Lagrangian Covariance Analysis of and x0

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Theoretical and Computational

Fluid Dynamics

Lagrangian Covariance Analysis of

β-Plane Turbulence

⋆

J.R. Ristorcelli

X Division, Los Alamos National Laboratory, University of California, Los Alamos, NM 87545, U.S.A.

A.C. Poje

Division of Applied Mathematics, Brown University, Providence, RI 02192, U.S.A.

Communicated by H.J.S. Fernando

Received 8 March 1999 and accepted 12 December 1999

Abstract. The effects of Rossby wave–turbulence interactions on particle dispersion are investigated in a Lagrangian analysis of the potential vorticity equation. The analysis produces several exact statistical results for fluid particle dispersion in barotropic turbulence on aβ-plane. In the inviscid problem the first integral time scale of the meridional velocity is found to be zero, as might occur in pure wave processes, and the meridional particle dispersion is bounded. The second integral time scale, which determines the magnitude of the bound, is shown to depend explicitly onβ, the enstrophy and the energy of the meridional velocity. Expressions relating the autocorrelation of the vorticity to the autocorrelation of the meridional velocity are derived and the Lagrangian integral time scale of the relative vorticity is diagnostically related to the meridional velocity correlation. The applicability of these predictions is verified in a series of numerical simulations. For a range ofβvalues, the meridional extent of quasisteady alternating zonally averaged jets occurring in the numerical solutions scales with a length scale given by the the standard deviation of the meridional particle dispersion.

1. Introduction

Homogeneous, barotropic turbulence on aβ-plane serves as a simple test case for isolating basic physical aspects of complex, large scale geophysical flows. As such, the problem has been the subject of a number of detailed studies. The occurrence of Rossby waves and their interaction with the turbulence cascade was initially investigated by Rhines [1] where the idea of a length scale, now commonly called the Rhines length, was introduced. For nondivergentβ-plane turbulence, kinetic energy at high wave numbers cascades to larger scales until the Rhines length is reached. At this point, the cascade is arrested and energy is radiated by Rossby waves. The energetics of this phenomena have received detailed consideration in the numerical simulations of Maltrud and Vallis [2] and Vallis and Maltrud [3]. For a wide class of initial conditions both forced and unforced flows reach a state characterized by a quasisteady anisotropic flow of alternating zonal jets [1]–[3]. The interaction of large scale Rossby waves with smaller scale turbulence and the emergence of these zonal flows substantially alters turbulent transport leading to a rapid asymptote of the meridional particle dispersion, as observed in [4].

⋆_{The first author was supported in part by the US Department of Energy Climate Change Prediction Program and the second author}

was Supported in part by the Office of Naval Research under ONR Grant No. N00014-93-I-0691. 1

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As a significant amount of spatial transport in large scale flows is accomplished by eddy fluxes, accounting forβ-effects on the fluctuations is an important issue. Studies of scalar turbulent transport modified byβ -effects can be found in [5] and [4]. One of the impelling reasons for studying eddy transport in this situation is that many numerical models for large scale atmospheric and oceanic flows account for the effects of unresolved scales in terms of isotropic eddy diffusivites. Such parameterizations are not only independent ofβbut are also inconsistent with the anisotropy imposed on the flow by variations in the ambient vorticity. Bartello and Holloway [5] address this problem and propose a parameterization of the meridional component of the diffusivity tensor,Kyy, which depends on the value ofβ. This parameterization reflects a striking

dependence of the meridional diffusivity on the size ofβ. Simulations, for a range of nondimensionalβ

parameters pertinent to ocean flows, indicate that the meridional turbulent flux can vary by an order of magnitude. This is what one might call the direct influence ofβ-effects on eddy anisotropy and transport.

There are also what might be termed indirect transport effects ofβin the form of mean flows induced by gradients of the eddy fields. The dynamics of this “rectified circulation on aβ-plane” have been addressed in Rhines [6]–[8]. Variants and experimental verification can be found in [9]–[11]. These studies are motivated, in large part, by the need to determine the effects ofβon eddies for the problem of transport by an eddy-generated mean flow. It should be noted that this is a case of energy transfer from the fluctuations to the mean flow. Current diffusive parameterizations for the eddy field account only for energy transfer from the mean to the fluctuations.

While the simplicity of two-dimensional, homogenous turbulence on aβ-plane stands in sharp contrast to the complexity of naturally occurring large scale geophysical flows, the model does allow for the study of Rossby wave–turbulence interactions. The goal of the present work is to provide exact statistical predictions of the meridional particle dispersion in this simplified problem as a step towards developing a statistical closure for turbulent transport in more complex situations. The suitability of such an eddy diffusivity approach has been studied in [4] and [5]. Exact results for limiting cases such as those considered here are invaluable for validating the performance of closure models used in realistic simulations. The aim is to deduce, directly from the governing potential vorticity equation, bounds on the meridional dispersion and determine how these bounds are related toβ, the enstrophy, the energy, and various Lagrangian time scales of the flow. Our development follows, conceptually, the Lagrangian methods introduced by Taylor [12], [13].

In the spirit of Taylor’s Lagrangian analysis of particle dispersion, the first integral of the potential vorticity equation is recognized as a formal solution for the relative vorticity. In Section 3 the assumptions of statistical stationarity are used in a moment analysis of the formal solution. Several statistical properties of theβ-plane turbulence are deduced. The first two integral time scales of the meridional velocity are determined. The results for the integral scales are applied, in Section 4, to Taylor’s dispersion analysis where all terms in the expression for the meridional dispersion are determined analytically. In short, a Lagrangian analysis of the potential vorticity equation provides the statistical information about the dynamics required to close Taylor’s kinematic dispersion result. The analytical results predict the behavior seen in numerical simulations (see [4]) where the meridional dispersion rapidly approaches an asymptote.

In an analytical context, applications of Lagrangian methods to ocean flows are not new. Rhines and Holland [4] give an overview of the assumptions and shortcomings of such analyses in the context of the gradient type transport models they produce. Lagrangian arguments have also been used to explain the rectified flows on aβ-plane [6], [7]. Experimental and theoretical discussions of Lagrangian methods and developments which include corrections for inhomogeneity have been given in papers by Davis [15]– [17]. Tennekes and Lumley [18] and Monin and Yaglom [19] can be referred to for additional informa-tion.

The statistical properties of homogeneousβ-plane turbulence deduced from the potential vorticity equa-tion are demonstrated numerically in Secequa-tion 6 for unforced (slowly decaying) homogeneous, barotropic

β-plane turbulence. Autocorrelations and integral time scales identified by the analysis are found by tracking ensembles of Lagrangian particles. The meridional dispersion of the ensemble is numerically computed and the numerical results are found to confirm the predictions of the analysis. Additional studies investigating the possibility of a relation between the Lagrangian statistics and the Eulerian structure of the flow are also performed. Specifically, an attempt is made to relate the Lagrangian dispersion length scale and the width of the zonal jets appearing in the Eulerian frame. Similar approaches to determining the scale selection of rectified zonal flows are found in the forced simulations of Panetta [21] and Nozawa and Yoden [22] and the unforced simulations of Cho and Polvani [23], as well as those of Vallis and Maltrud [24].

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2. Problem Statement and Background

Inviscid unforced barotropic flow on aβ-plane satisfies the following evolution equation:

Dtζ(x, t) =−βu2(x, t), (1)

whereu2(x, t) is the Eulerian meridional velocity. There is no mean flow and the flow is nondivergent as

corresponds to a fluid of uniform depth. Expression (1) is the equation for the relative vorticity,ζ, in the presence of a mean potential vorticity gradient induced solely by the meridional (northward) gradient of the vertical component of the planetary rotation rate,β. The streamfunction, velocity and relative vorticity are related:ζ =_∇2_{ψ, ζ}₌_u

2,1−u1,2,whereu1 =−ψ,2andu2=ψ,1.

In the Lagrangian frame the substantial derivative isD/Dt= d/dt. The Eulerian,ui, and Lagrangianvi

identification, by initial condition, is often assumed. The Lagrangian form of the vorticity equation, d

dtζ(t) =−βv(t), (2)

is used to infer a number of properties of the statistics of the solution and the mixing by the velocity field determined by (2).

An equation for the enstrophy,_hζζi, is easily derived from (2): d

dthζζi= 0, (3)

indicating that the the vorticity variance is stationary,_hζζi=_hζ0ζ0i=hζζi0. The angle brackets represent an

average over an ensemble of Lagrangian particle trajectories. The fact that for stationary stochastic processes the process and its derivative are uncorrelated,_hζ(t)v(t)_{i ∼ h}ζ(t)(d/dt)ζ(t)_i= 0, has been used.

2.1. The Dispersion Analysis of Taylor

The dispersion analysis of Taylor [12] is well known and its generalization to the anisotropic homogeneous problem is summarized in [24]:

hxi(t)xj(t)i=hx0ix0ji+ Z t

0 Z t1

[_hvi(t1)vj(t2)i+hvi(t2)vj(t1)i] dt1dt2. (4)

Taylor’s dispersion result is the formal consequence of the kinematic relationship, (d/dt)x(t|x0_{, t} 0) = v(t|x0_{, t}0_{), describing a specific fluid particles trajectory whose solution is given by the first integral.}

Omit-ting the dependence on the initial conditions, the short and long time results are_hxαxβi ∼ hvαvβiRαβ(0)t2

ast→0 and_hxαxβi ∼2hvαvβiTαβtast→ ∞where no sum is implied on the Greek subscripts. Several

things should be noted: in the short time, ballistic limit, the dispersion scales witht2_{. This is very much}

in accord with the short time numerical diffusion studies on a β-plane of [4]. In the long time limit the dispersion scales witht. As the long time scaling is that of a Brownian diffusion,_∼t, the long time limit is also called the diffusion limit. The long time limit depends, crucially, on the integral time scale_Tαβ. In

the studies of [4] the diffusion limit is approached in about about one eddy turnover time_∼ 1/ζrms. The

variance of the meridional dispersion is

hx2(t)x2(t)i= 2 Z t

0 Z t1

0 h

v2(t1)v2(t2)idt2dt1= 2hvvi Z t

(t− η)Rvv(η) dη. (5)

The definition of the autocorrelation, for a stationary process, has been used:

Rvv(τ) = h

v(t)v(t+τ)_i

hvvi .

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Tvv = Z ∞

Rvv(η) dη, (6)

T2 1vv = −

Z ∞

η Rvv(η) dη, (7)

the meridional dispersion becomes, in the diffusion limit [t→ ∞],

hx2(t)x2(t)i∞= 2hvvi[Tvvt+T

1vv]. (8)

This kinematic result indicates that asymptotic meridional dispersion is determined by the zeroth and first moments of the meridional velocity autocorrelation. As one of the objectives of this article is to understand the role thatβplays on the dispersal of fluid particles it is necessary to understand how the imposition of a mean vorticity gradient influences the integral time scales_TvvandT1vv.

3. Moment Analysis of the Potential Vorticity Equation

In this section several consequences of a moment analysis of (2) are derived. A Lagrangian analysis of the potential vorticity equation is used to obtain the necessary statistical information of the dynamics appearing in Taylor’s kinematic dispersion result.

3.1. Temporal Microscale ofζ(t)

The two-point temporal correlation of the vorticity, for a stationary process, can be written as

Rζζ(τ) = h

ζ(t)ζ(t+τ)_i

hζ(t)ζ(t)_i . (9)

The correlation is an even function and it is customary to define a time scale, the temporal microscale, by the curvature ofRζζat the origin: for smallτ,

Rζζ(τ) = 1−

τ2 τ2 ζ

+_·. (10)

Using the definition ofRζζa Taylor series is used to relateτζ2to the square of the temporal derivative of the

vorticity:

Rζζ(τ) = 1− τ 2

2_hζζi

d dtζ

+_·. (11)

Squaring and averaging the vorticity equation, (2), indicates that

h_ζ˙_ζ˙_i₌

d dtζ

=β2

hvvi=2hζζi

τ2 ζ

(12) and thus the temporal microscale is known in terms of single point second-order moments:

τ2

ζ =

2_hζζi

β2_h_vv_i. (13)

Note that curvature at the origin of the ζ autocorrelation, a “small scale” quantity, has been related to “large scale” second moment quantities,_hζζi,hvvi. Note also that in the limit of inviscid two-dimensional turbulence, [β→0], that the microscale, for statistically homogeneous stationary, is infinite.

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3.2. Integral Time Scale ofv(t)

The Lagrangian solution for the vorticity, (2), can be formally written as

ζ(t|x0_{, t}0_{) =}_ζ 0−β

Z t 0

v(t1|x0, t0) dt1. (14)

The meridional velocity that drives (2) is related to the vorticity through a Poisson equation. Thus expression (14), while not immediately useful as a solutionper se, allows several rigorous results for the statistical properties of the solution of (2) to be deduced.

The quantity ζ0 is a random variable denoting the vorticity at the tagging time and location, t0, x0i.

Multiplying (14) byζ(t) and averaging produces

hζ(t)ζ(t)_i=_hζ(t)ζ0i −β Z t

0 h

ζ(t)v(t1)idt1. (15)

The fluid particle identification has, for convenience, and without loss of generality, been dropped. As

t→ ∞,_hζ(t)ζ0i →0 and one finds that

hζ0ζ0i=−β Z ∞

0 h

ζ(t)v(t−η)_idη; (16)

the integral time scale of the joint process,_{ζ, v}, is finite and known in terms of the initial enstrophy and

β. The fact that the variance,_hζζi=_hζ0ζ0i, is stationary for the inviscid problem has been used.

The moment of (14) with the meridional velocity produces

hv(t)ζ0i −β Z t

0 h

v(t)v(t1)idt1 = 0, (17)

where again stationarity implies_hv(t)ζ(t)_i= 0. This can be rewritten as

hv(t)ζ0i=βhvvi Z t

Rvv(η) dη, (18)

which indicates that, in the diffusion limit, the first integral time scale of the meridional velocity is given by

Tvv = Z ∞

Rvv(η) dη= 0, (19)

since_hv(t)ζ0i →0 ast → ∞. This result and its consequences are the central result of this article. The

vanishing of the first integral time scale for finiteβimplies that the meridional particle dispersion is bounded (see [18]).

The above results are a rigorous consequence of (2),_hζvi= 0,stationarity, and that the statistics of the flow forget their initial condition, i.e.,_hζ(t)ζ0i →0 andhv(t)ζ0i →0 ast→ ∞.

3.3. Autocorrelation ofζ(t) and the Second Integral Time Scale ofv(t)

The formal solution, (14), can also be written as

ζ(t+τ|x0_{, t}0_{) =}_ζ₍_t

|x0_{, t}0₎

−β

Z t+τ t

v(t1|x0, t0) dt1. (20)

The moment withζ(t) is now used to obtain an equation for the autocorrelation. Dropping the particle labels and substitutingt1 =t+ηproduces

hζ(t+τ)ζ(t)_i=_hζ(t)ζ(t)_{i −}β Z τ

0 h

ζ(t)v(t+η)_idη. (21)

The solution for_hζ(t+τ)ζ(t)_irequires the two-time correlation_hζ(t)v(t+η)_i. Taking the moment with respect to velocity in (14) produces

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hv(t+η)ζ(t)_i=_hv(t+η)ζ0i −β Z t

0 h

v(t+η)v(t1)idt1. (22)

Use of the result (18) and defining a two-point meridional velocity autocorrelation allows the equation to be written as

hv(t+η)ζ(t)_i = βhvvi

Z t+η 0

Rvv(η1) dη1−βhvvi Z t+η

Rvv(η1) dη1 (23)

= βhvvi

Z η 0

Rvv(η1) dη1, (24)

which has the form, as might be expected, of a stationary process. Substitution into (21) produces the following relationship between the autocorrelations of thevandζprocesses:

Rζζ(τ) = 1−

β2_h_vv_i

hζζi

Z τ 0

Z η 0

Rvv(η1) dη1dη. (25)

The above result is a consequence of the vorticity equation, (2), and nothing else. Note, asβ→0,Rζζ(τ)→

1 as would occur in statistically homogeneous two-dimensional flows for which vorticity is materially conserved. Integrating by parts, the equation becomes

Rζζ(τ) = 1−

β2

hvvi hζζi

Z τ 0

(τ−η)Rvv(η) dη. (26)

The result for the autocorrelation, (26), has several consequences. The second integral time scale can be obtained by taking the limitτ → ∞; withRζζ(τ)→0 and asTvv = 0 one finds

T2 1vv =−

Z ∞

ηRvv(η) dη= h

ζζi

β2_h_vv_i. (27)

Thus while the first integral time scale of thevprocess is zero, the second integral time scale,_T1vv, is related

toβand two single-point second-order moments,_hvviand_hζζi. Additionally the second integral time scale ofvis related to the micro-time-scale ofζ:

T2 1vv =−

Z ∞

ηRvv(η) dη= hζζi

β2_h_vv_i = 1 2τ

ζ. (28)

The result (26) also leads to the differential equation d2

dτ2Rζζ(τ) =− β2_h_vv_i

hζζi Rvv(τ) (29)

relating the vorticity autocorrelation to the autocorrelation of the meridional velocity. Evaluating the differ-ential equation atτ = 0 reproduces the definition of the temporal microscale ofζ(t).

3.4. Integral Time Scale ofζ(t)

Equation (26) can also be used to determine the integral time scale of the vorticity in terms of second-order moments and the second moment of the velocity autocorrelation. Equation (26) is integrated to obtain

Tζζ= Z ∞

Rζζ(τ) dτ = Z ∞

dτ−β

2_h_vv_i

hζζi

Z ∞

0 Z τ

(τ−η)Rvv(η) dη. (30)

Using the definition of_T₁2_vvone of the boundary terms in the second integral cancels the first (unbounded) integral and the integral time scale of the vorticity is given by

Tζζ =

β2

hvvi

2_hζζi

Z ∞

0 τ2_R

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The integral time scale of the vorticity is related to the second moment of the autocorrelation of the meridional velocity.

4. Application of Taylor’s Dispersion Result

The Taylor analysis is now applied to the meridional mixing on theβ-plane. The meridional dispersion, in the diffusion limit, is

hx2(t)x2(t)i= 2hvvi Z ∞

(t−η)Rvv(η) dη= 2hvvi[Tvvt+T12vv]. (32)

It has been shown that_Tvv = 0 and as a consequence the meridional dispersion is finite and independent of

time:

hx2x2i= 2hvviT12vv= 2 hζζi

β2 . (33)

The meridional dispersion is bounded and the bound is determined by the enstrophy andβ. Note that the meridional dispersion bound does not depend on the energy of the meridional velocity,_hvvi, as might be expected from Taylor’s kinematic dispersion result.

The notion of a bounded meridional dispersion of a Lagrangian particle is not new; physical discussions of the rectified circulation on aβ-plane [6], [7] reflect a theoretical consideration of the possibility of bounded particle dispersion for wave motion. Here we have shown that the particle dispersion is bounded for turbulent flow interacting with Rossby waves. Theβ-plane simulations of [3], [4], or [5] present observations of the boundedness of the meridional dispersion. What we have here is, apparently, a mathematical explanation in terms of the first two integral scales of the autocorrelation, of the bounded meridional dispersion seen in β-plane simulations. Figure 12 of [4] shows a striking numerical example of the boundedness of the meridional particle dispersion predicted by this result.

4.1. Alternate Dispersion Bound Derivation

The relationship (33) has been arrived at by taking the diffusion limit of moments of the formal solution of (14). In that way several useful statistical results were derived. There is, however, a faster way, not using Taylor’s dispersion expression and without any results for integral scales, of deducing (33). The result (33) is recognizable as being a direct consequence of the first integral of (d/dt)ζ = ₋βv = ₋β(d/dt)x2 =

−β(d/dt)y. Taking the first integral and squaring and averaging produces

hζ(t)ζ(t)_{i −}2_hζ(t)ζ0i+hζ0ζ0i=β2h(y−y0)2i. (34)

For the inviscid unforced problem_hζζi=_hζ0ζ0iand the equation is rewritten as

2_hζζi[1₋Rζζ(t)] =β2h(y−y0)2i. (35)

If the flow is ergodic one can assumeRζζ(t)→0 ast→ ∞. That the flow is ergodic is not a straightforward

matter. Both inviscid two-dimensional turbulence or a random field of βwaves, asymptotic limits of the

β-plane turbulence, have nonfinite integral time scales and are therefore not ergodic (see [25] for additional viewpoints). In Section 6 we verify that the integral time scale of the ζ(t) process is finite which allows

Rζζ(t)→0 ast→ ∞to obtain

h(y−y0)2i= 2h ζζi

β2 , t→ ∞, (36)

i.e., the result (33). While the result (36) has a shorter derivation, it is important to recall that the initial derivation leading to (33) placed this result in the context of the Taylor dispersion analysis and produced several additional results describing the statistical properties of stationaryβ-plane turbulence.

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4.2. Approach to the Diffusion Limit

Achieving the dispersion bound predicted by (36) or (33) depends on how rapidlyRζζvanishes. The result

(36) can be also be written, using (26), as

hζ(t)ζ0i=hζ0ζ0i −β2hvvi Z t

(t − η)Rvv(η) dη. (37)

SinceRvv has an oscillatory character one anticipates thathζ(t)ζ0imay not vanish rapidly. If one takes the

two-time correlation ofζin theforcedβ-plane simulations of Figure 3 of [4]) as indicative of_hζζ0ione sees

thatRζζ(t) decreases slowly.

That the approach of Rζζ(t) → 0 may be slow can also be anticipated in the following way: in the

absence of theβ-effectζis materially conserved andRζζ(t) is constant. For theβ-plane,Rζζ(t) vanishes in

proportion to the size ofβ. Dividing through by_hζ0ζ0ithe approach to the diffusion limit goes as β2_h₍_x

2−x20)2i

2_hζ(t)ζ(t)_i = 1−Rζζ0(t). (38)

One might view (33) or (36) as an upper bound on the particle dispersion:Rζζ0 while positive for small

t changes sign becoming oscillatory as it decreases with increasing t. There is another useful way of understanding the intermediate time result (38). The autocorrelation of the vorticity can be computed from the meridional particle dispersion.

5. Summary of Results

The assumptions of statistical stationarity have allowed several important statistical properties ofβ-plane turbulence to be deduced. In anticipation of the next section, in which numerical simulations are used to verify and extend the predictions of the mathematical development, the results are summarized:

1. The micro-time-scale of the vorticity is

τζ2 =

2_hζζi

β2_h_vv_i. (39)

Thus the curvature at the origin of the autocorrelation, ostensibly a small scale quantity, is related to the large scale quantities of enstrophy and meridional energy. Note that for a two-dimensional (β = 0) inviscid homogeneous turbulence the microscale is not finite.

2. The integral time scale of the meridional velocity is zero,

Tvv = Z ∞

Rvv(η) dη= 0, (40)

as is characteristic of wave processes.

3. The second integral time scale of the meridional velocity is related to single point second-order quantities:

T2 1vv=−

Z ∞

ηRvv(η) dη= h

ζζi

β2_h_vv_i = 1 2τ

ζ. (41)

4. A relationship between the autocorrelation functions of the meridional velocity and the vorticity has been obtained:

Rζζ(τ) = 1− β 2

hvvi hζζi

Z τ 0

Z η 0

Rvv(η1) dη1dη (42)

= 1₋ β

hvvi hζζi

Z τ 0

(τ−η)Rvv(η) dη. (43)

The integral on the last line appears in Taylor’s [12] treatment of the dispersion process: the meridional par-ticle dispersion is diagnostically related to the vorticity autocorrelation. For an inviscid two-dimensional turbulence,β= 0, in which vorticity is materially conserved,Rζζ(τ) = 1.

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5. The integral time scale of the vorticity is related to the second moment of the velocity autocorrelation:

Tζζ=

β2_h_vv_i

2_hζζi

Z ∞

0 τ2_R

vv(τ) dτ. (44)

6. A diagnostic expression relating the autocorrelation of the vorticity to the time evolution of an ensemble of Lagrangian particles has been derived:

Rζζ(t) = 1−

β2_h_x

2(t)x2(t)i

2_hζζi . (45)

7. The diffusion limit of the diagnostic relationship above predicts a bound on the dispersion of an ensemble of Lagrangian particles. Unlike the classic Taylor dispersal problem, in which the particle diffusion length scales as a Brownian diffusion,_∼ t1/2_{, the meridional particle dispersion in a homogeneous} β-plane turbulence is bounded by

hx2x2i∞=

2_hζζi

β2 . (46)

This is due to the vanishing of the integral time scale of the meridional velocity [18] and the fact that an explicit expression for the second integral time scale ofv(t) is possible.

That the meridional particle dispersion does not scale as a Brownian diffusion means, from a practical point of view, that turbulent transport is not suitably parameterized by an eddy diffusivity. This statement is only valid on time and length scales appropriate to the diffusion limit.

6. Numerical Results

In order to demonstrate the applicability and range of the proceeding results, we have conducted a series of numerical simulations of unforced (decaying) two-dimensional turbulence on aβ-plane. The numerical scheme is a standard spectral method adopted from [26], [27] and used previously in [28], [29], and others (see [30] for a review). The Lagrangian particle dynamics are computed simultaneously using cubic-spline interpolation. The details of the implementation are described fully in [26], [27].

The numerical simulations solve the equation

D/Dtζˆ=₋βˆvˆ+νp∇pζ.ˆ (47)

The small scale dissipation mechanism is taken to be hyperviscous with the exponentpset to 4. The parameter ˆ

βisβL/ζc where the characteristic vorticity is taken as the rms of the initial enstrophy,ζc2 =hζ0ζ0i. The

length scaleLis the size of the basin (L= 2πthroughout). The Lagrangian results were also verified with

p= 2 simulations with grid resolutions 2562 _{and 512}2_{. As the Eulerian-Lagrangian studies are easier with}

thep= 4 model we show, for continuity of presentation, results for only thep= 4 simulations.

The hyperviscous simulations are conducted for different values ofβusing modest, 1282_{, resolution with}

the hyperviscosity fixed atνp = 10−13. In each simulation, the Eulerian field is seeded with an equally

spaced grid of 1089 passive Lagrangian particles at timet=t0. The initial conditions for the Eulerian fields

were chosen from a Gaussian distribution of random vorticity with fixed energy and enstrophy. The spectral evolution of the fields is shown in Figure 1 for low and high values ofβ. The steepening of both spectra and the decrease of the enstrophy decay for higherβvalues are apparent.

The simulated, freely decaying flows are neither statistically stationary nor inviscid. In order to approxi-mate the unforced-undamped dynamics best, as assumed in the analysis, statistics are gathered over periods of time during which the flow is approximately stationary. Over the time scale of any statistics computed we ensure that both the viscous time scale be large with respect to the integral time scale and also that the enstrophy decay rate is as small as possible. Fortunately, such restrictions are more easily met with increasing values of theβparameter. As stated above and shown in the following figures, increasingβleads to both a

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100 101 102 10−8

10−7 10−6 10−5 10−4 10−3 10−2 10−1 100

Energy

Initial Conditions Beta = 10 Beta = 80

100 ₁₀1 ₁₀2

10−5 10−4 10−3 10−2 10−1 100 101

Enstrophy

Initial Conditions Beta = 10 Beta = 80

Figure 1. The energy and enstrophy spectra forβ= 10 and 80 at timet= 10. The initial spectra, used in all simulations, are shown by

the bold line.

decrease in the enstrophy decay rate (due to spectral cascade inhibition) and to a reduction in Lagrangian time scales.

6.1. Autocorrelations and Integral Time Scales

The two-point Lagrangian correlation functions for the vorticity and the zonal and meridional velocity components are shown in Figure 2 for increasing values ofβ. Of the three correlations,Rvv is seen to go

negative most quickly as might be expected from the analysis. Recall that one of the central deductions of the analysis was that, unlike_Tuu andTζζ, the integral time scaleTvv is zero. The integral timescales are

shown in Figure 3. Increasingβappears to lead to a decrease in the vorticity timescale and in a more rapid approach to zero of the time scale of the meridional velocity. Relative to the time scale of the vorticity the time scale of the fluctuating zonal velocity increases withβbecoming commensurate with the time scale of the vorticity suggesting an increased stability of zonal flow structures and a stronger coupling to the vorticity. A direct comparison of the enstrophy covariance for three differentβvalues, Figure 4, shows the decrease of the microscale (increase of curvature) withβas seen in (39). Similarly, the decrease of the meridional microscale and the increase in the decay rate of the meridional macro-scale is shown. The increasingly

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wave-0 0.5 1 1.5 2 2.5 3 −0.2

0 0.2 0.4 0.6 0.8 1

t−t

Beta = 20

R_uu R_vv R

0 0.5 1 1.5 2 2.5 3

−0.2 0 0.2 0.4 0.6 0.8 1

t−t

Beta = 40

R_uu R_vv R

0 0.5 1 1.5 2 2.5 3

−0.2 0 0.2 0.4 0.6 0.8 1

t−t

Beta = 60

R_vv R_zz

0 0.5 1 1.5 2 2.5 3

−0.2 0 0.2 0.4 0.6 0.8 1

t−t

Beta = 80

R_vv R_zz

Figure 2. The Lagrangian correlation functions,Rζζ, Ruu, Rvv, for four values ofβ.

like nature of theRvv process is apparent in the computations for largerβvalues. Due to the increasingly

important effects of nonstationarity over the larger sampling times required for higher order statistics, moments such as_T₁2vv did not converge satisfactorily (for the given resolution and particle sample size).

Such experimental limitations indicate the utility of (41) which provides an exact relation between such higher-order statistics and lower-order ones.

6.2. Verification of the Meridional Particle Dispersion Result

It has been deduced that the meridional particle dispersion was bounded (33). This key result is verified in Figure 5. The dispersion curves in Figure 5 have been normalized byβ2_{/< ζζ >}_{Due to the viscous decay}

of enstrophy and to variations in the enstrophy decay rate withβ, the enstrophy scale,< ζζ >, in (46) was chosen as the average enstrophy, at eachβ, over the time taken for the particle dispersion to asymptote. The results, for the fixed value of Lagrangian-averaged enstrophy, unquestionably validate the analysis.

The early time departures from the asymptotic bound are due to the fact that the meridional dispersion asymptotes as quickly as the vorticity autocorrelation.Rζζis nonzero at short times and the relation

1₋Rζζ(t) = 1₂βˆ2hxˆ2(t) ˆx2(t)i (48)

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0 0.5 1 1.5 2 2.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 t−t 0 T

Beta = 20

T_vv T_zz

0 0.5 1 1.5 2 2.5 3

0 0.1 0.2 0.3 0.4 0.5 0.6 t−t 0 T

Beta = 40

T_vv T_zz

0 0.5 1 1.5 2 2.5 3

0 0.1 0.2 0.3 0.4 0.5 0.6 t−t 0 T

Beta = 60

T_uu T

0 0.5 1 1.5 2 2.5 3

0 0.1 0.2 0.3 0.4 0.5 0.6 t−t 0 T

Beta = 80

T_uu T

Figure 3. The Lagrangian time scales,Tζζ,Tuu,Tvv, for four values ofβ.

The effects of finite viscosity on the derived dispersion bound enter almost exclusively through the time dependence of the decaying enstrophy. The direct effects of viscous drain on the temporal behavior of _hx2x2iare apparently quite small since the long-time value of the unscaled meridional dispersion is

practically constant in all finiteβsimulations. To test the pointwise temporal applicability of (46), we have conducted similar experiments at a higher (2562) resolution and a lower (νp= 10−15) viscosity. The resulting

temporal dependence of

β2

hx2x2i(t)

2_hζζi(t)

is shown in Figure 6 for the same four values ofβ. As before, (46) provides the proper scaling of the meridional particle dispersion with bothβand the local time value of the Lagrangian averaged enstrophy. All four curves indicate an increase with time in the normalized dispersion due to the viscous decay of the particle enstrophy in the numerical simulations. Increasingβslows the enstrophy decay rate decreasing the exponential decay constant. To comprehend Figure 6 it is important to understand that all four simulations were begun at the same initial time. The initial time was determined by half the enstrophye-foldingtime of theβ = 20 flow. Thus the increase in the scaled meridional dispersion is more pronounced for the highest

βvalue since this flow sees the largest proportional change in enstrophy over the ten time units shown. While the mathematical results have been verified it is helpful to obtain a physical picture of the particle dispersion. To provide some idea of the Lagrangian particle behavior for which the above mathematical

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0

0.2 0.4 0.6 0.8 1

t−t 0 R zz

Beta = 20 Beta = 40 Beta = 80

0 0.5 1 1.5 2 2.5 3

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

t−t 0 T vv

Beta = 20 Beta = 40 Beta = 80

Figure 4. Comparison of the Lagrangian vorticity correlation function,Rzzand the meridional Lagrangian timescales,Tvv, for four

values ofβ.

results are relevant the flow was specially seeded with only a few particles. The particle trajectories are shown in Figure 7 over a period of a few eddy turnovers. The meridional boundedness is clearly seen.

Our primary result, the magnitude of the meridional particle dispersion bound, has been verified. We now ask whether there is a phenomenological connection between the scale of the rectified Eulerian flow and the meridional length scale imposed by the Lagrangian particle dispersion bound.

6.3. Length Scales of Zonally Averaged Eulerian Structure

The imposition of theβ-plane has a profound effect on the large scale Eulerian structure of the averaged flow. Most noticeable is the appearance of long lived coherent jets in the zonally averaged velocity field (see

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0

0.2 0.4 0.6 0.8 1 1.2 1.4

t−t

<yy> (Normalized)

Beta = 20 Beta = 40 Beta = 60 Beta = 80

Figure 5. Mean square meridional particle displacements normalized with bothβand the averaged value ofhζζi.

0 1 2 3 4 5 6 7 8 9 10

0 0.5 1 1.5

t−t 0

<yy> (Normalized with <zz>(t))

Beta = 20 Beta = 40 Beta = 60 Beta = 80

Figure 6. Mean square meridional particle displacements normalized with bothβand the instantaneous value ofhζζi. (Results for low

viscosity, 256×256 resolution experiments.)

Figures 8 and 9). Both the instantaneous and the time- or space-averaged fields indicate a marked decrease in the meridional scale of the Eulerian structures with increasing values ofβ. The possibility of a scaling relationship between the imposed value of planetary vorticity gradient and the characteristic width of the zonal flows is the subject of the present section.

The dispersion analysis considered in Section 4 indicates that a well-defined length scale exists for the meridional motion of a Lagrangian particle. Denoting this by

lL= 2hx2x2i1/2 = 2

(2_hζζi)1/2

β (49)

it is seen that the the Lagrangian length scale varies inversely withβ. This dependence is consistent with the scaling proposed by Holloway and Hendershott [20], who equate the turbulent “frequency,” (ζζ)−₁/₂

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0 1 2 3 4 5 6 0

1 2 3 4 5 6

X Y

Figure 7. Representative particle trajectories forβ= 80 showing the confinement of meridional spreading.

the Rossby wave speed to obtain

lHH∼

(_hζζi)1/2

β . (50)

Choosing the turbulent dispersion relation to depend on the local length scale,ωT ∼κucwhere the velocity

scaleucis fixed, the now classical Rhines scale [1] is given by

lR∼ s

2uc

β . (51)

(The factor of 2 is included for consistency with previous definitions.) Arguments by Vallis and Maltrud [3] based on more detailed equilibrium range assumptions for the turbulent “dispersion relation” produce intermediatel∼β−₃/₅

scalings. While these relations each indicate the functional dependence of the length scale on β, the Lagrangian analysis provides an exact measurable quantity, lL, independent of arbitrary

multiplicative constants.

significantly larger thanlRand, aside from ensuring that the curves pass through the origin atβ= 0,ljetdoes

not conclusively scale withβ−₁/₂

. Similarly, the results of [21] for two layer, quasigeostrophic flow driven by an imposed vertical shear indicate that whileljetscales well with a normalizedlR, the strong dependence

ofuconβdoes not allow an unambiguous conclusion regarding the scaling ofljetwithβalone. In the forced

[22] and decaying [23] simulations of barotropic and equivalent barotropic flow on a sphere scalings are less conclusive. In this more complicated geometry, the number of observed alternating jets initially scales inversely with some power of rotation rate (effectiveβ) but eventually, at sufficiently large rotation rates, coherent zonal structures exist only in the subpolar regions and the relation between jet scale and rotation rate is not obvious.

For the unforced Cartesian simulations conducted here,uc and, to a slightly lesser extent,hζζiare set

by the initial conditions, independent ofβ. This implies thatlR∼β−1/2whilelHHandlL∼β−1. The lack

of forcing leads however to some ambiguity in the practical identification and definition of the meridional extent of the zonal jets. As shown in Figures 8 and 9, for a given value ofβ, jets of varying sizes and strengths exist in the flow. For the unforcedβ-plane turbulence the exact location, the strength, and sometimes the total number of the jets is a function of the initial vorticity distribution. For example, a rotation of the initial

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0 1 2 3 4 5 6 0 1 2 3 4 5 6 X Y −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

0 1 2 3 4 5 6

0 1 2 3 4 5 6 X Y −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2

0 1 2 3 4 5 6

0 1 2 3 4 5 6 X Y −1 −0.5 0 0.5

0 1 2 3 4 5 6

0 1 2 3 4 5 6 X Y −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Figure 8. The Eulerian zonal velocity field forβ= 20 (left) andβ= 80 (panels). The instantaneous field is shown and the time averaged

field below.

vorticity field by 180◦

velocity. Numerical experiments with several different initial conditions indicates this to be the most robust scheme for characterizing jet width.

The behavior ofljetdefined by the zero crossings, along with both the numerically computed value of

2_hyyi1/2_{and the predicted values of}_l

L, is shown in Figure 10. The different data points forljet, for a givenβ,

reflect the modest degree of initial condition dependence mentioned: for the same initial energy, enstrophy andβwith different random initial fields the strongest jets have nominally different widths. The variability in jet width decreases with increasingβ as might be expected. We note that the two extremeβlimits are singular with respect to the inviscid Lagrangian analysis. The vorticity autocorrelation does not decay in the case of either a purely two-dimensional turbulence (β→0) or a purely linear Rossby wave field (β→ ∞). Numerical restrictions also limit the range ofβover which the Eulerian scaling can be explored. At lowβ, the weakness of the jets and their initial condition dependence hinder the robust measurement and definition ofljet. At highβ, viscous effects place a lower bound on the computed jet width. Nonetheless, for the range

ofβ shown in the figure, there is a remarkable quantitative agreement between rmsmeridional particle dispersion and the measured size of the zonal jets.

The unnormalized Rhines scale, computed using the initial (and nearly constant) velocity scaleuc = √

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5 10 15 20 25 30 0

1 2 3 4 5 6

beta=20

time

5 10 15 20 25 30

0 1 2 3 4 5 6

beta = 80

time

Figure 9. Time–distance maps of the zonally averaged zonal velocity forβ= 20 (upper) andβ= 40 (lower) indicating the width and

stability of the zonal jets. Only the westward (u <0) contours are shown. Contour intervals are 0.1.

Shown in Figure 10 isℓRnnormalized by 1/2.9 to match the observedljetatβ= 40. While this normalized

scaling does provide a better fit than the rms dispersion curve, the relatively small range ofβparameters examined limits our ability to say anything truly definitive, especially at highβvalues, about the functional dependence ofljet(β). The value of the bounded meridional particle dispersion or the more predictive results

of (46) do provide good quantitative agreement, without resort to an arbitrary scaling factor, for the scale of the jet widths.

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0 20 40 60 80 100 120 0

5 10 15 20 25

k jet

k−jet k−<yy> (observed) k−<yy> (predicted) k−Rhines (normalized)

Figure 10. Wave number of the observed zonal jets,κ= 2∗π/l, along with the wave numbers based on the observed meridional particle

dispersion, the predicted meridional particle dispersion, and the normalized Rhines scale.

7. Summary and Conclusions

A Lagrangian analysis of the inviscid two-dimensional homogeneous β-plane turbulence establishes a number of explicit predictions for statistical quantities relevant to the meridional dispersion of a fluid particle. In the diffusion limit, the analysis has shown that:

1. The integral time scale of the meridional velocity vanishes and, as a consequence, the meridional particle dispersion is bounded for all time [18]. The boundedness of the Lagrangian particles is something one might expect in a wave field; the equations indicate the dispersion is bounded for a turbulent field. 2. The second integral time scale of the meridional velocity is fully determined byβ2_{and the single point}

moments_hvviand_hζζi. Thus, using Taylor’s result, the bound on the meridional particle dispersion is given, in the inviscid case, byβand the enstrophy.

3. The autocorrelation of the vorticity is diagnostically related to the autocorrelation of the meridional velocity. At intermediate times, the meridional particle dispersion serves as a proxy for the vorticity autocorrelation.

The analytical predictions obtained are a rigorous consequence of the governing equation, (1), using only the assumptions of statistical stationarity of Lagrangian quantities. The results of the Lagrangian analysis, namely the vanishing of the meridional time scale and the scaling of the meridional dispersion bound with

β, have been unambiguously verified by numerical simulation. Taylor’s dispersion result,

hx2(t)x2(t)i∞= 2hvvi[Tvvt+T

1vv], (52)

is a kinematic result which depends on three quantities related to the dynamics of the flow, namely, the Reynolds stress and the first and second Lagrangian integral time scales. The formal solution to the potential vorticity equation has been used to deduce the values of_TvvandT₁2vv. As a consequence the form of Taylor’s

dispersion law appropriate to homogeneous stationaryβ-plane turbulence is

hx2x2i∞=

2_hζζi

β2 (53)

and it is mathematically demonstrated why the simplest of geophysical jets, the zonal flows on aβ-plane, are thin and persistent; their lateral integral time scale is zero and the meridional particle dispersion is bounded. While Taylor’s kinematic result (52) suggests a dependence on the energy of the meridional fluctuations,

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hvvi, it is interesting to note that the analysis shows the meridional particle dispersion to be independent of

hvvi.

Physically, the consequences of the analysis can be understood in the context of Rhines’ comments on geophysical jets [31], which are typically strong narrow flows that remain laterally compact for long distances. Such thin and persistent high Reynolds number structures stand in sharp contrast to engineering jets where the turbulence rapidly diffuses the jets momentum. The rapid spreading displayed by engineering jets can be appreciated with reference to Taylor’s dispersion result where_hxαxαi ∼tscales as a Brownian

diffusion and the corresponding turbulent diffusivity isKαα∼dhxαxαi/dt∼Const.

Rhines’ comment that the behavior of geophysical jets is “in apparent disregard for the laws of classical turbulence” is an allusion to the Brownian diffusion scaling of Taylor’s result. As shown here, this is not the case for inviscid barotropicβ-plane turbulence where, as a direct consequence of absolute vorticity conservation, the meridional particle dispersion,_hx2x2i, is bounded, implying that the effective meridional

turbulent diffusivity is zero.

Ancillary numerical simulations have suggested a predictive link between the width of Eulerian zonal flows and the Lagrangian particle dispersion length. While difficult to establish mathematically, an argument based on the eddy mixing of zonal mean momentum is physically plausible. Unfortunately, as is expected for unforced simulations, the computed jet widths, positions and strengths are less than robust quantities showing a modest dependence on initial conditions. Statements regarding a precise link between Eulerian flow structures and the Lagrangian dispersion bound remain, in the absence of a rigorous theoretical connection, tentative. This lack of uniqueness of the jet strengths and sizes necessary to establish a robust Eulerian – Lagrangian connection is substantially improved in the forced problem, the subject of current studies.

References

[1] Rhines, P.B. (1975). Waves and turbulence on aβ-plane.J. Fluid Mech.,69, 417.

[2] Maltrud, M.E., and Vallis, G.K. (1991). Energy spectra and coherent structures in forced two-dimensional andβ-plane turbulence. J. Fluid Mech.,228, 321.

[3] Vallis, G.K., and Maltrud, M.E. (1993). Generation of mean flows and jets on aβ-plane and over topography.J. Phys. Ocean.,23, 1346.

[4] Holloway, G., and Kristmannsson, S.S. (1984). Stirring and transport of tracer fields by geostrophic turbulence.J. Fluid Mech., 141, 27.

[5] Bartello, P., and Holloway, G. (1991). Passive scalar transport inβ-plane turbulence.,J. Fluid Mech.,223, 521.

[6] Rhines, P.B. (1977). The dynamics of unsteady ocean currents. InThe Sea(E.D. Goldberg, I.N. McGrave, J.J. O’Brein, and J.H. Steele, eds.) Vol. 6. Wiley, New York.

[7] Haidvogel, D.B., and Rhines, P.B. (1983). Waves and circulation driven by oscillating winds in an idealized ocean basin.Geophys. Astrophys. Fluid Dynamics,25, 1.

[8] Whitehead, J. (1974). Mean flow generated by circulation on aβ-plane.Tellus,29, 358.

[9] Colin De Verdiere, A. (1979). Mean flow generation by topographic Rossby waves.J. Fluid Mech.,94, 39.

[10] McEwan, A.D., Thompson, R.O.R.Y., and Plumb, R.A. (1980). Mean flow generation by topographic Rossby waves.J. Fluid Mech.,99, 655.

[11] Holland, W.R., and Rhines, P.B. (1980). An example of eddy induced ocean circulation.J. Phys. Ocean.,10, 1010. [12] Taylor, G.I. (1921). Diffusion by continuous movements.Proc. Roy. Lond. Math. Soc.,20, 196.

[13] Taylor, G.I. (1932). The transport of vorticity and heat through fluids in turbulent motion.Proc. R. Soc. London, Ser. A,135, 685. [14] Rhines, P.B., Holland, W.R. (1979). A theoretical discussion of eddy-driven mean flow.Dyn. Atmos. Oceans.,3, 289.

[15] Davis, R.E. (1983). Oceanic property transport, Lagrangian particle statistics and their prediction.J. Marine Res.,41, 163. [16] Davis, R.E. (1987). Modeling eddy transport of passive scalars.J. Marine Res.,45, 635.

[17] Davis, R.E. (1991). Observing the general circulation with floats.Deep Sea Res.,38, S531. [18] Tennekes, H., and Lumley, J.L. (1972).A First Course in Turbulence. MIT Press, Cambridge, MA. [19] Monin, A.S., and Yaglom, A.M. (1973). Statistical Fluid Mechanics. MIT Press, Cambridge, MA.

[20] Holloway, G., and Hendershott, M.C. (1977). Stochastic closure for nonlinear Rossby waves.J. Fluid Mech.,82, 747.

[21] Panetta, R.L. (1993). Zonal jets in wide baroclinically unstable regions: persistence and scale selection.J. Atmos. Sci.,50, 2073. [22] Nozawa, T., and Yoden, S. (1997). Formation of zonal bands structure in forced two-dimensional turbulence on a rotating sphere.

Phys. Fluids,9, 2081.

[23] Cho, Y.-K., and Polvani, L.M. (1996). The emergence of jest and vortices in freely evolving, shallow water turbulence on a sphere. J. Atmos. Sci.,50, 2073.

[24] Hinze, J.O. (1975). Turbulence. McGraw-Hill.

[25] Shepherd, T.G. (1987). Non-ergodicity of inviscid two-dimensional flow on a beta-plane and on the surface of a rotating sphere. J. Fluid Mech.,184, 289.

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[26] Babiano, A., Basdevant, C., Legras, B., and Sadourny, R. (1987). Vorticity and passive scalar dynamics in two-dimensional turbulence.J. Fluid Mech.,183, 379.

[27] Babiano, A., Basdevant, C., Le Roy, P., and Sadourny, R. (1990). Relative dispersion in two-dimensional turbulence.J. Fluid Mech.,214, 535.

[28] Provenzale, A., Babiano, A., and Villone, B. (1995). Single-particle trajectories in two-dimensional turbulence.Phys. Fluids,6, 2465.

[29] Elhma¨idi, D., Provenzale, A., and Babiano, A. (1993). Elementary topology of two-dimensional turbulence from a Lagrangian viewpoint and single-particle dispersion.J. Fluid Mech.,257, 533.

[30] Provenzale, A. (1999). Transport by coherent barotropic vortices. To appear,Ann. Rev. Fluid Mech. [31] Rhines, P.B. (1994). Jets.Chaos,4, 313.

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0 1 2 3 4 5 6

X Y

Figure 7. Representative particle trajectories forβ= 80 showing the confinement of meridional spreading.

the Rossby wave speed to obtain

lHH∼

(_hζζi)1/2

β . (50)

Choosing the turbulent dispersion relation to depend on the local length scale,ωT ∼κucwhere the velocity

scaleucis fixed, the now classical Rhines scale [1] is given by

lR∼ s

2uc

β . (51)

scalings. While these relations each indicate the functional dependence of the length scale on β, the Lagrangian analysis provides an exact measurable quantity, lL, independent of arbitrary multiplicative constants.

A number of previous studies have sought to connect the meridional extent of the zonal flows observed inβ-plane turbulencewith the Rhines scale and its variants. Figure 4 of [3] indicate that the jetwidthljetis significantly larger thanlRand, aside from ensuring that the curves pass through the origin atβ= 0,ljetdoes not conclusively scale withβ−₁/₂

. Similarly, the results of [21] for two layer, quasigeostrophic flow driven by an imposed vertical shear indicate that whileljetscales well with a normalizedlR, the strong dependence ofuconβdoes not allow an unambiguous conclusion regarding the scaling ofljetwithβalone. In the forced [22] and decaying [23] simulations of barotropic and equivalent barotropic flow on a sphere scalings are less conclusive. In this more complicated geometry, the number of observed alternating jets initially scales inversely with some power of rotation rate (effectiveβ) but eventually, at sufficiently large rotation rates, coherent zonal structures exist only in the subpolar regions and the relation between jet scale and rotation rate is not obvious.

For the unforced Cartesian simulations conducted here,uc and, to a slightly lesser extent,hζζiare set

by the initial conditions, independent ofβ. This implies thatlR∼β−1/2whilelHHandlL∼β−1. The lack of forcing leads however to some ambiguity in the practical identification and definition of the meridional extent of the zonal jets. As shown in Figures 8 and 9, for a given value ofβ, jets of varying sizes and strengths exist in the flow. For the unforcedβ-plane turbulence the exact location, the strength, and sometimes the total number of the jets is a function of the initial vorticity distribution. For example, a rotation of the initial

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0 1 2 3 4 5 6 0 1 2 3 4 5 6 X Y −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

0 1 2 3 4 5 6

0 1 2 3 4 5 6 X Y −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2

0 1 2 3 4 5 6

0 1 2 3 4 5 6 X Y −1 −0.5 0 0.5

0 1 2 3 4 5 6

0 1 2 3 4 5 6 X Y −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Figure 8. The Eulerian zonal velocity field forβ= 20 (left) andβ= 80 (panels). The instantaneous field is shown and the time averaged field below.

vorticity field by 180◦

will produce a set of jets at different locations with modestly different widths and strengths. For simplicity and repeatability, at eachβthe strongest westward jet in the flow is chosen to define the jet width. The jet width,ljet, is chosen as the distance between zero crossings of time-averaged zonal velocity. Numerical experiments with several different initial conditions indicates this to be the most robust scheme for characterizing jet width.

The behavior ofljetdefined by the zero crossings, along with both the numerically computed value of 2_hyyi1/2_{and the predicted values of}_l

L, is shown in Figure 10. The different data points forljet, for a givenβ, reflect the modest degree of initial condition dependence mentioned: for the same initial energy, enstrophy andβwith different random initial fields the strongest jets have nominally different widths. The variability in jet width decreases with increasingβ as might be expected. We note that the two extremeβlimits are singular with respect to the inviscid Lagrangian analysis. The vorticity autocorrelation does not decay in the case of either a purely two-dimensional turbulence (β→0) or a purely linear Rossby wave field (β→ ∞). Numerical restrictions also limit the range ofβover which the Eulerian scaling can be explored. At lowβ, the weakness of the jets and their initial condition dependence hinder the robust measurement and definition ofljet. At highβ, viscous effects place a lower bound on the computed jet width. Nonetheless, for the range ofβ shown in the figure, there is a remarkable quantitative agreement between rmsmeridional particle dispersion and the measured size of the zonal jets.

The unnormalized Rhines scale, computed using the initial (and nearly constant) velocity scaleuc = √

(3)

5 10 15 20 25 30

0 1 2 3 4 5 6

beta=20

time

5 10 15 20 25 30

0 1 2 3 4 5 6

beta = 80

time

Figure 9. Time–distance maps of the zonally averaged zonal velocity forβ= 20 (upper) andβ= 40 (lower) indicating the width and stability of the zonal jets. Only the westward (u <0) contours are shown. Contour intervals are 0.1.

Shown in Figure 10 isℓRnnormalized by 1/2.9 to match the observedljetatβ= 40. While this normalized scaling does provide a better fit than the rms dispersion curve, the relatively small range ofβparameters examined limits our ability to say anything truly definitive, especially at highβvalues, about the functional dependence ofljet(β). The value of the bounded meridional particle dispersion or the more predictive results of (46) do provide good quantitative agreement, without resort to an arbitrary scaling factor, for the scale of the jet widths.

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0 20 40 60 80 100 120 0

5 10 15 20 25

k jet

k−jet k−<yy> (observed) k−<yy> (predicted) k−Rhines (normalized)

Figure 10. Wave number of the observed zonal jets,κ= 2∗π/l, along with the wave numbers based on the observed meridional particle dispersion, the predicted meridional particle dispersion, and the normalized Rhines scale.

7. Summary and Conclusions

moments_hvviand_hζζi. Thus, using Taylor’s result, the bound on the meridional particle dispersion is given, in the inviscid case, byβand the enstrophy.

β, have been unambiguously verified by numerical simulation. Taylor’s dispersion result,

hx2(t)x2(t)i∞= 2hvvi[Tvvt+T 2

1vv], (52)

dispersion law appropriate to homogeneous stationaryβ-plane turbulence is

hx2x2i∞= 2_hζζi

β2 (53)

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hvvi, it is interesting to note that the analysis shows the meridional particle dispersion to be independent of

hvvi.

diffusion and the corresponding turbulent diffusivity isKαα∼dhxαxαi/dt∼Const.

References

[1] Rhines, P.B. (1975). Waves and turbulence on aβ-plane.J. Fluid Mech.,69, 417.

[2] Maltrud, M.E., and Vallis, G.K. (1991). Energy spectra and coherent structures in forced two-dimensional andβ-plane turbulence.

J. Fluid Mech.,228, 321.

[3] Vallis, G.K., and Maltrud, M.E. (1993). Generation of mean flows and jets on aβ-plane and over topography.J. Phys. Ocean.,23, 1346.

[4] Holloway, G., and Kristmannsson, S.S. (1984). Stirring and transport of tracer fields by geostrophic turbulence.J. Fluid Mech.,

141, 27.

[5] Bartello, P., and Holloway, G. (1991). Passive scalar transport inβ-plane turbulence.,J. Fluid Mech.,223, 521.

[6] Rhines, P.B. (1977). The dynamics of unsteady ocean currents. InThe Sea(E.D. Goldberg, I.N. McGrave, J.J. O’Brein, and J.H. Steele, eds.) Vol. 6. Wiley, New York.

[7] Haidvogel, D.B., and Rhines, P.B. (1983). Waves and circulation driven by oscillating winds in an idealized ocean basin.Geophys. Astrophys. Fluid Dynamics,25, 1.

[8] Whitehead, J. (1974). Mean flow generated by circulation on aβ-plane.Tellus,29, 358.

[9] Colin De Verdiere, A. (1979). Mean flow generation by topographic Rossby waves.J. Fluid Mech.,94, 39.

[10] McEwan, A.D., Thompson, R.O.R.Y., and Plumb, R.A. (1980). Mean flow generation by topographic Rossby waves.J. Fluid Mech.,99, 655.

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Lagrangian Covariance Analysis of and x0

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Dokumen yang terkait

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