lagrangian statistics of three rafos floatspsguest/oc3570/cdrom/... · floats (nps #80 & #81)...

Lagrangian Statistics of three RAFOS floats NPS #80, #81 and #82 in the California Undercurrent

LT Hsieh, Chung-Ping R.O.C Taiwan Navy

14 Sep 2006

2

LIST OF CONTENTS

TABLE OF FIGURES AND TABLES 3

1. INTRODUCTION 4

2. DATA AND ANALYSIS 5

2.1 Three RAFOS floats observations 5

2.2 Single Particle Dispersion 11

2.3 Two Particles Dispersion 13

2.4 Three Particles Dispersion 23

III. CONCLUSIONS 26

REFERENCES 28

3

LIST OF FIGURES AND TABLES

Figure 1. Temperature along NPS #80 trajectory. 6 Figure 2. Temperature and Pressure of NPS #80. 6 Figure 3. Velocity vector of NPS #80. 7 Figure 4. Temperature along NPS #81 trajectory. 8 Figure 5. Temperature and Pressure of NPS #81. 8 Figure 6. Velocity vector of NPS #81. 9 Figure 7. Temperature along NPS #82 trajectory. 10 Figure 8. Temperature and Pressure of NPS #81. 10 Figure 9. Velocity vector of NPS #81. 10 Figure 10. Velocity vs. Latitude in the CUC 11 Figure 11. Single particle dispersion Kuu and Kvv. 12 Figure 12. Pair 1(NPS #80 & #81) trajectories. 14 Figure 13. Separation and Square Separation of Pair 1. 15 Figure 14. Diffusivity of Pair 1. 15 Figure 15. The Comparison of alongshore and cross-shore component of pair 1. 16 Figure 16. Pair 2(NPS #80 & #82) trajectories. 17 Figure 17. Separation and Square Separation of Pair 2. 18 Figure 18. Diffusivity of Pair 2. 18 Figure 19. The Comparison of alongshore and cross-shore component of pair 2. 19 Figure 20. Pair 3(NPS #81 & #82) trajectories. 20 Figure 21. Separation and Square Separation of Pair 3. 21 Figure 22. Diffusivity of Pair 3. 21 Figure 23. The Comparison of alongshore and cross-shore component of pair 3. 22 Figure 24. Three particles trajectories and enclosed triangle area. 24 Figure 25. The triangle area growth of three RAFOS floats. 24 Figure 26. Base and Height of the triangle area. 25

Table 1. NPS #80 trajectory information. 5 Table 2. NPS #81 trajectory information. 7 Table 3. NPS #82 trajectory information. 9 Table 4. Single Particle Diffusivity result. 12 Table 5. Particle Pair Diffusivity result. 23

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1. INTRODUCTION

The California Undercurrent is a poleward flow adjacent to the continental slope

along the West coast of the United State. It transports warm, salty and low oxygen equatorial

water at the intermediate depth. The three RAFOS floats, NPS #80, #81 and #82 were

deployed between Jul 2000 and May 2002 above 400 dbar. After launch, they trapped in the

California Undercurrent (CUC) and moved along the coast, and then turn into the interior of

the Pacific Ocean at different time and trapped into the westward small scale cyclonic

California eddies (cuddy) close to the California Current (CU). After a period time, two

floats (NPS #80 & #81) resume the ploeward CUC movement, and then went back to the

interior again. When in the interior, the NPS #82 had two periods, one was in the larger

cyclonic eddy and the other one was in the smaller cyclonic eddy. The detail of these three

RAFOS floats will discuss in section 2.1.

The purpose of these RAFOS floats is using the Lagrangian measurement of the

trajectory of each float to determine the character of single float dispersion in the

undercurrent, the separation (distance) between two floats to determine the two particles

dispersion, and the triangle area of three RAFOS floats at the same time step to determine the

three particles dispersion.

In single particle dispersion, the comparison of alongshore and cross-shore velocity

diffusivity is the key to determine the dispersion. And the as expected that the alongshore

component dominated the total dispersion in the CUC (detailed in Sec 2.2). In two-particle

dispersion, the square separation is the main factor to find the correlation between the two

floats and compare to the Richardson growth regime (D2 ∝ t3). The results show that in pair 1

(#80 & #81) and pair 2 (#80 & #82), alongshore diffusivity dominated the total diffusivity.

But in Pair 3 (#81 &#82), each of alongshore and cross-shore diffusivity dominated the total

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diffusivity in two different periods (detailed in 2.3). The triangle area growth of three

RAFOS floats is the main idea to the three particles dispersion. But, there still have no exact

equation to calculate the diffusivity (detailed in 2.4).

2. DATA AND ANALYSIS

2.1 Three RAFOS floats observations.

The three RAFOS floats NPS #80, #81 and #82 launched on Jul 2000. The

deployment period of NPS #81 was longer than the other two floats. Each float movement

can be divided into two main period, CUC and interior period.

2.1.1 NPS #80

After launch on Jul 2000, NPS #80 was entrained in the CUC and moving ploeward

in first 30 days and then move into the interior. After 294 days, it resumed to the CUC and

then moved into the interior again (Table 1,Figure 1).

Table 1. NPS #80 trajectory information Launch Surface

Date φ °N λ °W Depth,m Date φ °N λ °W Sample/day26/07/00 36.63 122.48 2541 23/09/01 35.63 125.13 2

Pressure (dbar) and Temperature °C Mission Day Planned Actual Planned Actual

275 ---

348±53.4 7.31±0.4 425 425

Alongshore (CUC) Interior 1-30

295-340 31-294 341-425

6

From figure 1, the four periods of the float movement can be easily noticed. At the

first period, the temperature in the CUC was warmer than other periods due to the poleward,

warm and salty equator water. The temperature all over the period was 7.31 ± 0.4 °C and the

pressure was around 348 ± 53.4 dbar(figure 2). The direction of the movement in the CUC

was about 30° northwest, and resulted in the positive alongshore velocity component and

negative cross-shore velocity component (Figure 3).

(a)

(b)

Figure 1. Temperature along the trajectory of NPS 80.

Figure 2. (a) Temperature (green dot) and Pressure (blue dot) vs. Time. (b) Temperature vs. Pressure of NPS 80. The temperature was about 6 to 8 °C and pressure was 300 to 400 dbar.

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(a)

(b)

2.1.2 NPS #81

NPS #81 also entrained in CUC for 24 days after the launch and then entered into the

interior. After 168 days, it resumed to the CUC and then trapped in the interior again. This

float traveled the widest area (36 °N to 45 °N) and survived longer than the other two floats

(Table2, Figure 4). The temperature was around 7 to 8 °C and the pressure was about 300

dbar (Figure 5). Again, the northwest movement resulted in the positive alongshore velocity

and negative cross-shore velocity component (Figure 6).




275 ---

275.8±53.7 7.54±0.2 666 666

Alongshore (CUC) Interior 1-24

169-289 25-168 190-665

Figure 3. (a) Total, alongshore and cross-shore velocities in the CUC period 1(1-24 days). (b) Total, alongshore and cross-shore velocities in the CUC period (169-289 days) of NPS 80. The total velocity was moving in northwest direction in the CUC and had a result of negative cross-shore velocity and positive velocity.

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(a)

(b)



9

(a)

(b)

2.1.3 NPS #82

This float entrained in the CUC for first 88 days and then entered into the interior. In

the interior period, the float first trapped in the larger small-scale cyclonic eddy (California

Current eddy-cuddy) and then trapped into a smaller westward cuddy and made a series of

three cyclonic circles (Table 3, Figure 7).




275 ---

215.5±22.6 7.61±0.25 425 425

Alongshore (CUC) Interior 1-88 89-425

The temperature of whole period was between 7.61 ± 0.25 °C and the pressure was

215.5 ± 22.6 dbar (Figure 8). The northwest movement also resulted in the positive

alongshore velocity and negative cross-shore velocity (Figure 9).

Figure 6. (a) Total, alongshore and cross-shore velocities in the CUC period 1(1-24 days). (b) Total, alongshore and cross-shore velocities in the CUC period (169-889 days) of NPS 81. The total velocity was moving in northwest direction in the CUC and had a result of negative cross-shore velocity and positive velocity.

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(a)

(b)



Figure 9. Total, alongshore and cross-shore velocities in the CUC period (1-88 days) of NPS 82. The total velocity was moving in northwest direction in the CUC and had a result of negative cross-shore velocity and positive velocity.

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2.1.4 Velocity vs. Latitude

In figure 10, the RAFOS caught in the CUC had larger speed around Pt. Reyes. And

at the Cape Mendocino and Pt. Reyes, the floats entered or exited the CUC, it means around

here have lager upwelling and result in divergence of the water. Then the float was forced to

enter the interior and resume in the CUC while smaller upwelling effect.

When these three floats trapped in the CUC, the northwest motion affected the

characteristics of any single float, two floats and even three or more floats dispersion. So, the

alongshore component domination can be expected.

2.2 Single Particle Dispersion

In order to have a clear picture of alongshore and cross-shore component, the

conversion to the Cartesian coordinate and rotating about 30 ° was applied. The Largranian

correlation Rij(τ) is defined as [Taylor, 1921; Batchelor, 1952; Moffat, 1983; Collins et al,

Figure 10. Velocity vs. Latitude in the CUC. NPS80-1(blue solid line), NPS 80-2(blue dashed line). NPS 81-1(red solid line). NPS 81-2(red dashed line). NPS 82(green solid line)

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2004] and the diffusivity tensor is determined as :

The single particle diffusivity of all the RAFOS floats caught in the CUC had the

same result, the alongshore diffusivity (Kuu) was larger than the cross-shore diffusivity (Kvv).

This result proves that in the CUC, the cross-shore and alongshore diffusivity both exist and

affect the RAFOS float dispersion but the alongshore is the dominant of the total diffusivity

(Figure 11, Table 4).

Table 4. Single particle Diffusivity result

Diffusivity ×107 , cm2/s RAFOS NO. CUC Period (days) Kvv (alongshore) Kuu (cross shore) 1-30 0.8 0.4 NPS #80 295-340 0.045 0.028 1-24 2.4 0.5 NPS #81 169-189 0.62 0.1

NPS #82 1-88 0.65 0.3

(a)

(b)

r2i j = V0

2Ri j (τ ) = Vi (x0 , t)Vj (x0 , t + τ ) = limT→∞

1

TVi (x0 , t)Vj (x0 , t + τ )dt

t0

t0 +τ

∫ ..............................(1)

Ki j = r2i jdτ

0

∞

∫ = V02 Ri j (τ )

0

∞

∫ dτ ..................................................................................................(2)

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(c)

(d)

(e)

2.3 Two Particles Dispersion

In two particles dispersion, the distance (separation) between two RAFOS floats —

pair 1(NPS #80 & #81), pair 2 (NPS #80 & #82) and pair 3 (NPS #81 & #82) — is main

factor to determine the dispersion. Here, the diffusivity can be expressed as [Davis. 1985;

Garfield and Collins et al, 1998]:

α ij =1

4

d

dtSi 'Sj ' =

1

4

d

dt

1

NSi 'S j '∑

.....................................................................................(3)

2.3.1 Pair 1 (NPS #80 & #81)

a. Total Separation

The NPS #80 and #81 had initial separation of 33.3km. After launch, the two float

moving in the CUC and correlated, and then uncorrelated and moving independently (figure

Figure 11. Alongshore diffusivity Kvv (red line) and cross-shore diffusivity Kuu (blue line) of (a) NPS 80-1. (b) NPS 80-2. (c) NPS 81-1. (d) NPS 81-2. (e) NPS 82.

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12). They moved toward (separation decreased) in first 3.25 days and then moved away

(separation increased) until 19.25 days. After 19.25 days, the separation decreased and

increased randomly (figure 13 a). It indicated the two floats were moving independently and

not correlated after 19.25 days. The diffusivity was determined only during 3.25 and 19.25

days.

The logarithm plot of the square separation (figure 13 b) showed that, between 3.25 to

5.75 days the square separations was exponential growth (stretching) and then applied the

least square method to find the square separation growth rate with time (S2 /t β), here the

growth rate β = 2.25. If the two floats were perfectly dispersed, the exponent relation with

time (S2 / tβ) will match the Richardson regime (S2 /t3) and closed together as a straight

horizontal line.

Figure 12. NPS 80 (red line) and NPS 81 (blue line) trajectories. The two floats had initial separation 33.3km.

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(a)

(b)

The maximum total diffusivity of pair 1 is 6 × 10 7 from 5.75 to 12.75 days. And after

12.75 days the diffusivity decreased, then it could not be determine as dispersion. The total

dispersion was start from 5.75 to 12.75 days (figure 14).

(a)

(b)

b. Alongshore and cross-shore Separation

The alongshore and cross-shore separation, square separation and diffusivity were

showed in figure 15. In pair 1, the cross-shore component almost had no effect to the total

dispersion, and the alongshore component had almost the same result with the total

Figure 13. (a) Separation (b) Square Separation of pair 1(NPS 80 & 81). In (b) the red dot indicates the exponential growth, the purple line is the least square trend of the square separation. And at the right below corner is the plot of compensated Evolution of Square separation compare to the Richardson growth regime (S2 / t3).

Figure 14. (a) Diffusivity (red dot) vs. time (b) Diffusivity (red dot) vs. Separation of pair1 (NPS 80 & 81). The blue line is the least square trend of the diffusivity.

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dispersion. So the alongshore diffusivity component dominated the total diffusivity in the

Pair 1.

(a)

(b)

(c)

Figure 15. The left side is the cross-shore component, and the right side is the alongshore component of pair 1(NPS 80 & 81). (a) Separation (b) Square separation and (c) Diffusivity.

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2.3.2 Pair 2 (NPS #80 & #82)

a. Total Separation

The NPS #80 and #82 had initial separation of 26.7 km. After launch, the two float

moved in the CUC and correlated to each other, and then they uncorrelated and moved

independently (figure 16). They moved away (separation increased) from each other for

50.25 days after launch. After then the separation decreased and increased randomly (figure

17 a). It indicated the two floats were moving independently and not correlated after 50.25

days. The diffusivity was determined only during 1 to 4.75 and 24.25 to 50.25 days.

The logarithm plot of the square separation (figure 17 b) showed that, between day1

to 4.75 the square separation was exponential growth (stretching) and square separation

growth rate between day 24.25 to 50.25 with β = 5.5. It was larger than the Richardson

regime (S2 /t3) and the Compensated evolution of square separation lined together. The

maximum total diffusivity of pair 2 was 10 × 10 7 from day 24.25 to 50.25 (figure 18).

Figure 16. NPS 80 (red line) and NPS 82 (green line) trajectories. The two floats had initial separation 26.7 km.

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(a)

(b)

(a)

(b)



showed in figure 19. In pair 2, the cross-shore component also almost had no effect to the

total dispersion, and the alongshore component dominated the total dispersion.

Figure 18. (a) Diffusivity (red dot) vs. time (b) Diffusivity (red dot) vs. Separation of pair 2 (NPS 80 & 82). The blue line is the least square trend of the diffusivity.


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(a)

(a)

(b)

(c)

2.3.3 Pair 3 (NPS #81 & #82)

a. Total Separation

The NPS #81 and #82 had initial separation of 38.5 km. After launch, the two floats

correlated and moved along the CUC, and then turned into the interior. When trapped in the

cyclonic cuddies, they flowed uncorrelated and independently (figure 20). From the track, it


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can be seen clearly that at the beginning, the two floats move together to the north

(alongshore) with velocity of NPS #82 faster then the NPS #81, and then they turned into the

interior together. So we can expect that, when moved north in the CUC, the alongshore

component dominated and when leaving the CUC turning into the interior, the cross-shore

component dominated. They moved toward in first 3.75 days and then away from each other

till 64.25 days. After 64.25 days, the separation decreased and increased randomly (figure 21

a). It indicated the two floats were moving independently and not correlated after 64.25days.

The diffusivity was determined only during 3.75 to 13.75 and 20.25 to 60.25 days.

Figure 20. NPS 81 (blue line) and NPS 82 (green line) trajectories. The two floats had initial separation 38.5 km.

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(a)

(a)

The logarithm of the square separation (figure 21 b) showed that, the first period the

square separation had a growth rate of β=2.5 and the diffusivity α ~5×107, and then the

second period β = 3.7 and the diffusivity α ~15×107 (figure 22).

(a)

(b)



showed in figure 23. In pair 3, the alongshore component dominated the total diffusivity and

during the second period the cross-shore component dominated with larger diffusivity. This

result matched from the expectation from trajectories plot.

Figure 22. (a) Diffusivity (red dot) vs. time (b) Diffusivity (red dot) vs. Separation of pair 3 (NPS 81 & 82). The blue line is the least square trend of the diffusivity.


22

(a)

(b)

(c)

The pair of any two of the RAFOS floats can be correlated only a short period of time

after launch. During this period, the floats usually still trapped in the CUC and then the

alongshore diffusivity component dominated the total diffusivity can be expected (table 5). If

the two floats move to the interior at the same time, then the cross-shore diffusivity became

the dominant but it didn’t happen quite often based on 3 floats statistics (not enough floats).


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Table 5. Particle-Pair Diffusivity result

RAFOS Pair Diffusivity ×107 , cm2/s Power Exponent D2 ~ tβ

Total 5.9 2.3 Alongshore 6 1.9 NPS 80 & 81 Cross shore --- ---

Total 10 5.3 Alongshore 10 5.2 NPS 80 & 82 Cross shore --- ---

Total 4.5 15 2.3 3.7 Alongshore 4 --- 5.1 --- NPS 81 & 82 Cross shore --- 14 --- 4

2.4 Three Particles Dispersion

The three RAFOS data launched at the same time but with different record time.

After interpolation the time with six hours interval, they can be compared to each other and

determined the dispersion relation. The two particles dispersion was widely used in many

scientific papers but the three particles dispersion was not. The concept of the three particles

came from the two particles separation but the triangle area of three floats (figure 24).

The triangle area of these three RAFOS floats steadily increased within 60 days, then

it increased and decreased randomly. The dispersion relation needs to take the increasing

triangle area into account, so the only period can be determined as the three particles

diffusivity was the first 60 days (figure 25 a). And the triangle area can be calculated as

expressed:

∆area = s(s − a)(s − b)(s − c)...................................................................................(4)

s =1

2(a + b + c)............................................................................................................(5)

24

(a)

(b)

Because of no exact equation or coefficient can be applied to the triangle area growth

rate, the size of the triangle can be one way to determine how fast the area grew.

The size of a triangle was defined as [J.H. LaCasce and Carter Ohlmann, 2003].

R2 =1

3a2 + b2 + c2( ).....................................................................................................(6)

Figure 24. The three RAFOS floats trajectories (red line—NPS 80, blue line—NPS 81, green line—NPS 82) and the area (black enclosed lines) in 60 days

Figure 25 (a) The Triangle area of Three RAFOS floats increasing period. (b) Size of triangle (Root mean Square).

25

In figure 25 b, we can see the size of the triangle area increase with time of R~ t 5.5 in

the first 15 days, stayed the same size during 15 to 30 days and then increased with time of

R~ t9 till 60 days. It showed that the second period growth rate was larger than the first

period. There is another way to describe how big the size from the base—the longest leg of a

triangle and the height (figure 26 a). With the Aspect Ratio,base divided by the height, the

triangle area growth rate with time can also be determined. In figure 26 b, the first period, the

AR increased with time of AR~ t 0.28, and decreased in the second period of AR~ t -0.07. This

indicated that the base increased more than the height during first period, and meant there

were two of these three floats far away kept moved away from each other. Otherwise, during

the second period, the height increased more than base (height < base) and indicated that

there were two of three floats moved toward to each other.

(a)

(b)

All these determinations of three particles dispersion from the triangle area growth

are only an idea. The exact equation and coefficient need more RAFOS floats experiments

and the statistics to prove it. Now, the only way to see the three particles dispersion is to see

how the triangle area’s size and aspect ratio increase with time.

Figure 26 (a) Base (blue line) and height (red line). (b) Aspect ratio (Base / height).

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3. Conclusion

NPS #80, #81 and #82 launched at the same time and correlated between each other

only few days after the launch, usually still trapped in the California undercurrent.

When in the warm, salty poleward CUC, we can expect the alongshore component

will affect the characteristics of either single or two particles dispersion. From the data

analysis, the result showed both the cross-shore and alongshore diffusivity component

contributed to the total diffusivity, but when in the CUC, the alongshore diffusivity was

always larger than the cross-shore diffusivity due to the northwest motion along the coast.

In two particles dispersion, the alongshore diffusivity domination was still expected

due to the short period of time of correlation with any of these two floats. In pair 1 and pair 2,

the result met the expectation as alongshore domination. In pair 3, when in the CUC, the

alongshore diffusivity still dominated the total diffusivity, but when they turned in the

interior at the same time, the westward motion caused the larger cross-shore diffusivity and

dominated the total diffusivity during this time. But it only happened when the two float

moved westward at the same time.

The triangle area growth rate of three RAFOS floats is the idea to determine the three

particles dispersion. The size (root mean square of three legs) and the Aspect ration (base

divided by the height) can be applied to determine how the area grew with time. There still

have no exact coefficient and equation to calculated the relation. But with more and more

deployment of RAFOS floats and the data analysis to get the statistics, the three particles

dispersion can be determined more correctly and precisely.

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Acknowledgement

Without the help from professor Collins and Margolin, Tetynan, I can’t learn the

particles dispersion idea and methods to analysis the RAFOS data and of course can’t finish

the this important project. I thank them for devoting their time and patiently explaining all

the detail.

I also want to thank all the people who devoted all their effort on deploying the

RAFOS floats and processed the data. With the existence data information, I don’t need to

worry about I can’t get the desire data from the cruise. So I could learned more things, such

as how to cast the CTD and Rawinsonde, how to get the synoptic measurement of the ocean

data and how to read the atmosphere but not only on what I eager to get.

I also thank professor Guest and the assistant during the cruise to let me learn the

basic meteorology concept and how to be an operational researcher. And thanks for all the

crews on the RV. Pt Sur to let me have safe and sound trip in the San Francisco Bay and

California coast.

The last I want to thank are all the classmates in our class. With your effort doing the

wonderful and excellent presentation, and the team work during the cruise ,I can easily learn

new ideas from you without a pain.

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REFERENCES

Curtis A. Collins, Newell Garfield, Robert G. Paquette, and Everett Carter (1996), Lagrangian Measurement of subsurface poleward Flow between 38°N and 43°N along the west Coast of the United States during Summer, 1993, Geophysical Research Letters, vol. 23, No. 18, pages 2461-2464.

Newell Garfield, Curtis A. Collins, Robert G. Paquette, and Everett Carter (1996), Lagrangian Exploration of California Undercurrent, 1992-95, Journal of Physical Oceanography, Vol 29, pages 562-583.

Newell Garfield, Mathew E. Maltrud, Curtis A. Collins, Thomas A. Rago, Robert. Paquette (2001), Lagrangian flow in the California Undercurrent, an observation and model comparison, Journal of Marine Systems, Vol 29, pages 201-220.

Russ E. Davis (1985), Drifter Observations of Coastal Surface Current During CODE: The Method and Descriptive View, Journal of Geophysical Research, Vol 90, No C3, Pages 4741-4755.

Russ E. Davis (1985), Drifter Observations of Coastal Surface Current During CODE: The Statistics and Dynamical View, Journal of Geophysical Research, Vol 90, No C3, Pages 4756-4772.

Curtis A. Collins, Leonid M. Ivanov, Oleg V. Melnichenko, and Newell Garfield (2004), California Undercurrent variability and eddy transport estimated from RAFOS float observations, Journal of Geophysical Research, Vol 190, C05028, Pages 1-19.

J.H. LaCase and Carter Ohlmann (2003), Relative dispersion at the surface of the Gulf of Mexico, Journal of Marine Research, Vol 61, No. 3 Pages 285-312.

H.J. Freeland, P.B. Rhines and T. Rossby (1975), Statistics observations of the trajectories of neutrally buoyant floats in the North Atlantic, Journal of Marine Research, Pages 383-404.

J.H. LaCase and A. Bower (2000), Relative dispersion in the subsurface North Atlantic, Journal of Marine Research, Vol 58, Pages 863-894.

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