## 1. Introduction

Being a quantity of fundamental interest in the statistical characterization of a turbulent flow, the Lagrangian velocity correlation function (LVCF) has been the subject of detailed study (Hinze 1975; Monin and Yaglom 1971). Indeed, if we view the midlatitude free troposphere as being in a nonideal turbulent state (Shepherd 1987a, b), it is of interest to inquire into the nature of its LVCF. Similarly, apart from being a measure of transport and its close relation to the LVCF (Taylor 1921; Monin and Yaglom 1971), the absolute dispersion (AD) is known to yield information about the structure of the underlying flow (Bouchaud and Georges 1990; Leoncini and Zaslavsky 2002).

Of course there exist other measures, such as the relative dispersion (or Lyapunov exponents for smooth flows) and finite-scale analyses (or finite-sized-Lyapunov exponents), which are a part of the complete statistical characterization of a flow. In particular, finite-scale statistics are argued to be useful when the available range of scales is small and there is the possibility of crossover effects from differing regimes (Boffetta et al. 2000).^{1} The approach here is slightly different from earlier modeling studies of large-scale atmospheric dispersion that employ a Langevin equation, equivalently assuming an exponential LVCF, to parameterize the dispersion process [Gifford (1982, 1984); a discussion of such an approach can be found in Sawford (1984)]. In the present note, only the LVCF and AD will be considered, the focus will be on their characterization and complementary nature. Also, the accordance of these statistics with simpler physically relevant systems will be seen.

For computational purposes, daily pressure level global T63 (192 × 96, 17 levels) resolution wind data from the European Centre for Medium-Range Weather Forecasts (ECMWF) is used as the 3D advecting velocity field. We computed the trajectories (and record the velocity along these trajectories) of an ensemble of passive particles. The trajectories were computed in latitude, longitude, and pressure coordinates by a standard fourth-order Runge–Kutta scheme and the velocity fields were interpolated in a linear fashion. Averaging is done with respect to an ensemble (denoted by 𝗦) of passive particles that remain in the free troposphere, that is, particles that escape into the boundary layer or into the stratosphere are excluded from the statistics.^{2}

## 2. Lagrangian velocity correlation functions

### a. Zonal LVCF

*u*(

*λ*,

*ϕ*,

*p*,

*t*), the zonal LVCF

*R*(

_{u}*τ*) is defined as

**x**(

*t*) represents the trajectory of an individual member of 𝗦. To account for the inhomogeneity of the flow and to focus our attention the midlatitudes, 𝗦 is chosen to comprise

*N*members located randomly such that they initially satisfy 0° <

*λ*(𝗦) < 360°, 35° ≤

*ϕ*(𝗦) ≤ 55°, and 400 mb ≤

*p*(𝗦) ≤ 700 mb, where

*λ*,

*ϕ*,

*p*represent the longitude, latitude, and pressure coordinates. Note that no restriction is placed on the trajectories, that is, the statistics presented are not conditional on the particles remaining in the midlatitudes for all times.

As can be seen in the upper panel of Fig. 1, which shows log[*R _{u}*(

*τ*)] for the winter and summer seasons, the zonal LVCF is clearly nonexponential. This is in agreement with studies on 2D (Pasquero et al. 2001) and geostrophic (Pecseli and Trulsen 1997) turbulence, but stands in contrast to investigations of 3D turbulence (Mordant et al. 2001; Pope 2000). Further, for intermediate values of

*τ*(i.e., 2 <

*τ*< 10 days), it appears that (lower panel of Fig. 1)

*R*(

_{u}*τ*) ∼

*τ*

^{−α}with 0 <

*α*< 1 [

*α*= 0.32 and 0.45 for December–February (DJF) and June–August (JJA), respectively]. The issue of whether the correlation function is indeed a power law is known to be a delicate matter. Indeed any monotonically continuous function, such as the aforementioned power law, can be expressed (over a given range of scales) as the discrete sum of exponentials—see Berglund (2004) for definitions and details—a fact that has been used elegantly in the analysis of queues in communication networks (Feldmann and Whitt 1998; Starobinski and Sidi 2000). Whether the zonal LVCF is the sum of two (or more) Ornstein–Uhlenbeck processes with differing time scales—as is argued to be an effective parameterization of the LVCF in 2D turbulence (Pasquero et al. 2001)—or, if it indeed shows power-law scaling, is practically indistinguishable, especially if the behavior is seen over a small range of scales.

^{3}On the other hand, what is certain is the enhancement of the Lagrangian correlation time

*T*= ∫

_{u}^{∞}

_{0}

*R*(

_{u}*t*)

*dt*(Bouchaud and Georges 1990). Indeed, as will be observed, the persistent correlations in the zonal velocity field result in a superdiffusive AD regime.

### b. Meridional LVCF

Defined in a manner similar to Eq. (1), but using the meridional component of the velocity field *υ*(*λ*, *ϕ*, *p*, *t*), we compute the meridional LVCF *R _{υ}*(

*τ*). As is seen in Fig. 2,

*R*(

_{υ}*τ*) decays to zero in one week. Interestingly, we notice a pronounced anticorrelation, that is,

*R*(

_{υ}*τ*) < 0 before the final

*R*(

_{υ}*τ*) → 0 behavior. On comparison with 2D (Pasquero et al. 2001) and geostrophic turbulence (Pecseli and Trulsen 1997) this feature is found to be unique to the present situation. This is easily explained when one takes into account the latitudinal restriction imposed by the rotation of the planet (Rhines 1994). Specifically, in the absence of strong diabatic or frictional effects, the conservation of potential vorticity gives rise to large-scale Rossby waves (Pedlosky 1987). We argue that the meridional oscillatory motion implied by the Rossby waves is responsible for the aforementioned anticorrelation.

### c. Eddy and time mean LVCFs

To gain some insight into the connection between the nature of the LVCFs and the structure of the tropospheric flow, we partition the daily data into time mean and transient components. Specifically, the time mean is *û*(**x**) = (1/*T*) ∫^{T}_{0} *u*(**x**, *t*) *dt* (where *T* is the duration of the entire season) and the transient or eddy component is defined as *u*′(**x**, *t*) = *u*(**x**, *t*) − *û*(**x**). Even though the tropospheric flow does not possess a clear spectral gap, that is, there is a near continuum of active (temporal and spatial) scales, we expect the above decomposition to separate processes, which vary on scales that are farthest apart (Blackmon et al. 1984).

Figures 3 and 4 show the zonal and meridional LVCFs computed by exclusively utilizing the Eulerian time mean (upper panels) and Eulerian eddy fields (lower panels). In spite of the crudeness of the partition here, the similarity between the eddy LVCFs in both cases is evident. Indeed, apart from the slight anticorrelation retained in the meridional eddy LCVF, both *R _{u}*

_{′}(

*τ*) and

*R*

_{υ}_{′}(

*τ*) are rapidly decaying functions with a time scale of the order of a couple of days.

On the other hand, the zonal and meridional time mean LVCFs are strikingly different. Where *R _{û}*(

*τ*) is strongly correlated on long time scales—something to be expected from a slowly varying zonal jet—

*R*(

_{υ̂}*τ*) represents oscillatory motion as induced by a large-scale wave. Moreover, comparing the behavior of

*R*

_{υ}_{′}(

*τ*) and

*R*(

_{υ̂}*τ*) leads us primarily attribute the anticorrelation observed in

*R*(

_{υ}*τ*) to the time mean component of the flow. Indeed, behavior consistent with these results has been observed in studies of balloon trajectories in the Southern Hemisphere [Morel and Desbois (1974), see especially their Figs. 9 and 10, and the discussion regarding time scales involved in the definition of stationary and transient components of the flow].

## 3. Absolute dispersion

*τ*→ 0) of the above yields ballistic motion, whereas when

*T*is finite, the long time limit (i.e.,

_{u}*τ*≫

*T*) yields diffusive behavior (Hinze 1975). The presence of other exponents, that is, AD(t) ∼

_{u}*t*;

^{γ}*γ*≠ 1, is referred to as anomalous diffusion (Bouchaud and Georges 1990). Before displaying the results, let us fix some notation. We denote the total AD by

*A*(

*t*), that is,

*A*(

*t*) = 〈[

*x*(

*t*) −

*x*(0)]

^{2}+ [

*y*(

*t*) −

*y*(0)]

^{2}+ [

*z*(

*t*) −

*z*(0)]

^{2}〉𝘀, where

*x*,

*y*,

*z*are Cartesian coordinates. Further the individual components of the AD are denoted by

*A*(

_{i}*t*) where

*i*represents a coordinate, for example,

*A*(

_{x}*t*) = 〈 [

*x*(

*t*) −

*x*(0)]

^{2}〉

_{𝘀}.

*R*(

*τ*) ∼

*τ*

^{−α}⇒ AD(

*t*) ∼

*t*

^{2−}

*. Even though the nonideal nature of the present flow, especially its rich time mean structure (Shepherd 1987a, b), is likely to invalidate the direct applicability of Eq. (3); from the form of*

^{α}*R*(

_{u}*τ*) (whether one takes it to be a power law or a sum of exponentials, both being equivalent from the discussion in the previous section) it is reasonable to expect the zonal AD to exhibit anomalous behavior at intermediate time scales. Indeed as can be seen in Fig. 5, which shows

*A*(

_{x}*t*) as computed using the daily wind data,

^{4}

*A*(

_{λ}*t*) in Fig. 6. Now the anomalous regime (

*t*;

^{δ}*δ*= 1.6) lasts from

*T*

_{1}<

*t*<

*T*

_{2}days (

*T*

_{1}∼ 2 and

*T*

_{2}∼ 25), after which we see the beginning of an asymptotic diffusive regime (lower panel of Fig. 4).

^{5}It is worth mentioning that the anomalous behavior of

*A*(

_{λ}*t*) is in close agreement with laboratory experiments on quasigeostrophic flows (Weeks et al. 1996). Further, zonal superdiffusion has been identified in studies involving large-amplitude Rossby waves (Flierl 1981) and in more general potential vorticity (PV) -conserving flows [where the superdiffusion is along PV contours; LaCasce and Speer (1999)].

^{6}

*t*, as

*R*→ 0 quite rapidly. Once again, we use Eq. (3) to get a feel for the meridional AD at intermediate time scales. Qualitatively approximating

_{υ}*R*(

_{υ}*τ*) ≈

*e*

^{−τ/C1}cos(

*ωτ*) (Fig. 2), numerical integration of Eq. (3) yields the AD shown in Fig. 7. Apart from the two asymptotic regimes we notice transient subdiffusive scaling. This is in accord with results utilizing random shear flows, where anticorrelation in the LCVF was associated with subdiffusion and even complete trapping in extreme cases (Elliott et al. 1997). Indeed, the actual meridional AD,

*A*(

_{z}*t*) shown in Fig. 8, behaves in precisely the same manner,

## 4. Summary

Employing daily wind data from the ECMWF, we have estimated the zonal and meridional LVCFs of the midlatitude tropospheric flow. The zonal LVCF is seen to be nonexponential in character. Physically, given that the midlatitude tropospheric flow has a rich time mean structure along with an energetic eddy field (Shepherd 1987a, b), this observation is not entirely unexpected. Moreover, from this perspective our examination of *R _{û}*(

*τ*) and

*R*

_{u}_{′}(

*τ*) serves to clarify the roles of the time mean and eddy fields, respectively. Specifically, the eddy field by itself generates an almost exponential rapidly decaying LCVF whereas the time mean component—roughly a slowly varying unidirectional jet flow (Blackmon et al. 1984)—is seen to be strongly correlated.

Apart from decaying to zero in a relatively short time (≈1 week), the meridional LVCF exhibits an anticorrelation, that is, *R _{υ}*(

*τ*) < 0 before

*R*(

_{υ}*τ*) → 0. We attribute this anticorrelation to the presence of large-scale planetary waves—a basic consequence of PV conservation on a rotating planet. Examining

*R*

_{υ}_{′}(

*τ*), we see that the meridional eddy LCVF is very similar to its zonal counterpart; whereas

*R*(

_{υ̂}*τ*), a manifestation of the large-scale stationary waves, has an oscillatory character and indicates the time mean component to be primarily responsible for the above-mentioned anticorrelation in

*R*(

_{υ}*τ*).

In regard to the AD, the point that stands out is the simultaneous existence of superdiffusive and subdiffusive anomalous scaling in the zonal and meridional directions, respectively. It must be stressed that we lack a quantitative relationship between the LCVF and the AD in this nonideal situation. Nonetheless, a certain qualitative basis is provided by super and subdiffusive behavior in ideal turbulent fields with enhanced (power laws or sums of exponentials depending on one’s interpretation) and anticorrelated LVCFs respectively.^{7} Finally, given that similar behavior has been observed in drift–wave turbulence (Basu et al. 2003), we are led to speculate on the possible universality of this phenomenon in fields where (slow) jets and waves coexist with (fast) eddies.

## Acknowledgments

Comments by Dr. R. Saravanan are gratefully acknowledged. Also, comments by all three referees led to a significant improvement in the material presented. This work was carried out at the National Center for Atmospheric Research, which is sponsored by the National Science Foundation.

## REFERENCES

Basu, R., V. Naulin, and J. Rasmussen, 2003: Particle diffusion in anisotropic turbulence.

,*Commun. Nonlinear Sci. Numer. Simulation***8****,**477–492.Berglund, A., 2004: Nonexponential statistics of fluorescence photobleaching.

,*J. Chem. Phys.***121****,**2899–2903.Blackmon, M., Y. Lee, and J. Wallace, 1984: Horizontal structure of the 500-mb height fluctuations with long, intermediate, and short time scales.

,*J. Atmos. Sci.***41****,**961–979.Boffetta, G., A. Celani, M. Cencini, G. Lacorata, and A. Vulpiani, 2000: Nonasymptotic properties of transport and mixing.

,*Chaos***10****,**50–60.Bouchaud, J-P., and A. Georges, 1990: Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications.

,*Phys. Rep.***195****,**127–293.Elliott, F., D. Horntrop, and A. Majda, 1997: Monte Carlo methods for turbulent tracers with long range and fractal random velocity fields.

,*Chaos***7****,**39–48.Feldmann, A., and W. Whitt, 1998: Fitting mixtures of exponentials to long-tail distributions to analyze network performance models.

,*Perform. Eval.***31****,**245–279.Flierl, G., 1981: Particle motions in large-amplitude wave fields.

,*Geophys. Astrophys. Fluid Dyn.***18****,**39–74.Gifford, F., 1982: Horizontal diffusion in the atmosphere: A Lagrangian-dynamical theory.

,*Atmos. Environ.***16****,**505–512.Gifford, F., 1984: The random force theory: Application to meso- and large-scale atmospheric diffusion.

,*Bound.-Layer Meteor.***30****,**159–175.Hinze, J., 1975:

*Turbulence*. McGraw-Hill, 790 pp.LaCasce, J. H., and K. G. Speer, 1999: Lagrangian statistics in unforced barotropic flows.

,*J. Mar. Res.***57****,**245–274.Leoncini, X., and G. Zaslavsky, 2002: Jets, stickiness, and anomalous transport.

,*Phys. Rev. E***65****.**046216, doi:10.1003/PhysRevE.65.046216.Monin, A., and A. Yaglom, 1971:

*Statistical Fluid Mechanics*. Vol. 1, MIT Press, 769 pp.Mordant, N., P. Metz, O. Michel, and J-F. Pinton, 2001: Measurement of Lagrangian velocity in fully developed turbulence.

,*Phys. Rev. Lett.***87****.**214501, doi:10.1103/PhysRevLett.87 .214501.Morel, P., and M. Desbois, 1974: Mean 200-mb circulation in the Southern Hemisphere deduced from EOLE balloon flights.

,*J. Atmos. Sci.***31****,**394–407.Pasquero, C., A. Provenzale, and A. Babiano, 2001: Parameterization of dispersion in two-dimensional turbulence.

,*J. Fluid Mech.***439****,**279–303.Pecseli, H., and J. Trulsen, 1997: Eulerian and Lagrangian correlations in two-dimensional random geostrophic flows.

,*J. Fluid Mech.***338****,**249–276.Pedlosky, J., 1987:

*Geophysical Fluid Dynamics*. Springer-Verlag, 710 pp.Pope, S., 2000:

*Turbulent Flows*. Cambridge University Press, 771 pp.Rhines, P., 1994: Jets.

,*Chaos***4****,**313–339.Sawford, B., 1984: The basis for, and some limitations of, the Langevin equation in atmospheric relative dispersion modelling.

,*Atmos. Environ.***18****,**2405–2411.Shepherd, T., 1987a: Rossby waves and two-dimensional turbulence in a large-scale zonal jet.

,*J. Fluid Mech.***183****,**467–509.Shepherd, T., 1987b: A spectral view of nonlinear fluxes and stationary–transient interaction in the atmosphere.

,*J. Atmos. Sci.***44****,**1166–1179.Starobinski, D., and M. Sidi, 2000: Modeling and analysis of power-tail distributions via classical teletraffic methods.

,*Queueing Sys.***36****,**243–267.Taylor, G., 1921: Diffusion by continuous movement.

,*Proc. London Math. Soc.***20****,**196–211.Weeks, E., J. Urbach, and H. Swinney, 1996: Anomalous diffusion on asymmetric random walks with a quasi-geostrophic flow example.

,*Physica D***97****,**291–310.

Meridional LCVF; *R _{υ}*(

*τ*) < 0 before

*R*(

_{υ}*τ*) → 0.

Citation: Journal of the Atmospheric Sciences 62, 10; 10.1175/JAS3560.1

Meridional LCVF; *R _{υ}*(

*τ*) < 0 before

*R*(

_{υ}*τ*) → 0.

Citation: Journal of the Atmospheric Sciences 62, 10; 10.1175/JAS3560.1

Meridional LCVF; *R _{υ}*(

*τ*) < 0 before

*R*(

_{υ}*τ*) → 0.

Citation: Journal of the Atmospheric Sciences 62, 10; 10.1175/JAS3560.1

(top) Time mean zonal LCVF (DJF data). (bottom) Eddy zonal LCVF. Note the different time scales. Also, by about one week, the eddy correlations have almost completely died out, whereas the mean flow is still strongly correlated. JJA data (not shown) behave in a qualitatively similar manner.

Citation: Journal of the Atmospheric Sciences 62, 10; 10.1175/JAS3560.1

(top) Time mean zonal LCVF (DJF data). (bottom) Eddy zonal LCVF. Note the different time scales. Also, by about one week, the eddy correlations have almost completely died out, whereas the mean flow is still strongly correlated. JJA data (not shown) behave in a qualitatively similar manner.

Citation: Journal of the Atmospheric Sciences 62, 10; 10.1175/JAS3560.1

(top) Time mean zonal LCVF (DJF data). (bottom) Eddy zonal LCVF. Note the different time scales. Also, by about one week, the eddy correlations have almost completely died out, whereas the mean flow is still strongly correlated. JJA data (not shown) behave in a qualitatively similar manner.

Citation: Journal of the Atmospheric Sciences 62, 10; 10.1175/JAS3560.1

(top) Time mean meridional LCVF (DJF data). (bottom) Eddy meridional LCVF. Once again, note the different time scales. JJA data (not shown) behave in a qualitatively similar manner.

Citation: Journal of the Atmospheric Sciences 62, 10; 10.1175/JAS3560.1

(top) Time mean meridional LCVF (DJF data). (bottom) Eddy meridional LCVF. Once again, note the different time scales. JJA data (not shown) behave in a qualitatively similar manner.

Citation: Journal of the Atmospheric Sciences 62, 10; 10.1175/JAS3560.1

(top) Time mean meridional LCVF (DJF data). (bottom) Eddy meridional LCVF. Once again, note the different time scales. JJA data (not shown) behave in a qualitatively similar manner.

Citation: Journal of the Atmospheric Sciences 62, 10; 10.1175/JAS3560.1

Zonal AD. Ballistic → superdiffusive → saturation.

Citation: Journal of the Atmospheric Sciences 62, 10; 10.1175/JAS3560.1

Zonal AD. Ballistic → superdiffusive → saturation.

Citation: Journal of the Atmospheric Sciences 62, 10; 10.1175/JAS3560.1

Zonal AD. Ballistic → superdiffusive → saturation.

Citation: Journal of the Atmospheric Sciences 62, 10; 10.1175/JAS3560.1

(top) DJF longitudinal AD (ballistic → superdiffusive → diffusive); (bottom) *dA _{λ}*(

*t*)/

*dt*vs

*t*. Note that

*dA*(

_{λ}*t*)/

*dt*→ constant. ⇒ normal diffusion.

Citation: Journal of the Atmospheric Sciences 62, 10; 10.1175/JAS3560.1

(top) DJF longitudinal AD (ballistic → superdiffusive → diffusive); (bottom) *dA _{λ}*(

*t*)/

*dt*vs

*t*. Note that

*dA*(

_{λ}*t*)/

*dt*→ constant. ⇒ normal diffusion.

Citation: Journal of the Atmospheric Sciences 62, 10; 10.1175/JAS3560.1

(top) DJF longitudinal AD (ballistic → superdiffusive → diffusive); (bottom) *dA _{λ}*(

*t*)/

*dt*vs

*t*. Note that

*dA*(

_{λ}*t*)/

*dt*→ constant. ⇒ normal diffusion.

Citation: Journal of the Atmospheric Sciences 62, 10; 10.1175/JAS3560.1

(top) Synthetic LVCF *R*(*τ*) ≈ *e*^{−τ/C1} cos(*ωτ*) (*C*_{1} = 7, *ω* = 0.3). (bottom) Induced AD.

Citation: Journal of the Atmospheric Sciences 62, 10; 10.1175/JAS3560.1

(top) Synthetic LVCF *R*(*τ*) ≈ *e*^{−τ/C1} cos(*ωτ*) (*C*_{1} = 7, *ω* = 0.3). (bottom) Induced AD.

Citation: Journal of the Atmospheric Sciences 62, 10; 10.1175/JAS3560.1

(top) Synthetic LVCF *R*(*τ*) ≈ *e*^{−τ/C1} cos(*ωτ*) (*C*_{1} = 7, *ω* = 0.3). (bottom) Induced AD.

Citation: Journal of the Atmospheric Sciences 62, 10; 10.1175/JAS3560.1

(top) Meridional AD (JJA and DJF curves have been shifted for clarity). Ballistic → subdiffusive → diffusive; (bottom) *A _{z}* (

*t*)/

*t*∼

*t*→

*A*(

_{z}*t*)/

*t ∼ t*− 1 <

^{β}*β*< 0 →

*A*(

_{z}*t*)/

*t*∼ constant.

Citation: Journal of the Atmospheric Sciences 62, 10; 10.1175/JAS3560.1

(top) Meridional AD (JJA and DJF curves have been shifted for clarity). Ballistic → subdiffusive → diffusive; (bottom) *A _{z}* (

*t*)/

*t*∼

*t*→

*A*(

_{z}*t*)/

*t ∼ t*− 1 <

^{β}*β*< 0 →

*A*(

_{z}*t*)/

*t*∼ constant.

Citation: Journal of the Atmospheric Sciences 62, 10; 10.1175/JAS3560.1

(top) Meridional AD (JJA and DJF curves have been shifted for clarity). Ballistic → subdiffusive → diffusive; (bottom) *A _{z}* (

*t*)/

*t*∼

*t*→

*A*(

_{z}*t*)/

*t ∼ t*− 1 <

^{β}*β*< 0 →

*A*(

_{z}*t*)/

*t*∼ constant.

Citation: Journal of the Atmospheric Sciences 62, 10; 10.1175/JAS3560.1

^{1}

I thank referee A for the discussion of finite-scale statistics.

^{2}

Experiments were performed with an initial set of 20 000 and 30 000 particles, and no discernable difference in the statistics was noticed in these two situations.

^{3}

Comments of referee B are acknowledged as they provoked a more careful look at this issue.

^{4}

Here, *A _{y}*(

*t*) is virtually identical to

*A*(

_{x}*t*). Also,

*A*(

_{z}*t*) (shown later) =

*A*(

_{x}*t*), hence

*A*(

*t*) also behaves in the same fashion as

*A*(

_{x}*t*).

^{5}

Note that we should expect *δ* ≠ *γ* as *A _{λ}*(

*t*) only involves changes in

*λ,*whereas

*A*(

_{x}*t*) is sensitive to both

*λ*and

*ϕ*.

^{6}

I thank referee C for providing references on particle motion in PV-conserving systems.

^{7}

Note that this anomalous behavior is transient, i.e., it is flanked on either side by an asymptotic regime. Even though in the present situation this behavior is supported by the nature of the LVCF, it is worth keeping in mind that crossover effects (especially in the superdiffusive case) could play a role in determining the quantitative nature of the anomalous exponents (Boffetta et al. 2000).