^{−5/3}, |*k*|≥1. With this initial choice of the variance parameter *σ*_{vk} and the damping parameter *γ*_{vk}, we perform a series of experiments where we integrate the tracer variance statistics to a statistically steady state replacing
5.6
in a fashion consistent with the white noise limit discussed in §5*a* since *η*_{k}=*σ*_{vk}/*γ*_{vk} is held constant with . All numerical experiments are calculated with a large inertial range for the tracer so that there are 1000 spatial wavenumbers, 1≤|*k*|≤1000 in the tracer dynamics with small tracer diffusivity *κ*=10^{−8}, uniform damping *d*_{T}=0.1 and mean background gradient *α*=10. All experiments use the OU equations for the mean jet from (2.11) with parameters *γ*_{U}=0.04, *σ*_{U}=0.4, *f*_{U}=0.09 so that the mean jet with jet variance ; thus the physical requirement in (1.9) is satisfied. The *β*-plane Rossby dispersion relation *ω*_{vk}=*βk*/(*k*^{2}+*F*) is used with *β*=8.91 and *F*=16 while the values *γ*_{vk}=*d*_{v}+*νk*^{2} are used for the Rossby wave dissipation with the inertial range parameters *d*_{v}=0.032 for Ekmann friction and *ν*=10^{−8} for viscosity; these values together with imposing *E*_{k}=|*k*|^{−5/3} for the velocity spectrum in the initial simulation determine *σ*_{vk} via .

In figure 1, we show the tracer spectrum that emerged from four simulations with the above parameters with *r*=1, *r*=50, *r*=10^{3}, *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 wavenumber. The tracer variance spectra show a roughly *k*^{−3} spectrum for the first 100 wavenumbers for *r*=1, 50 and a steeper slope for higher wavenumbers. The spectral plot for *r*=10^{3} shows a definite roll-over of the spectrum for the large-scale wavenumbers 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 behaviour is more pronounced at large wavenumbers for *r*=10^{4}. As evident from the large scatter at large wavenumbers, it takes a very long time for the tracer statistics to equilibrate at very high wavenumbers.

## 6. Strongly intermittent probability density functions with fat exponential tails

As discussed in §1 in the paragraph surrounding (1.9), the uncorrelated 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 [27] involves deterministic time periodic mean flow, , with no random fluctuations, *U*′(*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 zeros of with enhanced transport.

*Regime B*. The regime discovered recently [24,26] 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 random shear model with atmospheric parameters for the Rossby waves with the jet constrained to satisfy physical requirements: the mean jet is non-negative, , and the standard deviation of the jet fluctuations also yields a positive jet, i.e. . 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 [5].

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*.

As already illustrated in §5, we use the uncorrelated velocity field model with dispersive Rossby waves for the shear together with simple numerical experiments to demonstrate these results.

### (a) Stronger tracer probability density function 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 *d*_{T}=0.1, and *κ*=0.001; as in §5 the *β*-plane Rossby dispersion relation *ω*_{vk}=*βk*/(*k*^{2}+*F*) is used for the random shear waves with the atmospheric values *β*=8.91 and *F*=16 with the dissipation values *γ*_{vk}=*d*_{v}+*νk*^{2}, with *d*_{v}=0.6 and *ν*=0.1 and the turbulent energy spectrum
6.1
where was used in the simulation. We fix the forcing *f*_{U}(*t*)=2 and the dissipation *γ*_{U}=0.1 in the OU process from (2.11) for the jet so that the statistically steady mean jet becomes . We systematically increase the variance of the random forcing driving the fluctuations of the jet *σ*_{U} from 1 to 8. Note that even for the largest value of *σ*_{U}=8, we have so the physical requirements for *Regime B* are satisfied.

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

In figure 1 from §5, we showed the transitions in the tracer variance spectrum 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, 10^{3}, 10^{4}. In figure 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*=10^{3}, 10^{4} 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 probability density function intermittency

Here, we study the scenario of intermittency described in *Regime A* [27]. 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 tracer is time dependent and spikes at the zeros of the cross-sweep. In figures 7–9, we show the time-averaged PDFs of the tracer for the deterministic mean jet *U*(*t*) given by (2.21) with the oscillatory forcing for the jet, , where *A*_{U}=1, 10, 1000 and *ω*_{U}=*π*/3. Note that the amplitude of the deterministic jet is proportional to *A*_{U}. The other parameters had the following values: *γ*_{U}=0.04, *d*_{T}=0.1, *κ*=0.01, *α*=1, *ω*_{vk}=*βk*/(*k*^{2}+*F*) with *β*=8.91 and *F*=16, *γ*_{vk}=*d*_{v}+*νk*^{2}, with *d*_{v}=0.032 and *ν*=0.002 and the turbulent energy spectrum for the waves is given by
6.2
Note that as we increase the amplitude of the cross-sweep, the intermittency becomes stronger. Also note that in figure 7 where the spatial PDF is hardly intermittent, the PDFs of the smaller-scale Fourier modes have significant intermittency; for *A*_{U}=10, both the largest scale Fourier mode and the spatial PDF have increased intermittency while for *A*_{U}=1000 all displayed Fourier modes are strongly intermittent. Finally, we consider a general cross-sweep with both zeros and randomness in the cross-sweep. We start with the purely deterministic cross-sweep with *A*_{U}=10 with tracer PDFs depicted in figure 8 and include randomness in the jet with *σ*_{U}=2 while keeping the rest of the parameters the same. Comparing figure 10 with figure 8, we see that the intermittency of the tracer PDF has been increased substantially due to the random jet fluctuations.

## 7. Concluding discussion

In the preceding sections, we have both motivated (§2) and developed (§§2 and 3) elementary models for turbulent diffusion with complex physical features, mimicking crucial aspects of laboratory experiments, atmospheric observations, etc. These simplified models have the advantage of analytic and simple numeric tractability in the understanding of subtle statistical properties of a tracer with a background mean gradient, including closed expressions for tracer eddy diffusivity which are non-local in both space and time (§4), theory and simple numerics for the tracer variance spectrum (§5), and tracer PDF intermittency (§6) in simple models satisfying physical constraints of the atmosphere yet mimicking actual observations. Comments throughout the text have been made which indicate how such unambiguous test models with complex realistic statistical features are useful for climate change science [24,29] and for contemporary issues of real-time filtering from sparse noisy observations [25,26,31,32,36–38].

It is desirable to have even more analysis of the present simplified models, including further analytic/asymptotic/numerical processing of the formulas for eddy diffusivity and tracer variance spectrum in §§2 and 3. An important challenge is a rigorous mathematical proof of the transition to PDF intermittency with fat exponential tails for the tracer in a background gradient as the variance of the mean jet fluctuations increases (§5). It is worth mentioning here that for simpler models of tracer PDF intermittency involving random uniform shear flows and a decaying tracer in all of space, rigorous mathematical results on intermittent fat tails have been established in the literature [40–44].

## Acknowledgements

The research of A.J.M. is partially supported by National Science Foundation grant no. DMS-0456713, the office of Naval Research grant nos. 25-74200-F6607 and N00014-05-1-0164, and the Defense Advanced Projects Agency grant no. N0014-07-1-0750. B.G. is supported as a post-doctoral fellow through the same agencies.

## Footnotes

One contribution of 13 to a Theme Issue ‘Turbulent mixing and beyond: non-equilibrium processes from atomistic to astrophysical scales I’.

- © 2012 The Author(s) Published by the Royal Society. All rights reserved.