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LAGRANGIAN ANALYSIS AND PREDICTION
OF COASTAL AND OCEAN DYNAMICS
Written by a group of international experts in their field, this book is a review
of Lagrangian observation, analysis and assimilation methods in physical and
biological oceanography. In recent years a large number of floating and
drifting research buoys have been deployed in the global oceans to study the
state of the ocean and its variation in terms of water mass properties, circula-
tion and heat transport. Lagrangian techniques are required to analyze the
data from these buoys.
This multidisciplinary text contains observations, theory, numerical simula-
tions, and analysis techniques. It presents new results on nonlinear analysis of
Lagrangian dynamics, the prediction of particle trajectories, and Lagrangian
stochastic models. It includes chapters on floats and drifters, Lagrangian-
based analysis methods and models in marine biology, the statistics of particle
trajectories in the ocean, numerical simulations and their relationship with
classical turbulence results, and nonlinear Lagrangian-based theory for study-
ing ocean transport and particle trajectories. The book contains historical
information, up-to-date developments, and speculation on future develop-
ments in Lagrangian-based observations, analysis, and modeling of physical
and biological systems.
Containing contributions from experimentalists, theoreticians, and model-
ers in the fields of physical oceanography, marine biology, mathematics, and
meteorology this book will be of great interest to researchers and graduate
students looking for both practical applications and information on the theory
of transport and dispersion in physical systems, biological modeling, and data
assimilation.
Cover illustration: The cover depicts the abrupt breakup of a large ocean
eddy in the Gulf of Mexico. Eddy Fourchon was tracked by assimilating
satellite data into the University of Colorado version of the Princeton Ocean

University of Miami
THOMAS ROSSBY
Graduate School of Oceanography
University of Rhode Island
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
First published in print format
ISBN-13 978-0-521-87018-4
ISBN-13 978-0-511-27414-5
© Cambridge University Press 2007
2007
Information on this title: www.cambrid
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e.or
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/9780521870184
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
ISBN-10 0-511-27414-9
ISBN-10 0-521-87018-6
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
hardback
eBook (EBL)

Oceanography
Woods Hole Oceanographic Institute
Woods Hole, MA 02543
USA
Annalisa Bracco
Department of Physical
Oceanography
Woods Hole Oceanographic
Institution
Woods Hole, MA 02543
USA
Giuseppe Buffoni
ENEA
Santa Teresa – Lerici
La Spezia I-19100
Italy
Jim Carton
University of Maryland
Stadium Drive
College Park, MD 20742-0001
USA
Luca R Centurioni
Scripps Institute of Oceanography
9500 Gilman Drive
La Jolla, CA 92093-0213
USA
Toshio Chin
RSMAS/MPO
University of Miami
4600 Rickenbacker Causeway

University of Miami
4600 Rickenbacker Causeway
Miami, FL 33149
USA
ISMAR/CNR
Forte Santa Teresa
La Spezia I-19036
Italy
Semyon Grodsky
Department of Meteorology
University of Maryland
College Park, MD 20742
USA
Gary L Hitchcock
RSMAS/MBF
University of Miami
4600 Rickenbacker Causeway
Miami, FL 33149
USA
Kayo Ide
Institute of Geophysics & Planetary
Physics
UCLA
Los Angeles, CA 90095-1567
USA
Christopher Jones
University of North Carolina at
Chapel Hill
CB #32350
UNC-CH

Busan National University
Busan 609-735
South Korea
Thomas N Lee
RSMAS/MPO
University of Miami
4600 Rickenbacker Causeway
Miami, FL 33149
USA
Rick Lumpkin
Atlantic Oceanographic &
Meteorological Lab
NOAA/AOML/PhOD
4301 Rickenbacker Causeway
Miami, FL 33149
USA
Svend-Aage Malmberg
Marine Research Institute
1 Hafrannsoknansofnunin
PO Box 1390
Skulgata 4
Reykjavik 121
Iceland
Arthur J Mariano
RSMAS/MPO
University of Miami
4600 Rickenbacker Causeway
Miami, FL 33149
USA
Maria Grazia Mazzocchi

USA
viii List of contributors
Nathan Paldor
Hebrew University of Jerusalem
Institute of Earth Sciences
Edmund Safra Campus, Givat Ram
Jerusalem 92509
Israel
Sara Pasquali
CNR-IMATI
via Bassini, 15
Milano I-20133
Italy
Claudia Pasquero
Earth System Science Dept.
University of California
3224 Croul Hall
Irvine, CA 92697-3100
USA
Mayra C Pazos
Atlantic Oceanographic &
Meteorological Lab
NOAA/AOML/PhOD
4301 Rickenbacker Causeway
Miami, FL 33149
USA
Leonid Piterbarg
University of Southern California
Kaprielian Hall, Room 108
3620 Vermont Avenue

Roma I-00060
Italy
List of contributors ix
Edward H Ryan
RSMAS/MPO
University of Miami
4600 Rickenbacker Causeway
Miami, FL 33149
USA
Vitalii A Sheremet
Graduate School of Oceanography
University of Rhode Island
Narrangansett, RI 02882
USA
Hedinn Valdimarsson
Marine Research Institute
1 Hafrannsoknansofnunin
PO Box 1390
Skulagata 4
121 Reykjavik
Iceland
Jeffrey B Weiss
Dept. of Atmospheric & Oceanic
Science
PAOS
University of Colorado
Boulder, CO 80309-0311
USA
Elizabeth Williams
RSMAS/MPO

maintain a good balance between historical and state-of-the-art developments
in Lagrangian-based observations, theory, numerical modeling and analysis
techniques.
This book seems to be a first of its kind because the central theme is the
Lagrangian viewpoint for studying the transport phenomena in oceanic flows.
Another unique and timely aspect of this book is its multidisciplinary nature
with contributions from experimentalists, theoreticians, and modelers from
diverse fields such as physical oceanography, marine biology, mathematics,
and meteorology.
The book starts with a historical perspective of the development and appli-
cation of Lagrangian methods, while more recent measurements and results
xi
are presented in Chapter 2. Some striking examples of Lagrangian trajectories
are depicted by a collection of authors in Chapter 3. A number of new
theoretical approaches to understand and describe particle motion are out-
lined in Chapters 4, 5, 6, and 9 . New methods for assimilating Lagrangian data
in ocean models to improve their forecast are described in Chapters 7 and 8.A
suite of applications of Lagrangian techniques to transport of biological
speciesaregiveninChapters10to12.Finally,weclosewithanextensive
observational and theoretical review of Lagrangian techniques that were pre-
sented in the three LAPCOD workshops held in 2000, 2002, and 2005.
We would like to express special thanks to Dr. Manuel Fiedeiro from the US
Office of Naval Research (ONR) for sponsoring much of the research pre-
sented in this book, while fostering collaboration between many groups of
researchers and initiating LAPCOD workshops. We also thank Dr. Jerry
Miller from ONR-London for supporting some of the LAPCOD workshops.
Special thanks are also due to Edward Ryan, who has dedicated countless days
to help organize this book. We also thank anonymous reviewers for many
useful suggestions to help improve the chapters, and for maintaining a quality
standard of scientific work. Finally, we thank all the scientists who have played

because for a very long time we did not have the tools to observe the ocean in
motion directly.
Fluid motion can be specified in two ways. The first, generally known as the
Eulerian method, specifies the velocity field as a function of location and time.
The other – Lagrangian – method specifies the position of labeled fluid parcels
as a function of time. The two methods have strengths and weaknesses.
Virtually all theoretical and numerical research of fluid dynamics uses the
Eulerian specification because of the clear separation of the independent
variables, space and time. In the Lagrangian frame, the spatial information
enters through the initial position of each and every particle. The dependent
Lagrangian Analysis and Prediction of Coastal and Ocean Dynamics, ed. A. Griffa, D. Kirwan, A. Mariano,
T.
¨
Ozg
¨
okmen, and T. Rossby. Published by Cambridge University Press. # Cambridge University Press
2007.
variables are the particles’ subsequent positions as a function of time. The
equations that describe this system can be solved only for certain, very simple
flows, and as a result are hardly ever used. However, from an observational
point of view the Lagrangian method has a major advantage in that it tells us
precisely how fluid parcels move about in space. This may seem like a self-
evident if not tautological statement, but it assumes special significance when
one considers how difficult it is to accurately determine the horizontal struc-
ture of ocean currents, particularly at depth.
How does one observe the spatial structure of currents and how well can this
be done? In the Eulerian frame resolution is set by the number of observation
points. In the Lagrangian frame it is set by the number of markers or tagged
parcels that are released. For synoptic applications such as tracking weather,
the Eulerian specification is the method of choice. The synoptic approach can

methods describes the evolution of the acoustic float technology; the second
2 T. Rossby
section discusses a range of platform-based in situ measurements that have
added significant value to the Lagrangian trajectory data. This is followed by a
brief review of the sound source technology and acoustic navigation techni-
ques. We then attempt a summary of lessons learned from the use of
Lagrangian methods, and the brief last section speculates on likely developments
in the near future. Common to the entire discussion given here is the acoustic
transparency of the ocean, the property that allows us to generate and detect
acoustic signals at great distances for the purposes of location and tracking.
1.2 History of floats
The first neutrally buoyant float designed to track water movements at depth
was developed by John Swallow, a British oceanographer. It consisted of
two aluminum pipes strapped together with a battery and timer circuit that
would excite a magnetostrictive transducer, a ‘‘pinger,’’ hanging underneath
(Swallow, 1955), Figure 1.1. The signals could be heard from a ship overhead.
Figure 1.1 This widely reproduced photo shows John Swallow on deck
preparing his float for launch. The float consisted of two aluminum tubes
strapped together. The toroidal pinger (barely visible to Swallow’s right)
hangs underneath. Note the ship’s cat paying close attention!
Evolution of Lagrangian methods in oceanography 3
Using acoustic triangulation the ship could determine the float’s position and
depth. The weight of the float was carefully trimmed so that it would float at a
desired depth. It is also essential that the float be less compressible than
seawater. Imagine that for some reason the float is displaced downward a
bit. Since it won’t compress as much as the surrounding water, it will be lighter
than the displaced water and thus return to its equilibrium depth.
The new Swallow float enjoyed great success with the discovery of a south-
flowing deep western boundary current (Swallow and Worthington, 1961) that
had been predicted by Stommel (1957). The direct observation of southward

potential for using the SOFAR channel for tracking subsurface drifters over
4 T. Rossby
great distances for extended periods of time. These early ideas remained on his
mind, as we shall see.
The SOFAR channel was discovered during World War II at the Woods
Hole Oceanographic Institution as part of its research into the acoustics of the
ocean, a very important aspect of antisubmarine warfare (Ewing and Worzel,
1948). The SOFAR channel is an acoustic waveguide due to a sound velocity
minimum below the warm upper ocean. The speed of sound is a positive,
nearly linear function of both temperature and pressure such that the speed of
sound first decreases with increasing depth due to the rapid decrease in
temperature. Beyond a certain depth below which temperature only slowly
decreases, the pressure effect dominates such that the speed of sound increases
with further increase in depth. This minimum in sound speed, typically about
1490 ms
À1
around 1000–1300 m below the surface in tropical and subtropical
waters, gives rise to a sound channel which tends to trap sound near these
depths, in short an acoustic waveguide. Figure 1.2 (widely reproduced from
the 1948 Ewing and Worzel paper) illustrates the trapping of sound in the
sound channel.
In 1965–1966, at Stommel’s initiative, M. J. Tucker, on sabbatical leave
from England at the Massachusetts Institute of Technology, and Douglas
Webb at the Woods Hole Oceanographic Institution (WHOI), used a low
frequency piezoelectric transducer, small enough for neutrally buoyant float
Figure 1.2 Ray diagram of sound propagation through a stratified ocean
showing both refracted and surface reflected rays (from Ewing and Worzel,
1948).
Evolution of Lagrangian methods in oceanography 5
applications, to evaluate its use for acoustic signaling in the SOFAR channel

patterns at much lower cost albeit without any spatial or temporal detail along
the trajectories. Ensembles of mean displacement vectors would nonetheless
provide independent information on the mean circulation at depths where the
dynamic method gives little insight. Building upon Stommel’s original idea of
using small explosives to study currents, but instead of a single float carrying
many charges, we used many small floats, each one with a single pressure-
actuated explosive. The charge was a standard SOFAR signaling device
secured inside a small glass flotation sphere. The sphere float had a small
port to allow it to be flooded at a certain time. This would cause it to fill until
6 T. Rossby
Figure 1.3 Gordon Volkmann (left) with acoustic transducer about to be
lowered to sound channel depth to test long-range transmission through the
ocean. The ring transducer visible below the shiny electronics module
operated at 778 Hz.
Evolution of Lagrangian methods in oceanography 7
Figure 1.4 Visual display of first signals at Bermuda from SOFAR float #1
January 20, 1968. The scales are 1 minute across and 0–24 hours top to
bottom. The bright line shows the one per minute transmissions starting at
0930 GMT. At first only the early off-axis arrivals show up because the float
has just been deployed and has not yet reached sound channel depth (Rossby
and Webb, 1970).
8 T. Rossby
Figure 1.5 The aluminum sphere SOFAR float with the transducer underneath
(mostly concealed by launch frame). See Plate 1 for color version.
Evolution of Lagrangian methods in oceanography 9
the pressure reached the point where the charge was hydrostatically armed and
subsequently triggered. An exploratory experiment to test the concept took
place with a dozen floats set to drift for three weeks. Unfortunately, due to a
leak in the flooding mechanism only one displacement vector was obtained.
While the test showed that the concept did work and could have been

mini-MODE program (Swallow et al., 1974). These floats were transponding
10 T. Rossby
Figure 1.6 The MODE SOFAR float with two resonator tubes mounted on
opposite sides of the flotation tube. See Plate 2 for color version.
Evolution of Lagrangian methods in oceanography 11