Elementary models for turbulent diffusion
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 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 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 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.
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 (Avellaneda 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
~ = κ∆T,
+ ~v · ∇T
∂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 twodimensional. 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 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.
E-mail: jonjon@cims.nyu.edu
Article submitted to Royal Society
TEX Paper
2Andrew J. Majda Corresponding author address: Andrew J. Majda, Department of Mathematics, and Center fo
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 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
~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)
(1.3)
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)
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 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
T (~x, t) = T 0 (~x, t) + αy,
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(1.4)
Elementary models for turbulent diffusion with complex physical features: eddy diffusivity, spectrum, and intermitt
so that with (1.1) and (1.2), T 0 satisfies
∂T 0
∂T 0
∂T 0
+ U (t)
+ v(x, t)
= κ∆T 0 − αv(x, t).
∂t
∂x
∂y
(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) = Ū (t) + U 0 (t),
(1.6)
where Ū (t) is a deterministic mean sweep and U 0 (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)
+P
, U (t) v(x, t) = Fv (x, t).
∂t
∂x
(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 T 0 (x, t) only depend on the x variable alone and dropping
the prime results in the simplified version of (1.5) given by
∂T
∂2T
∂T
+ U (t)
= −αv(x, t) + κ 2 − dT T.
∂t
∂x
∂x
(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 multidimensional 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) = Ū (t) with Ū (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 mechanism 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
Ū (t) > 0,
Ū 2 − V ar(U 0 (t)) > 0,
(1.9)
with V ar(U 0 (t)), the variance of U 0 (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
Article submitted to Royal Society
4Andrew J. Majda Corresponding author address: Andrew J. Majda, Department of Mathematics, and Center fo
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 (Gershgorin 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 diffusivity 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 numerical 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 ∂Ψ
,
∂y ∂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
~ = F (~x, t).
+ ~v · ∇Q
∂t
Article submitted to Royal Society
(2.4)
Elementary models for turbulent diffusion with complex physical features: eddy diffusivity, spectrum, and intermitt
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
F (~x, t)
= −U (t)y + ψ(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
= FU (U, t),
dt
(2.6)
dq
∂q
∂ψ
+ U (t)
+ (F U (t) + β)
= D(∆)ψ + Fq (x, t).
dt
∂x
∂x
(2.7)
and for q = ψxx − F ψ
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
−F
∂x2
−1
∂
.
∂x
(2.9)
∂
The symbol of R ∂x
at a given spatial wave number is R = − k2ik
+F . To get the
∂
equation for v, we apply R ∂x to (2.7) and use (2.9) to obtain the dynamics
∂v
∂v
+ U (t)
+ (F U (t) + β)R
∂t
∂x
∂
∂x
v = −dv v + ν
∂2v
+ fv (x, t) + σv Ẇv (t).(2.10)
∂x2
Special choices of the forcing Fq result in (2.10) so that fv (x, t) is deterministic
forcing and σv (x)Ẇv (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 −dv v
∂2v
and a small scale frictional viscosity ν ∂x
2 (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 forcing FU (U, t) in (2.6) has the special form of
Article submitted to Royal Society
6Andrew J. Majda Corresponding author address: Andrew J. Majda, Department of Mathematics, and Center fo
deterministic forcing fU (t), damping −γU and white noise forcing σU W˙(t) which
results in the dynamics
dU (t)
= −γU U (t) + fU (t) + σU ẆU (t).
dt
(2.11)
The evolution of the shear flow is given in general by the following pseudo-differential
equation
∂
∂v(x, t)
+P
, U (t) v(x, t) = Fv (x, t),
∂t
∂x
(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 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)
= (−γvk + iωvk )vk (t) + fvk (t) + σvk Ẇvk (t),
dt
(2.13)
where γvk is dissipation and ωvk 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 γvk does not depend on U (t)
ωvk = ak U (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
∂ ∂
∂
∂2
, U (t) = U (t)
+ F U (t) + β R
+ dv − ν 2 .
(2.15)
P
∂x
∂x
∂x
∂x
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 , ωvk = −ck,
where ν is the flow viscosity (Gershgorin and Majda, 2010c),
2. Uncorrelated Rossby waves: γvk = ν(k 2 +F ), ωvk = k2βk
+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),
Article submitted to Royal Society
Elementary models for turbulent diffusion with complex physical features: eddy diffusivity, spectrum, and intermitt
3. Correlated Rossby waves (a generalization of case 2 developed in (2.10) above):
γ vk
=
dv + νk 2 ,
(2.16)
ωvk
=
(2.17)
ak
=
bk
=
ak U (t) + bk ,
F
k
−1 ,
k2
βk
.
k2 + F
(2.18)
(2.19)
(c) Statistics of the velocity field
The solution for the cross-sweep becomes
U (t) = Ū (t) + U 0 (t),
(2.20)
where
Z
Ū (t)
t
=
GU (s, t)fU (s)ds,
(2.21)
t0
U 0 (t)
= GU (t0 , t)U0 + UW (t),
Z t
GU (s, t)dWU (s),
UW (t) = σU
(2.22)
(2.23)
t0
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
Z t
hU (t)i = GU (t0 , t) hU0 i +
GU (s, t)fU (s)ds,
(2.25)
t0
V ar(U (t))
= G2U (t0 , t)V ar(U0 ) +
2
σU
1 − G2U (t0 , t) .
2γU
(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 identities 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
1
hzeix i = hzi + iCov(z, x) eihxi− 2 V 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
Z t
Z t
vk (t) = Gvk (t0 , t)vk,0 +
Gvk (s, t)fvk (s)ds + σvk
Gvk (s, t)dWvk (s), (2.28)
t0
Article submitted to Royal Society
t0
8Andrew J. Majda Corresponding author address: Andrew J. Majda, Department of Mathematics, and Center fo
where vk,0 = vk (t0 ) and the Green’s function for the shear flow is defined as
Gvk (s, t)
=
Jk (s, t)
=
e−γvk (t−s)+iJk (s,t) ,
Z t
ωvk (s0 )ds0 .
(2.29)
(2.30)
s
For the case of correlated flow, we compute further
Z t
ωvk (s0 )ds0 = ak L(s, t) + bk (t − s),
Jk (s, t) =
s
Z t
U (s0 )ds0 = LD (s, t) + LW (s, t) + b0 (s, t)U0 ,
L(s, t) =
(2.31)
(2.32)
s
Z tZ
LD (s, t)
s
GU (r0 , s0 )fU (r0 )dr0 ds0 ,
Z tZ
UW (s0 )ds0 = σU
=
s
b0 (s, t)
= −
(2.33)
t0
t
Z
LW (s, t)
s0
=
s
s0
GU (s00 , s0 )dWU (s00 )ds0 ,
(2.34)
t0
1
(GU (t0 , t) − GU (t0 , s)).
γU
(2.35)
Next, we find the mean of vk
Z
t
hvk (t)i = hGvk (t0 , t)vk,0 i +
hGvk (s, t)ifvk (s)ds.
(2.36)
t0
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,0 i =
hGvk (s, t)i =
hJk (s, t)i =
V ar(Jk (s, t))
(hvk,0 i + iak b0 (t0 , t)Cov(vk,0 , U0 )) hGvk (t0 , t)i ,
−γvk (t−s)+ihJk (s,t)i− 21 V ar(Jk (s,t))
e
ak hL(s, t)i + bk (t − s),
,
= a2k V ar(L(s, t)),
hL(s, t)i =
V ar(L(s, t))
=
V ar(LW (s, t))
=
(2.37)
(2.38)
(2.39)
(2.40)
LD (s, t) + b0 (s, t) hU0 i ,
(2.41)
b20 (s, t)V ar(U0 ) + V ar(LW (s, t)),
2
σU
(−2 + 2γU |t − s| + 2e−γU |t−s|
2γU3
−G2U (t0 , t) − G2U (t0 , s)),
(2.42)
+ 2GU (2t0 , s + t)
(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
∂2
∂T
+ U (t)
= κ 2 T − dT T − αv.
∂t
∂x
∂x
Article submitted to Royal Society
(2.44)
Elementary models for turbulent diffusion with complex physical features: eddy diffusivity, spectrum, and intermitt
In Fourier space, this equation becomes
dTk
= (−γTk + iωTk )Tk − αvk ,
dt
(2.45)
where
γ Tk
= 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
Z t
Tk (t) = GTk (t0 , t)Tk,0 − α
GTk (s, t)vk (s)ds,
(2.48)
t0
where Tk,0 = Tk (t0 ) and the Green’s function for the tracer is given by
GTk (s, t)
=
L(s, t)
=
e−γTk (t−s)−ikL(s,t) ,
Z t
U (s0 )ds0 .
(2.49)
(2.50)
s
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
Z t
hTk (t)i = hGTk (t0 , t)Tk,0 i − α
hGTk (s, t)Gvk (t0 , s)vk,0 i ds
t0
Z tZ
s
−α
hGTk (s, t)Gvk (r, s)i fvk (r)drds,
t0
(2.51)
t0
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 hGvk i from (2.38)
1
.
hGTk (s, t)Gvk (r, s)i = e−γvk (r−s)−γTk (t−s)+i(hJk (s,t)i−khL(s,t)i)− 2 V ar(Jk (s,t)−kL(s,t))(2.52)
In the statistically steady state, the mean of the tracer becomes
Z t Z s
hTk (t)i = −α
hGTk (s, t)Gvk (r, s)i fvk (r)drds.
−∞
(2.53)
−∞
And the covariance is given by
Cov(Tj (t), Tk (t)) =
Z t Z t ∗
α2
GTj (s, t)G∗Tk (s0 , t)vj (s)vk∗ (s0 ) − GTj (s, t)vj (s) hGTk (s0 , t)vk (s0 )i dsds0 ,
−∞
−∞
(2.54)
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10Andrew J. Majda Corresponding author address: Andrew J. Majda, Department of Mathematics, and Center
where
GTj (s, t)G∗Tk (s0 , t)vj (s)vk∗ (s0 ) =
Z s Z s0
Gvj (r, s)G∗vk (r0 , s0 )GTj (s, t)G∗Tk (s0 , t) fvj (r)fv∗k (r0 )drdr0
−∞
−∞
+σv2k δkj
Z
min(s,s0 )
−∞
hGTk (s, t)G∗Tk (s0 , t)Gvk (r, s)G∗vk (r, s0 )idr
(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 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)
+P
v(x, t) = Fv (x, t),
∂t
∂x
(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 velocity 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 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
Article submitted to Royal Society
Elementary models for turbulent diffusion with complex physical features: eddy diffusivity, spectrum, and intermit
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
dvk (t)
dt
dTk (t)
dt
where for the example
= −γU U (t) + fU (t) + σU ẆU (t),
(3.2)
=
(−γvk + iωvk )vk (t) + fvk (t) + σvk Ẇvk (t),
(3.3)
=
(−γTk + iωTk )Tk (t) − αvk (t),
(3.4)
of uncorrelated Rossby waves
βk
,
+ Fs
= −kU (t),
ωvk
=
ωTk
k2
2
(3.5)
(3.6)
γ vk
= ν(k + Fs ),
(3.7)
γ Tk
= 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 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 ak ≡ 0 in (2.17). We use
the latter way and find
Z t
hTk (t)i = hGTk (t0 , t)Tk,0 i − α
Gvk (t0 , s) hGTk (s, t)vk,0 i ds
t0
Z
t
−α
hGTk (s, t)iV̄k (t0 , s)ds,
(3.9)
t0
where
Z
s
V̄k (t0 , s) =
Gvk (r, s)fvk (r)dr.
(3.10)
t0
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)
Z t
hTk (t)i = −α
hGTk (s, t)iV̄k (s)ds
(3.11)
−∞
Article submitted to Royal Society
12Andrew J. Majda Corresponding author address: Andrew J. Majda, Department of Mathematics, and Center
where
Z
s
V̄k (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)
Z t Z t 2
GTj (s, t)G∗Tk (s0 , t) hvj (s)vk∗ (s0 )i
Cov(Tj (t), Tk (t)) = α
−∞ −∞
∗
− GTj (s, t) hvj (s)i G∗Tk (s0 , t) hvk (s0 )i dsds0 ,(3.13)
where
hvj (s)vk∗ (s0 )i
s
Z
Z
s0
=
−∞
+σv2k δkj
Z
−∞
Gvj (r, s)G∗vk (r0 , s0 )fvj (r)fv∗k (r0 )drdr0
min(s,s0 )
Gvk (r, s)G∗vk (r, s0 )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
Z t Z t
2
σ2
e−(γ+κk )(2t−s−r) e−ikŪ (r−s) e−γk |s−r|+iωk (s−r)
Cov(Tk (t), Tl (t)) = α2 k δkl
2γk
−∞ −∞
×e−
k2
2
(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)) −
2 σU
−γU (t−s)
−γU (t−r)
−γU (s−r)
e
−
e
−
1
+
e
.
2γU3
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).
Article submitted to Royal Society
Elementary models for turbulent diffusion with complex physical features: eddy diffusivity, spectrum, and intermit
(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
0
vk (t) = v̄k|U (t) + vk|U
(t),
(3.16)
where v̄k|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 Ẇvk ; on the other
0
hand, vk|U
(t) denotes fluctuations of the shear around the conditional mean. From
(2.13) we find the dynamics of the conditional mean v̄k|U (t)
dv̄k|U (t)
= (−γvk + iωvk )v̄k|U (t) + fvk (t),
dt
(3.17)
and of the fluctuations around this mean
0
dvk|U
(t)
dt
0
= (−γvk + iωvk )vk|U
(t) + σvk Ẇvk (t).
(3.18)
It is very important to emphasize that for a general master model, v̄k|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 v̄k|U (t) becomes
deterministic and equal to the mean of the waves, v̄k|U (t) = v̄k (t). First, we define
0
the white noise limit for the fluctuations vk|U
(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, Ẇvk ,
the autocorrelation function of vk0 (t) becomes
0
0
0
Corrvk|U
(τ ) = hvk|U
(t + τ )vk|U
(t)∗ i =
σv2k −γv τ iJk (t,t+τ )
e k e
.
2γvk
(3.19)
Therefore, if we keep the following ratio fixed
ηk =
σ vk
= const,
γ vk
(3.20)
then the absolute value of the autocorrelation function formally approaches a deltafunction
0
|Corrvk|U
(τ )| →
ηk2
δ0 (τ ),
2
(3.21)
or, equivalently, the fluctuations vk0 (t) approach the white noise
0
(t) → ηk Ẇvk (t).
vk|U
(3.22)
Secondly, we proceed with the white noise limit for v̄k|U (t) from (3.17). Suppose
that the forcing fvk (t) grows as the dissipation γvk grows in the white noise limit
fvk (t) = γvk f¯vk (t),
Article submitted to Royal Society
(3.23)
14Andrew J. Majda Corresponding author address: Andrew J. Majda, Department of Mathematics, and Center
where f¯k (t) is independent of γvk . Then, for the value of v̄k|U (t) given by (2.28)
without the last term
Z t
v̄k|U (t) = Gvk (t0 , t)v̄k|U (t0 ) +
Gvk (s, t)fvk (s)ds,
(3.24)
t0
we find, using (3.23) in (3.24), the white noise limit as γvk → +∞
v̄k|U (t) → f¯vk (t).
(3.25)
Note that in the white noise limit, v̄k|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) → γvk f¯vk (t),
(3.26)
and then using the limiting approach to the delta-function of the following terms
γvk Gvk (s, t) →
δt (s),
(3.27)
σvk Gvk (s, t) →
ηk δt (s).
(3.28)
Then, in this white noise limit the master model is described by the equations
dU (t)
= −γU U (t) + fU (t) + σU ẆU ,
dt
dTk
= (−γTk + iωTk )Tk − α f¯vk (t) + ηk Ẇvk ,
dt
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
Z t
Z t
Tk (t) = GTk (t0 , t)Tk,0 − α
GTk (s, t)f¯vk (s)ds − αηk
GTk (s, t)dWvk (s). (3.29)
t0
t0
The mean of the tracer becomes
Z
t
hTk (t)i = hGTk (t0 , t)Tk,0 i − α
hGTk (s, t)if¯vk (s)ds.
(3.30)
t0
In the statistically steady state, the mean simplifies to
Z t
hTk (t)i = −α
hGTk (s, t)if¯vk (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)
Z t Z s
hTk (t)i = −α
hGvk (r, s)GTk (s, t)i γvk f¯vk (r)drds
−∞
t
Z
→
−α
−∞
Z t
=
−∞
s
Z
−α
−∞
Article submitted to Royal Society
δs (r) hGTk (s, t)i f¯vk (r)drds
−∞
hGTk (s, t)i f¯vk (s)ds
(3.32)
Elementary models for turbulent diffusion with complex physical features: eddy diffusivity, spectrum, and intermit
Next, we find the covariance in the white noise limit in the statistically steady state
= α2
Cov(Tj , Tk )
t
Z
Z
−∞
t
−∞
hGTj (s, t)G∗Tk (s0 , t)i − hGTj (s, t)ihG∗Tk (s0 , t)i f¯vj (s)f¯v∗k (s0 )dsds0
δj
+ k α2 ηk2 .
2γTk
(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 (s0 , t)vj (s)vk∗ (s0 ) =
Z s Z s0
Gvj (r, s)G∗vk (r0 , s0 )GTj (s, t)G∗Tk (s0 , t) f¯vj (r)f¯v∗k (r0 )drdr0
γ vk γ vj
−∞
+σv2k δkj
s
Z
−∞
min(s,s0 )
Z
−∞
s0
Z
→
−∞
Gvk (r, s)G∗vk (r, s0 )GTk (s, t)G∗Tk (s0 , t) dr
−∞
δs (r)δs0 (r0 ) GTj (s, t)G∗Tk (s0 , t) f¯vj (r)f¯v∗k (r0 )drdr0
+ηk2 δkj hGTk (s, t)G∗Tk (s0 , t)i
=
Z
min(s,s0 )
−∞
GTj (s, t)G∗Tk (s0 , t) f¯vj (s)f¯v∗k (s0 )
δs (r)δs0 (r)dr
+ ηk2 δkj δs (s0 )e−2γTk (t−s) .
(3.34)
Now, the covariance becomes
Cov(Tj , Tk ) =
Z t Z t ∗
GTj (s, t)G∗Tk (s0 , t)vj (s)vk∗ (s0 ) − GTj (s, t)vj (s) hGTk (s0 , t)vk (s0 )i dsds0
α2
−∞
t
→ α2
Z
−∞
t
Z
−∞
−∞
t
+α2 ηk2 δkj
=
α2
Z
t
Z
Z
Z
GTj (s, t)G∗Tk (s0 , t) − hGTj (s, t)ihG∗Tk (s0 , t)i f¯vj (s)f¯v∗k (s0 )dsds0
t
e−2γTk (t−s) δs (s0 )dsds0
−∞ −∞
t −∞ −∞
δj
+ k α2 ηk2
2γTk
GTj (s, t)G∗Tk (s0 , t) − hGTj (s, t)ihG∗Tk (s0 , t)i f¯vj (s)f¯v∗k (s0 )dsds0
(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
du2
dt
=
−γ1 u1 + f1 (t) + σ1 Ẇ1
=
(−γ2 + i(ω0 + a0 u1 ))u2 + f2 (t) + σ2 Ẇ2
Article submitted to Royal Society
(3.36)
16Andrew J. Majda Corresponding author address: Andrew J. Majda, Department of Mathematics, and Center
whereas the white noise limit of the master model is given by
dU 0 (t)
dt
dTk
dt
= −γU U 0 (t) + fU0 (t) + σU ẆU ,
=
(−γTk − ik(U0 + U 0 (t)))Tk − αv̄k (t) + αη Ẇk .
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
U 0 (t). The forcing of the fluctuation of the jet is denoted as fU0 (t). Here, the mean
shear flow −αv̄k (t) plays the role of the forcing of the tracer through the mean
gradient. Suppose that fU0 (t) and all or some of v̄k (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 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 (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 v̄k (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.
Article submitted to Royal Society
Elementary models for turbulent diffusion with complex physical features: eddy diffusivity, spectrum, and intermit
(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)
= −γU (t) + fU (t) + σU ẆU
dt
(3.38)
γU
→ +∞,
(3.39)
σU
σU
γU
fU (t)
γU
→ +∞,
when
=
ηU = const.
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) → Ū (t) + ηU ẆU (t),
(3.40)
with Ū (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)
= (−γTk − ik Ū (t))Tk − αf¯vk (t) − ikηU Tk (t) ◦ ẆU (t) − αηk Ẇvk . (3.41)
dt
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 nonvanishing 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) ◦ ẆU i 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
D
E
Rt
0
0
hGTk (s, t)i =
e−γTk (t−s)−ik s U (s )ds
D
E
η2
→ ḠTk (s, t) e−ik(WU (t)−WU (s)) = ḠTk (s, t)e− 2 (t−s) ,(3.42)
where the deterministic part of the Green’s function for the tracer is given by
ḠTk (s, t) = e−γTk (t−s)−ik
Rt
s
Ū (s0 )ds0
(3.43)
Now, the mean of the tracer becomes a white noise limit of (3.31)
Z
t
hTk (t)i = −α
−∞
Article submitted to Royal Society
e−
2
ηU
2
k2 (t−s)
ḠTk (s, t)f¯vk (s)ds.
(3.44)
18Andrew J. Majda Corresponding author address: Andrew J. Majda, Department of Mathematics, and Center
η2
Here we note the correction to the diffusivity (the eddy diffusivity) 2U that comes
from the diffusion-induced advection term that appears after rewriting the SDE
(3.41) in the Ito form (Gardiner, 1997)
dTk (t)
k2 2
= (−γTk − ik Ū (t) − ηU
)Tk − αf¯vk (t) − ikηU Tk (t)ẆU (t) − αηk Ẇvk(3.45)
.
dt
2
Here, the eddy diffusivity is local in space and time and is equal to
κe =
2
ηU
σ2
∼ U2 .
2
2γU
(3.46)
Next, the covariance, is a white noise limit of (3.33)
Cov(Tj (t), Tk (t))
= α
2
Z
t
Z
−∞
×e−
+
2
ηU
2
t
−∞
ḠTj (s, t)Ḡ∗Tk (s0 , t)f¯vj (s)f¯v∗k (s0 )
(j 2 (t−s)+k2 (t−s0 ))
2
0
(eηU kj(t−max(s,s )) − 1)dsds0
δkj 2 2
α ηk .
2γTk
(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
Z t Z t
2
ηU
2
2
0
Cov(Tj (t), Tk (t)) = α2
ḠTj (s, t)Ḡ∗Tk (s0 , t)e− 2 (j (t−s)+k (t−s ))
−∞ −∞
2
∗ 0
ηU
kj(t−max(s,s0 ))
× hvj (s)vk (s )ie
− hvj (s)ihvk∗ (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
= −γvk hvk (t)i + ihωvk (t)vk (t)i + fvk (t).
dt
(4.1)
Here, ωvk (t) is Gaussian, ωvk (t) = ak U (t) + bk , according to its definition in (2.17)
and we encounter a moment closure problem manifested in the term hωvk (t)vk (t)i.
Article submitted to Royal Society
Elementary models for turbulent diffusion with complex physical features: eddy diffusivity, spectrum, and intermit
We use the decomposition of both ωvk (t) and vk (t) into their respective deterministic means and random fluctuations around those means
ωvk (t)
vk (t)
= hωvk (t)i + ωv0 k (t),
= hvk (t)i + vk0 (t),
and find
dhvk (t)i
= (−γvk hvk (t)i + ihωvk (t)i)hvk (t)i + ihωv0 k (t)vk0 (t)i + fvk (t).
dt
(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
= −γTk hTk (t)i + ihωTk (t)Tk (t)i − αhvk (t)i.
dt
(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 + ωT0 k (t),
Tk (t)
=
hTk (t)i + Tk0 (t),
and find
hTk (t)i
= (−γTk hTk (t)i + ihωTk (t)i)hTk (t)i + ihωT0 k (t)Tk0 (t)i − αhvk (t)i.
dt
(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
Z t
hvk (t)i = hGvk (t0 , t)ihvk,0 i +
hGvk (s, t)i fvk (s)ds,
(4.5)
t0
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)
Jk (s, t)
= e−γvk (t−s)+iJk (s,t) ,
Z t
=
ωvk (s0 )ds0 ,
(4.6)
(4.7)
s
For the special case of atmospheric waves in the QG model, we have for example
Z t
F
βk
Jk (s, t) = k
−1
U (s0 )ds0 + 2
.
(4.8)
k2
k
+F
s
Article submitted to Royal Society
20Andrew J. Majda Corresponding author address: Andrew J. Majda, Department of Mathematics, and Center
We find the time derivative of the mean of vk
dhvk (t)i
dt
(−γvk + i hωvk (t)i)hvk (t)i + fvk (t)
=
−k 2 κve (k, t0 , t)hGvk (t0 , t)ihvk,0 i
Z t
κve (k, s, t)hGvk (s, t)ifvk (s)ds,
−k 2
(4.9)
t0
where the eddy diffusivity in the general case becomes
κve (k, s, t) =
1 ∂
V ar(Jk (s, t)).
2k 2 ∂t
(4.10)
Note that we factored out the k 2 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
Kev [hvk,0 i, fvk ]
=
κve (k, t0 , t)hGvk (t0 , t)ihvk,0 i
Z
t
+
κve (k, s, t)hGvk (s, t)ifvk (s)ds,
(4.11)
t0
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 Kev [hvk,0 i, fvk ]. (4.12)
We compare this equation with the formula in (4.2) to find
ihωv0 k (t)vk0 (t)i = −k 2 Kev [hvk,0 i, fvk ].
(4.13)
In physical space, the equation for the mean of the shear flow becomes
dhv(x, t)i
=P
dt
∂
∂2 v ∂
, hU (t)i hv(x, t)i +
K̃
, t0 , t ,
∂x
∂x2 e ∂x
(4.14)
where K̃ is given by its Fourier symbol in (4.11). For the QG model, we find
κve (k, s, t)
1
=
2
2
Z t
F
∂
0
0
−1
V ar
U (s )ds .
k2
∂t
s
The time derivative of the variance of
∂
V ar
∂t
Z
t
U (s0 )ds0
=
s
+
Rt
s
(4.15)
U (s0 )ds0 is given by
2 GU (t0 , s) − GU (t0 , t) GU (s, t)V ar(U0 )
γU
2 σU
1 − GU (s, t) − GU (2t0 , s + t) + G2U (t0 , t)2(4.16)
,
2
γU
where we used (2.42) and GU is defined in (2.24). Next, we study the role of each
component of the eddy diffusivity.
Article submitted to Royal Society
Elementary models for turbulent diffusion with complex physical features: eddy diffusivity, spectrum, and intermit
(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,0 i ,
(4.17)
and the differential equation for the mean becomes
dhvk (t)i
= (−γvk + i hωvk (t)i)hvk (t)i − k 2 κve (k, t0 , t)hvk (t)i
dt
(4.18)
with κve given explicitly by (4.10). For the QG model, the corresponding equation
in physical space becomes
∂
∂2 v ∂
dhv(x, t)i
κ̃
=P
, hU (t)i hv(x, t)i +
,
t
,
t
hv(x, t)i ,
(4.19)
0
dt
∂x
∂x2 e ∂x
where κ̃ve is 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
κve (k, s, t) into the product of non-local temporal and non-local spatial parts
κve (k, s, t) = κve,tm (s, t) κve,sp (k).
(4.20)
Then, we find for the QG model
κve,tm (s, t)
κve,sp (k)
Z t
1 ∂
V ar
U (s0 )ds0 ,
2 ∂t
s
2
F
−1 .
=
k2
=
(4.21)
(4.22)
In physical space, we find
κ̃ve,sp
∂
∂x
=
F
2
∂ −2
−
1
.
∂x−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
κve,sp (k) ≈ 1.
(4.24)
At wavelengths larger than LR genuine 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
κve,tm (s, t) =
2
σU
.
2γU2
(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.
Article submitted to Royal Society
22Andrew J. Majda Corresponding author address: Andrew J. Majda, Department of Mathematics, and Center
(ii) Waves in the statistically steady state
Next, we turn to the case of the statistically steady state with nonzero forcing for
the waves, fvk 6= 0, which induces a non-zero statistically steady mean, v̄k (t). Here
the initial conditions for hvk i 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
Z t
hvk (t)i =
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 hωvk i)hvk (t)i + fvk (t)
Z t
2 v
−k κe,sp (k)
κve,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
Z t
Hk [fvk ](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 hvk i 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,
κve,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
= (−γvk + i hωvk i)hvk (t)i + fvk (t) − k 2 κve,sp (k)Hk [fvk ](t).
dt
In physical space, this equation becomes
∂
∂2 v
∂
dhv(x, t)i
H̃k (x, t).
=P
, hU (t)i hv(x, t)i +
κ̃
dt
∂x
∂x2 e,sp ∂x
(4.29)
(4.30)
Note that, in the statistically steady state, the temporal part of the eddy diffusivity becomes
σ2 κve,tm = U2 1 − e−γU (t−s)
(4.31)
2γU
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 κve,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.
Article submitted to Royal Society
Elementary models for turbulent diffusion with complex physical features: eddy diffusivity, spectrum, and intermit
(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
Z t
hTk (t)i = hGTk (t0 , t)ihTk,0 i − α
hGTk (s, t)Gvk (t0 , s)ihvk,0 ids
t0
Z tZ
s
hGTk (s, t)Gvk (r, s)ifvk (r)drds.
−α
(4.32)
t0
t0
We find the effective equation for this mean of the tracer by differentiating (4.32)
dhTk (t)i
dt
=
(−γTk + ihωTk i)hTk (t)i − αhvk (t)i
−k 2 κTe (k, t0 , t0 , t)hGTk (t0 , t)ihTk,0 i
Z t
+k 2 α
κTe (k, t0 , s, t)hGTk (s, t)Gvk (t0 , s)ids hvk,0 i
t0
+k 2 α
Z tZ
t0
s
κTe (k, r, s, t)hGTk (s, t)Gvk (r, s)ifvk (r)drds, (4.33)
t0
where
κTe (k, r, s, t) =
1 ∂
V ar(L(s, t)) −
2 ∂t
F
−1
k2
∂
Cov(L(r, s), L(s, t)),
∂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
KeT [hvk,0 i, hTk,0 i, fvk ]
= κTe (k, t0 , t0 , t)hGTk (t0 , t)ihTk,0 i
Z t
−α
κTe (k, t0 , s, t)hGTk (s, t)Gvk (t0 , s)ids hvk,0 i
t0
Z tZ
s
−α
t0
κTe (k, r, s, t)hGTk (s, t)Gvk (r, s)ifvk (r)drds,
t0
(4.35)
highlighting the three separate contributions. Then, the equation for the mean of
the tracer takes a simple form
dhTk (t)i
= (−γTk + ihωTk i)hTk (t)i − αhvk (t)i − k 2 KeT [hvk,0 i, hTk,0 i, fvk ]. (4.36)
dt
By comparing this equation with the formula in (4.35) we find
ihωT0 k (t)Tk0 (t)i = KeT [hvk,0 i, hTk,0 i, 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.
Article submitted to Royal Society
24Andrew J. Majda Corresponding author address: Andrew J. Majda, Department of Mathematics, and Center
(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
∂hT (x, t)i
∂ 2 hT (x, t)i
− dT hT (x, t)i, (4.38)
+ hU (t)i
= (κ + κTe (t0 , t))
dt
∂x
∂x2
where
κTe (t0 , t) =
1 ∂
V ar
2 ∂t
Z
t
U (s)ds .
(4.39)
t0
With this simple special case, we demonstrate how the eddy diffusivity brings memory 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 nonvanishing which leads to the following differential equation for the mean of the
tracer
dhTk (t)i
dt
=
(−γTk + ihωTk i)hTk (t)i − αhvk (t)i
Z t
2
+k α
κTe (k, t0 , s, t)hGTk (s, t)Gvk (t0 , s)ids hvk,0 i, (4.40)
t0
where κTe 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 → −∞
Z t Z s
hTk (t)i = −α
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
= (−γTk + ihωTk i)hTk (t)i − αhvk (t)i − k 2 KeT [fvk ](t),
dt
Article submitted to Royal Society
(4.42)
Elementary models for turbulent diffusion with complex physical features: eddy diffusivity, spectrum, and intermit
where the eddy diffusivity functional for the tracer has the following form
Z t Z s
T
κTe (k, r, s, t)hGTk (s, t)Gvk (r, s)ifvk (r)drds,
Ke [fvk ](t) = −α
−∞
(4.43)
−∞
and the eddy diffusivity kernel in the statistically steady state is given by
F
σ2 .(4.44)
κTe (k, r, s, t) = U2
1 − e−γU (t−s) − 2 − 1 e−γU (t−s) − e−γU (t−r)
2γU
k
Note that KeT [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
∂ 2 K̃eT [fv ](x, t)
∂hT (x, t)i
∂ 2 hT (x, t)i
−
d
hT
(x,
t)i
+
− αhv(x, t)i.
(4.45)
+ Ū (t)
=κ
T
dt
∂x
∂x2
∂x2
The functional K̃eT [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
κTe =
2
σU
.
2γU2
(4.46)
By comparing (4.43) with (4.41) we find that with the constant kernel κTe 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
KeT [fvk ](t) =
2
σU
hTk (t)i .
2γU2
(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
∂hT (x, t)i
∂ 2 hT (x, t)i
+ Ū (t)
= (κ + κTe )
− dT hT (x, t)i − αhv(x, t)i,(4.48)
dt
∂x
∂x2
with a constant eddy diffusivity, κTe , given by (4.46). Note that here, unlike in the
case with the eddy diffusivity of the shear flow, the eddy diffusivity κTe 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 differential equation for the mean hTk (t)i from (4.33) by noting that the Green’s function
Article submitted to Royal Society
26Andrew J. Majda Corresponding author address: Andrew J. Majda, Department of Mathematics, and Center
for the velocity field in (2.29) becomes deterministic and the eddy diffusivity functional from (4.35) becomes
KeT [hvk,0 i, hTk,0 i, fvk ]
= κTe (t0 , t)hGTk (t0 , t)ihTk,0 i
Z t
− α
κTe (s, t)hGTk (s, t)iGvk (t0 , s)ds hvk,0 i
Z
t0
t
− α
κTe (s, t)hGTk (s, t)i
t0
Z
s
Gvk (r, s)fvk (r)dr ds.
t0
(4.49)
Thus the eddy diffusivity kernel is local in space and nonlocal in time
κTe (s, t) =
1 ∂
V ar(L(s, t)),
2 ∂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
κTe (s, t) ≈ keddy ,
(4.51)
then (4.49) becomes
∂hTk (t)i
− ihωTk ihTk (t)i ≈ −(κ + κeddy )k 2 hTk (t)i − γTk hTk (t)i − αhvk (t)i. (4.52)
∂t
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 − αf¯vk (t) − k 2 κTe (t0 , t)hGTk (t0 , t)ihTk,0 i
Z t
2
+k α
κTe (s, t)hGTk (s, t)if¯vk (s)ds,
t0
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) =
Article submitted to Royal Society
1 ∂
V ar(L(s, t)).
2 ∂t
(4.53)
Elementary models for turbulent diffusion with complex physical features: eddy diffusivity, spectrum, and intermit
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 (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 numerics 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 straightforward 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 f¯vk = 0,
V ar(Tk ) =
α2 ηk2
.
2γTk
(5.2)
Also recall that ηk is a fixed ratio of σvk and γvk as they both go to infinity in the
σ
white noise limit, ηk = γvvk . Now formally in the white noise limit starting with
k
the initial steady state velocity spectrum in (5.2), as shown in section 3b), we have
k
ηk2 = 2E
γv and the limiting velocity field converges to
k
vk (t) = ηk Ẇk (t),
(5.3)
yielding the steeper white noise limiting velocity spectrum
V ar(vk ) =
Article submitted to Royal Society
ηk2
2Ek
=
.
2
γ vk
(5.4)
28Andrew J. Majda Corresponding author address: Andrew J. Majda, Department of Mathematics, and Center
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 + κk 2 )−1 V 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 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 wave numbers
with small molecular diffusivity, so that κk 2 1, for a substantial range of wave
numbers, the tracer variance spectrum in the white noise limit is proportional to
2
the velocity spectrum with constant, dαT .
(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 section 2; this model amounts to setting
∂
, U (t) in (2.15) and this model has been utilized
U (t) ≡ 0 in the formula for P ∂x
elsewhere by the authors recenty in turbulent regimes for both climate change science (Majda and Gershgorin, 2010) and for a filtering test model (Gershgorin and
Majda, 2010c). 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 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)
σ
since ηk = γvvk is held constant with γvk → +∞. All numerical experiments are
k
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 Ū = γU−1 fU = 2.25 with jet
variance
2
σU
2γU
= 2; thus the physical requirement in (1.9) is satisfied. The β-plane
Rossby dispersion relation ωvk = k2βk
+F is utilized with β = 8.91 and F = 16 while
2
the values γvk = dv + νk are used for the Rossby wave dissipation with the inertial
range parameters dv = 0.032 for Ekmann friction and ν = 10−8 for viscosity; these
Article submitted to Royal Society
Elementary models for turbulent diffusion with complex physical features: eddy diffusivity, spectrum, and intermit
values together with imposing Ek = √
|k|−5/3
√ for the velocity spectrum in the initial
simulation determine σvk via σvk = 2Ek dv + νk 2 .
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 = 104 , 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 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 (Bourlioux and Majda, 2002) involves
deterministic time periodic mean flow, Ū (t) = AU sin(ωU t), with no random fluctuations, U 0 (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 Ū (t) with enhanced transport.
Regime B) The regime discovered recently (Majda and Gershgorin, 2010; Gershgorin 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 random shear model with atmospheric parameters for the Rossby waves with the jet
U (t) = Ū (t) + U 0 (t) constrained to satisfy physical requirements: the mean jet Ū
is non-negative, Ū (t) > 0, and the standard deviation of the jet fluctuations also
yield a positive jet, i.e., Ū 2 (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).
As already illustrated in section 5, we use the uncorrelated velocity field model
Article submitted to Royal Society
30Andrew J. Majda Corresponding author address: Andrew J. Majda, Department of Mathematics, and Center
with dispersive Rossby waves for the shear together with simple numerical experiments 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 = k2βk
+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 dv = 0.6 and ν = 0.1 and the turbulent
energy spectrum

 1 , for |k| ≤ 5
2
V ar(vk ) = 1 k −θv
(6.1)

, for |k| > 5.
2 5
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 Ū = 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 Ū 2 − V ar(U 0 ) = 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 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, 103 , 104 . 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
Article submitted to Royal Society
Elementary models for turbulent diffusion with complex physical features: eddy diffusivity, spectrum, and intermit
tracer is time dependent and spikes at the zeros of the cross-sweep. In Figs. 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, fU (t) = AU sin(ωU t), where
AU = 1, 10, 1000 and ωU = π3 . Note that the amplitude of the deterministic jet
is proportional to AU . The other parameters had the following values: γU = 0.04,
2
dT = 0.1, κ = 0.01, α = 1, ωvk = k2βk
+F with β = 8.91 and F = 16, γvk = dv + νk ,
with dv = 0.032 and ν = 0.002 and the turbulent energy spectrum for the waves is
given by

100, for |k| ≤ 5
−θv
V ar(vk ) =
(6.2)
100 k
, for |k| > 5.
5
Note that as we increase the amplitude of the cross-sweep, the intermittency becomes stronger. Also note that in Fig. 7 where the spatial PDF is hardly intermittent, the PDFs of the smaller scale Fourier modes have significant intermittency;
for AU = 10, both the largest scale Fourier mode and the spatial PDF have increased intermittency while for AU = 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 AU = 10 with tracer PDFs depicted in Fig. 8 and include randomness in the
jet with σU = 2 while keeping the rest of the parameters the same. Comparing Fig.
10 with Fig. 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 (section 2) and developed (section 2, 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 nonlocal in both space and time (section 4), theory and simple numerics
for the tracer variance spectrum (section 5), and tracer PDF intermittency (section
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 (Gershgorin and Majda, 2010b; Majda and Gershgorin, 2010) and for contemporary issues of real time filtering from sparse noisy
observations (Gershgorin and Majda, 2008, 2010a,c; Majda et al., 2010; Gershgorin
et al., 2010a,b; Harlim and Majda, 2010).
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 section 2, 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 (section 5). It is worth mentioning here that for
simpler models of tracer pdf intermittency involving random uniform shear flows
Article submitted to Royal Society
32Andrew J. Majda Corresponding author address: Andrew J. Majda, Department of Mathematics, and Center
r=1
0
r=50
10
0
Spectrum of Tk
Spectrum of Tk
10
−5
10
numerical
−5
10
~k−5/3 WN limit
−3
~k
−10
10
0
2
0
10
2
10
10
k
k
r=1000
r=1000
0
Spectrum of Tk
Spectrum of Tk
10
10
0
10
−5
10
0
2
10
10
k
0
2
10
10
k
Figure 1. Variance spectrum of the tracer in the approach to the white noise limit of the
shear flow with r = 1, 50, 1000, 10000 in (5.6). The dashed line shows the theoretical
prediction of the tracer spectrum slope according to (5.5)
and a decaying tracer in all of space, rigorous mathematical results on intermittent
fat tails have been established in the literature (Majda, 1993b; McLaughlin and
Majda, 1996; Bronski and McLaughlin, 2000a,b; Vanden-Eijnden, 2001).
Acknowledgment
The research of A.J. Majda is partially supported by National Science Foundation grant DMS-0456713, the office of Naval Research grants 25-74200-F6607, and
N00014-05-1-0164, and the Defense Advanced Projects Agency grant N0014-071-0750. Boris Gershgorin is supported as a postdoctoral fellow through the same
agencies.
Article submitted to Royal Society
Elementary models for turbulent diffusion with complex physical features: eddy diffusivity, spectrum, and intermit
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Article submitted to Royal Society
36Andrew J. Majda Corresponding author address: Andrew J. Majda, Department of Mathematics, and Center
p(T )
p(T )
1
2
0
0
10
10
−2
−2
10
10
−4
10
−1
−4
−0.5
0
0.5
1
10
−0.5
0
p(T )
p(T )
3
4
0
0
10
10
−2
−2
10
10
−4
10
−0.4
0.5
−4
−0.2
0
0.2
0.4
10
−0.4
−0.2
0
0.2
0.4
P(T(x))
1
10
0
10
−1
10
−2
10
−3
10
−4
10
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
Figure 2. PDF of the tracer (solid line) in Regime B) from section 6, the cross-sweep is
random with the noise strength σU = 1: the upper four panels in Fourier space for the first
four spatial wave numbers of the tracer and the lower panel in physical space; the dashed
line shows Gaussian distribution with the same mean and variance. Note the logarithmic
scale ofsubmitted
the y-axis.
Article
to Royal Society
0.15
Elementary models for turbulent diffusion with complex physical features: eddy diffusivity, spectrum, and intermit
p(T )
p(T )
1
2
0
0
10
10
−2
−2
10
10
−4
10
−2
−4
−1
0
1
10
2
−1
−0.5
p(T )
1
0.5
1
4
0
0
10
10
−2
−2
10
10
−4
−1
0.5
p(T )
3
10
0
−4
−0.5
0
0.5
10
1
−1
−0.5
0
P(T(x))
1
10
0
10
−1
10
−2
10
−3
10
−4
10
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
Figure 3. PDF of the tracer (solid line) in Regime B) from section 6, the cross-sweep is
random with the noise strength σU = 2: the upper four panels in Fourier space for the first
four spatial wave numbers of the tracer and the lower panel in physical space; the dashed
line shows Gaussian distribution with the same mean and variance. Note the logarithmic
scale ofsubmitted
the y-axis.
Article
to Royal Society
0.25
38Andrew J. Majda Corresponding author address: Andrew J. Majda, Department of Mathematics, and Center
p(T )
p(T )
1
2
0
0
10
10
−2
−2
10
10
−4
10
−10
−4
−5
0
5
10
10
−5
0
p(T3)
p(T4)
0
0
10
10
−2
−2
10
10
−4
10
−4
5
−4
−2
0
2
10
4
−4
−2
0
2
4
P(T(x))
1
10
0
10
−1
10
−2
10
−3
10
−4
10
−1.5
−1
−0.5
0
0.5
1
Figure 4. PDF of the tracer (solid line) in Regime B) from section 6, the cross-sweep is
random with the noise strength σU = 4: the upper four panels in Fourier space for the first
four spatial wave numbers of the tracer and the lower panel in physical space; the dashed
line shows Gaussian distribution with the same mean and variance. Note the logarithmic
Article
to Royal Society
scale ofsubmitted
the y-axis.
1.5
Elementary models for turbulent diffusion with complex physical features: eddy diffusivity, spectrum, and intermit
p(T )
p(T )
1
2
0
0
10
10
−2
−2
10
10
−4
10
−10
−4
−5
0
5
10
10
−4
−2
p(T3)
0
4
6
0
10
−2
−2
10
10
−4
−4
2
p(T4)
10
10
0
−4
−2
0
2
10
4
−4
−2
0
2
4
P(T(x))
1
10
0
10
−1
10
−2
10
−3
10
−4
10
−1.5
−1
−0.5
0
0.5
1
Figure 5. PDF of the tracer (solid line) in Regime B) from section 6, the cross-sweep is
random with the noise strength σU = 8: the upper four panels in Fourier space for the first
four spatial wave numbers of the tracer and the lower panel in physical space; the dashed
line shows Gaussian distribution with the same mean and variance. Note the logarithmic
Article
to Royal Society
scale ofsubmitted
the y-axis.
1.5
40Andrew J. Majda Corresponding author address: Andrew J. Majda, Department of Mathematics, and Center
r=50
r=1
2
10
0
0
pdf(T(x))
pdf(T(x))
10
10
−2
10
−2
10
−0.4
p(T(x))
Gaussian
−0.2
0
−4
0.2
10
0.4
r=1000
0
−1
0
1
2
r=10000
0
10
10
−2
pdf(T(x))
pdf(T(x))
−2
10
−4
−2
10
−4
10
10
−2
0
2
−2
0
Figure 6. PDF of the tracer in the approach to the white noise limit of the shear flow
with r = 1, 50, 1000, 10000 in (5.6). The dashed line shows Gaussian distribution with
the same mean and variance. Note the logarithmic scale of the y-axis.
Article submitted to Royal Society
2
Elementary models for turbulent diffusion with complex physical features: eddy diffusivity, spectrum, and intermit
p(T )
p(T )
1
0
10
10
−2
−2
10
10
−4
−4
10
10
−6
10
−100
2
0
−6
−50
0
50
10
−100
100
−50
p(T )
100
20
40
4
0
10
10
−2
−2
10
10
−4
−4
10
10
−6
10
−100
50
p(T )
3
0
0
−6
−50
0
50
10
−40
100
−20
0
P(T(x))
0
10
−1
10
−2
10
−3
10
−4
10
−5
10
−6
10
−7
10
−40
−30
−20
−10
0
10
20
30
Figure 7. PDF of the tracer (solid line) in Regime A) from section 6, the cross-sweep is
deterministic and given by (2.21) with fU (t) = sin π3 t : the upper four panels in Fourier
space for the first four spatial wave numbers of the tracer and the lower panel in physical
space; the dashed line shows Gaussian distribution with the same mean and variance. Note
Article
submitted scale
to Royal
Society
the
logarithmic
of the
y-axis.
40
42Andrew J. Majda Corresponding author address: Andrew J. Majda, Department of Mathematics, and Center
p(T1)
0
10
−40
p(T2)
0
10
−20
0
20
40
−20
−10
0
p(T )
10
20
4
0
10
−20
20
p(T )
3
0
10
10
−10
0
10
20
−20
−10
0
P(T(x))
1
10
0
10
−1
10
−2
10
−3
10
−4
10
−8
−6
−4
−2
0
2
4
6
Figure 8. PDF of the tracer (solid line) in Regime A) from section 6, the cross-sweep
is deterministic and given by (2.21) with fU (t) = 10 sin π3 t : the upper four panels in
Fourier space for the first four spatial wave numbers of the tracer and the lower panel
in physical space; the dashed line shows Gaussian distribution with the same mean and
Article submitted to Royal Society
variance.
Note the logarithmic scale of the y-axis.
8
Elementary models for turbulent diffusion with complex physical features: eddy diffusivity, spectrum, and intermit
p(T )
p(T )
1
0
10
−10
2
0
10
−5
0
5
10
−5
p(T3)
0
0
10
p(T4)
0
10
5
10
−5
0
5
10
−10
−5
0
5
10
P(T(x))
1
10
0
10
−1
10
−2
10
−3
10
−4
10
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Figure 9. PDF of the tracer (solid line) in Regime A) fromsection 6, the cross-sweep
is deterministic and given by (2.21) with fU (t) = 1000 sin π3 t : the upper four panels in
Fourier space for the first four spatial wave numbers of the tracer and the lower panel
in physical space; the dashed line shows Gaussian distribution with the same mean and
Article submitted
to logarithmic
Royal Societyscale of the y-axis.
variance.
Note the
2.5
44Andrew J. Majda Corresponding author address: Andrew J. Majda, Department of Mathematics, and Center
p(T1)
0
10
10
−2
−2
10
10
−4
−4
10
10
−6
10
−100
p(T2)
0
−6
−50
0
10
−40
50
−20
p(T )
40
20
40
4
0
10
10
−2
−2
10
10
−4
−4
10
10
−6
10
−40
20
p(T )
3
0
0
−6
−20
0
20
10
−40
40
−20
0
P(T(x))
0
10
−1
10
−2
10
−3
10
−4
10
−15
−10
−5
0
5
10
Figure 10. PDF of the tracer (solid line) in a scenario that combines both Regime A) and
Regime B) from section 6, the random part of the cross-sweep has noise strength σU = 2
and the deterministic part of the cross-sweep is given by (2.21) with fU (t) = 10 sin π3 t .
Note the logarithmic scale of the y-axis.
Article submitted to Royal Society
15