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The use of simulated drifters to estimate vorticity
Journal of Marine Systems 29 Ž2001. 125–140 www.elsevier.nlrlocaterjmarsys
The use of simulated drifters to estimate vorticity Dylan D. Righi ) , P. Ted Strub College of Oceanic and Atmospheric Sciences, Oregon State UniÕersity, 104 Ocean. Admin. Bldg 104, CorÕallis, OR 97331-5503, USA Received 15 October 1999; accepted 23 August 2000
Abstract A primitive equation ocean model is used to generate trajectories of simulated clusters of drifters in the California Current ŽCC. region. These trajectories allow us to evaluate a least squares ŽLS. method of estimating vorticity and vertical velocity along a cluster’s path. Two clusters provide examples of successful and less successful estimates of vorticity and vertical velocity. Our analysis quantifies the dependence of estimate quality on several parameters that can be used as error predictors in the LS estimate of vorticity: cluster separation, number of drifters in a cluster, and cluster shape. A combination of cluster separation and ellipticity shows the most promise as an indicator of quality for the vorticity estimate. q 2001 Elsevier Science B.V. All rights reserved. Keywords: Drifters; Vorticity; Vertical velocity
1. Introduction Estimates of vorticity and vertical velocity in the upper ocean are difficult to make but are important fields for a number of reasons. The dynamics of the regional circulation are often more readily diagnosed in terms of vorticity, especially around frontal structures ŽShearman et al., 1999.. Vertical velocity also plays a strong role in current dynamics and is even more relevant for biological reasons: the vertical movement of nutrients and organisms can change the nature of entire ecological systems. Drifters, measuring platforms that attempt to follow the same parcel of water, are ideally designed to
) Corresponding author. Tel.: q1-541-737-5751; fax: q1-541737-2064. E-mail address: [email protected] ŽD.D. Righi..
provide estimates of changes along the trajectory of the water parcel. If there are a number of nearby drifters Ža cluster., the relative motions of these drifters can be used to estimate vorticity, divergence and ultimately, vertical velocities. The more drifters in a cluster, the more certain these estimates will be Žin a statistical sense.. However, it has been rare to find large numbers of drifters seeded in clusters, and the chance occurrence of drifter clusters is rarer still. This may change as the relative cost of drifters Žand tracking. continues to decrease in comparison to ship time. Motivated by this, we are carrying out a study of a least squares ŽLS. method that uses the relative motions of drifters in a cluster to predict vorticity. We generate simulated drifter trajectories at a fixed depth in a primitive equation model of the California Current ŽCC. system. Through a comparison of the LS estimates and the data saved along simulated
0924-7963r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 7 9 6 3 Ž 0 1 . 0 0 0 1 3 - 6
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drifter trajectories ŽU, V, z ., we estimate how well the cluster methods predict vorticity. Our particular interest here is how the accuracy of the prediction depends on parameters that can be derived from the drifter data alone, such as cluster separation, number of drifters in a cluster, and cluster shape. The LS method of estimating vorticity, divergence, and deformation rates from groups of current-following drifters for oceanic application has been described by Okubo and Ebbesmeyer Ž1976., Okubo et al. Ž1976. and, independently, by Molinari and Kirwan Ž1975.. Molinari and Kirwan Ž1975. apply the method to clusters of three drifters in the
western Caribbean Sea and achieve mixed results: two clusters provide reasonable vorticity estimates, while results from two other clusters are unrealistic. Niiler et al. Ž1989. use triads of drifters to estimate vorticity and divergence in the CC and conclude that the OkuborMolinari cluster method results in good qualitative agreement Žcorrect sign. with hydrographic estimates. Paduan and Niiler Ž1990. and Swenson et al. Ž1992. use clusters of three to five drifters deployed in the CC region to estimate vorticity and vertical velocity in the strong CC jets: both result in LS vorticity estimates that agree with the sense of the cluster rotations. Kirwan Ž1988. dis-
Fig. 1. POMrPWC model relative vorticity on s-level 1 for model date August 17, 1996.
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cusses limitations inherent in the LS method and outlines possible error sources. Results from some of these studies are discussed below and compared to our results. This paper is organized as follows. We describe the Pacific West Coast model and its use to provide simulated drifter trajectories in Sections 2 and 3. Section 4 describes the OkuborMolinari LS method and defines the parameters we used to predict the quality of the LS estimate. In Section 5 we present the results of the LS estimate for two example
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clusters and in Section 6 we discuss error indicators based on cluster characteristics. 2. Pacific West Coast model The model we are using to generate the simulated drifter trajectories is a version of the Princeton Ocean Model ŽBlumberg and Mellor, 1997. that has been implemented for the Pacific West Coast at the Naval Research Laboratory ŽClancy et al., 1996.. The
Fig. 2. Spaghetti plot for the entire drifter data set. Also shown are the release points of the 11-drifter clusters Žblack circles.. Drifters are released every 2 months.
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1
8 grid that 12 covers the ocean from 308N to 498N and from the coast out to 1358W. Vertically, the model has 30 sigma levels which are concentrated in the surface and bottom to resolve the boundary layers. The bottom depths are from the Navy DBDB5 bathymetry database. Depths shallower than 10 m are considered to be land. The open boundaries are defined by 1 nesting POMrPWC within the global 8 Navy Lay4 ered Ocean Model. We have 6.5 years of model runs from July 1993 to December 1999. As an example of the model POMrPWC model is set up on a
velocity structure, Fig. 1 shows the relative vorticity at the surface for model day August 17, 1996. This snapshot shows a fully developed southward flowing CC jet system with strong offshore meanders Žthe jet structure shows up as areas of strong gradients between the positive vorticity on-shore side of the jet and negative vorticity offshore side..
3. Simulated drifters To simulate the trajectories of drifters in the CC region, we seed the POMrPWC model with simu-
Fig. 3. Trajectories for the inner six drifters in cluster a76 and a287 Žlower left panel.. Each trajectory represents 60 days of position data. Cluster outlines are shown at 1-day intervals for C76 Žlower right panel. and C287 Župper panel. Žnote that the size of the cluster outlines have been increased four times to make it possible to see the changes in cluster shapes..
D.D. Righi, P.T. Strub r Journal of Marine Systems 29 (2001) 125–140
lated drifters at a depth of 15 m. The trajectories are calculated from the model velocity fields using a 4th order Runge–Kutta integration and are saved eight times per model day, which replicates the frequency of ARGOS data collection. Also output and saved are the U and V velocities, vorticity Ž z . and temperature at each drifter position. These data are interpolated from the model fields to the position of each drifter. It is important to stress here the difference between the U, V, and z data interpolated from the model fields, and the estimates that we will present later: the estimates of U, V, and z will be derived from only the drifter positions and their changes in time, the same data as is available from real drifters. The simulated drifter trajectories are initiated in clusters of 11 drifters each. The initial cluster shape is two nested pentagons around a center drifter Žas seen in the upper-left panel Fig. 4.. The outer and inner pentagons have diameters of ; 12 km and
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; 6 km, respectively. Most of the results discussed here are based only on the trajectories from the six inner drifters of each cluster. The initial positions of the clusters are shown overlayed in Fig. 2. Each AdotB in this figure represents 11 drifters. There are 60 clusters of 11 drifters for a total of 660 drifters. The clusters are re-initialized on this grid every 2 months. Drifters are AdeadB when they get within one grid cell length of the coast or the open boundaries. The drifter results presented in this paper are from the 1999 POMrPWC model year run. For the LS study, we use only trajectories from clusters whose members remain alive for an entire 2-month period. This gives 307 clusters Ž3377 drifters. for 60 days each, for a total of over 2 = 10 5 drifter days of trajectory data. Fig. 2 shows the drifter trajectory spaghetti plot for the entire cluster data set from January to December 1999. Most of the drifters are
Fig. 4. Principle component ellipses fitted to the positions of drifters in Ca287 at 0, 20, 40 and 60 days after the release of the cluster. Note that the axes ranges change.
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swept southward and offshore in the CC. There are a few drifters that are moved northward off Oregon and Washington in the winter months when the Davidson current is present. The trajectories that do show northward flow are concentrated near the coast. This replicates the northward movement of real drifters released off Oregon in winter ŽJ. Barth, personal communication.. As an example of cluster trajectories and to serve as background for discussions of cluster orientations, Fig. 3 shows the trajectories of the inner six drifters in clusters a76 and a287 ŽC76 and C287.. C76 is initialized in the model on March 1, 1999 and C287 on October 27, 1999. Cluster 287 is released at 39.58N, farther offshore than C76, but quickly moves towards the coast, sharply turning to the south at 398N and again at 37.58N. The upper panel of Fig. 3 shows the outline of the cluster at 1-day intervals over 60 days. Before and after the first turn, the cluster’s outline plot shows the cluster being stretched out by the velocity shear around the turn. The
stretching causes C287 to lose some of its AclusternessB, instead looking like a line of drifters. Cluster a76 is released offshore of Cape Mendicino and is carried to the south before turning eastward at 388N and then southward again and continuing down the coast to a position offshore of Point Conception. Peak speeds experienced by drifters in C76 are 40 cm sy1 . The cluster outline plot Žlower right panel. shows that, in contrast to C287, C76 stays compact for most of its lifetime as it follows the CC jet. The purpose of this work is to determine the characteristics of clusters such as C76 and C287, which can be used as indices of the quality of the estimated vorticity. These characteristics include parameters such as number of drifters in a cluster, cluster separation, area, and shape. Estimates of vorticity will ultimately be used to calculate vertical velocity, but here we focus the discussion on vorticity and only briefly discuss vertical velocity estimates.
Fig. 5. Top: comparison of LS estimate of relative vorticity Žthin. and the along-track relative vorticity Žbold. for Ca76. The uncertainty estimate from the LE method is used to plot error-bars at 1-day intervals. Bottom: the outline of the six-drifter cluster plotted as a function of time. Outlines are plotted at 1-day intervals, and the dot represents the same drifter in each outline. The reference line at the left represents a 10-km length.
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4. Methods A primary motivation for accurately estimating vorticity is to determine vertical velocity along drifter paths. Before discussing the details of the vorticity estimates, we review the methods used to estimate vertical velocity. This will determine the accuracy needed in the vorticity estimates. The most obvious and straight-forward approach for estimating vertical velocity from a drifter cluster is to measure the divergence of the cluster and use a conservation of mass argument to estimate vertical velocity from relative drifter motions: Eu
EÕ q
Ex
1 EA s
Ey
1 ED sy
A Et
1 s
D Et
D
wyD
Ž 1.
where A is the area of a cluster and D is the depth of the drogue or mixed layer. Inherent in this esti-
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mate is the assumption that the horizontal velocity is independent of depth above D. In near-geostrophic flow, calculating divergence directly is impractical because the equation compares two values of approximately equal magnitude and different sign. A scale analysis of the potential vorticity equation ŽBryden and Fofonoff, 1977. concludes that the velocity gradients must be accurate to 1% of their magnitudes to achieve quality divergence estimates. The area bounded by the drifter positions at each time step is easily calculated for clusters of three drifters by taking the cross product of position vectors of two of the drifters in relation to the third ŽMolinari and Kirwan, 1975.. By the correct vector rotation transforms, the vorticity, and shear and stretching deformations can also be calculated. However, this approach is sensitive to drifter tracking precision and the necessity of taking a time derivative of cluster area magnifies estimate errors.
Fig. 6. Cluster statistics as a function of time for the inner six drifters of cluster a76. The top panel compares the LS relative vorticity estimate Žthin line. and the along-track relative vorticity Žthick line.. The second panel shows the cluster maximum length Ždark. and area Žthin.. Shown in the bottom panel are the cluster ellipticity, s 1 rs 2 Žthick. and the LS estimate uncertainty Žthin..
D.D. Righi, P.T. Strub r Journal of Marine Systems 29 (2001) 125–140
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An alternate estimate of vertical velocities can be derived from diagnosis of the vorticity budget. Following a water parcel the vertical component of vorticity is dH dt
Ew
Ž zqf . y Ž zqf . Ez
Ez
Ew Eu
s yw
q Ez
Ew EÕ
Ž 2.
y E y Ez
Ex Ez
where Žd H rdt . is the horizontal total derivative. We assume that the terms on the right-hand side are small since the vertical scale of the horizontal velocities is large compared with the mixed-layer depth ŽSwenson et al., 1992.. Integrate the remaining terms vertically over the surface layer depth to arrive at an estimate for w: w Ž z s D . s yD Ž z q f .
y1
dH dt
Žz .
Ž 3.
ui s u q
Ex
Eu x i) q
EÕ Õi s Õ q
Ex
Ey EÕ
x i) q
Ey
yi) q uXi
Ž 4.
yi) q ÕiX
Ž 5.
x i) s x y x
1 2
As
Õ EÕ
Ex Eu
yi) s yi y y
Ex EÕ
Bs
Ey x 1) x 2) . . . x n)
1 1 Rs
y 1) y 2)
Ž 8.
yn)
u1 u2 Us . . . un
Õ1 Õ2 Vs . . . Õn
uX1 uX2 Es . . . uXn
ÕX1 ÕX2 Fs . . . ÕnX
Ž 9.
Then, Eqs. Ž4. and Ž5. can be expressed: U s RA q E V s RB q F And the LS solutions will be:
T
y1
RT U
y1
T
is1
Ž 10 . Ž 11 . Ž 12 .
B s Ž R R. R V Ž 13 . If there are four or more drifters in a cluster, it is possible to calculate the uncertainty in the LS estimates: 1 T
y1
T
y1
sA Ž t . s diag Ž R R .
su Ž t .
n
Ý Ž uXi2 q ÕXi 2 .
Ž 7.
Ey
A s Ž RT R .
Here, and are the distances from the cluster center to individual drifters, uXi and ÕiX represent higher order terms of the expansion, and the ŽO. operator represents a mean over all drifters in a cluster. Motions with a length scale smaller than the cluster scale will have a small-scale kinetic energy represented by: KE s
u Eu
1
ŽNote that we are ignoring the beta effect here.. The above vertical velocity estimation method requires calculating time derivatives of vorticity along a cluster’s path, making accurate estimates of vorticity a prime necessity. Molinari and Kirwan Ž1975. and Okubo and Ebbesmeyer Ž1976. estimate vorticity and other differential kinematic properties ŽDKPs. of a water parcel by using a LS regression model for cluster velocities and velocity gradients. First, expand each individual drifter’s velocity about the cluster mean velocity in a Taylor series: Eu
By substituting Eqs. Ž4. and Ž5. into Eq. Ž6., and minimizing the small-scale KE, we can solve in a LS sense for the cluster mean velocity and gradients. For clusters of four or more drifters, we can also estimate uXi and ÕiX . For three-drifter clusters, these terms are identically zero and no information about the energy of the small-scale motions can be extracted. The solution method is easily described by defining:
2
Ž 14 .
1
Ž 6.
sB Ž t . s diag Ž R R .
sÕ Ž t .
2
Ž 15 .
D.D. Righi, P.T. Strub r Journal of Marine Systems 29 (2001) 125–140
where sA and sB are vectors containing the uncertainties of the terms in A and B. The uncertainty estimates for z estimates shown later Ž zstd . are then the third component of sA plus the second component of sB . In the above equations su and sÕ are defined as:
su Ž t . s
sÕ Ž t . s
1 ny3 1 ny3
1
n
uXi
Ý Ž t.
2
is1 1
n
ÕiX
Ý Ž t.
2
is1
It is important that the cluster length scale be shorter than the length scales of the velocity field the cluster is embedded in. Another parameter that is of importance is the orientation or shape of the cluster. A cluster that becomes stretched into a linear shape will lose its ability to measure velocity gradients in the direction perpendicular to the cluster axis. To quantify this cluster stretch or ellipticity we use a
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principal component transformation of x ) , y ) Žthe displacement of drifters from the cluster centroid.. By fitting a principal component ellipse to the positions of drifters in a cluster at each time, we can use the ratio of the major and minor axes, s 1rs 2 , as a measure of cluster ellipticity. Fig. 4 shows an example of fitted ellipses for selected times in C287’s trajectory. For reference, the values of s 1rs 2 at these four times Ždays 0, 20, 40 and 60. are 1.0, 25.0, 13.8 and 2.1. As will be shown, a cluster’s orientation can become extremely elongated, resulting in large uncertainties and errors in the estimates of vorticity. Other authors Žfor example, Paduan and Niiler, 1990. have presented the cluster area as a criteria of a cluster’s ability to provide good estimates. They set an upper limit on area as defined by the local ocean current length scales. The lower limit is designed to eliminate clusters that are too small, and those that have become stretched into a linear formation. We will show area estimates for the simulated clusters,
Fig. 7. Comparison of LS estimate of relative vorticity Žthin. and the along-track relative vorticity Žbold. for Ca287. The uncertainty estimate from the LE method is used to plot error-bars at 1-day intervals. Bottom: the outline of the six-drifter cluster plotted as a function of time. Outlines are plotted at 1-day intervals, and the dot represents the same drifter in each outline. The reference line at the left represents a 10-km length.
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calculated by a line-integral of the polygon defined by the drifters in a cluster. We will also discuss the cluster maximum length scale, defined as the largest distance between any two drifters in a cluster. Kirwan Ž1988. points out that the cluster method relies on three key assumptions: Ž1. the translation velocity is common to all drifters, or alternatively, that the cluster length scale is shorter than the length scales of the current structure; Ž2. the velocity gradient is common to all drifters, i.e., there is a homogeneous deformation field; and Ž3. the cluster center corresponds to the Aflow center.B Certain flow situations will not satisfy these assumptions. Paduan and Niiler Ž1990. present a cluster that violates the second assumption Žthe drifters span a velocity jet. and contend that errors in the LS estimate are due to this violation. Thus, the cluster method should be used with an awareness of its limitations. The assumption that the cluster center corresponds to the flow center Že.g., points of diver-
gencerconvergence or vortex points. is a weakness inherent in the OkuborMolinari LS method. Other researchers have advanced the theory and defined methods that solve for the position and translation of the flow center. Kirwan et al. Ž1988. and Kirwan Ž1988. present an approach that uses the trajectories of single drifters to study the kinematics of eddies in the Gulf of Mexico. This method is sensitive to drifter trajectory measurement error since it uses higher-order time derivatives and hence requires smoothing after each differentiation. Halide and Sanderson Ž1993. develop a two-step regression scheme that uses clusters of drifters to solve for the differential kinematic properties and center of a flow. They have success with the method in simulated flow fields, but the results are more ambiguous when applied to real drifters. They postulate that this is because the ocean flow field is a product of possibly more than one flow center, making the method’s approach inadequate.
Fig. 8. Same as Fig. 6, but for cluster a287. The shading marks the times when the combined error predictor discussed in Section 6 is greater than 25.
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5. Results 5.1. LS z estimates We start the presentation of the results with estimates from the more well behaved C76. In Fig. 5, the top panel shows, as a function of time, the LS estimate of relative vorticity using the six inner drifters of the cluster compared to the relative vorticity of the model fields averaged over the cluster AfootprintB area. Also shown are the uncertainty estimates from the LS method ŽEqs. Ž14. and Ž15.., plotted at 1-day intervals. The mean along-track relative vorticity has a maximum of 0.1 f and changes sign numerous times. The LS estimate does well for C76, with a correlation coefficient of 0.94 between the LS estimate and the along-track relative vorticity. The estimate is significantly different from zero and follows the along-track z closely.
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The bottom panel of Fig. 5 shows the outline of the six inner drifters in C76 as a function of time. The outlines are plotted only at 1-day intervals. C76 stays compact for most of its path with the cluster maximum length never greater than 20 km. Fig. 6 shows what happens to the cluster along its trajectory in more detail. The cluster length scale and area Žseen in the second panel. are steady for the entire record. The bottom plot of Fig. 6 shows the time series of the uncertainty of the LS estimate and the ratio of ellipse axes Ž s 1rs 2 .. Both are well constrained and only show small increases around day 8 and from days 14 to 20. In contrast, the LS estimate of relative vorticity for C287 is not as well behaved ŽFig. 7.. The along-track z has a magnitude of the same order as C76 and shows a couple of strong sign changes. The error Ždifference between the estimated and alongtrack z . of the LS estimate increases around day 20
Fig. 9. Same as Fig. 6, but for all 11 drifters of cluster a287. The shading marks the times when the combined error predictor discussed in Section 6 is greater than 25.
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and then, much more significantly, at day 28. From days 28 to 38, the LS estimate is the opposite sign from the along-track z . The correlation coefficient for the C287 LS estimate is 0.26 between the LS estimate and the along-track relative vorticity. Looking in detail at the time plots for C287 at day 28 ŽFig. 8., the cluster length is increasing Žbut is still less than 40 km. while at the same time, the area goes towards zero. The uncertainty estimate and s 1rs 2 Žlower panel. also increase sharply at day 28. These indices combined suggest that the reason that the LS estimate fails around day 28 is that the cluster has been stretched into a line of drifters and has lost its ability to measure the velocity gradient in the cross-cluster direction due to the cluster shape. Increasing the number of drifters will reduce the 1 ., but as uncertainty in the LS estimate ŽA ny3 seen in Fig. 9 will not increase the accuracy of the LS estimate in all cases. Here, all 11 drifters in C287 are used. The LS estimate of z shows the same error at days 28 to 40, returning a value of a different sign than the along-track vorticity. Also, the cluster length increases strongly from day 25 to 32 and plays a part in the estimate error. The estimate still suffers from a poor cluster ellipticity and cannot provide a good
estimate of the vorticity in this time period. In contrast, Molinari and Kirwan Ž1975. compare LS estimates from cluster subsets of three and four drifters in the Caribbean Sea. For two of their clusters the LS estimate from three drifters shows strong magnitude and sign changes over certain time periods, while increasing the number of drifters to four stabilizes the estimates. Here, the extra drifter improves the cluster shape, or ellipticity, resulting in a AbetterB cluster for the LS method. Estimates of vertical velocity using the inner six drifters of C287 are presented in Fig. 10. Using Eq. Ž1. and calculating the changes in cluster area gives a very noisy vertical velocity estimate Žthin line.. Also shown are estimates from the conservation of vorticity argument ŽEq. Ž3.. using the LS relative vorticity Žthick dash-dot line. and the along-track relative vorticity Žthick line.. All three estimates use a depth scale of D s 15 m. Paduan and Niiler Ž1990. and Swenson et al. Ž1992. apply the LS approach to clusters in the CC region and calculate vertical velocity estimates on the order of 20 m dayy1 , which are larger than the values here Ž; 1 m dayy1 .. The w estimate based on the LS z shows significant error at days 20 and 28, reflecting the times when the LS estimate of z itself is unreliable. Magnitudes of the
Fig. 10. Vertical velocity along the Ca287 trajectory estimated using cluster area Žthin line., the LS estimate of relative vorticity Žthick dash-dot line., and the along-track relative vorticity Žthick line.. Estimates are scaled using D s 15 m. The shading marks the times when the combined error predictor discussed in Section 6 is greater than 25.
D.D. Righi, P.T. Strub r Journal of Marine Systems 29 (2001) 125–140
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Fig. 11. Same as Fig. 10, for Ca76.
LS z and along-track z w estimates compare well during other periods but the correlation is very low. For C76, the vertical velocity estimates derived from LS relative vorticity estimates and along-track relative vorticity show a higher correlation ŽFig. 11.. As for C287, the vertical velocity estimate based on the area of this cluster is noisy, reflecting the error-prone nature of this approach. Other authors have similar problems when attempting to estimate vertical velocities using cluster data. For example, Mariano and Rossby Ž1989. estimate the terms in a Lagrangian vorticity budget study following clusters of three SOFAR drifters at 700 and 1300 m depth during the POLYMODE Experiment in the Gulf Stream region. To estimate vertical velocities along cluster trajectories, they calculate time derivatives by fitting a linear slope to objectively chosen periods during which the vorticity signals show a definite trend. This reflects the difficulty of taking time derivatives of noisy LS vorticity estimates. 6. Discussion Both the length scale and shape of a cluster play a role in determining whether the LS estimate will provide satisfying results. The length scale must be smaller than or comparable to the scales of the local
currents. One can easily envision examples where clusters with length scales too long will miss vorticity features: a cluster with drifter members straddling a jet flow will estimate no vorticity, missing the jet entirely. Cluster shape will also affect the LS calculation. A cluster in a long stretched-out orientation that does not allow estimates of gradients in the direction perpendicular to the cluster will not provide good estimates. This is seen in the examples for C287 shown above. Other authors have used cluster area as a measure of a cluster’s usefulness Že.g. Paduan and Niiler, 1990.. The area of a cluster is determined by both cluster length scale and shape. Thus, using a cluster’s area does not point directly to either parameter as the source of estimate failure. We propose that the use of the ellipticity ratio Ž s 1rs 2 . and length scale is an improvement. These parameters separate the error contributions of cluster shape and length scale. The LS estimate for C287 around day 28 serves as an example. The ellipticity of the cluster Žbottom panel of Fig. 8. shows that the cluster is very stretched out around day 28, with the major axis at least 20 times greater than the minor axis from days 27 to 33. This indicates that the LS estimate at this time for C287 will be prone to significant error. The cluster area Žsecond panel of Fig. 8., on the other hand, shows a decrease right at day 28, but stays
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within reasonable bounds. Using only cluster area as a measure of estimate value, we might be tempted to believe the LS relative vorticity estimate. It is obvious that the LS cluster method will not always provide good estimates. Clusters will always diverge or become oriented such that they cannot resolve the gradients in the surrounding ocean flow. However, we can define an approach to maximize the amount of reliable estimates following a cluster. We suggest a recursive approach using the ellipticity and length scale of a cluster as error parameters to determine the best estimate at each time. For the entire cluster data set Ž307 clusters. the correlation between the LS z estimate and the along-track z is 0.67 for clusters made up of the
inner six drifters. For the full 11 drifter clusters the correlation coefficient is 0.68. We can increase the correlations by tossing estimates out based on error predictors such as the cluster length scale, the LS uncertainty estimate, or the cluster ellipticity. Fig. 12 shows the dependence of the correlation coefficient on these error predictors for the inner six drifter clusters. Each panel shows the correlation of all data points with values of the error predictor below the threshold shown on the x axis. Also, the number of data in each bin are plotted. As the threshold is increased, more points are used and the correlation decreases. The upper left panel shows the dependence on the LS uncertainty estimate. The correlation Ždark line.
Fig. 12. Dependence of the correlation coefficients Žthick lines. between the LS z estimate and the along-track z on error predictors for the six inner drifter clusters. Only data at times where the error predictor is below the threshold shown on the x-axis are included in the correlation calculation. The error predictors used are the LS uncertainty estimate Župper left., the cluster maximum length Župper right., the cluster ellipticity Žlower left., and a parameter combining the length and ellipticity Žlower right.. Also shown in each panel is the number of data points Žthin lines. included in the calculation of each correlation coefficient.
D.D. Righi, P.T. Strub r Journal of Marine Systems 29 (2001) 125–140
increases slightly for smaller values of zstdrf and has a maximum of only 0.81. The number of data Žthin line. shows that not many data points are thrown out until values of zstdrf below 0.05. Using the maximum length of a cluster as an error predictor gives better results Župper right panel of Fig. 12.. The correlation goes to above 0.9 for cluster lengths less than 20 km. The number of data drops below 8 = 10 4 for clusters with lengths shorter than 20 km, but only starts to fall off for L less than 60 km. Using the ellipticity of the cluster Žlower left panel. as an error predictor also provides marked improvement in the estimate quality. Using only clusters with ellipticities less than 50 increases the correlation to 0.86 and greater. But the ellipticity as error predictor also reduces the number of data points sharply.
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The best error predictor would improve the estimate quality while removing the least number of data points. The results from the length and ellipticity plots suggest using a combination of these two. The lower right panel shows the dependence of .r2. Using a correlation on the quantity Ž L q s 1 sy1 2 cut-off value equal to 50 increases the correlation to 0.90 and leaves slightly more than 8 = 10 4 data points. Here, the correlation reaches a maximum of 0.95. Similar results for the full 11 drifter clusters are shown in Fig. 13. Using 11 drifters will reduce the LS uncertainty and this is reflected in the upper left panel: less data points are thrown out based on zstdrf and the correlation only improves to 0.78. The ellipticity, interestingly, does not improve the estimates as strongly for the larger clusters as it did for the
Fig. 13. Same as Fig. 12, but for the full 11 drifter clusters.
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smaller clusters, while using the length scale seems to still be effective. We suggest that this is due to the fact that larger clusters are more likely to diverge, ending up separated by large distances. When this happens, the ellipticity does not necessarily suffer: large drifter separations can still be in an orientation that results in a near circular formation. Thus, using ellipticity to filter out bad estimates might not work, while length scales still will. The effect of the combined error predictor for the example clusters presented earlier is shown by the shading in Figs. 8–10. In each plot, the times when .r2 is greater than a threshold of 25 are Ž L q s 1 sy1 2 shaded. In Fig. 8 both shaded time periods correspond to the times when the LS z estimate has the largest errors. The first period, centered around day 20 is governed by the peak in the cluster’s ellipticity. At day 28 the error predictor becomes larger than the cut-off threshold due the combined increase of cluster length and ellipticity. But the shaded area does not cover the entire time period when the LS z exhibits significant error Ždays 28–38., suggesting that a lower threshold for this particular error predictor should be chosen. The error predictor for the larger 11 drifter cluster is over the threshold for longer times ŽFig. 9. due to the longer cluster length. Ideally, we would define a cluster parameter to be used to assess the quality of the LS estimate for that cluster. Then, after a cluster has failed our parameter tests, we will test the drifter subsets. By eliminating one or more drifters from a cluster, we can reduce the maximum cluster length scale and cluster ellipticity. If a subset can pass the defined parameter filters, its LS estimates can be used to fill the missing data times. We believe that the combination of ellipticity and cluster length scale provides a promising index of the expected error in the vorticity estimate, although further work is needed to refine this index and define strategies to use it to recover estimates of vorticity from subsets of the drifters.
References Blumberg, A.F., Mellor, G.L., 1997. A description of a three-dimensional coastal circulation model. In: Heaps, N. ŽEd.., Three-Dimensional Coastal Circulation Models. American Geophysical Union, Washington, DC, USA. Bryden, H.L., Fofonoff, N.P., 1977. Horizontal divergence and vorticity estimates from velocity and temperature measurements in the MODE region. J. Phys. Oceanogr. 7, 329–337. Clancy, R.M., deWitt, P.W., May, P., Ko, D.S., 1996. Implementation of a coastal ocean circulation model for the west coast of the United States. Conference on Coastal Oceanic and Atmospheric Prediction, Jan. 28–Feb. 2, 1996, Atlanta, GA. American Meteorological Society, Boston, MA, USA, pp. 72–75. Halide, H., Sanderson, B.G., 1993. Determining flow field singularities from difter trajectories. J. Geophys. Res. 98, 8413– 8423. Kirwan, A.D., 1988. Notes on the cluster method for interpreting relative motions. J. Geophys. Res. 93, 9337–9339. Kirwan, A.D., Lewis, J.K., Indest, A.W., Reinersman, P., Quintero, I., 1988. Observed and simulated kinematic properties of loop current rings. J. Geophys. Res. 93, 1189–1198. Mariano, A.J., Rossby, T., 1989. The Lagrangian potential vorticity balance during POLYMODE. J. Phys. Oceanogr. 19, 927– 939. Molinari, R., Kirwan, A.D., 1975. Calculation of differential kinematic properties from Langrarian observations in the western Carribean Sea. J. Phys. Oceanogr. 5, 483–491. Niiler, P.P., Poulain, P.-M., Haury, L.R., 1989. Synoptic three-dimensional circulation in an onshore-flowing filament of the California Current. Deep-Sea Res. 36, 385–405. Okubo, A., Ebbesmeyer, C.C., 1976. Determination of vorticity, divergence, and deformation rates from analysis of drogue observations. Deep-Sea Res. 23, 349–352. Okubo, A., Ebbesmeyer, C.C., Helseth, J.M., 1976. Determination of Langrangian deformations from analysis of current followers. J. Phys. Oceanogr. 6, 524–527. Paduan, J.D., Niiler, P.P., 1990. A Langrangian description of motion in northern California coastal transition filaments. J. Geophys. Res. 95, 18095–18109. Shearman, R.K., Barth, J.A., Kosro, P.M., 1999. Diagnosis of the three-dimensional circulaton associated with mesoscale motion in the California Current. J. Phys. Oceanogr. 29, 651–670. Swenson, M.S., Niiler, P.P., Brink, K.H., Abbott, M.R., 1992. Drifter observations of a cold filament off Point Arena, California. J. Geophys. Res. 97, 3593–3610.
The use of simulated drifters to estimate vorticity Dylan D. Righi ) , P. Ted Strub College of Oceanic and Atmospheric Sciences, Oregon State UniÕersity, 104 Ocean. Admin. Bldg 104, CorÕallis, OR 97331-5503, USA Received 15 October 1999; accepted 23 August 2000
Abstract A primitive equation ocean model is used to generate trajectories of simulated clusters of drifters in the California Current ŽCC. region. These trajectories allow us to evaluate a least squares ŽLS. method of estimating vorticity and vertical velocity along a cluster’s path. Two clusters provide examples of successful and less successful estimates of vorticity and vertical velocity. Our analysis quantifies the dependence of estimate quality on several parameters that can be used as error predictors in the LS estimate of vorticity: cluster separation, number of drifters in a cluster, and cluster shape. A combination of cluster separation and ellipticity shows the most promise as an indicator of quality for the vorticity estimate. q 2001 Elsevier Science B.V. All rights reserved. Keywords: Drifters; Vorticity; Vertical velocity
1. Introduction Estimates of vorticity and vertical velocity in the upper ocean are difficult to make but are important fields for a number of reasons. The dynamics of the regional circulation are often more readily diagnosed in terms of vorticity, especially around frontal structures ŽShearman et al., 1999.. Vertical velocity also plays a strong role in current dynamics and is even more relevant for biological reasons: the vertical movement of nutrients and organisms can change the nature of entire ecological systems. Drifters, measuring platforms that attempt to follow the same parcel of water, are ideally designed to
) Corresponding author. Tel.: q1-541-737-5751; fax: q1-541737-2064. E-mail address: [email protected] ŽD.D. Righi..
provide estimates of changes along the trajectory of the water parcel. If there are a number of nearby drifters Ža cluster., the relative motions of these drifters can be used to estimate vorticity, divergence and ultimately, vertical velocities. The more drifters in a cluster, the more certain these estimates will be Žin a statistical sense.. However, it has been rare to find large numbers of drifters seeded in clusters, and the chance occurrence of drifter clusters is rarer still. This may change as the relative cost of drifters Žand tracking. continues to decrease in comparison to ship time. Motivated by this, we are carrying out a study of a least squares ŽLS. method that uses the relative motions of drifters in a cluster to predict vorticity. We generate simulated drifter trajectories at a fixed depth in a primitive equation model of the California Current ŽCC. system. Through a comparison of the LS estimates and the data saved along simulated
0924-7963r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 7 9 6 3 Ž 0 1 . 0 0 0 1 3 - 6
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drifter trajectories ŽU, V, z ., we estimate how well the cluster methods predict vorticity. Our particular interest here is how the accuracy of the prediction depends on parameters that can be derived from the drifter data alone, such as cluster separation, number of drifters in a cluster, and cluster shape. The LS method of estimating vorticity, divergence, and deformation rates from groups of current-following drifters for oceanic application has been described by Okubo and Ebbesmeyer Ž1976., Okubo et al. Ž1976. and, independently, by Molinari and Kirwan Ž1975.. Molinari and Kirwan Ž1975. apply the method to clusters of three drifters in the
western Caribbean Sea and achieve mixed results: two clusters provide reasonable vorticity estimates, while results from two other clusters are unrealistic. Niiler et al. Ž1989. use triads of drifters to estimate vorticity and divergence in the CC and conclude that the OkuborMolinari cluster method results in good qualitative agreement Žcorrect sign. with hydrographic estimates. Paduan and Niiler Ž1990. and Swenson et al. Ž1992. use clusters of three to five drifters deployed in the CC region to estimate vorticity and vertical velocity in the strong CC jets: both result in LS vorticity estimates that agree with the sense of the cluster rotations. Kirwan Ž1988. dis-
Fig. 1. POMrPWC model relative vorticity on s-level 1 for model date August 17, 1996.
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cusses limitations inherent in the LS method and outlines possible error sources. Results from some of these studies are discussed below and compared to our results. This paper is organized as follows. We describe the Pacific West Coast model and its use to provide simulated drifter trajectories in Sections 2 and 3. Section 4 describes the OkuborMolinari LS method and defines the parameters we used to predict the quality of the LS estimate. In Section 5 we present the results of the LS estimate for two example
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clusters and in Section 6 we discuss error indicators based on cluster characteristics. 2. Pacific West Coast model The model we are using to generate the simulated drifter trajectories is a version of the Princeton Ocean Model ŽBlumberg and Mellor, 1997. that has been implemented for the Pacific West Coast at the Naval Research Laboratory ŽClancy et al., 1996.. The
Fig. 2. Spaghetti plot for the entire drifter data set. Also shown are the release points of the 11-drifter clusters Žblack circles.. Drifters are released every 2 months.
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1
8 grid that 12 covers the ocean from 308N to 498N and from the coast out to 1358W. Vertically, the model has 30 sigma levels which are concentrated in the surface and bottom to resolve the boundary layers. The bottom depths are from the Navy DBDB5 bathymetry database. Depths shallower than 10 m are considered to be land. The open boundaries are defined by 1 nesting POMrPWC within the global 8 Navy Lay4 ered Ocean Model. We have 6.5 years of model runs from July 1993 to December 1999. As an example of the model POMrPWC model is set up on a
velocity structure, Fig. 1 shows the relative vorticity at the surface for model day August 17, 1996. This snapshot shows a fully developed southward flowing CC jet system with strong offshore meanders Žthe jet structure shows up as areas of strong gradients between the positive vorticity on-shore side of the jet and negative vorticity offshore side..
3. Simulated drifters To simulate the trajectories of drifters in the CC region, we seed the POMrPWC model with simu-
Fig. 3. Trajectories for the inner six drifters in cluster a76 and a287 Žlower left panel.. Each trajectory represents 60 days of position data. Cluster outlines are shown at 1-day intervals for C76 Žlower right panel. and C287 Župper panel. Žnote that the size of the cluster outlines have been increased four times to make it possible to see the changes in cluster shapes..
D.D. Righi, P.T. Strub r Journal of Marine Systems 29 (2001) 125–140
lated drifters at a depth of 15 m. The trajectories are calculated from the model velocity fields using a 4th order Runge–Kutta integration and are saved eight times per model day, which replicates the frequency of ARGOS data collection. Also output and saved are the U and V velocities, vorticity Ž z . and temperature at each drifter position. These data are interpolated from the model fields to the position of each drifter. It is important to stress here the difference between the U, V, and z data interpolated from the model fields, and the estimates that we will present later: the estimates of U, V, and z will be derived from only the drifter positions and their changes in time, the same data as is available from real drifters. The simulated drifter trajectories are initiated in clusters of 11 drifters each. The initial cluster shape is two nested pentagons around a center drifter Žas seen in the upper-left panel Fig. 4.. The outer and inner pentagons have diameters of ; 12 km and
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; 6 km, respectively. Most of the results discussed here are based only on the trajectories from the six inner drifters of each cluster. The initial positions of the clusters are shown overlayed in Fig. 2. Each AdotB in this figure represents 11 drifters. There are 60 clusters of 11 drifters for a total of 660 drifters. The clusters are re-initialized on this grid every 2 months. Drifters are AdeadB when they get within one grid cell length of the coast or the open boundaries. The drifter results presented in this paper are from the 1999 POMrPWC model year run. For the LS study, we use only trajectories from clusters whose members remain alive for an entire 2-month period. This gives 307 clusters Ž3377 drifters. for 60 days each, for a total of over 2 = 10 5 drifter days of trajectory data. Fig. 2 shows the drifter trajectory spaghetti plot for the entire cluster data set from January to December 1999. Most of the drifters are
Fig. 4. Principle component ellipses fitted to the positions of drifters in Ca287 at 0, 20, 40 and 60 days after the release of the cluster. Note that the axes ranges change.
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swept southward and offshore in the CC. There are a few drifters that are moved northward off Oregon and Washington in the winter months when the Davidson current is present. The trajectories that do show northward flow are concentrated near the coast. This replicates the northward movement of real drifters released off Oregon in winter ŽJ. Barth, personal communication.. As an example of cluster trajectories and to serve as background for discussions of cluster orientations, Fig. 3 shows the trajectories of the inner six drifters in clusters a76 and a287 ŽC76 and C287.. C76 is initialized in the model on March 1, 1999 and C287 on October 27, 1999. Cluster 287 is released at 39.58N, farther offshore than C76, but quickly moves towards the coast, sharply turning to the south at 398N and again at 37.58N. The upper panel of Fig. 3 shows the outline of the cluster at 1-day intervals over 60 days. Before and after the first turn, the cluster’s outline plot shows the cluster being stretched out by the velocity shear around the turn. The
stretching causes C287 to lose some of its AclusternessB, instead looking like a line of drifters. Cluster a76 is released offshore of Cape Mendicino and is carried to the south before turning eastward at 388N and then southward again and continuing down the coast to a position offshore of Point Conception. Peak speeds experienced by drifters in C76 are 40 cm sy1 . The cluster outline plot Žlower right panel. shows that, in contrast to C287, C76 stays compact for most of its lifetime as it follows the CC jet. The purpose of this work is to determine the characteristics of clusters such as C76 and C287, which can be used as indices of the quality of the estimated vorticity. These characteristics include parameters such as number of drifters in a cluster, cluster separation, area, and shape. Estimates of vorticity will ultimately be used to calculate vertical velocity, but here we focus the discussion on vorticity and only briefly discuss vertical velocity estimates.
Fig. 5. Top: comparison of LS estimate of relative vorticity Žthin. and the along-track relative vorticity Žbold. for Ca76. The uncertainty estimate from the LE method is used to plot error-bars at 1-day intervals. Bottom: the outline of the six-drifter cluster plotted as a function of time. Outlines are plotted at 1-day intervals, and the dot represents the same drifter in each outline. The reference line at the left represents a 10-km length.
D.D. Righi, P.T. Strub r Journal of Marine Systems 29 (2001) 125–140
4. Methods A primary motivation for accurately estimating vorticity is to determine vertical velocity along drifter paths. Before discussing the details of the vorticity estimates, we review the methods used to estimate vertical velocity. This will determine the accuracy needed in the vorticity estimates. The most obvious and straight-forward approach for estimating vertical velocity from a drifter cluster is to measure the divergence of the cluster and use a conservation of mass argument to estimate vertical velocity from relative drifter motions: Eu
EÕ q
Ex
1 EA s
Ey
1 ED sy
A Et
1 s
D Et
D
wyD
Ž 1.
where A is the area of a cluster and D is the depth of the drogue or mixed layer. Inherent in this esti-
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mate is the assumption that the horizontal velocity is independent of depth above D. In near-geostrophic flow, calculating divergence directly is impractical because the equation compares two values of approximately equal magnitude and different sign. A scale analysis of the potential vorticity equation ŽBryden and Fofonoff, 1977. concludes that the velocity gradients must be accurate to 1% of their magnitudes to achieve quality divergence estimates. The area bounded by the drifter positions at each time step is easily calculated for clusters of three drifters by taking the cross product of position vectors of two of the drifters in relation to the third ŽMolinari and Kirwan, 1975.. By the correct vector rotation transforms, the vorticity, and shear and stretching deformations can also be calculated. However, this approach is sensitive to drifter tracking precision and the necessity of taking a time derivative of cluster area magnifies estimate errors.
Fig. 6. Cluster statistics as a function of time for the inner six drifters of cluster a76. The top panel compares the LS relative vorticity estimate Žthin line. and the along-track relative vorticity Žthick line.. The second panel shows the cluster maximum length Ždark. and area Žthin.. Shown in the bottom panel are the cluster ellipticity, s 1 rs 2 Žthick. and the LS estimate uncertainty Žthin..
D.D. Righi, P.T. Strub r Journal of Marine Systems 29 (2001) 125–140
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An alternate estimate of vertical velocities can be derived from diagnosis of the vorticity budget. Following a water parcel the vertical component of vorticity is dH dt
Ew
Ž zqf . y Ž zqf . Ez
Ez
Ew Eu
s yw
q Ez
Ew EÕ
Ž 2.
y E y Ez
Ex Ez
where Žd H rdt . is the horizontal total derivative. We assume that the terms on the right-hand side are small since the vertical scale of the horizontal velocities is large compared with the mixed-layer depth ŽSwenson et al., 1992.. Integrate the remaining terms vertically over the surface layer depth to arrive at an estimate for w: w Ž z s D . s yD Ž z q f .
y1
dH dt
Žz .
Ž 3.
ui s u q
Ex
Eu x i) q
EÕ Õi s Õ q
Ex
Ey EÕ
x i) q
Ey
yi) q uXi
Ž 4.
yi) q ÕiX
Ž 5.
x i) s x y x
1 2
As
Õ EÕ
Ex Eu
yi) s yi y y
Ex EÕ
Bs
Ey x 1) x 2) . . . x n)
1 1 Rs
y 1) y 2)
Ž 8.
yn)
u1 u2 Us . . . un
Õ1 Õ2 Vs . . . Õn
uX1 uX2 Es . . . uXn
ÕX1 ÕX2 Fs . . . ÕnX
Ž 9.
Then, Eqs. Ž4. and Ž5. can be expressed: U s RA q E V s RB q F And the LS solutions will be:
T
y1
RT U
y1
T
is1
Ž 10 . Ž 11 . Ž 12 .
B s Ž R R. R V Ž 13 . If there are four or more drifters in a cluster, it is possible to calculate the uncertainty in the LS estimates: 1 T
y1
T
y1
sA Ž t . s diag Ž R R .
su Ž t .
n
Ý Ž uXi2 q ÕXi 2 .
Ž 7.
Ey
A s Ž RT R .
Here, and are the distances from the cluster center to individual drifters, uXi and ÕiX represent higher order terms of the expansion, and the ŽO. operator represents a mean over all drifters in a cluster. Motions with a length scale smaller than the cluster scale will have a small-scale kinetic energy represented by: KE s
u Eu
1
ŽNote that we are ignoring the beta effect here.. The above vertical velocity estimation method requires calculating time derivatives of vorticity along a cluster’s path, making accurate estimates of vorticity a prime necessity. Molinari and Kirwan Ž1975. and Okubo and Ebbesmeyer Ž1976. estimate vorticity and other differential kinematic properties ŽDKPs. of a water parcel by using a LS regression model for cluster velocities and velocity gradients. First, expand each individual drifter’s velocity about the cluster mean velocity in a Taylor series: Eu
By substituting Eqs. Ž4. and Ž5. into Eq. Ž6., and minimizing the small-scale KE, we can solve in a LS sense for the cluster mean velocity and gradients. For clusters of four or more drifters, we can also estimate uXi and ÕiX . For three-drifter clusters, these terms are identically zero and no information about the energy of the small-scale motions can be extracted. The solution method is easily described by defining:
2
Ž 14 .
1
Ž 6.
sB Ž t . s diag Ž R R .
sÕ Ž t .
2
Ž 15 .
D.D. Righi, P.T. Strub r Journal of Marine Systems 29 (2001) 125–140
where sA and sB are vectors containing the uncertainties of the terms in A and B. The uncertainty estimates for z estimates shown later Ž zstd . are then the third component of sA plus the second component of sB . In the above equations su and sÕ are defined as:
su Ž t . s
sÕ Ž t . s
1 ny3 1 ny3
1
n
uXi
Ý Ž t.
2
is1 1
n
ÕiX
Ý Ž t.
2
is1
It is important that the cluster length scale be shorter than the length scales of the velocity field the cluster is embedded in. Another parameter that is of importance is the orientation or shape of the cluster. A cluster that becomes stretched into a linear shape will lose its ability to measure velocity gradients in the direction perpendicular to the cluster axis. To quantify this cluster stretch or ellipticity we use a
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principal component transformation of x ) , y ) Žthe displacement of drifters from the cluster centroid.. By fitting a principal component ellipse to the positions of drifters in a cluster at each time, we can use the ratio of the major and minor axes, s 1rs 2 , as a measure of cluster ellipticity. Fig. 4 shows an example of fitted ellipses for selected times in C287’s trajectory. For reference, the values of s 1rs 2 at these four times Ždays 0, 20, 40 and 60. are 1.0, 25.0, 13.8 and 2.1. As will be shown, a cluster’s orientation can become extremely elongated, resulting in large uncertainties and errors in the estimates of vorticity. Other authors Žfor example, Paduan and Niiler, 1990. have presented the cluster area as a criteria of a cluster’s ability to provide good estimates. They set an upper limit on area as defined by the local ocean current length scales. The lower limit is designed to eliminate clusters that are too small, and those that have become stretched into a linear formation. We will show area estimates for the simulated clusters,
Fig. 7. Comparison of LS estimate of relative vorticity Žthin. and the along-track relative vorticity Žbold. for Ca287. The uncertainty estimate from the LE method is used to plot error-bars at 1-day intervals. Bottom: the outline of the six-drifter cluster plotted as a function of time. Outlines are plotted at 1-day intervals, and the dot represents the same drifter in each outline. The reference line at the left represents a 10-km length.
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D.D. Righi, P.T. Strub r Journal of Marine Systems 29 (2001) 125–140
calculated by a line-integral of the polygon defined by the drifters in a cluster. We will also discuss the cluster maximum length scale, defined as the largest distance between any two drifters in a cluster. Kirwan Ž1988. points out that the cluster method relies on three key assumptions: Ž1. the translation velocity is common to all drifters, or alternatively, that the cluster length scale is shorter than the length scales of the current structure; Ž2. the velocity gradient is common to all drifters, i.e., there is a homogeneous deformation field; and Ž3. the cluster center corresponds to the Aflow center.B Certain flow situations will not satisfy these assumptions. Paduan and Niiler Ž1990. present a cluster that violates the second assumption Žthe drifters span a velocity jet. and contend that errors in the LS estimate are due to this violation. Thus, the cluster method should be used with an awareness of its limitations. The assumption that the cluster center corresponds to the flow center Že.g., points of diver-
gencerconvergence or vortex points. is a weakness inherent in the OkuborMolinari LS method. Other researchers have advanced the theory and defined methods that solve for the position and translation of the flow center. Kirwan et al. Ž1988. and Kirwan Ž1988. present an approach that uses the trajectories of single drifters to study the kinematics of eddies in the Gulf of Mexico. This method is sensitive to drifter trajectory measurement error since it uses higher-order time derivatives and hence requires smoothing after each differentiation. Halide and Sanderson Ž1993. develop a two-step regression scheme that uses clusters of drifters to solve for the differential kinematic properties and center of a flow. They have success with the method in simulated flow fields, but the results are more ambiguous when applied to real drifters. They postulate that this is because the ocean flow field is a product of possibly more than one flow center, making the method’s approach inadequate.
Fig. 8. Same as Fig. 6, but for cluster a287. The shading marks the times when the combined error predictor discussed in Section 6 is greater than 25.
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5. Results 5.1. LS z estimates We start the presentation of the results with estimates from the more well behaved C76. In Fig. 5, the top panel shows, as a function of time, the LS estimate of relative vorticity using the six inner drifters of the cluster compared to the relative vorticity of the model fields averaged over the cluster AfootprintB area. Also shown are the uncertainty estimates from the LS method ŽEqs. Ž14. and Ž15.., plotted at 1-day intervals. The mean along-track relative vorticity has a maximum of 0.1 f and changes sign numerous times. The LS estimate does well for C76, with a correlation coefficient of 0.94 between the LS estimate and the along-track relative vorticity. The estimate is significantly different from zero and follows the along-track z closely.
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The bottom panel of Fig. 5 shows the outline of the six inner drifters in C76 as a function of time. The outlines are plotted only at 1-day intervals. C76 stays compact for most of its path with the cluster maximum length never greater than 20 km. Fig. 6 shows what happens to the cluster along its trajectory in more detail. The cluster length scale and area Žseen in the second panel. are steady for the entire record. The bottom plot of Fig. 6 shows the time series of the uncertainty of the LS estimate and the ratio of ellipse axes Ž s 1rs 2 .. Both are well constrained and only show small increases around day 8 and from days 14 to 20. In contrast, the LS estimate of relative vorticity for C287 is not as well behaved ŽFig. 7.. The along-track z has a magnitude of the same order as C76 and shows a couple of strong sign changes. The error Ždifference between the estimated and alongtrack z . of the LS estimate increases around day 20
Fig. 9. Same as Fig. 6, but for all 11 drifters of cluster a287. The shading marks the times when the combined error predictor discussed in Section 6 is greater than 25.
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and then, much more significantly, at day 28. From days 28 to 38, the LS estimate is the opposite sign from the along-track z . The correlation coefficient for the C287 LS estimate is 0.26 between the LS estimate and the along-track relative vorticity. Looking in detail at the time plots for C287 at day 28 ŽFig. 8., the cluster length is increasing Žbut is still less than 40 km. while at the same time, the area goes towards zero. The uncertainty estimate and s 1rs 2 Žlower panel. also increase sharply at day 28. These indices combined suggest that the reason that the LS estimate fails around day 28 is that the cluster has been stretched into a line of drifters and has lost its ability to measure the velocity gradient in the cross-cluster direction due to the cluster shape. Increasing the number of drifters will reduce the 1 ., but as uncertainty in the LS estimate ŽA ny3 seen in Fig. 9 will not increase the accuracy of the LS estimate in all cases. Here, all 11 drifters in C287 are used. The LS estimate of z shows the same error at days 28 to 40, returning a value of a different sign than the along-track vorticity. Also, the cluster length increases strongly from day 25 to 32 and plays a part in the estimate error. The estimate still suffers from a poor cluster ellipticity and cannot provide a good
estimate of the vorticity in this time period. In contrast, Molinari and Kirwan Ž1975. compare LS estimates from cluster subsets of three and four drifters in the Caribbean Sea. For two of their clusters the LS estimate from three drifters shows strong magnitude and sign changes over certain time periods, while increasing the number of drifters to four stabilizes the estimates. Here, the extra drifter improves the cluster shape, or ellipticity, resulting in a AbetterB cluster for the LS method. Estimates of vertical velocity using the inner six drifters of C287 are presented in Fig. 10. Using Eq. Ž1. and calculating the changes in cluster area gives a very noisy vertical velocity estimate Žthin line.. Also shown are estimates from the conservation of vorticity argument ŽEq. Ž3.. using the LS relative vorticity Žthick dash-dot line. and the along-track relative vorticity Žthick line.. All three estimates use a depth scale of D s 15 m. Paduan and Niiler Ž1990. and Swenson et al. Ž1992. apply the LS approach to clusters in the CC region and calculate vertical velocity estimates on the order of 20 m dayy1 , which are larger than the values here Ž; 1 m dayy1 .. The w estimate based on the LS z shows significant error at days 20 and 28, reflecting the times when the LS estimate of z itself is unreliable. Magnitudes of the
Fig. 10. Vertical velocity along the Ca287 trajectory estimated using cluster area Žthin line., the LS estimate of relative vorticity Žthick dash-dot line., and the along-track relative vorticity Žthick line.. Estimates are scaled using D s 15 m. The shading marks the times when the combined error predictor discussed in Section 6 is greater than 25.
D.D. Righi, P.T. Strub r Journal of Marine Systems 29 (2001) 125–140
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Fig. 11. Same as Fig. 10, for Ca76.
LS z and along-track z w estimates compare well during other periods but the correlation is very low. For C76, the vertical velocity estimates derived from LS relative vorticity estimates and along-track relative vorticity show a higher correlation ŽFig. 11.. As for C287, the vertical velocity estimate based on the area of this cluster is noisy, reflecting the error-prone nature of this approach. Other authors have similar problems when attempting to estimate vertical velocities using cluster data. For example, Mariano and Rossby Ž1989. estimate the terms in a Lagrangian vorticity budget study following clusters of three SOFAR drifters at 700 and 1300 m depth during the POLYMODE Experiment in the Gulf Stream region. To estimate vertical velocities along cluster trajectories, they calculate time derivatives by fitting a linear slope to objectively chosen periods during which the vorticity signals show a definite trend. This reflects the difficulty of taking time derivatives of noisy LS vorticity estimates. 6. Discussion Both the length scale and shape of a cluster play a role in determining whether the LS estimate will provide satisfying results. The length scale must be smaller than or comparable to the scales of the local
currents. One can easily envision examples where clusters with length scales too long will miss vorticity features: a cluster with drifter members straddling a jet flow will estimate no vorticity, missing the jet entirely. Cluster shape will also affect the LS calculation. A cluster in a long stretched-out orientation that does not allow estimates of gradients in the direction perpendicular to the cluster will not provide good estimates. This is seen in the examples for C287 shown above. Other authors have used cluster area as a measure of a cluster’s usefulness Že.g. Paduan and Niiler, 1990.. The area of a cluster is determined by both cluster length scale and shape. Thus, using a cluster’s area does not point directly to either parameter as the source of estimate failure. We propose that the use of the ellipticity ratio Ž s 1rs 2 . and length scale is an improvement. These parameters separate the error contributions of cluster shape and length scale. The LS estimate for C287 around day 28 serves as an example. The ellipticity of the cluster Žbottom panel of Fig. 8. shows that the cluster is very stretched out around day 28, with the major axis at least 20 times greater than the minor axis from days 27 to 33. This indicates that the LS estimate at this time for C287 will be prone to significant error. The cluster area Žsecond panel of Fig. 8., on the other hand, shows a decrease right at day 28, but stays
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within reasonable bounds. Using only cluster area as a measure of estimate value, we might be tempted to believe the LS relative vorticity estimate. It is obvious that the LS cluster method will not always provide good estimates. Clusters will always diverge or become oriented such that they cannot resolve the gradients in the surrounding ocean flow. However, we can define an approach to maximize the amount of reliable estimates following a cluster. We suggest a recursive approach using the ellipticity and length scale of a cluster as error parameters to determine the best estimate at each time. For the entire cluster data set Ž307 clusters. the correlation between the LS z estimate and the along-track z is 0.67 for clusters made up of the
inner six drifters. For the full 11 drifter clusters the correlation coefficient is 0.68. We can increase the correlations by tossing estimates out based on error predictors such as the cluster length scale, the LS uncertainty estimate, or the cluster ellipticity. Fig. 12 shows the dependence of the correlation coefficient on these error predictors for the inner six drifter clusters. Each panel shows the correlation of all data points with values of the error predictor below the threshold shown on the x axis. Also, the number of data in each bin are plotted. As the threshold is increased, more points are used and the correlation decreases. The upper left panel shows the dependence on the LS uncertainty estimate. The correlation Ždark line.
Fig. 12. Dependence of the correlation coefficients Žthick lines. between the LS z estimate and the along-track z on error predictors for the six inner drifter clusters. Only data at times where the error predictor is below the threshold shown on the x-axis are included in the correlation calculation. The error predictors used are the LS uncertainty estimate Župper left., the cluster maximum length Župper right., the cluster ellipticity Žlower left., and a parameter combining the length and ellipticity Žlower right.. Also shown in each panel is the number of data points Žthin lines. included in the calculation of each correlation coefficient.
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increases slightly for smaller values of zstdrf and has a maximum of only 0.81. The number of data Žthin line. shows that not many data points are thrown out until values of zstdrf below 0.05. Using the maximum length of a cluster as an error predictor gives better results Župper right panel of Fig. 12.. The correlation goes to above 0.9 for cluster lengths less than 20 km. The number of data drops below 8 = 10 4 for clusters with lengths shorter than 20 km, but only starts to fall off for L less than 60 km. Using the ellipticity of the cluster Žlower left panel. as an error predictor also provides marked improvement in the estimate quality. Using only clusters with ellipticities less than 50 increases the correlation to 0.86 and greater. But the ellipticity as error predictor also reduces the number of data points sharply.
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The best error predictor would improve the estimate quality while removing the least number of data points. The results from the length and ellipticity plots suggest using a combination of these two. The lower right panel shows the dependence of .r2. Using a correlation on the quantity Ž L q s 1 sy1 2 cut-off value equal to 50 increases the correlation to 0.90 and leaves slightly more than 8 = 10 4 data points. Here, the correlation reaches a maximum of 0.95. Similar results for the full 11 drifter clusters are shown in Fig. 13. Using 11 drifters will reduce the LS uncertainty and this is reflected in the upper left panel: less data points are thrown out based on zstdrf and the correlation only improves to 0.78. The ellipticity, interestingly, does not improve the estimates as strongly for the larger clusters as it did for the
Fig. 13. Same as Fig. 12, but for the full 11 drifter clusters.
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smaller clusters, while using the length scale seems to still be effective. We suggest that this is due to the fact that larger clusters are more likely to diverge, ending up separated by large distances. When this happens, the ellipticity does not necessarily suffer: large drifter separations can still be in an orientation that results in a near circular formation. Thus, using ellipticity to filter out bad estimates might not work, while length scales still will. The effect of the combined error predictor for the example clusters presented earlier is shown by the shading in Figs. 8–10. In each plot, the times when .r2 is greater than a threshold of 25 are Ž L q s 1 sy1 2 shaded. In Fig. 8 both shaded time periods correspond to the times when the LS z estimate has the largest errors. The first period, centered around day 20 is governed by the peak in the cluster’s ellipticity. At day 28 the error predictor becomes larger than the cut-off threshold due the combined increase of cluster length and ellipticity. But the shaded area does not cover the entire time period when the LS z exhibits significant error Ždays 28–38., suggesting that a lower threshold for this particular error predictor should be chosen. The error predictor for the larger 11 drifter cluster is over the threshold for longer times ŽFig. 9. due to the longer cluster length. Ideally, we would define a cluster parameter to be used to assess the quality of the LS estimate for that cluster. Then, after a cluster has failed our parameter tests, we will test the drifter subsets. By eliminating one or more drifters from a cluster, we can reduce the maximum cluster length scale and cluster ellipticity. If a subset can pass the defined parameter filters, its LS estimates can be used to fill the missing data times. We believe that the combination of ellipticity and cluster length scale provides a promising index of the expected error in the vorticity estimate, although further work is needed to refine this index and define strategies to use it to recover estimates of vorticity from subsets of the drifters.
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