Is the Loop Current a Chaotic Oscillator?

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Is the Loop Current a Chaotic Oscillator? ALEXIS LUGO-FERNÁNDEZ Environmental Sciences Section, Gulf of Mexico Outer Continental Shelf Region, Minerals Management Service, New Orleans, Louisiana (Manuscript received 11 April 2006, in final form 15 September 2006) ABSTRACT Dynamical systems theory is employed to study the irregular Loop Current in the Gulf of Mexico using a short database of shedding periods and north–south positions of the current. Two independent tests based on these data suggest that the Loop Current is not chaotic but behaves as a nonlinear driven and dampened oscillator with a very short memory. It is suggested that this current varies around a limit-cycle elliptical attractor. It was found that the amplitude and period of the oscillation vary at time scales of 3–5 yr, a time scale that agrees with those of the North Atlantic Oscillation (NAO) and/or ENSO; however, it is proposed that NAO provides the link between these systems. The proposed mechanism is the ITCZ changes caused by NAO, which affects the wind strength and the transport across the Yucatan Channel. A forecasting scheme that allows for prediction of the next eddy-shedding period from knowledge of the last shedding event, a condition caused by the short memory of the system, is provided.

1. Introduction The Loop Current (LC) enters the Gulf of Mexico through the Yucatan Channel from the Caribbean Sea and exits through the Straits of Florida, dominating the circulation in the eastern Gulf. This current is also the main source of energy for the mesoscale variability of the Gulf of Mexico and is a component of the meridional circulation of the North Atlantic Ocean (Nowlin et al. 2000; Coats 1992). Characteristics of the LC include a year-round warm (T ⬃ 25°–26°C) and salty (36.7– 36.8) source of water for the Gulf, transport near 23–27 Sv (1 Sv ⬅ 106 m3 s⫺1; Badan et al. 2005; Hamilton et al. 2005), and a baroclinic flow structure with the bulk of the transport above 800 m (Sheinbaum et al. 2002). The most notable characteristic of the LC, however, is its north–south inrushing into the Gulf, which generally culminates in an eddy shedding at its maximum northward position (Maul and Vukovich 1993). The eddy-shedding cycle begins with an LC slightly extended into the Gulf that exits rapidly through the Straits of Florida, followed by a period of northward

Corresponding author address: Alexis Lugo-Fernández, Environmental Sciences Section (MS 5433), 1201 Elmwood Park Blvd., Gulf of Mexico OCS Region, Minerals Management Service, New Orleans, LA 70123-2394. E-mail: [email protected]

DOI: 10.1175/JPO3066.1

JPO3066

inrushing, then an eddy shedding, and ending with a rapid retraction of the LC. Sometimes, reattachments of the eddy to the LC occur. The resulting eddy is a large mass of warm and salty water with diameters of 300–400 km that rotates anticyclonically (Elliot 1982). The time interval between eddy-shedding events is commonly called the eddy-shedding period. Earlier investigators thought that eddy-shedding was annual (Maul 1977); however, subsequent studies (Vukovich 1995, 2005; Sturges and Leben 2000; Leben 2005) have shown this to be incorrect. Since 1995, the range of the shedding period increased to over 18 months (Leben 2005; Vukovich 2005); see Fig. 1. Because eddy shedding is irregular, it suggests that the LC could be chaotic (Nowlin et al. 2000). The eddy-shedding cycle described above suggests an analogy between eddy and drop shedding from a leaky faucet. This resemblance can be visualized by noting that the eddy-shedding cycle begins with an LC slightly extended into the Gulf and exiting through the Straits of Florida (water surface just outside the faucet), a period of northward penetration (growth of the drop), the eddy shedding (drop breaking), and then a period of rapid retraction of the LC to its position where it usually turns directly into the Straits of Florida (water surface returning to the faucet). Sometimes this sequence is complicated by reattachments, but this is not considered here. The behavior of drop shedding from a leaky

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FIG. 1. Loop Current eddy-shedding period observed from 1973 to 2003. Notice the irregular behavior of the eddy-shedding period, which suggests chaotic behavior. Data are from Leben (2005).

or dripping faucet depends on the flow magnitude and suggests that the inflow through the Yucatan Channel is a fundamental quantity to understand eddy shedding. This inflow is precisely the least understood aspect of the LC system, however. There is no implication in the analogy that the inflow plays the same role in both systems. This analogy, however, suggests that the interval between eddy shedding could reveal information about the LC system similar to the interval between drops in a leaky faucet (Shaw 1984). The cause of eddy shedding, even today, remains largely elusive. Hurbult and Thompson (1980) showed that eddy shedding occurs at constant inflows, but that a barotropic inflow at the Yucatan Channel stops eddy shedding. Pichevin and Nof (1997) proposed that eddy shedding results from a northward current entering an ocean and turning eastward because of vorticity conservation to balance the momentum. Recently, Candela et al. (2002) and Oey et al. (2003) showed that vorticity input and transport fluctuations at the Yucatan Channel affect eddy shedding. These results provide two potential routes to chaos in the LC: 1) inflow variations at the Yucatan Channel, which represent variations in the driving force, and 2) variations of input vorticity from the Caribbean Sea, representing initial conditions of the vorticity equation in the Gulf. Shedding of an LC ring is a drastic event with important repercussions on the Gulf’s circulation and over the entire water column. The interaction of the LC with the west Florida shelf at the time of eddy shedding is believed to be a source of bottom-trapped topographic Rossby waves that predominate in the deep Gulf circulation (Hamilton 1990; Hamilton and Lugo-Fernández

2001; Oey and Lee 2002). Ocean models suggest that LC rings induce near-bottom eddies that affect the Gulf’s deep circulation, and that large vertical but spatially localized particle excursions occur during eddy shedding (Welsh and Inoue 2000, 2002). The LC also affects many activities in the Gulf of Mexico, such as oil and gas exploration and production, navigation, and hurricane activity (Lewis et al. 1991; Shay and Uhlhorn 2006). Forecasting of the LC and eddy shedding has been an elusive goal of the oceanographic community of the Gulf. Leben (2005) recently developed a forecasting scheme for the LC eddy-shedding period on the basis of the retreat latitude as the independent variable. Thus, whether the LC is chaotic or not has large implications for prioritizing of resources, guiding research directions, and forecasting possibilities of the system. It has been found that certain fluid systems exhibit chaos through the nonlinear interaction of a few degrees of freedom (Swinney and Gollub 1986). Under these circumstances, dynamical systems theory is a perfect approach to study such complicated systems without actually solving the differential equations (Shaw 1984; Glazier and Libchaber 1988; Dijkstra and Ghil 2005). In the Gulf of Mexico, several studies have applied dynamical systems theory to study circulation and dispersion using model outputs and Lagrangian observations (Toner et al. 2003; Poje et al. 2002; Kirwan et al. 2003; Kuznetsov et al. 2002). A common aspect of these works has been the use of numerical models output to increase their database. This study also examines the LC from a dynamical system viewpoint, but the analysis is based solely on actual observations that, while span-

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ning several decades, are few in number because of the time scales involved. This work is not an update or extension of Leben (2005) and Vukovich (2005) who present statistical summaries of various aspects of the LC. Observations that suggest chaotic behavior of the LC include 1) variability and failure of the shedding period to converge with an increasing database, 2) equations of motions that contain nonlinear terms even if this is not a guarantee for chaotic behavior (Baker and Gollub 1996), 3) an LC system that has at least three degrees of freedom [two spatial coordinates (east–west and north– south), and the north–south velocity], and 4) resemblance of eddy shedding to a leaky or dripping faucet, which is known to be a chaotic system under certain flow values (Shaw 1984). This work examines the possibility that the LC and eddy shedding are chaotic by calculating several metrics using observations of the shedding period and north–south positions of the LC. If the LC is chaotic, using eddy shedding or its northward penetration behavior to gauge the performance of models may not be a valid test or comparison. A chaotic LC implies that forecasting eddy shedding may only be feasible for short times.

2. Methods Viewing eddy shedding as a casting of drops in which the time interval between eddies as the observable parameter is a discrete process or map. Maps are analyzed somewhat differently than flows. Flows, on the other hand, are continuous processes and in this work the north–south positions of the LC represent a flow. These differences need to be considered when analyzing the databases available. The shedding period database consists of 39 events spanning from 1973 to 2003 (Leben 2005). Sturges and Leben (2000) report uncertainties of 0.25–1.25 months (1–5 weeks) with an rms uncertainty of 0.9 months (3.5 weeks) in the eddy-shedding period. LC-shedding periods before 1993 were derived from infrared images, while latter values are derived from satellite altimetry. Infrared images can introduce a bias because of the lack of contrast in summer, the result of isothermal conditions of the Gulf, a condition that makes detection of the LC nearly impossible (Vukovich 1988). The north–south positions are from Vukovich (1995, 2005) and represent monthly averages of the northward LC front from 30°N during 1977 to 2003, resulting in a time series of 324 observations. Each year of north–south positions was detrended by removing its annual mean such that the variations represent the eddy-shedding process and not interannual variations of the northward penetration of the LC. Details of how

these data were generated are found in Vukovich (1995). This work is essentially an exploratory data analysis in the time domain. It employs actual observations, unlike analyses of equations or synthetic time series characteristic of many works of chaos theory. However, using real data introduces difficulties that need to be evaluated. The main difficulties in using real data are that the number of points is usually limited, noise is present, and there are problems associated with sampling. The first two limitations affect the accuracy of the results because they affect the signal-to-noise ratio of the parameters. The sampling time, which equals 1 month, resulting from the satellite’s repeat periods of 10 and 35 days should resolve most of the LC processes of interest to this study (eddy shedding, annual and interannual cycles), but this may affect resolving the LC rapid retraction that occurs in about 1 month. This sampling interval also yields noisy standard plots. A shorter sampling time would result in a larger number of points, which should reduce the errors and yield smoother standard plots. Thus, the analyses conducted were selected because of their simplicity and low demand of data. Higher-order analyses, such as the correlation integral (Ding et al. 1993), were evaluated but precluded from use because the small database available renders them inaccurate. Analyses conducted include calculation of basic descriptive statistics, autocorrelation analyses, the Hurst exponent, and the Shannon entropy (Sprott 2003; Baker and Gollub 1996). Because of the short and noisy database used, several metrics were calculated with the expectation that a robust pattern revealing chaos would emerge. Besides the many typical plots employed, this work uses fairly extensively the standard or return plot, which is very useful in chaos theory. A standard plot consists of plotting a value against its previous value. The Hurst exponent measures the smoothness of a time series and the analysis consisted of examining the quantity R/␴. ⌯ere R is the range (of cumulative sums) given by the difference between the maximum and minimum value of the trace Yn, defined as n

Yn ⫽

兺 共X ⫺ X兲 i

共1兲

i⫽1

over the first n intervals and X is the average of the time series (Sprott 2003). Sigma (␴) is the standard deviation of the Xi time series. The Hurst exponent H is the slope of the curve log(R/␴) versus log n. From the Hurst exponent, the dimension of the trajectory can be calculated as H ⫺1; the other parameter estimated from H is the slope ␣ of the trace spectrum as 2H ⫹ 1.

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The Shannon entropy, defined as

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TABLE 1. Descriptive statistics of the Loop Current eddy-shedding period.

n

H⫽⫺

兺 p lnp , i

共2兲

i

i⫽1

was calculated from histograms of different bin sizes. The frequency pi for each bin was estimated as ni /N, where ni is the number of observations per bin and N is the total number of observations. Entropy can be an indicator of a system’s chaotic behavior. The autocorrelation analysis consisted of calculating this function at different lags as

Parameter

Total

First half

Second half

Mean Std error Median Mode Std dev Sample variance Min Max Count Confidence level (95%)

9.38 0.7 9.0 6.00 4.2 17.5 0.5 18.5 39 1.4

9.37 0.6 9.0 9.00 2.7 7.2 6.0 15.0 19 1.3

9.40 1.2 9.5 11.5 5.3 28.1 0.50 18.5 20 2.5

N⫺k

兺 共X

n

C共k兲 ⫽

⫺ X兲共Xn⫹k ⫺ X兲

n⫽1

共3兲

N⫺k



共Xn ⫺ X兲2

␪共t兲 ⫽ ct ⫹

n⫽1

to estimate parameters of the system such as the correlation time scale, memory of the system, and rate at which its predictability decreases. The global Lyapunov exponent ␭ of the shedding periods is estimated as (Sprott 2003)

␭ ⫽ lim

N→⬁

1 N

N⫺1

兺 lnⱍf ⬘共X 兲ⱍ, n

共4兲

n⫽0

where f ⬘(Xn) is the derivative with respect to Xn, N is the number of observations, and ln(x) is the natural logarithm function. In this work, an expression of the form f (Tn) was found that allowed evaluating the derivative and estimating ␭ directly from the shedding time intervals. The north–south positions’ time series yn was transformed to zn by zn ⫽

yn ⫺ y , ␴y

共5兲

where ␴y is the standard deviation of the series and y is the mean of the series. This transformation recasts the data as fractions of the standard deviation. The north–south velocity of the LC was estimated by using a simple difference, ␷ ⫽ (zi ⫺ zi⫹1)/⌬t, where the zs represent two consecutive values and ⌬t (⫽1 month in this study) is the time step. These two quantities were used to construct standard plots and the phase diagram of the Loop Current. A new test for chaos developed by Gottwald and Melbourne (2004) based on the north–south position yn, and independent of the dimension of the system, was applied. This test has the additional advantage that is independent of the equations of motion of the system. The equations employed in this test are

p共t兲 ⫽





t

共6兲

yn共s兲 ds,

0

t

yn共s兲 cos关␪共s兲兴 ds,

0

1 T→⬁ T

M共t兲 ⫽ lim



and

关 p共t ⫹ ␶兲 ⫺ p共␶兲兴2 d␶.

共7兲 共8兲

These integrals were evaluated numerically using the trapezoidal method, with ⌬s ⫽ 1 month and T ⫽ 324 months. The constant c could be any value ⬎ 0; I selected c ⫽ 1.7, which is the value employed by Gottwald and Melbourne (2004). A plot of log M(t) versus log t is constructed and its slope K is estimated by least squares regression. If the system is not chaotic, K ⫽ 0; if the system is chaotic, K ⫽ 1.

3. Results a. The shedding period map Eddy-shedding periods ranged from 0.5 to 18.5 months over 39 events (Leben 2005); furthermore, close examination of Fig. 1 reveals no temporal trend, suggesting that the LC eddy shedding is a stationary process (Davis 1973). Table 1 shows, along with other descriptive statistics, that the average shedding period is 9.4 months with a standard deviation of ⫾4.2 months and standard error of the mean of ⫾0.7 months. The table also shows basic statistics of the first and second halves of the shedding period database. All means are similar within the standard errors and the 95% confidence interval, but the variances for the first and second halves are different at the same confidence level. Because of its importance, the stationary test as described in Bendat and Piersol (1986) was applied to the shedding time intervals. The outcome was that the shedding intervals are stationary at 99% confidence level. Leben (2005) reached a similar conclusion for the periods and

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FIG. 2. Entropy of the Loop Current system estimated from the eddy-shedding periods (filled squares), the north–south positions (filled triangles), and maximum entropy (filled diamonds). Notice the nearly identical behavior of entropy even when based on two different aspects of the Loop Current.

other metrics of the LC, including the north–south position time series. The stationary condition is important because this is an underlying assumption of several of the analyses conducted. More important, a stationary process implies that the data employed should be a robust representation of this current. Figure 2 presents the entropy estimated from the shedding periods and the north–south positions along with the theoretical maximum entropy [Hmax ⫽ ln(M ⫽ number of bins)]. Two aspects of this figure are noticeable. First, the entropies estimated from the data are nearly identical in magnitude and shape. Second, both curves approach the maximum entropy. The shape of the curves can be understood by noting that, at a large number of bins, the slope tends to zero or a horizontal curve because ⳵H/⳵M ⫽ 1/M. Also, if the uncertainty of the entropy is estimated as ⌬H ⫽ 1/M, it is easy to see that both curves are equal within the error. High entropy could be interpreted as the amount of disorder in the system is near its maximum because the observations fall in all possible bins. If the observations would have fallen in just a few bins, the entropy of the system would be near zero. Another interpretation of entropy is the amount of information missing to specify the state of the system. For the situation depicted in Fig. 2, both variables contain similar information and the amount of information needed is near its maximum. The curves also suggest that the data are distributed nearly evenly over all bins since the curves are close to the maximum entropy. Because chaotic systems tend to have entropies close to the maximum (Baker and Gollub 1996), Fig. 2 suggests that the LC is a chaotic system; however, this is not a sufficient condition for chaotic behavior. The shedding periods’ correlogram, not shown, re-

veals correlation values not different from zero for all lags except at one and six. This result indicates that the shedding periods are mostly a series of uncorrelated values. Parameters estimated from the autocorrelation are in Table 2. The first parameter is the correlation time scale or the lag at which the autocorrelation reaches a value of 0.37. The estimated correlation time is 0.57 events, which is very short as expected, since natural phenomena exhibit short-term correlations. The second parameter estimated is the “memory” of the system, which is 1.2 events and implies a very short memory in the LC after a shedding event. Put another way, each shedding event is influenced little by the previous event. The final parameter estimated from the autocorrelation is the average rate at which predictability is lost. For eddy shedding, this rate is large (0.87), which means that predictability is not possible after one shedding event. This conclusion also agrees with the finding that shedding events are nearly independent of each other. Results of the Hurst analysis for the shedding periods are in Table 3. The Hurst exponent has a value of 0.8 and implies a smooth series or data without large variations. This value of H ⬎ 0.5 indicates persistence or the tendency for the direction of the curve to continue in its current direction and produces enhanced diffusion. AnTABLE 2. Parameters of the LC system estimated from the autocorrelation analysis. Parameter

Periods

Positions

Correlation “time” Memory Mean rate of predictability loss

0.6 events 1.2 events 0.87

8.4 months 16.8 months 0.06

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TABLE 3. Parameters of the LC system estimated from Hurst analysis. Parameter

Periods

Positions

H exponent Dimension of trajectory Slope ␣ of spectrum

0.80 1.25 2.60

0.30 1.70 1.59

other important parameter derived from the Hurst exponent is the dimension of the trajectory, which is estimated as D ⫽ H ⫺1. The dimension estimated for the shedding period is 1.25, which implies an attractor between a line and a surface. This last result indicates that the periods define a one-dimensional map, a result that allows estimating the Lyapunov exponent. The last parameter estimated from the Hurst analysis was the slope of the trace’s spectrum ␣, which is 2.6 for the shedding periods such that the spectrum follows a power law of the form f ⫺2.6. For comparison, in 3D fully developed turbulence for almost any turbulent flow the spectrum falls as f ⫺1.67 (Frisch 1995) and indicates that the trace’s spectrum falls faster. Figure 3 shows the standard or return plot from the eddy-shedding periods. The top left panel represents the standard plot using all 39 events. There seems to be some structure present on this plot, but nothing resembling the maps obtained from the dripping faucet (Shaw 1984) as expected from the analogy of the LC to a leaky faucet. Of course, the amount of data needed to correctly delineate the attractor is far more than the 39 events available. Standard plots based on fewer data points reveal closed loops (L1–L7) consisting of four– seven events per loop, with an average number of five per loop. These loops exhibit clockwise rotation (the rectangle marks the initial point). If the analogy of an LC behaving like a leaky faucet will hold true, these diagrams should have exhibited period doublings (points alternating between two locations in the standard plot), parabolic sections like a Henon map, or a Rössler attractor depending on the input flow conditions (Shaw 1984). Period doubling seems to occur at the beginning of each loop, but then it stops. Other behaviors present in these loops are rotation and stretching. An important behavior exhibited by the loops is size variation; initial expansion (L1–L3), a contraction (L4), and another expansion (L5–L7). The behaviors observed in these loops are explained using the previous results of the correlation and Hurst analyses. First, the memory of the system was one event, which means little correlation between two consecutive events and could explain why the period doubling ceases rapidly. The loops are not lines nor are they surfaces, which is what should be expected from a dimension

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between 1 and 2 estimated here. Last, the closeness of the loops, changes in size, and sense of rotation are explained by a Hurst exponent greater than 0.5. In this case there is persistence, which means a tendency for the trajectories to continue in their current direction (clockwise) to create closed loops and enhanced diffusion (size variation). The number of four to seven points composing each loop agrees with the nonzero correlation found at a lag of six. Thus, while the analogy with the dripping faucet seems reasonable initially, these results show that it is not a good model to describe the LC system. From the standard or return map of all 39 shedding events, it was found by least squares fitting that the describes these obpower relation Tn⫹1 ⫽ 9.831T ⫺0.0901 n servations with a large scatter and a very small r 2. However, this relationship leads to very important results. First, this power relationship helps to calculate the global Lyapunov exponent ␭ from Eq. (4) because it allows evaluating the derivative using the shedding period data and the exponent of the power relation. The estimated ␭ is ⫺2.42 for ␤ ⫽ ⫺0.0901. Using the 95% confidence interval of ␤ from the regression, the largest value of ␭ (⫽⫺0.861) is still negative. This result needs to be viewed as a zero-order estimate given the very low r 2 of the power relation. A negative global exponent implies that chaos is not possible because neighboring points do not diverge when ␭ is negative. Put another way, the Lyapunov exponent, which measures the system’s sensitivity to initial conditions and is a necessary condition of chaos, has to be positive to allow divergence. A negative global exponent prevents divergence and indicates that the attractor is either a point or a limit cycle (Sprott 2003).

b. The north–south flow Analysis of the maximum northward penetration of the LC over 11.5 yr (between 1977 and 2003) yields the following statistics: an average of 383 km, a maximum of 529 km, and a minimum of 222 km. Leben’s (2005) analysis of daily northward positions yielded an average northward position of 26.2°N or a distance of 244 km from 24°N; the maximum and minimum northward penetration from 24°N were 456 and 11 km, respectively. The differences in both sets of statistics result from the facts that 1) the first set is a subset of the second one and 2) they cover different time spans. Results of the entropy analysis for the north–south positions are shown in Fig. 2. These results are very similar to those of the shedding periods and will not be discussed further except to say that they confirm the need for more information to study the system. A plot of the LC-transformed (zn) north–south posi-

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FIG. 3. Standard plots of the eddy-shedding periods showing seven loops. Note the changes in orientation brought about by rotation, variation in size, and the number of points delineating each loop.

tions (not shown) displays nearly periodic variations of varying amplitudes that resemble a modulation similar to that observed by Maul and Vukovich (1993). There are 28 cycles over 324 months, yielding an average period of 11.6 months. The correlogram of the north–south positions (also not shown) resembles that of a sinusoidal process that contains noise. The plot shows positive correlations that are different from zero at 36- and 47-month lags. A peak of positive but not significant correlation occurs at 12 months, which agrees with the finding of little power at the annual frequency by many (Leben 2005; Sturges 1992, 1994; Vukovich 1995, 2005). Previous analyses

also found evidence of energy at periods of 25–30 months, which support the significant correlation at 36 months (Sturges 1992; 1994; Sturges and Leben 2000). Because this autocorrelation displays oscillations, Sprott (2003) recommends performing the analyses by using the envelope of the function. The results of the analysis for these positions are shown in Table 2. For the north–south positions, the correlation time is 8.4 months and the memory of the system is about 16.8 months, meaning that after about 17 months the north– south positions of the LC become unrelated to previous positions. It is interesting to note that 17 months is very close to the maximum eddy-shedding period observed.

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The rate of predictability loss for the north–south positions is only 0.06, much smaller than that of the periods, and suggests that there is a high possibility of predicting the system’s near future. The Hurst exponent for the north–south positions (see Table 3) is 0.30 and implies a “rough time series” or presence of large fluctuations in this series. Since the Hurst exponent (0.28 ⱕ H ⱕ 0.31) is less than 0.5, it implies antipersistence, which means that the trajectory tends to return to the point from which it started and suppresses diffusion. The dimension of the position’s attractor is 1.70, which indicates it is somewhere between a line and a surface. The slope of the trace’s spectrum ␣ for the north–south positions has a value of 1.59, which implies that the spectrum follows a power law of the form f ⫺1.59 and falls about the same as the spectrum of fully developed turbulent. Standard plots of zn reveal 25 loops (L1–L25), all with similar characteristics. For purposes of illustration and discussion, Fig. 4 shows the first 15 loops (L1–L15). These loops resemble ellipses with major axis oriented diagonally (near 45°) and suggesting an anticyclonic or clockwise rotation (squares mark the initial points). Some loops are closed and others are not, but the same tendencies are evident in all loops. Close loops, or one where the final point tries to return to its initial position, are expected from a Hurst exponent ⬍ 0.5. These loops vary in size, similar to the loops in the shedding period data. The square end of the loops is probably caused by the large sampling interval (1 month) in these data. Figure 4 resembles the standard plot of a sinusoidal process, which is an ellipse with clockwise rotation, which is expected from the quasi-periodic variations of zn. Furthermore, the inclination of the major axis can be shown to be near 45°, as is evident in these plots. The distance along the vertical axis is ⌬y ⫽ 2A sin(⌬␪), where ⌬␪ ⫽ 2␲/N and N is the number of points describing the ellipse; N is also the period of the motion because ⌬t ⫽ 1 month. Similar but horizontally oriented ellipses are obtained when the north–south velocity of the LC is plotted against the position (Fig. 5). The amplitude and period of the motion estimated from these ellipses are shown in Fig. 6. The plot reveals five–six cycles composed of three–five events that are highly correlated (correlation coefficient ⫽ 0.73; significant at 95%), which suggests that the LC north–south cycle repeats every three–five events. A complex demodulation analysis (Bloomfield 1976) of the north– south positions (not shown) depicts 3–5-yr variations of the amplitudes. Maul and Vukovich (1993) found similar variations of the amplitude of the north–south LC variations. The average period from these loops is 11.9 months, which is very close to the mean shedding pe-

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riod (Sturges and Leben 2000; Vukovich 1995, 2005). However, the standard error of the mean (standard deviation/squared root of the number of observations) shows that this period is not significantly different than 12 months, a result similar to Maul and Vukovich (1993). Recall that the annual period has been found not to be a robust estimate (Leben 2005; Sturges 1992, 1994; Vukovich 1995, 2005). The interannual cycle of three–five events agrees with the significant correlation time scales of 36–47 months. Recall that the loops described by the shedding periods also consisted of five events on average (four–seven events per loop). These oscillations could represent adjustments of the LC system to vorticity and transport changes (Oey et al. 2003; Candela et al. 2002) or to deepwater volume variations as suggested by Candela et al. (2003). Figure 7 shows that the period of the north–south oscillation is a nonlinear function of the amplitude as indicated by the quadratic polynomial fitted to the data (r 2 ⫽ 0.65). This result indicates that the LC behaves as large amplitude or nonlinear oscillator (e.g., Becker 1954; Sprott 2003). It is well known that such nonlinear oscillators can exhibit chaos (e.g., Baker and Gollub 1996; Sprott 2003). An interpretation of Figs. 6 and 7 is that of an oscillatory system that overshoots and tries to correct, but overshoots, never reaching a stable orbit but moving around it. Such a system is called a limit cycle (Sprott 2003) and agrees with the finding of a negative global Lyapunov exponent. An example of this type of system is the Van der Pol oscillator (Sprott 2003). Furthermore, limit cycles, while depicting irregular oscillations, are not chaotic, since the trajectories never diverge, but are bounded. This indicates that limit cycles do not exhibit sensitivity to initial conditions; however, they do have an attractor. The phase diagram or zn– ␷ plot (Fig. 5) is very revealing in that its shape is an ellipse instead of a circle. If the LC were a linear oscillator, its phase diagram would have been circles of different radii corresponding to varying initial conditions; if it were a dampened system, the attractor should be the origin. The elliptical phase diagram suggests that the LC is behaving like a dampened and driven oscillator with an elliptical attractor whose shape depends on the damping constant, the frequency, and amplitude of the driving force. Regardless of the initial conditions, the motion would be attracted to this ellipse (Sprott 2003). Of course, these parameters are unknown at this time. The results of the Gottwald and Melbourne (2004) test shows that M(t) is nearly constant at ⬃ 5.1 over the range t ⱕ 30 months and decreases in a steplike manner to ⬃4.5 afterward. The nearly constant values satisfies the condition that T (⫽324) k t (⫽30) is required by the test. The estimated slope is K ⬃ 0, a more accurate

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FIG. 4. Standard plot of the transformed (zn) north–south positions showing ellipses oriented at 45°. Note the variation of the size and number of points in the ellipses.

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FIG. 5. Phase diagram of the Loop Current oriented along the horizontal axis. The ordinate is speed (month⫺1) and the abscissa is zn. Note the changes in size.

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FIG. 6. Variations of amplitude A and period N estimated from the 25 loops described by the north–south positions of the LC with a high correlation (␳ ⫽ 0.73). Note the five cycles of three–five events.

result, and indicates that the LC is not chaotic. This finding agrees with the result from the Lyapunov exponent estimated from the eddy-shedding periods.

4. Discussion Because the LC and eddy shedding vary at relatively long temporal scales, O(1 yr), and because the tools needed to synoptically observe this current—that is, remote sensing sensors—became available just a few decades ago, the number of observations available is not extensive. The data employed herein consist of 39 eddy-shedding events and a 27-yr (324 months) time series of the north–south position of the LC. Consequences of the short shedding period database consist of the inability to determine the dimension of the attractor using higher-order analyses and not being able to delineate the attractor from the shedding periods; however, the errors from using the longer time series are not large. The strategy to minimize these errors was to estimate several metrics to obtain a robust pattern that would increase our confidence in the results. Despite the small number of observations, I undertook this analysis because its outcome outweighs it shortcomings and helps the oceanography of the Gulf of Mexico. Based on the longest series available, 324 points, the dimension of a strange attractor that can be resolved is ⬃1.3. Thus, these results need to be viewed as zero-order estimates. The autocorrelation and Hurst analyses are very revealing of the dynamics of the LC. The autocorrelation

shows that the eddy-shedding periods are a series of uncorrelated values. This analysis also reveals that the system’s memory is ⬃10–18 months. This time frame is equal to the time it takes for the LC to complete an entire cycle consisting of northward inrushing, shedding, and returning to near its initial position. When considered as a whole, these results suggest that eddyshedding events are independent realizations. If a relation exists, it seems to be weak and unresolved by these analyses. This finding agrees with the prediction scheme of Leben (2005), which can only predict one event into the future. The result also agrees with the finding that the rate of predictability loss is high for the shedding periods. The rate of predictability loss for the north–south positions is small, which reflects the fact that they describe ellipses that are predictable. The Hurst analysis reveals that the shedding periods and north–south positions behave differently; the period’s series exhibit persistence or loops that are not close, while the positions exhibit closed ellipses. An important finding from the Hurst analyses is that the dimension of the attractor is between 1 and 2, establishing that this is a low-dimensional system; however, this result needs to be confirmed with more data in the future. This result provides a justification for using dynamical systems theory to analyze the LC system. Last, the K ⬃ 0 value obtained from the Gottwald and Melbourne (2004) test, and the negative Lyapunov exponent from the shedding period map indicate that the LC is not chaotic. A negative Lyapunov exponent suggests an attractor that is either a point or a limit cycle. These

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FIG. 7. The period of the Loop Current’s north–south positions against amplitude, revealing the nonlinear relationship among them.

results also show that the analogy of the LC to a leaky faucet does not hold under close scrutiny despite its initial promise as a model to guide this work. Figures 4 and 5 reveal that the LC north–south motion can be described by a sinusoidal function and is acting like a dampened, driven oscillator as indicated by the ellipses observed in both plots. The damping is also revealed by the negative Lyapunov exponent found, a necessary condition for a dissipative system, an expected result based on considerations of the physics of the system. An interesting result is presented in Fig. 6, which shows that the amplitude and period of the LC undergo variations of about five–six cycles. The complex demodulation analysis confirms these amplitude variations. Because each cycle averages ⬃9 months, this variation represents a time scale of 3–5 yr. Candela et al. (2003) suggested an oscillation with period of several years to explain the volume changes noted in the deep Gulf of Mexico. These 3–5-yr cycles match the time scales observed in Fig. 3. This result is revealing because it suggests that the driving force of the LC, the Yucatan Current, must also be varying at similar interannual time scales. A new result is that the period of the LC is a quadratic or nonlinear function of the amplitude (Fig. 7). This relation is typical of oscillators whose amplitude is not small or nonlinear oscillators. Recall that the north–south excursions of the LC, which are of O(300 km), are not small! If the LC behaves like a driven and dampened oscillator, there exists the possibility of some degree or resonance. Resonance would explain the LC large amplitude or north–south excursions observed; however, again information about the

system is lacking to determine if resonance is occurring. Resonance also implies that the period of the motion reflects that of the driving force because they should be equal. Another inference from these results is that the periods in the standard plots could be those of the driving force and not the natural period of the system. The finding that the LC behaves like a driven, damped-oscillator should not be surprising. The result of interest is finding that the LC is not chaotic but behaves like a nonlinear oscillator. This result raises the possibility that the LC is in a state of resonance. Its short memory requires that it reset itself every cycle; the irregular behavior could represent a response to changing initial conditions in the Yucatan Channel in terms of transport or vorticity input from the Caribbean Sea (Oey et al. 2003; Candela et al. 2002). Hurbult and Thompson (1980) found that the LC spinup is reinitiated after an eddy is shed in agreement with this work. Damping could also be responsible for the LC observed irregularity. If damping is nonlinear and depends, say, on the LC displacements, when the system experiences short displacements, it grows, but when the displacements are large, then damping is large. While the Gulf’s basin constrains the north–south incursions of the LC, this current can extend westward to increase its displacement as observed recently; see plate 4 in Leben (2005). This would force the LC to move toward its attractor, which is a limit cycle in the form of an ellipse, as suggested by the tests conducted. If the LC were chaotic, we should abandon hope of predicting its evolution, as predictability in a chaotic system consists of predicting where the points may fall,

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FIG. 8. Short-term prediction scheme for the shedding period of the Loop Current warm eddies, on the basis of 39 shedding events. Data are from Leben (2005).

not on their exact value. However, since the system is not chaotic, it should be predictable. In fact, Leben (2005) presents the first empirical scheme for predicting eddy shedding by the LC. In this work and true to its exploratory data spirit, another scheme for predicting the eddy shedding is developed. It is based on the fact that the standard plot of the periods is described by Tn⫹1 ⫽ AT ␤n . Dividing this expression by Tn, one gets the following recursive expression: Tn⫹1ⲐTn ⫽ AT n␤⫺1. Figure 8 shows the plot of Tn⫹1/Tn versus Tn; the data collapse and are described by the above relationship. Division by Tn transforms the power relationship, Tn⫹1 ⫽ AT ␤n , to a weak inverse proportion, that is, Tn⫹1/Tn ⬇ T ⫺1 n , because ␤ (⫽⫺0.0901) is small. This figure provides a scheme for predicting the future eddyshedding period on the basis of the last period. The above relationship explains only 54% of the variance, revealing that other relevant processes are not captured by this forecasting scheme. This scheme (Fig. 8) agrees with the fact that the system’s memory is very short and predictions can only be made for short times, and it is similar to Leben’s (2005) in the sense that the future shedding period appears to depend mainly on the previous one. Last, the 3–5-yr variation present in Fig. 6 calls for an explanation. Our current understanding of the Yucatan Current and vorticity input from the Caribbean Sea precludes us from providing a dynamical explanation of this cycle. However, examination of known climatic signals, which vary at interannual time scales, provides the best chance of offering a tentative explanation for the 3–5-yr cycle. This signal matches two well-known

climatic variations, El Niño–Southern Oscillation (ENSO) and the North Atlantic Oscillation (NAO). The ENSO period is about 5.2 yr (Dijkstra and Ghil 2005) and suggests a possible connection between the ENSO and the LC. However, the more plausible influence is from the NAO since it is known that the North Atlantic circulation and the Gulf Stream respond strongly to NAO variability (Czaja et al. 2003). The NAO variability has a period of 6–7.5 yr (Czaja et al. 2003; Arzel et al. 2006) and is close enough to the observed time scales for partial resonance to occur. Furthermore, the NAO exhibits behavior consistent with a damped oscillation and a “resonant timescale” of O(10 yr) (Czaja et al. 2003). The suggested mechanism is that NAO affects the intertropical convergence zone, which in turn, affects the strength of the trade winds (Czaja et al. 2003); these in turn affect the transport across the Yucatan Channel, similarly to that described by Oey et al. (2003). Obviously, these ideas need to be confirmed with models and detail analyses.

5. Conclusions In this work, I examine the LC of the Gulf of Mexico through a dynamical systems approach to determine if this current is chaotic. The analyses are based on a limited database available from published sources that produced results of a preliminary nature. The main findings of this work are as follows: 1) The memory of the LC system is very short and limits forecasting to one shedding cycle into the future (Fig. 8). 2) The LC is not chaotic, as the Gottwald and Melbourne (2004) test and negative Lyapunov exponent indicate.

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3) This system’s behavior is consistent with a nonlinear, driven, and dampened oscillator whose period depends on the amplitude of the motion and is known as a limit cycle. 4) The irregular behavior is not the result of sensitivity to initial conditions, but rather of the response of the nonlinear oscillator to varying conditions at Yucatan Channel and its tendency toward the attractor. 5) The attractor of the LC is an ellipse of only two dimensions. 6) The amplitude and period of the oscillations vary at a time scale of 3–5 yr, which matches the NAO and/ or ENSO time scales. 7) The analogy of the LC to a leaky faucet is not supported by this work. Acknowledgments. The author appreciates the support of the Gulf of Mexico OCS Region, Minerals Management Service, U.S. Department of the Interior, during the preparation of this manuscript. The opinions expressed by the author are his own and do not necessarily reflect the opinion or policy of the U.S. government. The author thanks Dr. F. M. Vukovich for providing the revised and extended data of the north–south positions and two anonymous reviewers for providing helpful comments. REFERENCES Arzel, O., T. Huck, and A. C. de Verdiere, 2006: The different nature of the interdecadal variability of the thermohaline circulation under mixed and flux boundary conditions. J. Phys. Oceanogr., 36, 1703–1718. Badan, A., J. Candela, J. Sheinbaum, and J. Ochoa, 2005: Upperlayer circulation in the approaches to Yucatan Channel. Circulation in the Gulf of Mexico: Observations and Models, Geophys. Monogr., Vol. 161, Amer. Geophys. Union, 57–69. Baker, G. L., and J. P. Gollub, 1996: Chaotic Dynamics: An Introduction. Cambridge University Press, 256 pp. Becker, R. A., 1954: Introduction to Theoretical Mechanics. McGraw-Hill, 420 pp. Bendat, J. S., and A. G. Piersol, 1986: Random Data Analysis and Measurement Procedures. 2d ed. John Wiley and Sons, 566 pp. Bloomfield, P., 1976: Fourier Analysis of Time Series: An Introduction. John Wiley and Sons, 258 pp. Candela, J., J. Sheinbaum, J. Ochoa, A. Badan, and R. Leben, 2002: The potential vorticity flux through the Yucatan Channel and the Loop Current in the Gulf of Mexico. Geophys. Res. Lett., 29, 2059, doi:10.1029/2002GL015587. ——, S. Tanahara, M. Crepon, B. Barnier, and J. Sheinbaum, 2003: Yucatan Channel flow: Observations versus CLIPPER ATL6 and MERCATOR PAM models. J. Geophys. Res., 108, 3385, doi:10.1029/2003JC001961. Coats, D. A., 1992: The Loop Current: The physical oceanography of the U.S. Atlantic and eastern Gulf of Mexico, Vol. II, J. D. Milliman and E. Imamura, Eds., OCS Rep. 92-0003,

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