Journal of Oceanography, Vol. 60, pp. 681 to 687, 2004

Short Contribution

Correspondence between Lagrangian and Eulerian Velocity Statistics at the ASUKA Line N IKOLAI MAXIMENKO* International Pacific Research Center†, SOEST, University of Hawaii, Honolulu, HI 96822, U.S.A.; also at P.P. Shirshov Institute of Oceanology, Russian Academy of Sciences, Moscow, Russia (Received 11 April 2003; in revised form 27 August 2003; accepted 30 September 2003)

The 20–30% differences between drifter- and altimetry-derived mean velocities reported at the ASUKA line (Affiliated Surveys of the Kuroshio off Cape Ashizuri) are explained in terms of the non-Eulerian character of statistics of drifter trajectories’ crossovers with the TOPEX/Poseidon track. Larger mean drifter velocities are shown to result from the fact that more water crosses the line when velocity is large, so that the “line” average velocity corresponds to the Eulerian quantity /, where V n is a velocity component perpendicular to the line. Practical methods are suggested to correct the bias. General properties of the “line” statistic and improvement of dynamical balance by correct accounting for the acceleration term are discussed.

Keywords: ⋅ Lagrangian drifters, ⋅ Eulerian statistic, ⋅ velocity bias, ⋅ statistic correction, ⋅ convergence.

At the same time, UIH98 revealed notably (up to 30%) larger mean velocities computed from the drifter ensemble compared to time-average altimetric velocities referenced to in situ observations. They concluded that the discrepancy is because “of the buoy’s tendency to sample preferentially in the high-velocity Kuroshio”. This conclusion can be interpreted as an assumption of the peculiar role played by horizontal convergence along the Kuroshio axis, which results in the majority of the buoys gathering in the regions of large current velocities. If true, this might require a reconsideration of a number of previous studies based on Lagrangian statistic (e.g., Hsueh et al., 1996; Maximenko et al., 1997). However, the problem has an alternative explanation lying in the definition of the statistic used by UIH98, namely, the average over the events when the Eulerian line is crossed by a Lagrangian particle. This statistic is neither Eulerian nor Lagrangian. The probability of a particular velocity value at the line depends on how many particles (and equivalently how much water) crossed the line, and this number is proportional to the velocity component normal to the line. Apparently, no particle may cross the line at zero velocity, which is thus completely absent from the “line” statistics. In the present study, drifter data (the same as those used by UIH98 but supplemented by recent observations) are applied to a simple theoretical model to evaluate the relation between Eulerian and “line” statistics and to suggest methods for correction of the latter. The paper is or-

1. Introduction Drifter velocity and satellite altimetry observations are essential components of modern oceanographic data. Enhanced accuracy and the large volume of data acquired during the past decade make the two datasets suitable for straightforward quantitative evaluation of surface currents, especially their temporal variability. Uchida et al. (1998), hereafter UIH98, conducted a comprehensive analysis of drifter velocity, TOPEX/Poseidon (T/P) sea level anomaly, and in situ observations along the ASUKA line (Affiliated Surveys of the Kuroshio off Cape Ashizuri, Fig. 1) oriented along one of the T/P tracks and almost perpendicular to the direction of mean currents. Using data of 28 WOCE/SVP (World Ocean Circulation Experiment/Surface Velocity Program) drifters with their drogues attached at 15 m depth, and selecting appropriate along-track filter parameters for altimetry, UIH98 revealed a remarkably good correlation between the observed drifter velocity component normal to the line and the corresponding geostrophic component derived from altimetry. The correlation was 0.92 and even higher than that when the centrifugal term was taken into account.

* E-mail address: [email protected] †

International Pacific Research Center is partly sponsored by Frontier Research System for Global Change. Copyright © The Oceanographic Society of Japan.

681

34

2

x

33

1 32

a)

Latitude N

31 30 29 28

100 km 27 26

y 25 130

132

134 136 Longitude E

138

33

2

b)

32.8 32.6

Latitude N

32.4

shown in Fig. 1a. In addition to drifters with the drogues attached at 15 m depth, it also includes drifters that lost their drogues but whose data were corrected to the downwind slip (Niiler, 2001; Pazan and Niiler, 2001). The format of the data is the same as that used by UIH98. To compare results with UIH98 and to show that they are not specific to the dataset, seven 24.8-km wide bins were selected close to those used by UIH98 (Fig. 1(b)), and all computations (when possible) were performed both for the complete dataset and for the UIH98 subset. In total, 97 drifters have crossed the ASUKA line within the seven bins. Six of them have done so twice, and the other two have crossed three times while looping in the large eddy. Thus, the full dataset contains 109 crossovers (22 of these crossovers provided by 20 drifters where used by UIH98). Drifter coordinates and velocities were transformed into x-y coordinates (Fig. 1), where y is offshore distance and x is the distance from the line, positive in the eastward direction. Crossovers of drifter trajectories and the ASUKA line were calculated by linear interpolation of initial 6-hourly drifter positions.

32.2 32

3. Relation between “Line” and Eulerian Statistics Consider an ensemble of Lagrangian particles of spatial density n(x, y, t) advected by time-variable, two-dimensional flow V(x, y, t). The number of particles crossing a small segment of line x = 0 of length δ y centered at y during a short time δt is

50 km 31.8 31.6 31.4

1 m/s

31.2 132.5

133

133.5 Longitude E

134

134.5

Fig. 1. (a) Location of the ASUKA line south of Japan and (b) enlarged study area. Dots mark 6-hourly positions of the complete SVP dataset. Rectangle in (a) locates the area occupied by (b). Vectors in each box of (b) are (north to south) mean velocities for the complete dataset averaged in boxes and bins (bold) and bin-averaged mean velocities for the UIH98 subset. Numbers denote Kyushu (1) and Shikoku (2).

NL(y, t, δ y, δt) = n(0, y, t)·|Vn(0, y, t)|·δy·δt,

(1)

ganized as follows. Section 2 describes the drifter dataset and its correspondence with the dataset used by UIH98. Section 3 contains a simple theoretical model of the “line” statistic and describes its deviation from the Eulerian one. Section 4 suggests methods to correct this deviation. Section 5 summarizes results of this study and suggests an additional source of discrepancy between drifter and altimetry observations, an unaccounted part of acceleration.

where Vn is a velocity component perpendicular to the line. In good agreement with (1), Fig. 2(a) (dots and circles) illustrates a clear tendency for a larger number of drifters to cross the ASUKA line at locations where the mean speeds are higher. As UIH98 pointed out, this tendency can be due to drifter convergence toward the Kuroshio axis. As drifters always stay at the sea surface, such a convergence would increase their density n in the region of large Kuroshio velocities, and together with the non-uniform scheme of the drifters’ original release, this would work against horizontal eddy diffusion (Davis, 1991), the effect of which would make n more homogeneous. Figure 2(b) illustrates that estimates of n found according to (1) do not reveal any systematic growth with the mean velocity and it is, thus, likely that the greater number of crossovers occur at large velocities because of larger values of water transport there rather than because of horizontal inhomogeneity of the drifter ensemble.

2. Data The recent version of the SVP drifter dataset obtained from the NOAA Atlantic Oceanographic and Meteorological Laboratory contains data of 180 drifters in the area

4. Methods for Correcting the “Line” Statistics This section describes alternative statistics and offers methods to transform the “line” statistics into the Eulerian statistic.

682

N. Maximenko

Note that, unlike NL in (1), NB is totally defined by the distribution n of drifters and does not depend on V. This is easy to understand: although fast moving particles have a greater chance of entering the box, they stay within the box for a shorter time. An evident difference from the “line” statistic is that zero velocities are accounted for in the box average but are absent from the “line” average (a motionless particle cannot cross the line). The probability density function (PDF) of the velocity component V n normal to the ASUKA line in the fixed box can be estimated as

0.35

a)

0.3

Ni/sum(Ni)

0.25 0.2 0.15 0.1 0.05 0 40

Ni/sum(Ni)/i, s/cm

3.5

60

x 10

80

100

120

140

3

PB (Vn0 ) = ∑ N B / ∑ N B / δVn ,

3

V

2.5 2

where ∑ V indicates a summation over time intervals δt when Vn0 ≤ Vn ≤ Vn0 + δ Vn, and ∑ T(NB) is a total number of observations. From (2) it follows that

1.5 1 0.5 0 40

(3)

T

b) 60 80 100 120 Bin average speed, cm/s

PB (Vn ) = ∑ n ⋅ δt / ∑ n ⋅ δt / δVn .

140

V

Fig. 2. (a) Relative number of drifters having crossed the ASUKA line in the bins of Fig. 1(b) and (b) those values scaled by the mean absolute velocity component normal to the line versus mean speed for UIH98 (circles) and complete (dots) datasets. Crosses represent analogous box average values for the complete dataset with the exception that in (b) scaling is by the overall seven-boxes-mean |Vn|.

4.1 Box average Box averaging of Lagrangian data is the simplest way to obtain gridded mean velocities. This technique is widely used to evaluate mean surface circulation in many regions (e.g., Hsueh et al., 1996; Poulain et al., 1996; Maximenko et al., 1997; Fratantoni, 2001). This section shows that the box average provides a statistic that is closer to the Eulerian one than the “line” average. Square boxes were positioned (Fig. 1(b)) at bin locations and have the same size of 24.8 km. To prevent fast moving drifters from slipping through the boxes, all trajectories were linearly interpolated onto hourly intervals. Seven boxes contained 1307 hourly drifter observations provided by 98 drifters (only one drifter that visited the boxes did not cross the ASUKA line). The small size of the boxes ensures that the Eulerian statistic of the velocity in the strip formed by the boxes is close to that for the ASUKA line. The number of reports coming from the drifters within the square box of size δ y during time δt is NB(y, t, δy, δt) = n(0, y, t)·δy2·δt.

(2)

( 4)

T

And, hence, for n = const (particles distributed uniformly in space and time), PB corresponds to the local Eulerian PDF indicating what fraction of time velocity is within the selected range. The distribution of drifters in the boxes (crosses in Fig. 2) shows no tendency for drifters to converge toward the Kuroshio axis (actually, their number decreases slightly in the boxes with the strongest currents). The correspondence between box average and true Eulerian statistics is a complex issue influenced by a non-uniform spatiotemporal distribution n (e.g., Davis, 1991). However, the “line” statistic carries all the problems of the box statistic plus an additional one: bias in data distribution toward larger velocities. The PDFs of the “line” (bin) and box average |Vn| shown in Fig. 3(a) computed for the complete dataset and for the UIH98’s subset all demonstrate the same tendencies. Namely, the very weak velocities are missing in the “line” statistics, and the line-averaged values (average over the seven bins of Fig. 1(b)) of |Vn| (90.9 cm/s for the complete dataset and 94.8 cm/s for the UIH98 subset) greatly exceed the corresponding box averages (57.6 and 72.8 cm/s, respectively). Figure 1(b) illustrates that lineaverage Vn’s are also larger than corresponding box-average values for each box/bin. 4.2 Analytical method Analogously to (3)–(4), the expression for the PDF of Vn in line statistic is PL (Vn0 ) = ∑ n ⋅ Vn0 ⋅ δt / ∑ n ⋅ Vn0 ⋅ δt / δVn . V

(5)

T

Lagrangian vs. Eulerian Velocity Statistics

683

and

Probablility density, s/cm

0.015

a)

B = 1/L.

0.01

(11)

From (8) it follows that 0.005

0

0

50

100 |Vn|, cm/s

150

200

L = B/B,

(12)

L = B/B = B·(1 + B/(B)2),

(13)

and 3

L = B/B

2.5

b)

PL/PB

2

(14)

(where |Vn|(B,L)′ = |Vn| – (B,L)), which leads to the “line” dispersion

1.5 1

L = B – (B/B)2 + B/B. (15)

0.5 0

0

0.5

1

1.5 |Vn|/B

2

2.5

3

Fig. 3. Probability density functions (a) of absolute value of velocity component normal to the ASUKA line for “line” (P L, dashed) and box (P B) statistics for the complete dataset (thick) and the UIH98 subset (thin) and (b) ratio PL/PB versus |V n|/ B (dots and circles for the two datasets, correspondingly).

By noticing that ∑ V(n·|Vn0|·δ t) ≈ |Vn0|·∑ V(n·δt) and using (4) P L(Vn) = PB(Vn)·|Vn|/B,

(6)

L/B ≈ 1 + (σ/V 0)2.

and P B(Vn) = PL(Vn)/(|Vn|·L).

(7)

The validity of (6) and its applicability to our dataset is confirmed by Fig. 3(b). “Line” and box averages of any function F(V) are then related as L = B/B,

(8)

B = L/L.

(9)

B = L/L,

(10)

and

Curiously,

684

Hence (from (13)), L is always larger than B by the value determined by the relative dispersion of |Vn|, and, in case of zero skewness of the Eulerian probability density of the velocity, bin-based dispersion of |Vn| is smaller (15) than the corresponding box average value. At the ASUKA line, velocity variability is largely affected by lateral shifts of the Kuroshio axis, and the ratio of the two rightmost terms of (15) B·B/(B)2 varies between –0.4 in the northern boxes of Fig. 1(b) to 1.5 in the southern boxes. Using (12) it can be shown that if box-PDF PB = PB(Vn, V0, σ ), where V0 = B and σ = stdB(Vn), provides finite B and B, then for σ B/ ∂V 0 = ∂∫V n·|V n |(P B ·|V n – V 0|, σ )·dV n/∂V 0 = ∫∂[(V n + V0)·|Vn + V0|]·PB(|Vn|, σ)·dVn/∂V0 = 2·B and for V0

Short Contribution

Correspondence between Lagrangian and Eulerian Velocity Statistics at the ASUKA Line N IKOLAI MAXIMENKO* International Pacific Research Center†, SOEST, University of Hawaii, Honolulu, HI 96822, U.S.A.; also at P.P. Shirshov Institute of Oceanology, Russian Academy of Sciences, Moscow, Russia (Received 11 April 2003; in revised form 27 August 2003; accepted 30 September 2003)

The 20–30% differences between drifter- and altimetry-derived mean velocities reported at the ASUKA line (Affiliated Surveys of the Kuroshio off Cape Ashizuri) are explained in terms of the non-Eulerian character of statistics of drifter trajectories’ crossovers with the TOPEX/Poseidon track. Larger mean drifter velocities are shown to result from the fact that more water crosses the line when velocity is large, so that the “line” average velocity corresponds to the Eulerian quantity /, where V n is a velocity component perpendicular to the line. Practical methods are suggested to correct the bias. General properties of the “line” statistic and improvement of dynamical balance by correct accounting for the acceleration term are discussed.

Keywords: ⋅ Lagrangian drifters, ⋅ Eulerian statistic, ⋅ velocity bias, ⋅ statistic correction, ⋅ convergence.

At the same time, UIH98 revealed notably (up to 30%) larger mean velocities computed from the drifter ensemble compared to time-average altimetric velocities referenced to in situ observations. They concluded that the discrepancy is because “of the buoy’s tendency to sample preferentially in the high-velocity Kuroshio”. This conclusion can be interpreted as an assumption of the peculiar role played by horizontal convergence along the Kuroshio axis, which results in the majority of the buoys gathering in the regions of large current velocities. If true, this might require a reconsideration of a number of previous studies based on Lagrangian statistic (e.g., Hsueh et al., 1996; Maximenko et al., 1997). However, the problem has an alternative explanation lying in the definition of the statistic used by UIH98, namely, the average over the events when the Eulerian line is crossed by a Lagrangian particle. This statistic is neither Eulerian nor Lagrangian. The probability of a particular velocity value at the line depends on how many particles (and equivalently how much water) crossed the line, and this number is proportional to the velocity component normal to the line. Apparently, no particle may cross the line at zero velocity, which is thus completely absent from the “line” statistics. In the present study, drifter data (the same as those used by UIH98 but supplemented by recent observations) are applied to a simple theoretical model to evaluate the relation between Eulerian and “line” statistics and to suggest methods for correction of the latter. The paper is or-

1. Introduction Drifter velocity and satellite altimetry observations are essential components of modern oceanographic data. Enhanced accuracy and the large volume of data acquired during the past decade make the two datasets suitable for straightforward quantitative evaluation of surface currents, especially their temporal variability. Uchida et al. (1998), hereafter UIH98, conducted a comprehensive analysis of drifter velocity, TOPEX/Poseidon (T/P) sea level anomaly, and in situ observations along the ASUKA line (Affiliated Surveys of the Kuroshio off Cape Ashizuri, Fig. 1) oriented along one of the T/P tracks and almost perpendicular to the direction of mean currents. Using data of 28 WOCE/SVP (World Ocean Circulation Experiment/Surface Velocity Program) drifters with their drogues attached at 15 m depth, and selecting appropriate along-track filter parameters for altimetry, UIH98 revealed a remarkably good correlation between the observed drifter velocity component normal to the line and the corresponding geostrophic component derived from altimetry. The correlation was 0.92 and even higher than that when the centrifugal term was taken into account.

* E-mail address: [email protected] †

International Pacific Research Center is partly sponsored by Frontier Research System for Global Change. Copyright © The Oceanographic Society of Japan.

681

34

2

x

33

1 32

a)

Latitude N

31 30 29 28

100 km 27 26

y 25 130

132

134 136 Longitude E

138

33

2

b)

32.8 32.6

Latitude N

32.4

shown in Fig. 1a. In addition to drifters with the drogues attached at 15 m depth, it also includes drifters that lost their drogues but whose data were corrected to the downwind slip (Niiler, 2001; Pazan and Niiler, 2001). The format of the data is the same as that used by UIH98. To compare results with UIH98 and to show that they are not specific to the dataset, seven 24.8-km wide bins were selected close to those used by UIH98 (Fig. 1(b)), and all computations (when possible) were performed both for the complete dataset and for the UIH98 subset. In total, 97 drifters have crossed the ASUKA line within the seven bins. Six of them have done so twice, and the other two have crossed three times while looping in the large eddy. Thus, the full dataset contains 109 crossovers (22 of these crossovers provided by 20 drifters where used by UIH98). Drifter coordinates and velocities were transformed into x-y coordinates (Fig. 1), where y is offshore distance and x is the distance from the line, positive in the eastward direction. Crossovers of drifter trajectories and the ASUKA line were calculated by linear interpolation of initial 6-hourly drifter positions.

32.2 32

3. Relation between “Line” and Eulerian Statistics Consider an ensemble of Lagrangian particles of spatial density n(x, y, t) advected by time-variable, two-dimensional flow V(x, y, t). The number of particles crossing a small segment of line x = 0 of length δ y centered at y during a short time δt is

50 km 31.8 31.6 31.4

1 m/s

31.2 132.5

133

133.5 Longitude E

134

134.5

Fig. 1. (a) Location of the ASUKA line south of Japan and (b) enlarged study area. Dots mark 6-hourly positions of the complete SVP dataset. Rectangle in (a) locates the area occupied by (b). Vectors in each box of (b) are (north to south) mean velocities for the complete dataset averaged in boxes and bins (bold) and bin-averaged mean velocities for the UIH98 subset. Numbers denote Kyushu (1) and Shikoku (2).

NL(y, t, δ y, δt) = n(0, y, t)·|Vn(0, y, t)|·δy·δt,

(1)

ganized as follows. Section 2 describes the drifter dataset and its correspondence with the dataset used by UIH98. Section 3 contains a simple theoretical model of the “line” statistic and describes its deviation from the Eulerian one. Section 4 suggests methods to correct this deviation. Section 5 summarizes results of this study and suggests an additional source of discrepancy between drifter and altimetry observations, an unaccounted part of acceleration.

where Vn is a velocity component perpendicular to the line. In good agreement with (1), Fig. 2(a) (dots and circles) illustrates a clear tendency for a larger number of drifters to cross the ASUKA line at locations where the mean speeds are higher. As UIH98 pointed out, this tendency can be due to drifter convergence toward the Kuroshio axis. As drifters always stay at the sea surface, such a convergence would increase their density n in the region of large Kuroshio velocities, and together with the non-uniform scheme of the drifters’ original release, this would work against horizontal eddy diffusion (Davis, 1991), the effect of which would make n more homogeneous. Figure 2(b) illustrates that estimates of n found according to (1) do not reveal any systematic growth with the mean velocity and it is, thus, likely that the greater number of crossovers occur at large velocities because of larger values of water transport there rather than because of horizontal inhomogeneity of the drifter ensemble.

2. Data The recent version of the SVP drifter dataset obtained from the NOAA Atlantic Oceanographic and Meteorological Laboratory contains data of 180 drifters in the area

4. Methods for Correcting the “Line” Statistics This section describes alternative statistics and offers methods to transform the “line” statistics into the Eulerian statistic.

682

N. Maximenko

Note that, unlike NL in (1), NB is totally defined by the distribution n of drifters and does not depend on V. This is easy to understand: although fast moving particles have a greater chance of entering the box, they stay within the box for a shorter time. An evident difference from the “line” statistic is that zero velocities are accounted for in the box average but are absent from the “line” average (a motionless particle cannot cross the line). The probability density function (PDF) of the velocity component V n normal to the ASUKA line in the fixed box can be estimated as

0.35

a)

0.3

Ni/sum(Ni)

0.25 0.2 0.15 0.1 0.05 0 40

Ni/sum(Ni)/i, s/cm

3.5

60

x 10

80

100

120

140

3

PB (Vn0 ) = ∑ N B / ∑ N B / δVn ,

3

V

2.5 2

where ∑ V indicates a summation over time intervals δt when Vn0 ≤ Vn ≤ Vn0 + δ Vn, and ∑ T(NB) is a total number of observations. From (2) it follows that

1.5 1 0.5 0 40

(3)

T

b) 60 80 100 120 Bin average speed, cm/s

PB (Vn ) = ∑ n ⋅ δt / ∑ n ⋅ δt / δVn .

140

V

Fig. 2. (a) Relative number of drifters having crossed the ASUKA line in the bins of Fig. 1(b) and (b) those values scaled by the mean absolute velocity component normal to the line versus mean speed for UIH98 (circles) and complete (dots) datasets. Crosses represent analogous box average values for the complete dataset with the exception that in (b) scaling is by the overall seven-boxes-mean |Vn|.

4.1 Box average Box averaging of Lagrangian data is the simplest way to obtain gridded mean velocities. This technique is widely used to evaluate mean surface circulation in many regions (e.g., Hsueh et al., 1996; Poulain et al., 1996; Maximenko et al., 1997; Fratantoni, 2001). This section shows that the box average provides a statistic that is closer to the Eulerian one than the “line” average. Square boxes were positioned (Fig. 1(b)) at bin locations and have the same size of 24.8 km. To prevent fast moving drifters from slipping through the boxes, all trajectories were linearly interpolated onto hourly intervals. Seven boxes contained 1307 hourly drifter observations provided by 98 drifters (only one drifter that visited the boxes did not cross the ASUKA line). The small size of the boxes ensures that the Eulerian statistic of the velocity in the strip formed by the boxes is close to that for the ASUKA line. The number of reports coming from the drifters within the square box of size δ y during time δt is NB(y, t, δy, δt) = n(0, y, t)·δy2·δt.

(2)

( 4)

T

And, hence, for n = const (particles distributed uniformly in space and time), PB corresponds to the local Eulerian PDF indicating what fraction of time velocity is within the selected range. The distribution of drifters in the boxes (crosses in Fig. 2) shows no tendency for drifters to converge toward the Kuroshio axis (actually, their number decreases slightly in the boxes with the strongest currents). The correspondence between box average and true Eulerian statistics is a complex issue influenced by a non-uniform spatiotemporal distribution n (e.g., Davis, 1991). However, the “line” statistic carries all the problems of the box statistic plus an additional one: bias in data distribution toward larger velocities. The PDFs of the “line” (bin) and box average |Vn| shown in Fig. 3(a) computed for the complete dataset and for the UIH98’s subset all demonstrate the same tendencies. Namely, the very weak velocities are missing in the “line” statistics, and the line-averaged values (average over the seven bins of Fig. 1(b)) of |Vn| (90.9 cm/s for the complete dataset and 94.8 cm/s for the UIH98 subset) greatly exceed the corresponding box averages (57.6 and 72.8 cm/s, respectively). Figure 1(b) illustrates that lineaverage Vn’s are also larger than corresponding box-average values for each box/bin. 4.2 Analytical method Analogously to (3)–(4), the expression for the PDF of Vn in line statistic is PL (Vn0 ) = ∑ n ⋅ Vn0 ⋅ δt / ∑ n ⋅ Vn0 ⋅ δt / δVn . V

(5)

T

Lagrangian vs. Eulerian Velocity Statistics

683

and

Probablility density, s/cm

0.015

a)

B = 1/L.

0.01

(11)

From (8) it follows that 0.005

0

0

50

100 |Vn|, cm/s

150

200

L = B/B,

(12)

L = B/B = B·(1 + B/(B)2),

(13)

and 3

L = B/B

2.5

b)

PL/PB

2

(14)

(where |Vn|(B,L)′ = |Vn| – (B,L)), which leads to the “line” dispersion

1.5 1

L = B – (B/B)2 + B/B. (15)

0.5 0

0

0.5

1

1.5 |Vn|/B

2

2.5

3

Fig. 3. Probability density functions (a) of absolute value of velocity component normal to the ASUKA line for “line” (P L, dashed) and box (P B) statistics for the complete dataset (thick) and the UIH98 subset (thin) and (b) ratio PL/PB versus |V n|/ B (dots and circles for the two datasets, correspondingly).

By noticing that ∑ V(n·|Vn0|·δ t) ≈ |Vn0|·∑ V(n·δt) and using (4) P L(Vn) = PB(Vn)·|Vn|/B,

(6)

L/B ≈ 1 + (σ/V 0)2.

and P B(Vn) = PL(Vn)/(|Vn|·L).

(7)

The validity of (6) and its applicability to our dataset is confirmed by Fig. 3(b). “Line” and box averages of any function F(V) are then related as L = B/B,

(8)

B = L/L.

(9)

B = L/L,

(10)

and

Curiously,

684

Hence (from (13)), L is always larger than B by the value determined by the relative dispersion of |Vn|, and, in case of zero skewness of the Eulerian probability density of the velocity, bin-based dispersion of |Vn| is smaller (15) than the corresponding box average value. At the ASUKA line, velocity variability is largely affected by lateral shifts of the Kuroshio axis, and the ratio of the two rightmost terms of (15) B·B/(B)2 varies between –0.4 in the northern boxes of Fig. 1(b) to 1.5 in the southern boxes. Using (12) it can be shown that if box-PDF PB = PB(Vn, V0, σ ), where V0 = B and σ = stdB(Vn), provides finite B and B, then for σ B/ ∂V 0 = ∂∫V n·|V n |(P B ·|V n – V 0|, σ )·dV n/∂V 0 = ∫∂[(V n + V0)·|Vn + V0|]·PB(|Vn|, σ)·dVn/∂V0 = 2·B and for V0