Theoret. Comput. Fluid Dynamics 2000 14: 1–20
Theoretical and Computational
© Springer-Verlag 2000
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 a β-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 on β, 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 of β values, 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 a β-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 nondivergent
β-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
2 J.R. Ristorcelli and A.C. Poje
As a significant amount of spatial transport in large scale flows is accomplished by eddy fluxes, accounting for
β-effects on the fluctuations is an important issue. Studies of scalar turbulent transport modified by β- 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
of β but 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,
K
yy
, which depends on the value of β. This parameterization reflects a striking
dependence of the meridional diffusivity on the size of β. Simulations, for a range of nondimensional β
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 of
β-effects on eddy anisotropy and transport. There are also what might be termed indirect transport effects of
β in the form of mean flows induced by gradients of the eddy fields. The dynamics of this “rectified circulation on a
β-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 of β on 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 a
β-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 to
β, 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 the β-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 a
β-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 homogeneous β-plane turbulence deduced from the potential vorticity equa-
tion are demonstrated numerically in Section 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].
Lagrangian Covariance Analysis of
β
-Plane Turbulence 3
2. Problem Statement and Background
