OCS Study 

MMS 2003-074 

 

Modeling and Data Analyses 

of Circulation Processes 

in the Gulf of Mexico 

Final Report 

U.S. Department of the Interior

Minerals Management Service 

Gulf of Mexico OCS Region 
U.S. Department of the Interior

Minerals Management Service 

Gulf of Mexico OCS Region 

OCS Study 

MMS 2003-074 

Modeling and Data Analyses 

of Circulation Processes 

in the Gulf of Mexico 

Final Report 

Authors 

L.-Y. Oey 

Princeton University 

P. Hamilton 

Science Applications International Corporation 

H.-C. Lee 

Princeton University 

Prepared under MMS Contract 

1435-01-99-CT-31028 

by 

Science Applications International Corporation 

615 Oberlin Rd., Suite 100 

Raleigh, North Carolina 27605 

Published by 

New Orleans

February 2004 
iii 

DISCLAIMER 

This report was prepared under contract between the Minerals Management Service (MMS) and 

Science Applications International Corporation. This report has been technically reviewed by 

the MMS, and it has been approved for publication. Approval does not signify that the contents 

necessarily reflect the views and policies of the MMS, nor does mention of trade names or 

commercial products constitute endorsement or recommendation for use. It is, however, exempt 

from review and compliance with the MMS editorial standards. 

REPORT AVAILABILITY 

Extra copies of the report may be obtained from the Public Information Office (Mail Stop 5034) 

at the following address: 

 U.S. Department of the Interior 

 Minerals Management Service 

 Gulf of Mexico OCS Region 

 Public Information Office (MS 5034) 

 1201 Elmwood Park Boulevard 

 New Orleans, Louisiana 70123-2394 

 Telephone Number: 1-800-200-GULF or 

 504-736-2519 

CITATION

Suggested citation: 

Oey, L.-Y., P. Hamilton, and H.-C. Lee. 2004. Modeling and data analyses of circulation 

processes in the Gulf of Mexico: Final report. U.S. Dept. of the Interior, Minerals 

Management Service, Gulf of Mexico OCS Region, New Orleans, LA. OCS Study MMS 

2003-074. 140 pp. 

ABOUT THE COVER 

Wave rays (in blue; light blue for wavelength < 30 cm) traced using the topographic Rossby 

wave dispersion relation showing the deep energy paths in the northern Gulf of Mexico for 

waves originating under the Loop Current (LC) and Loop Current eddies (LCE). The 10-year 

assemblage of the outer boundary of the LC and LCE are shown by the thinner red lines. 
v 

TABLE OF CONTENTS 

 Page

LIST OF FIGURES .............................................................................................................vii 

LIST OF TABLES ...............................................................................................................xiii 

ABSTRACT..........................................................................................................................xv 

1.0 INTRODUCTION......................................................................................................1-1 

 1.1 Background........................................................................................................1-2 

 1.2 Objectives ..........................................................................................................1-3 

 1.3 Outline of the Report .........................................................................................1-3 

2.0 METHODS .................................................................................................................2-1 

 2.1 Basic Model Description....................................................................................2-1 

 2.1.1 Satellite Data Assimilation ......................................................................2-5

 2.1.2 Data Assimilation in the Princeton Ocean Model....................................2-6 

 2.2 Observational Database .....................................................................................2-9 

 2.2.1 Currents....................................................................................................2-10 

 2.2.2 Hydrography ............................................................................................2-14 

 2.3 Analysis Tools ...................................................................................................2-15 

 2.3.1 Topographic Rossby Waves and Ray Tracing.........................................2-15 

 2.3.2 Hydrographic Analyses............................................................................2-24 

 2.3.3 SVD Analysis...........................................................................................2-25 

3.0 TOPOGRAPHIC ROSSBY WAVES.......................................................................3-1 

 3.1 Model Investigations..........................................................................................3-1 

 3.1.1 Introduction..............................................................................................3-1 

 3.1.2 Setup of the Model for Deep Flow Experiments .....................................3-2 

 3.1.3 Model Data Processing ............................................................................3-5 

 3.1.4 Deep Flow Eddy Kinetic Energy .............................................................3-5 

 3.1.5 Discussion................................................................................................3-24 

 3.2 Comparisons with Deep Current Measurements ...............................................3-31 

 3.2.1 Statistics and Spectra ...............................................................................3-33 

4.0 SLOPE EDDIES.........................................................................................................4-1 

 4.1 Introduction........................................................................................................4-1 

 4.2 Observations ......................................................................................................4-1 

 4.3 The Model..........................................................................................................4-6 

 4.4 Model Prediction of Observational Conditions during Period-B ......................4-7 

 4.5 Flow Energetics .................................................................................................4-7 

5.0 CONTRIBUTIONS TO OTHER STUDIES ...........................................................5-1 

 5.1 DeSoto Canyon Comparison..............................................................................5-1 

 5.2 Assimilation of Drifter Data ..............................................................................5-7

 5.3 Causes of Loop Current Variability...................................................................5-8 

6.0 SUMMARY AND RECOMMENDATIONS...........................................................6-1 

 6.1 Summary............................................................................................................6-1 

 6.2 Recommendations for Future Model Development...........................................6-2 

 6.3 Recommendations for Future Process Studies...................................................6-3

7.0 REFERENCES...........................................................................................................7-1 
vii 

LIST OF FIGURES 

Figure Page

2.1-1a. The model orthogonal curvilinear grids encompass the Gulf of 

Mexico, the Caribbean Sea, and a portion of the Atlantic Ocean. ..................2-2 

2.1-1b. Oey and Lee's (2002) northwest Atlantic Ocean model domain 

and the nested doubled-resolution Gulf of Mexico region where 

calculations described in this report are conducted.........................................2-3 

2.1-2. The maximum speed that results from a one-year test calculation 

using initially level density field with perturbation: ρ(z) = ρr.+ ρ', 

 where ρr = area-averaged and annual-mean climatological density, 

 and ρ' = -0.1 kg/m3

 x exp(z/1,000 m) and zero forcing...................................2-4 

2.1-3. An example of the regional, near-surface circulation obtained from 

the data-assimilated run...................................................................................2-7 

2.2-1. Locations of deepwater moorings used in this study. .....................................2-11 

2.2-2. Station position maps for F13SQUIR/F14LEDDY and GULFCET07...........2-16 

2.3-1. A schematic that describes the relations between three vectors: the 

gradient of water depth ∇h, wavenumber vector K, and the group 

velocity Cg, in the northern hemisphere and when NHK/f is O(1) or 

larger................................................................................................................2-19 

2.3-2. (a) Time (7 years) and depth (below 27.5 σt surface) -averaged 

Brunt-Väisälä frequency, N (10-4

S

-1). (b) Smoothed compared with 

 model topography............................................................................................2-21 

2.3-3. The modeled currents averaged over the last seven years of the 

ten-year simulation, at 200 m above the bottom in the Gulf...........................2-22 

3.1-1a. Ninety-day snapshots of relative vorticity (ξ/f: 5-day averaged) at 

z=-50 m showing LCE shedding, southwestward propagation and 

decay................................................................................................................3-3 

3.1-1b. Time/Latitude contours of free-surface elevation η along 90°W, 

showing passages of Loop Current Eddies when η maximizes to 

~ 0.3 m.............................................................................................................3-4 
viii 

LIST OF FIGURES (continued) 

Figure Page

3.1-2. A time-averaged cross-sectional contour plot of density at 90°W 

that describes isopycnal division of model's three-dimensional field 

into four layers: layer-1 from surface to 27σt, layer-2 from 27σt to 

27.5σt, layer-3 from 27.5σt, and layer-4 from 27.7σt to bottom. ....................3-6 

3.1-3. A comparison of the time series of lower-layer (layer #4) kinetic 

energy (LOKE; thin curve) at a station just west of the LC (88°W, 

25.8°N) with the corresponding 20-100 day band-passed LOKE 

(LOKE 20-100d, thick curve)..............................................................................3-7 

3.1-4. The lower-layer kinetic energy in the 20-100 day periods 

(LOKE 20-100d)..................................................................................................3-9 

3.1-5. Contours of the minimum period (in days) P=2π/σ≤ tanh(NhK/f)/(N|∇|) 

allowed by the TRW dispersion relation, for N given in Figure 2.3-2a 

and 2p/K=110 km............................................................................................3-10 

3.1-6. Top panel: contours of maximum lagged correlation (at 95% significance 

level) in the vicinity of the 3,000-m isobath (CI-500 m), CGKE band 

(see text), from 86°W to 92°W. Lower panel: the corresponding lags 

 in days..............................................................................................................3-12 

3.1-7a. Top panel: wave rays (in blue; lighter blue for wavelength < 30 km) 

traced using the TRW dispersion relation indicating deep energy paths 

in the Gulf that originate under the Loop Current and Loop Current 

Eddies. Bottom panel: the 20-100 day deep energy from Figure 3.1-4. ........3-13 

3.1-7b. Various TRW and ambient properties along ray #2 of Figure 3.1-7a.............3-14 

3.1-8. TRW dispersion curves corresponding to peak spectral periods of 32 

days (left-most solid and dotted curves), 42.7 days, 51.2 days and 

 64 days.............................................................................................................3-16 

3.1-9. (a) Five locations along ray #2 (see Figures 3.1-7a) where vertical 

profiles of the square root of kinetic energy in the 20 to 100-day band are 

plotted (b). .......................................................................................................3-19 

3.1-10. Vector stick plots at location 'A' (a) and 'D' (b) of Figure 3.1-9a, upward 

from the near-bottom depth level where the LOKE 20-100d shows a 

maximum (bottom panel). The top panel shows vectors at a 

representative near-surface level. ...................................................................3-20 
ix 

LIST OF FIGURES (continued) 

Figure Page

3.1-11. Sixty-four (64) day period TRW rays traced from station #2 using 

benchmark N-field of Figure 2.3-2 and initial 2π/K=100 km (ray 

on down-slope side and 0.625×benchmark N and initial 2π/K-150 km 

(up-slope side). ................................................................................................3-23 

3.1-12. Vorticity maps at 5-day interval in an enlarged region of the LC during 

a period of its northward extrusion before an eddy-shedding event 

(on day 1150, not shown). ...............................................................................3-26 

3.1-13. Time-series at station #2 (i.e. the '+' point in Figure 3.1-12). .........................3-27 

3.1-14. Modeled eddy kinetic energy at 188 m above the bottom on day 1345 

(top-left panel) when the northern edge of LC touches point #1 and 

a TRW ray is initiated there; on day 1370 when the northwestern 

edge of LC touches point #2, and a second TRW ray is initiated there; 

on day 1385, and day 1405..............................................................................3-30 

3.2-1. Positions of model-saved daily profiles of velocity, temperature, and 

salinity. ............................................................................................................3-32 

3.2-2. Standard deviation ellipses for velocity data below 1,000 m from 

observations, and the regular model run. ........................................................3-34 

3.2-3. Standard deviation ellipses for velocity data below 1,000 m from 

observations, and the assimilation model run. ................................................3-35 

3.2-4. Eight-year time averaged velocity spectra expressed in m s-1 at sigma-level 

6 (a

100 m below the surface) as a function of the spatial scale for model 

region north of 26° N but deeper than the 500 m isobaths..............................3-36 

3.2-5. Kinetic energy spectra for mooring sites G, I1, FF and GG, and R from 

observed currents.............................................................................................3-37 

3.2-6. Kinetic energy spectra for mooring sites G, I1, GG, and R using 

assimilated model currents. .............................................................................3-38 

3.2-7. Coherence squared and phase differences between indicated depth 

levels and velocity components for sites G(lhs), I1 (middle) and R (rhs).......3-40 

3.2-8. TRW ray tracing using observed estimated periods and wavelengths............3-41 
x 

LIST OF FIGURES (continued) 

Figure Page

4.1-1. Depth of the 25.50 sigma-t surface (i.e. the main pycnocline) during 

the observation period December 5-12, 1993: high-pass filtered to 

remove scales larger than about 150 km. ........................................................4-2 

4.2-1. The 200 m sigma-t during the two observational periods (A) 

December 5-12, 1993 and (B) December 16-23, 1993. ..................................4-3 

4.2-2. The 200-780 m dynamic height (dynamic cm) during the two 

observational periods (A) December 5-12, 1993 and (B) 

December 16-23, 1993. ...................................................................................4-4 

4.2-3. The 200-780 m geostrophic velocity during the two observational 

periods (A) December 5-12, 1993 and (B) December 16-23, 1993 ................4-5 

4.3-1. The model sea-surface height (SSH) just prior to nudging of hydrographic 

measurements from observational Period-A. ..................................................4-8 

4.4-1. The observed geostraphic velocities at 200 m during the two periods, 

and the modeled velocities on December 14 (at end of nudging), 

December 23 (at end of observation Period B during 

unassimilated run), and January 1, 1994 (at end of unassimilated run). .........4-9 

4.4-2. Left panels: the observed along-slope geostrophic velocities U (colored, 

 m s

-1) and temperature (light contours, C) at the cross-slope vertical 

section shown in Figure 4.4-1, during the two indicated observational 

 periods. ............................................................................................................4-10 

4.5-1. The sectional averaged (section length =Ly, where the section is the 

northern half of the cyclone shown in Figure 4.4-1) EKE and PEP, 

∫(EKE)dy/Ly (upper panel), and ∫(PEP)dy/Ly (lower panel), respectively, 

as functions of depth (z;m) and time (εt:days). ...............................................4-13 

4.5-2. Time-averaged (period P=5 days from December 15-30) EKE and PEP, 

∫ (EKE)dt/P, and ∫(PEP)dt/P, respectively, as functions of depth (z;m) 

and latitude for the northern half of the section cutting across the 

cyclone shown in Figure 4.4-1 ........................................................................4-14 

4.5-3a. Sectional (y) and depth (z = -H to -50 m) averaged values of the terms 

in equations (4.5-2) and (4.5-3).......................................................................4-15 

4.5-3b. Sectional (y) and depth (z = -1 to -50 m) averaged values of the terms 

in equations (4.5-2) and (4.5-3).......................................................................4-16 
xi 

LIST OF FIGURES (continued) 

Figure Page

4.5-4. Sverdrup potential vorticity (PV) for layer between the near-surface 

and the depth of the 26.5 sigma-t surface (top panel) for Period-A................4-18 

4.5-5. Sverdrup potential vorticity (PV) for layer between the near-surface 

and the depth of the 26.5 sigma-t surface (top panel) for Period-B. ...............4-19 

4.5-6. The Q-vector divergence at 200 m overlaid with the 200-m geostrophic 

velocity vectors for Period-A. .........................................................................4-20 

5.1-1. Mean currents and variances at mooring locations from observations 

and model. .......................................................................................................5-3 

5.1-2. First and second spatial patterns of observed and model currents. .................5-4 

5.1-3. First and second spatial patterns of observed and satellite-derived 

currents. ...........................................................................................................5-6 

5.2-3. Observed and simulated trajectories for drifter #0012 and #0019 in the 

DeSoto Canyon group. ....................................................................................5-9 

5.3-1. Histograms of Loop Current eddy shedding periods for Experiments B, 

C, CS and D (see Table 5.3-1).........................................................................5-10 

5.3-2. Loop Current eddy detachment and reattachment, experiment CS: 

Eulerian trajectories launched for 5 days centered around the indicated 

date in each panel, and from every 10th grid point at the first sigma level 

(i.e. surface), superimposed on color image of surface elevation 

(red for values > 0.4 m, blue < -0.6 m). ..........................................................5-11 

5.3-3. The second EOF mode (20%) from satellite SSH anomaly (1992-1999), 

and first EOF mode (26%) from experiment CS, and the corresponding 

time series........................................................................................................5-12 

6.2-1. Positions of the moored instrument array in the MMS Exploratory 

Program (April 2003 to April 2004)................................................................6-4 

6.3-1. A schematic depiction of the 'violent' interaction of a warm eddy with 

a continental slope...........................................................................................6-7 
xiii 

LIST OF TABLES 

Table Page

2.2-1. Deep Current Meter Moorings.......................................................................2-10 

5.3-1. Model Experiments........................................................................................5-13 
xv 

ABSTRACT 

The objective of this study is to describe and understand circulation processes in the Gulf of 

Mexico through modeling and data analysis. The main focus is on two topics, topographic 

Rossby waves (TRW’s) and slope eddies, which are contained in Chapters 3 and 4 of this report. 

Another section (Chapter 5) reports briefly on other research funded in part by this contract 

through cooperation with other MMS studies. This study has resulted in four published papers: 

Oey and Lee (2002); Oey et al. (2003a); Ezer et al. (2003); Wang et al. (2003); and two 

manuscripts: Oey and Zhang (2003) and Fan et al. (2003). 

The TRW work extends Hamilton (1990) and Oey (1996). Based on the results of a highresolution model, the analysis concentrates more precisely on regions in the Gulf where TRW’s 

are likely to occur, their possible generation site(s) and mechanism(s), and also propagation 

paths. It is found that TRW’s in the 20~100 day periods are excited by small-scale peripheral 

eddies or meanders around the LC or propagating Loop Current Eddies (LCE’s). The TRW’s 

propagate westward and the energy is predominantly confined seaward of the 3,000-m isobath in 

the central and western Gulf. It is also concluded that the deep Gulf circulation is cyclonic, and 

that the deep (horizontal) shear in part accounts (together with sloping topography) for the 

seaward confinement of the TRW rays. These findings are contained in Oey and Lee (2002). 

The slope-eddy study focuses on the following scientific problem. Given an observed slope 

eddy field (spatial scales ≈ 50~150 km) over the north-central Gulf, can a model generate its 

temporal and spatial evolution, hence deduce its properties (e.g. eddy energy, stability etc), and 

check the ‘prediction’ against observations? Two surveys are taken from LATEX-C and 

GulfCet-I measurements (Hamilton et al., 2002) for the observed eddy fields: (A) Dec/03-

14/1993 and (B) Dec/16-23/1993. Data from Period A is used to initialize the model, while that 

from Period-B is used to check model’s prediction. To initialize the model, the larger-scale 

circulations of LC and LCE’s are generated by assimilating satellite altimeter and SST data using 

an optimal interpolation scheme over the whole model domain. The observed temperature and 

salinity measurements from Period A are ‘injected’ by nudging them into the heat/salt equations. 

The model ‘predicts’ growth of the eddy energy of this initial field. The growth agrees with the 

observed increase of the kinetic energy of the geostrophic flow from Period-A to Period-B. 

Stability analysis indicates that the growth is caused by mixed barotropic and baroclinic 

instabilities. The success of this use of hydrographic surveys in a model assimilation scheme is a 

first step along the path that will allow the incorporation of in-situ, ocean observing systems into 

real-time model predictions. 

One of the cooperative studies compares modeled and observed surface currents in the Desoto 

Canyon for the period April 1997 through April 1999. The model used here is an earlier version, 

with coarser grid spacing, than was used for the TRW and eddy work. The model shows good 

agreements with observations in terms of the means and variance ellipses. An SVD (Single 

Value Decomposition) analysis shows that the model can capture the more dominant mode 1 

observations, but misses the mode 2. The latter failure is most likely due to the coarse resolution 

used in this early model, which may not simulate well the small-scale cyclone-anticyclone pair 

that exists in mode 2. This study is summarized in Wang et al. (2003). 
xvi 

A drifter comparison study examines the possibility of assimilating observed drifters into the 

model. The 1998 NEGOM drifters were used to test the scheme. It is shown that a combination 

of both satellite data and drifter assimilations yields the least RMS errors in drifter positions. 

The two assimilations are complementary: satellite gives Gulf-wide coverage of large-scale LC 

and LCE’s, while drifters enable more realistic simulations of localized (small-scale) eddy field. 

By comparing the modeled and in-situ currents over the west Florida shelf, it is also shown that 

the anomalously intense southward shelf currents in the summer of 1998 were driven by LC 

forcing as the latter ‘brushed’ against the shelfbreak in the Florida Straits. This study is reported 

in a manuscript by Fan et al. (2003). 

One of the most outstanding problems in Gulf circulation studies is the cause of the observation 

that the LC sheds eddies at irregular intervals from 3~17 months. It is shown that wind-induced 

transport fluctuations through the Greater Antilles passages can cause shedding at shorter 

intervals (≈ 3~7 months). Anticyclonic Caribbean eddies cause shedding at longer periods (≈

14~16 months) through potential vorticity conservation, which indicates that Caribbean eddies 

tend to deter northward extension of the LC into the Gulf. This leads to longer periods between 

eddy shedding. Fluctuating inflow at the Yucatan Channel that is associated with winds and/or 

Caribbean eddies can cause a LC eddy to temporarily (~ 1 month) detach from, and then reattach 

back to the LC, a phenomenon often observed. Model results also suggest that southwest of 

Hispaniola, warm eddies are spun up by the local wind stress curl. This type of eddy drifts 

southwestward, then westward after merging with the Caribbean Current, and then northward as 

it progresses towards the Yucatan Channel. These eddies significantly affect the shedding 

behavior of warm-core rings. The time scale for spin-up and drift from Hispaniola is about 100 

days. Satellite data indicate the existence of these eddies in the real ocean. These findings are 

reported in Oey et al. (2003a). 
1-1 

1. Introduction 

The circulation of the Gulf of Mexico is dominated by processes related to the periodic extension 

of the Loop Current (LC) and the shedding of large (~300 to 400 km diameter) anticyclonic 

eddies. These warm LC eddies translate westward or southwestward, at speeds of ~3 to 6 km 

day-1, across the deep water of the central and western Gulf until they collide with the western 

Mexican slope. During this passage across the Gulf, LC eddies may generate subsidiary 

cyclones or frontal eddies, and interact with existing cyclones and anticyclones. On the western 

Mexican slope, LC eddies undergo complex interactions with topography and the existing eddy 

field. Dissipation of LC eddies in the west is a very complex process that may involve the 

generation of smaller scale cyclones and anticyclones. A consequence of the complexity of eddy 

generation processes is that observations of upper-layer circulation show complex fields of both 

warm and cold eddies of varying scales ranging from ~ 40 to 50 km up to LC eddy diameters 

over most regions of the deep Gulf, including the northern and western continental slopes. 

Observations of vertical current and density structures have shown that the deep waters of the 

Gulf are essentially a two layer system with the upper layer dominated by the LC and eddies of 

varying scales. Below the transition at about 800 to 1,200 m, limited current measurements have 

shown the flows to be almost depth independent with some bottom intensification. Current 

fluctuations have periods of between ~10 and 100 days and are usually decoupled from the 

upper-layer eddy flows. These types of lower-layer fluctuations have been characterized as 

topographic Rossby waves (TRW) with wavelengths of ~ 100 to 200 km and propagations 

speeds of ~ 10 to 20 km day-1. It has been speculated, in analogy with studies of the deep Gulf 

Stream region of the Northwest Atlantic, that the LC fluctuations, and eddy interactions with 

other eddies and topography are responsible for deep TRW currents. Model studies have also 

shown lower-layer eddy-like flows that are associated with the translation of a large LC eddy 

across the Gulf. At the present time, the limited observational database in deep water has not 

confirmed the existence of these lower-layer eddy flows. 

Over the last two decades, the Gulf of Mexico has been one of the principal laboratories for the 

development of meso-scale, ocean circulation, hydrodynamic models. Studies began with the 

LC eddy shedding problem, using two-layer models, and have progressed to investigations of 

boundary conditions affecting the LC, including interaction of the Gulf of Mexico with the 

Caribbean Sea and the wider western Atlantic, and the assimilation of observations (principally 

satellite altimetry maps of sea-surface height (SSH) anomalies, and sea-surface temperatures 

(SST)) to model eddy fields that correspond with the real world. Modern ocean models use 

three-dimensional high-resolution grids in both the horizontal and vertical. Because of the 

complexity of circulations in the Gulf of Mexico, numerical models are essential tools for the 

understanding of the dynamics of the, necessarily limited, observed flows. One of the aims of 

this study is to examine specific processes, known to be important for energetic sub-surface 

currents, through analysis of model simulations and observed data, so that an understanding of 

model dynamics is achieved. This in turn will be a basis for improving the ability of models to 

simulate Gulf of Mexico circulation processes, which will lead to more accurate hindcasts and 

forecasts that are important for industry and public uses of this sea. 
1-2 

1.1 Background 

When this study was initiated, 3-D numerical models had been applied successfully to the Gulf. 

Open boundaries were placed at a variety of positions (e.g. the Yucatan, the mid-Atlantic ridge, 

etc.) and a variety of forcing mechanisms were applied at the sea-surface and open boundaries. 

These modeling studies all reproduced the basic LC extension and eddy-shedding cycle. 

Observational studies had shown that the upper-layer over the northern slope and in the deep 

basin contained a rich field of eddies, with a range of scales and including cyclones as well as 

anticyclones, not just the dominant LC rings. The higher resolution models also showed eddies 

at varying scales, but little attempt was made to compare simulations of the eddy fields with the 

limited findings of the observational studies. It was realized that in order to make quantitative 

comparisons with observations, models would need to assimilate observational data. A model 

with forcing applied at the boundaries could only be expected to attempt to reproduce the 

statistics of eddies because of the stochastic nature of the flows. Assimilation schemes primarily 

used satellite observations of SSH anomaly and SST because these data types were readily 

available on regular schedules. SSH is a dynamic quantity and studies had shown that mapping 

methods could be used to resolve mesoscale circulations in the upper layer at spatial and time 

scales of about 100 km and 10 days, respectively. SSH is used to infer density anomalies deeper 

in the water column and thus, constrain the models’ subsurface flow fields. Therefore, advances 

in modeling Gulf of Mexico eddy fields were expected to come from assimilation schemes for 

various data types and more detailed comparisons with observations. The latter investigations 

would be expected to show whether the dynamics of the models were reasonable, and whether 

the scales and eddy interactions were realistic or not. Model improvements would be expected to 

follow. In turn, the model could provide information on the genesis, evolution and decay of the 

eddy fields that is not readily available from experimental studies. 

Below 1,000 m in water depths greater than ~ 2,000 m, observations have shown the flows to be 

dominated by propagating planetary waves (TRW’s) that have quite different characteristics to 

the upper-layer eddies. These TRW’s are quite energetic with amplitudes greater than 10 cm/s, 

periods between ~ 7 and 50 days, and wavelengths of ~ 100 km. They have been observed in all 

regions of the deep Gulf where current measurements have been made. It is important that 

models are able to simulate these deep currents and thus, the generation and propagation of these 

waves. Again, if the models were able to reproduce the general characteristics of observed 

TRW’s, the simulations would be expected to provide information on the dynamics of the 

transfer of energy from upper-layer eddies to the lower-layer. With the exception of Oey (1996), 

the ability of Gulf models to generate and propagate deep TRW’s has not been investigated. 

Below 1,000 m, assimilation of direct observations is not practicable at present, and SSH has 

little influence. Since TRW currents at any given position will have propagated from remote 

source regions, the future ability models to hindcast such events will depend crucially on 

simulating the processes responsible for generation of the wave trains. It is expected that this 

will also depend on correctly simulating the dynamics of the upper-layer eddy field and the 

mechanisms of energy transfer to the lower layer. 
1-3 

1.2 Objectives 

Based on the above discussion, the basic objectives of this study are to improve the ability of 

numerical models to simulate the dynamics of the Gulf by investigating two important 

phenomena. They are the generation and propagation of TRW’s in the deep basin, and evolution 

of eddy fields through the alteration of vorticity through eddy frontal instabilities, eddy-eddy, 

and eddy-topography interactions. These processes are investigated by a combination of 

modeling and high-level analyses of both data and model output. Process-oriented modeling 

uses increasingly realistic simulations of the Gulf, along with model experiments, to evaluate the 

dynamics of TRW’s and eddies. The model processes are investigated relative to the dynamics 

revealed by the data analyses. 

1.3 Outline of the Report 

Chapter 2 contains descriptions of the models used for this study, and specific analysis tools used 

for the model and observational data. The observational database is also given. Chapters 3 and 

4 concentrate on the analysis and simulation of topographic Rossby waves and slope eddies, 

respectively. Since this study was one of four modeling investigations, funded by the Minerals 

Management Service (MMS), there was a good-deal of interaction between this and other 

projects. Chapter 5 briefly summarizes contributions of this study to other projects that involved 

model-data comparisons and analysis of circulation processes. Finally, Chapter 6 summarizes 

the report and makes recommendations on future strategies for advancing modeling of the Gulf 

of Mexico. 
2. Methods 

2.1 Basic Model Description 

Past experiences (Oey, 1996) indicate that, in order to unambiguously study the Gulf’s

circulation, it is necessary to allow a free dynamical interaction between the Gulf of Mexico and 

the Caribbean Sea. Oey’s (1996) model of the Gulf, for example, includes the northwestern 

portion of the Caribbean Sea. In the present work, Oey’s model domain is further enlarged to 

include a portion of the North Atlantic Ocean west of 55o

W and from 6 to 50oN, shown in Figure 

2.1-1. In some calculations, a nest is placed over the Gulf of Mexico to effect double resolution. 

The model is based on the primitive-equation Princeton Ocean Model as described in more detail 

in Oey and Lee (2002). Time-independent total transports are specified at 55o

W according to

Schmitz (1996). Except in experiments in which we assimilate sea-surface height anomaly from 

the satellite, the steady transports effectively filter out propagating signals from the region east of

55o

W. These transports determine the two-dimensional depth-integrated velocities at the open 

boundary and are meant to account for the large-scale transports, caused by the wind-stress curl 

and thermohaline forcing, through 55o

W. The open-boundary conditions are a combination of 

these transport specifications along with radiation and advection as detailed in Oey and Chen 

(1992). For example, the temperature and salinity fields are advected using a one-sided 

difference scheme when flows are eastward (that is, outflow), and are prescribed from either the 

annual or monthly-mean temperatures and salinities from the Generalized Digital Environmental 

Model (GDEM) climatology (Teague et al., 1990) when flows are westward. These openboundary specifications also set the baroclinic structure, which in the present case is largely 

geostrophic through the thermal-wind balance. All fluxes are zero across closed boundaries. At

the sea-surface, climatological heat and salt fluxes are specified, and momentum flux (windstress) is specified from the six-hourly ECMWF data. 

The variable model grid spacing ranges from ∆ ≈ 10 km in the northwestern Caribbean/Yucatan 

Channel, to ∆ ≈ 5 km in the eastern and northern Gulf of Mexico. There are 25 vertical sigma

levels with finer resolution over the upper and lower 500-1,000 m of the water column (Oey and 

Lee, 2002). For all experiments, the Smagorinsky’s (1963) mixing coefficient is set to 0.1, and 

the ratio of (horizontal) diffusivity to viscosity is 0.1. 

The sigma-level pressure gradient error (Haney, 1991) in the model is not large when compared 

with the physically meaningful modeled currents. The error is reduced by removing the basinaveraged density profile (i.e. <ρ> is a function of z only) from the time-dependent density field 

before evaluating the pressure gradient terms (Mellor et al. 1998). A one-year test calculation 

using an initially level density field with perturbations (see Mellor et al., 1998) and zero forcing 

was conducted. Figure 2.1-2 shows that the maximum error asymptotes to ≈ 0.15 cm/s, which is 

relatively small in comparison to, say, the Loop Current speeds ≈ 1 m/s. Moreover, this 

maximum occurs off Cape Hatteras, under the Gulf Stream, and thus removed from the region of 

particular interest to this study. 

The model has been described in a number of publications: Oey and Lee (2002), Oey et al. 

(2003a), Wang et al. (2003) and Ezer et al. (2003). Oey et al. (2003b) in particular address the 

sigma-level pressure gradient error, satellite data and assimilation scheme, and also model 

2-1
Figure 2.1-1a

The model orthogonal curvilinear grids encompass the Gulf of Mexico, the 

Caribbean Sea, and a portion of the Atlantic Ocean. Time-independant inflow and 

outflow that account for the large scale transports (Sverdrup + themohaline) are 

specified across the open boundary at 55 W as a function of latitude.

2-2

-6000

-5000

-5000

-4000

-4000

-40

-4000

00

-3000

-3000

-3000

-2000

-2000

-1000

-200

95W 90W 85W 80W 75W 70W 65W 60W 55W

10N

15N

20N

25N

30N

35N

40N

45N

50N

20Sv

35Sv

38Sv

93Sv
Figure 2.1-1b

Oey and Lee's (2002) northwest Atlantic Ocean model domain and the nested 

doubled-resolution Gulf of Mexico region where calculations described in this 

report are conducted. The "parent" grid lines are shown at every seventh grid 

point, and the approximate distribution of doubled resolution grid sizes is 

indicated

2-3

95W 90W 85W 80W 75W 70W 65W 60W 55W

10N

15N

20N

25N

30N

35N

40N

45N

50N

20Sv

35Sv

38Sv

93Sv

Gulf of Mexico

Caribbean Sea

Nested Region

Yucatan

Channel

Straits of

Florida

Atlantic Ocean

D~5 km

D~3 km

D~10 km
Figure 2.1-2. The maximum speed that results from a one-year test calculation using initially 

level density field with perturbation: ρ(z) = ρr + ρ', where ρr = area-averaged and 

annual-mean climatological density, and ρ' = -0.1 kg/m3 • exp (z/1000 m) and 

zero forcing (see Mellor et al., 1998). For this perturbation, the maximum speed 

asymptotes to 1.45x10-3 m s-1 in about 60 days. Only the first 60 days is shown 

in the plot. The error is approximately proportional to the amplitude of 

perturbation.

2-4

Umax.

0.0000

0.0005

0.0010

0.0015

0.0020 (m/s)

10 20 30 40 50 60

(day)
sensitivity to grid resolution. Satellite data assimilation is the basic method for hindcasting Gulf

of Mexico circulations and is discussed below. Assimilation of hydrography is discussed in 

Chapter 4. 

2.1.1 Satellite Data Assimilation 

The satellite data are assimilated into the model following the methodology given in Mellor and 

Ezer (1990). The model is integrated without assimilation for 8 years, forced by 6-hourly 

European Centre for Medium-Range Weather Forecast (ECMWF) winds, and surface heat and

salt fluxes. The correlations between SSH anomaly δη and subsurface temperature and salinity 

(T/S) are calculated from the model results. Given the satellite SSH anomaly, δηsa, the model

subsurface temperature anomaly δT is calculated as: 

δT(x,y,z ,t) = FT(x,y,z) δηsa(x,y,t) (2.1-1) 

where the correlation factor is 

FT = <δT δη>/<δη2

>, (2.1-2a)

and the corresponding correlation coefficient is 

 CT = <δT δη>/[<δT2

><δη2>]1/2. (2.1-2b) 

After each assimilation time step ∆tA (= 1 day), the model temperature T is replaced by the 

assimilated temperature TA: 

 TA = T + [2 RACT

2

/(1 + 2 RACT

2 − CT2

)] (TO − T) (2.1-3) 

where RA is the ratio of ∆tA to the de-correlation time scale ∆tE of the model eddy field (≈ 30 

days), and TO is the ‘observed’ temperature inferred from satellite SSH anomaly, which from 

(2.1-1) is: 

 TO = <T> + FT δηsa. (2.1-4) 

In (2.1-4), <T> = TC, the climatological mean temperature. The assimilation effect is such that 

TA ≈ TO in regions where the correlation is high, but TA ≈ T, where the correlation is small. Also, 

to minimize potential satellite errors near the coast, the assimilation is restricted to regions where 

water depths are > 500 m, thus excluding the shelves. 

Satellite altimeter data, AVISO (Archiving, Validation, and Interpretation of Satellites

Oceanographic), are obtained from the French Space Agency. The data product was created by 

merging TOPEX/Poseidon (T/P) and ERS-1 & -2 altimeter measurements (Ducet et al., 2000). 

The combined, inter-calibrated altimeter data are interpolated in time and space using a global 

objective analysis. The length scale of the interpolation varies with latitudes, and is about 200 

km at mid-latitudes. The e-folding time scale is set at 10 days in the tropics and 15 days 

elsewhere. The resulting satellite product has a spatial resolution of 0.25o

 x 0.25o and is 

2-5
provided at 10-day intervals. The merged T/P + ERS-1&2 SSH anomaly maps provide reduced 

and more homogeneous mapping errors than either individual data set, and thus, more realistic 

statistics. Fratantoni (2001) compared AVISO- and drifter-derived kinetic energy and found 

reasonable agreements. Wang et al. (2003) checked the AVISO data against two-year direct 

current measurements in the DeSoto Canyon, and found that their first two SVD (Single Value

Decomposition; Bretherton et al. 1992) modes agreed. Oey et al. (2003) found that in the Gulf 

the AVISO data agreed well with those obtained from the Colorado Center for Astrodynamics 

Research (Leben et al. 2002). In this work, the AVISO data from 1992/October through 1999 

are used. Figure 2.1-3 shows an example of the assimilated result. 

2.1.2 Data Assimilation in the Princeton Ocean Model 

POM has a number of options for different data assimilations. The various forms are all based

on standard optimal interpolation (or Multivariate OI or MVOI; e.g. Daley, 1991). They differ

only in the details in which the various data are utilized and how the error covariance matrix is 

defined. The data (at present) consist of satellite sea-surface height anomaly (SSH), sea-surface 

temperature (SST), (described in the previous section), moored temperatures and currents, 

hydrography, and drifters. One can in principle also incorporate other data (e.g. CODAR, 

SOFAR etc). Given the observed (anomaly) vector , the model ‘forecasts’ (or first-guess) 

at N grid points, ‘i’, that is corrected to obtain an analyzed vector according to: 

o wβ

f wia wi

∑ ∑ = =

= + −

M N

i

f

i i

o

i

f

i

a wi w K w H w

1 1

[ ( )]

β

β β β . (2.1-5) 

Greek subscripts denote the M observational locations, is the Gain matrix (or weights), and 

interpolates modeled values onto the observational point. is obtained by minimizing the 

square of the analysis error , where is the ‘truth’ and the angle brackets

denote (time or ensemble) averaging. Thus, 

Kiβ

Hβ Kiβ

=< − > 2 ( )i

a

i

a Ei w w wi

α

β

β βα i

f

M

f T o Ki (HP H P ) (P H)

1

∑ + = =

, (2.1-6a)

where 

=< ( − )( − ) > (2.1-6b) Pαβ wα wα wβ wβ

f f f

Pαβ =< (wα − wα)(wβ − wβ ) > o o o (2.1-6c)

2-6
LCE

Loop

Caribbean Current

Current

Florida Current

ACE

Cyclone

LCE Atlantic Ocean

Eddy Field

Shelf Current

Shelf

Current

Yucatan

Current

Gulf of 

Mexico

Caribbean Sea

CARE

Cyclonic

Gyre

Cyclonic

Gyre

CARE

YC: Yucatan Current CARE: Caribbean Eddy

LC: Loop Current ACE: (Smaller) Anticyclone

LCE: Loop Current Eddy

Cyclone

ACE

Cyclone

Greater Antilles

Venezuela

Lesser Antilles

Windward 

Passage

Mona

Passage

Columbian 

Basin

Cayman 

Basin

Anegada

Passage

Nicaragua 

Rise

U.S.A

Mexico

Jamaica

Netherlands

Antilles

0.0 0.2 0.4 0.6 0.8

1.0

(m/s)

Hispaniola

ure 2.1-3.An example of the regional, near-surface circulation obtained from the data-assimilated run (from Oey, et al., 2003a). 

Plotted are ten-day Eulerian trajectories x = xo + ∫ u dt, where x and u are position and velocity vectors respectively, 

and the integration is over ten days. The trajectories are launched from July 19 through July 28, 1997, and from every 

8th grid point at the first sigma level (i.e. surface). Colors indicate speeds such that greenish blue is ≈ 0.5 m s-1 and red 

is ≥ 1 m s-1. 
are the space-time correlations of the ‘forecast’ error and observational error, respectively. Since 

is usually not known, various levels of approximation must be made for and , hence 

. 

wα

f Pαβo Pαβ

Kiβ

Level 1: 

Here the is analytically specified (and in general not “optimum”), and Kiβ

 and , (2.1-7a) Kiβ = ΓijΛjβ i j ij Λ H = I β β

where 

∑=

− − ∆ Γ =

M

ij i

d

i

a

ij M

t

t t

t

t

1

0 exp( ) ( )/ 2

β

δ λ β , (2.1-7b)

∑=

Λ = M

i

i

i

β 1

β

β

β

λ

λ

(2.1-7c)

exp( ) 2

2

2

2

2

2

z

i

y

i

x

i

i

R

z

R

y

R

x β β β λ β = − − − , (2.1-7d)

Iij is the unit matrix, δ ij is the Kronecker delta function, ∆t is the assimilation time step, is 

the assimilation time scale (shorter for stronger assimilation), is the most recent time when 

observation is available at point ‘i,’ td is the time influence of the observational data, (x,y,z)iβ is

the distance between the model and observational grid points, and Rx,y,z are (x, y, z) scales, 

respectively, that measure the radius of influence of observation and M is the number of

observational locations. Typically (ta, td) ≈ hours to days, and (Rx, Ry, Rz) ≈ (10’s km, 10’s km, 

10’s m), taken to be independent of grid point ‘i.’. To put in more ‘physical flavor,’ it can be 

shown that if = model time step (using a ‘leap-frog’ scheme), (2.1-5) and (2.1-7) lead to: 

at

i t0

∆t

∑=

= + Γ − ∂

∂ N

j

f

j

o

ij j

f

i physics w w

t

w

1

) ~ ( ) ( (2.1-8a)

∑ ∑ = =

=

M M

i

o

i

o wi w

1 1

( )/( ) ~

β β

λ β β λ β , (2.1-8b)

which is the nudging method. Fan et al. (2003) used this scheme to assimilate observed drifters 

(see Chapter 5). 

2-8
Level 2 (“Standard OI”): 

Here, is diagonal and is Gaussian: o Pαβ

f Pαβ

Pαβ

o = σ2

δαβ (2.1-9a)

exp( ) 2

2

2

2

2

2

2

2

2

x y z t

f

R

t

R

z

R

y

R

x

P w αβ αβ αβ αβ

αβ =< δ α > − − − − , (2.1-9b) 

where σ is the standard deviation of observational error (assumed constant), is the 

(model) variance, and as before xαβ = xα − xβ etc., and Rx etc. are scales. This (equations 2.1-9, 

2.1-2 and 2.1-1) is the scheme used in POM for assimilating the satellite SSH and SST (Mellor 

and Ezer, 1991; Wang et al., 2003; Oey et al., 2003a). 

< >2 δwα

Level 3a: 

Here, is given by (2.1-9a) but is computed from (2.1-2b) by assuming the model and 

observed climatologies are the same and ‘truths.’ Then is just the model covariance.

o Pαβf Pαβ

f Pαβ

Level 3b: 

Here, the observed are assumed to be ‘truths.’ Thus and is computed from (2.1-2b) 

with . 

= 0 o Pαβ

f Pαβ

o wα = wα

Level 3c: 

Here, a complete hindcast is first made with the Level-1 scheme. This is then used in Level-3b to

pre-compute . This hindcast is then repeated but at each time step, the pre-computed , 

hence from (2.1-2), are used to correct the first-guess . Dong and Oey (2003; 

unpublished work) found this scheme to give good ‘prediction’ at points where no data were 

assimilated.

f Pαβf Pαβ

Kiβ

f wi

2.2 Observational Database 

The observational database used for analysis and comparison with model simulations consists

mainly of time series of currents and temperatures from moored instruments in deep water, and 

hydrographic data from surveys of northern slope waters and LC eddies. The moored data is 

from a number MMS programs beginning in 1983, and the hydrographic data is primarily from 

the GulfCet I and LATEX C surveys, from the early 1990’s, of the northern slope. In Chapter 5, 

near-surface Lagrangian drifter data, deployed in the region of the DeSoto Canyon as part of the 

MMS studies in the Northeast Gulf (Hamilton et al., 2000; Jochens et al., 2002), were used for

2-9
model assimilation studies. These drifter data are described in Chapter 5.2, as they were not part 

of the main study. The current and hydrography databases are described in the next two sections. 

2.2.1 Currents 

The positions of moorings, in deep water, with deployments longer than 6 months are given in 

Figure 2.2-1. This is not the complete inventory of deepwater moorings, but only the ones that

were used for this study. The depths and time periods of the current data are given in Table 2.2-

1. All these data were available from the SAIC database management system and therefore, had 

been processed so that deployments had been concatenated, and time series had been qualityassured and filtered with 3 and 40-hour low-pass Lanczos kernels, with decimations to 1 and 6-

hour time steps, respectively. In some cases a 7-day low pass filter, with decimation to 1-day 

intervals, was also used for analysis of longer-period fluctuations. 

Table 2.2-1 Deep Current Meter Moorings 

Mooring ID 

(Reference) 

Deployment Period Water 

Depth (m) 

Current Meter 

Depth (m) 

Instrument 

Type 

MMS Gulf of Mexico Physical Oceanography Program: Years 1, 2 and 4 

A 

(SAIC, 1987) 

Jan 1983-Feb 1986 1,700 172 

400 

738 

1,100 

1,600 

NWCM 

“ 

“ 

“ 

“ 

G 

(SAIC, 1987) 

Jan 1984-Jan 1986 3,200 177 

397 

703 

1,565 

2,365 

3,174 

NWCM 

“ 

“ 

“ 

“ 

“ 

MMS Gulf of Mexico Physical Oceanography Program: Year 3

P 

(SAIC, 1988) 

Jun 1985-May 1986 2,000 30 

300 

1,000 

1,500 

RCM 

“ 

“ 

“ 

Q 

(SAIC, 1988) 

Jun 1985-May 1986 3,000 100 

300 

1,000 

1,500 

RCM 

“ 

“ 

“ 

R 

(SAIC, 1988) 

Jun 1985-May 1986 3500 100 

300 

1,000 

1,500 

3,000 

RCM 

“ 

“ 

“ 

“ 

2-10
200 m

1000 m

2000 m

3000 m

200 m

Years 1,2 & 4

Year 3

Year 5

DeSoto Cyn Eddy

DeSoto Extension

Figure 2.2-1.Locations of deepwater moorings used in this study. Moorings are color coded by 

program. "Years" refer to data collection years of the Gulf of Mexico Physical 

Oceanography Program.

2-11
Table 2.2-1 (continued) 

Mooring ID 

(Reference) 

Deployment Period Water 

Depth (m) 

Current Meter 

Depth (m) 

Instrument 

Type 

S 

(SAIC, 1988) 

Jun 1985-May 1986 1,500 100 

300 

1,000 

RCM 

“ 

“ 

T 

(SAIC, 1988) 

Jun 1985-May 1986 2200 100 

300 

1,000 

RCM 

“ 

“ 

MMS Gulf of Mexico Physical Oceanography Program: Year5

EE 

(SAIC, 1989) 

Apr 1987-Nov1988 845 100 

300 

725 

NWCM 

“ 

“ 

FF 

(SAIC, 1989) 

Apr 1987-Nov1988 1750 100 

300 

725 

1,650 

NWCM 

“ 

“ 

“ 

GG 

(SAIC, 1989) 

Apr 1987-Nov1988 3,000 100 

300 

725 

1,650 

2,500 

NWCM 

“ 

“ 

“ 

“ 

DeSoto Canyon Eddy Intrusion Study 

A1 

(Hamilton et 

al., 2000) 

Mar 1997-Apr 1999 100 12-72 

95 

ADCP 

(4 m cells) 

S4 

A2 

(Hamilton et 

al., 2000) 

Mar 1997-Apr 1999 500 12-72 

200 

300 

490 

ADCP 

(4 m cells) 

RCM 

“ 

S4 

A3 

(Hamilton et 

al., 2000) 

Mar 1997-Apr 1999 1300 12-68 

500 

1290 

ADCP 

(4 m cells) 

RCM 

NWCM 

B1 

(Hamilton et 

al., 2000) 

Mar 1997-Apr 1999 100 12-72 

95 

ADCP 

(4 m cells) 

S4 

B2 

(Hamilton et 

al., 2000) 

Mar 1997-Apr 1999 500 12-72 

200 

300 

490 

ADCP 

(4 m cells) 

RCM 

“ 

NWCM 

2-12
Table 2.2-1 (continued)

Mooring ID 

(Reference) 

Deployment Period Water 

Depth (m) 

Current Meter 

Depth (m) 

Instrument 

Type 

B3 

(Hamilton et 

al., 2000) 

Mar 1997-Apr 1999 1300 12-68 

500 

1290 

ADCP 

(4 m cells) 

RCM 

NWCM 

C1 

(Hamilton et 

al., 2000) 

Mar 1997-Apr 1999 100 12-72 

95 

ADCP 

(4 m cells) 

S4 

C2 

(Hamilton et 

al., 2000) 

Mar 1997-Apr 1999 500 12-72 

200 

300 

490 

ADCP 

(4 m cells) 

RCM 

“ 

S4 

C3 

(Hamilton et 

al., 2000) 

Mar 1997-Apr 1999 1300 12-68 

500 

1,290 

ADCP 

(4 m cells) 

RCM 

NWCM 

D1 

(Hamilton et 

al., 2000) 

Mar 1997-Apr 1999 100 12-68 

95 

ADCP 

(4 m cells) 

S4 

D2 

(Hamilton et 

al., 2000) 

Mar 1997-Apr 1999 500 12-72 

200 

300 

490 

ADCP 

(4 m cells) 

RCM 

“ 

S4 

D9 

(Hamilton et 

al., 2000) 

Mar 1997-Apr 1999 200 21-152 ADCP 

(9 m cells) 

E1 

(Hamilton et 

al., 2000) 

Mar 1997-Apr 1999 100 12-72 

95 

ADCP 

(4 m cells) 

S4 

2-13
Table 2.2-1 (continued)

Mooring ID 

(Reference) 

Deployment Period Water 

Depth (m) 

Current Meter 

Depth (m) 

Instrument 

Type 

DeSoto Canyon Eddy Intrusion Extension 

I1 

(Hamilton et 

al., 2003) 

Sep 1999-Sep 2001 2,000 12-68 

122-250 

256-368 

372-588 

658-1,202 

800 

1,000 

1,200 

1,400 

1,600 

1,800 

1,989 

ADCP (4m)

“ 

“ 

ADCP (8m)

“ 

RCM 

“ 

“ 

“ 

“ 

“ 

NWCM 

I2 

(Hamilton et 

al., 2003) 

Sep 1999-Sep 2001 1998 1,600 

1,800 

1,989 

RCM 

“ 

NWCM 

I3 

(Hamilton et 

al., 2003) 

Sep 1999-Sep 2001 2175 1,775 

1,975 

2,164 

RCM 

“ 

NWCM 

I4 

(Hamilton et 

al., 2003) 

Feb 2001-Sep 2001 1957 1,557 

1,757 

NWCM 

RCM 

J1 

(Hamilton et 

al., 2003) 

Sep 1999-Aug 2000 1373 972 

1,172 

1,361 

RCM 

“ 

NWCM 

ADCP – RDI Acoustic Doppler Current Profiler (Narrow & Broad Bands, LongRanger) 

RCM - Aanderra Current Meter (versions 4,5,7,8 & 9) 

NWCM – General Oceanics Niskin Winged Current Meter (Mks I, II & III) 

S4 – Inter Ocean S4 Current Meter 

2.2.2 Hydrography 

The slope aircraft surveys of LATEX C (Berger et al. 1996; Hamilton et al., 2002), and the shipbased surveys of GulfCet I (DiMarco, Personal Communication), are the primary data sources. 

The former used AXBT’s and the latter XBT’s and CTD’s. All the (A)XBT data were converted 

to CTD’s as discussed by Hamilton et al. (2002). The (A)XBT’s are corrected for depth offsets, 

relative to a CTD, using the formulae given by Singer (1990), then salinities are estimated from 

nearby CTD stations that were taken within one month of the XBT data. In some cases the CTD 

data was from different years than the (A)XBT data. This is a reliable procedure because of the 

tight T-S relation in the Gulf for temperatures less than about 16°C. The scatter in the T-S in the 

surface layers leads to errors in the surface dynamic height, relative to 780dbar (the deepest 

depth of the stretched XBT profiles), of ~ 2 and 5 dynamic cm if the CTD data used are from the 

same or different years, respectively. Details are given in Hamilton et al. (2002). 

2-14
The dynamic height calculations for the slope region employ the method given by Csanady 

(1979) for integrating the bottom density profile across the isobaths. Only stations with bottom

depths less than 780 m on the upper slope require the integration of bottom densities from south 

to north along the cross-slope transects. The method is considered accurate if the bottom σt

isolines are approximately parallel to the isobaths, so that integration along a transect is 

independent of position. This assumption is usually reasonable on the upper slope. The assumed 

level of no motion at 780 dbar, though dictated by the XBT profiles, is also reasonable, given 

that the region of minimum energy in the current profile is between 800 and 1,200 m (Hamilton 

et al., 2000). Dynamic height and derived σt profiles are the basis of the higher order analyses 

given in the next section. 

A total of 21 AXBT flights and 7 GulfCet cruises were processed and the resulting fields mapped 

to grids. The station positions for two AXBT and one GulfCet surveys are shown in Figure 2.2-

2. Stations from more than one survey are often combined to increase the area of coverage if

they were taken within about a week of each other. The surveys in Figure 2.2-2 are extensively

used in Chapter 4. Some of the aircraft surveys deployed AXCP’s, which provide velocity shear 

profiles. AXCP velocity data was treated as in Hamilton et al. (2002), by smoothing with leastsquare splines with a length scale of 50 m, and removing a barotropic offset. The latter was 

estimated from regions near the bottom of the velocity profiles that exhibit no shear, as in

Sandford et al. (1987). 

The processed salinity, temperature and σt profiles were mapped to horizontal grids using the

statistical interpolation method given by Pedder (1993), which is an iterative method similar to 

the Barnes (1964) two-pass scheme and its successors. The length scale used by the Gaussian 

influence function is estimated as the mean separation of the station positions over the grid. This 

is about 40 and 50 km for the AXBT and GulfCet surveys, respectively. The method is iterated 

to convergence so that the differences between interpolated and observed fields at the station 

positions are minimized. However, for noisy calculations that difference the gridded fields (e.g. 

relative vorticity), some smoothing is often useful and this is achieved by limiting the number of 

iterations (Pedder, 1993). The interpolation method to a regular horizontal grid can employ 

either a equally spaced or the model’s curvilinear grid (see section 2.1). The former is used for 

analyses, and the latter for the assimilation of observations into the model runs. 

2.3 Analysis Tools 

2.3.1 Topographic Rossby Waves and Ray Tracing 

The fluctuations of observed lower-layer currents have a wave-like nature consisting of the 

nearly barotropic flows in the lower 1,000 m of the water column. Rhines (1970) first gave the 

theory of topographic Rossby waves using quasi-geostrophic (QG) dynamics (Pedlosky 1979).

Important features of TRW theory are: 

2-15
5 14 23 32 41 50 59 16 9

200 m

2000 m

46

49

63

53 54 75

76

96

97

115

117

134

136

200 m

2000 m

265

55

264

58

62

64

Figure 2.2-2.Station position maps for F13SQUIRT/F14LEDDY (top panel) and GULFCET07 

(bottom panel). Station types are X - (A)XBT, Solid Square - CTD and Solid 

Diamond (AXCP). In the top panel the F14LEDDY stations are purple.

2-16
1. The wave motion is bottom intensified; thus current amplitudes increase in magnitude

towards the bottom.

2. There is no phase difference between currents at different depths; therefore at any 

particular wave frequency, the motion is columnar. 

3. The maximum frequency or cutoff frequency for TRW's is Nα where α is the bottom 

slope, defined as normal to the general trend of the isobaths, and N is the Brunt-Väisälä 

frequency of the lower water column. Bottom slopes are defined over scales similar to 

the wavelengths of TRW's (~ 50 to 100 km). The effect of small or large perturbations of 

the bottom slope on TRW propagation is not covered by present linear theories. The 

wave frequency, ω, is proportional to the bottom slope and the cosine of the angle that

the wavevector makes with the isobaths. At the highest allowed frequency, the phase 

velocity is parallel to the isobaths such that shallow water is on the right of the direction 

of wave propagation. 

4. At the highest allowed frequency, wave motions are rectilinear and perpendicular to the

phase vector and isobaths. At lower frequencies the wave motions become more parallel 

to the isobaths. Longer wave-length motions are also less bottom trapped than high 

frequency waves. 

5. TRW's are dispersive and it can be shown that if the phase vector is directed into deeper 

water, then the energy flux has component towards the shallower water (Thompson, 

1977). TRW's are refracted by changing topography. These ray-paths can be calculated 

by WKB theory which assumes that the environment (e.g. bottom slopes) change “slowly

” compared to the wavelength of the motions. This is rarely strictly true in practice. 

TRW ray paths in this study are calculated using the QG dispersion relation. The basis of the

method is given in Meinen et al. (1993) and used by Pickart (1995) to calculate TRW ray paths 

generated by the deep Gulf Stream in the Middle Atlantic Bight. The dispersion relation for 

TRW’s is given by the coupled equations (Pickart, 1995; Oey and Lee, 2002): 

μ

2

 = (K2 + βk/σ ) ( N/f )2 (2.3-1a)

μ tanh( μh ) = N2/(σf ) ( K x ∇h)z (2.3-1b)

where h is the water depth, 

N = (-g/ρ0 ∂zρ)

1/2 is the Brunt-Väisälä frequency (assumed constant), 

f is the Coriolis parameter using the β-plane assumption, 

K = (k, l) is the wavenumber vector in east and north (x, y) coordinates, 

σ is the wave frequency, 

 1/μ is the vertical trapping scale of the wave, and 

∇ (= i ∂x + j ∂y ) is the horizontal gradient operator. 

The subscript ‘z’ is the vertical component of the vector, thus (K × ∇h)z=(khy– lhx). For typical 

N/f≈10 and maximum slope gradients |∇h| < ≈0.05 used in the model, it can be shown that the

QG dispersion is valid for wave periods ≥ 20 days. The model analyses focus solely on waves

2-17
with periods > 20 days. When topographic slope dominates, βTopo = f|∇h|/h >> β, and the β-term

can be dropped. Then μ can only be real and it becomes the inverse vertical trapping scale of the 

wave. Equations (2.3-1) then give, 

σ tanh(NhK/|f|)/N = sgn(f) (nK×∇h)z (2.3-2) 

where nK=K/K is the wavenumber unit vector, and sgn(f) = sign of ‘f’. Without loss of 

generality, σ can be taken to be >0, so that (nK×∇h)z must have the same sign as sgn(f). Thus the 

phase propagation direction must lie to the right (left) of the direction of increasing water depth 

in the northern (southern) hemisphere. In other words, the wavenumber vector makes a clockwise 

(anti-clockwise) angle θ with respect to the direction of steepest topographic decent (i.e. with 

∇h/|∇h|). For NhK/|f| ≈ O(1) or larger (which typically requires wavelengths < 200 km),

tanh(NhK/|f|)≈1 or at most a weak function of “NhK/|f|,” and (2.3-2) shows that the frequency σ

is independent of the magnitude of the wavenumber, and depends only on the angle that the 

wavenumber vector makes with the x or y-axis. It follows then that the component in the

direction of K of the rate of change of frequency in the wavenumber space, nK•∇Kσ, where 

∇K=(∂/∂k, ∂/∂l), must be zero, since this direction is by definition fixed and σ itself depends on it 

only, and not on the magnitude K. Thus, the wavenumber vector and group velocity Cg (=∇Kσ)

must be perpendicular to each other. Moreover, (2.3-1b) (with tanh(NhK/|f|)≈1) gives 

Cg = N nK•∇h ( l, -k)/K2, (2.3-3) 

so that Cg × K = NnK•∇h = N|∇h|cos(θ). Thus Cg is directed clockwise (upslope) with respect to 

K when the latter points downslope, 0<θ<π/2, and anti-clockwise (downslope) when K points 

upslope, π/2<θ<π (Figure 2.3-1). 

Under the WKB approximation, where changes in wave amplitude and phase caused by the 

environment are assumed to vary on scales larger than the local wavelength, the equations 

governing the path of a wave and its wavenumber are (LeBlond and Mysak, 1978; Lighthill,

1978): 

Dt x = ∂σ / ∂K = Cg (2.3-4) 

Dt K = ∑ - ∂σ / ∂γi ∇γi (2.3-5)

where Dt = ∂ / ∂t + Cg . ∇

is the derivative following the wavegroup, x is the path of the ray, and Cg is the group velocity,

defined above. The γi are the environmental parameters that cause refraction of the wave. There 

are three such parameters for TRWs: h (water depth), ∇h (bottom slope), and N (Brunt-Väisälä 

frequency). Though the wavenumber vector undergoes refraction as the environmental

2-18
K (f > 0)

Cg

—h

q

2500 m 3000 m

3500 m

Figure 2.3-1. A schematic that describes the relations between three vectors: the gradient of 

water depth —h, wavenumber vector K, and the group velocity Cg, in the northern 

hemisphere and when NhK/f is O(1) or larger (see text). q is the clockwise angle 

that K makes with —h. Cg is very nearly perpendicular to K. Moreover, Cg x K a

cos(q), so that it is positive (negative) and Cg points upslope (downslope) when K 

points downslope (upslope) when 0<q<p/2 (p/2<q<p). The figure shows the case 

0<q<p/2.

2-19
parameters change along a path, the frequency remains constant, because by 2.3-4 and 2.3-5, Dtσ

= 0. The WKB assumption is marginal though it is often used under conditions that have sharp

changes in the environmental parameters. Therefore, the topography must be smoothed over at 

least the wavelength scale for the method to apply. Equations, (2.3-4) and (2.3-5), are solved 

using 4th order Runge-Kutta with adaptive step size methods (Press et al., 1992) to determine ray 

paths and the change in the wavenumber vector along the rays. The condition that the frequency 

must be constant along a ray path serves as a check on the numerical method. 

The method requires specifications of four environmental fields: N(x,y,z,t), h(x,y), and the two 

components of ∇h(x,y). The topography was smoothed using a Gaussian-type interpolator (Oey 

et al., 2001) with a ‘radius of influence’ of 0.6°

, comparable to the filter width of 150 km used by 

Pickart (1995). Figure 2.3-2 compares the smoothed topography with the original one used in the 

numerical simulation (see Chapter 3). The smoothing removes short-scale topographic

irregularities but leaves the large-scale features (and also those of ∇h, not shown) essentially 

intact. To obtain a smoothed N, we performed time and depth averaging over the last seven 

years of the ten-year simulation, and also for depths below the 27.5σt surface (i.e. in layers 3 and

4). The resulting contours of N(x,y) are shown in Figure 2.3-2. Note that over the deep basin,

N≈10−3

 s−1. These smoothed values (N and h) are interpolated onto a longitude/latitude grid with

constant grid sizes (0.01o

×0.01o), on which ∇h is calculated using finite differences. Note that 

the uniform grid is used for calculating the environmental fields only, and does not define the

actual coordinates of rays. 

In addition to N(x,y,z,t), h(x,y) and ∇h(x,y), deep currents udeep also affect ray paths (Oey and 

Lee, 2002). Figure 2.3-3 shows the modeled currents averaged over the last seven years of the 

ten-year simulation, at 200 m above the bottom in the Gulf (Oey et al., 2002). The modeled deep 

mean flow, udeep=(u1, u2), is generally cyclonic around the deep Gulf. Between the 3,000 and 

3,500 m contours in the northern Gulf, maximum speed reaches 0.07 m/s, but in general it is 

more sluggish ~0.03 m/s. Given TRW group speeds of about 10 km day-1, the effects of udeep can 

be as much as 50% or more at some location. To assess the effects, the ray equations (2.3-4 and 

2.3–5) are modified to account for udeep as follows (Lighthill, 1978). Firstly, the absolute 

frequency σa is connected to the relative frequency σ (i.e. relative to medium at rest) by the 

Doppler’s relation: 

σa = σ + u . K (2.3-6)

Thus, the group velocity Cga becomes 

Cga = ∇Kσa = ∇Kσ + u = Cg + u, (2.3-7) 

where Cg is derived from (2.3-1). Thus, the ray path is defined by 

Dt x = Cg + u. (2.3-8)

2-20
Figure 2.3-2.

(a) Time (7 years) and depth (below 27.5 σt surface) -averaged Brunt-Väisälä 

frequency, N (10-4 s-1). (b) Smoothed (solid) compared with model (dashed) 

topography. Smoothed fields were used in ray tracing calculations.

8

8

10

10

10

10

10

10

10

10

10

10

10

10

10

12

12

12

12

12

12

12

12

12

14

14

14

14

14

14

14

14

14

16

16

16

16

16

16

16

16

18

18

18

18

18

18

20

20

20

22

22

24

Brunt-Vaisala Frequency (x10-4 s-1), 27.5-Bot.

95°W 90°W 85°W

20°N

25°N

30°N

-3000

-3000 -3000

-3000

-2000

-2000

-2000

-1000

-1000

95°W 90°W 85°W

20°N

25°N

30°N

a)

b)

2-21
Figure 2.3-3.

The modeled currents averaged over the last seven years of the ten-year 

simulation, at 200 m above the bottom in the Gulf from (Oey et al., 2002).

95 90 85

20

25

30

1000

1000

2000

2000

2000

3000

3000

3000

500

500

3000

3000

3000

3500

3500

95 90 85

20

25

30

0.3 m/s

95°W 90°W 85°W

20°N

25°N

30°N

2-22
Similarly, the absolute frequency is used for the derivation of (2.3-5), to obtain 

Dt K= ∑ - ∂σ / ∂γi ∇γi − K . ∇u (2.3-9) 

Equations (2.3-8) and (2.3-9) are used (in place of (2.3-4) and (2.3-5)) to trace rays when effects 

of mean flow are to be included. Multiply the x component of (2.3-9) by k and assume zonal h 

and N (or locally functions of the diabathic coordinate ‘y’ so that positive ‘y’ points in the 

opposite direction to the vector ∇h), and also udeep = (udeep(y),0), an estimate is obtained of how 

wavelength changes along rays embedded in a mean sheared flow: 

Dt (k

2

/2) = −l sin(θ) ∂(|hy|N)/∂y − kl ∂udeep/∂y. (2.3-10) 

This shows that the effect of udeep is that, for positive kl, the wave shortens as the ray propagates 

into a region of cyclonic sheared current ∂udeep/∂y < 0.

Using (2.3-6), (2.3-8) and (2.3-9), it can be readily shown that Dt σa = 0 along a ray. It follows 

then from (2.3-6) that σ + u . K is constant on a ray. Assuming a zonal shear flow and using 

(2.3-2), it can be estimated how rays are bent (i.e. how θ changes along rays) by the sheared 

current: 

N|∇h|sin(θ) = −udeep(y) k. (2.3-11) 

Since k < 0, rays propagating upslope into a region of decreasing parabathic current (i.e. cyclonic 

shear), the effect of cyclonic shear alone is to bend rays back towards down-slope (i.e. θ

decreases). Note that if furthermore N|∇h| increases up-slope, then Dt (ki

2

/2) > 0 from (2.3-10)

with udeep = 0, and it is only necessary that udeep < 0 for it to have the effect of bending rays 

down-slope. 

Note that the above tacitly assumes |∂udeep/∂z| << 1, i.e. small vertical shears. This is consistent 

with slowly-varying N (over TRW wavelengths and periods) required by the use of the theory of 

bottom-trapped TRW; i.e. the dispersion relation (2.3-1). The results presented in chapter 3 

suggest that in comparison to topographic vorticity gradient, gradients in N play a relatively 

minor role. While the theory is self-consistent in this regard, the question as to how a nonslowly varying N will affect ray paths, including the possible energy coupling of near-bottom 

with layers above, is beyond the scope of this study. 

For the model, average bottom currents are readily available. Similar to the calculations for an

average, horizontally varying N, a seven-year averaging was performed on the modeled currents 

at 200m above the bottom (Figure 2.3-3). The resulting udeep is used in the ray calculations

based on (2.3-8) and (2.3-9) to assess the effects of deep mean flows on TRW propagation. 

2-23
2.3.2 Hydrographic Analyses 

In classical analysis of hydrographic surveys, the distribution of potential vorticity (PV) on 

isopycnal surfaces is an important diagnostic of the dynamics. This is because PV is a conserved

quantity in the absence of dissipation and external forces (Ertel, 1942) and can be used to 

analyze dynamical processes. To first order, water particles move along surfaces of constant

potential density (i.e. isopycnal surfaces), and relative changes in the components of PV 

distributions on isopycnal surfaces occur during the formation of eddies. The discussion and

derivations below follow the work of Allen and Smeed (1996) and Pollard and Regier (1992). 

The Rossby-Ertel potential vorticity on an isopycnal surface is given as:

PV = (f + ζ ) (∂ρ/∂z)/ρ ⏐ρ = const (2.3-12)

Where the relative vorticity, ζ = ∂v/∂x - ∂u/∂y, and potential density, ρ, are defined on an

isopycnal surface. For horizontal scales greater than the local Rossby radius, ζ << f and the PV 

simplifies to: 

PV = f (∂ρ/∂z)/ρ, (2.3-13)

which is usually referred to as the Sverdrup PV and depends primarily on the vertical distance 

between two isopycnal surfaces. For mesoscale flows in the Gulf of Mexico, |ζ| ≈ 0.3f, and 

though the Sverdrup term is dominant, the relative vorticity is an important component. 

Therefore, for the calculation of the Ertel PV, using (2.3-12), ζ is estimated from the geostrophic

velocity components: 

(U, V) = 1/(ρ0 f ) ( -∂P/∂y, ∂P/∂x ) (2.3-14) 

where the horizontal pressure differences are calculated from dynamic height. The horizontal 

dynamic height finite differences are readily calculated on sloping constant isopycnal surfaces 

with the assumption that the potential density surfaces are parallel over the scale of the 

difference estimation (usually a few kilometers). This is a good assumption in practice. 

The three-dimensional velocity field can be separated into geostrophic and ageostrophic

components. Thus, 

(u, v, w) = (U, V, 0) + (ua, va, w), (2.3-15) 

where |ua| << |U|. The equations of motion become

Dt U – f (V + va) = -1/ρ0 ∂P/∂x (2.3-16a)

Dt V + f (U + ua) = -1/ρ0 ∂P/∂y (2.3-16b)

2-24
and the conservation of mass as 

Dt ρ + w ∂ρ/∂z = 0, (2.3-17) 

which can be rewritten using the hydrostatic approximation as 

N2 w = -1/ρ0 Dt (∂P/∂z). (2.3-18)

The acceleration term, Dt , is approximated by using the geostrophic velocities (U, V), so that 

Dt = ∂/∂t + U∂/∂x + V∂/∂y . 

Note that using (2.3-15) and (2.3-16), the ageostrophic velocity components are defined as

proportional to the geostrophic acceleration terms as 

Dt U = f va ; Dt V = -f ua . 

Using an f-plane assumption, differentiating (2.3-18) with respect to x and y and eliminating P 

from (2.3-16) generates equations for (ua, va) in terms of w, U and V. The ageostrophic 

velocities are eliminated through continuity so that the omega or Q-vector equation becomes 

f

2 ∂2

zw + ∇2

 (N2 w) = ∇ . Q, (2.3-19) 

where 

Q = [ 2f (Vx Uz + Vy Vz), -2f (Ux Uz + Uy Vz ) ]. (2.3-20) 

For a horizontal surface, the geostrophic shear terms can be rewritten as horizontal density 

gradients (Wang and Ikeda, 1997), so that 

Q = [ 2g/ρ0 (Vx ρy + Ux ρx ), 2g/ρ0 (Vy ρy + Uy ρx ) ]. (2.3-21) 

Thus, Q can be derived from coincident interpolated dynamic height, and σt or σθ fields at any 

given depth level using (2.3-21). Equation (2.3-19) states that given suitable boundary 

conditions, the vertical velocity field can be diagnosed from the geostrophic velocity field.

However, the vector Q and its divergence or convergence can be used directly to infer where upand down-welling velocities, respectively, are likely to occur, without directly solving (2.3-19). 

2.3.3 SVD Analysis 

As one of the contributions to other studies, this project provided the current data from the

DeSoto Canyon Eddy Intrusion Study (see Table 2.2-1 and Figure 2.2-1), as well as model runs, 

to a project that investigated how well the POM, with assimilation, could reproduce the dominant

flow patterns in the DeSoto Canyon. The results were reported in Wang et al. (2003). The

statistical technique used by Wang et al. (2003) to see if simulated and observed flow patterns 

had similar characteristics is called Singular Value Decomposition (SVD) analysis, which was 

introduced by climate studies, where similar problems of comparing model output with

2-25
observations where the flow fields are not fully deterministic arise (Bretherton et al. 1992; Lau 

and Weng, 2001; Robertson et al., 2000). The approach explores the possibility that model 

results may contain comparable spatial and temporal patterns with the observed currents, even if 

local (point-wise) correlations between the model and observations are low. The approach is 

slightly different from using Composite Principal Component Analysis (CPCA), which applies

EOF analysis to combined spatial series from the model and observations, and Bretherton et al. 

(1992) have shown that SVD gives a better representation of the significant coupled patterns than 

CPCA (see also Wallace et al., 1992). Since this is an important method that should be 

increasingly applied to the comparison of models with spatially dense observations, where nondeterministic flows such as eddies are predominant, the basics of the SVD analysis are repeated

here from Wang et al. (2003). 

Both model and observational currents are mapped to coincident grids. In Wang et al. (2003), 

the mapping was performed using the Universal Co-Kriging (UCK) method of Pedder (1989) 

because daily maps of geopotential were required for comparison with SSH anomalies from 

altimeter data. Following Bretherton et al. (1992), the spatial vector time series s(t) and z(t), can 

be expanded in terms of a set of patterns (NS and NZ are the vector dimensions in s(t) and z(t)): 

∑=

← =

NS

k

k pk s t s t a t

1

( ) ( ) ~ ( ) (2.3-22a)

∑=

← =

NZ

k

k qk z t z t b t

1

( ) ( ) ~ ( ) (2.3-22b)

The time series and are called the left and right expansion coefficients and the vectors

are and the corresponding spatial patterns. It can be shown that if and are the k-th left

and right singular vectors of the covariance matrix , the covariance between the k-th left and 

right expansion coefficients, , is equal to the corresponding k-th singular value. The

SVD expansion can be ordered such that the first mode corresponds to the largest singular value, 

and so on. The percentage of the covariance explained by the k-mode, the squared covariance 

fraction (SCF), is, 

a (t) k b (t) k

pk qk pk qk

Csz

< akbk >

2

2

C

a b SCF k k

k

< > = (2.3-23)

where the squared Frobenius matrix norm is the total amount of squared covariance summed

over all entries in C: 

∑∑= =

=

NS

i

NZ

j

C Cij

1 1

2 2

. (2.3-24)

Obviously, SCF1 > SCF2, and the first mode extracts most of the spatial covariance (as defined 

by the Frobenius matrix norm). 

2-26
3. Topographic Rossby Waves 

This chapter is divided into two sections: the first concerns model investigations of TRW 

characteristics and dynamics that are based on Oey and Lee (2002), and the second is about the 

ability of the model to simulate the observed characteristics of the waves in various parts of the 

deep Gulf. The latter uses the deep current measurements listed in section 2.2. 

3.1 Model Investigations 

3.1.1 Introduction 

Hamilton (1990) first gave observational evidence of TRW’s in the Gulf of Mexico (GOM) 

through analysis of long-term current measurements below 1,000 m depth made in the late 

1980's. Based on deep moorings over the slope and rise around the Gulf, east from Florida to 

northern and western Gulf, he found energy spectral peaks at 25, 45 and 100 days, within the 

range of periods expected for TRW’s. The motions were characterized as columnar, with

amplitudes decreasing with increasing height off the seabed. The waves are transverse and

rectilinear such that the principal major axis is at an angle to the general trend of the isobaths.

More detailed analyses showed that these waves have wavelengths of about 110-300 km, phase

velocities that were offshore and energy propagation that were westward with speeds of 

approximately 9 km/day. More recent measurements (Hamilton and Lugo-Fernandez, 2001) 

made at the base of the slope (2,000 m), just south of the Mississippi delta at 90°W, showed high 

amplitude (~ 50 cm/s), short period (~10-12 days) period wave trains regularly passing through 

the site (Hamilton et al., 2003). These waves seem to originate on the west side of an extended 

LC and appear to be refracted back into deep water by the steep Sigsbee escarpment, since they

are not observed further west in the basin. These short period waves are not likely to be resolved

by the model grid and most of the attention in this section is on waves with periods of 20 to 100 

days. 

The Loop Current (LC), with speeds that can exceed 2 m/s near the surface and O (10cm/s) at 

1,000 m depths, constitutes a major forcing to flow fluctuations in the Gulf of Mexico. On time

scales from weeks to years, the LC can extrude as far northward as the Alabama-Mississippi 

continental rise near the foot of the De Soto Canyon, and retract southward to near the Yucatan 

Channel (Vukovich et al., 1979). This LC pulsation is often accompanied, at periods that range 

from 6 to 20 months, by shedding of Loop Current Eddies (LCE) - energetic warm-core rings

(similar to those found in the Gulf Stream system) with diameters which typically range from 

200 to 300 km and near-surface swirling speeds that exceed 1.5 m/s (Kirwan et al., 1988; 

Forristall et al., 1992; Hamilton et al., 1999). In addition, through some kind of dynamical

instability, or otherwise a cascade mechanism that is not yet fully understood, smaller-scale 

eddies (cyclones in particular) and frontal meanders are believed to be generated from the largerscale LC and LCEs (Paluszkiewicz et al., 1983; Hamilton, 1990, 1992; Hamilton et al., 2000). 

Given their ubiquitous signatures, it seems logical to hypothesize that, through some

mechanisms, the Loop Current and LCE’s force deep-flow eddy kinetic energy (EKE) and 

TRW’s in the Gulf. While field evidence is hard to come by, (numerical) models of the Gulf, 

forced by time-independent forcing, suggest that this might in fact be the case (e.g. Oey, 1996).

3-1
Motivated by Hamilton’s (1990) work, Oey attempted to identify TRW’s from his calculation of

the LC and LCEs in the Gulf. Similar to what Hamilton found, the calculations yielded columnar 

motions for depths deeper than 1,500m at locations over the slope and rise around the Gulf,

along-isobath motions that intensified near the bottom, spectral peaks within the 20 to 100 day 

periods, and westward energy propagation with speeds of about 12 km/day. Because these 

model runs used steady forcing, these findings established a direct link between LC and LCEinduced variability and bottom EKE, though they fell short of establishing what that variability 

might be, and of proving that the deep motions were indeed TRW’s. Moreover, it is of interest

to study, if TRW’s do exist, what their paths are in the Gulf and whether or not (and how) they

can account for EKE recorded (in the model and/or field experiments) at locations that seem 

distant from direct LC and LCE influences. These issues are addressed using an improved 

version of the primitive-equation (PE) model used by Oey (see section 2.1). It will be shown

that, in a model forced by steady transport from the Caribbean Sea and Atlantic Ocean, the deepflow EKE over certain parts of the slope and rise in the Gulf are TRW’s, that these TRW’s 

originate from regions beneath the LC and from locations coincident with the paths of

propagating LCE’s, and that the link between the energy of the large-scale, i.e. those of the LC

and LCE’s, and the near-bottom energy is surface high-frequency, short-wavelength disturbances 

produced around the LC and LCE’s. 

3.1.2 Setup of the Model for Deep Flow Experiments 

The model, described in section 2.1, is initialized with an ocean at rest, and run in diagnostic 

mode for one month, during which the density field (ρ) is fixed at its annual-mean distribution. 

A (predominantly) geostrophically-adjusted velocity field is established during this time. The

prognostic calculation is then initiated from this balanced field and a quasi-equilibrium state, in 

which regular, nearly-periodic LCE shedding occurred, is established in about 2 years. The 

integration is then continued through the tenth year. Figure 3.1-1a gives snapshot examples 

(every 90 days) of contours of ζ/f (relative vorticity divided by local Coriolis parameter) at 

z=−50 m, showing LCE shedding and westward propagation. Figure 3.1-1b plots time/latitude 

contours of free-surface elevation η along 90o

W, showing passages of LCEs when η maximizes

to ~ 0.3 m. Various characteristics of the LC and LCEs are as follows. The shedding period is 

very nearly constant at 10 months (Figure 3.1-1b) and LCE diameters are about 300 km. Once

shed, modeled eddies traverse across the Gulf in a southwestward direction at speeds of

approximately 4 to 5 km day-1, and decay eventually in the southwestern corner of the Gulf. In 

this constant-inflow experiment, the basic shedding mechanics and eddy kinematics follow 

closely those described in Hurlburt and Thompson (1980) (c.f. Sturges et al., 1993; Oey, 1996). 

Typical swirl speeds and ς/f around an eddy are 1.2 m s-1 and −0.4, respectively, while the

corresponding values at the western edge of the LC in the Yucatan Channel are 1.5 m s-1 and 0.7.

The maximum swirl speeds are weaker than those typically observed, about 1.5 to 2 m s-1

(Kirwan et al., 1988; Forristall et al., 1992). The resulting weaker forcing will likely result also

in a weaker deep response. This should not however, seriously jeopardize our attempt to 

diagnose and study TRW’s, which basically are linear waves. On the other hand, the maximum 

modeled swirl speeds represent improvements over those found in Oey (1996), which gives 

3-2
Figure 3.1-1a

Ninety-day snapshots of relative vorticity (ζ/f; 5-day averaged) at z=-50 m 

showing LCE shedding, southwestward propagation, and decay.

3-3
Figure 3.1-1b

Time/Latitude contours of free-surface elevation η along 90°W, showing passages 

of Loop Current Eddies when η maximizes to ~ 0.3 m. Regions where η > 0 are 

shaded.

3-4

24

26

28 (Latitude)

400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500

(day)

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values of 0.76 m s-1 (ς/f ≈ -0.25). Because the forcings are similar in the two calculations, the 

improvements are a result of increased grid resolution (doubled) in the present case (c.f. Oey, 

1998). 

While the near-periodic shedding and constancy of direction of LCE propagation are idealized

settings not found in the real ocean, they represent dynamically consistent forcing for (deep) 

flow EKE in the Gulf. By understanding how TRW’s develop in this simplified system, we hope

to develop hypotheses and ideas for future analyses of more realistic models, and of observations 

as well. 

3.1.3 Model Data Processing 

For the purpose of identifying TRW’s in the PE model, it is useful to divide the model’s results

into four isopycnal layers, with layer 1 from surface to 27σt (~300 m thick), layer 2 from 27σt to 

27.5σt (~ 500 m thick), layer 3 from 27.5σt to 27.7σt (~ 500 m thick), and layer 4 from 27.7σt to 

bottom (thickness ≈ 1,000 m or more). Figure 3.1-2 shows an example of this division at 90o

W. 

Because TRW motions are columnar (vertically coherent) at depths below about 1,000 to 1,500 

m, EKE is examined in the fourth layer, i.e. below the 27.7σt surface (note that this lies

approximately 1,000 to 1,500 m below the free surface). The modeled currents are daily means, 

which are then depth-averaged in each layer. The depth averaging in the fourth layer precludes 

possible biases of high EKE that might occur in some region if a fixed z-level were used to 

search for TRW’s. It has the added nicety of also eliminating any grid-point noise. On the other 

hand, the depth-averaging reduces by at least 50% the values of the near-bottom kinetic energy. 

The last 7 years of the 10-year run were then spectrally analyzed, and the results in the 20 to 100

day, TRW band examined. 

3.1.4 Deep Flow Eddy Kinetic Energy 

The model regions, where TRW’s are active in the Gulf, are determined in this section. To do

this we first search for areas where a significant (60%) part of the total energy falls in the TRW

period range of 20-100 days, where the deep energy is above a certain threshold, and where 

bottom intensification exists. These regions are then checked so that the topographic slopes and 

stratification are such that the 20 to 100-day TRW’s can be supported. The spatial correlations 

are then calculated so as to examine, in an ad hoc way, wave propagation in these regions of

significant deep energy. The TRW dispersion equation (2.3-1) is utilized and the ray equations 

(2.3-4) and (2.3-5) (or (2.3-8) and (2.3-9) when deep mean currents are included) are integrated

to show that rays originate in the LC and LCE’s, and are confined in these significant deep 

energy regions. 

Figure 3.1-3 compares the time series of lower-layer (i.e. layer 4) kinetic energy fluctuations 

(LOKE) at a station just west of the LC (88°W, 25.8°N) with the corresponding 20-100 day 

band-passed LOKE (henceforth referred to as LOKE 20-100d). At this station the amplitude and

phase of the band-passed series generally follow those of the total series. The ratio of their 

standard deviations, LOKE 20-100d/LOKE (note that here the same notation ‘LOKE’ is used for 

standard deviation) ≈ 0.68. Observations of deep currents show that, where TRW’s are observed

3-5
Figure 3.1-2.

A time-averaged cross-sectional contour plot of density at 90°W that describes 

isopycnal division of model’s three-dimensional field into four layers: layer-1 

from surface to 27σt

, layer-2 from 27σt to 27.5σt, layer-3 from 27.5σt to 27.7σt, 

and layer-4 from 27.7σt

 to bottom.

3-6

Average density profile along 90°W
0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

(day)

ure 3.1-3A comparison of the time series of lower-layer (layer #4) kinetic energy (LOKE; thin curve) at a station just west of the 

LC (88°W, 25.8°N) with the corresponding 20 to 100-day band-passed LOKE (LOKE20-100d, thick curve).
over the continental slope of the Gulf, they generally account for some 95% of the total nearbottom EKE. In the present case in which the energy is averaged over the lower layer (thickness

~ 1,000 to 2,000 m assumed > TRW trapping scales; see below), the appropriate percent-value, 

assuming TRW exponential decay with height above the bottom (from a maximum of 95% at

bottom), is ≈ (1 − e-1)×0.95 ≈ 60%. Thus Figure 3.1-3 suggests that, at least at this location, 

TRW’s may exist. Therefore, only those regions where LOKE 20-100d/LOKE > 60% are

considered. To avoid regions where the ratio is artificially inflated, a second, minimum-LOKE

constraint is imposed such that the local LOKE exceeds 10-3 m2

 s-2, the Gulfwide-averaged 

standard deviation of LOKE. A third constraint is also imposed such that only those regions 

where the kinetic energy shows intensification near the bottom (Hamilton, 1990) are considered. 

These constraints are probably over-restrictive in that they will likely eliminate potential sites

(for TRW activities) where both TRW and strong locally-forced, non-TRW components of the 

EKE coexist, under the LC for example. Thus, while the minimum-LOKE constraint is satisfied,

the ratio and bottom-intensification requirements may not be. On the other hand, the constraints 

ensure that what survive are robust features of the model that may have better chance of being 

observed in the real ocean. 

Figure 3.1-4 gives LOKE 20-100d after the above three constraints are imposed. It shows a 

significant LOKE 20-100d region across the Gulf over approximately the 3,000-m isobath. This 

along-3,000 m-isobath band of LOKE 20-100d will be refered to it as the “CGKE,” or “central-Gulf 

LOKE 20-100d” band, and will be the focus of the investigations. While there are other significant

LOKE 20-100d regions in the model Gulf (e.g. 88°W, 28°N; Figure 3.1-4), the central-Gulf band 

suggests a simpler cause-and-effect scenario: i.e. forcing under the LC and southwestward 

propagating LCE’s, and near-bottom energy that spreads westward. It is noted that the CGKE 

band not only indicates regions where LOKE 20-100d/LOKE > 60%, it also coincides well with 

areas where bottom intensification exists in the model Gulf. In other words, the extent and shape 

of the band in Figure 3.1-4 are essentially unchanged if only the bottom-intensification and 

minimum-LOKE constraints are imposed. 

The CGKE band is now checked that it resides in a region where TRW’s can be supported. 

Equation (2.3-2) is used to plot contours of the minimum period P=2π/σ≤tanh(NhK/f)/(N|∇h|). 

Figure 3.1-5 shows P for 2π/K=110 km, and using N(x,y) derived from the model as explained 

below. Except for the relatively gentle topography of the deep central Gulf where only longperiod (>100 days) TRW’s can be supported, waves of shorter periods can in theory exist almost 

anywhere in the Gulf. It is therefore, not immediately apparent why LOKE 20-100d is confined

only along the narrow band over approximately the 3,000-m isobath in Figure 3.1-4. In other 

words, why is there not a more expansive north/south spread of energy that exhibits TRW 

characteristics to other regions of the Gulf? The answer lies in the way that TRW energy is

refracted. In the following, two methods are used to attempt the linking of the CGKE band with 

TRW’s: firstly a somewhat ad-hoc phase and correlation analysis, and secondly a more precise 

ray (energy) tracing calculation. Based on these calculations, possible sources of these nearbottom EKE are discussed. 

3-8
Figure 3.1-4.

The lower-layer kinetic energy in the 20 to 100-day periods (LOKE 20-100d). 

Regions where the ratio of LOKE 20-100d to total LOKE is less than 60%, where 

there is no bottom intensification, and where LOKE does not exceed the Gulfwide 

average of 10-3 m2 s-2 are omitted as discussed in the text.

95 W 90 W 85 W

20 N

25 N

30 N

1000

1000

2000

2000

2000

3000

3000

3000

0.0 0.4 0.8 1.2 1.6 2.0 2.4

EKE(20d-100d)(σo:27.7-Bot.)

20 N

30 N

(10-3 m2 s-2)

20 N

30 N

3-9
Figure 3.1-5

Contours of the minimum period (in days) P=2π/σ ≤ tanh(NhK/f)/(N|∇h|) allowed 

by the TRW dispersion relation, for N given in Figure 2.3-2a and 2p/K=110 km. 

Regions that cannot support TRWs with periods shorter than 100 days are shaded.

95

W

90

W

85

W

20°N

25

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3-10
Phase and Correlation Analysis

Time-lagged correlations were computed of the LOKE 20−100d time series at 88°W, 25.8°N (e.g. 

Figure 3.1-3) with all other grid points, chosen because it is near the generation site of TRW’s 

(see below). The top panel of Figure 3.1-6 shows contours of the lagged correlation (which

differ from zero at the 95% confidence level) in the CGKE region, from 86°W to 92°W, and the 

lower panel, the lags in days. These calculations show that the CGKE band coincides with region 

of significant correlation, with fair values = 0.45 at approximately 92.2°W, 25.8°N, a distance 

some 400 km west of the ‘X’ point (the 95% significance level is ≈ 0.15 at this western point). 

In the vicinity of the ‘X’ point, the time-lag contours suggest a phase propagation from northeast 

to southwest. Strictly speaking, time-lag contours and phase lines are equivalent only for 

monochromatic waves, however, TRW spectra are often dominated by a single peak (Hamilton,

1990). Since the isobaths are approximately east/west in this region, the southwestward phase 

propagation is consistent with a southwest-directed or down-slope wavenumber vector, hence 

northwest-directed or up-slope group velocity that one would deduce from the TRW dispersion 

relation (see Section 2.3 and Figure 2.3-1). 

Ray-Tracing Calculations

The approximate coincidence of the region of significant correlation with the CGKE band 

suggests a ‘channeling’ effect of near-bottom energy over the 3,000-m isobath, perhaps related to 

refraction of TRW’s. To study this, wave rays were traced using the QG dispersion relation 

(Section 2.3). In addition to yielding information on various wave properties along the paths, the

calculations also helped locate sources of TRW’s. In the Gulf Stream region, the ray-tracing

method has been used by Pickart (1995) to identify TRW’s and TRW sources. 

Initial Positions and Wavenumber Vectors of the Rays

To integrate equations (2.3-4) and (2.3-5) (or (2.3-8) and (2.3-9) if mean flows are included), 

initial position (xo, yo) and wavenumber vector (ko, lo), as well as the frequency σ defined for 

each ray, must be assigned. By integrating the equations backward (inverse ray-tracing; Pickart,

1995) from positions in the CGKE band, test calculations with various (ko, lo) (at period = 64 

days, −0.125 < lo < −0.03 km-1 and |ko| << |lo|, where here the direction of lo is opposite to that of 

the local ∇h; thus hodograph ellipses are predominantly aligned with the local isobaths,

Hamilton (1990)) satisfying the dispersion equation indicate that rays consistently trace back to 

locations under the LC and around the northern edges of the southwestward propagating LCE’s. 

These tests suggest that LC and LCE variability are the prime driver of the lower-layer 

fluctuations found in the PE model. With these tests serving as a rough guidance, six initial 

positions, (xo, yo), shown as ‘+’ in Figure 3.1-7a, are chosen, and the ray equations are then 

integrated forward with a set of (ko, lo) determined from the PE model outputs as follows. For

convenience, the stations and corresponding rays will be referred to consecutively from east to 

west as station/ray #1 through station/ray #6. 

In the vicinity of each initial position, energy spectra at a cluster of ‘n’ stations (n=10, including 

(xo, yo) was used), hence their phases at each frequency, are computed. Any three stations then 

give at least two linearly independent equations for the two unknowns k and l. The set of k and l

3-11
ure 3.1-6.Top panel: contours of maximum lagged correlation (at 95% significance level) in the vicinity of the 3000 m isobath 

(CI=500 m), CGKE band (see text), from 86°W to 92°W. Lower panel: the corresponding lags in days.
Figure 3.1-7a.

Top panel: wave rays (in blue; lighter blue for wavelength < 30 km) traced using 

the TRW dispersion relation indicating deep energy paths in the Gulf that 

originate under the Loop Current and Loop Current Eddies. The outer rims of LC 

and LCEs are indicated in red by the 10-year assemblage of the ζ/f=-0.2 contours 

at z=-50 m obtained from the PE model. The green arrows indicate wavenumber 

vectors, plotted every 10 days along the path, with lengths equal to wavelengths 

(the 100 km scale is shown below the panel). Bottom panel: the 20 to100-day 

deep energy from Figure 3.1-4. Superimposed is a comparison of rays from the 

top panel (thin solid) with those derived by including effects of deep mean flow 

(thick green curves; using equations (2.3-8) and (2.3-9)).

3-13
Figure 3.1-7b.

Various TRW and ambient properties along ray #2 of Figure 3.1-7a: Top to bottom 

left-side panels: 1) east/west (solid) and north/south (dashed) group velocity 

components, and corresponding speed (dash-dot); 2) Brunt-Väisälä frequency; 3) 

topographic gradient (solid; left y-axis) and ratio of topographic to planetary beta, 

βTopo/β = f|∇h|/(hβ) (dash; right y-axis); and 4) wavenumber vector (k,l) 

variation. Top to bottom right-side panels: 1) wavelength; 2) water depth (solid; 

left y-axis), trapping scale (dash; right y-axis), and estimates from the numerical 

model results at five locations along ray #2 (full circles; see Figure 3.1-9); 3) 

lower-layer EKE in the 20 to 100-day periods; and 4) distance along the ray as a 

function of longitude. The dotted-curves show respective properties 

corresponding to the ray that includes effects of deep mean currents (group speed 

in the top-left panel).

Cgx(solid), Cgy(dashed) .

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Wavelength.

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100 (km))

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Depth .

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KE(20-100d) .

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(X 10-3 m 2 s-2)

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k

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-96 -95 -94 -93 -92 -91 -90 -89 -88 -87

Longitude

3-14
thus solved from all (non-redundant) triad permutations are then averaged. Figure 3.1-8 shows

an example, at Station 2 (see Figure 3.1-7a), of four such solutions plotted on the dispersion

curves for four frequencies. The frequencies were chosen to correspond to spectral peaks at this 

and its surrounding nine stations, at periods ≈ 32, 42.7, 51.2 and 64 days (c.f. Oey, 1996). The 

results at other stations are similar. The wavelength values range from 63 to 210 km and they are 

averaged for each period. Figure 3.1-8 shows that the averaged solution (ko, lo) pairs correspond 

to wavelengths 2π/K ≈ 85 to 150 km for the four selected periods. Moreover, they reside within 

the linear portion of the dispersion curves, i.e., they approximately satisfy equation (2.3-2) with 

β≈0 and when tanh(NhK/f)≈constant. Thus, for each frequency, only the ratio ko/lo (i.e. the angle

θ) is relevant. While group speeds increase with increasing wavelengths (equation (2.3-3)), raypaths are relatively insensitive to wavelength (see below). In the following, 2π/K = 110 km is 

used for the ‘benchmark’ calculation so that equation (2.3-2) then yields a first guess value of 

either ko or lo, and is used in an iterative process to solve (2.3-1) for more precise values of ko

and lo. Hamilton (1990) estimates from observations (his Table 2) wavelengths from 110 km to 

300 km for TRW periods from 18 to 300 days. Thus the wavelength values estimated from the

PE model are on the lower end of the observed. However, more recent observations (Hamilton 

et al., 2003) with more closely spaced current meter arrays indicate wavelengths of ≈ 60 to 150 

km for shorter period TRW’s. The sensitivity of the ray solution to larger wavelengths is

evaluated below. 

Results and Interpretations

In Figure 3.1-7a, an example of six rays emanating from initial stations marked ‘+’ is shown 

superimposed on ten-year assemblage contours of ζ/f=−0.2 at z=−50 m. This value of ζ/f is used

as it indicates well the outer rim of the LC and LCE’s. The contours show the predominantly 

southwestward propagation of LCE’s from their initial birth-place in the LC. Each ray is

integrated for 100 days, using a period of 64 days, and on each one, the wavenumber vector

direction, with length proportional to its wavelength, is plotted at 10-day intervals. Rays at 

periods 32, 42.7 and 51.2 days (Figure 3.1-8) have also been tested and the results will be 

discussed later. 

Rays 1 and 2 originate from under the region of active LC north-south pulsation and LCE 

shedding. At station #1, the topographic slope is weak, and βTopo/β = f|∇h|/(hβ) < 1. Waves that 

emanate from here are at first predominantly planetary, and propagate slowly (~ −3 km/day). It 

later converges with ray #2, which is detailed next. 

At station #2, βTopo/β ≈ 6, and ray #2 follows closely the 3,000-m isobath, crosses over it, but

remains just inshore of it as the ray continues to the western Gulf. Its path, together with that of

ray #1, coincides closely with areas of significant LOKE 20-100d, from about 87°W through 

approximately 92°W (Figure 3.1-7a, lower panel). West of 92o

W, the wavelength shortens to 

values < 30 km, which is too short to be resolved by the PE model (this portion of the ray is of

lighter shade in Figure 3.1-7a, top panel). The lower panel of Figure 3.1-7a (see also Figure 3.1-

4) also shows areas along the 3,000 m-isobath where LOKE 20-100d attains local maxima near 

station #2 or the ‘source,’ and near 91°W, 94°W and 95°W where rays converge. The ray 

equations can be used to explain the ‘channeling’ effect (of LOKE 20-100d) over the 3,000 misobath, mentioned previously, and also why TRW wavelengths shorten westward. Specifically, 

3-15
Figure 3.1-8.

TRW dispersion curves corresponding to peak spectral periods of 32 days (leftmost solid and dotted curves), 42.7 days, 51.2 days and 64 days. Solid and dotted 

curves are for N=10-3 s-1 and 6.25×10-4 s-1, respectively. The symbols denote 

averaged values derived from the numerical model results at station #2 of Figure 

3.1-7a: filled circle for 32 days, open circle for 42.7 days, filled square for 51.2 

days and an open square for 64 days 

-0.10

-0.08

-0.06

-0.04

-0.02

-0.00 l(1/km)

-0.08 -0.06 -0.04 -0.02 0.00 0.02

k(1/km)

3-16
it can be explained why the ray must bend from propagating northwestward to southwestward at 

about 90o

W, and why further west it stays inshore of the 3,000-m isobath, and does not veer 

northward to cross the 2,000 m isobath. Since βTopo/β remains >> 1 (see Figure 3.1-7b and 

discussion of it below), equation (2.3-2) is an excellent approximation to the dispersion relation. 

Thus, 

σ = N|∇h|sin(θ)/tanh(NhK/|f|) ≈ N|∇h|sin(θ), (3.1-1) 

where θ is the clockwise angle the wavenumber vector makes with the direction of steepest 

topographic decent (i.e. with ∇h/|∇h|; Figure 2.3-1). Also for NhK/|f| ≈ O(1) or larger, the ‘tanh’ 

is a weak function of ‘NhK/|f|’, and the first form of (3.1-1) is good with ‘tanh’ ≈ constant. Since 

σ is constant along a ray, sin(θ), and hence θ (which remains in the first quadrant, i.e. θ < 90o), 

must decrease as the ray enters regions of steeper topographic slopes and/or stronger

stratification. In other words, the wavenumber vector must become more perpendicular to, and

the ray path (which points in the direction of the group velocity; see discussion in Section 2.3)) 

more aligned with, the local isobaths. Figure 2.3-2b shows that the topographic slope becomes 

steep between the 2,000 m and 3,000-m isobaths west of 90o

W. This, coupled also with 

increasing (though slight) stratification to the north (Figure 2.3-2a), explains why the ray must

turn from northwestward to southwestward around 90o

W, and must be confined between the 

2,000 and 3,000-m isobaths further west. The agreement of this ‘channeling’ or ‘focusing’ effect 

of the ray path and the CGKE band lends support to the idea that the latter is a manifestation of 

TRW’s contained in the PE model. 

It can be explained also how wavelengths are shortened by refraction as TRW’s propagate 

northwestward into regions of increasing topographic slope and stratification. Multiply (2.3-5) 

by ki and use (3.1-1): 

d(ki

2

/2)/dt = − l sin(θ) ∂(|hy|N)/∂y, (3.1-2) 

in which it is assumed for simplicity that contours of h and N are predominantly zonal. It is clear 

from Figures 2.3-2 that both ∂|hy|/∂y and ∂N/∂y are positive inshore of the 3,000-m isobath and 

since l <0, d(ki

2

/2)/dt >0 and TRW wavelength shortens along the ray. Hamilton (1990) also

noted westward shortening of TRW’s from observations. 

Figure 3.1-7b gives various properties along the ray path #2. The magnitude of the group 

velocity (dotted curve in first left panel) varies from about 8 km/day at the initial location

(station #2) to over 10 km/day as the ray turns southward along the Mexican/Texas slope, with 

an average of about 9 km day-1. These values of the group speed are consistent with those 

reported by Hamilton (1990) and Oey (1996). The x-component (Cgx) is negative through the 

entire ray-path as wave energy propagates westward, while the y-component (Cgy) attains both 

positive and negative values as the ray undulates in accordance with the location of steep

topographic gradient as explained above. This undulation is also reflected by the changing sign 

of the x-component of the wavenumber vector k, shown on the bottom left panel, since this is 

very nearly perpendicular to Cgy. The TRW is continuously being refracted along the ray-path 

and its wavelength shortens from an initial value of 110 km to 30 km (first right panel). In 

addition to Gulf processes (Hamilton 1990), Pickart (1995) also found wave-shortening over the

3-17
slope off Cape Hatteras. As noted above, topographic slope steepens (Figure 3.1-7b: third left 

panel – solid line) and topographic beta dominates on ray #2 as βTopo/β increases to over 40 near 

the western Gulf (third left panel – dashed line). The solid curve in the second right panel

indicates decreasing water depth along the ray, while the dotted curve shows the vertical trapping 

scale μ

-1 of the wave. Along the ray, this is very nearly proportional to the wavelength, as can be 

seen by comparing the first and second right panels. This follows directly from the linear theory

for which the vertical structure ~ cosh(μz) ≈ cosh(NKz/f) from (2.3-1a). The values vary from

about 1,000 m at the beginning of the ray to 300 m some 300 km to the west. The trapping 

scales from the PE model profiles may be estimated in Figure 3.1-9. Here the vertical profiles of

the square root of the 20 to 100-day kinetic energy at five locations from east (‘A’) to west (‘E’) 

are plotted along ray #2. Excluding the very near-bottom boundary layer (<50 m), where the 

energy is low, the profiles all show energy decay with height from the bottom, to a minimum that 

probably represents the combined effect of the upper-ocean energy decay (i.e. the first baroclinic 

mode) and TRW decay from below. Apart from this complication, one may assume an 

exponential decay from each profile’s maximum to its minimum, and estimate the corresponding

e-folding decay height. The solid-circle symbols in the second right panel of Figure 3.1-7b show 

the five estimated values. Though values differ, the PE model results also show westward 

decreasing trend of the trapping scales (from 600 to 300 m) in agreement with the TRW solution. 

Finally, the third right panel shows LOKE 20-100d, the accumulated energy of not only those 

TRW’s that originate at station #2 and propagate along ray #2 at the 64-day period, but also 

those of other rays. It illustrates energy increases due to ray-focusing at the previously

mentioned locations: 91°W, 94°W and 95°W. It is noted that these local maxima in LOKE 20-100d

coincide with minima in wavelength (first right panel), a feature consistent with the linear wave 

theory, where energy varies as the square of the wavenumber. 

To get a sense of the bottom motions, Figure 3.1-10 shows examples of vector stick plots at 

locations ‘A’ and ‘D’ of Figure 3.1-9a, upward from the near-bottom depth level where the 

LOKE 20-100d shows maximum (Figure 3.1-9b). The top panel shows vectors at a representative

near-surface level. The up- or y-directed sticks in these plots denote motions parallel to local 

isobath, positive clockwise around the Gulf (i.e. generally eastward). Both locations show 

columnar (i.e. in-phase) structures that decay with height in the lower 1,000 m or so of the water 

column, and fluctuations in the 20 to 100-day periods can be seen. At location ‘A’, the motion 

appears to be uncorrelated with the near surface. Therefore, their relation, if any, cannot be seen 

in this plot of the whole motion (see, however, below), because LC and LCE fluctuations 

dominate the near-surface motions. At location ‘D’, away from the direct influence of the LC 

and LCE’s, the near-bottom and near-surface motions are more correlated, yet the former over a 

height of about 1,000 m from the bottom is evidently more topographically controlled, i.e. more 

aligned with isobaths. Note that in the mean, near-surface currents at these and other locations

are anticyclonic, while near-bottom currents are cyclonic around the Gulf (see Figure 2.3-3). To 

assess the effects of deep currents on ray properties, section 2.3 extends the ray equations (2.3-4) 

and (2.3-5) to include the ambient mean currents of Figure 2.3-3. One important effect is to

change the velocity of the ray, equation (2.3-8), which in general increases the westward

propagation speed. The green curves in the lower panel of Figure 3.1-7a show that with deep 

currents, rays #2 and #3 have looped around the western portion of the model Gulf. The 

averaged group speed increases from 9 to 12 km day-1 (Figure 3.1-7b, top left panel). The latter 

value agrees with Oey (1996), who also found (from correlation and time-lag 

3-18
Figure 3.1-9.

(a) Five locations along ray #2 (see Figure 3.1-7a) where vertical profiles of the 

square root of kinetic energy in the 20 to 100-day band are plotted (b).

(a)

(b)

1000

2000

3000

3000

E

95W 90W 85W

24N

26N

28N

D

95W 90W 85W

24N

26N

28N

C

95W 90W 85W

24N

26N

28N

B

95W 90W 85W

24N

26N

28N

A

95W 90W 85W

24N

26N

28N

95°W 90°W 85°W

24°N

26°N

28°N

3-19
Figure 3.1-10.

Vector stick plots at location ‘A’ (a) and 'D' (b) of Figure 3.1-9a, upward from the 

near-bottom depth level where the LOKE 20-100d shows a maximum (bottom 

panel). The top panel shows vectors at a representative near-surface level. Up or 

y-directed sticks in these plots denote motions parallel to the local isobath, 

positive clockwise around the Gulf (i.e. generally eastward).

(a)

(b)

-10

0

10

(cm/s)

650m

-10

0

10

2150m

-10

0

10

2350m

-10

0

10

2550m

-10

0

10

2750m

-10

0

10

500 1000 1500 2000 2500

Time(day)

2950m

-10

0

10

(cm/s)

350m

-10

0

10

1900m

-10

0

10

2100m
analysis) a value of 12 km day-1. A less obvious effect is that current shears (and divergences) 

also diffract rays (Lighthill, 1978; see below) as can be seen from equation (2.3-9). These tend 

to ‘bend’ rays into the main CGKE region especially in the western Gulf, thus improving the

overlapping of rays with areas of more significant LOKE 20-100d obtained from the numerical 

simulation (Figure 3.1-7a, lower panel; see also Figure 3.1-7b where changes in ray properties of

ray #2 are given). 

Ray #3 originates from a location just outside the direct influence of the LC north/south 

pulsation, but affected by LCE sheddings and passages. Its path is similar to that of ray #2. It 

propagates upslope, and after crossing the 3,000-m isobath, converges with ray #2 westward 

from approximately the 90°W. It is concluded, therefore, that rays #1, 2 and 3 represent waves 

that originate from a region of active LC and LCE fluctuations. The paths traced by these rays 

coincide well with the eastern portion (~ from 86°W to 92°W) of the CGKE band. East of 

approximately 89°W, the band is likely a result of both locally forced (by LE and LCE’s) and 

TRW components. West of 89°W, free TRW’s are likely as the direct influence of LC and 

LCE’s is almost nil. 

Rays # 4, 5 and 6 were chosen to originate from the northern rims of propagating LCE’s. Ray #4 

first propagates northwestward, turns southwestward then westward to remain just offshore of 

the 3,000-m isobath before crossing it at approximately 92°W. Being initially over a relatively 

gently-sloping portion of the central Gulf, ray #4 starts off as a predominantly planetary Rossby 

wave (group velocity ~ −5 km day-1), then quickly becomes TRW with its group speed 

increasing to ~10 km day-1. Rays #5 and 6, and another ray further west (not shown), in part

account for the moderate LOKE 20-100d near the western Gulf (~93 to 94.5°W, 25°N). They exist 

as mixed topographic and planetary Rossby waves as βTopo/β remains below 10 through most of 

their paths. Rays #4, 5 and 6 were also traced backward (from stations 4, 5 and 6). The results 

(not shown) indicate that they do not connect back to the LCE shedding zone in the eastern Gulf. 

This confirms that the energy along these rays originate from propagating LCE’s in the central 

Gulf. 

Depth-profile analysis similar to that conducted for ray #2, was performed for the other rays (c.f. 

Figures 3.1-9 and 3.1-10) and the near-bottom intensification of currents was confirmed. The

wavenumber analysis (c.f. Figure 3.1-8), based on the PE model results, also confirms that the 

(k, l) pairs at locations on ray paths are consistent with the TRW dispersion curves. These, the 

phase and correlation analysis (Figure 3.1-6), and the coincidence of ray paths with the CGKE

band (Figure 3.1-7), strongly support that TRW’s exist in the PE model results. In summary, it is

concluded that two types of TRW’s can account for the CGKE band. The first is a 

predominantly topographic Rossby-wave type that originates under the region of active LC and 

LCE fluctuations in the eastern Gulf, and that tends to be ‘channeled’ between the 2,000 and 

3,000-m isobaths. Though the CGKE band and rays extend into the western Gulf, it is cautioned 

that wave-shortening due to refraction means that the waves are under-resolved (~west of 92-

93°W) with the present (PE) model’s resolution. The second is a mixed topographic and 

planetary Rossby-wave type that originates from under LCE’s that propagate southwestward 

over the deeper waters of the central Gulf. These waves account for LOKE 20-100d found offshore 

of the 3,000-m isobath in the western Gulf. 

3-21
Effects of Wavelength, Stratification, Periods and Deep Currents on Rays

It was mentioned previously that the averaged TRW wavelengths deduced from the simulation 

results are shorter than Hamilton’s (1990) estimates from observations, but more consistent with 

later observations from the base of the slope at 90°W (Hamilton et al., 2003). Hamilton reported 

wavelengths from 110 to 300 km, while estimates from the numerical simulation range from 63 

to 210 km. An average of 110 km was used as the initial condition for ray tracing. The rather

wide range suggests that the WKB assumption of slowly varying (time and space) N (and other 

environmental variables) may not be strictly satisfied in the simulation results. It is also found 

that N varies depending on the depth-range used for averaging. Thus, N ≈ 2.5×10-4 s-1 over the 

CGKE region if only the lowest 500 m is used. Figure 3.1-8 also plots, for each of the four 

periods, the dispersion curves corresponding to N= 6.25×10-4 s-1 (dotted curves), an average of

the benchmark N ≈ 10-3 s-1 (approximately over the 3,000-m isobath) and N = 2.5×10-4 s-1. Thus, 

the estimated (ko, lo) pairs are within the range of uncertainty in N. This uncertainty is also 

reflected in the value used by Hamilton, N=5×10-4 s-1, an averaged value based on CTD casts for 

1,500 to 3,000 m depth in the central Gulf. In view of the rather wide ranges in which both N 

and 2π/K can vary, it is of interest to examine how they affect the ray-paths. 

The effects of K, N and σ on rays can be assessed from equations (2.3-3) and (3.1-1) (again

assuming βTopo/β >> 1 which is a good approximation). The former shows that |Cg| = N|∇h|/K, 

so that group speed increases with wavelength but decreases with N. The ray path also depends 

on the direction of Cg, or on the requirement that σ is constant (or σa=constant when udeep is 

included, see Section 2.3) along the ray. For NhK/|f| ≈ 1, the “tanh” term changes slowly for 

changes in its argument and can be assumed constant. Equation (3.1-1) shows then that for fixed 

N and σ, increasing the wavelength (from the 110 km used in Figure 3.1-7a) will merely increase

the group speeds of the rays, otherwise the ray-paths are similar to those shown in Figure 3.1-7a 

(not shown). This insensitivity of the ray-path to wavelength has been found to be valid for

2π/K up to ~250 km. However, if N (and/or period) is decreased, the angle θ would increase and

rays would tend to propagate more upslope than those shown in Figure 3.1-7a. Though the

relation is nonlinear (equations 2.3-4 and 2.3-5), we find that these deductions based on (2.3-3) 

and (3.1-1) provide a fairly accurate picture of the effects of changes in wavelength, stratification

and period on the ray path. In the top panel of Figure 3.1-11, two rays are plotted, starting from 

station #2, the original ray from Figure 3.1-7a with N=10-3 s-1 and 2π/K=110 km, and a more 

upslope ray for N=6.25×10-4 s-1 and 2π/K=150 km (both rays are virtually unchanged when 2π/K 

is increased to, say, 250 km – when the time variable is scaled by ratio of the original K to new 

K to account for the corresponding increase in the group speed). Since only the ratio σ/N matters 

in equation (3.1-1), it is found that for N = 10-3 s-1, rays propagate upslope for 2π/σ ≈ 43 days or 

shorter, similar to the upslope ray of Figure 3.1-11. It appears, therefore, that the CGKE band 

seen in the simulation cannot be explained by shorter-period TRW’s (<43 days) and/or TRW’s at 

lower possible ranges of N < 10-3 s-1. 

 

The situation is quite different when udeep is included. The generally westward flow with 

cyclonic shear between the 1,000-2,000 m and 3,500 m isobaths (Figure 2.3-3), coupled with 

increasing N|∇h|, constitute sufficiently strong constraints for rays to bend down slope (see

Section 2.3, equation (2.3-11)). Intuitively, rays are ‘blown down wind’ or westward. This is 

3-22
Figure 3.1-11.

Sixty four-day period TRW rays traced from station #2 using the benchmark Nfield of Figure 2.3-2 and initial 2π/K=100 km (ray on down-slope side), and 

0.625×benchmark N and initial 2π/K=150 km (up-slope side). Top panel does not 

include udeep, while lower panel does. In both panels, the LOKE 20-100d contours 

from Figure 3.1-4 are superimposed.

95°W 90°W 85°W

25

1000

1000

1000

1000

1000

2000

2000

2000

2000

2000

2000

2000

3000 3000

3000

3000

3000

3000

3000

95 90 W 85 W

25

95 90 W 85 W

25

100 km

95°W 90°W 85°W

25°N

95°W 90°W 85°W

25

1000

1000

1000

1000

1000

2000

2000

2000

2000

2000

2000

2000

3000 3000

3000

3000

3000

3000

3000

95 90 W 85 W

25

95°W 90°W 85°W

25°N

0.0 0.4 0.8 1.2 1.6 2.0 2.4

(10-3 m2 s-2)

3-23
more so in the upper than lower slope region if a cyclonic shear exists. The lower panel of

Figure 3.1-11 shows this effect, and shows how rays are confined to remain in the vicinity of the 

3,000-m isobath. Extensive tests verify that this confinement of rays is valid also for rays of

other periods, 32, 42.7 and 51.2 days, for longer wavelengths up to ≈ 250 km, and also for other 

stations. Figure 3.1-11 shows that the down slope turning of the ray is first affected near 

90~91°W in a region of cyclonic shear (ς ≈ 0.05f s-1). The amount of turning caused by current 

shear as ray propagates up-slope can be estimated from equation (2.3-11): 

δθ|ς≠0 ≈ − ∫ray |k| ς/(|∇h|N) Cg2 dt, (3.1-3) 

which, with appropriate values (from the PE model) |k| ≈ 0.01 km-1, ς ≈ 0.05f s-1, |∇h|N ≈ 5×10-6

s

-1 and ∫rayCg2dt ≈ 10 km, gives δθ|ς≠0 ≈ −4o

. This value, while reasonable, is likely to be an

under-estimated one. As explained in Section 2.3, since d(|∇h|N)/dt > 0, dki

2

/dt also > 0 (from

equation (3.1-2) or (2.3-10)), and the mere existence of a deep current (without shear) in the 

direction of k (<0) would cause rays to turn down slope, i.e. contributes to additional (more 

negative) δθ. An upper-bound (i.e. maximum possible turning) estimate of this latter effect is 

obtained from equations (2.3-10) and (3.1-2) with ς=0: 

δθ|ς=0 = ∫ray (d|k|/dt)(|∇h|N)-1 udeep dt > ∫ray (∂|∇h|/∂y)|∇h|-1 udeep dt , (3.1-4) 

which, with (∂|∇h|/∂y)|∇h|-1 ≈ 2×10-2 km-1 and ∫rayudeepdt ≈ −5 km, gives 0 > δθ|ς=0 > -6o

 or 

roughly the same contribution as that from current shear alone. It is concluded therefore that the 

presence of deep currents over the 2,000-3,500 m isobaths (Figure 2.3-3) explains why in the PE 

model simulation there is little LOKE upslope, i.e. why the LOKE is predominantly confined 

within the CGKE band. This is particularly so for TRW rays with periods shorter than about 43 

days, and the ray-confinement is more effective when the current also contains a cyclonic shear.

In the region of the base of the slope at 90°W where the PE model shows strong, deep cyclonic 

flows, the recent current observations (Hamilton et al. 2003) also show that TRW energy does 

not penetrate upslope of the Sigsbee Escarpment. Hamilton et al., (2003) speculate that

reflection of upslope propagating TRW’s by the steep escarpment may account for the lack of

energy over the lower slope, but the refraction mechanisms discussed here may also contribute to 

the insulation of the slope from energetic TRW motions. 

3.1.5 Discussion 

The model LC executes a south-north-south vacillation approximately every 5-6 months, and 

sheds a LCE every 10 months or so. These periods are too long for a direct forcing of the 

TRW’s. The vorticity maps of Figure 3.1-1 show the appearance of small-scale disturbances

around the LC and shedding and propagating LCE’s. The study of these meanders (i.e. the 

mechanism of how they are produced) clearly deserves a separate treatment (see 

recommendations in Chapter 6). However, it is shown, in the following, how these short-scale 

fluctuations can provide the necessary linkage for energy transfer from the LC and LCE’s to 

deep motions. An example of a specific TRW event is given and the implications of what raypaths may mean to the existence of deep currents, discussed. 

3-24
Sources of TRW Energy

The example focuses, on station #2, the origin (energy source) for ray #2. Results at other 

stations are similar. Figure 3.1-12 shows, at 5-day intervals, vorticity maps in an enlarged region

of the LC during a period of its northward extrusion before an eddy-shedding event. It shows

clearly the appearance of high cyclonic vorticity perturbations that propagate around the LC and 

into the Florida Straits (e.g. day 901-921, and also day 931-956). The source of these cyclonic 

disturbances appears to be along the western edge of the LC in the Yucatan Straits. The meander 

begins as these disturbances traverse across the Yucatan shelf and slope into the deep Gulf (e.g. 

day 921, 926 and 931). Because of the color contrast, cyclones are most visible in these maps, 

but propagating anticyclones are also present. These cyclones and anticyclones constitute the

propagating meanders that were referred to above. By noting the recurrences of the meanders

from these and similar maps, it is possible to infer time scales (~ periods) of approximately 20 to 

60 days, and spatial scales (~ wavelengths) about 100 to 200 km. To examine more closely how

these shorter-scale fluctuations near the surface excite deep currents, time-series analyses of 

various dynamical variables was performed at several stations. An example at station 2 of Figure 

3.1-7 (shown as the ‘+’ point in Figure 3.1-12) is given in Figure 3.1-13. The top two panels 

give time variations of the layer-1 depth and (relative) vorticity, respectively, that cover

approximately five periods of northward LC extrusion and eddy-shedding. LCE shedding occurs 

at approximately model day 450, 750, 1,150, 1,500 and 1,800 (denoted by the symbol ‘E’ in each 

panel; c.f. Figure 3.1-1b), each time at the ‘foot’ of a sudden drop in layer-1 depth. The drop is 

followed by a trough (i.e. a low) that lasts approximately 100 days, then a relatively slower rise 

that crests in approximately 200 days, that signifies the northward extrusion of LC past the 

station. On these LC-extrusion and LCE-shedding time scales, the vorticity is highly 

anticorrelated with layer-1 depth: it becomes more negative (i.e. anti-cyclonic) during layer-1’s 

rise and maximizes to about zero vorticity during layer-1’s trough. Apparent in the vorticity 

time-series is the existence of high-frequency fluctuations during periods of LC extrusion (when 

ζ/f becomes more negative and layer-1 deepens), and the absence of these fluctuations when the 

LC retreats southward of the station after each shedding. The layer-1 depth time-series shows 

similar, though less transparent, characteristics. It is clear that these fluctuations are associated 

with propagating meanders and other LC variability when the latter extends northward (Figure

3.1-12). 

The existence of these fluctuations in the 20 to 100-day, TRW periods can be most clearly seen 

in the band-passed time series. The middle two panels of Figure 3.1-13 show 20 to 50-day bandpassed kinetic energy in layers 1 and 4, respectively, and lower two panels show same in the 50 

to 100-day periods. The plots show clearly the amplification of these shorter-period (i.e. periods 

less than LC north-south vacillation and LCE-shedding periods) fluctuations near the surface 

during the LC’s northward extrusion phase, and also how they almost disappear when the LC

retracts following a shedding event. The fluctuations transmit to the lower layer (#4), where the 

energy of the combined 20 to 50-day and 50 to 100-day intervals account for approximately 64%

of the total lower-layer energy. Thus longer-period LC vacillations and LCE sheddings near the 

surface have little direct signature in the lower layer (they actually account for less than 10% of 

the lower-layer energy, with the remaining, approximately 25%, residing in the almost steady 

bottom currents), while the short-scale parasite fluctuations they produce, which account 

3-25
Figure 3.1-12.

Vorticity maps at 5-day interval in an enlarged region of the LC during a period of 

its northward extrusion before an eddy-shedding event (on day 1150, not shown). 

The ‘+’ symbol indicates origin of ray #2 of Figure 3.1-7a. Dashed contours 

denote the 3000 m and 3500 m isobaths.

3-26
Figure 3.1-13.

Time-series at station #2 (i.e. the ‘+’ point in Figure 3.1-12). Top two panels: time 

variations of the layer-1 depth and (relative) vorticity; middle two panels: 20 to 

50-day band-passed kinetic energy in layers 1 and 4; bottom two panels: 50 to 

100-day band-passed kinetic energy in layers 1 and 4. The symbol ‘E’ indicates 

times when LCEs are shed in the model. Ratio (in panels 3 to 6) = (Energy in 

respective band)/ (Energy over the entire spectrum).

3-27
for only 37% of the near-surface kinetic energy (Figure 3.1-13), have profound impact on the 

lower-layer motions. 

To infer the inter-connections between the upper and lower-layer fluctuations, the time-lagged 

correlations of various quantities were calculated, focusing on the 20 to 100-day periods. Let hi, 

ζi, and KEi be the fluctuations in the layer-depth, relative vorticity and kinetic energy, 

respectively, of the ith layer. Typical for these time series, correlations which exceed 0.15 differ 

from zero at the 95% confidence level. It is found that h1 and ζ1 are negatively correlated (= -

−0.5) with the former leading slightly by 1 day. Because potential vorticity is conserved, the 

negative correlation (and non-zero lag) suggests the importance of advection in layer 1 where 

currents due to the LC are strong, typically of O(1 m-s). On the contrary, h4 and ζ4 are positively 

correlated = 0.5 with zero lag, which suggests excellent local conservation of potential vorticity: 

∂[(f+ζ4)/h4]/∂t ≈ 0. The link between near-surface and near-bottom motions comes through h1

and h4, which are negatively correlated (= −0.6) with h1 leading h4 by about 3 days. Thus, 

bottom energy in the 20 to 100-day periods originates from regions under active LC and LCE 

variability, and its excitation by LC and LCE’s, therefore, goes as follows. Through dynamical 

instabilities of some kind, propagating meanders develop around the perimeter of these largerscale features, where currents are strong (~1 m-s) and strongly sheared. These meanders have 

length scales of O (100~200 km) and temporal scales of about 20 to 60 days (Figure 3.1-12). 

Though not as readily identified from the vorticity maps, there also exist fluctuations of 

comparable energy in the 50 to 100-day band (Figure 3.1-13). Variations that cause layerstretching (shrinking) near the surface force shrinking (stretching) of the near-bottom layer, 

which through potential vorticity conservation induces anticyclonic (cyclonic) near-bottom 

vorticity. It is not entirely trivial however, how this near-bottom vorticity fluctuations 

transforms into the kinetic energy of TRW’s (included in KE4). Indeed, the conditions under 

which a transfer of energy from the surface meanders to TRW motions might be possible, have 

not been studied. The transfer may be caused by the phase-matching mechanism proposed by 

Pickart (1995; due originally to Malanotte-Rizzoli; see also Pedlosky, 1977), especially in the 

vicinity of station #2 where the 3,000-m isobath tilts from southwest to northeast (Figure 3.1-12; 

similarly also for station #1 on the 3,200 m isobath). This tilt allows eastward TRW phase

propagation (Figure 2.3-1) that can couple with the eastward meanders that propagate past this 

station, so that energy radiation becomes possible. In Pickart’s study, this coupling mechanism 

explains that the TRW energy found off Cape Hatteras was forced by eastward Gulf Stream 

meanders. A similar mechanism maybe at work also in the present PE model simulation (see

below), though clearly this needs to be studied in more detail, preferably within a more idealized

model setting. At the I1 site (see Section 2.2), north of station #2, Hamilton et al. (2003) found a 

few events where current fluctuations were visually coherent through the 2,000 m water column. 

These events were associated with cyclonic eddies on the LC or a LCE front. Thus, there is 

some, but not conclusive, observational support for this type of coupling mechanism.

Though weaker in strengths, similar excitations of TRW’s by meanders around the LC as 

described above also exist around detached LCE’s that propagate southwestward in the PE

model (not shown). In this case, the combined planetary beta and (gentler) sloping topography 

in the deeper portion of the central Gulf (beyond the 3,000-m isobath) contribute to the west/ 

northwestward spread of TRW energy, as discussed previously for rays #4-6 in Figure 3.1-7a. 

3-28
Robustness of the Calculation and Example of a Specific Event

The excitation mechanism suggested herein is apparently more akin to Pickart (1995) than to that 

described in Hogg (1981) and Louis et al. (1982). These latter works suggest forcing of the 

TRW’s through response to specific ring shedding event, that is, being more of an impulse 

response, rather than a near-periodic forcing problem by propagating meanders. In the Gulf of

Mexico, Hamilton (1990) found correlations between pairs of eastern and western moorings that 

were related to LCE-shedding events. On the other hand, secondary small-scale features such as 

meanders owe their existence to forcing by LC and LCE’s, the movement of which over a 

specific area (topographic slope in particular) can therefore be considered as ‘impulsive’ (~1−2 

weeks). While costs and logistics generally dictate that observations (of TRW’s) are for specific 

events, modelers can afford longer-term (model) data and are compelled to demonstrate the 

occurrence of more than one or two specific events only, as these can arise due to specialized 

forcing, initial and boundary conditions, and sometimes even to the particular model grid being 

used. Therefore, an analysis of the deep EKE, as a seven-year ensemble of specific excitation

and propagation events of TRW’s (Figures 3.1-4 and 3.1-7), has been performed. The existence

of a band of deep EKE across the central Gulf (the CGKE) provides some evidence that the 

simulated events are robust model features that are repeatable (this is clear from animation of 

LOKE). To further ascertain this, the 10-year simulation has been repeated by doubling the 

horizontal grid resolution. It is found that this doubled-resolution experiment also gives a very 

similar (last seven year) composite map of deep EKE containing the CGKE band as in Figure

3.1-4, and which also is made up of individual TRW excitation and propagation events. The 

difference is that the deep EKE is now more intense (Oey, 1998). Therefore, an example is

given of a typical TRW excitation and propagation event taken from the doubled-resolution 

seven-year ensemble. 

Figure 3.1-14 plots images of EKE at 188 m above the bottom, where the EKE is dominated by 

energy in the 20 to 100-day band, and contours of EKE|20-100d are virtually identical. The EKE 

(model sigma-level 17; green-red-yellow, values less than 1.4×10-2 m2

 s-2 are omitted) is 

superimposed on 200 m Eulerian trajectories (tracked for 5 days and launched at every 12th grid 

point) that are colored with local values of ς/f (dark blue for cyclone ≥ 0.4 and red for 

anticyclone ≤ −0.4). These trajectories indicate the locations of the near-surface LC and eddies. 

For example, the LC front is delineated by the transition from red to blue trajectories. On days 

1,345 and 1,370, the front passes over point #1 and point #2, respectively. Point #1 is the same

as station #1 of Figure 3.1-7, and point #2 is slightly south (by 0.3o

Lat) of station #2. Note also 

that meanders typically have cross-frontal amplitude of about 50 km (i.e. ≈ 10 grids), and 

therefore cannot be readily distinguished in trajectories launched every 12th point. They are 

nonetheless easily seen in maps of relative vorticity as in Figure 3.1-12. On day 1,370, high 

EKE region is seen stretching westward and onshore from point #1, following approximately the 

3,000-m isobath. High EKE is also seen at point #2. On days 1385 and 1405, these high EKE 

progressed westward to point ‘R’, the ray-convergence zone discussed previously in conjunction 

with Figure 3.1-7. Despite the rich (surface) eddy structures west of 88°W, deep EKE rarely

rises above the 1.4×10-2 (m s-1)

2

 cut-off except near the 3,000-m isobath when the high EKE 

arrives, a phenomenon strongly suggestive of TRW propagation as discussed previously. Figure 

3.1-14 also shows two TRW rays launched from point #1 using properties at day 1345 as initial

conditions, and also from point #2 using day 1370 as initial conditions. Both calculations also

3-29
Day 1345.

0.50 1.20 1.90 2.60 3.30 4.00

1

2

R

92W 90W 88W 86W 84W

22N

24N

26N

28N

30N

Deep EKE x 100 (m s-1)2 & 200 m Traj+Vor/f

Day 1370.

0.50 1.20 1.90 2.60 3.30 4.00

1

2

R

92W 90W 88W 86W 84W

22N

24N

26N

28N

30N

Day 1385.

0.50 1.20 1.90 2.60 3.30 4.00

1

2

R

92W 90W 88W 86W 84W

22N

24N

26N

28N

30N

Day 1405.

0.50 1.20 1.90 2.60 3.30 4.00

1

2

R

92W 90W 88W 86W 84W

22N

24N

26N

28N

30N

3-30

Figure 3.1-14. Modeled eddy kinetic energy (green-red shade, values ≤ 1.4x10-2 m2 s-2 are 

omitted) at 188 m above the bottom on day 1345 (top-left panel) when the northern 

edge of LC touches point #1 and a TRW ray (green) is initiated there; on day 1370 

(top-right) when the northwestern edge of LC touches point #2, and a second TRW

ray is initiated there (black); on day 1385 (bottom-left), and day 1405 (bottomright). The arrowed curves are Eulerian trajectories at 200 m below the surface, 

released at every 12th model point and each tracked for 5 days (scale = 0.5 m s-1

per 100 km), and colors on them indicate local values of V/f; dark-blue is cyclonic 

≥ 0.4, and red is anticyclonic ≤ -0.4. Dotted curves are the 200-m and 3000-m 

isobaths. Point ‘R’ marks the location where TRW rays converge. Note that 

despite rich (surface) eddy structures west of 88°W, deep EKE rarely rises above 

the 1.4x10-2 m2 s-2 cut-off, except near the 3000-m isobath where TRW's pass.
include deep mean (seven-year) currents (see previous section). Despite the many 

simplifications inherent in the TRW theory, the rays are seen to track the high EKE progression 

fairly well. In particular, both rays would eventually converge near 90° to 91°W. Also, the 

velocity of progression of the EKE feature, ≈ −8.3 km day

-1, matches well the average of the 

group velocities of the two rays (≈9 km day

-1). Similar events of excitation and propagation of

TRW’s are also found to be triggered by southwestward propagating LCE’s in the western Gulf 

(not shown). 

Ray-Paths and the Existence of Deep Currents

It has been showed that the generally westward deep currents in the model Gulf increase the 

TRW group speeds, and also confine rays to the vicinity of the 3,000-m isobath in more

accordance with the simulated CGKE band (Figures 3.1-7 and 3.1-11). Thus, knowledge of 

group speeds and ray-paths may provide information on deep currents. Based on a time-lagged 

correlation analysis between two deep moorings, one in the eastern (Mooring G at (85°30’W,

25°36’N) on 3,200 m; see Section 2.2) and another one in the western Gulf (Mooring Q at

(94°53’W, 25°52’N) on 3,000 m) Gulf, Hamilton (1990) obtained a lag-time of 106 days, and 

estimated a lower bound group velocity of −9 km day-1 by assuming a straight-line distance 

between the two moorings. If, as suggested by our ray calculation, the ray-path is curved

approximately following the 3,000-m isobath, the additional distance of 140 km would yield a 

group velocity of about −10.3 km day-1. Hamilton also used the dispersion relation to arrive at

an estimate of |Cg| ≈ 8.5 km day-1 based on a wavelength value of about 189 km. Thus, there is a 

difference in group velocity of about −2 km day

-1. Hamilton’s Cg values can be compared with 

the present averaged values (after applying a small 5.5% adjustment to account for the 

differences in wavelength and N between Hamilton’s and this work) of −11.3 and −8.5 km day-1, 

with and without udeep respectively. In the present case, the difference, about −3 km/day, is 

entirely caused by udeep. The −2 km day-1 (≈−2 cm s-1) difference in Hamilton’s case may also be 

due to the existence of deep currents. His analysis, and also that of Hamilton et al. (2003) for 

site I1, suggests that the observed TRW’s do not stray far up-slope, which may also imply 

existence of deep shear flows. 

3.2 Comparisons with Deep Current Measurements 

This section discusses the characteristics of the deep velocity field from the model run, described 

above, and the same model run (i.e. the same grid, forcing and boundary conditions) with 

assimilation of satellite altimetry and SST, using the methods of section 2.1.1. The assimilation

scheme synchronizes the upper-layer eddy field with reality and therefore, the interest is in 

whether the energy levels of the TRW’s are improved over the regular non-assimilated run. The

assimilation method does not directly influence the model fields below ~ 500 m. Profiles of 

velocity, temperature and salinity were extracted from the model runs at the positions shown in 

Figure 3.2-1. The last three years of the runs were used with a time interval of 3 hours. The

positions in Figure 3.2-1 include all the mooring sites in Table 2.2-1, as well as other deep-water

sites taken from the SAIC database, and a set of slope positions at regular intervals around the 

Gulf. 

3-31
1000

1000

1000

1000

1000

1000

1000

2000

2000

2000

2000

2000

2000

3000

3000

3000

3000

3000

3000

4000

4000

Figure 3.2-1.Positions of model-saved daily profiles of velocity, temperature and salinity.

3-32

95°W 90°W 85°W 80°W

95°W 90°W 85°W 80°W

30°N

25°N

20°N

30°N

25°N

20°N
3.2.1 Statistics and Spectra 

Figure 3.2-2 shows the standard deviation ellipses for the observations and from the regular run. 

The observations are generally within about 200 m from the bottom and the model data is from 

the nearest sigma-level. The last two years of the model run were used, and the observations 

were 40-HLP filtered and vary in length from about 2.5 to 0.5 years. The model underestimates 

the kinetic energy (KE) at almost all sites for reasons discussed above. The largest differences

are under the LC (site G) and south of the delta (site I). The latter seems to be an exceptional 

area for high energy TRW’s (Hamilton et al. 2003) which are characterized by short periods (~

10 to 12 days) and short wavelengths (~ 60 to 100 km). The model is not expected to be able to

resolve such motions and thus the comparison at this site is not particularly valid. The model 

shows rather more homogeneous distribution of KE than the data, though the observations are

taken over different periods. The model data are also more polarized along the isobath direction 

than the observations, and again this is probably because the model favors the longer period and 

longer wavelength waves. However, the model does show an increase in energy from the base of 

the slope into deeper water, as observed, because of the CGKE band. When the assimilation 

model results are compared (Figure 3.2-3), it is immediately apparent that the model deep KE 

has increased over the regular run at all sites. Except for under the LC (site G), the deeper sites

(GG and R) have the largest increases indicating that the CGKE band is more active. Site I also

shows a large relative increase and it is speculated that a more accurate pulsating of the LC may 

be generating TRW’s that are propagating into this region south of the delta. A comparison of 

upper-layer eddy speeds as a function of horizontal eddy scale, between model runs, shows that 

the assimilation model generates similar higher speeds at scales similar to the model with double 

resolution, but without assimilation (Figure 3.2-4). Therefore, the higher speeds of upper-layer 

eddy flows, in the assimilation model, are reflected in the lower KE increases. Despite the 

improvements, the assimilation model still underestimates the observed KE. 

The KE spectra, in variance preserving form where equal area under the curve represent equal 

variances in frequency space, for the observed currents at five of the deep-water sites are given 

in Figure 3.2-5. All the moorings show bottom intensification where more than one

measurement is available below 1,000 m. The highest energies are in the east (G and I1) and

apart from G, which is under the LC, all sites show a prominent peak at about 20 to 30 days. 

Under the LC at G, the highest energies are at longer periods of 30 to 50 days and around 100 

days. At I1, there is high energy at higher frequencies (periods of 7 to 12 days) that is not in 

evidence at the other sites. The bottom left-hand plot in Figure 3.2-5 shows the large increase in 

energy between the base of the slope (FF) and the deep basin (GG). There is a similar contrast 

between P and R and A and G in the western and eastern Gulf, respectively (not shown – see 

Hamilton, 1990). Similar KE spectra for the same sites (except FF), but using data from the 

assimilated model, are shown in Figure 3.2-6. Apart from the differences in the variance scales,

noted in Figure 3.2-3, the model spectra also show bottom intensification at most frequencies and 

similar spectral peaks. Site G, under the LC, is more barotropic than other sites and its 

prominent spectral peak is at about 20 to 30 days, similar to the other model sites. Site I1,

however, does show an 8 to 10-day peak as well as the one at longer periods, indicating that the 

model is generating some of the higher frequency TRW energy on the northwest side of the LC. 

The model I1 and G also have an energetic peak at periods greater than 100 days. This is 

probably related to LC pulsation and if it occurs at all in the observations at G and I1 that can 

3-33
40-HLP Observations

Model Results

Standard Deviation Ellipses for Velocity Data Below 1000-m

2000-m

3000-m

200-m

Figure 3.2-2.Standard deviation ellipses for velocity data below 1000 m from observations (top 

panel), and the regular model run.

G A

I

GG

FF

P

Q R

3-34
40-HLP Observations

Standard Deviation Ellipses for Velocity Data Below 1000-m

2000-m

3000-m

200-m

G A

I

GG

FF

P

Q R

Figure 3.2-3.Standard deviation ellipses for velocity data below 1000 m from observations (top 

panel), and the assimilation model run.

Assimilation Model Results

3-35
Figure 3.2-4.

Eight-year, time-averaged velocity spectra expressed in m s-1 at sigma-level 6 

(100 m below the surface) as a function of the spatial scale for model region north 

of 26°N but deeper than the 500 m isobath. Green - regular model, blue - model 

with assimilation, and red - regular model with double resolution.

3-36

Averaged Velocity Spectra at Level 6

1000 500 100 50 10

Eddy Diameter (km)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

STD (m/s)

FGE

Effects of Grid & Data Assimilation

CGE

CGE+Assim
Figure 3.2-5.

Kinetic energy spectra for mooring sites G (top left), I1 (top right), FF (red) and 

GG (blue) (bottom left), and R (bottom right) from observed currents. Depth 

levels and time periods are indicated in the labels.

3-37
Figure 3.2-6.

Kinetic energy spectra for mooring sites G (top left), I1 (top right), GG (bottom 

left), and R (bottom right) using assimilated model currents. Depth levels are 

indicated in the labels.

3-38
resolve such long periods, it is much less apparent. Sites GG and R in the central and western 

Gulf show model energy at longer periods (40–100 days) that is not observed at least at GG. The

observations at R are not long enough to resolve properly such long periods. Thus, the 

agreement of the spectral peaks and the evidence of bottom intensification between model and 

observations is encouraging and support the notion that TRW’s are being generated and 

propagated in the model as discussed in the previous section. 

One of the most important characteristics of the observed deep currents is the high coherence of 

the columnar flows. Therefore, it is useful to examine the model output to determine how depthindependent flows are in the lower water column. Figure 3.2-7 shows the results of calculating 

the coherence squared and phase differences between widely spaced depth pairs for sites G, I1 

and R. The model data shows the high coherence and zero phase differences, similar to the 

observations at G and I1. The coherence levels are well above the 95% significance levels for 

these sites. At R, however, the model shows much less coherence than the observations, which 

still have a strong columnar character. Similar differences are found at P and Q. Therefore, the 

model seems to have difficulty in maintaining the barotropic nature of flows in the lower water 

column as the TRW’s propagate into the western Gulf. Therefore, there is the unusual situation 

that the model results show more noise than the observations as far as the vertical structure is

concerned. 

The observed current data has been used to estimate wavelengths and dominant periods of 

TRW’s (Hamilton, 1990; Hamilton et al., 2003). The ray-tracing equations (2.3-4) and (2.3-5) 

were used to trace wave paths backwards and forwards from the mooring site. The results are 

given in Figure 3.2-8, which should be compared with Figure 3.1-7. Rays traced from GG and R 

are refracted to the south along the Mexican slope and originate in the deep central Gulf. This 

implies that their origin is more likely to be associated with propagating eddies than the LC.

However, longer wavelengths and periods can originate in the east, because the group velocities 

become more parallel to the isobaths (e.g. Figure 3.1-7). These calculations were performed 

with no background advection by the mean, deep flows that may significantly increase the 

southward refraction as discussed above. This latter effect may be particularly important for I1,

where there is a steep escarpment and little of the high energy associated with the short 

wavelength TRW’s appears to penetrate north and west of the site. Hamilton et al. (2003) deal

with this by artificially reflecting the waves at the escarpment. A westward directed mean slope 

current that decreases in the offshore direction may achieve the same effect, and this is supported 

by the observations. Mean currents for a two-year period at 200 m from the bottom are shown 

(Figure 3.2-8) for I1 and a site ~ 25 km offshore (I3) that show a westward, cyclonically sheared, 

mean (~ 3 cm/s at I1). Whatever the mechanism, it is clear that 10 to 14 day waves, at I1,

probably originate on the west side of the LC where they have the scales and periods that have a 

similar match to propagating cyclones on the LC front (c.f. Pickart, 1995). Moreover, these short 

period waves do not appear to penetrate very far to the west and this explains why little 10-day 

energy is observed at GG (Figure 3.2-5). 

3-39
Figure 3.2-7.

Coherence squared and phase differences between indicated depth levels and 

velocity components for sites G (lhs), I1 (middle) and R (rhs). The U and V

components are across and along isobath directions, respectively. Top row is from 

the assimilation model run, and the bottom is from the observations. The 95% 

significance level is indicated.

Assimilation Model Run

Observations

3-40
I1

GG

R

25 day

120 km

25 day

175 km

60 day

200 km

10 day

70 km

14 day

60 km

200 m

2000 m

3000 m

TRW Ray Tracing Using Observed Periods and Wavelengths

Figure 3.2-8.TRW ray tracing using observed estimated periods and wavelengths. Rays are 

traced backwards and forwards from the mooring sites. Arrow heads are every 5 

days. The purple arrows represent two-year mean currents (9/1999 to 9/2001), 

200 m from the bottom. 

5 cm/s

3-41
4. Slope Eddies 

4.1 Introduction 

In general, LCE's do not collide directly onto the northern Gulf slope as they rarely intrude

northwards of about the 1,000-m isobath. As a result of extensive hydrographic surveys by ship 

and aircraft (LATEX C and GulfCet; Berger et al. 1996), the northern slope has been shown to 

be populated by a complex eddy field of smaller (than typical LCE's) cyclones and anticyclones 

with diameters ranging from ~ 20 to 150 km. An example is given in Figure 4.1-1. Eddy scales 

seem to be related to water depth, probably through the baroclinic Rossby radius, so that larger 

eddies tend to be observed on the lower slope and smaller on the upper. These slope eddies are 

ubiquitous and their occurrence is found at all positions along the slope from west Texas to the 

De Soto Canyon. The larger scale eddies have rotational periods of ~ 10 to 20 days, which are 

longer than typical LCE's (7 to 10 days in the central Gulf). Extensive drifter deployments in the 

SCULP-1 study showed that the eddies interact, with the smaller translating around the larger in 

the direction of the swirl velocities of the larger, more energetic eddies. This also applies to the

interaction of slope eddies with eddies in the deep basin. Thus, the leading edge of a westward 

propagating LC anticyclone can move a peripheral cyclone onto the slope, where it may become

stranded and remain on the slope as the LCE continues to the west. Similarly, a large deep-basin

eddy can move smaller lower-slope eddies into deeper water (Hamilton et al., 2003). Drifters 

can show the short-term (a few weeks at most) evolution of specific slope eddies. However, the

long-term evolution of the slope eddy field is not understood. There is limited evidence in a 

couple of cases that slope cyclones of ~ 100 km diameter can persist for longer than 6 months

(Hamilton, 1992; Hamilton et al., 2003). Evidence will be presented that in at least one case, 

energy of the eddy field appears to increase with time scales of days. 

The technical problem is then to initialize the model with hydrographic measurements of smallscale eddies over a limited area of the Gulf (slope), and at the same time also account for the 

large-scale eddy field. It is noted that the scales of the slope eddies in Figure 4.1-1 are not, at 

present, well resolved by satellite altimetry. The north-central Gulf slope during two periods of 

the LATEX C and GulfCet observations (Hamilton et al., 2003): (A) December 3-14, 1993 and 

(B) December 16-23, 1993, will be the focus of this work. The scientific problem is therefore,

given the slope eddy field, what can be deduced about its evolution? An attempt is made to 

address this problem using model simulations with and without data assimilation. 

4.2 Observations 

The measurement locations during the two observational periods (Period A and Period B) are 

shown in Figure 2.2-2. The corresponding 200-m sigma-t, the 200 to 780-m dynamic height and 

200 to 780-m geostrophic velocity are shown in Figures 4.2-1, 4.2-2 and 4.2-3, respectively. 

Period A is characterized by a series of anticyclonic and cyclonic eddies spanning from west to 

east over the northern slope of the Gulf. The northern portion of an LCE can be seen from 92°W 

~ 90.5°W. The magnitude of the 200 m geostrophic velocity (relative to 780 m) is about 0.25 m 

s

-1. Period B is over the north-central slope only from 94.5°W ~ 90.2°W. This region will be the 

primary focus of the model analyses (below). When comparing with Period A, it is clear that the

4-1
High-Pass Filtered 

Scales 50 - 150 km

Figure 4.1-1.Depth of the 25.50 sigma-t surface (i.e. the main pycnocline) during the 

observation period December 5-12, 1993; high-pass filtered to remove scales 

larger than about 150 km.

200 m

2000 m

4-2
Figure 4.2-1.

The 200-m sigma-t during the two observational periods (A) December 5-12, 

1993 (upper panel) and (B) December 16-23, 1993 (lower panel).

4-3
Figure 4.2-2.

The 200 to 780-m dynamic height (dynamic cm) during the two observational 

periods (A) December 5-12, 1993 (upper panel) and (B) December16-23, 1993 

(lower panel). 

4-4
25 cm/s

40 cm/s

Figure 4.2-3.The 200 to 780-m geostrophic velocity during the two observational periods (A) 

December 5-12, 1993 (upper panel) and (B) December16-23/1993 (lower panel). 

4-5
eddy field has evolved significantly during the intervening ~1 week period between Period A and 

B. The map of dynamic height (Figure 4.2-2) and the 200 m geostrophic velocity for Period B 

now show the appearance of smaller-scale features. The geostrophic velocities are some 70% 

more intense in Period B than Period A, and it is necessary to investigate how much of this 

change could arise from the differing resolutions of the two surveys. 

The average influence radii of the statistical mapping method (Pedder, 1993) are 50 and 40 km

for A and B, respectively. The cross-slope station spacing of the two surveys is very similar, so

the main differences are in the along-slope spacing. The 200 m dynamic height field for the 

Period-B observations was re-calculated with an increased along-slope influence scale, so that 

the average influence scale was 50 km. The resulting geostrophic velocities were about 20% less 

in magnitude than those in Figure 4.2-3. Therefore, it is concluded that the increase in intensity

of the eddy flows between periods A and B is real and not an artifact of the analysis method. 

Thus, one aspect of this study is the explanation of these changes through model-data analyses. 

The hydrographic fields were prepared for assimilation by the model through use of the

statistical, horizontal mapping scheme (Pedder, 1993) at 25 m depth intervals between the 

surface and 780 m. For each depth level, the station data were mapped to the model’s curvilinear 

grid positions where they overlapped with the region of the survey. The horizontal area covered 

varies with depth because of the bottom slope. Hydrographic fields mapped to the model grid 

include temperature, salinity, sigma-t, and components of geostrophic velocity calculated relative 

to 780 dbars. 

4.3 The Model 

The model domain, topography and approximate grid distribution are shown in Figure 2.1-1a.

Figure 2.1-1b also shows the nested Gulf of Mexico region where the grid resolution is doubled 

as described in Section 2.1. This section concentrates on the particular problem of initializing 

the model using observations from Period A, and on model evolution through Period B. 

The model is initialized with climatological T/S and forced with six-hourly ECMWF wind 

beginning January 1, 1992. Satellite data-assimilation begins on October 22, 1992 and updated 

daily by interpolating between two 10-day base data. SST from the weekly AVHRR-MCSST

dataset is also assimilated, but tests show that SST assimilation has little effects on the model

solution (Wang et al., 2003). On December 3, 1993, hydrographic data (temperature) from 

observational Period A is ‘injected’ into the simulation. A simple nudging scheme is used for 

this purpose. Thus, the observed temperature To in the upper 780 m in the region shown in

Figure 4.1-1 is used as a source term in the temperature equation: 

 DT/Dt = ∂(KH∂T/∂z)/∂z + F − λ(x,y,z,t) (T − To), (4.3-1) 

where KH is the vertical eddy diffusivity and F represents the horizontal mixing. λ is a nudging 

coefficient of the order of 1 day-1 that decays exponentially away from the (three-dimensional) 

observational region. λ also varies like a tanh function in time such that nudging is in full effect 

over a two-day transition, e.g. December 3 to 5, and returns to zero (i.e. is turned off) also in two 

days; December 12 to 14. In this way, the model is allowed to adjust to the observed 

4-6
hydrography during the intervening interval (December 5 to 12) when the full nudging is in 

effect. 

Nudging (or for that matter, any other more sophisticated assimilation scheme) amounts to 

introducing sources and sinks into model thermodynamics and dynamics. The situation is 

difficult in the present case since the data are neither continuous in time nor in space. The 

method works only if the initial model field (before nudging) approximates the observed 

conditions. Figure 4.3-1 shows the model sea-surface height (SSH) on December 3, 1993, and

represents the model’s best estimate of the observed conditions just prior to the hydrographic 

measurements. This can be compared with the observed dynamic height in Figure 4.2-2. Both 

figures show an anticyclone at 90~91°W and a cyclone to the west at 92~93o

W. The model SSH 

shows another anticyclone further west around 94o

W. Observations did not cover this western 

region but the eastern arm of the anticyclone can be seen. The upshot is that the model mimics 

quite well the observed conditions during Period A. 

4.4 Model Prediction of Observational Conditions during Period B 

After the above initialization of the T/S fields, the satellite data-assimilation was turned off and

the model was allowed to evolve according to its own dynamics. This unassimilated, prognostic 

run continued from December 14, 1993 through January 1,1994. 

Figure 4.4-1 shows the observed geostrophic velocities at 200 m during the two intervals (left 

panels) and compares them with the modeled velocities (right panels) on December 14 (at end of 

nudging), December 23 (at end of observation Period B during the unassimilated run) and

January 1, 1994 (at end of unassimilated run). The observations show increased current intensity 

and meanders from Period A to Period B as was noted earlier in Figures 4.2-1-4.2-3. The model

also shows a similar increase. Figure 4.4-2 indicates these increases more clearly with vertical 

sectional contours of the along-slope velocities and temperature at the cross-slope transect that

cuts across the cyclone as shown in Figure 4.4-1. Observations show an increase of eastward 

velocity from 0.2 m s-1 during Period A to 0.8 m s-1 during Period B and westward velocity from

−0.1 m s-1 to −0.3 m s-1. Note that these geostrophic velocities are relative to an assumed level of

no-motion of 780 m. The increases occur mostly in the upper layers but are also significant at 

depth. The model also shows increases, from 0.4 (−0.4) m s-1, on December 14, to 0.8 (−0.8) m 

s

-1 on December 23. A close examination of the density field (temperature in Figure 4.4-2, and 

others not shown) indicates more small-scale (~50 km; but resolvable by the model grid) 

structures during Period B. Because of the similarity of the model fields and their behaviors to 

the observations, the energetics of the model were investigated further. 

4.5 Flow Energetics 

Each flow variable is separated into its time mean and fluctuation (perturbation), u = u + u', 

v = v + v' , ρ = ρ + ρ' etc., where the over bar denotes some appropriate time mean to be defined 

4-7
Figure 4.3-1.

The model sea-surface height (SSH) just prior to nudging of hydrographic 

measurements from observational Period A. The study region is shown in the 

rectangle.

4-8

Study

Region

cm

94W 90W 86W 82W

14N

18N

22N

26N

30N

Model SSH (cm) on December 3, 1993
Figure 4.4-1.

The observed geostrophic velocities at 200 m during the two periods (left panels), 

and the modeled velocities (right panels) on December 14 (at end of nudging), 

December 23 (at end of observation Period-B during the unassimilated run), and 

January 1, 1994 (at end of unassimilated run). The vector scale is 1 m s-1. 

Contours are the 500 (dotted), 1,000, 2,00 and 3,000-m isobaths. The line across 

the cyclone near 93.5°W indicates the transect where variables are analyzed in 

more details (see Figure 4.4-2, also the text).

4-9

1 m s-1
Figure 4.4-2.

Left panels: the observed along-slope geostrophic velocities, U (colored, m s-1) 

and temperature (light contours, °C) at the cross-slope vertical section shown in 

Figure 4.4-1, during the two indicated observational periods. The view is towards 

west so that positive U indicates eastward flow. Right panels: the corresponding 

modeled along-slope velocities on December 14 (at end of nudging), December 

23 (at end of observation Period B during the unassimilated run) and January 1, 

1994 (at end of unassimilated run).

4-10
shortly. The total (i.e. potential + kinetic) perturbation energy equation is then (Orlanski and 

Cox, 1972; Brooks and Niiler, 1977): 

} '

2 ( ' ' ) 2

1

{ 2

0

2

2

2

2 2

ρ

ρ

N

g u v

dt

d

+ + ( ' '/ ) ρ 0 v p r = −∇ ⋅ ( ' ' ' ' ' ' ) 2 2

x

v

u v

y

u

u v

y

v

v

x

u

u

∂

∂

+

∂

∂

+

∂

∂

+

∂

∂ −

 ) ' ' ' ' ( 2 2

0

2

2 2

0

2

N y

g v

N x

g u

∂

∂

+

∂

∂ − ρ

ρ

ρ ρ

ρ

ρ ( ' ' ' ' )

z

v

w v

z

u

w u

∂

∂

+

∂

∂ − . 

 (4.5-1)

The various symbols are standard, i.e. g is acceleration due to gravity, N the Brunt-Väisälä

frequency, etc. Equation (4.5-1) may be written symbolically as: 

dt

d (EKE + PEP) ( ' '/ ) ρ 0 v p r = −∇ ⋅ + BT + BC + KH. (4.5-2) 

Here, EKE is the kinetic energy of the perturbation (often referred to as the eddy kinetic energy) 

and PEP is the potential energy of the perturbation. The BT is the barotropic conversion term 

that, if positive, drains energy from the mean horizontal shears to the eddy field. The BC is the 

baroclinic conversion term that, if positive, drains energy from the mean available potential 

energy field (i.e. horizontal density gradients) to the eddy field. The KH is the Kelvin-Helmholtz 

instability term that depends on the mean vertical shears and Reynolds stresses in the vertical 

plane. Thus, both the BT and KH can release (if positive) the mean kinetic energy, while BC (if

positive) can release the mean potential energy. The pressure work term on the right side of 

equation (4.5-2) (or 4.5-1) would vanish if integrated over a section with side-walls. In the

present case this is not true (see below). There is also an exchange term, PKC, that appears with

opposite sign on the right side of the separate equation for EKE and PEP: 

 ( 'w') g PKC

o

ρ

ρ = − . (4.5-3)

A positive PKC drains energy from PEP and into EKE, and vice versa for a negative PKC. 

The individual terms in equation (4.5-1) were computed, using six-hourly averaged model fields 

from December 15 through December 30 at the cyclone section shown in Figure 4.4-1, as 

follows. Each of the terms is assumed to evolve with a fast time “t” when significant 

perturbation growth (or decay) occurs, and a slow time “εt” (ε << 1; i.e. long period) over which 

time averaging can be meaningfully defined. This long period is taken to be 5 days, i.e. the

overbar (.) is for 5 days. Thus, each of the terms in equation (4.4-1) evolves slowly with a time 

scale of O(εt), and we can examine how they vary in the intervening 15 days after the 

observation Period A. 

4-11
Figure 4.5-1 shows the sectional averaged (section length = Ly) EKE and PEP, 

and , respectively, as functions of depth (z; m) and time (εt; days). 

For reasons that will become clear later, the section is taken as spanning the northern half of the

cyclone (i.e. over the slope; Figure 4.4-1) only. Large eddy energies are mostly in the upper 300 

m of the water column. A thin layer (~30 m) near the surface has particularly large energy,

especially EKE, that is due to the eddy stirring by the wind. From December 15 through around 

December 20, there is a clear increase in both the EKE and PEP. This coincides with the 

observed increase in current intensity and ‘meander’ activity from Period A to Period B (Figures 

4.2-1 to 4.2-3). Thereafter there is a general decline in PEP, but a second increase in EKE

occurs around December 25. 

∫ EKE dy Ly ( ) / ∫ PEP dy Ly ( ) /

Figure 4.5-2 shows the time-averaged (period P = 15 days from December 15 to 30) EKE and 

PEP, and , respectively, as functions of depth (z; m) and latitude at 

the cross-slope section shown in Figure 4.4-1. Once again the concentration of eddy energy in

the upper 300 m is clearly seen, however, the EKE appears to penetrate deeper than the PEP. 

Apart from the wind-induced eddy energy in the very near-surface layers, most of the subsurface 

eddy energy appears in more offshore regions (27°N~27.25°N), where the depth of water is 

approximately 1,000 m or greater. 

∫(EKE)dt / P ∫(PEP)dt / P

The terms in equations (4.5-2) and (4.5-3) were averaged over y as well as z, e.g., <EKE> = 

, where <.> denotes the double-averaging, where the averaging over 

depth excludes the near-surface wind-forced layer. The conclusions below are not changed by

this choice. Figure 4.5-3a shows plots for the individual terms as a function of time. In this 

figure, the <EKE Source> = <BT> + <KH> + <PKC>, while <PEP Source> = <BC> − <PKC>; 

i.e. they are the source terms (except for the pressure work term which also appears in the 

equation for EKE) in the EKE and PEP equations respectively. The Kelvin-Helmholtz term, 

<KH>, is weak (note change in its scale) throughout the whole period. It is clear that from

December 15 through 20, there exists both barotropic and baroclinic instabilities, but the latter 

dominates. There is a transfer of energy from PEP to EKE during this period, i.e. <PKC> is

positive. This suggests that the observed increases in the intensity of the geostrophic velocity 

and meanders from Period A to Period B may be a result of mixed barotropic and baroclinic 

instabilities. During the middle period, December 20 through 25, there are only weak energy

exchanges, between <EKE> and <PEP>, or between the eddy and mean energies. At later 

times, December 25 through 28, the instability appears to be barotropic. The <PKC> is negative, 

and there is a transfer of <EKE> to <PEP>. However, the flow appears to be baroclinically

stable (<BC> is negative) as the cyclone moves off-slope (Figure 4.4-1). 

∫ ∫

−

−

( ) /(250 )

50

y

m

H

EKE dydz L

The northern half of the cyclone is situated over the sloping topography (Figures 4.4-2 and 4.5-

2), which should be destabilizing because the slope and isopycnals are of opposite signs. It is 

expected that analysis of the southern half might yield different results, which are shown in 

Figure 4.5-3b. Here <BC> is predominantly negative (except for the small values near the 

beginning), and there is a drain of <PEP> to <EKE> (i.e. <PKC> is positive). However, despite

<BT> that is also positive (albeit small), the <EKE> actually decreases. A more detailed analysis

4-12
Figure 4.5-1. The sectional averaged (section length = Ly, where the section is the northern half 

of the cyclone shown in Figure 4.4-1) EKE and PEP, (upper panel), 

and (lower panel), respectively, as functions of depth (z; m) and 

time (εt; days).

∫ L

y (EKE )dy /

∫ L

y (PEP)dy /

4-13
Figure 4.5-2. Time-averaged (period P = 5 days from Dec 15th-30th) EKE and PEP, 

(upper panel) and, (lower panel), respectively, as functions of depth 

(z; m) and latitude for the northern half of the section cutting across the cyclone 

shown in Figure 4.4-1.

∫(EKE)dt / P

∫(PEP)dt / P

4-14
Figure 4.5-3a.

Sectional (y) and depth (z = -H to -50 m) averaged values of the terms in equations 

(4.5-2) and (4.5-3). The section is over the northern half across the cyclone as 

shown in Figure 4.4-1. The <EKE Source> = <BT> + <KH> + <PKC>, while 

<PEP Source> = <BC> - <PKC>. Note that <KH> is small and its scale is 

magnified by almost a factor of ten. See text for details.

4-15
Figure 4.5-3b.

Sectional (y) and depth (z = -H to -50m) averaged values of the terms in equations 

(4.5-2) and (4.5-3). The section is over the southern half across the cyclone as 

shown in Figure 4.4-1. The <EKE Source> = <BT> + <KH> + <PKC>, while 

<PEP Source> = <BC> - <PKC>. Note that <KH> is small and its scale is 

magnified by almost a factor of ten. See text for details.

4-16
reveals that the pressure work term: (v' p'/ )dydz ρ 0

r − ∇ ⋅ ∫∫ is not insignificant and there is a

‘leak’ of eddy energy through the offshore end of the section. 

The higher-order hydrographic analyses, given in Section 2.3.2, confirm the development of the

instabilities between Periods A and B. For Period A, Figure 4.5-4 shows the Sverdrup and Ertel 

PV of the surface layer, which is essentially inversely proportional to the depth of the 26.5

sigma-t surface, overlaid with the geostrophic velocity field on the 25.5 sigma-t surface. The 

later is a marker for the center of the pycnocline. By comparing the PV fields, it is apparent that 

the relative vorticity is an important component of the PV and serves to homogenize the PV 

within eddies, and sharpen the gradients between them. This is not unexpected if the centers of 

eddies are almost in solid body rotation. The strongest geostrophic velocities approximately 

follow the contours of constant (Ertel) PV, though there are significant PV perturbations, with 

about a 50 km scale, along the paths of strongest velocities that indicate incipient meanders. For 

example, along ~ 92°W and ~ 93°W, or the boundaries traced out by the 0.8 and 0.1 x 10-9 s

-1

contours in the lower panel of Figure 4.5-4. The same PV plots (Figure 4.5-5) for Period B 

shows that: (1) the relative vorticity has increased relative to the Sverdrup PV, (2) the 

geostrophic flow field is more chaotic, though the basic eddy structures are identifiable from 

Figure 4.5-4, and (3) the (Ertel) PV perturbations, along the paths of the largest geostrophic 

velocities, have increased in amplitude (e.g. east and west of the cyclone centered ~ 92.5°W). 

Even though the calculation of ζ in the Ertel PV (Equation 2.3-12) is inherently noisy for 

observational data, it is considered that the increases in activity between Periods A and B in the 

PV maps are significant even if all the details cannot be relied upon. 

The Q-vector divergence was calculated for the 200 m depth level, which is at about the deepest 

descent of the 26.5 sigma-t surface, i.e., just below the pycnocline layer (Figure 4.5-6). Positive

and negative Q-vector divergence roughly corresponds to upwelling and downwelling, 

respectively, and again is an inherently noisy calculation involving horizontal gradients of

geostrophic velocity and sigma-t. The Period A map shows alternate patches of divergence and 

convergence around the peripheries of the eddies where the velocities are largest. If a

streamline, corresponding to the position of the strong currents, is followed around the cyclone 

situated between 92 and 94°W, then as the flow enters a region of increasing PV (Figure 4.5-4),

there is a tendency to encounter a region of positive divergence or upwelling. Similarly, if the 

flow along a streamline enters a region of decreasing PV, there is a tendency for convergence or 

downwelling. If the Q-vector divergence map is compared with the 50-150 km filtered 

pycnocline depth in Figure 4.1-1, then, where the up- and down-welling are significant on the 

periphery of the eddies, has a rough correspondence to up (< 0) and down (> 0) displacements, 

respectively, of the pycnocline. This implies that the smaller scales embedded in the eddy field 

are directly linked to (or generated by?) the perturbations around the peripheries of eddies which 

are in turn a result of the instability processes as discussed above. The same kind of patterns are 

found in the Period B maps, but are much harder to discern because of the complexity of the

flow field. The increase in perturbation energy between Periods A and B and the coupling of 

eddy scales to PV perturbations in regions of strong flow imply that small-scale eddies can 

develop rapidly over the northern slope and may account for the complexity of the mapped eddy 

fields discussed in Hamilton et al. (2003). 

4-17
40 cm/s

Geostrophic Velocities

on 25.5 Sigma-t Surface

40 cm/s

Geostrophic Velocities

on 25.5 Sigma-t Surface

Figure 4.5-4. Sverdrup potential vorticity (PV) for layer between the near-surface and the depth 

of the 26.5 sigma-t surface (top panel) for Period A. The lower panel shows the 

Ertel PV for the same layer. Geostrophic velocities for the center 25.5 sigma-t 

surface are overlaid on both maps.

4-18
75 cm/s

Geostrophic Velocities

on 25.5 Sigma-t Surface

75 cm/s

Geostrophic Velocities

on 25.5 Sigma-t Surface

Figure 4.5-5. Sverdrup potential vorticity (PV) for layer between the near surface and the depth 

of the 26.5 sigma-t surface (top panel) for Period B. The lower panel shows the 

Ertel PV for the same layer. Geostrophic velocities for the center 25.5 sigma-t 

surface are overlaid on both maps. Note the change in velocity scale from Figure 

4.5-4.

4-19
25 cm/s

200 m Geostrophic Velocity

Figure 4.5-6. The Q-vector divergence at 200 m overlaid with the 200-m geostrophic velocity 

vectors for Period A.

4-20
In summary, it has been shown that high-resolution hydrographic measurements over the 

northwest central Gulf may be merged with a model field of currents and density assimilated 

with satellite SSH. Both observations and model show increased current intensity and ‘eddy

activity’ (meanders) from Period A (December 5 to 12, 1993) to Period B (December 16 to 23, 

1993). The latter period in particular shows a cyclone over the slope (Figure 4.2-1). The growth

in current intensity from Period A to Period B was shown to be due to mixed barotropic and 

baroclinic instabilities, especially over the on-slope side of the cyclone where isopycnals and

topography are of opposite signs. On the off-slope side the flow is stabilizing and there is a

leakage of eddy energy through the fluctuating pressure work term. 

4-21
5. Contributions to Other Studies 

As part of the modeling and data investigations undertaken by this project, a number of 

contributions were made to other studies that were conducting research into model-data 

assimilation and physical circulation processes in the Gulf of Mexico. The purpose of this 

chapter is to provide summaries of this work as it has direct relationships with the assimilation 

studies reported above. The reader is referred to the appropriate journal papers for details of

these ancillary studies, which include comparison of current data from the DeSoto Canyon Study 

with model simulations and satellite altimetry (Wang et al., 2003), the assimilation of surface 

lagrangian drifter data into the model (Fan et al., 2003), and an investigation into the

mechanisms that contribute the irregular shedding of LCE’s from the LC (Oey and Lee, 2002). 

These three topics are discussed below. Contributions were also made to a companion modeling 

study into the high-speed subsurface jet phenomena that was undertaken by S. DiMarco at Texas 

A&M University (TAMU). This work included providing POM output to TAMU, and analyzing 

output from the model runs, used by this project, for the occurrence of jets. This later 

contribution is discussed in TAMU’s report (DiMarco et al., 2003), and so is not reported here. 

5.1 DeSoto Canyon Comparison 

This study evaluates a data-assimilated model simulation of near-surface circulation in DeSoto 

Canyon (DSC), Gulf of Mexico, with emphasis on analyzing moored current-meter observations 

and comparing them with satellite data and model results. The study period was for two years

from April 1997 to April 1999 and a description of the data is given by (Hamilton et al., 2000). 

The model results are from a high-resolution Gulf of Mexico model forced by analyzed wind and 

surface heat flux that is described in Section 2.1. The assimilation of satellite observations of 

sea-level anomaly (SLA) and SST by the model used the techniques given in Section 2.1.1. Two 

types of data were used to deduce near-surface circulation: moored current meters at thirteen 

locations in the DSC (see Section 2.2), and satellite altimeter-derived, sea level anomaly. The 

moored currents were mapped through multivariate objective analysis to produce surface 

currents and surface geopotentials, against which satellite and model derived sea surface heights 

and geostrophic currents are compared. Coupled patterns between the observations, model 

results, and satellite data, were obtained using the Singular Value Decomposition (SVD) analysis

that is outlined in Section 2.3.3. There were two dominant modes: a ‘single-eddy’ mode where 

currents were concentrated at the foot of the canyon and an ‘eddy-pair’ mode where one eddy 

was at the foot of the canyon and the other, a counter-rotating eddy, was over the head of 

canyon. Mode 1 appears to have been associated with the mesoscale eddy travelling around the 

Loop Current crest and trough, and mode 2 was associated with the intrusion of Loop Current 

crest and trough over the west Florida shelf. The observed and model currents are in good 

agreements about the means and variances. The model currents also appear to be well 

constrained by the steep topography. However, the model velocity field contains only the first 

mode. The satellite-derived velocity field, on the other hand, contains both the first and second 

modes; though, the satellite field does not adequately resolve the velocity structures over the

slope. 

The prediction of shelf edge currents in the DSC requires that the model must be able to 

represent realistically the LC and LCE’s, and include the indirect stochastic effects of frontal 

5-1
eddies. This study presents comparisons of the results from the high-resolution ocean prediction 

experiment, discussed in Chapters 3 and 4, with data from a comprehensive network of moored 

current meter observations. An ocean prediction system, in general, has three basic components: 

an observational network, a dynamical model, and a data assimilation scheme. In the Gulf of

Mexico, satellite remote sensing of sea-level anomaly (SLA) and sea surface temperature (SST) 

provides the only basin-wide observational network. Results from a model experiment, which 

assimilates both satellite SST and SLA and is forced by ECMWF winds and surface heat fluxes, 

were used to compare the near-surface currents and sea surface heights derived from both the 

moored and satellite altimetry data. The study period was two years, covering several major

eddy events. While the utility of satellite altimeter in mapping eddies in the open ocean has long 

been established, relatively little has been explored on assimilating altimeter data on the shelf 

edge. This study, with the DSC as an example, is a major step towards a better formulation of 

ocean prediction systems for shelf break and slope regions and complements the more recent

studies with the slope hydrographic surveys that are given Chapter 4. 

Model and observed mean currents and their corresponding variance (rms velocity) ellipses are 

shown in Figure 5.1-1. All observational data has been 15-day, low-passed filtered so as to be

compatible with satellite SLA maps. Since the slope currents were dominated by eddies, such 

low-passed current observations contain 75 to 85% of the variance of the unfiltered daily data. 

Mean currents were eastward and their magnitudes were small (~5 cm/s) compared to the

fluctuations (~15 cm/s). Both the means and variance ellipses tend to follow the bottom

topography. The model mean currents are generally in good agreements with the observed 

currents, having the same general direction and comparable magnitude. However, the model

means are much too strong (>10 cm/s) near the head of the canyon. The model variances also 

agree well with the observed, however, the model ellipses are narrower than the observations,

suggesting that the model currents are more strongly constrained by the bottom topography. 

SVD (Section 2.3.3) was used to extract patterns in the model currents that are coherent with 

observed currents (means removed). The conventions are that the left data field is assigned to 

observed currents and the right data field is assigned to model currents. The first mode alone

explains 96% of the covariance between observed and model currents (that is, SCF1 = 0.96), and 

the second mode explains 2%. The method captures almost the entire covariance structure in a 

single mode. Figure 5.1-2 shows the first and second spatial patterns of the observed and model

currents. The mode 1 spatial patterns of observed and model currents are comparable in that 

both indicate a single-eddy structure located at the foot of the canyon. (The term “eddy” is used 

here to describe unidirectional flows over the shelf edge. Since the geostrophic flow was nondivergent, the unidirectional flow must somehow veer around the shelf edge, turning either 

cyclonic or anticyclonic.) Also, both the observed and model eddies appear to have been 

‘blocked’ by the steep canyon wall. This shows that the model results contain a high degree of 

reality, as both the model and observed currents indicate similar spatial (and temporal; not 

shown) structures for the dominant fluctuations. The mode 2 spatial pattern consists of an

isolated eddy at the head of the canyon, and the correlation (γ) between observed and model

second modes is 0.51. The fractional variances explained by mode 2 are 10% and 12% 

respectively of the observed and model currents. 

5-2
Figure 5.1-1.

Mean currents (left) and variances (right) at mooring locations from observations 

(upper) and model (lower). Note that the velocity scale for means is 3 times larger. 

Dotted lines are isobaths at 2000, 1000, 200 and 100 m.

5-3

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x3

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x3

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Figure 5.1-2.

First (left) and second (right) spatial patterns of observed (upper) and model 

(lower) currents. The velocity scale is arbitrary. Dotted lines are isobaths at 2000, 

1000, 200 and 100 m.

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The observed currents were also compared with the geostrophic currents derived from satellite

SLA. The 0.25o

 x 0.25o SLA data are first mapped onto the analysis grid, and SVD was used to 

find coupled patterns between observed and satellite-derived currents. The first modes of 

observed and satellite-derived currents are well correlated with γ = 0.83. The first spatial pattern 

(Figure 5.1-3) of observed currents is the same as in the earlier case, showing a single eddy

confined to the deeper part of the canyon. The first spatial pattern of satellite-derived currents 

also indicates a single eddy; however, the eddy seems to penetrate well onto the head of the 

canyon. In other words, the satellite-derived currents were less constrained by the steep canyon 

wall. This is expected since satellite SLA was spatially smoothed with a length scale of 200 km. 

However, it is encouraging that the satellite-derived currents could resolve the eddy structure.

The second spatial pattern of observed currents suggests two counter-rotating eddies splitting 

roughly at the center of the canyon, one at the head of the canyon and the other at the foot. The 

mode 2 velocity pattern of observed currents again seems to be guided by the topography. The 

second spatial pattern of satellite-derived currents also reveals an eddy pair, and again its

velocity structure seems to show more cross-slope motion. 

The first mode explains 87% of the covariance between observed and satellite-derived currents, 

and accounts for 48% and 40% respectively of the observed and satellite-derived variances. The 

second mode explains 10% of the covariance, and accounts for 16% and 18% respectively of the

observed and satellite-derived velocity variances. The first mode is the dominant fluctuation.

The second mode, whose temporal and spatial structures are quite different from the second 

mode derived between observed and model currents, also seems to be quite meaningful. This is 

supported by the fact that the first two EOF modes of observed currents are almost identical to 

the first two modes of observed currents of the SVD analysis. 

The results of this SVD analysis to the two-year observed and satellite-derived near-surface 

currents in DSC revealed two dominant modes of low frequency (“eddy”) fluctuations. Mode 1 

consists of a single eddy trapped near the foot of the canyon, and mode 2 displays an eddy-pair 

with counter-rotating eddies at both ends of the canyon. Satellite SLA is consistent with the 

surface geopotential calculated from the observed currents, in the sense that it contains both

modes. It is expected that satellite SLA is able to reproduce the first mode because satellite data 

is of good quality over the deep part of the Gulf. The presence of the second mode in satellite 

SLA is surprising, because despite the poor local correlations between satellite-derived and 

observed currents, satellite SLA retained the coherent spatial patterns. This suggests the 

possibility of assimilating altimetry measurements, which will have major impacts on coastal 

ocean prediction experiments in the shelf and slope region. 

The dominant mode 1 fluctuations are associated with both cyclonic and anticyclonic eddies. 

The cold cyclonic eddy (LC frontal eddy) has been well documented. In contrast, the 

anticyclonic eddy remains rather obscure. Wiseman and Dinnel (1988) found frequent presence 

of warm water and associated strong eastward currents on the Mississippi-Alabama shelf, 

however, these events were not usually associated with the deep northward extension of LC or 

LCE. It is suggested that those warm intrusions were associated with the mode 1 anticyclonic 

eddy, which according to analysis of the larger scale distribution of SLA (see Wang et al. 2003),

is often present in this region when the LC is at its normal southern position. The model runs 

include assimilation of both SLA and SST, with the model means and current ellipses agreeing

5-5
Figure 5.1-3.

First (left) and second (right) spatial patterns of observed (upper) and satellitederived (lower) currents. The velocity scale is arbitrary. Dotted lines are isobaths 

at 2000, 1000, 200 and 100 m.

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fairly well with the observations. The model currents contain realistic single-eddy structure that 

accounts for about half of the total velocity variance. Moreover, the model currents, unlike the 

corresponding satellite-derived currents, are constrained by the steep slope topography. This is 

encouraging since a principal benefit of employing data assimilation is to supplement partial 

observations with model dynamics. The model currents, however, do not agree as well as the 

satellite-derived currents. This suggests that the data assimilation scheme has not taken full 

advantage of the satellite information (i.e. is not optimal). One solution may be to relax the

(arbitrary) constraint that was imposed on assimilating the satellite SSH only in regions where

the water depths > 500 m. A more serious shortcoming is that the degree of data nudging (see 

equation 2.1-1) was determined by pre-computed correlations between model sea-level anomaly 

and subsurface temperatures. Unfortunately, the correlations are low over the northern Gulf

slope and rise, and the data assimilation has little direct impact in the DSC. To improve the precomputed correlations, the model will need to generate more realistic LC frontal eddies in the 

eastern Gulf. Alternatively, it may be advantageous to use correlations between SSH and 

subsurface temperatures derived from observations such as the DSC study and other large-scale 

deep basin studies (see Chapter 6). Such experimentally derived correlations, supplemented by 

some empiricism, may be more widely applicable than for the immediate region (and time

period) of the observational study. 

5.2 Assimilation of Drifter Data 

Accurate predictions or hindcasts of surface currents is an important goal for models of the Gulf

of Mexico. Improvements in simulated surface currents will allow more accurate prediction of 

the positions and flows of eddies and other flow phenomena that will improve the prediction of 

surface dispersion of pollutants and assist in search and rescue operations. One source of surface 

current measurements is drifter trajectories, which can be tracked in near real-time using 

satellites. This source of data would be expected to complement schemes for the assimilation of

SLA, SST, and hydrography that were discussed above. The interest is in the extent to which 

observed trajectories and circulation may be more accurately modeled when satellite and drifter 

information are used. In the study reported by Fan et al. (2003), a simple nudging scheme was

used to assimilate observed drifters into a circulation model of the Northeastern Gulf of Mexico. 

The scheme was used in conjunction with the OI methods that assimilate satellite SSH and SST 

data, which were discussed above. The results were then compared with a model experiment

that had no assimilation and other experiments with various permutations of SSH, SST and 

drifter assimilations. The study also demonstrated that data assimilation could effect the correct 

placement of Loop Current at the shelf edge, which in turn (remotely) forces the shelf currents. 

Eight surface drifter trajectories were obtained for the period May to June 1998. They were all

deployed in the Northeast Gulf as part of the MMS Northeastern Gulf of Mexico Shelf Physical 

Oceanography Program (NEGOM, DiMarco et al., 2001). Drifter trajectories were smoothed 

using a Gaussian-filter scale of 24 hours to eliminate tidal and inertial currents, and were subsampled at 3-hour intervals (Hamilton et al., 1999). Velocity components were then estimated 

from centered finite differences. The assimilation scheme used nudging of the momentum 

equations to the velocities of the drifters with an equation similar to (2.1-8). In one experiment 

the drifter velocity nearest the grid point was used to nudge, and in another all the drifters were 

used to make a weighted estimate of the nudging factor for each grid point. The 60-day drifter 

5-7
trajectories were divided into 10-day segments to increase sample size and estimate the forecast 

errors over 10-day periods. 

The model experiments were with no assimilation (A), with satellite SSH and SST assimilations

(B), and other various permutations of SSH, SST and drifter assimilations (C, D and E). 

Experiment E has the satellite SSH, SST and weighted drifter assimilation, which gave the best 

results. Numerical trajectories that start and end at the times coincident with the beginnings and 

endings of the 10-day drifter trajectories were calculated using model surface velocities. The

model prediction errors were then calculated as differences between the observed and modeled 

trajectories. Comparison of prediction errors shows that assimilation with satellite SSH, SST 

and drifters yielded the least error. Drifter position errors range from 30 ~ 80 km with a mean of

about 60 km in 10 days, comparable to the errors obtained by Castellari et al. (2001). In 

particular, drifter assimilation can yield small-scale (diameter ~ 100 km) eddies that are 

generally not resolved by satellite data. An example is given in Figure 5.2-1 for two drifters in 

the DeSoto Canyon. Solid lines indicate the observed trajectories, and thin, medium and heavy 

dotted lines are modeled trajectories for experiments A, B and E, respectively. The observed

trajectories indicate a small cyclone in the canyon, and the drifter assimilation (e.g. experiment 

E) could simulate this small current feature while it was missed in the experiments without

drifter assimilation. This study also investigated the effect of data assimilation over the deep 

ocean region and showed that improving the remote forcing leads to a better simulation of 

observed shelf currents (Fan et al., 2003). 

5.3 Causes of Loop Current Variability 

One aspect of the Gulf’s circulation which has thus far remained a mystery is why the Loop 

Current sheds eddies at such irregular, wide-ranging intervals, 3~17 months according to Sturges

and Leben (2000). By systematically examining the model’s response to forcing of varying 

complexities using the ECMWF wind and satellite data (Table 5.3-1), it was shown that windinduced transport fluctuations through the Greater Antilles passages (see Figure 2.1-3) cause 

shedding at shorter intervals (≈ 3~7 months). On the other hand, Caribbean eddies 

(anticyclones) cause shedding at longer periods (≈ 14~16 months). This result is summarized in

Figure 5.3-1. Other significant results from this study are: (1) LCE’s sometimes detach and 

reattach from the LC (Figure 5.3-2), as have also been observed, and (2) there is evidence from 

satellite and model results that southwest of Hispaniola in the Caribbean Sea, anticyclones are 

fairly regularly spun up with periods of about 100 days (Figure 5.3-3). In Figure 5.3-2, the first 

three panels show a LC on the verge of shedding. The LCE is shed or detaches on 1999/8/18. A 

Caribbean eddy (CARE) can be seen to arrive at the channel from 1999/8/18-28, carrying with it

the white-colored trajectories that indicate strong anticyclonic vorticity (ς/f < −0.4), causing the 

LC to make a tight right-hand turn into the Straits of Florida. As the CARE completes its

passage through the channel, part of its mass ‘leaks’ along the Cuban northern coast (1999/9/2-

7), however, the remaining (main) portion interacts with the shed eddy (1999/9/7-12), and the 

two eddies eventually merge (1999/9/12-27). These results are described in more detail in Oey 

et al. (2003a). 

5-8
Figure 5.2-1.

Observed and simulated trajectories for drifter #0012 (upper panel) and #0019 

(lower panel) in the DeSoto Canyon group. Solid lines indicate the observed 

trajectories. Thin (blue), medium (red) and heavy (green) dotted lines are 10-day 

re-sample modeled trajectories of experiments A (no assimilation), B (SST & 

SSH assimilation) and E (SST, SSH & drifters), respectively. 

5-9

Drifter negom20012

89W 88W 87W 86W

28N

29N

30N

Drifter negom20019

89W 88W 87W 86W

28N

29N

30N
Figure 5.3-1.

Histograms of Loop Current eddy shedding periods for Experiments B, C, CS and 

D (see Table 5.3-1). The abscissa is time in months between shedding and the 

ordinate is the number of shed eddies. The result for Experiment C is from a 32-

year time series, while others are from 16-year time series. 

Experiment B : Steady Transport at 55oW

0

2

4

6

8

10

Number of eddies

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Experiment C : ECMWF

0

2

4

6

8

10

Number of eddies

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Experiment CS : Mean ECMWF

0

2

4

6

8

10

Number of eddies

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Experiment D : Assimilated Caribbean Eddies

0

2

4

6

8

10

Number of eddies

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Month

5-10
Figure 5.3-2.

Loop Current eddy detachment and reattachment, experiment CS: Eulerian 

trajectories launched for 5 days centered around the indicated date in each panel, 

and from every 10th grid point at the first sigma level (i.e. surface), superimposed 

on color image of surface elevation (red for values ≥ 0.4 m, blue < -0.6 m). Colors 

on trajectories indicate ζ/f such that black/dark-blue through yellow indicate ζ/f 

from 0 through -0.4, and white for ζ/f < -0.4. Cyclonic trajectories are omitted for 

clarity. Dark contours are the 200-m and 2000-m isobaths. Time interval between 

panels is 5 days.

5-11
Figure 5.3-3.

The second EOF mode (20%) from satellite SSH anomaly (1992-1999; top panel), 

and first EOF mode (26%) from experiment CS (middle panel), and the 

corresponding time series (lower panel; solid: satellite, dotted: model). The region 

is southwest of Hispaniola, as shown. The model time series has been phaseshifted by p in order to show the similarity in periods of fluctuations between the 

model and satellite. Both show periods a 100 days.

5-12
Table 5.3-1. Model Experiments 

All experiments had steady transport specified at 55°W, taken from Schmitz (1996). 

Experiments A, B, CS and D were for 16 years, while experiment C was for 32 years. 

Experiments A and B were initialized from the 10-year run of Oey and Lee (2002), while the 

other experiments were initialized from the eighth year of Experiment B. Note that the 

experiments were designed so that only one forcing at one time was changed. The “−” means 

that the particular forcing was not imposed, i.e. is zero. 

Forcing 

Experiments 

Boundary 

T/S @ 55°W 

Wind Satellite

Assimilation

A Steady

* - -

B Monthly

# - -

C Monthly 6-hourly 

ECMWF - 

CS Monthly

Mean 

ECMWF - 

D Monthly - SSH Anomaly

* Annual Climatology 

# Monthly Climatology 

5-13
6. Summary and Recommendations 

6.1 Summary 

This study has investigated two main issues concerning modeling circulation processes in the 

Gulf of Mexico. They are the characterization of TRW’s in the lower –layer, and the evolution 

of the upper-layer northern-slope eddy field. The former involves the genesis, propagation and 

refraction of TRW’s and how well model TRW’s compare with observed lower-layer currents. 

The latter involves assimilating limited area, but high resolution, hydrographic fields into a 

model and investigating the physical processes of the eddy interactions as the model evolves. A

theme of this study has been the use of observations to improve the performance of models 

through assimilation and comparison with physical processes described by data. 

One aim of this study was to attempt to understand the mechanism(s) by which bottom-trapped 

topographic Rossby waves (TRW’s) over the continental slope and rise of the Gulf of Mexico 

are generated by Loop Current (LC) and Loop Current Eddy (LCE) variability. In Chapter 3 and 

Oey and Lee (2002), evidence for finding TRW’s from a primitive-equation (PE), circulation

model of the Gulf of Mexico, and in locating their possible energy sources was given. A tenyear model simulation forced only by constant transport from the Atlantic was conducted so that 

regular LCE sheddings occured. Deep-layer analyses were then performed to band-pass motions 

with 20 to 100-day periods, a range that corresponds to that found for the observed deep EKE at 

the few available current-meter locations in the Gulf. It was found that in certain well-defined 

regions, over the 3,000-m isobath across the central Gulf in particular, spectral peaks occur 

within these periods; the band-passed EKE is significant (i.e. exceeds a Gulf-wide standard

deviation of 10-3 m2

s

-2), exhibits bottom intensification, and accounts for over 60% of the deep 

energy (Figure 3.1-4). These deep motions, moreover, occur in regions where TRW’s can be

supported, and the phase and correlation analysis suggests offshore and down-slope phase 

propagation consistent with the TRW dispersion relation (Figure 3.1-6). The wavenumber 

vectors computed from the PE model results are consistent with the dispersion relation (Figure

3.1-8), and yield wavelengths 63-210 km. These model-derived wavelengths are shorter than

Hamilton’s (1990) estimates of 110-300 km based on the observations, but more in-line with 

later estimates from more closely spaced arrays (Hamilton et al., 2003). A possible source of 

discrepancy is due to the uncertainty in the Brunt-Väisälä frequency. By utilizing the TRW

dispersion relation to track energy pathways (ray-tracing), it is found that the rays coincide with 

the above region of significant EKE over the 3,000-m isobath (Figure 3.1-7a). East of 91°W, the

source of EKE comes from the LC and LCE-shedding region, while over the western Gulf 

additional source is from the southwestward propagating LCE’s. Since, in a topographically 

dominated region, ray directions depend primarily on the angles between wavenumber 

components, the computed ray-paths are relatively insensitive to wavelengths, so that the (PE) 

modeled region of significant EKE survives despite the possible discrepancy between modeled 

and observed wavelengths. The vertical trapping scales (of the 20 to 100-day bottom motions) 

are estimated to decrease (from ≈ 600 m to 300 m) with distance to the west. While exact values 

differ, the trend agrees with the ray solution (Figure 3.1-7b). These rays indicate that refractions

by topographic gradients and by deep currents with cyclonic shearing play an important role in 

confining the possible energy paths just inshore of the 3,000-m isobath in the central Gulf. In 

regions where rays are focused, the simulated EKE’s increase (Figure 3.1-7a at 91°W, 94°W and 

6-1
95°W). Refraction causes wave shortening, which suggests dissipation as TRWs propagate

westward (Hamilton, 1990). The presence of deep (westward) currents increases the TRW group 

speeds, by 2-3 km day-1. This and the constraints on allowable ray-path imposed by current 

shear may be used to infer deep currents in the Gulf. The ray calculations also suggest that 

energy sources for the TRW’s, found in the PE model, are from short-scale propagating 

meanders (~ 100 to 200 km wavelengths and <100 day periods) around the LC and LCE’s. 

Direct comparison of modeled deep currents with observations shows that energy levels were 

generally underestimated though they improved when assimilation and higher resolution grids 

were used. Comparison of KE spectra shows that the model generates a similar spectral

character to the observations with 20 to 100-day KE being prominent. Bottom intensification, 

characteristic of TRW’s, is also found in the model where bottom currents are energetic. A 

slightly suprising finding is that model currents in the lower water column are less coherent

through the vertical than were the observations. 

In Chapter 4, the report on a first attempt to assimilate directly hydrographic data into a model

that was already using assimilation of Gulf-wide SSH and SST at coarser resolution was given. 

The hydrographic data only covered a limited area of the northern slope region and thus, had to 

be injected into evolving model fields. This was successfully accomplished for a survey of the 

northern Gulf slope during December 5 to 12, 1993. The evolution of the slope eddy field was 

followed for a few weeks and compared with a partial survey of the same region during 

December 16 to 23, 1993. Both the model and observations showed an increase in energy

associated with increased meandering of the shear regions between cylones and anticyclones.

An analysis of the model energetics showed that mixed barotropic-baroclinic instabilities were

feeding the eddy flows. These instabilities were more effective on the northern side of a slope 

cyclone, where the topographic and isopycnal slopes were of opposite sign. On the southern side 

of the cyclone, the analysis showed that the flow is stabilizing and the eddy energy was being 

transferred to deeper water. Hydrographic analysis of PV and Q-vector divergence showed that 

the meandering of PV fronts was related to up- and down-welling on the periphery of the eddies, 

and thus, up and down displacements of the pycnocline. This is an indication of how the

instabilities may develop over the course of a few days to a week time period. 

6.2 Recommendations for Future Model Development 

While progress has been made in the modeling and data analyses of various circulation processes 

in the Gulf, some issues have come out, both in the ability to model and also in new hypotheses 

that may be raised. This subsection describes improvements that can be made to the model and 

also future modeling studies that should be useful to the overall objective of obtaining more

realistic hindcast currents in the Gulf. The next subsection discusses various plausible 

mechanisms and hypotheses that may be explored. 

Although the model can reproduce quite well the near-surface and mid-level energy (as seen

previously in DeSoto Canyon comparison study by Wang et al. (2003), and also in our 

simulation of the slope eddies over the central Gulf, in Chapter 4), it tends to underestimate the 

near-bottom energy. This was discussed previously in conjunction with data from the DeSoto 

Canyon extension study. It is believed that the deficiency in this case is related to the rather poor

6-2
vertical resolution that we have used, ∆z ≈ 350 m outside the bottom boundary layer (BBL).

With a vertical decay scale of TRW’s of about 1,000 ~ 1,500 m, the energy is bound to be 

underestimated. It may be thought that some of the BBL grid points may be relaxed (i.e. less 

grid points in BBL) and be better used to resolve the TRW’s. The situation is not as straight

forward, however, and it possible that the BBL plays a prominent role in subsurface dynamics

over a slope (see below). A better course of action would be to transit fully to the newlyimplemented MPI version of the PROFS code, which has 40 vertical levels and an averaged 

horizontal resolution of about 7 km in the Gulf of Mexico. 

In Chapter 4, a procedure was described through which satellite-SSH assimilation can be 

combined with nudging of the density field from hydrographic surveys to obtain fairly realistic 

features of the slope eddies including the ‘prediction’ of their evolution. The algorithm in this

case is simple and may be expanded. The hydrographic fields can be included as part of the 

overall MVOI (Multi-Variate Optimum Interpolation) scheme already used for the satellite SSH 

(and SST) assimilations (Section 2.1.2). The MVOI should ideally be tested using the MMS

Exploratory Physical Oceanography Study presently being conducted over the central Gulf 

(Figure 6.2-1). This Exploratory Study will map eddies and deep currents over a substantial 

region of the north central Gulf using a combination of PIES (Inverted Echo Sounders with 

Pressure that effectively measure the density profile through the water column (Hamilton et al., 

2003)), current meter moorings and deep RAFOS lagrangian floats. Thus, the study will 

examine rather thoroughly currents and density throughout the water column in a region where 

the study of Oey and Lee (2002) has indicated potentially large energy as a result of LC 

pulsations and LCE passages. Here, it would be possible to explore thoroughly the feasibility of

using the various types of data at time intervals of the order of one day. For example, one can 

check the accuracy of the simulation results if only floats are included as part of the MVOI 

assimilation, if only the moorings data are used, or if only satellite data (SSH) are used, etc. The 

skill and benefits of assimilating more observations, velocity, drifters, hydrography, in addition

to satellite information, can be assessed. It would also be useful to explore how models with and 

without assimilation ‘predict’ energetic deep currents, and how well they compare with 

observations. This type of assimilative parametric studies can yield useful information 

especially with regard to optimal use of observations in hindcasting the currents. It can also

suggest the type of data that are most useful for accurate hindcasts. Therefore, the capability can 

now be built in which future MMS observational studies could first be ‘simulated’ so as to 

determine the optimal placement of data points. The potential benefits of such an intelligent 

system, both in terms of its scientific reward and cost saving, are obvious. 

6.3 Recommendations for Future Process Studies 

In addition to the improvements in assimilative modeling that are discussed above, it is important 

that model-data analyses are performed to investigate specific processes that must be accurately 

reproduced by the model if predictions are to be reliable. As more deep-basin observations 

become available from comprehensive experimental studies, such as the ongoing Exploratory 

study, noted above, and other large scale experiments that have been proposed by MMS and 

other agencies, model predictions can be tested. For example, if TRW’s are produced as 

6-3
30 30

29 29

28 28

27 27

26 26

25 25

-95

-95

-94

-94

-93

-93

-92

-92

-91

-91

-90

-90

-89

-89

-88

-88

-87

-87

N6 N5

M2

N4

O3

M1

N3

O2

Q2

N2

O1

O4

L2

L4

L1

L3

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

SS1

SS3

SS2

3000 m

1000 m

Unlabelled Isobaths

1500, 1750, 2500, 3500 m

Exploratory Instrument Sites

 US EEZ

 Full-Depth Moorings

 Short Moorings

 New Short Moorings

 PIES

 Sound Source

2000 m

 Cocodrie 

Figure 6.2-1.Positions of the moored instrument array in the MMS Exploratory Program (April 

2003 to April 2004). Full-depth moorings are instrumented with current and 

temeperature sensors through the water column, short moorings have instruments 

100 and 500 m from the bottom, and PIES are inverted echo sounders with 

pressure that can be used to generate 3-D temperature and salinity maps, and 

bottom pressure maps. The sound sources are used to locate deep RAFOS floats.

6-4
suggested by the model analyses, i.e. through excitation by peripheral frontal eddies (Oey and 

Lee, 2002), then can these processes be verified (or refuted) by detailed analyses of new data. 

Such a finding is clearly potentially significant, as it basically defines the way one must conduct 

future modeling studies, i.e. not only must the LCE’s, be resolved, but also eddies at smaller 

scales O(50 km). Other processes that are suggested by the analyses in Chapters 3 and 4, and 

other model studies, are discussed below. 

The observed erratic motions of the LCE’s suggest that eddy-eddy interaction must play a role

(Hamilton et al., 1999). Moreover, some of the eddies do not have much surface expression and 

therefore, can not be detected by satellite. It has been shown that these subsurface eddies are 

smaller than the LCE’s but are potentially intense. Observationally well-known examples of 

such deeply intensified features are the cyclones that develop east of the LC, and that sometimes 

appear to ‘cut across’ the LC, leading to shedding. The physical mechanisms that produce these 

subsurface, smaller eddies are not well understood and there is little direct observational data. 

Genesis of peripheral eddies through dynamical instability may be one mechanism, but that alone 

may not be sufficient. The ‘violent’ interaction of LCE’s with continental slope may be another, 

potentially important mechanism, as Oey and Zhang (2003, submitted, CSR) have found in a 

process study. In this study, the warm ring initially ‘sits’ over a slope with an adjoining shelf in 

a periodic channel, and its subsequent evolution is examined. The ‘inviscid’ response is cyclonic 

‘peeling-off’ of the on-slope portion of the warm ring. The cyclone propagates away (to the left 

looking on-slope) from the warm ring, and is bottom-intensified as well as slope-trapped (crossslope scale ≈ Rossby radius). The near-surface flow ‘leaks’ further onto the shelf while 

subsurface currents are blocked by the slope. The ‘viscous’ response consists of the formation of 

a bottom boundary layer (BBL) with a temporally and spatially dependent displacement 

thickness. The BBL ‘lifts’ the strong along-slope (leftward) current or jet (> 0.5 m s-1) away

from the bottom. The jet, coupled with weak stratification within the BBL and convergence due

to downwelling across the slope, becomes supercritical. Super-inertial disturbance in the form of

a hydraulic jump or front, with strong upwelling and downwelling cell, and the jet, propagate 

along the slope as well as off-slope and upward into the water column. The upward propagation 

is halted at z ≈ ztrap when mixing smoothes out the ‘jump’ to an along-slope scale λtrap that allows

the ambient jet to bend the propagation path horizontal. At this ‘matured’ stage, ztrap ≈ −250 m, 

λtrap ≈ 50 km, and the jet’s cross-slope and vertical scales are ≈ 30 km and 50 m respectively. A 

sketch of the interactions of such flows with the slope is given in Figure 6.3-1. Oey and Zhang 

(2003) also gives an example that illustrates the process under a more realistic setting in the Gulf 

of Mexico when the Loop Current impinges upon the west Florida slope is given. The 

phenomenon may be relevant to recent oil industry’s measurements in the Gulf, which at times 

indicate jets at z ≈ −150 m through −400 m over the slope. Such intense development of cyclone 

and subsurface jet (together with the potentially important offshore and upward leakage of the 

otherwise slope-bound energy) should be more thoroughly studied in the future. 

Finally, there is an considerable interest in understanding why the LC sheds eddies at such 

irregular intervals, and perhaps more importantly from a practical standpoint, if it can be 

predicted. The study by Oey et al. (2003a) provides a tentative answer. They point out the 

important roles played by velocity and vorticity distributions at the Yucatan Channel. An 

expanded study can be conducted with the recently acquired Yucatan Channel dataset 

(DeepStar). One can combine these data with a series of assimilative model experiments and 

6-5
attempts to understand the response of the modeled LC. One fundamental question is the role of 

Caribbean eddies that squeeze through the channel in affecting the timing of shedding. Another 

relates to the intense cyclonic vorticity that develops at the western portion of the LC in the 

channel, for which models often show downstream amplification. Do these cyclonic

perturbations account for the development of peripheral eddies around the LC, and do the 

cyclonic eddies amplify when the LC impacts upon the west Florida continental slope. These 

issues are both of scientific and practical interests. 

 

6-6
Surface Flow of 

a Warm Eddy

Subsurface Flow 

of a Warm Eddy

Secondary 

Surface Flow

The Coast The Shelf

~ 1000 m

The Slope

x

z

y

The Slope

Main 

Surface 

Anticyclone

Deep 

Anticyclone

Subsurface 

Anticyclone

Deep Cyclone

Weaker 

Subsurface 

Cyclone

Weak 

Surface 

ëCycloneí

Trapping 

Height

Mixing Front

Jet's Path

BBL

The Slope

z = 0

ure 6.3-1.A schematic depiction of the ‘violent’ interaction of a warm eddy with a continental slope (from Oey et al., 

unpublished). See text for descriptions.
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The Department of the Interior Mission

As the Nation's principal conservation agency, the Department of the Interior has responsibility 

for most of our nationally owned public lands and natural resources. This includes fostering 

sound use of our land and water resources; protecting our fish, wildlife, and biological diversity; 

preserving the environmental and cultural values of our national parks and historical places; 

and providing for the enjoyment of life through outdoor recreation. The Department assesses 

our energy and mineral resources and works to ensure that their development is in the best 

interests of all our people by encouraging stewardship and citizen participation in their care. 

The Department also has a major responsibility for American Indian reservation communities 

and for people who live in island territories under U.S. administration. 

The Minerals Management Service Mission 

As a bureau of the Department of the Interior, the Minerals Management Service's (MMS) 

primary responsibilities are to manage the mineral resources located on the Nation's Outer 

Continental Shelf (OCS), collect revenue from the Federal OCS and onshore Federal and Indian 

lands, and distribute those revenues. 

Moreover, in working to meet its responsibilities, the Offshore Minerals Management Program 

administers the OCS competitive leasing program and oversees the safe and environmentally 

sound exploration and production of our Nation's offshore natural gas, oil and other mineral 

resources. The MMS Minerals Revenue Management meets its responsibilities by ensuring the 

efficient, timely and accurate collection and disbursement of revenue from mineral leasing and 

production due to Indian tribes and allottees, States and the U.S. Treasury. 

The MMS strives to fulfill its responsibilities through the general guiding principles of: (1) being 

responsive to the public's concerns and interests by maintaining a dialogue with all potentially 

affected parties and (2) carrying out its programs with an emphasis on working to enhance the 

quality of life for all Americans by lending MMS assistance and expertise to economic 

development and environmental protection.