Observing the general circulation with floats

1l19tH114'11'11 S3,(NJ + (J (M' Pergamon Pre~~ pic

Def!p-Sea Rflearch, Vol 3N, Suppl I. pp S531-S57I , 19'J1

Pnnted

In

© 1'191

Great Bntam

Observing the general circulation with floats Russ E.

DAVIS*

(Received 10 March 1989; in revised form 30 March 1990; accepted 5 June 1990)

Abstract-The prospect for describing advection and eddy transport of tracers in the general circulation using floats is examined. This is done within the context of a recently proposed generalization of the advection-diffusion equation for passive scalars in which eddy transport is governed by a time-dependent eddy diffusivity tensor K(t) which at large t approaches the constant 1("" appropriate to pure advection-diffusion models, A minimal description of the general circulation would include the Eulerian mean velocity Vex) and the diffusivity K(t). Given sufficient numbers of current-followers which adequately follow ideal fluid particles, both the horizontal components of U and the purely horizontal components of K could be measured. The connection between these transport parameters and statistics of ideal particles shows that if the mean density of the float array is nonuniform then the mean velocity deduced from it will be in error by an array bias produced by downgradient diffusion of floats; this same phenomenon is responsible for the bias of Lagrangian mean velocity away from U toward high eddy diffusivity. A bias also affects diffusivity estimates when the sampling array is not uniform in the mean, The effects of nonuniform sampling make it difficult to piece together an accurate description of the general circulation from floats deployed in localized regional arrays, Both horizontal and vertical separations develop between initially paired floats and fluid particles because floats do not follow vertical water motion, The effect of this on the V and I( measured with floats is examined and is found to be negligible outside of strong currents. With floats, Eulerian statistics must be estimated from a combination of space and time averaging. The uncertainty, dU, ofa measured V then depends on the spatial averagingscaleA. The trade-off between accuracy and resolution is at the analyst's control by adjusting the averaging scale, but the product dU· A is fixed by the sampling density and eddy field properties. The measurement uncertainty of the diffusivity I(t) increases with t, even after K has reached its asymptote I(~. Numerical simulation of particle motion is used to test the generalized advection-diffusion equation upon which the development is based, to study how the diffusivity depends on properties of the eddy field, and to explore problems in mapping the general circulation in the presence of statistically inhomogeneous eddies, boundaries and strong currents. When the mean float density is reasonably uniform, then measurements of mean flow and the fully horizontal components of the eddy diffusivity are accurate and are equally useful in strong boundary currents and in broad interior flows.

1. INTRODUCTION

technology of current-following drifting buoys is developing rapidly, and this is making possible extensive regional studies which are cumulatively beginning to describe the general circulation (FREELAND et al., 1975; RICHARDSON, 1983, 1985; RISER and ROSSBY, 1983; McNALLY et al., 1983; COLIN DE VERDIERE, 1983; PRICE, 1983; HOFFMAN,

THE

• Scripps Institution of Oceanography, La Jolla, CA 92093, U.S.A. S531

S532

R. E.

DAVIS

1985; KRAUSS, 1986; KRAUSS and BONING, 1987). Since drifters provide an economical way of obtaining the spatial and temporal coverage needed to observe the general circulation, it is likely that they will continue to be a major tool for obtaining the needed velocity observations. The purpose of this paper is to examine how well advection and eddy transport can be determined by observing trajectories of current-following floats and drifters. A complete description of the general circulation will require reconciliation of velocity and tracer field observations within the framework of a model of advection and eddy transport. For the general circulation, the conventional Reynolds-averaged transport equations based on the time average () are a reasonable starting point. It is here assumed that time averages over some interval of the order years to decades are meaningful and that mean and fluctuating concentration and velocity fields can be defined with respect to such averages in the usual way: e = «(), ()' = 0 - 0, U = (u) and u' = u - U. Description of the general circulation involves determining both the advective flux US and the eddy flux (u'O'). hopefully without measuring every field variable 0 on the eddy scale. Advection presumably can be specified given enough observations of the velocity, be these from moored or drifting instruments. While advection might be specified using a Lagrangian description. for most purposes this seems fundamentally more complicated because in the Lagrangian frame spatial variation becomes confused between the labeling coordinates and time. There are several approaches to describing eddy transport. The most conventional is to parameterize the flux using a flux vs ve law and to tunc the diffusivities until realistic simulation of observed mean field is achieved. A second approach is to observe directly eddy fluxes of properties such as heat (cf. NOWLIN et al., 1985) and to tune parameterizations to these observations. Another approach is to compare observed velocity statistics with eddy-resolving models which, when validated. could be used to predict the eddy transport of various properties; this is in the spirit of the SCHMITZ and HOLLAND (1982) comparison of oceanic eddy statistics with their model counterparts. The approach examined here is to use current-following buoys to directly measure the purely lateral components of the diffusivity. The term lateral transport here means transport along surfaces of approximately constant potential density referenced to a representative pressure but, while it is critical to separate vertical and diapycnal transport, lateral and horizontal transport can generally be interchanged. Unfortunately. the motion of practicable floats provides no information about diapycnal transport. The focus of this paper is characterizing lateral mean oceanic transport using statistics of quasi-Lagrangian current-following buoys. A critical point is that, without approximation, evolution of the mean field of a conserved passive scalar is fully determined by the mean initial and source fields and the statistics of single Langrangian particles (cf. DAVIS, 1987. Section 2.1). These statistics are substantially easier to observe and model than the multiparticle (e.g. pair separation) statistics required to describe relative dispersion or the typical spread of individual property clouds. The approach suggested here is to map, using Eulerian averaging, the mean velocity and single-particle diffusivity. Both quantities are based on particle displacements, but being Eulerian averages they gracefully describe spatial variation, overcoming the indirect connection between location and labeling coordinate which makes Lagrangian averages so ungainly in the face of inhomogeneity. The start.ing point for the discussion here is the elaborated advection-diffusion equation developed In DAVIS (1987) for describing evolution of the mean concentration of a passive scalar:

S533

Observing the general circulation with floats

a at

- sex, t) + U(x) . VS(x, t) = Q(x, t) + v .

ft dr dK(X ' r) . V0(x, t 0

ar

T)

(1.1)

which is based on the flux vs history-of-gradient relation

(Un'(x, t)()'(x, t » =

-ft d r OKnm(X, 1") a8(x, t o

aT

aXm

T) ,

(1.2)

where the repeated-index summation convention is in force. In (1.1) U(x) is the Eulerian mean velocity and Q is the mean source of O-stuff, including convergence of molecular fluxes. Initial conditions are applied at t == O. In arriving at (1.1) and (1.2) it is necessary that the source (of which Q is the average) and the initial concentration field (at t = 0) both be statistically independent of u'; it is this requirement that defines t = O. The diffusivity K is not a phenomenological parameter but rather an observable Lagrangian velocity statistic defined below in (2.3). It is a generalization to circumstances with mean flow of the single-particle diffusivity introduced by TAYLOR (1921). In general K depends on time but for l much greater than some time T, K(X, t} = KOO(X). The time scale T is a reflection of the finite time and space scales of the dispersing fluid motions. If Tis short compared with the scale over which 0 evolves, then K(X. t) quickly approaches KOO(X) and (1.1) and (1.2) are well approximated by familiar eddy diffusion laws with time-invariant diffusivity K"" . Some of the ways which the finite time scale in (1.1) makes its solutions differ from those of the advection-diffusion equation are discussed by DAVIS (1987). This limit may adequately describe transport in the general circulation, but it is inadequate for describing how float and drifter statistics are related to U and K. The purposes of this paper are to test the flux law (1. 2), to show that (1.1) can be used to describe particle transport and to discuss how U and the purely horizontal components of K can be measured from floats and drifters. In Section 2 the model (1.1) is used to relate particle motion statistics to the (Eulerian) mean velocity U(x) and the diffusivity K. In determining U(x) the strategy is simply to use current-followers as moving current meters whose velocities are averaged. The rub is that the probability of a particle being at x depends on, among other factors, the velocity there. Consequently, velocity averages computed from particles can be biased. In Section 2 it is shown that velocity estimates made from particles passing through an area are proportional to the total flux, advection plus eddy flux, so that estimates of mean velocity made from particle motion can be biased by eddy diffusion. The well-known effect of the mean motion of particles released at a point being toward high diffusivity is an example of such bias. This bias is, according to the generalized advection-diffusion equation, proportional to the mean gradient of particle concentration. Consequently, accurate velocity mapping requires using arrays with approximately uniform average buoy density. Utility of (1.1) depends on an ability to account for molecular effects in Q. In particular, effects like shear dispersion (YOUNG et al., 1982) which depend on molecular diffusion must either be unimportant or be parameterized before the advection-diffusion equation can be predictive. Utility of realizable floats and drifters in characterizing U and K rests in their statistics being accurate approximations of the statistics of ideal particles which exactly follow the continuum flow. Surface drifters are subject to wind and wave forces which prevent them from following flow exactly; these effects must be minimized. Subsurface floats do not accurately follow vertical flow and this causes floats to disperse differently from particles and to be differently advected by generalized Stokes Drifts. In

S534

R. E.

DAVIS

Section 3 the statistical differences between molecules and continuum particles and between particles and floats are examined. Molecular effects are negligible in the mean concentration field on the scales of interest. Except in strong currents with substantial vertical motion, even isobaric fioats follow the flow well enough that horizontal components of U and the horizontal components of K can be measured. Float displacements inherently average velocity, but further averaging is generally required to achieve stable means. The sampling requirements for mean velocity and diffusivity, discussed in Section 4, differ substantially. There is a natural tension between the desires for statistical reliability and spatial resolution which, along with the need to avoid bias, must be balanced in designing float sampling arrays. In Section 4 a simple formula is used to describe the statistical precision and spatial resolution of float arrays. The critical array parameter is the average concentration of fioats times the averaging time. It is a persistent myth that current-followers are unsuitable for describing strong currents because they are swept through the region of interest too quickly to provide good sampling. This is true if floats are deployed in the current and observed only on their first short passage through it. But if particles are seeded uniformly in the interior, then new particles are drawn into the strong current as fast as they are expelled and, because the sampling density is the same everywhere, the strong currents are as well measured as the interior. In the final section various considerations in mapping the general circulation are summarized using numerically simulated particle trajectories in random velocity fields. These are used to test the flux gradient law (1.1) and the mean velocity biases predicted by it. Conjectures on how the diffusivity is related to scales of the velocity field are also examined. Finally, simulations are used to examine various factors which make it difficult to map mean flow and diffusivity, including inhomogeneous eddy fields, boundaries and strong currents. 2. PARTICLE STATISTICS AND TRANSPORT PARAMETERS

Here we consider how observable particle motion statistics are related to the transport parameters U and K in the generalized advection-diffusion equation (1.1). The following notation is employed. Fields can be described using either Eulerian or Lagrangian coordinates. In Eulerian coordinates u(x. t) is the velocity at position x at time t. In Lagrangian coordinates u(t\xo, to) is the velocity at time r of the particle passing through position Xo at time to. The coordinates following the bar in a Lagrangian reference are the particle label, here taken to be any position and time on the particle's trajectory~ this labeling is ambiguous since a particle can be labeled by any position and time along its trajectory . Eulerian averages, like U(x) = (u(x ,t)), are defined over an ensemble of randomly timed observations at one point or, equivalently, over time. Lagrangian averages, like V(tlx) = (o(t + tolx, to)} are taken over an ensemble of particles randomly deployed at x at different deployment times to; the requirement that particles be labeled by their random deployment time and position removes ambiguity in a particle's label. In what follows, U and V will refer to Eulerian and Lagrangian mean velocities, respectively, while u ' and Vi will refer to the departures from the means U and V, respectively. The fluctuation field u/, like velocity itself, can be referenced to either Eulerian or Lagrangian coordinates but, in either case, it is a departure from the Eulerian mean. Even though Eulerian statistics are

Observing the general circulation with floats

S535

stationary, Lagrangian statistics generally depend on time since deployment; for example R(tlx), the Lagrangian average of ret + tolx, to), evolves with t away from R(Olx) = x. Lagrangian averaging is complicated by the fact that the velocity field being observed also affects where and how often observations are made. There is, for example, generally a difference between averaging over particles randomly deployed at x or over particles encountered at the same place. Suppose that a certain fraction of all fluid particles are marked so they can be observed. The probability of encountering a particle at x is then proportional to the fluid density there. In that case averages over an ensemble of particles encountered at x are dominated by periods of high density while particles randomly deployed at x sample all densities equally. The average of any property statistically related to density will differ between the two ensembles. If the fraction of particles marked is uniform throughout the fluid and the flow is nondivergent, then the probability of encounter does not vary so that random-deployment and random-encounter statistics are equal. In that case, a particle can be relabeled whenever it passes through x and then added to the ensemble of particles 'deployed' at x. This is equivalent to averaging over the labeling time when particles pass through x while holding the time since relabeling fixed. However, when particles are not uniformly distributed through the field, there is generally a difference between random-deployment and random-encounter statistics even if the flow is nondivergent. The most familiar example is the Stokes Drift of particles in a field of irrotational surface waves. In that case the mean velocity of particles at a fixed depth is zero, particles randomly encountered on the surface have mean drift Vs equal to the usual Stokes Drift, particles randomly deployed on the surface have Lagrangian means velocity Vet) which varies from iVs at t = 0 to Vs at large t, and the Lagrangian mean velocity of particles randomly deployed at a depth varies from zero at t = 0 to Vs at large t. The diffusion and array biases discussed below are other examples of Lagrangian mean velocities caused by nonuniform particle distributions. The positions of ideal particles evolve according to arr(tIXo. to) = u[r(tIXo, to), t]. The Lagrangian mean velocity, V(tlx)

= -ala R(tlx) = (u[r(t + tolx, to), t + to),

(2.1)

generally depends on t, the time after particles were at x, and differs from U(x). By definition r(tolx, to) = " so that the Eulerian mean U(x) equals the Lagrangian mean V(Olx); both are simply the mean particle velocity at x. Thus U(x) =

[~ R(flx)] i)t

1=0

(2.2)

and, although an Eulerian statistic, U(x) can be determined from particle trajectories. The single particle diffusivity appearing in (1.1) is defined

(2.3a) There are some subtleties in this definition. Because r(tolx, 10 ) = x it follows that u'(x to) could be replaced by v'(tolx, In) and that the mean product of u' and [x - R] vanish'es. Consequently an alternate definition of the diffusivity is

S536

R. E.

DAVIS

Kjk(X, t) = -(vj(/olx. to)r,,(tO - tlx, tu».

(2.3b)

While these two forms are formally equivalent, it is shown in Section 4 that when perfect averaging cannot be achieved the latter is to be preferred. This form can be expressed Kjk(X.

t)

= f~, drPjk(r) ,

(2.3c)

where P is the time-lagged Lagrangian velocity covariance Pjk(t) = (v;(t"lx. to)Vk(t + tnl x • to»·

(2.4)

Since it can be defined in terms of r' and v' , both departures from Lagrangian means, K is a pure Lagrangian statistic. It is the generalization of TAYLOR'S (1921) single-particle diffusivity to fields with inhomogeneous statistics. Note that K(X, t) is defined for both signs of t. In the more common problem of evolution of 0 from an initial condition it is the t > 0 part which appears in (1.1). In this case K is the mean product of u' at x and the particle displacement undergone to reach that point; it is the history of the particle arriving at x that determines how much O-stuff it is transporting. The t < 0 part is appropriate to the question "What fraction of particles found at Xo were at x at an earlier time?" For t < 0, K is the mean product of u' at x and the negative of the displacement of particles leaving that point. For I > 0 it can be seen from (2.3) that the diagonal elements of the K tensor are typically positive while for t < 0 they are negative; the significance of K < 0 for t < 0 is discussed below. The time-lagged Lagrangian velocity covariance in (2.4) generally does not have the symmetry of its Eulerian counterpart with respect to changing the sign of I. This is so because the first index refers to a velocity observed at x while the second does not. However, in a statistically homogeneous field, where velocities at all positions are statistically equivalent,

(2.5) In this case the diagonal elements of K are odd functions of t. In what foHows the relation between Lagrangian statistics and transport parameters is examined from two perspectives. The first and more conventional is modeled on an ensemble of particles deployed at a point. The second, more appropriate to inhomogeneous fields, is an Eulerian mapping of the statistics of particles found in different regions.

2.1. Deployment at a point Here transport properties are related to statistics of particles randomly deployed at Xo. As noted above, when particles are uniformly distributed throughout a nondivergent velocity field, these statistics also can be achieved by relabeling particles when they arrive at XI). To relate particle motion statistics and property transport it is noted that the probability density of r(tlxo, to), the position of particles deployed at xo. to, is C(x, I - II), xo) = (o[x - r(t\X(h to)]).

(2.6)

In physical terms c(x, t) =: ~[x - r(/!X(h to)] is the concentration of particles in a realization of a particle being released at Xu, to. The probability density C(x, t - 10, Xo) is then also the average concentration at time t of particles deployed at Xu. to and, according to (1.1), evolves as .

S537

Observing the general circulation with floats

a - C(x, t, Xu)

at

+ U(x) . VC(x, t, Xu) =

V·

ft 0

dr

aK(X r) ' . VC(x, t - r, Xo) aT

(2.7)

from the initial condition C(x, 0, x.J) = o(x - Xo)· To derive (1.1) it is necessary that the initial concentration be statistically independent of the flow. Thus (2.7) applies only when particles are either randomly deployed at Xo or selected from a uniform mean concentration in a nondivergent velocity field; otherwise particle density will generally be related to the flow at t = toTwo simple observable particle statistics describing particles deployed at x are the mean and mean square displacements:

D,(llx) = (r,(1 + 411x, 10) - x,) =

L

drV, (rlx)

= Jdr[rk -

x,]C(r, t, x)

(2.8)

and Sjk(tlx) = ([r,(t

=

+ tolx, to)

- Xj][rA;(t

+ tolx, to) - xA;l)

fdr[rj - Xj)[rk - xk]C(r, t, x).

(2.9)

In principle, and in practice when V· u = 0 and particles are uniformly distributed, D and S are defined for both positive and negative t describing particles arriving at and departing from x. Note that Sdescribes the size of the mean field resulting from a sequence of point releases, not the typical size of the individual fields averaged to make that mean field; describing the latter requires particle-pair statistics. The time derivatives of (2.8) and (2.9) involve atc given by (2.7); using this, spatial integration by parts and the fact that IC dx = 1 provides relations to U and K:

a -Dk(tlx) at

= Jdr[UA;(r)C(r,t,x) + Jt

0

a a drC(r.t- r,X)--Kkm(r,r)] arm ar

(2.10)

(2.11) where the repeated index summation convention is in force and U, k -+ k, j] denotes the same terms with k, j replacing j, k. In general, for time lags greater than some Tthe velocity covariance in (2.3) will vanish and K(X, t) will approach asymptotic values K(X, + (0) or K(X, - (0) depending on the sign of t. The size of T depends on the finite scales of the dispersing flow. If these scales are small (as in Brownian motion) T will be small and the time variation of everything except Kcan be neglected in the r integrations in (2.10) and (2.11). These integrations are then easily done and give results identical to those obtained directly from a pure advection-diffusion equation with time-invariant diffusivity equal to the appropriate asymptotic K depending on the sign of t.

S538

R. E. DAVIS

Equations (2.10) and (2.11) are most useful when particle displacements are smaller than the scalesofU and K. Then C(r, t, x) is concentrated nearr = x and U(r) and K(r) can be expanded in Taylor series centered at the labeling position x. Integrating by parts and noting that the integral of C is unity converts (2.10) to

(2.12) The analogous result for S(t) obtained from (2.11) is most useful for becomes independent of t and

Itl »

T when

K

(2.13) where SJk == Sjk - DjD k is the covariance of displacements and K"" is the appropriate asymptotic K depending on the sign of t. Mean particle displacement is, according to (2.12), produced both by U and by spatial variation (divergence) of K. Particle dispersion is. according to (2.13), produced by the processes measured by K and amplified by mean flow shear. Intuition based on molecular diffusion is inadequate to understand the implications of (2.12) and (2.13) and they deserve comment. In doing this it is useful to separate the diffusivity tensor into a symmetric part I(s (K~ = K~) and an antisymmetric part KA; these components have quite different effects on transport and have different observabilities. It is also useful to consider the case of weakly inhomogeneous eddy fields for which the symmetry condition (2.5) applies. The symmetric part of the diffusivity, K S , is analogous to the diffusivity appearing in the classic gradient-diffusion equation where the diffusivity tensor is symmetric. According to (2.13), only K S leads to position variance and it predicts that the particle variance grows explosively if there is substantial mean shear. As a practical matter, this makes it difficult to determine the diffusivity from particle dispersion unless the mean shear is accurately known. All components of K vanish at t = 0, and (2.5) shows that I(s(x, t) is typically negative for t < 0 and that it is an odd function of t in weakly inhomogeneous eddy fields. The physical significance of a negative K S follows from (2.13). Sitlx) is the position variance (uncertainty) of particles which pass through x at t = O. This variance must be positive and generally increases with tIt for both signs of t~ according to (2.13). Kjj{X. t) must then be negative for t < O. In quasihomogeneous eddy fields, when KS is approximately an odd function of t, the displacement variance SJ) grows at the same rate with increasing It I regardless of the sign of I; the uncertainty of where a particle will go and where it came from are equal at the same Itl. According to (2.12). spatial variation of K causes the mean velocity of particles passing through x to differ from Eulerian velocity by what could be called a 'diffusion bias.' As discussed above (2.2). the definitions of the Eulerian average U(x) and the Lagrangian average V(otx) are equivalent and the two must be equal. This poses no inconsistency with (2.12) because K(X,O) = O. Consequently. the mean velocity deduced from particle displacements over short time intervals will be the Eulerian mean. But ifU(x) is estimated from particle displacements over a time interval t of 0(1) (so that K(I) is significant) the result would be V(Tlx) averaged between T = () and T == t and could be a biased estimate of U.

Observing the general circulation with floats

8539

(1975) were the first to comment on the 'diffusion bias' offtoat velocities produced by K , and they pointed out how floats deployed at a point tend to migrate toward high eddy energy (high K S). It will be noted from (2.3c) that KS(t) = - ~(-t) so that the opposite migration is observed for particles arriving at x and those leaving the same point. For t < 0 KS is generally negative so that it becomes more positive moving toward low eddy activity. Consequently, the time-mean velocity of particles arriving at x generally differs from U(x) by a bias toward low eddy activity. In weakly inhomogeneous cases the arrival and departure biases are equal and opposite and V(t)x) + V( -tlx) = 2U(x). This suggests that, so far as the effects of K S are concerned, U(x) is best measured using particle displacements over an interval centered on the time of arrival at x. The antisymmetric KA appears in the flux law (1.2) but the divergence of the associated flux vanishes unless KA varies spatially; thus adding a constant to ~ does not affect the evolution of a passive tracer field. And then KA does vary, its effect on transport in (1.1) is more like advection than diffusion since, for example FREELAND et al. S

oo

d,[Kt djC]

= [d,*¢""][djC].

In fact, if the time scale of the scalar field is long compared with the time T (so that the advection-diffusion equation applies) the effect of variable ~ is indistinguishable from advection by the velocity -V· KA. According to (2.11), or its special case (2.13), ~ does not affect dispersion. as measured by S. It may, consequently, be inappropriate to call KA a 'diffusivity', but by any name it has an important effect on transport. The special role of ~ was first noted by RHINES and HOLLAND (1979) using a short time approximation for particle motion. They noted, for example, that the classical Stokes Drift of a particle in a surface wave field is described by (2.12) with the diffusion bias coming from vertical variation of KA. In two-dimensional flow KA is determined by the difference 1(12 - 1(21' From (2.3) this is (u' x r'), where is a unit vertical vector; thus the difference of diagonal terms is invariant to the coordinate system. This difference describes a veering tendency of the fluctuating field such that a particle starting off in one direction can be expected to turn one way rather than the other. Particle paths under surface waves or in unsteady quasi-circular eddies are examples of this kind of motion. Because KA enters the transport equation (1.1) as a pseudo-velocity, it cannot be determined from the evolution of a tracer field without independent knowledge of U; for the same reason it does not affect the displacement covariance S and cannot be determined from particle dispersion . The only way to measure KA is from the difference between U and the effective advection velocity of a tracer (a tricky calculation at best) or directly from the definition (2.3). Neither can the diffusion bias coming from divergence of ~ be easily minimized, as can the bias from K S • For example, in a quasi-homogeneous eddy field the symmetry relation (2.5) shows KA to be an even function of t. Thus the diffusion bias is approximately an even function of t and no clever selection of sampling interval (other than taking a very short one) can eliminate V - U.

- z.

z

2.2. Eulerian averages of particle motion

The discussion above focuses, in the customary way, on Lagrangian statistics computed from an ensemble of particles deployed at one point, that is Lagrangian averages. Such averages are not very useful in describing a flow whose typical properties, like mean flow

S540

R. E . DAVIS

or eddy activity, vary spatially. The connection between position and Lagrangian label is too indirect, particularly as the time since 'deployment' becomes large, for spatial structures to be well described by Lagrangian statistics. When addressing a fundamentally inhomogeneous flow, such as the general circulation, it is therefore desirable to adopt an Eulerian perspective. Here we examine the process of estimating Eulerian averages from floats. This examination discloses another mean flow bias. 'array bias', and provides an alternate perspective on 'diffusion bias'. Consider estimating U and K directly from (2.1) and (2.3) using spatial averaging over some region A and over an ensemble of deployments in each of which particles are placed on the same initial array at t = O. This type of deployment leads to a temporally evolving mean concentration, C(x. t), of particles, and this affects the resulting statistics. In a later section random sampling errors in such estimates will be examined; here only biases of the mean are considered. The simplest estimate of U in A would be the ensemble-space average of all particle velocities in A at each time t: U(t)

=

(I f dx o[x - r(tln)lu(x. t»)I(~f n

A

/I

dx o[x - r(tln )]).

(2.14)

A

In this procedure all particles in A are averaged and the cumbersome Lagrangian labeling conventions are lost. In (2.14) the os ensure that only those particles in A at t are counted. As discussed below (2.6), the sum over n of these bs is the concentration field c(x, t) of particle mass. Thus (2.14) can be written V(t) =

L

It

dx(c(x,t)u(x,l»

(2.15a)

dxC(x,I),

where C(x, t) is the ensemble-space average of c(x. t). Note that this spatial average estimate ofU is essentially the flux of marked particles divided by the mean concentration. Separating c = C + c' and using the flux vs gradient-history law (1.2) for (u ' c' ) gives V(t) =

t

dx [U(x) C(x , I) - [dr aK~; r) VC(x, I - r)]I

L

dxC(x, I).

(2.1 Sb)

The estimate iJ is defined by the flux of marked particles and identically equals the area average of U only when the concentration c is equal to its mean. According to (2.15), Uis biased away from the spatial average of U by a term representing the eddy flux of particles which, since this flux is itself driven by the gradient of C, vanishes only when the sampling array (concentration of marked particles) is uniform. Because this bias is caused by an average nonuniformity of the sampling array, it can reasonably be called 'array bias'. Array bias is apparently distinct from the better-known diffusion bias, the former being caused by spatial variation of concentration while the latter depends on variation of the diffusivity. On closer inspection, however, diffusion bias is a special ease of the more general relation (2.15). The relationship of the two biases is made clearer by examining two extreme examples. . First, suppos~ the initi~l array is a point and the aver~ging region A is large enough to mclude all partl~les. 11us corresponds to computing U as the average velocity of an ensemble of partlcles deployed at a specific spot . the same physical situation used above to

S541

Observing the general circulation with floats

define Lagrangian statistics such as the particle statistics D and S. Integrating (2.15) by parts and noting that K(X, 0) = 0 gives

Ok(t) =

I dx [uk(X)C(X, t) + It dr a2Kkm(X, r) C(x, t - r)]II BrBxm A

0

dx C(x. t - r) A

(2.16) where the approximation follows from neglecting spatial variation of U and VK. Note that in this case 'array bias' has become the 'diffusion bias' identified in (2.12). In deployments at a point, variation of K causes mean particle motion because particles which move toward high K are apt to move farther than particles which move toward low K. From the Eulerian averaging perspective, this same effect results from preferential diffusion down the C gradient which is, itself, made asymmetric by variation of K. Second, suppose that a velocity estimate U is made by averaging the area average Vet) in (2.14) over a time interval much longer than T during which the mean particle concentration is approximately steady. This is a model for how mean velocity might be estimated by seeding an ocean basin fairly uniformly with floats and observing them over many eddy time scales. If the averaging time is long enough, (2.15) simplifies to

o=

L

dx[U(x)C(x) -

K

oo

IL

(x) . VC(x)1

dx C(x),

(2.17)

where KOC(X) is the t -+ 00 limit of K(X, t). In this case the mean velocity estimated from particles is the concentration-weighted spatial average of U plus a bias produced by the downgradient diffusion of particles. This bias is eliminated if the mean particle density C is uniform; it might be corrected if K<:X> and VC were known. If account is not taken of this effect, erroneous conclusions could be reached from average particle velocities. For example, if floats were repetitively deployed in a small region the mean velocity determined from their motion around this region, where C is decreasing outward, would be biased toward a divergent flow by downgradient diffusion. Unfortunately, similarly flawed analysis procedures do appear in the literature. It is also possible to estimate the diffusi~ity K directly from (2.3) using an area averaging procedure similar to that used to find U. Surprisingly, there is a difference between estimating the diffusivity in (2.3a) and that in (2.3b); although these are equivalent when averages are taken over all fluid particles passing the labeling point, they differ when only a 'marked' subset of particles can be observed. Applying the area averaging procedure to (2.3b) gives

,,(I) = =

(~

L

LdX[K(X,

I

1) (~ Ldx b[x -

dx b[x - r(loln )]u' (x, (0)[ - r' (10 - lin) r)C(x. I)

+

r(loln)])

r: dr(e' (X. to)V' {lolx. 4.)v' (4. + rlx, to) 11Ldx C(x, t,,). (2.18)

Applying the same procedure to (2.3a) gives

I1(t) = l1(t) -

L

dx(u' (x. to)e' (x.to)O(tlx)

IL

dxC(x. (0 ).

(2.19)

S542

R. E.

DA.vIS

where D is the mean displacement defined in (2.8) and the flux (o'e') could be related to C using t1.2). The diffusivityestimate Kin (2.18) is based on the form Ie = -(u'r')from (2.3b). It is a Cweighted area average of the true diffusivity Ie plus a bias proportional to (c'v'v'). I am unaware of any closure model with which even the size of this mean triple product can be predicted reliably. When the marked particles are uniformly distributed this bias must vanish since this situation is physically equivalent to the definition of Ie in (2.3) where an average is to be taken over all particles. However, I know of no straightforward mathematical manipulation to show this. In any case, avoiding biased diffusivity estimates, like avoiding biased mean velocity estimates. depends on making the array of marked particles (floats) uniform so that they are representative of all fluid particles. The estimate Kin (2.19) is based on Ie = (u'[x - r]) of (2.3a). It is equal to K plus an additional bias proportional to the mean displacement D = (x - r) and the eddy flux of particles. This additional bias vanishes when the sampling array is, on average, uniform because the eddy flux factor, proportional to the recent history of the mean gradient VC, then vanishes. But when the array has gradients of mean density and there are mean particle displacements, this additional bias could be significant. For this reason, it is clearly better to estimate diffusivities using the form (2.3b) for K, and this is the approach used here. It is, of course, not generally possible to know D exactly, but use of any reasonable estimate is to be preferred to making no correction, particularly as the error in Ie produced by an error in D is second order in sampling error (since (v') = 0).

2.3. Summary The relationship of the Eulerian statistics U(x) and Ie(x, l) to statistics of particle trajectories has been examined. The Lagrangian mean velocity V(tlx) generally depends on t, the time since the particle was at x, even when the Eulerian statistics are stationary. Under the convention that particles are randomly placed at t == 0, the initial Lagrangian mean velocity V(Oix) is also the Eulerian mean U(x). The particle diffusivity appearing in the passive scalar evolution equation (1.1) is lejk(X,

t) = -(u;(x, lo)rk(to - lix, (0»'

(2.20)

For t > 0, the case appearing in the transport law (1.1), -r/(to - tlx, to) is a statistical fluctuation of the displacement from time to - t to to experienced by the particle arriving at x at time to. The diffusivity is also defined for t < 0 when it describes uncertainty of source location of the particles arriving at a point. Recall from the discussion about (2.3) that the Eulerian u'(x, to) in (2.20) is, by definition, equal to the Lagrangian v'(/nlx, (0). In general the statistics of an ensemble of particles depend on both the flow in which they are embedded and the sampling array. Particle statistics like the mean and variance of particle displacement were examined in the case of repeated deployments at a point, a case where the mean concentration of particles in the sampling array is highly nonuniform. In that case the Lagrangian mean particle velocity is a concentration-weighted area average of U plus a bias (called diffusion bias) proportional to the spatial gradient of K. The time increase of particle displacement variance is proportional to a concentration-weighted average of the symmetric (with respect to direction indices) part of Ie plus a bias introduced by mean shear. The effect of shear amplifies the displacement variance and makes it difficult to infer Ie directly from displacements; it is better to measure K directly from its

Observing the general circulation with floats

S543

definition (2.20). The antisymmetric part of K is impossible to measure from the displacement variance since this part of K affects particles and mean scalar fields exactly like advection. Consequently ~ is only observable from the definition (2.20). The Eulerian average analysis of Section 2.2 applies to a more general class of particle arrays than the point release. It shows that, in general, spatial mean particle velocities are biased from the Eulerian mean by an array bias proportional to the mean concentration of particles. Diffusion bias is a special case of this array bias which pertains when the averaging area is large and all particles start from a single point; the diffusivity gradient enters according to how it affects the mean concentration which drives the bias. A practical method of using floats to determine U and K emerges from this analysis. First, U can be determined from the average particle velocity in a region so long as either the mean particle concentration is uniform or the diffusivity is known and corrections are made for array bias. Second, K can be found by area averaging the construct in (2.20) and this form is preferred over alternates which are equivalent when all fluid particles are included in the average. Diffusivity estimates are also subject to an array bias whose size is hard to estimate. The most general conclusion of this analysis is that biases of mean flow and diffusivity, and probably any quantity intended to represent properties typical of all particles, are poorly measured when the sampling array is nonuniform. This is not a mysterious statistical effect but rather something easily understood in terms of not being able to measure what is not observed. For example, an array bias of mean velocity would be an obvious effect of putting a large number of floats inside an area A and none outside. If one looked just outside the edge of A almost all the floats observed would be leaving A because that is the most likely place from where they could come. Fluid particles would be going both ways, but most of the ones that could be observed would be leaving and hence the float-based velocity estimate would be divergent. 3. MOLECULES, PARTICLES AND FLOATS

Ultimately, success of using the generalized advection-diffusion transport equation (1.1) with U and K determined from float observations depends on the relation of float behavior to the motion of molecules. First, utility of (1.1) requires that molecular effects either be negligible or be parameterized into Q. Second, it is necessary that float motion be adequately related to the motion of ideal particles so that they can be used to measure the transport properties U(x) and K(X, t). In this section molecular effects and the differences between floats and ideal particles are examined in order to find when float observations can be used to determine lateral advection and eddy transport of molecular species in the general circulation. Surface drifters are, of course, subject to instrumental errors associated with slippage through the water under wind and wave forcing; these errors are beyond the scope of this study. It is probably safe to assume that realizable subsurface floats do follow the horizontal currents at their position, but they clearly do not exactly follow vertical fluid motion. This will cause float dispersion and advection to differ from that of ideal particles. To the extent that floats stay on constant depth surfaces their motion is more directly related to the Eulerian average velocity (also an average at fixed depth) and less directly related to the diffusivity which measures the random motions of true continuum particles which move vertically. In what follows the differences between the motion of ideal

S544

R. E.

DAVIS

particles and floats caused by small-scale processes like internal waves and by lowfrequency vertical motion are considered separately. It is important to note, in this context, that U and K are themselves averages so that only a statistical similarity of floats and particles is required for float observations to be accurate. It is not necessary that a float follow its paired particle in any realization of the flow. Because isobaric floats are confined to a level, their concentration varies with the horizontal divergence V· Uu associated with internal waves. This causes particles to preferentially sample certain phases of the internal waves and makes the mean float velocity different from the Eulerian mean velocity. Outside surface and bottom boundary layers, lOW-frequency flows are nearly geostrophic and have little horizontal divergence. Consequently, there is no bias of float-derived U arising from the mesoscale or larger. Because the diffusivity K involves the displacement of ideal particles which follow both horizontal and vertical flow, both internal waves and low-frequency motions can cause errors in float-based estimates of the purely horizontal components of K. Floats are, of course, useless in observing components of K involving vertical velocity or displacement. 3.1. Molecular effects

In (1.1) the term Q(x, t) is the Lagrangian average convergence of molecular fluxes on particles at x, t. This is not simply the convergence computed from the molecular diffusivity and the mean 8 gradient, because molecular and turbulent fluxes are inseparably connected in the Lagrangian frame. As YOUNG et al. (1982) have shown, the dispersion of molecular species in sheared flow can greatly exceed the spread of fluid (continuum) particles plus the spread due to molecular diffusion alone. If (1.1) is to be useful in the face of shear dispersion, Q must be predictable from quantities measured by floats and not depend on molecular properties. YOUNG et al. (1982) considered advection and diffusion in a velocity field with vertically sheared horizontal currents. In their model these currents, modeled on inertial motions, had infinite horizontal extent and a nonzero minimum Eulerian frquency . There is no true dispersion of fluid particles becuase the particle diffusivity is proportional to the Lagrangian frequency spectrum of velocity at zero frequency (cf. DAVIS, 1983) and in the model both the Eulerian and Lagrangian frequency spectra vanish at zero frequency. On the other hand, molecules which can diffuse are dispersed. For realistic estimates of the internal wave shear field, however, the lateral diffusivity associated with shear dispersion is much less than 0.1 m 2 S-I. Does the difference between molecules and particles indicated by shear dispersion make the mean transport description gained from floats less useful? The large-scale diffusivity induced by shear and molecular effects is several orders of magnitude smaller than the lateral single particle oceanic diffusivities found by FREELAND et al. (1975), PRICE (1983) or by KRAUSS and BONING (1987) and is negligible in mean property transport. Further, ifthe Young et at. analysis were expanded to include finite horizontal scales of the horizontal motion, then fluid particles would also disperse; it is shown below that this dispersion is of the same magnitude as shear dispersion. Although shear dispersion is negligible in evolu~ion on the ~en~~al circulation scale of mean scalar fields, it can be quite important in the dispersal of m~lvldual property clouds. In this case the appropriate comparison is between the shear-mduced diffusivity and the particle-pair-separation diffusivity which describes the spread of particle clouds.

S545

Observing the general circulation with floats

3.2. Internal wave bias olU If floats do not follow the vertical motion of internal waves then vertical variation of horizontal flow will lead to horizontal velocity differences between initially paired floats and particles. RISER (1982) considered this quasi-Lagrangian nature of floats using an analysis based on simulating motion in fields of a few energetic modes. Here w~ examine the effect using an analysis based on a linearization of the particle motion equation much like that used to describe Stokes Drift. Let ~(tlxo, to) be the position of the float initially at xo, to and let r(tIXo, to) be the position of the particle initially paired with it. Since U and K are statistics, it is the statistics of r - ~ which are important. The mean velocity difference between an ideal particle and a point following the Eulerian mean U is the Stokes Drift. Using a Taylor series expansion around the labeling point, v = u(ro) + [r - ro] . Vu + ... , this is easily shown to be

(3.1) While (3.1) is only the linear approximation to Stokes Drift, comparisons (DAVIS, 1982, 1983) suggest it is accurate for all but very nonlinear flows when it is likely to be an overestimate. Isobaric floats move laterally like ideal particles but do not move vertically. Consequently, only the effect modeled in the second term of (3.1) contributes to the error (3.2a)

airH - ~II) = (r z • azUH)'

Unlike the full Stokes Drift, this term does not depend on surface and bottom reflection coupling up- and downgoing waves into modes. So long as the WKB approximation for vertical structure holds, the contribution of single wave component is easily shown (using internal wave relations between OzUH and r z) to be (r z ' 0zUH) =

-k

N 2 -w 2 2 2 w(~), w

(3.2b)

-I

where k is the horizontal wavenumber, w is the wave frequency, N is the buoyancy frequency and I is the CorioUs parameter. The internal wave bias can be estimated from a Garrett-Munk empirical spectrum (e.g. MUNK, 1981) modified to permit the horizontal anisotropy without which the bias vanishes:

<,',> ~ fb2

N ; E

r 0

iu

fN

dO ~ f dwD( O)H(j)

(2 OJ

--;"f> liZ , 2

(3.3)

where j is the vertical mode number, H(j) is the modal spectrum function in the model spectrum, 0 is the horizontal wavenumber direction and f dOD(O) = ~ R(j) = 1. The empirical constants suggested are scale height b = 1.3 km, surface-extrapolated buoyancy frequency No = 5 X 10- 3 S-1 and energy E = 4 X 10- 5 . The largest drift occurs if all wave numbers are parallel, say to the 1 axis; then with 1« N

(r z • azUl)

= -bNEn

I jH(j). iu

j=1

Because of the factor j, this sum is sensitive to the high vertical wavenumber part of H(j) and to the cut-off iu; for the GM79 form given by MUNK (1981) the sum depends

S546

R. E.

DAVIS

logarithmically onju' For perfectly collimated waves and i" = 250 (roughly a vertical scale of 1 m), the mean float-partjcle separation velocity near the surface (where N = No) is il,(rH - ~H) = 9rrbNo E = 7 mm s-'.

The Garrett-Munk spectra are based on the observation that the internal wave spectrum is nearly isotropic; even the still highly directional 2:rD(O) = 1 + cos 8 would reduce the near-surface drift to 3 mm S-I. Since the drift is approximately proportional to N, it decreases with depth becoming approximately 20% of the above values at 2 km depth. Consequently, even if the internal wave spectrum is quite directional in some places, the associated bias of float velocities will be negligible.

3.3. Internal wave dispersion The failure of floats to follow vertical currents causes them to disperse differently than water particles and might lead to erroneous estimates of the horizontal components of K. It is conservative to estimate the diffusivity error as being the same size as the total diffusivity induced by internal waves. As discussed ahove. if the Lagrangian frequency spectrum of velocity were the same as the Eulerian spectrum and vanished at zero frequency, then there would be no particle dispersion. But the nonlinear relation between partial and substantive time derivatives allows transfer of energy across the Lagrangian spectrum and leads to particle dispersion. Simulations by DAVIS (1982, 1983) have shown how particle dispersion occurs in wave fields with no energy in the Eulerian spectrum at zero frequency. Although still speculative, these studies suggest this dispersion can be predicted from a combination of Corrsin's conjecture that particle displacements arc poorly correlated with velocity plus the assumption that displacements are normally distributed. For particles passing through a point in an isotropic and homogeneous velocity field these assumptions lead to 2

d? d (rt) = 2(Ul(OIO, O)Ul(t\O, 0) =2 fdX(Ul(O, O)u,(x, t))C(x, t),

(3.4)

where Ut(tIO,O) is a horizontal component of particle velocity and C is the (assumed Gaussian) mean particle density at time I. Initially d(r~)/dt = O. With the Eulerian velocity statistics known. (3.4) provides a prescription for evolving (ri.). The theory also provides an analogous equation for (r;), but I believe it is inappropriate~ clearly stable density stratification opposes fluid particles dispersing vertically by introducing position-velocity correlations which are neglected in Corrsin's conjecture. It seems more appropriate to use the linear approximation

(r;(/» = 2

f' dt, J' dt (uz{O, t,)uz(O, '2»'

(3.5)

2

"

!)

Combining (3.4) and (3.5), expressing the velocity covariances in terms of the wavenumber-frequency spectrum for velocity. <1>, and carrying out the integral over x give

~ (ri) = 2 Jdw Jdk lI(k. w) cos (wI) exp (- ; (r2) ::: z

fdw fdk

<))

ww

(k • w ) 1 -cos(wt) 2' (J)

(ri) - ;

:~

=;: (~)].

(3.6)

Observing the general circulation with floats

S547

For the GM79 spectrum in (3.3) the k integrals in (3.6) can be performed analytically. The remaining w integrals and the differential equation were integrated numerically. The resulting horizontal diffusivity oscillates for approximately 20 inertial periods while approaching its asymptotic value (for N = No)

lim ddt
, _ 00

The persistence of the inertial oscillations in the model indicates that internal waves are weakly nonlinear, and indeed the zero-frequency Lagrangian velocity spectrum is much smaller than the Eulerian spectrum for w > f. Dispersion is dominated by lateral motion so a float's failure to follow r z may not affect its dispersion much. In any case, the total lateral dispersion produced by internal waves is a very small fraction of the observed lateral dispersion and of the same magnitude as that produced by shear dispersion. The K error caused by failure of floats to follow vertical motion should be even smaller. 3.4. Low-frequency vertical motion

Low-frequency vertical motions will cause the horizontal positions of paired floats and particles to diverge in the presence of low-frequency, vertically sheared horizontal currents. Consequently, floats do not follow water particle trajectories over long periods. In the transport equation (1.1) advection is represented by U, an Eulerian average at fixed depth. Averages of isobaric float velocities are not biased from U by their absence of vertical motion; indeed the lateral motion of ideal particles is biased from U by this effect. The diffusivity K is determined by the displacement of ideal particles which also move vertically. Thus failure of floats to follow vertical water motion will cause float-based diffusivities to be in error. The position difference between particle and float, or = r - ;, initially grows because r z does not follow the vertical velocity, i.e. orz = t· Uz • In the presence of vertically sheared flow, Or z then causes a horizontal displacement difference orH = t 2 . Uz • dzUH' The associated diffusivity error is of the order OK =u' orB, where u' is a typical horizontal velocity fluctuation. For large times this is to be compared with K/X) = u,2T, where T is the integral time scale. As shown in the following section, the smallest error in estimating K oo is obtained by using the smallest time t at which K approaches its asymptote. Assuming this time is of order T, then the minimum relative error is [

dK] K

=

Tu~,ZUH

(3.7)

MIN

In what follows, two separate approaches to estimating u z , and hence OK, are followed. First, suppose that density conservation UH· V'HP + uiJzp = 0 and the thermal wind relation r x (\UH = -gVHP both hold. Substituting Vp from the latter into the former specifies uiluu and then (3.7) leads to

[-dK] K

where

(J

. (0) la UHI UH = fT sm 2-' Z

MIN

is the angle between UH and

N

dzUH'

2

U'

(3.8a)

R.E.D ...V1S

S548

Second, suppose that changes of potential vorticity following a particle are of the sam.e order as those induced by the planetary gradient p = dfldy so that fil="l.' = O(puu)· This allows the vertical velocity U z to be estimated and (3.7) evaluated, giving

[

6K] K

0

mm

=

o[h/JTUI,I IdzUlIl]. f U

(3.8b)

where h is the vertical scale of U z' The relative error of K depends on the geographically variable factors sin(O) or It, the ratio of mean flow velocity to eddy velocity and the Lagrangian integral time scale T. As an order of magnitude example intended to be extreme even for strong currents or eddies, consider the case f= 10- 4 S-I, N 2 = 10- 4 S-2, /J 10- 11 m- I 5- 1 , drUII = 10- 3 S~I, sin (0) = 0.1, and h = 1 km which have been adjusted to make (3.Ha) and (3.8b) agree. The associated vertical velocity is proportional to UII and for aim 5- current is 0.1 mm S-1 (10 m day-I). The relative error in K is, from (3.7). proportional to Tand the ratio ulI/u'. Tofthe order 10 days (FREELAND el al .. 1975; PRICE. 19R3). u"lu' = 10 and the parameters above give a 6KI K = O( 1). Of the errors considered here. this is clearly the most serious. Still, it is significant only in the strongest currents or eddies.

=

0

\

3.5. Summary Even when accelerated by shear, the lateral dispersion produced by molecular diffusion is negligible compared with that produced by mesoscale motions. Thus parameterization of the source Q in (1.1) appears feasible without knowledge on the molecular scale. The bias of float-measured U caused by internal wave effects related to Stokes Drift is negligible in all plausible cases. Thus Eulerian mean velocities can he measured accurately with floats. Lateral particle dispersion caused by internal waves is comparable to shear dispersion and is also mueh less than the pOlrticlc diffusivities produced by mesoscale motions. Low-frequency vertical motions in highly sheared currents or eddies can cause substantial errors in the horizontal components of K estim~lted from floats. The diffusivity can be measured accurately in all but the strongest currents. 4. SAMPLlNG ERRORS

The intrinsic instrumental errors associated with floats. and the effect of these errors on measurements of U and K, were examined in Section 3. The conclusion was that both these quantities, themselves statistics, can be measured with useful accuracy from floats because the statistics of particles and floats do not differ greatly. Here we address the question of how many float data are required before U and K can he accurately determined. nut before turning to this it is necessary to re-evaluate the utility of the basic 'time averagc' framcwork upon which (1.1) is based. 4.1. Mean value or low-frequency filter?

To anticipate, many observations are required to accurately observe U and K and it will probably take many years to collect them. Little is known of ocean variability on these time scales, but observations now appearing (OWENS et al .• 19RH; ZENK and MOLI.ER. 1l)88~

S549

Observing the general circulation with floats

1989) suggest a substantial variation of even mid-ocean mid-depth currents occurring on the several-year time scale. Then how useful is any conceptual or analysis framework like (1.1) which is based on temporal averages? The approach taken here is the pragmatic one. It is not useful to define the average separating U from u' as the mean over all geologic time. Rather it must be an average over some finite time representative of a particular ocean climate and one for which averages can be measured. From an operational perspective, there is no difference between estimating an infiniteinterval mean from data taken over a period of length L or estimating from the same data the mean over a finite averaging interval, TA, so long as T A> L. The difference is in the accuracy with which the sample approximates the desired mean and what kind of data are required to know this accuracy. The averaging interval is crucial because uncertainty of averages is caused by the longest period variations in that interval, and in a typically red geophysical spectrum this variability increases rapidly with the averaging period. It is variability at zero frequency which affects estimates of the hypothetical infinite-interval averages of the statistics text. To know how well the mean over any period is determined requires many records of that length. Present instrumental records are largely limited to a few years duration, so it is realty only possible to ask how well sample means approximate the few-year mean U or Kin the presence of typical eddy variability on shorter scales. As longer records become available the statistical description of many-year variability will improve and it will become possible to ask how well longer-term means can be determined. It is likely, however, that the more important use of longer records will be to resolve variations between means over different few-year intervals rather than to estimate longer period averages. Basically, mean field equations like (1.1) are introduced because detailed fields cannot be observed but mean fields might be. In this context the observational objective is to use the shortest averaging interval for which the associated low-frequency filtered fields can be accurately measured. To assess the difficulty of determining sampling errors it is useful to review the process of estimating averages over period T A from records of length L < T A' If U is the true average for infinite TA and u is a stationary process with integral time scale T, then the typical error of the sample average over period L » Tis SCHMITZ,

«( 0 - Uf) = =

~ JL dt JL dr
()

1. Joc L

-00

0

dt(u'(O)u'(t)

= T (U,2) = 27[' ¢(w = 0), L

L

(4.1)

where is the double-sided spectrum of u as a function of radian frequency w. Although the rule that the L-Iong sample is equivalent to LIT independent samples is familiar, it disguises the fact that estimating T, or equivalently <1>(0), is as difficult as estimating the mean itself. The spectra of sea level and temperature records appear to increase without bound as w -+ 0, and one must suspect that velocity spectra are also poorly defined at low frequency. If spectra are sufficiently red infinite-interval time averages cannot even be defined and even if such averages can be defined, very long records will be needed to overcome the sampling noise described in (4.1). On the other hand, estimating the average over a finite time interval, T A, is more nearly feasible. Let U(TA) be the true average over period TA .

5550

R. E.

DAVIS

Then, in a manner similar to that leading to (4.1), the mean square difference between U( TA) and estimates 0 made from records of length L < TA is found to be

<[0 = U(TA)f> =

oo

J

dw (c.u)[sinc (wT A/21r) - sinc (wLI21r)f """ 2

J7lIL

dw (w),

~~

-00

(4.2)

where sinc (x) = sin (1rx)hrx. For spectra which decrease with increasing w the mean square error of U(TA) is less than (21rIL)(nITA); for red spectra this is significantly smaller than the error in estimating the infinite-period mean. In what follows, the sampling errors in estimating U and K are examined. In this, the notation of averages over an infinite interval is retained, but it is to be understood that U and K would, in practice, be averages over finite intervals and primed quantities would be deviations from these low-frequency components. In that case U and K, and the statistics required to determine their sampling errors, are substantially easier to determine than formulae for infinite-period means would indicate.

4.2. Estimating the mean velocity The Eulerian mean velocity U can be estimated as the average velocity of all floats in an averaging area A. If the total time of observation is L, this estimate could be written

iJ =

I J dt J n

dx o[x - r(t\n)]u(x, t)/2:

L A n

J dt J dx o[x L

r(t\n)],

(4.3)

A

where r(tln) is the position of float n and the ()s simply ensure that only those floats in A are counted in the sum over n. The relation of this estimate's true mean (iJ) (obtained from many realizations of the same sampling process) and the true Eulerian mean U is given in (2.17), where C(x) is the mean concentration of floats averaged over the time interval L. This discloses the array bias associated with K. It also shows that if X() is the center of A, then (iT) differs from U(xo) by an amount which depends on variations of U over A and on C within A. If C is uniform these biases vanish and if the diffusivity and mean concentration are known, they can be corrected. An appropriate correction would be to subtract from (4.3), something like _K

oo •

V(ln C)

+ JA dr C(r)[r - xol . VU /

L

dr C(r),

where gradients are evaluated at X(,. Implementing this would require the variation of U to be deduced from averages over regions around A. Although appropriate corrections can make iJ an unbiased estimate of U(Xo), this estimate is still subject to sampling errors which vanish only when the record length L becomes very large. The error variance is

(dU.l' =

(I~ J dl L

L

dx d[x - r(lln)]uHx, I)

/~

LL dl

dx d[x - r(llnnl)

(4.4)

The size of this complex statistic can be easily estimated only under simplifying conditions. In what follows a sequence of enumerated simplifications are introduced and (4.4) is

Observing the general circulation with fioats

S551

successively reduced to the simple (4.7). Intermediate results given allow use of more complex but less restrictive forms. (1) Let the statistics of velocity and float concentration be stationary over Land homogeneous over A. (2) Let the number of float trajectory segments through A be large. Then the denominator of (4.4), the total record length in A (float-years), can be approximated by its mean CAL. (3) Let Corrsin's conjecture apply. CORRSIN'S (1960) conjecture is the ad hoc, but apparently very accurate (cf. DAVIS, 1983), assumption that in computing statistics like (4.4) a particle's position and its velocity can be considered as independent random variables. Restrictions (1)-(3) reduce (4.4) to

(dUk)'

=

LLL L dl

dt

x

II n

dx

di [Ekk(X - i, t-I)

(o[x - r(tln)]o[i -

r(~m)])l/ (CALf,

(4.5)

m

where Ejk(x, t) is the time and space-lagged covariance of the j and k components of u' . (4) Let the record length L and the dimensions of A be large compared with the time and length scales of E. This reduces (4.5) to

J dx Ekk(x, t) I

N

(I>Uk)2 =

Joo -00

dt

A

I

N

n=1 m=l

(o[x - Snm(t)]) / CAL,

(4.6)

where the snm(t) is the separation between particle n and particle m at time lag t and the averaged delta function is its probability density. The sum of the terms with m = n in the integrand in (4.6) is the expression for the singleparticle Lagrangian time-lagged covariance derived from Corrsin's conjecture (DAVIS, 1983). The integral would then give (OUk )2 = lIL f~oo dtPkk(t) , where P is the Lagrangian time-lagged covariance defined in (2.4); this corresponds to the familiar rule that the number of degrees of freedom is the record length divided by the (here Lagrangian) integral time scale. From (2.3c) this expression can also be written (OUkk )2 = 2Kkk1L, showing that the same zero-frequency variability produces single-particle dispersion and variability of sample means. The terms with m ¢ n represent loss of sampling efficiency when the velocities of different particles are correlated, that is when particles are within an eddy correlation length of each other. (5) Let the concentration of particles be uniform and particle positions be roughly statistically independent. Noting the diffusivity's definition in (2.3c), defining the integral time scale Tk by (u,?)Tk = 2K':k =

J:ao dt Pkk(t)

and defining the integral length scale Ak by

(ui/)TkA~ = J:ao dt Jdx Ekk(X, t)

S552

R. E.

DAVIS

leads to the simple result for sampling error ( bU )2

= 2K:k

k

CAL

[1 +

(CA - I)

CA

li]. A

(4.7)

Here LIT is the equivalent number of independent samples obtained from a single float record of length L, CA is the number of t10ats typically in A, and AlA? is the maximum number of effectively independent sample~ which can he made in A at one time. The most efficient sampling is obtained from low concentration arrays with ;,2e« 1, when floats are separated hy more than an integral length scale. Then the number of equivalent independent ~amplcs is CA LIT. where CA L is the total record length in A, say as float years. To a first approximation the error hU varies as l/YA, the inverse of the averaging length. In the limit of large Cthe sampling error ceases to depend on the number of floats in A ~ in this limit the equivalent numher of independent observations which can be made at any moment has reached its limit At),:!. The resolution scale VA. of course. is determined in the analysis stage and is not a property of the array itself. Thus the analy!\t can select a posteriori A to achieve various resolution vs accuracy comhinations. Dut all combinations have the same ~U· vA product determined hy the sampling array conccntmtion and record length. 4.3. Estimating Ihe diffu.'iivity

The eddy diffusivity is defined in (2.3) as K,k(X. I) = -(v;(/ulx,/lI)dk( -llx.

tu)I>.

(4.8)

where d(rlx. III) = r(t

+ IlIlx. III) - r(tllix. 'll)

is the particle displacement from its lahcling point in time I; recall from the discussion about (2.3) that the Lagrangian v'(/nlx, 10 ) in (2.20) is, by definition, equal to the Eulerian u'(x. '0)' The basic approach to estimating this from float dahl is similar to that employed for velocity: the average is estimated hy summing over all floats and all In for which r(/lIlx. to) is in an averaging area A. Analysing the sampling errors of this construct is complicated because (a) K(/) has different errors at different I, even when it has reached its asymptotic value K" and (b) relating the typical sampling error to familiar two-variable statistics of particle velocity and displacement is difficult bec~lUse K is itself a product. Thc results above for sampling U simplify determining the uncertainty of I( estimated from an array of floats given the uncertainty of K made fmm a single trajectory. According to (4.7), if the array is sparse the uncertainty from a uniform array is approximately (CA) 1/2 times the uncertainty ohtained from a single float residing in A for the same time period L. Suppressing directional indices and letting a subscript I denote the estimate made from a single float found in A for () < t < L. the diffusivity estimate is ,(1(/)

= LI

fl. dq v'(q)d'( -tlq). ..

(4.9)

S553

Observing the general circulation with floats

where v(q) is the particle's velocity at time q and d(tlq) is its displacement from time q to time q + t. Note that (4.8) and (4.9) involve d'. rather than d. If it were not for sampling errors, either d or d' would lead to the same results but, as discussed below, there are two reasons for insisting on calculating with d' in practice. Following the procedure for U estimates, the expression for sampling variance is simplified in steps: (1) Let the distribution of v' and d' be approximately joint-normal. Then using the same procedure leading to (4.5), the mean square uncertainty of is

"1

[oK,(t)f

=

A JL dp JL dq [(v'(P)v'(q»(d'( -tlp)d'( -tlq» L (l

()

+(v'(P)d'( -tlq»(v'(q)d'( _·t\p»].

(4.10)

The first term in the integrand dominates for large t. The second term is bounded by the square of Kat), but for Ip - q\ « I the first grows as for small I and as t for t» T; for Ip - ql » T, (d'( -t\p)d'( -lip» vanishes. Note that the displacement covariance (d' d') appears in the first term ofthe integrand. If the total displacement d had been used in place of d' in (4.8) and (4.9), this would be replaced by the mean product of displacements and would grow as il whereas the covariance grows only as I. It was shown in (2.19) that the sampling error bias to the mean of estimated diffusivities is minimized if the mean displacement (d(t/q» = D(t) is subtracted from displacements before sample area averages are formed. Here (4.10) shows that the sampling variability is also minimized by working with d'. Even when the mean displacement is removed the term involving the displacement covariance grows with t. oo Thus it is essential to accurate estimation of K that (a) D be subtracted from displacements and (b) K(t) be estimated at the smallest t for which K(t) approximates Kat). (2) Let the Lagrangian time-lagged velocity covariance be stationary so

r

(v'(t + q)v'(q»

= (v,2)p(t),

T=

J:"", dt P(t) ,

K"'"

= (v,2)T/2.

(3) Let the observation period be L » T. Then (4.10) simplifies to

[dKI{t)f _ 4 1

[K""f -

L

'fl

Joo

-at) dr

[Jr

r-t

r+ f pcp) dp r

t

P(q) dq

+ 2P(r)r J:+t P(P) dp + Per)

J:::

(2r - p)P(P) dp

J.

(4.11)

The last two terms in (4.11) result from the mean squared displacement term in (4.10) and dominate the large time behavior. The sampling error from an array in the large t limit, as when estimating K'X- = K(t), is then lim [oK\(t)f = ~~ 1..... 00

(KOCf

CA L'

(4.12)

where K is an order unity constant which depends on the form of E(l) or equivalently the Lagrangian frequency spectrum of velocity. Figure 1 shows how sampling error increases with t for two particular time-lagged covariances.

S554

R. E. nAVIS

1.5

0.5

~

/

/

/

/

/

/

/

/

/

/

/

/

/

/

/

/

/

Fig. I . Standard deviation . ()lel(t). of ~ampling error for the diffusivity /(/) estimated from a single sample as computed from (4 . II) . The solid curve is for time-lagged correlation E(I) = exp( _;rI2) and the da<;hed IS f(lf E(I) == 1111 + Jr~t21. Beyond r "" 2 the curves arc parallel and proportional to 1 showing how the ..ampling error IOcrCll~C" with I even though Ie has reached its asymptote K '1 •

Beginning with the pioneering study hy fREELAND ft al. (1975), measurements of the diffusivity have been uncertain hecause K(t) often appears not to approach a constant for large I . For example. KRAUSS and B6NING (1987) present several plots of the Lagrangian time-lagged correlation (normalized forms of P(t» obtained from surface drifters in the North Atlantic. All these plots show rapid variation for I < 10 days with much slower variation around zero at larger time . But the approach to P = 0 at large I is ambiguous, leading the investigators to compute diffusivities hy integrating P not over the full record length. but rather to shorter times where it was hclieved the true covariance had reached a value of zero . If this were not done the associated diffusivities would continue to vary at all t. It is, of course. possible that the velocity fields contained some very weak. long time scale variability which causes K(/) to vary slowly at large t. It seems more likely that, in this case and many others exhibiting the same qualitative behavior. the large I behavior is a result of sampling errors. First. as Krauss and Boning note, the number of observations of K(t) generally decreases as I increases. making K less certain. Second, as shown here, even ifthe number of samples were constant the uncertainty of K(t) would continue to increase with t. With these facts in mind. the expediency of truncating the covariance at a finite time where sampling errors are still small seems quite reasonahle .

4.4. Summary An infinite-period mean velocity or diffusivity may not even exist for the ocean. The ~ncert~inty of any estimate ~f these hypothetical means is, in practice, large and Imposslbl~ to measure because It depends on very long period variability which cannot be

charactenzed. If, however, means are defined over observable lime spans then U,

K

and

Observing the general circulation with floats

S555

uncertainty estimates can all be determined. Thus the basis for averaged transport equations such as (1.1) is both conceptually sound and practically useful if the associated averages are defined over observable time periods. The procedure for estimating mean velocity from floats is straightforward: U(x) is approximated by the average velocity of all particles found in an area A around x. Corrections can be introduced to eliminate the 'array bias' discussed in Section 2. The sampling error is given approximately by (4.7); the mean square uncertainty varies inversely as the observational record length, the area A and, in typical oceanographic situations, the density of floats. The diffusivity K can be estimated in the same way and the sampling error of the asymptotic K oo is given approximately by (4.12). Accurate measurement requires using displacement fluctuations d', from which the mean displacement has been removed, rather than the full displacement in the definition (4.8). The most accurate determinations oo of the asymptotic diffusivity K = limt-.ooK(t) are made by measuring K(t) at the smallest t at which K(t) = K«J. Even when K(l) has reached its asymptote, sampling errors continue to grow as t 1/2 ; this is one reason why observed diffusivities often do not appear to approach an asymptote. 5. SIMULATION EXPERIMENTS

Numerical simulations of particle motion in complex random velocity fields were used to test the fundamental transport theory upon which the foregoing analysis is based and to provide a test of how well particle tracking can describe advection and eddy transport in a simulated general circulation. There are essentially two elements of the eddy flux parameterization in basic transport theory: (a) the flux's dependence on the history integral of the mean gradient and (b) the definition of the eddy diffusivity in terms of particle statistics given by (2.3). Both of these have been tested using two simulations closely connected with array and diffusion biases of mean particle velocities. The same general simulation procedure has been used to investigate the ability of floats to describe the circulation of a highly idealized circulation gyre including a model western boundary current and eddy inhomogeneity, The simulations employ two-dimensional non-divergent velocity fields representing inhomogeneous unconfined random velocity fields and random flows in confined regions with mean circulation. The questions addressed are (a) How is the diffusivity related to the Eulerian scales of the random velocity fields? (b) Does the model (1.1), upon which the analysis here is based, describe the transport of particles? (c) Are diffusion bias and array bias properly predicted? (d) How well can the mean velocity and eddy diffusivity be measured in flow inhomogeneous eddies and a concentrated boundary current?

5.1. The simulation procedure To simulate unconfined random motion, statistically homogeneous streamfunctions l/JH were generated from Fourier series employing 32 x 32 equally spaced wavenumbers periodic over distance 2.7l. The Fourier amplitudes evolved independently according to a first order Markov process with time constant TM and forced by normally distributed white noise whose energy was adjusted to provide the desired wavenumber spectrum. The

S556

R. E.

DAVIS

vector-wavenumber frequency spectrum of kinetic energy for the homogeneous eddy field IS:

(5.1) where the wavenumber and frequency integral of is the kinetic energy. For unconfined runs, intended to represent spatially 'red' spectra, v was zero and the peak wavenumber ko ranged from one to three (the fundamental wavenumber is unity). To simulate statistically inhomogeneous eddies, the velocity was derived from a streamfunction 11' obtained by modulating the statistically homogeneous streamfunction by a linear function of x according to 1/J = (1 + ax) . 1/JH' To simulate random velocity fields in a confined region, a basic eddy streamfunction tfJlI was generated from a Fourier sine series of 64 x 64 equally spaced wave numbers appropriate to the domain 0 < x < 1 and 0 < y < 1. Amplitudes were generated as in the unconfined example to have the spectrum (5.1), where the area average kinetic energy of 11'11 is the wavenumber and frequency integral of ell. Thinking of the confined flow as representing a gyre, v = 1 was used to force a 'mesoscale' spectral peak at ko = 32.7l' (a wavelength twice the smallest in the Fourier series). To simulate western intensified ·eddies, the eddy velocity was computed from the streamfunction 1JJ = exp ( - x12) . 'l/JII' To simulate gyre-like mean flows the total stream function was taken as 1/J + 'V where the mean-flow stream function was

'I'(x, Y) = VII sin (1Cy)ll -

xU 1 -

exp (-X/AU)]'

(5.2)

5.2. Parameterizing I( Can the diffusivity I«t) be related to the scales of the random velocity field? A number of suggestions for how the purely lateral components of the asymptotic diHusivity 1(00 depend on observable oceanic parameters have appeared in the literature. ROSSBY et al. (1983) suggest K
ku

II

TM

V'U

Wu

3

5 5 5

(J.2 1l.4

0.4H 1I.4H

3.4

n,411

S

2

3.4 3.4

0.09 0.16 0.26 0.29

3.4 3.4 4.3

n.}O 0.31

0.24

0.50

3.n 2.4

0.26

(J.Y)

1.3 4.3 3.0

0.40

2.4

0.39

1.3

0.68

3 3

3 3 3

5 5 4

}

(,

2

3

5 5 4

3

(,

2

5 S

}

4 ~

O.4R OAR O.4X

O.4S

n.M

~

0.45

~

0.50

QC QC

n.M

0.93

},4

K

n.32 0.26 0.27

considered here. Rather attention is focused on what might be called unconstrained oo turbulence and how the asymptotic K and the time dependence of K are related to the scales of the energetic part of the spectrum. The parameters ko, v and p, in (5.1) determine the characteristic length scales of the random velocity field while the inverse r.m.s. vorticity lIwo, the eddy turnover time TE = tprlu~ and the Eulerian time scale fM are characteristic time scales. The characteristics ofthe velocity fields for which diffusivities were measured are summarized in Table 1. oo The velocity variance and K define the integral time scale by T = 4K oo I~ (T is the double-sided integral scale and uf, is twice the variance of either velocity component). Figure 2 shows the time dependence of K(t)/KOC as a function of tiT for the extreme cases examined. All curves are similar, whether the Lagrangian time scale is set by a small Eulerian time scale TM, or by advection through slowly evolving ed~y motions as when TE « I'M' Furthermore, the time dependence can be crudely modeled as K(t) = KQO[l - exp( -tl1)]. This is significant because, as shown in DAVIS (1987), for a K of this form the integro-differential (1.1) becomes a differential equation involving only first and second time derivatives. If the energy-containing scales are most important in determining the asymptotic oo diffusivity then K should depend primarily on the r.m.s. streamfunction tJI O, the r.m.s. speed U(h the r.m.s. vorticity lV(h and the Eulerian frame time scale TM' On dimensional grounds then

u;,

(5.3) where/is a dimensionless function, W = 1/'(JwJu~and TE = 1/'(/u~is the eddy turnover time. The parameter W is a measure of the wavenumber spectrum's width or, equivalently. the

S558

R. E. D.\VIS t.~

,rr Fig. 2. The time variation of (he dl(fu .....vity ICC, I for randum "cI~ity fiekll with homogeneous and isotropic statisticll and VaflOUl\ Eulerian "pecHa . K( r) Willi cnmpuccd frnm (2.20).1(" estimated from

large t valuell and the intcgraillme T determined from ,,~ and 1\ .. 11 >. Run"lOdudcd have (kel'~.' TM ) trjpler~ 0.5.0.2). 2».0.4.1). (J .~.I ). P.4. ~) and (\. .. . ~) which rcpre~nc the ("treme value .. of 1"1 ;tnd flllI'I',/lI f. e",amlned (OoCC 'I ;ll1lc I).

n.5.

spread between the scales characterizing the energy- and enstrophy-containing scales. The ratio TM/Tr: is a measure of nonlincarity which hecomes large when particle velocity is changed more rapidly by advection acro~s !lipatia' ~tructurc!\ than hy time changes at a point; when this ratio is small the linearizing approximation iJlilI » u . V is accurate. If K'JO is indeed determined hy the cnergy-containin~ scales, K'#> IV!n should dcpcnd only on W and TM/rf~ ' To test this. Fig. :\ shows ~ '11>/'1'0 multiplied hy W to the powers -1.0 and 1 as a function of TMltr. The scaling 1(21. - 'I'f). that is no dependence on \V, scems to minimize scatter, but over the range of W examined the other choices "fC nearly as good; in any case the dependence on the spectral width parameter lV is small. at least over the small range of that variable which can he achieved with rca!\onahlc spectral shapes. Clearly. the Eulerian time scale also has a Mrong effect on ~ "" when TM is small compared with the eddy turnover time TF.' The most important point is that K'" docs appear to he determined by the energy-containing scales. particularly as they affect ~!(l and TM/r, .. None of the proposed seatings for oceanic diffusivities discussed ahove includes dependence on an Eulerian time analogous to TM' They must. then, be regarded as proposals only for fully nonlinear turbulence where the characteristic Eulerian time scale is determined by the parameters considered, say Uo and IPn. In situations dominated by wave-like motion or eddies generated hy rapid flow nver topography one would expect these forms to significantly over-estimatc the diffusivity. The suggestions K(X) -- lIJo. K~ -- uoL" and K¥. -- u~T/( are made equivalent hy taking Llf = UJ.I'I'II and TIC = fr:. The distinction is simply on how much LIC and TIC vary over the conditions encountered: Since these scales are all connected by W, which is a weak function of spectral shape, it is likely that all are reasonably similar in predictive ahility.

S559

Observing the general circulation with floats

0

0

0

o a

PI·l

0

x

i

•

I~

• 10- 1

. ,. •

•

•

cP

0

D

•

. :-

a

D

•~

0

D

=0

PI

A

4

A~ 4

A

Ae

A

" =-\

4

A

A

10

'tM/'t£

2 10

Fig. 3. Variation of ,,00 with the Eulerian time scale TM. the eddy turnover time TE = 1JIo/~ and the spectral width parameter W = 1jJoWo/u~ . "is scaled by the r .m.s. streamfunction lPo . The n = 0 points cluster well indicating I/° "" tpn so long as TM exceeds TE, that is the flow is reasonably nonlinear.

5.3. Diffusion bias The diffusivity K is defined in terms of measurable quantities by (2.3) or (2.20) and the diffusion bias of Lagrangian mean velocities is predicted by (2.12) or (2.16). Simulation of particle motion in statistically inhomogeneous random velocity fields then provides an opportunity (a) to test the basic transport model (1.1), including the effects of the timevarying diffusivity, upon which much of the analysis here has been based, and (b) to examine the counter-intuitive behavior predicted for negative diffusivities, K(t), when t< O. Simulations were based on statistically homogeneous streamfunctions from (5.1) with ko = 3, I-t = 5 and "l'M = 1. This stream function was modulated by 1 + ax. Since this modulation does not change Wand "l'MI'rE is large, the results shown in Fig. 3indicate that the diffusivity then varies as I«x, t) = (1 + ax)K(O, t). Particles were labeled at a particular Xo and tracked forward and backward in time to define the relative position r(to + tlX{h to) - 10. This was averaged over to and over 0 < xo < :;r/4 to obtain the mean relative position R(tlxn) - R(Olxo) which is D(tlxo) of (2.8). According to (2.12) the mean particle velocity a,Re,) differs from the Eulerian mean velocity (which here vanishes) by the gradient of K. For the particular imposed inhomogeneity

Rx(tIXo) = a

Jt

de K.u(O, 1),

(5.4)

()

showing that for particles passing through x = 0 the mean source and destination positions (before and after passing through x = 0) are both in the positive x halfplane. In the simulations the imposed spatial modulation causes particle displacements to have a very skewed probability distribution, and many records were required to achieve stable

S560

R. E. DAVIS 0.15

§

0.10

I.

/

,

CC"

I

...

......

0:."

O.O~

Fig. 4. The mean relative position RAt) - RAO) of particles in a random velocity field with energy which Increases with increasing x. See the text for a description of the experiment. The solid curve is the average taken from the simulations and the dashed curve is predicted by (5.4). Expected sampling error exceeds the difference.

measurements of mean particle displacements. It was also found important that particles be uniformly distributed well outside the labeling region because much of the diffusion bias of R is caused by a small fraction of the particles with the largest displacements both away from Xo and to it. Figure 4 shows the mean RAt) - RAO) based on a record of total length ('fioat years') of 120,000 TE, or 105 eddy turnover times, and the mean displacement predicted from (5.4). It is immediately clear from this that the simple description "particle motion is biased toward high K" is only half right. The mean position of particles passing through x 0, t 0 is on the high K side of x = 0 both before and after 1=0. This is predicted by (5.4) since KxX
=

=

5.4. Array bias and eddy fluxes As was discussed in conjunction with (2.14) and (2.15), the average velocity Oofmarked particles in a region A is related to the total flux of marked particles by

Observing the general circulation with floats

S561 (5.S)

where c is the concentration of particles (a field of delta functions) and C = (c) A is the area avera¥e and time average concentration. If the Eul~rian average fluid veloci~y vanishes then U is a direct measure of the eddy flux. Accord1Og to (1.2) the eddy flux 10 a steady concentration field is (

,

U:C I

')

=

00

ac

-K -/ c - . ) aXk

(5.6)

Because the area average particle velocity Udoes not vanish when V C is finite, it is a biased estimate of the Eulerian mean U, that is

OJ =

Uj

-

Kfk i.lnc.

ak

(5.7)

On the other hand, this array bias permits direct measurement of the eddy flux. In the present case, it also permits another test of the fundamental model, (1.1) and (1.2), of eddy transport. To test (5 .5), (5.6) and (5.7), unconfined homogeneous velocity fields with zero mean and ko = 3, f.1 = 5 and t'M = 1 were used. A time-invariant gradient of particle concentration was generated by removing any particle which crossed into x < n/16 and, at the same time, adding a particle at x = 3l.7r/16 (although the velocity field is periodic over c5x = 2.7r, the concentration field is not). The area average velocity 0 and concentration C were computed over different x-ranges of width n/16 using an averaging procedure equivalent to that which would be followed in mapping a mean flow from floats. Since the Eulerian mean U is zero, any mean Lagrangian velocity is due to what has been called array bias. The results (Fig. 5) were a quite uniform gradient ac/ax, consistent with the (xindependent) flux being proportional to the gradient, and an approximately constant cO, consistent with (2.15). The mean particle velocity 0 is downgradient and varies with x, reflecting the prediction that 0 is proportional to -a Jnc. Also shown in Fig. 5 is the flux predicted by a flux-gradient law based on the x-average mean gradient and the diffusivity defined by (2.3); this agrees very well with CU. The agreement with theory in this experiment supports the flux-gradient law with the diffusivity given by (2.3). This is apparently the first direct test that the large time limit of the TAYLOR (1921) diffusivity is the ratio of flux and gradient in a steady-state case. But perhaps more importantly, it demonstrates how mean particle motion in the Lagrangian frame is related to eddy flux in the Eulerian frame. Clearly the concentration of particles times their mean velocity (CO) is the flux of marked particles. From an Eulerian view this must be CU + (c'u') so that 0 - U is a direct measure of the eddy flux. From another perspective, U(x) is the average velocity of all particles as they pass through x, whereas Uis the average velocity of a particular class of labeled particles which may have a concentration gradient and therefore a flux through material surfaces moving at the velocity U.

5.5. Describing a gyre To provide a realistic test of the ability to measure general circulation transport processes with floats, a gyre circulation with a model western boundary current and

S562

R.E.DAVIS 0.4

0.02

cO

0.3

I \ I!,\/- , -'\./ V "V THEORY --~......-:;-::;:;L':-~'t;----::l 'I 1\

1',

~

\

0.2

c

I \

I

..,..1

I

I

I

I

\

0,01

~J

x Fig. S. Eddy flux and mean particle motion with a mean concentration gradient. The random velocity field is homogeneous, has periodic wavelength 2.7T and (kll' /-l, fM) = (3.5,1). Solid curve is the mean concentration C produced hy removing particles from x> 311l'116 and replacing them at 1l'/16. 0 is the mean particle velocity in small x-ranges. cO, the total particle flux. is approximately independent of x and approximately equal to predicted flux (marked THEORY) from a fluxgradient law and the diffusivity K oo of (2.3). The scale for 0 and C is to the left and for fluxes is to the right.

inhomogeneous eddy field was simulated as described in Section 5.1. The eddy energy parameter Q in (5.1) was adjusted so that the maximum eddy velocities outside the western boundary region were approximately (U /2 ) = (v/2) = 1. The mean-flow boundary layer thickness and the mean-flow amplitude in (5.2) were taken as Au = 1116 and Uo = 1/5, giving a maximum boundary layer velocity of 16/5. The Eulerian Markov time rM was taken to be 0.01, corresponding to approximately 2iE' To allow direct measurement of sampling noise, 16 separate runs of length L = 4 were made in each of which 64 'floats' were initially deployed on a regular 8 x 8 grid with spacing 118. For descriptive purposes, Fig. 6 shows a few trajectories in the southwest corner of the gyre. Despite the fact that the simulation has no dynamical fidelity. the general character of individual trajectories is similar to that of oceanic float trajectories. They variously show periods of aimless wandering, segments of relatively direct transit and occasional quasiclosed eddies; track A, and to a lesser extent track n, even show a model boundary current 'ring.' Typical of the full data set, a large fraction of the trajectories pass through the boundary current; but because the speeds are high there, each trajectory adds few float years to the data base for the region. As discussed below, the resultant particle concentration is about the same in the boundary current and outside it. The general pictures of flow presented in Fig. 6a and b are quite different; the flow in Fig. 6a has a tendency toward southward flow not found in Fig. 6b. The examples in the two figures were selected to demonstrate how relatively small-scale eddies can produce trajectory perturbations which have large time and space scales. In fact, the correlation of long time displacement and instantaneous velocity which produces a nonzero diffusivity is

SS63

Observing the general circulation with fioats

.4~--~------------------------------------~



c

B

.3

,

.2

.1

o~~~~~~~~~~~~~~~~~~~~~

0.

.1

.2

.4

.:5

.5

.6

r

,....

't '

.1

., \

.f....

D

(b) 0 0

.1

.2

.3

z

B .4

.5

.6

Fig. 6. Typical trajectories in the southwest corner of the simulated gyre. Trajectories begin at the lettered positions. Trajectories A, Band C in (a) were selected as tending to the north while D, E and F tend to the south . Note the large-scale differences in the trajectories introduced by small eddies and how trajectories beginning at the same position differ substantially.

also a measure of how eddy velocities tend to produce long-lasting perturbations to trajectories. But if the two maps in Fig. 6 were obtained from separate time periods, most analysts would conclude the large-scale flow differed between them. An alternate perspective on the same phenomenon is provided by contrasting the trajectories A and E, which begin at the same site, or Band F which also have a common origin. Figure 7 shows the result of estimating the 'general circulation' flow by averaging float velocities over spatial regions. The arrows with heads indicate the velocity averaged over all 16 runs and over the (non-overlapping) 3/64 by 3/64 areas centered at t~e arrow tails.

S564

R. E.

DAVIS

.5r---------------------, -----

U:I

%~----~---~--~----~~--~--~ I

Fig. 7. Estimating mean velocity from averages of float velocity in spatial regions. The arrows with heads are the mean velocity estimated from all floats; these estimates evidence no bias because the float concentration is uniform. Mean velocity was also estimated from the subset of floats 'deployed' in () < x < n.6 and 0 < y < 0.15. The concentration of these floats is contoured (in arbitrary units). The headless lines radiating from the arrow tails are the mean velocity of this subset minus the mean from all floats.

For this calculation the average velocity was based on all floats, regardless of where they were 'deployed.' The concentration field from the totality of floats was relatively uniform, so that array bias was minimal and indeed no bias from the area average true mean flow was detectable. For contrast, mean velocities were computed in the same manner but using only those floats 'deployed' in the region 0 < x < 0.6 and 0
K;

Observing the general circulation with floats

S565

0.01

\ '4,-

\.............

o Fig. 8. Profiles of components of the asymptotic diffusivity /c"" across y = 112 of the gyre simulation. The inequality of "xx and /Cyy near x = 0 is the result of the kinematic boundary condition u' = 0 at x = O. The small but significant variation of 0 near x = 0 is predicted to lead to a Stokes Drift too small to be measured.

a larger antisymmetric K:yrx; confined to the boundary layer. These result from the interaction of the eddy field, with its nearly isotropic Eulerian structure, and the boundary current shear. The quite noticeable oxt4r is, according to (2.12), associated with a Stokes Drift of the order V = 0.01, but this is too small to be observed in the face of sampling noise. Figure 9 shows three different measures of the mean flow U perpendicular to the boundary along y = 112. The solid curve of O(x) = 1I2t[RxCto + tlx, to) - R.lto - tlx. to)] corresponds to centered time differences and shows no significant deviation from the true Eulerian mean U = O. The curves for Ut(x) = lIt[RxCto + tlx, to) - xo] and U2(x) = lIt[xo - RxCto - tlx, to)], however, show significant, and opposite sense, deviations at both boundaries resulting from diffusion bias. As seen in Fig. 8, because Kxx vanishes at the boundaries it has substantial near-boundary gradients. As shown in (2.10), this causes the mean displacement leaving x to be toward high Kxx (hence UI is away from the boundary), while the mean displacement of particles arriving at x is away from high IKxxl (U2 is toward the boundary). The effect is smaller near x = 1 because the diffusivity gradient is small there. Figure 10 shows the east-west profile of northward flow V at y = 112. The true imposed Eulerian mean, the centered-difference particle velocity Vex) = 1I2t[Ry(to + tlx. to) Rito - tlx, to)], and 10 times the difference are shown. The difference varies little across the region, showing that errors in measuring a strong boundary current are no greater than those in measuring a weak broad-scale interior flow because the float concentration is about the same in both regions. If particles were followed only for a short period after being released it would be more difficult to measure the strong current because the particles in the fast-moving flow would be useful for only a short time. But in the present

S566

R. E .DAVIS 0.2

,,

I

- O.20.!-U...a...LLL.J.~~""""''''''''''..j..LJL..L..I..u.JL..L..I...I..I..J~.&....L..t....~~.u.J'''''''''~·

Fig. 9. Three measures of the mean flow V along y = 112. The solid curve is the centered time-difference O(x) ;:::: [R ..(ro + r\x, to) - R ..(to - t\x, to)]l2t with t = 11256. The long dash curve is based on displacements from x, VI (x) "" [R"(41 + tlx, 41) - xol/t with t ~ 1116. The short dash curve is based on displacements arriving at x, U2 (x) = [.to - RAto - rlx, to)]lt also with t ;:::: 1116. 0 does not significantly differ from U = 0 but there are significant diffusion biases of the other two ~urves near the boundaries at x = 0 and x = 1 where 1(.... varies rapidly.

2

x

Fig. to.

Measures of V(x) at y == 112. The solid curve is the centered time-difference ~(x) == [Ry(to + tlx, to) - Rx(to - fix, f01/2t with t = 1/256. The short dash curve is the true Euler-

ian mean. The long dash curve is 10 times the absolute value of the difference and is primarily the result of sampling error.

S567

Observing the general circulation with floats

0.02

o \

o

\

o

~,

"

0,

o

0

o

0.01 o

0 0

__ , 0

o

\

o

IJ

0

o 000

o

0

8

\

,

0

0

0°

tlO

o

...

CC\/

oCS

0

x

Fig. 11. Sampling error of mean velocities along y = 112. The variance of 0 and 9 over eight runs of duration L are plotted as symbols; «60)2) are squares and «oli)2) are circles. The sampling errors. 2K""/L. predicted by (4.7) are shown as curves, solid for 0 and dashed for 9.

case the total observational record length is roughly the same in all regions although a much larger number of particles pass through the boundary current per unit time. As (4.7) indicates, sampling error is determined primarily by the mean particle concentration times the record length which is not directly influenced by current speed. Figure 11 shows profiles of the sampling errors for 0 and If as deduced from variation between the eight runs of duration L and the sampling error variance 2K
5.6. Conclusion The results of Section 5.2 suggest that the time variation of eddy diffusivities in different flows may have a relatively universal form well approximated by exp( -2ItllT). The approximately exponential form is significant because for such a diffusivity the history integral form of (1.1) can be simplified, leading to a tracer evolution equation of purely differential form with one order higher time derivative than the advection~iffusion equation. It is also shown that, at least for the flows examined, the diffusivity depends primarily on the energy-containing scales of the eddy field including the Eulerian time scale. Thus accurate predictions of the diffusivity must involve both the eddy field's Eulerian time scale and the more commonly employed eddy turnover time. The results of Sections 5.3 and 5.4 serve to verify the theoretical transport equation (1.1) upon which the predictions here are based. They confirm both the flux vs history-of-mean-

R. E. O"VI!'

S568

0.006

o

0 A

6

o

0004

C 6

o

o

0.002

o IE)

00

~

A

AOA 6

0

0

A

OO~A8

o ~~6a C CQ,.,aII 0 0

~ dJ'o ~rf

6

o

ao

A

0

6-

00

0.001

0.002

0.003

.... ·(IIL)'12

Fig . 12. Sampling error of eddy diffusivities along y = 112 . The standard deviations of K(I) over eight runs of duration L are plotted agamst (he associated K" (II L) 112. Typically K varies by a factor of three ovcr the profile. Square!. are for K .. {r = \/32). ,irdes {m K\'V(r = \/32), lriangicll for Ku
gradient form and the diffusivity'sdefinition in (2.3). The specific predictions from (\ .1) of diffusion bias and array bias of mean particle velocities are confirmed. These biases are minimized when velocities at x are based on position differences over an interval centered on the time of arrival at x and when this interval is small . Section 5.5 provides an example of floats being used to measure the mean transport processes in a simulated gyre. Particle trajectories are qualitatively similar to oceanic float tracks and disclose large-scale differences hetween trajectories produced by the cumulative effects of small-scale eddies. Strong concentrated boundary currents nre found to present no particular problem to float-based observations beyond the basic trade-off between resolution and sampling uncertainty such that (dU)2 is inversely proportional to CA L, the total record length in a resolution cell of area A. The simulations show the sampling error estimates of Section 4 to be accurate and, particularly, show that errors in estimating the asymptotic eddy diffusivity are minimized by using the smallest value of t for which K(t) =K"". The simulations also demonstrate the forms of mean velocity bias predicted theoreticany . Figure 7 shows a qualitative example of how array bias can distort mean velocity maps if there are strong gradients of float concentration. Figure 10 shows how diffusion bias. caU!~ed by gradients of the diffusivity, differs for particles arriving at a point and for those leaving the same point. It also shows how this bias is minimized by using a small

Observing the general circulation with floats

S569

sample interval to define velocity from position and by centering that interval on the labeling time at which the particles averaged are at the same position.

6. DISCUSSION

This study arose out of a number of questions colleagues have raised about using floats to observe transport processes in the general circulation. Certainly the most basic and most interesting question was "Isn't there some better way to use fioats than as moving current meters?" The best answer I can give is only a start. Essential to the concept of a general circulation is the notion that the typical characteristics of the overall process can be described without describing all the details. The quantitative descriptions of 'typical' known to me are all statistics taken in some combination of an Eulerian or Lagrangian coordinate frame. After years of worrying about this problem, I believe that Eulerian statistics are intrinsically simpler than Lagrangian statistics. The basic reason is that Lagrangian coordinates are essentially redundant because a particle can be labeled by any time/position pair along its trajectory. As a consequence, Lagrangian statistics are cumbersome and Eulerian measures are to be preferred where they suffice. For example, in a statistically steady flow. advection is described by a relatively simple map of the Eulerian mean velocity U(x). The Lagrangian mean velocity V(/\X), on the other hand, depends on both x and time since the particle passed through that labeling point. The Eulerian mean velocity is probably the most efficient description of advection, but this does not mean that fixed point observations are to be preferred over float-based observations in describing advection. If the Eulerian mean involves only an average over time at a fixed point, then moored observations are probably the only way to determine U(x). But it would be perhaps more consistent with the notion of a general circulation to include some spatial filtering in the definition of U so as to remove small-scale topographically linked features. Floats are ideally suited to estimating such a space-time average, so part of the answer to how to use floats is that measuring velocity with moving current meters is appropriate and may be preferred to using true fixed-point instruments. The basic transport processes in a flow are most easily described in the Lagrangian frame, so much so that the Lagrangian description of processes like advection and dispersion approach being trivial. In particular, it is necessary to adopt a Lagrangian perspective to describe the effects of eddy variability on mean transport. The transport model developed in DAVIS (1987) and discussed in Section 2 provides a physically direct connection between the Lagrangian transport process and mixed Eulerian-Lagrangian statistics of particle motion, specifically the TAYLOR (1921) diffusivity. I am unaware of any comparably direct diagnosis of a flow's transport capacity which can be made using fixedpoint observations. Thus another part of the answer on how to use floats is that only a psuedo-Lagrangian approach can describe the effect of eddy processes on mean transport and that at least the single-point diffusivity should be included in any description of the general circulation. Thus the use of floats as space- and time-averaging current meters sho~ld be expanded to use their quasi-Lagrangian character to estimate the diffusivity. It is also frequently asked if realize able floats are adequate to describe ocean transport since they do not follow the trajectory of a true material particle. The discussion of Section 3 addresses this for the cases of Eulerian mean flow and the Taylor single-particle

S570

R. E.DAVIS

diffusivity. Basically, realize able floats are adequate for measuring horizontal or isopycnal transport because the statistics of floats and particles are reasonably similar. Present-day floats are clearly inadequate for direct determination of anything about cross-isopycnal transport or low-frequency vertical advection. They do not follow particle paths for long periods, so in a laminar flow involving vertical motion they cannot adequately describe advection. Even in a highly turbulent flow the vertical separation between paired floats would differ consistently from that of paired fluid particles. Dut for the lateral transport process.es determining mean transport in an oceanic parameter range where dispersion is large, the imperfections of floats do not significantly prejudice their measurement ability. Finally, it is frequently asked in connection with both Eulerian and float-based measurements "How many data are required to measure the general circulation?" There are two distinct aspects to this que~tion. The narrow aspect is dealt with in Section 4 where the relation of sampling error, the scales of the flow and the nature of the sampling array are examined. This shows that the squared sampling error from float arrays varies inversely to the total 'float years' in a resolution cell just as it varies inversely to the record length in a time series. The discussion also explains why observations of the time-varying oceanic diffusivity 1«(/) rarely show the theoretically expected asymptote to 1(0fJ . First, fewer data arc generally llvailable for large' than for small I . Second, as shown in Section 4. for a fixed record length the sampling error of 1«(/) increases roughly as ,1f2. Consequently, measurements of 1«/) typically hccome progressively less accurate as t increases and the best estimates of the asymptotic diffusivity are ohtained at the smallest 1 for which 1«/)

2::;

1('6,.

However, the more troubling aspect of the sampling problem is the fundamental question of how to define and use time avenlges in a red-spectrum process such as the ocean circulation appears to he. This is, of course, not a prohlem of float observations but rather of the attempt to quantitatively describe 'typical' features of the general circulation using statistics. It is suggested here that the notion of the 'mean' be replaced by that of a low-frequency. or secular-scale. component defined hy a spatial smoothing appropriate to filtering out local topographic variahility and a temporal smoothing appropriate to ocean circulation climates. Operationally, measuring this sccuhlr scale is equivalent to measuring a temporal mean until enough ohservations have heen mmJe to detect the secular variation . But adopting this secular-scale perspective makes it possihle to at least define statistics with which to dcscrihe the typical and. in principle. fcasihlc to attempt such a description in the foreseeahle future . Acknowlt'dgt'mt'nts-This work was 'iupported hy the Natinnal Science Foundation under grant OCE·87·09423 and by the Office of Naval Research contract NOOIll4·Xt)·J.J(l4n.

REFERENCES COLIN DE VEIlI>IEJE

A. (19143) Lagrangian eddy !IIatistics from surface drifter.. in the eastern North Atlantic.

Journar of Marint Rt"tQrcl1. 41. 3'7S-31JI-I.. COllSIN S. (I%() Progrc!ls report on some turbulent diffusion rCM:arch . In: fmc. Symp . 0"

Atmospht'ric

Di/fwion and Air Pollul/on. Academic Prcss. New York. pp. lflI-I64 . DAVIS R. E . (19M2) On relating Eulerian and Lagrangian velucity statistic,: IlInglc particles in homogeneous flows. Journal of FlUId M~cha",cs. 114. 1-20. DAVIS R. E. (19M3) Occamc propcny transport. lagrangian particle !\tatl'itK~. and their prediction . Journal of Murine Rt~ea,ch. 41. 1ft).. I !}.. ,

Observing the general circulation with floats

S571

DAVIS R. E. (1987) Modeling eddy transport of passive tracers . Journal of Marine Research. 45, 635-666. FREELAND H. J., P. RHINES and H. T. ROSSBY (1975) Statistical observations of trajectories of neutrally buoyant ftoats in the North Atlantic. lournal of Marine Research , 33, 383-404. HOFFMAN E. E. (1985) The large-scale horizontal structure of the Antarctic Circumpolar Current from FGGE drifters . lournal of Geophysical Research, 90,7087-7097. KRAuss W. (1986) The North Atlantic Current. lournal of Geophysical Research, 91,5061-5074. KRAUSS W. and C. W. BONING (1987) Lagrangian properties of eddy fields in the northern North Atlantic as deduced from satellite-tracked buoys. Journal of Marine Research , 45 , 259-291. KEffER T. and G. HOLLOWAY (1988) Estimating Southern Ocean eddy ftux of heat and salt from satellite altimetry. Narure, 332. 624-626. McNuuG. J., W. C. PATZERT, A. D. KIRWAN, Jr and A. C. VASTANO (1983) The near-surface circulation ofthe North Pacific using satellite tracked drifting buoys. Journal of Geophysical Research , 88, 7507-7518. MUNK W. (1981) Internal waves and small-scale processes. In: Evolution of physical oceanography, B. A. WARREN and C. WUNSCH , editors, MIT, Boston, pp . 2~291. NOWLIN W. D .. Jr. S. J . WORLEY and T. WHITWORTH III (1985) Methods for making point estimates of eddy heat ftux as applied to the Antarctic Circumpolar Current. lournal of Geophysical Research , 90, 3305-3324 . OWENS W . B., P. L . RICHARDSON . W. J. ScHMrrz. H. T. RossBY and D. C. WEBB (1988) Nine-yeartrajectoryofa SOFAR float in the southwestern North Atlantic. Deep-Sea Research, 35, 1851-1857. PRICE J. F. (1983) Particle dispersion in the western North Atlantic (unpublished manuscript) . Results quoted in chapters 4 and 5 of Eddies in marine science. A . R. ROBINSON. editor. Springer-Verlag, Berlin, 609 pp. RiCHARDSON P. L. (1983) Eddy kinetic energy in the North Atlantic from surface drifters. Journal of Geophysical Research. 88.4355-4367. RICHARDSON P. L. (1985) Average velocity and transport of the Gulf Stream near SSW . Journal of Marine Research, 43, 83-111. RISER S. C. (1982) The quasi-Lagrangian nature of SO FAR floats . Deep-Sea Research. 29.1587-1602. RISER S. C. and H. T. ROSSBY (1983) Ouasi-Lagrangian structure and variability of the subtropical western North Atlantic circulation. Journal of Marine Research. 41. 127-162. ROSSBY H. T .• S. C. RISER and A . G. MARIANO (1983) The western North Atlantic-a Lagrangian viewpoint . In: Eddies in marine science, A . R. ROBINSON, editor, Springer-Verlag, New York, pp. 6Cr91. SCHMITZ W . J. Jr (1989) The MODE site revisited. Journal of Marine Research, 47,131-151. SCHMITZ W. J., Jr and W. R. HOLLAND (1982) A preliminary comparison of selected numerical eddy-resolving general circulation experiments with observations. Journal of Marine Research, 40, 75-117. TAYLOR G. J. (1921) Diffusion by continuous movements. Proceedings of the London Mathematical Society, 20. 196-212. YOUNG W. R., P. D. RHINES and C . J. R. GARRElT (1982) Shear-flow dispersion. internal waves and horizontal mixing in the ocean. Journal of Physical Oceanography, 12.515-527. ZENIC W. and T. J. MOLLER (19HK) Seven-year current meter record in the eastetn North Atlantic. Deep-Sea Research. 35. 1259-1268.