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rsta.royalsocietypublishing.org
Elementary models for
turbulent diffusion with
complex physical features:
eddy diffusivity, spectrum
and intermittency
Andrew J. Majda and Boris Gershgorin
Research
Cite this article: Majda AJ, Gershgorin B.
2013 Elementary models for turbulent
diffusion with complex physical features: eddy
diffusivity, spectrum and intermittency. Phil
Trans R Soc A 371: 20120184.
http://dx.doi.org/10.1098/rsta.2012.0184
One contribution of 13 to a Theme Issue
‘Turbulent mixing and beyond:
non-equilibrium processes from atomistic
to astrophysical scales I’.
Subject Areas:
applied mathematics, atmospheric science,
geophysics
Keywords:
turbulent diffusion, eddy diffusivity,
intermittency, exactly solvable model,
white noise limit
Author for correspondence:
Andrew J. Majda
e-mail: [email protected]
Department of Mathematics, Center for Atmosphere Ocean Science,
Courant Institute, New York University, 251 Mercer Street, New York,
NY 10012, USA
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 realtime 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
∂T
+ v · ∇T = κT,
∂t
(1.1)
where κ > 0 is molecular diffusion and the velocity
field v is incompressible, div v = 0. For simplicity of
c 2012 The Author(s) Published by the Royal Society. All rights reserved.
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(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
⎫
(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 [20,22,23].
⎪
⎪
⎪
⎪
⎪
⎪
(B) The nature of the sustained turbulent spectrum for scalar variance
⎪
⎬
with a background gradient for the tracer [30].
⎪
⎪
⎪
⎪
(C) Fat tail PDFs for anthropogenic and natural tracers with highly
⎪
⎪
⎪
intermittent exponential tails in observations of the present climate[5]. ⎪
⎪
⎪
⎪
⎪
⎪
⎪
(D) Eddy diffusivity approximations for tracers in climate change
⎪
⎪
⎭
science [3,4,24].
(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
T(x, t) = T (x, t) + αy,
(1.4)
∂T
∂T
∂T
+ U(t)
+ v(x, t)
= κT − αv(x, t).
∂t
∂x
∂y
(1.5)
so that with (1.1) and (1.2), T satisfies
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 (t),
(1.6)
......................................................
v(x, t) = (U(t), v(x, t)),
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exposition, we assume here that x = (x, y) is two-dimensional. When v(x, t) is a turbulent velocity
field, the statistical properties of solutions of (1.1) such as their large-scale effective diffusivity,
energy spectrum and probability density function (PDF) are all important in applications. These
range from, for example, the spread of pollutants or hazardous plumes in 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]
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∂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 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 U(t) = Ū(t), with
Ū(t) 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
⎫
Ū(t) > 0 ⎬
(1.9)
and
Ū2 − Var(U (t)) > 0,⎭
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 βplane 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.
......................................................
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 (x, t)
depend only on the x variable alone and dropping the prime results in the simplified version of
(1.5) given by
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where Ū(t) 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)
∂
∂v(x, t)
+P
, U(t) v(x, t) = Fv (x, t).
(1.7)
∂t
∂x
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(a) Physical motivation of the master model
The equations of β-plane QG flow [33,34] involve a stream function
with velocity
Ψ = −U(t)y + ψ(x, t),
(2.1)
∂Ψ ∂Ψ T
,
,
v= −
∂y ∂x
(2.2)
Q = FU(t)y + ψ − Fy + βy,
(2.3)
and potential vorticity
linked by the conservation of potential vorticity
∂Q
+ v · ∇Q = F (x, t).
∂t
(2.4)
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)–(2.3) consisting of
a zonal jet U and one-dimensional Rossby waves with the form compatible with (1.2) given by
Ψ = −U(t)y + ψ(x, t)
(2.5)
and
F (x, t) = FyFU (U, t) + D()ψ + Fq (x, t).
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)
∂q
∂ψ
dq
+ U(t)
+ (FU(t) + β)
= D()ψ + Fq (x, t).
dt
∂x
∂x
(2.7)
and for q = ψxx − Fψ
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 2π -periodic which is used as the unit length. The velocity v is recovered from the
two identities
(2.8)
v = ψx and q = ψxx − Fψ,
through the non-local pseudo-differential operator
∂
q
v=R
∂x
and R =
∂2
−F
∂x2
−1
∂
.
∂x
(2.9)
The symbol of R(∂/∂x) at a given spatial wavenumber is R = −ik/(k2 + F). To get the equation for
v, we apply R(∂/∂x) to (2.7) and use (2.9) to obtain the dynamics
∂v
∂
∂v
∂ 2v
+ U(t)
+ (FU(t) + β)R
v = −dv v + ν 2 + fv (x, t) + σv Ẇv (t).
(2.10)
∂t
∂x
∂x
∂x
......................................................
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.
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2. Elementary models for turbulent tracers: physical motivation and exact
statistics for the mean and variance
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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 FU (U, t) in (2.6) has the special form of deterministic forcing fU (t),
damping −γU and white noise forcing σU Ẇ(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)
∂
(2.12)
+P
, U(t) v(x, t) = Fv (x, t),
∂t
∂x
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
P
∂
∂
∂2
∂
, U(t) = U(t)
+ (FU(t) + β)R
+ dv − ν 2 .
∂x
∂x
∂x
∂x
(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, 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 + νk2 , ωvk = −ck, where ν is the
flow viscosity [26].
2. Uncorrelated Rossby waves: γvk = ν(k2 + F), ωvk = βk/(k2 + F), by directly plugging in
the dispersion relation for the Rossby waves, where ν denotes the large-scale selective
damping diffusivity, say eddy diffusivity. Here, F = L−2
R and LR is the Rossby deformation
radius, β is the tangent approximation to the local Coriolis forcing. This version of
the master model was introduced and used recently as a test model in quantifying
uncertainty in climate change science [24] together with the tracer equation in (1.8).
......................................................
(b) Velocity field in the master model
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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 [35], which is readily represented
below by Fourier series [25]. The natural dissipative mechanisms for v are a combination of
Ekmann damping −dv v and a small-scale frictional viscosity ν(∂ 2 v/∂x2 ) [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).
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3. Correlated Rossby waves (a generalization of case 2 developed in (2.10)):
2
6
ωvk = ak U(t) + bk ,
F
ak = k 2 − 1
k
(2.17)
(2.18)
βk
.
k2 + F
(2.19)
U(t) = Ū(t) + U (t),
(2.20)
bk =
and
(c) Statistics of the velocity field
The solution for the cross-sweep becomes
where
Ū(t) =
and
t
t0
GU (s, t)fU (s)ds,
(2.21)
U (t) = GU (t0 , t)U0 + UW (t)
t
UW (t) = σU
GU (s, t)dWU (s),
(2.22)
(2.23)
t0
and we use the shortcut notation for the initial condition, U0 = U(t0 ), and for 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
t
U(t) = GU (t0 , t)U0 + GU (s, t)fU (s)ds
(2.25)
t0
and
Var(U(t)) = G2U (t0 , t)Var(U0 ) +
σU2
(1 − G2U (t0 , t)).
2γU
(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:
zeix = (z + i Cov(z, x)) eix−(1/2)Var(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
t
t
vk (t) = Gvk (t0 , t)vk,0 + Gvk (s, t)fvk (s)ds + σvk
Gvk (s, t)dWvk (s),
(2.28)
t0
t0
where vk,0 = vk (t0 ) and Green’s function for the shear flow is defined as
Gvk (s, t) = e−γvk (t−s)+iJk (s,t)
and
Jk (s, t) =
t
s
ωvk (s )ds .
(2.29)
(2.30)
......................................................
(2.16)
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γvk = dv + νk ,
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For the case of correlated flow, we compute further
LD (s, t) =
LW (s, t) =
and
t
s
ωvk (s )ds = ak L(s, t) + bk (t − s),
(2.31)
U(s )ds = LD (s, t) + LW (s, t) + b0 (s, t)U0 ,
(2.32)
t s
s t0
t
s
b0 (s, t) = −
GU (r , s )fU (r )dr ds ,
UW (s )ds = σU
t s
s t0
(2.33)
GU (s , s )dWU (s )ds
1
(GU (t0 , t) − GU (t0 , s)).
γU
(2.34)
(2.35)
Next, we find the mean of vk
vk (t) = Gvk (t0 , t)vk,0 +
t
t0
Gvk (s, t)fvk (s)ds.
(2.36)
Note that 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
[26,29,31,32] to find
Gvk (t0 , t)vk,0 = (vk,0 + iak b0 (t0 , t) Cov(vk,0 , U0 ))Gvk (t0 , t),
−γvk (t−s)+iJk (s,t)−(1/2)Var(Jk (s,t))
Gvk (s, t) = e
,
(2.38)
Jk (s, t) = ak L(s, t) + bk (t − s),
(2.39)
Var(Jk (s, t)) = a2k Var(L(s, t)),
(2.40)
L(s, t) = LD (s, t) + b0 (s, t)U0 ,
(2.41)
Var(L(s, t)) = b20 (s, t)Var(U0 ) + Var(LW (s, t))
and
Var(LW (s, t)) =
σU2
(−2 + 2γU |t − s| + 2 e−γU |t−s|
2γU3
(2.37)
(2.42)
+ 2GU (2t0 , s + t)
− G2U (t0 , t) − G2U (t0 , s)).
(2.43)
(d) Passive tracer in a mean gradient and tracer statistics in the master model
For completeness, we repeat the equation for the dynamics of a passive tracer with a mean
gradient in the y-direction from (1.8) which is given by
∂T
∂2
∂T
+ U(t)
= κ 2 T − dT T − αv.
∂t
∂x
∂x
(2.44)
In Fourier space, this equation becomes
dTk
= (−γTk + iωTk )Tk − αvk ,
dt
(2.45)
γTk = dT + κk2
(2.46)
ωTk = −U(t)k.
(2.47)
where
and
......................................................
L(s, t) =
s
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Jk (s, t) =
t
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where Tk,0 = Tk (t0 ) and Green’s function for the tracer is given by
GTk (s, t) = e−γTk (t−s)−ikL(s,t)
and
L(s, t) =
t
s
U(s )ds .
(2.49)
(2.50)
Note that the shear flow vk (s) is governed by the dynamics discussed above in (2.13). The mean
of the tracer is given by
Tk (t) = GTk (t0 , t)Tk,0 − α
−α
t s
t0 t0
t
t0
GTk (s, t)Gvk (t0 , s)vk,0 ds
GTk (s, t)Gvk (r, s)fvk (r)dr ds,
(2.51)
where GTk (s, t)Gvk (r, s) is a characteristic function of a Gaussian and can be computed
analytically in a similar fashion to Gvk from (2.38)
GTk (s, t)Gvk (r, s) = e−γvk (r−s)−γTk (t−s)+i(Jk (s,t)−kL(s,t))−(1/2)Var(Jk (s,t)−kL(s,t)) .
(2.52)
In the statistically steady state, the mean of the tracer becomes
Tk (t) = −α
t
s
−∞ −∞
GTk (s, t)Gvk (r, s)fvk (r)dr ds.
(2.53)
And the covariance is given by
Cov(Tj (t), Tk (t)) = α 2
t
t
−∞ −∞
(GTj (s, t)G∗Tk (s , t)vj (s)vk∗ (s )
− GTj (s, t)vj (s)GTk (s , t)vk (s )∗ )ds ds ,
(2.54)
where
GTj (s, t)G∗Tk (s , t)vj (s)vk∗ (s )
s s
Gvj (r, s)G∗vk (r , s )GTj (s, t)G∗Tk (s , t)fvj (r)fv∗k (r )dr dr
=
−∞ −∞
j
+ σv2k δk
min(s,s )
−∞
GTk (s, t)G∗Tk (s , t)Gvk (r, s)G∗vk (r, s ) dr.
(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.
......................................................
t0
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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
t
(2.48)
Tk (t) = GTk (t0 , t)Tk,0 − α GTk (s, t)vk (s)ds,
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(a) Uncorrelated velocity field
9
and
dU(t)
= −γU U(t) + fU (t) + σU ẆU (t),
dt
dvk (t)
= (−γvk + iωvk )vk (t) + fvk (t) + σvk Ẇvk (t)
dt
dTk (t)
= (−γTk + iωTk )Tk (t) − αvk (t),
dt
(3.2)
(3.3)
(3.4)
where for the example of uncorrelated Rossby waves
βk
,
+ Fs
(3.5)
ωTk = −kU(t),
(3.6)
γvk = ν(k2 + Fs )
(3.7)
γTk = dT + κk2 .
(3.8)
ω vk =
and
k2
Note that other forms of the dispersion relation for the waves can be studied since the formulas
are general [26].
......................................................
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:
rsta.royalsocietypublishing.org Phil Trans R Soc A 371: 20120184
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
ωvk . In physical space, the shear flow is given by
∂
∂v(x, t)
+P
v(x, t) = Fv (x, t),
(3.1)
∂t
∂x
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−α
t0
t0
GTk (s, t)V̄k (t0 , s)ds,
where
s
V̄k (t0 , s) =
(3.9)
Gvk (r, s)fvk (r)dr.
t0
(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 ak = 0 in (2.53)
t
Tk (t) = −α
GTk (s, t)V̄k (s)ds,
(3.11)
−∞
where
V̄k (s) =
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)
t t
Cov(Tj (t), Tk (t)) = α 2
(GTj (s, t)G∗Tk (s , t)vj (s)vk∗ (s )
−∞ −∞
− GTj (s, t)vj (s)G∗Tk (s , t)vk (s )∗ )ds ds ,
where
vj (s)vk∗ (s ) =
s
s
−∞ −∞
j
+ σv2k δk
(3.13)
Gvj (r, s)G∗vk (r , s )fvj (r)fv∗k (r )dr dr
min(s,s )
−∞
Gvk (r, s)G∗vk (r, s )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
t t
σ2
2
Cov(Tk (t), Tl (t)) = α 2 k δkl
e−(γ +κk )(2t−s−r) e−ikŪ(r−s) e−γk |s−r|+iωk (s−r)
2γk
−∞ −∞
× e−(k
2
/2)(Var(J(s,t))+Var(J(r,t))−2Cov(J(s,t),J(r,t)))
ds dr,
(3.15)
where for s > r
Cov(J(s, t), J(r, t)) = Var(J(s, t)) −
σU2
2γU3
(e−γU (t−s) − e−γU (t−r) − 1 + e−γU (s−r) ).
From equation (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 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 vk is
constant in time [29].
......................................................
t
10
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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
t
Tk (t) = GTk (t0 , t)Tk,0 − α Gvk (t0 , s)GTk (s, t)vk,0 ds
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(b) White noise limits of the master model
11
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
(t),
vk (t) = v̄k|U (t) + vk|U
(3.16)
where v̄k|U (t) is a conditional mean of the shear flow for given realizations of the cross-sweep,
(t) denotes
i.e. the average over the noise of the waves given by Ẇvk ; on the other hand, vk|U
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),
(3.17)
dt
and of the fluctuations around this mean
(t)
dvk|U
dt
= (−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) =
(t) from (3.18). To achieve this,
v̄k (t). First, we define the white noise limit for the fluctuations vk|U
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 vk (t) becomes
(t + τ )vk|U
(t)∗ =
Corrvk|U (τ ) = vk|U
σv2k
2γvk
e−γvk τ eiJk (t,t+τ ) .
(3.19)
Therefore, if we keep the following ratio fixed:
ηk =
σv k
= const.,
γvk
(3.20)
then the absolute value of the autocorrelation function formally approaches a delta-function
|Corrvk|U (τ )| →
ηk2
2
δ0 (τ ),
(3.21)
or, equivalently, the fluctuations vk (t) approach the white noise
(t) → ηk Ẇvk (t).
vk|U
(3.22)
Second, 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 f̄vk (t),
(3.23)
where f̄k (t) is independent of γvk . Then, for the value of v̄k|U (t) given by (2.28) without the last term
t
v̄k|U (t) = Gvk (t0 , t)v̄k|U (t0 ) + Gvk (s, t)fvk (s)ds,
(3.24)
t0
......................................................
(i) White noise limit for the shear flow
rsta.royalsocietypublishing.org Phil Trans R Soc A 371: 20120184
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].
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we find, using (3.23) in (3.24), the white noise limit as γvk → +∞
12
(3.25)
fk (t) → γvk f̄vk (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)
and
Then, in this white noise limit the master model is described by the equations
dU(t)
= −γU U(t) + fU (t) + σU ẆU
dt
and
dTk
= (−γTk + iωTk )Tk − α(f̄vk (t) + ηk Ẇvk ).
dt
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 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
t
t
(3.29)
Tk (t) = GTk (t0 , t)Tk,0 − α GTk (s, t)f̄vk (s)ds − αηk GTk (s, t)dWvk (s).
t0
t0
The mean of the tracer becomes
Tk (t) = GTk (t0 , t)Tk,0 − α
t
t0
GTk (s, t)f̄vk (s)ds.
In the statistically steady state, the mean simplifies to
t
GTk (s, t)f̄vk (s)ds,
Tk (t) = −α
−∞
(3.30)
(3.31)
where we took the limit t0 → −∞. 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)
t s
Gvk (r, s)GTk (s, t)γvk f̄vk (r)dr ds
Tk (t) = −α
−∞ −∞
→ −α
= −α
t
t
s
−∞ −∞
−∞
δs (r)GTk (s, t)f̄vk (r)dr ds
GTk (s, t)f̄vk (s)ds.
(3.32)
Next, we find the covariance in the white noise limit in the statistically steady state
t t
Cov(Tj , Tk ) = α 2
(GTj (s, t)G∗Tk (s , t) − GTj (s, t)G∗Tk (s , t))
−∞ −∞
j
× f̄vj (s)f̄v∗k (s )ds ds +
δk
2γTk
α 2 ηk2 .
(3.33)
......................................................
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
rsta.royalsocietypublishing.org Phil Trans R Soc A 371: 20120184
v̄k|U (t) → f̄vk (t).
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Similarly, we can find this expression by taking formally the white noise limit in equation (2.54)
directly. First, we find
j
+ σv2k δk
→
s
min(s,s )
−∞
s
−∞ −∞
j
Gvk (r, s)G∗vk (r, s )GTk (s, t)G∗Tk (s , t) dr
δs (r)δs (r )GTj (s, t)G∗Tk (s , t)f̄vj (r)f̄v∗k (r )dr dr
+ ηk2 δk GTk (s, t)G∗Tk (s , t)
min(s,s )
−∞
δs (r)δs (r)dr
= GTj (s, t)G∗Tk (s , t)f̄vj (s)f̄v∗k (s ) + ηk2 δk δs (s ) e−2γTk (t−s) .
j
(3.34)
Now, the covariance becomes
Cov(Tj , Tk )
t t
= α2
−∞ −∞
→ α2
t
t
−∞ −∞
j
+ α 2 ηk2 δk
= α2
t
(GTj (s, t)G∗Tk (s , t)vj (s)vk∗ (s ) − GTj (s, t)vj (s)GTk (s , t)vk (s )∗ )ds ds
t
t
(GTj (s, t)G∗Tk (s , t) − GTj (s, t)G∗Tk (s , t))f̄vj (s)f̄v∗k (s )ds ds
t
−∞ −∞
−∞ −∞
e−2γTk (t−s) δs (s )ds ds
(GTj (s, t)G∗Tk (s , t) − GTj (s, t)G∗Tk (s , t))f̄vj (s)f̄v∗k (s )ds ds
j
+
δk
2γTk
α 2 ηk2 .
(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
⎫
du1
⎪
⎪
= −γ1 u1 + f1 (t) + σ1 Ẇ1
⎬
dt
(3.36)
⎪
du2
⎪
⎭
= (−γ2 + i(ω0 + a0 u1 ))u2 + f2 (t) + σ2 Ẇ2 ,
and
dt
whereas the white noise limit of the master model is given by
dU (t)
= −γU U (t) + fU (t) + σU ẆU
dt
and
dTk
= (−γTk − ik(U0 + U (t)))Tk − α v̄k (t) + αηẆk ,
dt
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 (t). The forcing of the fluctuation
of the jet is denoted as fU (t). Here, the mean shear flow −α v̄k (t) plays the role of the forcing of
the tracer through the mean gradient. Suppose that fU (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
......................................................
−∞ −∞
rsta.royalsocietypublishing.org Phil Trans R Soc A 371: 20120184
GTj (s, t)G∗Tk (s , t)vj (s)vk∗ (s )
s s
Gvj (r, s)G∗vk (r , s )GTj (s, t)G∗Tk (s , t)f̄vj (r)f̄v∗k (r )dr dr
= γvk γvj
13
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As established in the study of Gershgorin & Majda [29], 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 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)
dU(t)
= −γ U(t) + fU (t) + σU ẆU
dt
when
γU → +∞,
σU → +∞,
σU
= ηU = const.
γU
and
(3.38)
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎪
fU (t)
⎪
is independent of γU .⎭
γU
(3.39)
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 stochastic differential equation (SDE) for the Fourier modes of the tracer:
dTk (t)
= (−γTk − ikŪ(t))Tk − α f̄vk (t) − ikηU Tk (t) ◦ ẆU (t) − αηk Ẇvk .
dt
(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 T(t) ◦ ẆU = 0 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
......................................................
— 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?
14
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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.
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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
s
U(s )ds
→ ḠTk (s, t)e−ik(WU (t)−WU (s)) = ḠTk (s, t) e−(η
2
/2)(t−s)
,
(3.42)
where the deterministic part of Green’s function for the tracer is given by
ḠTk (s, t) = e−γTk (t−s)−ik
t
s
Ū(s )ds
.
(3.43)
Now, the mean of the tracer becomes a white noise limit of (3.31)
Tk (t) = −α
t
e−(ηU /2)k
2
−∞
2
(t−s)
ḠTk (s, t)f̄vk (s)ds.
(3.44)
2 /2 that comes from the
Here we note the correction to the diffusivity (the eddy diffusivity) ηU
diffusion-induced advection term that appears after rewriting the SDE (3.41) in the Ito form [35]
k2 2
dTk (t)
= −γTk − ikŪ(t) − ηU Tk − α f̄vk (t) − ikηU Tk (t)ẆU (t) − αηk Ẇvk .
dt
2
(3.45)
Here, the eddy diffusivity is local in space and time and is equal to
2
σ2
ηU
∼ U2 .
2
2γU
κe =
(3.46)
Next, the covariance is a white noise limit of (3.33)
Cov(Tj (t), Tk (t)) = α
2
t
t
−∞ −∞
× e−(ηU /2)(j
2
2
ḠTj (s, t)Ḡ∗Tk (s , t)f̄vj (s)f̄v∗k (s )
(t−s)+k2 (t−s ))
(eηU kj(t−max(s,s )) − 1)ds ds
2
j
+
δk
2γTk
α 2 ηk2 .
(3.47)
Moreover, the white noise limit in U(t) of the second-order statistics of Tk (t) can be found for a
general case of the shear flow, not just in the case of the white noise limit of vk (t). The mean is
given by the same equation (3.44). The covariance has the form
Cov(Tj (t), Tk (t)) = α 2
t
t
−∞ −∞
ḠTj (s, t)Ḡ∗Tk (s , t) e−(ηU /2)(j
2
2
(t−s)+k2 (t−s ))
× (vj (s)vk∗ (s )eηU kj(t−max(s,s )) − vj (s)vk∗ (s ))ds ds .
2
(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
......................................................
t
rsta.royalsocietypublishing.org Phil Trans R Soc A 371: 20120184
GTk (s, t) = e−γTk (t−s)−ik
15
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ωvk (t) = ωvk (t) + ωv k (t)
and
vk (t) = vk (t) + vk (t),
and find
dvk (t)
= (−γvk vk (t) + iωvk (t))vk (t) + iωv k (t)vk (t) + 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 §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
dTk (t)
= −γTk Tk (t) + iωTk (t)Tk (t) − αvk (t),
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) = ωTk (t) + ωT k (t)
and
Tk (t) = Tk (t) + Tk (t),
and find
Tk (t)
= (−γTk Tk (t) + iωTk (t))Tk (t) + iωT k (t)Tk (t) − αvk (t).
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, equation (2.36), to find eddy diffusivity
approximation of the shear flow
t
(4.5)
vk (t) = Gvk (t0 , t)vk,0 + Gvk (s, t)fvk (s)ds,
t0
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)
Gvk (s, t) = e−γvk (t−s)+iJk (s,t)
and
Jk (s, t) =
t
s
ωvk (s )ds .
For the special case of atmospheric waves in the QG model, we have for example
t
F
βk
.
Jk (s, t) = k 2 − 1
U(s )ds + 2
k
k
+F
s
(4.6)
(4.7)
(4.8)
......................................................
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 ωvk (t)vk (t). We use the
decomposition of both ωvk (t) and vk (t) into their respective deterministic means and random
fluctuations around those means
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the issue of eddy diffusivity, we take the governing equation for the mean of the shear flow by
averaging (2.13)
dvk (t)
= −γvk vk (t) + iωvk (t)vk (t) + fvk (t).
(4.1)
dt
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We find the time derivative of the mean of vk
17
(4.9)
t0
where the eddy diffusivity in the general case becomes
κev (k, s, t) =
1 ∂
Var(Jk (s, t)).
2k2 ∂t
(4.10)
Note that we factored out the k2 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
Kev [vk,0 , fvk ] = κev (k, t0 , t)Gvk (t0 , t)vk,0 +
t
t0
κev (k, s, t)Gvk (s, t)fvk (s)ds,
(4.11)
so that the differential equation for the eddy diffusivity of the shear flow becomes
dvk (t)
= (−γvk + iωvk (t))vk (t) + fvk (t) − k2 Kev [vk,0 , fvk ].
dt
(4.12)
We compare this equation with the formula in (4.2) to find
iωv k (t)vk (t) = −k2 Kev [vk,0 , fvk ].
(4.13)
In physical space, the equation for the mean of the shear flow becomes
∂
∂2
∂
dv(x, t)
=P
, U(t) v(x, t) + 2 K̃ev
, t0 , t ,
dt
∂x
∂x
∂x
(4.14)
where K̃ is given by its Fourier symbol in (4.11). For the QG model, we find
t
2
∂
1 F
Var
.
−
1
U(s
)ds
2 k2
∂t
s
t
The time derivative of the variance of s U(s )ds is given by
κev (k, s, t) =
∂
Var
∂t
(4.15)
t
2
U(s )ds =
(GU (t0 , s) − GU (t0 , t))GU (s, t)Var(U0 )
γU
s
+
σU2
γU2
(1 − GU (s, t) − GU (2t0 , s + t) + G2U (t0 , t)2 ),
(4.16)
where we used (2.42) and GU is defined in (2.24). Next, we study the role of each component of
the eddy diffusivity.
(i) Unforced waves
First, we consider the case of unforced waves, i.e. fvk (t) ≡ 0. Then, we have
vk (t) = Gvk (t0 , t)vk,0 ,
(4.17)
and the differential equation for the mean becomes
dvk (t)
= (−γvk + iωvk (t))vk (t) − k2 κev (k, t0 , t)vk (t)
dt
(4.18)
......................................................
− k2 κev (k, t0 , t)Gvk (t0 , t)vk,0 t
− k2 κev (k, s, t)Gvk (s, t)fvk (s)ds,
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dvk (t)
= (−γvk + iωvk (t))vk (t) + fvk (t)
dt
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v
v
(s, t)κe,sp
(k).
κev (k, s, t) = κe,tm
Then, we find for the QG model
1 ∂
Var
2 ∂t
v
κe,tm
(s, t) =
and
v
κe,sp
(k) =
In physical space, we find
v
κ̃e,sp
∂
∂x
t
s
U(s )ds
(4.20)
2
F
−1 .
k2
(4.21)
(4.22)
2
∂ −2
= F −2 − 1 .
∂x
(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
v
(k) ≈ 1.
κe,sp
(4.24)
At wavelengths larger than LR , genuine spatially non-local effects persist. Moreover, if the crosssweep 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
v
κe,tm
(s, t) =
σU2
2γU2
.
(4.25)
Note that this is exactly the white noise limit in the cross-sweep for the eddy diffusivity from
§3b. In general, this example represents an interesting test case with eddy diffusivity, which is
non-local in time and space.
(ii) Waves in the statistically steady state
Next, we turn to the case of the statistically steady state with non-zero forcing for the waves,
fvk = 0, which induces a non-zero statistically steady mean, v̄k (t). Here the initial conditions for
vk are irrelevant but inhomogeneous forcing dominates. Next, we show a non-local 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
t
Gvk (s, t)fvk (s)ds.
(4.26)
vk (t) =
−∞
Now, the differential equation for the mean of vk becomes
dvk (t)
= (−γvk + iωvk )vk (t) + fvk (t)
dt
t
v
v
− k2 κe,sp
(k)
κe,tm
(s, t)Gvk (s, t)fvk (s)ds.
−∞
The integral in the last expression is a functional of the forcing of the shear flow
t
v
Hk [fvk ](t) =
κe,tm
(s, t)Gvk (s, t)fvk (s)ds.
−∞
(4.27)
(4.28)
......................................................
where κ̃ev 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 κev (k, s, t) into the
product of non-local temporal and non-local spatial parts
18
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with κev given explicitly by (4.10). For the QG model, the corresponding equation in physical
space becomes
∂
dv(x, t)
∂
∂2
(4.19)
=P
, U(t) v(x, t) + 2 κ̃ev
, t0 , t v(x, t),
dt
∂x
∂x
∂x
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(4.29)
In physical space, this equation becomes
∂
dv(x, t)
∂
∂2 v
H̃k (x, t).
=P
, U(t) v(x, t) + 2 κ̃e,sp
dt
∂x
∂x
∂x
(4.30)
Note that, in the statistically steady state, the temporal part of the eddy diffusivity becomes
v
=
κe,tm
σU2
2γU2
(1 − e−γU (t−s) ).
(4.31)
This part of the eddy diffusivity introduces temporal memory through (4.28) into the equation in
v (∂/∂x) makes the contribution of the
(4.30) and makes it non-local in time, and again the term κe,sp
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
Tk (t) = GTk (t0 , t)Tk,0 − α
−α
t s
t0 t0
t
t0
GTk (s, t)Gvk (t0 , s)vk,0 ds
GTk (s, t)Gvk (r, s)fvk (r)dr ds.
(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)
dTk (t)
= (−γTk + iωTk )Tk (t) − αvk (t)
dt
− k2 κeT (k, t0 , t0 , t)GTk (t0 , t)Tk,0 t
2
T
+k α
κe (k, t0 , s, t)GTk (s, t)Gvk (t0 , s) ds vk,0 t0
+ k2 α
t s
t0 t0
κeT (k, r, s, t)GTk (s, t)Gvk (r, s)fvk (r)dr ds,
where
κeT (k, r, s, t) =
1 ∂
Var(L(s, t)) −
2 ∂t
F
∂
Cov(L(r, s), L(s, t)),
−
1
∂t
k2
(4.33)
(4.34)
......................................................
dvk (t)
v
= (−γvk + iωvk )vk (t) + fvk (t) − k2 κe,sp
(k)Hk [fvk ](t).
dt
19
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It is important to emphasize that Hk [fvk ] is not equal to the mean wave vk , but instead it is
obtained using convolution of the forcing with the same Green’s function Gvk as for vk but also
v (s, t), in (4.28), i.e. H [f ](t) carries
multiplied by the temporal part of the eddy diffusivity, κe,tm
k vk
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
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t0
−α
t s
t0 t0
κeT (k, r, s, t)GTk (s, t)Gvk (r, s)fvk (r)dr ds,
(4.35)
highlighting the three separate contributions. Then, the equation for the mean of the tracer takes
a simple form
dTk (t)
= (−γTk + iωTk )Tk (t) − αvk (t) − k2 KeT [vk,0 , Tk,0 , fvk ].
dt
(4.36)
By comparing this equation with the formula in (4.35), we find
iωT k (t)Tk (t) = KeT [vk,0 , Tk,0 , fvk ],
(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
∂T(x, t)
∂ 2 T(x, t)
dT(x, t)
+ U(t)
= (κ + κeT (t0 , t))
− dT T(x, t),
dt
∂x
∂x2
where
κeT (t0 , t) =
1 ∂
Var
2 ∂t
t
U(s)ds .
(4.38)
(4.39)
t0
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
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 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:
dTk (t)
= (−γTk + iωTk )Tk (t) − αvk (t)
dt
t
2
T
+k α
κe (k, t0 , s, t)GTk (s, t)Gvk (t0 , s) ds vk,0 ,
(4.40)
t0
where κeT 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.
......................................................
KeT [vk,0 , Tk,0 , fvk ] = κeT (k, t0 , t0 , t)GTk (t0 , t)Tk,0 t
−α
κeT (k, t0 , s, t)GTk (s, t)Gvk (t0 , s) ds vk,0 20
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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
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(iii) Statistically steady state
21
Now, we find the differential equation for the mean of the tracer which is a special case of (4.36)
dTk (t)
= (−γTk + iωTk )Tk (t) − αvk (t) − k2 KeT [fvk ](t),
dt
where the eddy diffusivity functional for the tracer has the following form:
t s
κeT (k, r, s, t)GTk (s, t)Gvk (r, s)fvk (r)dr ds,
KeT [fvk ](t) = −α
−∞ −∞
and the eddy diffusivity kernel in the statistically steady state is given by
σ2
F
κeT (k, r, s, t) = U2 (1 − e−γU (t−s) ) − 2 − 1 (e−γU (t−s) − e−γU (t−r) ) .
k
2γU
(4.42)
(4.43)
(4.44)
Note that KeT [fvk ](t) is a non-local linear functional of the history of the mean shear forcing, fvk (t).
In physical space, we have the following equation:
∂T(x, t)
∂ 2 T(x, t)
∂ 2 K̃eT [fv ](x, t)
dT(x, t)
+ Ū(t)
=κ
− dT T(x, t) +
− αv(x, t).
2
dt
∂x
∂x
∂x2
(4.45)
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
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
κeT =
σU2
2γU2
.
(4.46)
By comparing (4.43) with (4.41), we find that with the constant kernel κeT 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) =
σU2
2γU2
Tk (t).
(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:
∂T(x, t)
dT(x, t)
∂ 2 T(x, t)
+ Ū(t)
= (κ + κeT )
− dT T(x, t) − αv(x, t),
dt
∂x
∂x2
(4.48)
with a constant eddy diffusivity, κeT , given by (4.46). Note that here, unlike in the case with the
eddy diffusivity of the shear flow, the eddy diffusivity κeT 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 Tk (t) from (4.33) by noting that Green’s function for the velocity field in (2.29) becomes
......................................................
−∞ −∞
rsta.royalsocietypublishing.org Phil Trans R Soc A 371: 20120184
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 t0 → −∞
t s
GTk (s, t)Gvk (r, s)fvk (r)dr ds.
(4.41)
Tk (t) = −α
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deterministic and the eddy diffusivity functional from (4.35) becomes
22
−α
t
t
t0
t0
κeT (s, t)GTk (s, t)Gvk (t0 , s)ds vk,0 κeT (s, t)GTk (s, t)
s
t0
Gvk (r, s)fvk (r)dr ds.
(4.49)
Thus, the eddy diffusivity kernel is local in space and non-local in time
κeT (s, t) =
1 ∂
Var(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
κeT (s, t) ≈ keddy ,
(4.51)
∂Tk (t)
− iωTk Tk (t) ≈ −(κ + κeddy )k2 Tk (t) − γTk Tk (t) − αvk (t).
∂t
(4.52)
then (4.49) becomes
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 [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)
dTk (t)
= (−γTk + iωTk (t))Tk (t) − α f̄vk (t) − k2 κeT (t0 , t)GTk (t0 , t)Tk,0 dt
t
+ k2 α κeT (s, t)GTk (s, t)f̄vk (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
κeT (s, t) =
1 ∂
Var(L(s, t)).
2 ∂t
(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
......................................................
−α
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KeT [vk,0 , Tk,0 , fvk ] = κeT (t0 , t)GTk (t0 , t)Tk,0 Downloaded from http://rsta.royalsocietypublishing.org/ on June 17, 2017
(a) Spectrum of the tracer in the white noise limit of the shear waves in the master equation
Recall from §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 equation (3.33) with f̄vk = 0:
Var(Tk ) =
α 2 ηk2
2γTk
.
(5.2)
Also recall that ηk is a fixed ratio of σvk and γvk as they both go to infinity in the white noise
limit, ηk = σvk /γvk . Now formally in the white noise limit starting with the initial steady-state
velocity spectrum in (5.2), as shown in §3b, we have ηk2 = 2Ek /γvk and the limiting velocity field
converges to
(5.3)
vk (t) = ηk Ẇk (t),
yielding the steeper white noise limiting velocity spectrum
Var(vk ) =
ηk2
2
=
2Ek
.
γvk
(5.4)
Thus with (5.2) and (5.4) in the white noise limit, the tracer variance spectrum is given by the
exact formula
Var(Tk ) = α 2 (dT + κk2 )−1 Var(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 wavenumbers with κk2 1, the scalar dissipation regime, the
limiting tracer spectrum is steeper than the limiting velocity spectrum by (κk2 )−1 . On the other
hand, for a large inertial range of wavenumbers with small molecular diffusivity, so that κk2 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 /dT .
(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.
......................................................
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 §3b [2,9,13] and then check these spectral predictions in a family of instructive
numerical experiments.
23
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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 = Var(vk ),
⎪
⎪
⎬
(5.1)
what is the corresponding variance spectrum for the tracer,
⎪
⎪
⎭
Var(Tk ), in a statistically steady state with a mean tracer gradient?
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(a) 1
(b)
24
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spectrum of Tk
1
10–5
10–5
numerical
~k–5/3 WN limit
~k–3
10–10
spectrum of Tk
(c)
(d)
1
1
10–5
102
1
102
1
k
k
Figure 1. Variance spectrum of the tracer in the approach to the white noise (WN) limit of the shear flow with (a) r = 1,
(b) r = 50, (c) r = 1000 and (d) r = 10 000 in (5.6). The dashed line shows the theoretical prediction of the tracer spectrum
slope according to (5.5).
To mimic the white noise limit of the shear velocity field discussed in §3b, we first perform
an initial experiment with a prescribed energy spectrum for the shear waves, here chosen to be
the Kolmogorov spectrum, Var(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
σv k
by rσvk ,
γvk
by rγvk ,
as r → ∞
(5.6)
in a fashion consistent with the white noise limit discussed in §5a since ηk = σvk /γvk is held
constant with γvk → +∞. 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 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 σU2 /2γU = 2;
thus the physical requirement in (1.9) is satisfied. The β-plane Rossby dispersion relation ωvk =
βk/(k2 + F) is used with β = 8.91 and F = 16 while the values γvk = dv + νk2 are used for the
Rossby wave dissipation with the inertial range parameters dv = 0.032 for Ekmann friction and
Ek = |k|−5/3 for the velocity spectrum
ν = 10−8 for viscosity; these values together with imposing
√
in the initial simulation determine σvk via σvk = 2Ek dv + νk2 .
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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,
Ū(t) = AU sin(ωU t), 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 Ū(t) 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 U(t) = Ū(t) + U (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 yields a positive jet, i.e. Ū2 (t) − Var(U (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 [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 dT = 0.1, and κ = 0.001; as in §5 the β-plane Rossby
dispersion relation ωvk = βk/(k2 + F) is used for the random shear waves with the atmospheric
values β = 8.91 and F = 16 with the dissipation values γvk = dv + νk2 , with dv = 0.6 and ν = 0.1
and the turbulent energy spectrum
⎧
1
⎪
⎪
⎨2,
Var(vk ) =
1 k −θv
⎪
⎪
⎩
,
2 5
for |k| ≤ 5
(6.1)
for |k| > 5,
......................................................
6. Strongly intermittent probability density functions with fat exponential tails
25
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In figure 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 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 = 103 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 = 104 .
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.
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p(T1)
(a)
p(T2)
(b)
26
10–4
–1.0
–0.5
0
0.5
1.0
p(T3)
(c)
0
–0.5
0.5
p(T4)
(d)
1
10–2
10–4
–0.4
–0.2
0
0.2
0.4
–0.4
–0.2
0
0.2
0.4
P(T(x))
(e)
10
1
10–1
10–2
10–3
10–4
–0.20
–0.15
–0.10
–0.05
0
0.05
0.10
0.15
Figure 2. PDF of the tracer (solid line) in Regime B from §6, the cross-sweep is random with the noise strength σU = 1: (a–d)
in Fourier space for the first four spatial wavenumbers of the tracer and (e) in physical space; the dashed line shows Gaussian
distribution with the same mean and variance. Note the logarithmic scale of the y-axis.
where θv = − 35 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 Ū2 − Var(U ) =
60 > 0 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 σU = 1, 2, 4 and 8.
The PDFs for the tracer are clearly Gaussian for σU = 0 and are essentially Gaussian for σU = 1;
......................................................
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(a)
(b)
p(T1)
p(T2)
27
1
–1
0
1
2
–1.0
p(T3)
(c)
–0.5
0
0.5
1.0
0
0.5
1.0
0.15
0.20
p(T4)
(d)
1
10–2
10–4–1.0
–0.5
0
0.5
1.0
–1.0
–0.5
P(T(x))
(e)
10
1
10–1
10–2
10–3
10–4
–0.25
0.20
–0.15
–0.10
–0.05
0
0.05
0.10
0.25
Figure 3. PDF of the tracer (solid line) in Regime B from §6, the cross-sweep is random with the noise strength σU = 2: (a–d)
in Fourier space for the first four spatial wavenumbers of the tracer and (e) in physical space; the dashed line shows Gaussian
distribution with the same mean and variance. Note the logarithmic scale of the y-axis.
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 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, 103 , 104 . 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 = 103 , 104
......................................................
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10–2
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p(T1)
(a)
p(T2)
(b)
28
1
–5
0
5
–5
10
p(T3)
(c)
0
5
p(T4)
(d)
1
10–2
10–4
–4
–2
0
2
4
–4
–2
0
2
4
P(T(x))
(e)
10
1
10–1
10–2
10–3
10–4
–1.5
–1.0
–0.5
0
0.5
1.0
1.5
Figure 4. PDF of the tracer (solid line) in Regime B from §6, the cross-sweep is random with the noise strength σU = 4: (a–d)
in Fourier space for the first four spatial wavenumbers of the tracer and (e) in physical space; the dashed line shows Gaussian
distribution with the same mean and variance. Note the logarithmic scale of the y-axis.
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
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p(T1)
(a)
p(T2)
(b)
29
1
–5
0
5
10
–4
p(T3)
(c)
–2
0
2
4
6
p(T4)
(d)
1
10–2
10–4
–4
–2
0
2
4
–4
–2
0
2
4
P(T(x))
(e)
10
1
10–1
10–2
10–3
10–4
–1.5
–1.0
–0.5
0
0.5
1.0
1.5
Figure 5. PDF of the tracer (solid line) in Regime B from §6, the cross-sweep is random with the noise strength σU = 8: (a–d)
in Fourier space for the first four spatial wavenumbers of the tracer and (e) in physical space; the dashed line shows Gaussian
distribution with the same mean and variance. Note the logarithmic scale of the y-axis.
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, fU (t) = AU sin(ωU t), where AU = 1, 10, 1000 and ωU = π/3. Note
that the amplitude of the deterministic jet is proportional to AU . The other parameters had the
following values: γU = 0.04, dT = 0.1, κ = 0.01, α = 1, ωvk = βk/(k2 + F) with β = 8.91 and F = 16,
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–10
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(a)
102
(b)
30
PDF(T(x))
1
10−2
10−2
−0.4
P(T(x))
Gaussian
−0.2
0
0.2
0.4
(c)
PDF(T(x))
......................................................
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1
10−4
−2
1
(d)
1
10−2
10−2
10−4
10−4
−2
0
2
−1
−2
0
0
1
2
2
Figure 6. PDF of the tracer in the approach to the white noise limit of the shear flow with (a) r = 1, (b) r = 50, (c) r = 1000
and (d) r = 10 000 in (5.6). The dashed line shows Gaussian distribution with the same mean and variance. Note the logarithmic
scale of the y-axis.
γvk = dv + νk2 , with dv = 0.032 and ν = 0.002 and the turbulent energy spectrum for the waves is
given by
⎧
⎪
for |k| ≤ 5
⎪
⎨100, k −θv
Var(vk ) =
, for |k| > 5.
⎪100
⎪
⎩
5
(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 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 figure 8 and include randomness in the jet with σU = 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
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p(T1)
(a) 1
p(T2)
(b)
31
10−6
−100
(c)
−50
0
50
−100
100
p(T3)
−50
50
100
20
40
p(T4)
(d)
1
0
10−2
10−4
−6
10−100
(e)
−50
0
50
100
−40
−20
0
P(T(x))
1
10−1
10−2
10−3
10−4
10−5
10−6
10−7
−40
−30
−20
−10
0
10
20
30
40
Figure 7. PDF of the tracer (solid line) in Regime A from §6, the cross-sweep is deterministic and given by (2.21) with fU (t) =
sin((π/3)t): (a–d) in Fourier space for the first four spatial wavenumbers of the tracer and (e) in physical space; the dashed
line shows Gaussian distribution with the same mean and variance. Note the logarithmic scale of the y-axis.
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
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(a)
20
40
p(T3)
1
−20
0
−10
0
−20
−10
20
(e)
10
20
10
20
p(T4)
(d)
10
0
32
−20
−10
0
P(T(x))
10
1
10−1
10−2
10−3
10−4
−8
−6
−4
−2
0
2
4
6
8
Figure 8. PDF of the tracer (solid line) in Regime A from §6, the cross-sweep is deterministic and given by (2.21) with fU (t) =
10 sin((π/3)t): (a–d) in Fourier space for the first four spatial wavenumbers of the tracer and (e) in physical space; the dashed
line shows Gaussian distribution with the same mean and variance. Note the logarithmic scale of the y-axis.
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
......................................................
−20
p(T2)
(b)
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−40
(c)
p(T1)
1
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(a)
(b)
p(T1)
p(T2)
33
1
0
5
10
−5
p(T3)
(c)1
−5
0
0
−10
10
10
p(T4)
(d)
5
5
−5
0
5
10
1.5
2.0
2.5
P(T(x))
(e)
10
1
10−1
10−2
10−3
10−4
−2.5
−2.0
−1.5
−1.0
−0.5
0
0.5
1.0
Figure 9. PDF of the tracer (solid line) in Regime A from §6, the cross-sweep is deterministic and given by (2.21) with fU (t) =
1000 sin((π/3)t): (a–d) in Fourier space for the first four spatial wavenumbers of the tracer and (e) in physical space; the
dashed line shows Gaussian distribution with the same mean and variance. Note the logarithmic scale of the y-axis.
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].
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(a)
p(T1)
1
p(T2)
(b)
34
10–6
–100
–50
0
–40
50
p(T3)
(c) 1
–20
0
20
40
20
40
p(T4)
(d)
10–2
10–4
10–6
–40
–20
0
20
(e)
–40
40
–20
0
P(T(x))
1
10–1
10–2
10–3
10–4
–15
–10
–5
0
5
10
15
Figure 10. PDF of the tracer (solid line) in a scenario that combines both Regime A and B from §6, the random part of the crosssweep has noise strength σU = 2 and the deterministic part of the cross-sweep is given by (2.21) with fU (t) = 10 sin((π/3)t).
Note the logarithmic scale of the y-axis.
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.
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