Abstract
The SOund Fixing And Ranging (RAFOS) floats were deployed by the Naval Postgraduate School (NPS) near California coast from 1992 to 2004 at depth between 150 and 600 m (http://www.oc.nps.edu/npsRAFOS/). Each drifter trajectory is adaptively decomposed using the empirical mode decomposition (EMD) into lowfrequency (nondiffusive, i.e., current) and highfrequency (diffusive, i.e., eddies) components. The identified eddies are mostly anticyclonic with total 203 anticyclonic and 36 cyclonic spirals. Eddy characteristics are determined from the time series of individual RAFOS float trajectory. They are the current velocity scale, eddy radial scale, eddy velocity scale, eddy Rossby number, and eddycurrent kinetic energy ratio. The California Current System is found an eddyrich system with the overall eddycurrent kinetic energy ratio of 1.78. It contains submesoscale and mesoscale eddies. The horizontal length scale of 10 km is a good threshold of the eddy radial scale (Leddy) to separate the two kinds of eddies. The mean eddy Rossby number is 0.72 for the submesoscale eddies and 0.06 for the mesoscale eddies. The currenteddy kinetic energy ratio is similar between submesoscale and mesoscale eddies. This may imply similar currenteddy kinetic energy transfer for submesoscale and mesoscale eddies. Statistical characteristics and interannual variability of current velocity scale and eddy characteristic parameters are also presented.
Descriptors:
Empirical mode decomposition; Eddy radial scale; Eddy velocity scale; Eddy rossby number; Eddycurrent kinetic energy ratio.
INTRODUCTION
The California Current system is one of the most productive and wellstudied eastern boundary upwelling systems with high mesoscale activity, frequent formation of eddies, fronts, and filaments throughout the year. Numerous mesoscale eddies are generated by surface and subsurface currents. Cyclonic eddies are mainly generated at the surface with a surface core (in the upper 150 m), whereas anticyclonic eddies are predominantly generated at subsurface with a subsurface core (~400 m) (Simpson and Lynn, 1990SIMPSON, J. J. & LYNN, R. J. 1990. A mesoscale eddy dipole in the off¬shore California Current. Journal of Geophysical Research, 95(C8), 1300913022.; Huyer et al., 1998HUYER, A., BARTH, J. A., KOSRO, P. M., SHEARMAN, R. K. & SMITH, R. L. 1998. Upperocean water mass characteristics of the California Current, summer 1993. Deep Sea Research Part II: Topical Studies in Oceanography, 45(89), 14111442.; Collins et al, 1996COLLINS, C. A., PAQUETTE, R. G. & RAMP, S. R. 1996. Annual variability of ocean currents at 350m depth over the continental slope off Point Sur, California, CalCOFI Report, 37. La Jolla: California Cooperative Oceanic Fisheries Investigations, pp. 257263., 2000COLLINS, C. A., GARFIELD, N., RAGO, T. A., RISERMILLER, F. W. & CARTER, E. 2000. Mean structure of the inshore countercurrent and California Undercurrent off Point Sur, California. Deep Sea Research Part II, 47, 765782.; Garfield et al., 1999GARFIELD, N., COLLINS, C. A., PAQUETTE, R. G. & CARTER, E. I999. Lagrangian exploration of the California undercurrent, 199295. Journal of Physical Oceanography, 29, 560583.).
Trajectories of a set of N Lagrangian drifters at time t are represented by {x^{(n)}(t) = [x^{(n)}(t), y^{(n)}(t)], u^{(n)}(x, y, t), n = 1, 2, …, N } with (x, y) the horizontal position. They measure fluid characteristics along fluid particles’ trajectories (i.e., Lagrangian trajectories), and make water masses distinguishable in terms of origin and/or destination and traced, while its characteristics are continually changing. Eulerain velocity field can be calculated from Lagrangian trajectories using routine ocean data assimilation systems such as the optimal spectral decomposition (OSD) [Chu et al., 2003 aCHU, P. C., IVANOV, L. M., KORZHOVA T. P., MARGOLINA, T. M. & MELNICHENKO, O. M. 2003a. Analysis of sparse and noisy ocean current data using flow decomposition. Part 1: theory. Journal of Atmospheric and Oceanic Technology, 20(4), 478491.,bCHU, P. C., IVANOV, L. M., KORZHOVA T. P., MARGOLINA, T. M. & MELNICHENKO, O. M. 2003b. Analysis of sparse and noisy ocean current data using flow decomposition. Part 2: application to Eulerian and Lagrangian data. Journal of Atmospheric and Oceanic Technology, 20, 492512.]. Some phenomena were identified such as fallwinter recurrence of current reversal from westward to eastward on the TexasLouisiana continental shelf from near surface drifting buoy and current meter [Chu et al. 2005CHU, P. C., IVANOV, L. M. & MELNICHENKO, O. M. 2005. Fallwinter current reversals on the TexasLouisiana continental shelf. Journal of Physical Oceanography, 35, 902910.], and propagation of long baroclinic Rossby waves at middepth (around 1,000 m deep) in the tropical north Atlantic from the Argo floats (Chu et al. 2007CHU, P. C., IVANOV, L. M., MELNICHENKO, O. V. & WELLS, N. C. 2007. On long baroclinic Rossby waves in the tropical North Atlantic observed from profiling floats. Journal of Geophysical Research, 112(C5), C05032, DOI: https://doi.org/10.1029/2006JC003698
https://doi.org/10.1029/2006JC003698...
).
Three major approaches are availble to identify dynamic characteristics of currents and eddies from Lagrangian drifters without using ocean data assimilation: (a) particle dispersion, (b) Eulrian mean velocity, and (c) empirical mode decomposition. The first approach is the particle dispersion relative to their center,
The variances of locations of N drifters at time instance t,
With (D_{x}, Dy) grow linearly with time and the corresponding constant spreading rates (K_{x}, K_{y}) are referred as the 2D turbulent diffusivity of eddies [Rypina et al. 2012RYPINA, I. I., KAMENKOVICH, I., BERLOFF, P. & PRATT, L. J. 2012. Eddyinduced particle dispersion in the nearsurface North Atlantic. Journal of Physical Oceanography, 42(12), 22062228.].
The second approach is to estimate the Eulerian mean velocity from a group of N drifters:
and subtraction of U(x, y) from the velocity identified from Lagrangian drifters, u^{(n)}(x, y, t), leads to the “residue” velocities,Here, the Eulerian mean velocities are computed as an ensemble average in the defined geographic region that velocities from different floats and times [e.g., Davis 1991DAVIS, R. E. 1991. Observing the general circulation with floats. Deep Sea Research, 38(Suppl 1), S531S571.; Collins et al., 2004COLLINS, C. A., IVANOV, L. M., MELNICHENKO, O. B. & GARFIELD, N. 2004. California Undercurrent variability and eddy transport estimated from RAFOS float observations. Journal of Geophysical Research, 109(C5), DOI: https://doi.org/10.1029/2003JC002191
https://doi.org/10.1029/2003JC002191...
; Chu et al., 2007CHU, P. C., IVANOV, L. M., MELNICHENKO, O. V. & WELLS, N. C. 2007. On long baroclinic Rossby waves in the tropical North Atlantic observed from profiling floats. Journal of Geophysical Research, 112(C5), C05032, DOI: https://doi.org/10.1029/2006JC003698
https://doi.org/10.1029/2006JC003698...
; Chu and Fan, 2014CHU, P. C. & FAN, C. W. 2014. Accuracy progressive calculation of Lagrangian trajectory from gridded velocity field. Journal of Atmospheric and Oceanic Technology, 31(7), 16151627.] or as binned velocities with cubic splines [Bauer et al., 1998BAUER, S., SWENSON, S., GRIFFA, A., MARIANO, A. J. & OWENS, K. 1998. Eddymean decomposition and eddydiffusivity estimates in the tropical Pacific Ocean. Journal of Geophysical Research, 103(C13), 855871.].
Both particle dispersion and Lagrangian residue velocities need sufficient number (N) of drifters in the calculation at then same time t. If there is very few codeployed drifters such as less than three SOund Fixing And Ranging (RAFOS) floats available at same timeperiods by the Naval Postgraduate School (NPS) in the California coasts (see website: http://www.oc.nps.edu/npsRAFOS/)? One option is to combine drifters from different timeperiods together as if they were deployed at the same timeperiod. Such treatment cannot identify temporal variability of eddy charcteristics.
The third approach is to use the empirical mode decomposition (EMD) (Huang et al. 1998HUANG, N., SHEN, Z., LONG, S. R. WU, M. C., SMITH, H. H., ZHENG, Q., YEN, N., TUNG, C. C. & LIU, H. H. 1998. The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis. Proceedings of Royal Society London, 454, 903995.) to separate a Lagrangian trajectory x(t) into deterministic [low frequency mode representing the currents, x_{cur}(t)] and stochastic [nonlow frequency mode representing eddy, x_{eddy}(t)] trajectories (Chu 2018CHU, P. C. 2018. Steepest ascent low/nonlow frequency ratio in empirical mode decomposition to separate deterministic and stochastic velocities from a single Lagrangian drifter. Journal of Geophysical Research, 123(3), 17081721, DOI: https://doi.org/10.1002/2017JC013500
https://doi.org/10.1002/2017JC013500...
)
Such Lagrangiantype decomposition changes the traditional approaches (particle dispersion and Lagrangian mean) of using N drifters [x^{(n)}(t), n = 1, , …, N] to get currents (i.e., mean flow) and eddies from Lagrangian trajectory data. The trajectories of 54 RAFOS floats deployed near California coast by C.A. Collins of the Naval Postgraduate School from 1992 to 2004 at depth between 150 and 600 m (http://www.oc.nps.edu/npsRAFOS/) (Fig. 1) show combination of deterministic (low frequency) and stochastic (high frequency) components.
Trajectories of 54 RAFOS floats in the California coast by the Naval Postgraduate School between 1992 and 2004 (http://www.oc.nps.edu/npsRAFOS/). The thick black trajectory is the Float N073.
This paper uses the deterministicstochastic EMD to separate current and eddy from a single Lagrangian drifter. This procedure gives current velocity scale as well as eddy characteristic parameters such as the eddy radial scale, eddy velocity scale, eddy Rossby number, and eddycurrent kinetic energy ratio. The rest of the paper is organized as follows. Section 2 describes the procedure of the deterministicstochastic EMD. Section 3 and 4 present the identified current and eddy characteristic parameters for each RAFOS float and their statistics for 54 RAFOS floats. Section 5 describes the temporal variability of the current and eddy characteristic parameters. Section 6 presents the results.
DETERMINISTICSTOCHASTIC EMD
The EMD (see ^{Appendix A}
APPENDIX A
EMPIRICAL MODE DECOMPOSITION
This appendix is duplicated from Section 2 of Chu et al. [2012]. Let x(t) represent the time series with fluctuations on various time scales [see Fig. A1]. The empirical mode decomposition (EMD) method is depicted as follows. First, the local minima and maxima of the signal x(t) are identified. Second, the local maxima are connected together by interpolation, forming an upper envelope emax(t). The same is done for local minima, providing a lower envelope emin(t). Third, the mean of the two envelopes are calculated (A1)m1(t)=[emax(t)+emin(t)]/2.
Fourth, the mean is subtracted from the signal, providing the local detail (A2)h1(t)=x(t)−m1(t).
The time series h1(t) is checked if it has local minima and local maxima. If yes, h1(t) is considered as the first IMF and denoted (A3)C1(t)=h1(t)
It is subtracted from the original signal and the first residual, (A4)r1(t)=X(t)−h1(t) is taken as the new series in step 1. If h1(t) is not an IMF, a procedure called “sifting process” is applied as many times as necessary to obtain an IMF. In the sifting process, h1(t) is considered as the new data, and the same procedure applies. The IMFs are orthogonal, or almost orthogonal functions (mutually uncorrelated). Because this method does not require stationarity and linearity of the data and is especially suitable for nonstationary and nonlinear time series analysis.
Figure A1
Procedure of the empirical mode decomposition (EMD).
By construction, the number of extrema decreases when going from one residual to the next. The application of the above algorithm ends when the residual has only one extrema, or is constant. In this case no more IMF can be extracted. The complete decomposition is then achieved in a finite number of steps. The signal c(t) is finally written as the sum of IFMs cp(t) and the trend r(t): (A5)c(t)=∑p=1Pcp(t)+r(t) where c(t) represents x(t) and y(t). There is no any oscillation (i.e., nonexistence of both maximum and minimum envelopes) in the trend r(t), which should represent the trend. Obviously, successfulness of the EMD depends on accurate determination of upper and lower envelopes.
) decomposes a Lagrangian trajectory into the intrinsic mode functions (IMFs) regardless of their linearity, stationarity, and stochasticity (Huang et al., 1998HUANG, N., SHEN, Z., LONG, S. R. WU, M. C., SMITH, H. H., ZHENG, Q., YEN, N., TUNG, C. C. & LIU, H. H. 1998. The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis. Proceedings of Royal Society London, 454, 903995.; Chu et al., 2012CHU, P. C., FAN, C. W. & HUANG, N. 2012. Compact empirical mode decomposition  an algorithm to reduce mode mixing, end effect, and detrend uncertainty. Advances in Adaptive Data Analysis, 4(3), 1250017, DOI: https://doi.org/10.1142/S1793536912500173
https://doi.org/10.1142/S179353691250017...
, 2014CHU, P. C., FAN, C. W. & HUANG, N. 2014. Derivativeoptimized empirical mode decomposition for the HilbertHuang transform. Journal of Computational and Applied Mathematics, 259(Pt A), 5764.). The key point to perform this decomposition is the sifting process (with four steps), which decomposes a Lagrangian drift trajectory x(t) into [see (A5) in ^{Appendix A}
APPENDIX A
EMPIRICAL MODE DECOMPOSITION
This appendix is duplicated from Section 2 of Chu et al. [2012]. Let x(t) represent the time series with fluctuations on various time scales [see Fig. A1]. The empirical mode decomposition (EMD) method is depicted as follows. First, the local minima and maxima of the signal x(t) are identified. Second, the local maxima are connected together by interpolation, forming an upper envelope emax(t). The same is done for local minima, providing a lower envelope emin(t). Third, the mean of the two envelopes are calculated (A1)m1(t)=[emax(t)+emin(t)]/2.
Fourth, the mean is subtracted from the signal, providing the local detail (A2)h1(t)=x(t)−m1(t).
The time series h1(t) is checked if it has local minima and local maxima. If yes, h1(t) is considered as the first IMF and denoted (A3)C1(t)=h1(t)
It is subtracted from the original signal and the first residual, (A4)r1(t)=X(t)−h1(t) is taken as the new series in step 1. If h1(t) is not an IMF, a procedure called “sifting process” is applied as many times as necessary to obtain an IMF. In the sifting process, h1(t) is considered as the new data, and the same procedure applies. The IMFs are orthogonal, or almost orthogonal functions (mutually uncorrelated). Because this method does not require stationarity and linearity of the data and is especially suitable for nonstationary and nonlinear time series analysis.
Figure A1
Procedure of the empirical mode decomposition (EMD).
By construction, the number of extrema decreases when going from one residual to the next. The application of the above algorithm ends when the residual has only one extrema, or is constant. In this case no more IMF can be extracted. The complete decomposition is then achieved in a finite number of steps. The signal c(t) is finally written as the sum of IFMs cp(t) and the trend r(t): (A5)c(t)=∑p=1Pcp(t)+r(t) where c(t) represents x(t) and y(t). There is no any oscillation (i.e., nonexistence of both maximum and minimum envelopes) in the trend r(t), which should represent the trend. Obviously, successfulness of the EMD depends on accurate determination of upper and lower envelopes.
],
The RAFOS subsurface data downloaded from http://www.oc.nps.edu/npsRAFOS/ contains 61 RAFOS floats. Among them 7 floats (N001, N009, N012, N042, N046, N049, N068) have too many missing data inside the time series. They are not included in the computation. The date rate varies from general 3/day to around 22/day (N030 18 May  10 June 1994). For each float, the time series [x(t_{i}), y(t_{i})] are decomposed into IMFs and trend using the EMD method (see ^{Appendix A} APPENDIX A EMPIRICAL MODE DECOMPOSITION This appendix is duplicated from Section 2 of Chu et al. [2012]. Let x(t) represent the time series with fluctuations on various time scales [see Fig. A1]. The empirical mode decomposition (EMD) method is depicted as follows. First, the local minima and maxima of the signal x(t) are identified. Second, the local maxima are connected together by interpolation, forming an upper envelope emax(t). The same is done for local minima, providing a lower envelope emin(t). Third, the mean of the two envelopes are calculated (A1)m1(t)=[emax(t)+emin(t)]/2. Fourth, the mean is subtracted from the signal, providing the local detail (A2)h1(t)=x(t)−m1(t). The time series h1(t) is checked if it has local minima and local maxima. If yes, h1(t) is considered as the first IMF and denoted (A3)C1(t)=h1(t) It is subtracted from the original signal and the first residual, (A4)r1(t)=X(t)−h1(t) is taken as the new series in step 1. If h1(t) is not an IMF, a procedure called “sifting process” is applied as many times as necessary to obtain an IMF. In the sifting process, h1(t) is considered as the new data, and the same procedure applies. The IMFs are orthogonal, or almost orthogonal functions (mutually uncorrelated). Because this method does not require stationarity and linearity of the data and is especially suitable for nonstationary and nonlinear time series analysis. Figure A1 Procedure of the empirical mode decomposition (EMD). By construction, the number of extrema decreases when going from one residual to the next. The application of the above algorithm ends when the residual has only one extrema, or is constant. In this case no more IMF can be extracted. The complete decomposition is then achieved in a finite number of steps. The signal c(t) is finally written as the sum of IFMs cp(t) and the trend r(t): (A5)c(t)=∑p=1Pcp(t)+r(t) where c(t) represents x(t) and y(t). There is no any oscillation (i.e., nonexistence of both maximum and minimum envelopes) in the trend r(t), which should represent the trend. Obviously, successfulness of the EMD depends on accurate determination of upper and lower envelopes. ). Seven IFMs and a trend are identified in the (x, y) directions (Fig. 2). It clearly shows that the high frequency motion dominates the low IFM modes. Frequency reduces as the IFM mode from the lowest (IMF1) to the highest (IMF7). The trend (no oscillation) is of course the part of the deterministic motion, but not all.
The IMFs and trend of (a) x(t) and (b) y(t) of the RAFOS float N073 from 21 November 1999 to 12 February 2001. Combination of IMF1 to IMF4 is the highfrequency component (eddy) and combination of IMF5 to IMF7 and the trend is the lowfrequency component (i.e., mean flow).
The IMFs are separated into the deterministic and stochastic parts, using the steepest ascent low/nonlow frequency ratio (see ^{Appendix B} APPENDIX B STEEPEST ASCENT LOW/HIGH FREQUENCY RATIO To separatec(t) into deterministic and stochastic components, we have to determine the sth IMF mode, in which the stochastic signal combination of the 1st, 2nd, …, sth IMFs constitutes, and the combination of (s+1)th, (s+2)th, …, Pth IMFs. The trend constitutes the deterministic signal. To do so, the Fourier transform is conducted on each IMF cp(t): (B1)fpk=1J∑j=0J−1cp(tj)exp[(−i2πk/J)tj],k=0,2,…,J−1 which is a set of complex coefficients in the frequency space. Here, and J is the length of the time series. For each IFM cp(t), tthe phase spectrum (Rios and de Mello 2016; Chu 2018) is calculated by (B2)θp(k)=arctanIm(fpk)Re(fpk) and the amplitude is computed up to the Nyquist frequency (B3)ap(k)=fpkf^pk/J,k=1,2,…,J/2 Let m represent lowest αfraction (α<1) of the frequency domain, i.e., m = [α × J / 2] with the bracket indicating the integer part. The powers of lowest αfraction frequency (Eα) and total frequency (EJ) are defined by (Chu 2018): (B4)Ep,α=∑k=1map2(k),Ep,J=∑k=1J/2ap2(k) The low/nonlow frequency ratio is defined by (B5)Rp,α=Ep,αEp,J−Ep,α which indicates the low frequency (αfraction) dominance of the pth IMF. The sth IMF mode is determined by the ratio of the low frequency dominance parameter for the (p+1)th IMF to the pth IMF (B6)Γp,α=Rp+1,αRp,α which represents increase of strength of the low frequency component. Since the low frequency enhances as the mode number p increases, separation of deterministic and stochastic signals is at the IMF with the maximum value of the ratio (B7)Γs,α=max(Γp,α∣p=1,2,…,P−1) The combination of the 1st, 2nd, …, sthIMFs constitutes the stochastic signal, and the combination of (s+1)th, (s+2)th, …, Pth IMFs, and the trend constitutes the deterministic signal. ). For RAFOS N073, combination of IMF1 to IMF4 is the highfrequency component (eddy) and combination of IMF5 to IMF7 and the trend is the lowfrequency component (i.e., mean flow). After the separation of Lagrangian drifter‘s trajectory into deterministic and stochastic components with the total number of position points of J,
Figure 3 shows the observed (red), deterministic (blue), and stochastic (green) trajectories of RAFOS N073. The deterministic trajectory represents the mean flow, i.e., the current. The stochastic trajectory indicates the eddy. The eddy is transported along the deterministic trajectory. The eddy in Figure 3 does not represent its real position and is separated from the deterministic trajectory arbitrarily just for illustration of eddylike motion.
Observed (red), deterministic (blue), and stochastic (green) trajectories of RAFOS N073. The deterministic trajectory represents the mean flow. The stochastic trajectory indicates the eddy. The center of the eddy is transported by the mean flow (i.e., along the deterministic trajectory), and is put away from the deterministic trajectory arbitrarily just for illustration of eddylike motion.
EDDY AND CUURENT CHARACTERISTICS IDENTIFIED FROM AN INDIVIDUAL RAFOS FLOAT
After the deterministic and stochastic trajectories and velocities are obtained from an individual RAFOS float, the eddy characteristics can be easily identified during the float ‘s drifting period. The eddy radial scale is defined by the root mean square of the stochastic trajectory
The eddy velocity scale (V_{eddy}) is defined by
The eddy velocity kinetic energy per unit mass, eddy angular velocity scale (), and eddy Rossby number (R_{eddy}) are defined by
where f is the Coriolis parameter, which is evaluated at 40oN here. The current velocity scale (V_{cur}) is defined by
Table 1 shows the current velocity scale and eddy characteristic parameters identified from the RAFOS floats with N073 float (21 November 1999 to 12 February 2001) as an example,
Identified current and eddy parameters from individual RAFOS float: radial scale (km), velocity scale (cm/s), and Rossby number.
The current kinetic energy per unit mass (K_{cur}) and the eddy/current kinetic energy ratio (r) are defined by
Due to eddy’s circular motion (green trajectory in Figure 3), the time series of [x_{sto}(t_{j}), v_{sto}(t_{j})] or [y_{sto}(t_{j}), u_{sto}(t_{j})] determine the types of eddy (cylcynic or anticyclonic),
Table 2 shows the eddycurrent kinetic energy ratio as well as the number of cyclonic/anticylonic spirals. For the RAFOS float N073 float, we have
Identified 36 cyclonic (denoted by ‘C’) spirals and 203 anticyclonic (denoted by AC) spirals and eddy to current kinetic energy ratio from 54 RAFOS floats.
STATISTICS OF EDDY AND CURRENT CHARCTERISTIC PARAMETERS
Histograms of current velocity scale (V_{cur}) and eddy characteristic parameters such as radial scale (L_{eddy}), velocity scale (V_{eddy}), Rossby number (R_{eddy}), and eddycurrent kinetic energy ratio (r) are constructed from 54 RAFOS floats shown in Tables 1 and 2. All the histograms are nonsymmetric, high dispersive, and positively skewed (Figure 4). The California Undercurrent is an eddyrich system [i.e., overall more kinetic energy for eddies than for currents with mean K_{eddy} /K_{cur} ratio of 1.78, standard deviation of 2.00, skewness of 2.79, and kurtosis 12.00 (Table 3). For N007 float (7 July  5 September 1993), the eddycurrent energy ratio is 0.13 with the eddy velocity scale of 6.91 cm/s and the current velocity scale of 19.16 cm/s, which is the minimum ratio to be observed. For N024 float (17 May  9 June 1994), the eddycurrent energy ratio is 11.18 with the eddy velocity scale of 33.94 cm/s and the current velocity scale of 10.15 cm/s, which is the maximum ratio to be observed. Besides, Table 3 shows the statistics of other eddy parameters. For example, the eddy radial scale (L_{eddy}) has a mean of 18.37 km, minimum of 1.12 km, maximum of 102.21 km, standard deviation of 21.33 km, skewness of 2.31, and kurtosis of 8.50. The eddy velocity scale has a mean of 11.98 cm/s, minimum of 2.72 cm/s, maximum of 44.17 cm/s, standard deviation of 8.65 cm/s, skewness of 1.74, and kurtosis of 5.82.
Histograms of current and eddy characteristic parameters identified from 54 RAFOS floats: (a) eddy radial scale (km), (b) eddy velocity scale (cm/s), and (c) eddy Rossby number, (d) current velocity scale (cm/s), and (e) eddy and current kinetic energy ratio.
Among 54 RAFOS floats, 24 RAFOS floats (44%) represent the submesoscale eddies with L_{eddy} < 10 km (Table 1): 2.12, 6.88, 1.12, 7.98, 1.26, 0.90, 1.41, 6.09, 1.33, 9.26, 6.86, 1.38, 6.43, 9.45, 1.81, 2.56, 3.89, 2.42, 5.14, 3.81, 5.06, 6.48, 2.73, and 9.13 km, with a mean of 4.40 km and a standard deviation of 2.89 km. Their eddy velocity scales are 8.54, 9.80, 6.91, 10.08, 44.17, 29.22, 33.94, 10.99, 30.43, 3.43, 7.31, 4.74, 9.71, 8.32, 5.27, 4.99, 7.16, 6.28, 6.46, 8.02, 2.72, 6.13, 2.81, and 5.06 cm/s with a mean of 11.35 cm/s and a standard deviation of 11.04 cm/s. Their eddy Rossby numbers (R_{eddy}) are 0.45, 0.16, 0.70, 0.14, 3.99, 3.70, 2.76, 0.20, 2.61, 0.04, 0.12, 0.40, 0.15, 0.11, 0.33, 0.21, 0.20, 0.30, 0.14, 0.22, 0.06, 0.10, 0.11, and 0.06 with the mean value of 0.72 and the standard deviation of 1.20.
The rest 30 RAFOS floats (56%) represent the mesoscale eddies with L_{eddy} > 10 km (Table 1): 33.49, 13.67, 28.35, 19.53, 16.36, 35.09, 29.87, 15.57, 24.35, 17.29, 83.97, 37.95, 17.46, 11.61, 85.09, 57.07, 11.43, 25.95, 102.21, 10.75, 18.56, 13.59, 20.21, 36.53, 28.13, 15.04, 12.91, 22.00, 17.10, and 25.13 km with a mean of 29.54 km and a standard deviation of 23.12 km. The eddy velocity scales (V_{eddy}) are 28.92, 22.55, 17.63, 25.04, 16.93, 17.15, 11.99, 10.82, 12.97, 10.18, 22.44, 9.76, 4.55, 4.85, 10.95, 11.59, 7.37, 8.40, 9.54, 12.14, 4.94, 7.14, 13.76, 19.56, 6.89, 13.37, 7.13, 7.02, 7.30, and 11.80 cm/s with a mean of 12.49 cm/s and a standard deviation of 6.28 cm/s. The eddy Rossby numbers (R_{eddy}) are 0.09, 0.17, 0.07, 0.13, 0.12, 0.05, 0.04, 0.07, 0.06, 0.07, 0.03, 0.03, 0.03, 0.04, 0.01, 0.02, 0.06, 0.03, 0.01, 0.12, 0.03, 0.06, 0.07, 0.06, 0.03, 0.10, 0.06, 0.03, 0.05, and 0.05 with a mean of 0.06 and a standard deviation 0.04. The overall eddy velocity scales are comparable between the submesoscale eddies (mean: 11.35 cm/s, standard deviation: 11.04 cm/s) and the mesoscale eddies (mean: 12.49 cm/s, standard deviation: 6.28 cm/s). This may imply that the California Undercurrent has submesoscale and mesoscale eddies and 10 km is a good threshold of the eddy radial scale (L_{eddy}) to separate the two. The eddy Rossby numbers are more than an order of magnitude larger for the submesoscale eddies (mean: 0.72, standard deviation: 1.20) than for the mesoscale eddies (mean: 0.06, standard deviation: 0.04).
The eddycurrent kinetic energy ratio (Table 3) ratios are 2.57, 1.27, 0.13, 0.76, 1.96, 4.20, 11.19, 0.71, 5.59, 0.48, 0.52, 0.50, 1.67, 3.94, 0.63, 0.95, 1.30, 0.20, 0.52, 0.59, 0.23, 0.56, 0.18, and 0.43 for L_{eddy} < 10 km, representing submesoscale eddies, with a mean of 1.71 and a standard deviation of 2.47. The eddycurrent kinetic energy ratios are 1.01, 1.70, 1.71, 1.09, 2.03, 1.94, 1.39,1.64, 2.38, 1.65, 8.13, 1.61, 0.31, 0.36, 2.70, 1.26, 1.08, 1.08, 1.91, 2.03, 0.66, 0.66, 3.01, 5.44, 0.72, 3.18, 0.40, 1.19, 1.19, and 1.65 for L_{eddy} > 10 km, representing mesoscale eddies, with a mean of 1.84 and a standard deviation of 1.57 for the rest 30 floats. Both mean and standard deviation are comparable for the submesoscale and mesoscale eddies. This may imply that the currenteddy kinetic energy transfer is similar for submesoscale and mesoscale eddies.
TEMPORAL VARIABILITY OF EDDY AND CURRENT CHARACTERISTIC PARAMETERS
All the identified eddy and current parameters (L_{eddy}, V_{eddy}, R_{eddy}, V_{cur}, r) have evident temporal variabilities (Figure 5). Large dispersion is found in L_{eddy} before August 1998 from 1.12 km (7 July  5 September 1993, N007) to 102.21 km (25 February 1997  8 July 1998, N051). Small dispersion in L_{eddy} is found after August 1998 with a maximum of 36.53 km (21 November 1999  12 February 2001, N073) (Figure 5a) and a minimum of 1.81 km (5 May 1999  18 May 2000, N069). Large dispersion is found in V_{eddy} before 1995 from 6.91 cm/s (7 July  5 September 1993, N007) to 44.17 cm/s (19 May 10 June 1994, N021). Small dispersion is found after December 1994 with a maximum of 19.56 cm/s (21 November 1999  12 February 2001, N073) and a minimum of 2.72 cm/s (11 September 2000  9 July 2002, N084) (Figure 5b). The eddy Rossby number (R_{eddy}) is mostly less than 0.5 (Figure 5c). Large disperse is found before 1995 with large values of 3.99 (N021), 3.70 (N022), 2.76 (N024), during 17 May  10 June 1994 and 18 May10 June 1994 and small value of 0.03 during 7 August  30 December 1994 (N031). Small dispersion is found after 1995 with a maximum of 0.33 during 5 May 1999  18 May 2000 (N069) (Figure 5c). The current velocity scale (V_{cur}) has larger dispersion before 1995. It has a maximum of 31.59 cm/s during 19 May  10 June 1994 (N021) and a minimum of 5.33 cm/s during 12 August  11 September 1992 (N002). It has smaller dispersion after 1995 with a maximum of 14.14 cm/s during 21 November 1999  12 February 2001 (N075) and a minimum of 4.19 cm/s during 29 April 1998  24 June 1999 (N065) (Figure 5d). The K_{eddy} /K_{cur} ratio has larger dispersion before 1995 with a maximum of 11.18 during 17 May  9 June 1994 (N024) and a minimum of 0.13 during 7 July  5 September 1993 (N007). It has smaller dispersion after 1995 with a maximum of 5.45 during 21 November 1999  12 February 2001 (N073) and a minimum of 0.23 values during 11 September 2000  9 July 2002 (N084) (Figure 5e).
Temporal variation of identified current and eddy characteristic parameters from 54 RAFOS floats: (a) eddy radial scale (km), (b) eddy velocity scale (cm/s), and (c) eddy Rossby number, (d) current velocity scale (cm/s), and (e) eddy and current kinetic energy ratio.
CONCLUSION
The deterministicstochastic EMD is used to decompose a RAFOS float trajectory. Time differentiation of the deterministic and stochastic trajectories leads to the Lagragian current and eddy velocities. Application of this method to analysis of 54 RAFOS floats deployed by NPS from 1992 to 2004 at depth between 150 and 600 m leads to the fact that the California Undercurrent is an eddyrich system with the overall eddycurrent kinetic energy ratio of 1.78. These eddies are mostly anticylconic with total 203 anticyclonic and 36 cyclonic spirals. Both submesoscale and mesoscale eddies exist with the mean eddy Rossby number of 0.72 for the submesoscale eddies and 0.06 for the mesoscale eddies. The overall eddy velocity scales are comparable between the submesoscale eddies (mean: 11.35 cm/s) and the mesoscale eddies (mean: 12.49 cm/s). The currenteddy kinetic energy transfer is similar between the two. The horizontal length scale of 10 km is a good threshold of the eddy radial scale (L_{eddy}) to separate the submesoscale and mesosclate eddies.
The identified eddy and current parameters (L_{eddy}, V_{eddy}, R_{eddy}, V_{cur}, r) have evident temporal variabilities. Large dispersion is found in L_{eddy} before August 1998 and small dispersion in Leddy is found after August 1998. Large dispersion of (V_{eddy}, R_{eddy}, V_{cur}, r) is found before 1995 and small dispersion is found after 1995. Further studies are needed on physical mechanisms to cause such a temporal variability.
ACKNOWLEDGMENTS
It is our pleasure to contribute to a special volume honoring Professor Afonso Mascarenhas. He was a visiting professor at NPS through the National Research Council Senior Visitor Program during October 2006  March 2007. The authors learned a lot from him on the circulation in the Gulf of California. Besides, the authors would like to thank Professor (emeritus) Curt Collins at NPS for collecting RAFOS data and unselfishly posting the data at the website.
APPENDIX A EMPIRICAL MODE DECOMPOSITION
This appendix is duplicated from Section 2 of Chu et al. [2012]CHU, P. C., FAN, C. W. & HUANG, N. 2012. Compact empirical mode decomposition  an algorithm to reduce mode mixing, end effect, and detrend uncertainty. Advances in Adaptive Data Analysis, 4(3), 1250017, DOI: https://doi.org/10.1142/S1793536912500173
https://doi.org/10.1142/S179353691250017...
. Let x(t) represent the time series with fluctuations on various time scales [see Fig. A1]. The empirical mode decomposition (EMD) method is depicted as follows. First, the local minima and maxima of the signal x(t) are identified. Second, the local maxima are connected together by interpolation, forming an upper envelope e_{max}(t). The same is done for local minima, providing a lower envelope e_{min}(t). Third, the mean of the two envelopes are calculated
Fourth, the mean is subtracted from the signal, providing the local detail
The time series h_{1}(t) is checked if it has local minima and local maxima. If yes, h_{1}(t) is considered as the first IMF and denoted
It is subtracted from the original signal and the first residual,
is taken as the new series in step 1. If h_{1}(t) is not an IMF, a procedure called “sifting process” is applied as many times as necessary to obtain an IMF. In the sifting process, h_{1}(t) is considered as the new data, and the same procedure applies. The IMFs are orthogonal, or almost orthogonal functions (mutually uncorrelated). Because this method does not require stationarity and linearity of the data and is especially suitable for nonstationary and nonlinear time series analysis.
By construction, the number of extrema decreases when going from one residual to the next. The application of the above algorithm ends when the residual has only one extrema, or is constant. In this case no more IMF can be extracted. The complete decomposition is then achieved in a finite number of steps. The signal c(t) is finally written as the sum of IFMs c_{p}(t) and the trend r(t):
where c(t) represents x(t) and y(t). There is no any oscillation (i.e., nonexistence of both maximum and minimum envelopes) in the trend r(t), which should represent the trend. Obviously, successfulness of the EMD depends on accurate determination of upper and lower envelopes.APPENDIX B STEEPEST ASCENT LOW/HIGH FREQUENCY RATIO
To separatec(t) into deterministic and stochastic components, we have to determine the sth IMF mode, in which the stochastic signal combination of the 1^{st}, 2^{nd}, …, sth IMFs constitutes, and the combination of (s+1)th, (s+2)th, …, Pth IMFs. The trend constitutes the deterministic signal. To do so, the Fourier transform is conducted on each IMF c_{p}(t):
https://doi.org/10.1002/2017JC013500... ) is calculated by
Let m represent lowest αfraction (α<1) of the frequency domain, i.e., m = [α × J / 2] with the bracket indicating the integer part. The powers of lowest αfraction frequency (E_{α}) and total frequency (E_{J}) are defined by (Chu 2018CHU, P. C. 2018. Steepest ascent low/nonlow frequency ratio in empirical mode decomposition to separate deterministic and stochastic velocities from a single Lagrangian drifter. Journal of Geophysical Research, 123(3), 17081721, DOI: https://doi.org/10.1002/2017JC013500
https://doi.org/10.1002/2017JC013500...
):
The low/nonlow frequency ratio is defined by
which indicates the low frequency (αfraction) dominance of the pth IMF. The sth IMF mode is determined by the ratio of the low frequency dominance parameter for the (p+1)th IMF to the pth IMF which represents increase of strength of the low frequency component. Since the low frequency enhances as the mode number p increases, separation of deterministic and stochastic signals is at the IMF with the maximum value of the ratioThe combination of the 1^{st}, 2^{nd}, …, sthIMFs constitutes the stochastic signal, and the combination of (s+1)th, (s+2)th, …, Pth IMFs, and the trend constitutes the deterministic signal.
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Publication Dates

Publication in this collection
08 Aug 2022 
Date of issue
2022
History

Received
26 July 2021 
Accepted
18 May 2022