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1
FRICTION-DOMINATED WATER EXCHANGE IN A FLORIDA ESTUARY
By
KIMBERLY ARNOTT
A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE
UNIVERSITY OF FLORIDA
2009
2
© 2009 Kimberly Arnott
3
To my father
4
ACKNOWLEDGMENTS
I thank my advisor and chair, Dr. Arnoldo Valle-Levinson, for the guidance and
support needed to complete this project. I also thank Dr. Thieke for being on my
committee, as well as Dr. Valle-Levinson’s group of research students, who gave me
insightful comments and suggestions throughout this study.
5
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS .................................................................................................. 4
TABLE OF CONTENTS .................................................................................................. 5
LIST OF FIGURES .......................................................................................................... 6
LIST OF ABBREVIATIONS ............................................................................................. 8
ABSTRACT ................................................................................................................... 10
CHAPTER
1 INTRODUCTION .................................................................................................... 12
Motivation ............................................................................................................... 12 Estuarine Background............................................................................................. 12 Circulation ............................................................................................................... 14 Turbulence .............................................................................................................. 15 Turbulent Kinetic Energy Dissipation Theory .......................................................... 17
2 METHODS .............................................................................................................. 21
Study Area .............................................................................................................. 21 Data Collection ....................................................................................................... 22 Data Processing ..................................................................................................... 23
Tidal Variability ................................................................................................. 24 Subtidal Structure ............................................................................................. 27
3 RESULTS ............................................................................................................... 29
Tidal Variability ....................................................................................................... 30 Exchange Flow ....................................................................................................... 32 Ekman- Kelvin Solution ........................................................................................... 32 Hydrographic Variables ........................................................................................... 33 TKE Dissipation ...................................................................................................... 36
4 DISCUSSION ......................................................................................................... 60
5 CONCLUSION ........................................................................................................ 64
LIST OF REFERENCES ............................................................................................... 65
BIOGRAPHICAL SKETCH ............................................................................................ 67
6
LIST OF FIGURES
Figure page 2-1 Map of Hillsborough Bay Estuary, showing transect line and five
hydrographic stations. ........................................................................................ 28
3-1 Along Estuary Tidal Flow (cm/s) for February 24. A) Transect 1. B) Transect 2. C) Transect 3. D) Transect 4. E) Transect 5. F) Transect 6............................ 38
3-2 Along Estuary Tidal Flow for February 24. A). Transect 7. B) Transect 8. C) Transect 9. D) Transect 10. E) Transect 11. F) Transect 12. ............................. 39
3-4 Across Estuary Tidal Flow for February 24. A) Transect 1. B) Transect 2. C) Transect 3. D) Transect 4. E) Transect 5. F) Transect 6. ................................... 40
3-5 Across Estuary Tidal Flow for February 24. A). Transect 7. B) Transect 8. C) Transect 9. D) Transect 10. E) Transect 11. F) Transect 12. ............................. 41
3-6 Tidal Current Amplitude (cm/s) and Phase (radians) for Along Channel Flow in February 24, as calculated from the least squares fit to the semi-diurnal tide…………….. .................................................................................................. 42
3-7 Tidal Current Amplitude (cm/s) and Phase (radians) for Across Channel Flow in February 24, as calculated from the least squares fit to the semi-diurnal tide………… ....................................................................................................... 43
3-8 Depth Averaged Along Estuary Tidal Flow (cm/s) for Time versus Distance Across for February 24, 2009. ............................................................................ 44
3-9 Residual Along and Across Channel Flow (cm/s) for February 24, as calculated using least squares fit to semi-diurnal tidal cycle. .............................. 45
3-10 Results from Ekman Kelvin Model for Along Estuary Residual Flow using low, middle, and high Ekman numbers ............................................................... 46
3-11 Results from Ekman Kelvin Model for Across Estuary Residual Flow using low, middle, and high Ekman numbers. .............................................................. 47
3-12 Temperature (Celsius) for February 24. A) Transect 1. B) Transect 2. C) Transect 3. D) Transect 4. E) Transect 5. F) Transect 6. The “x” symbols represent the hydrographic stations. .................................................................. 48
3-13 Salinity (psu) for February 24. A) Transect 1. B) Transect 2. C) Transect 3. D) Transect 4. E) Transect 5. F) Transect 6. The “x” symbols represent the hydrographic stations. ........................................................................................ 49
7
3-14 Density Anomaly (kg/m3) for February 24. A) Transect 1. B) Transect 2. C) Transect 3. D) Transect 4. E) Transect 5. F) Transect 6. The “x” symbols represent the hydrographic stations. .................................................................. 50
3-15 Mean Temperature (Celsius), Salinity (psu), and Density Anomaly (kg/m3) Contours for February 24. The “x” symbol represents the five hydrographic stations. .............................................................................................................. 51
3-16 Potential Energy Anomaly (J/m3). A) Transect 1. B) Transect 2. C) Transect 3. D) Transect 4. E) Transect 5. F) Transect 6. G-I) Bathymetry. ....................... 52
3-17 Potential Energy (J/m3) Contours for Time versus Distance Across for February 24. ....................................................................................................... 53
3-18 Mean Potential Energy Anomaly (J/m3) and Bathymetry for February 24. ......... 54
3-19 Turbulent Kinetic Energy Dissipation (m2/s3) using 128 scans for February 24. A) Transect 1. B) Transect 2. C) Transect 3. D) Transect 4. E) Transect 5. F) Transect 6. The “x” symbols represent the stations. .................................. 55
3-20 Turbulent Kinetic Energy Dissipation using 256 scans for February 24. A) Transect 1. B) Transect 2. C) Transect 3. D) Transect 4. E) Transect 5. F) Transect 6. The “x” symbols represent the hydrographic stations. ..................... 56
3-21 Mean Turbulent Kinetic Energy Dissipation using 128 and 256 scans for February 24. The “x” symbol represents the five hydrographic stations. ............ 57
3-22 Time series contours for Station 2. A) Richardson Number. B) TKE Dissipation using 128 Scans. C) TKE Dissipation using 256 Scans. .................. 58
3-23 Comparison of Friction and Coriolis Momentum Balance Terms. ....................... 59
8
LIST OF ABBREVIATIONS
Density (kg/m3)
Velocity gradient
Vertical eddy viscosity
Kinematic viscosity
Shear stress
B Basin width
Bi Body force
cics An O(1) constant
cvc Constant related to spectrum in viscous subrange
cw An O(1) constant
DT Diffusivity of heat
f Coriolis
F Fourier transform of
F* Complex conjugate
H Water depth (m)
k Max wavenumber
K Kelvin number
k Rad m-1
k Wavenumber (rad m-1)
kB Batchelor wavenumber
kk Kolmogorov wavenumber
L Length in describing flow
9
q Universal constant
rad Radians
Re Reynolds Stresses
Ri Internal Rossby Radius
T’ Temperature fluctuation
To Temperature center of region
u Sensor velocity relative to water
U Velocity of flow
w Internal waves
z’ Vertical ordinate
α Dimensionless wavenumber
γ & β Are constants
10
Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science
FRICTION-DOMINATED WATER EXCHANGE IN A FLORIDA ESTUARY
By
Kimberly Arnott
December 2009
Chair: Arnoldo Valle-Levinson Major: Coastal and Oceanographic Engineering
The pattern of net exchange flow typically observed in estuaries consists of a
vertically sheared distribution with outflow at the surface and inflow at depth. Theoretical
results of exchange flows dominated by frictional effects under lateral variations in
bathymetry, however, display a laterally sheared distribution with inflow occupying the
deepest portion of the cross-section and outflow over the shoals. There is little
observational evidence to support those theoretical results. Nonetheless, numerical
results in Hillsborough Bay, a branch of Tampa Bay, suggested that the net exchange
flow pattern is consistent with theoretical results for a flow dominated by friction. The
main purpose of this investigation was then to obtain observational evidence that
supported theoretical and numerical results. A 12-hour field survey was conducted on
February 24, 2009, where current velocity measurements and profiles of temperature
and electrical conductivity were collected. Observations from Hillsborough Bay were
compared to numerical model results and to an analytical solution. The along-estuary
tidal currents had amplitudes < 30 cm/s, which were relatively weak when compared to
other estuaries. Tidal current amplitudes were largest at the surface of the channel and
weakest over the shoals. The isotachs mimicked the bathymetry, indicative of frictional
11
influences from the bottom. The tidal current phase distributions showed that the
currents at the bottom and at depth lead the currents at the surface. The observed
residual exchange flow showed a horizontally sheared pattern, with net volume inflow in
the channel and outflow over the shoals. This residual exchange flow compared
favorably with the numerical model results. Given that the theoretical results indicated a
friction-dominated flow pattern, the friction term of the momentum equation was
compared against Coriolis acceleration and plotted over bathymetry. The friction term
was one order of magnitude higher than the Coriolis term, showing that the flow is
dominated by friction. The density distribution showed that the greatest stratification was
in the channel and more mixed conditions were over the shoals. The mixed water
conditions over the shoals are caused by friction from the bottom affecting the entire
water column. Due to the depth of the channel, frictional influences do not affect the
entire water column, allowing for stratification to occur. The distribution of turbulent
kinetic energy dissipation showed that the highest values were in the channel and near
the bottom. The strongest currents and greatest stratification took place in the channel.
Even though this estuary has weak tidal currents, observational evidence showed that
there can still be considerable frictional effects, resulting in a frictionally dominated
exchange flow.
12
CHAPTER 1 INTRODUCTION
Motivation
Pritchard (1956) proposed that the hydrodynamics of a coastal plain estuary is a
balance between pressure gradient and friction. Utilizing a flat bathymetry, this analysis
resulted in a two layer, vertically sheared estuarine exchange flow. This exchange
pattern is characterized by inflow of denser ocean water at depth and outflow of less
dense water at the surface. Wong (1994) revisited this concept using a triangular
bathymetry. This variation in bathymetry created an exchange flow pattern of inflow in
the middle and outflow over the shallow sides. The exchange flow typically observed in
estuaries is a combined vertically and laterally sheared distribution with inflow at depth
and outflow at the surface and on the sides. Numerical results in Hillsborough Bay
(Meyers et al., 2007) showed that the exchange flow pattern was horizontally sheared:
inflow in the entire water column of the channel and outflow over the shoals. This
laterally sheared exchange flow pattern is a highly frictional theoretical condition. There
is little observational evidence that supports this pattern, which motivated an
investigation at the Hillsborough Bay Estuary. The purpose of this analysis is to
compare field observations from Hillsborough Bay with the results of the numerical
model as well as with Valle Levinson (2008)’s analytical solution.
Estuarine Background
The region encompassing the meeting point between the ocean and river is
loosely defined as an estuary. Typically composed of brackish water, estuaries can be
classified by their circulation and can be categorized into four groups: highly stratified,
fjords, partially mixed, and homogeneous. Estuaries are typically described in along-
13
and across-estuary components of momentum where the along-estuary component
runs parallel to the main motion of flow, while the across component runs perpendicular
to the principal axis. Equation 1-1 describes the full momentum equation for the along
estuary component.
(1-1)
This equation is comprised of a balance of local acceleration (first term on the left-hand
side – l.h.s- of the equation), advection (second, third and fourth terms on the l.h.s.),
Coriolis forcing (fifth term on the l.h.s), barotropic (first term on the right-hand side –
r.h.s- of the equation) and baroclinic pressure gradients (second term on the r.h.s.), and
horizontal and vertical mixing (third, fourth, and fifth terms on the r.h.s.) (Valle-Levinson,
2009a). The parameters u, v, and w represent the along, across, and vertical
components of velocity, while f, g, ρ, and Ax, y, z stand for the Coriolis acceleration,
gravity, density, and vertical eddy viscosity in the x, y, and z components. In a partially
mixed estuary, the following assumptions can be made: steady state, linear motion, no
rotation, with friction only occurring in the vertical with a constant AZ (Valle-Levinson,
2009a). With these assumptions, Equation 1-1 reduces to Equation 1-2.
(1-2)
Equation 1-2 demonstrates a balance between pressure gradient and friction. Equation
1-3 represents the mean dynamical balance for the across-estuary component of
momentum.
(1-3)
14
This equation consists of local acceleration (first term on the l.h.s), advection (second,
third and fourth terms on the l.h.s.), Coriolis forcing (fifth term on the l.h.s.), total
pressure gradient (first term on the r.h.s.), and horizontal and vertical mixing in the
lateral direction(second, third, and fourth terms on the r.h.s.). Equation 1-4
demonstrates becomes a geostrophic balance between Coriolis and pressure gradient.
(1-4)
This is the reduction of Equation 1-3, utilizing the following assumptions: steady state,
frictionless, and linear motion. This is the dynamical framework established by Pritchard
(1956) to study estuaries. The water circulation occurring in estuaries is the next
concept to be discussed.
Circulation
Estuarine circulation is the residual movement of water, after the tidal effects have
been removed. Typical estuarine circulation is a density-driven flow, characterized by
denser ocean water entering the estuary along the bottom and less dense freshwater
moving at the surface, toward the ocean. However, circulation can differ depending on
parameters such as basin width, friction, and the effect of Coriolis (Valle-Levinson,
2008). Estuarine circulation can be characterized as vertically or horizontally sheared.
Vertical shearing is defined as outflow of the less dense water at the surface and the
inflow of denser water below. Horizontally sheared exchange flow is described as inflow
occurring in the channel and outflow over the shoals. The transition from vertically
sheared to horizontally sheared exchange has been explained by Valle-Levinson
(2008). Valle-Levinson (2008)’s model is a semi-analytical solution that solves density-
driven exchange flows in terms of Ekman, Ek, and Kelvin, K, numbers:
15
(1-5)
(1-6)
The parameters in the previous two equations are vertical eddy viscosity, Az,
Coriolis forcing, f, water column height, H, basin width, B, and internal Rossby Radius,
Ri. The internal Rossby Radius is a scale where the rotational effects become as
significant as buoyancy effects. The model solves for the along and across estuary
component residual flows. These flows are produced by pressure gradients and are
assumed to only be affected by friction and Coriolis, ignoring advective effects. The
most appropriate way to represent friction in the momentum balance is through
turbulence, which is explained next.
Turbulence
Turbulence is the unstable flow of a fluid and is characterized by random property
changes (McDowell & O’Conner, 1977, pp. 48-51). Turbulent fluid can be thought of as
a collection of eddies which are created by flow instabilities, bed irregularities and wind
and wave action. Turbulent eddies are distorted by velocity gradients, which can
increase the length of the vortex tube while decreasing the area, subsequently causing
the eddy to rotate faster. These distorted eddies are continuously reduced in size until
viscous friction between layers of varying velocities damp out the eddy motion. In this
process, the kinetic energy of the eddy is converted to heat energy (Lewis, 1997, pp.
94-97). Three main sources of velocity shears exist in estuaries: shears from wind,
shears from bottom stresses, and internal shears from velocity gradients in the water
column. In shallow tidal areas, vertical shear commonly occurs from the frictional drag
16
of the bed, with the greatest magnitudes in the principal direction of flow. Strong shears
can occur during the turn of the tide, when differences in phase result in distortion of the
direction of the current over depth.
Turbulence is representative of the non linear terms of the momentum equation
(Hughes & Brighton, 1999, p. 248). The momentum equation used to derive turbulence,
Equation 1-7, assumes the flow is incompressible and the viscosity is constant.
(1-7)
The parameters ρ, ui,j, p, μ, and Bi represent the density, velocity components,
pressure, viscosity, and body force per unit volume. The turbulent kinetic energy
equation (Equation 1-8) is achieved by multiplying the flow by the turbulent flow
momentum equation (Pielke, 2002, p. 167).
(1-8)
(1- 9) The total change turbulent kinetic energy (term on the l.h.s. of the equation) can be
thought of as the balance of the transport of turbulent kinetic energy by advection (first
two terms on the r.h.s of the equation), the shear production (third term on the r.h.s.
side of the equation), viscous dissipation (fourth term on the r.h.s. of the equation) and
the buoyancy production (fifth term on the r.h.s. of the equation). The parameters e, uj,
ui, θ, g, w represent the turbulent kinetic energy, velocity shear, subscale velocity fluxes,
potential temperature, gravity, and vertical velocity. Equation 1- 10 represents turbulent
dissipation.
17
(1-10)
The following section will discuss the theory behind how the turbulent kinetic
energy dissipation, used to investigate the frictional influences in the water column, is
measured with a microstructure profiler
Turbulent Kinetic Energy Dissipation Theory
Turbulent kinetic energy dissipation, ε, is estimated by fitting a theoretical form of
the temperature gradient spectrum to observed data (Soga & Rehmann, 2004). The
observed data are measured with a microstructure profiler that samples at 100 Hz. The
temperature gradient is a physical quantity describing the direction and rate of the
temperature change. A temperature gradient contains five portions: fine structure,
internal waves, inertial convective subrange, Batchelor spectrum, and noise spectra
(Luketina & Imberger, 2001). The higher wave number of the temperature gradient
spectrum is a function of ε and the dissipation of the temperature variance, χT. The fine
structure is observed when the field instrument vertically travels through a stationary
fluid stratified by density. The following vertical temperature profile equation represents
the case where heat is causing stratification.
(1-11)
The variables γ and β are constants, z’ is the vertical ordinate with the center at the
origin of interest, and To is the temperature at the center of the origin (Luketina &
Imberger, 2001). The temperature gradient can then be given by Equation 1-12.
(1-12)
18
Equation 1-13 then becomes the one-sided finestructure power spectrum of the
temperature gradient.
(1-13) F is the Fourier transform of the temperature gradient, the asterisk signifies the
complete conjugate, and k represents the wavenumber. Equation 1-14 is
representative of the temperature gradient spectrum where the internal waves have a
wavelength smaller than the internal Rossby radius.
(1-14) The variable cw denotes the wave speed of the internal waves. The internal waves are
bounded by a max frequency of N which is shown in the following equation.
(1-15)
The fluid density is represented by ρ. The wave number can then be calculated using
the maximum wavenumber, as shown in Equation 1-16.
(1-16) The sensor velocity relative to the water is denoted by u. The inertial convective
subrange portion of the temperature gradient is present for scales that are big enough
to be influenced by viscosity, yet smaller than the maximum wavenumber. The following
equation represents the inertial convective subrange portion of the temperature
gradient.
(1-17)
Cics is a constant and χt is the dissipation of the temperature variance. Equation 1-18
represents the dissipation of temperature variance due to turbulence.
19
(1-18)
(1-19)
DT is the diffusivity of heat, T’ is the temperature, cvc is a constant, and ν is the fluid
viscosity. The Batchelor spectrum segment of the temperature gradient is a derived
temperature gradient spectrum with the assumptions that for high Reynolds turbulence,
the small scale components of the temperature distribution are statistically
homogenous, steady, and isotropic (Soga & Rehmann, 2004). The one-dimensional
Batchelor spectrum is represented by Equation 1-20 (Luketina & Imberger, 2001).
(1-20)
(1-21)
The variable kB represents the Batchelor wavenumber and α is a dimensionless
wavenumber. Equation 1-22 is representative of the normalized Batchelor spectrum.
(1-22)
The final part of the temperature gradient is the noise spectra. This section is created
from noise associated with the sensors or the processing circuitry.
Presently, there are three ways of fitting the observed temperature gradient to
the Batchelor spectrum, from which turbulent dissipation can be estimated. The first
method involves making a graphical fit of the temperature gradient data to the
nondimensionalized Batchelor spectrum (Luketina & Imberger, 2001). A second is to
make a nonlinear least squares fit method of the Batchelor spectrum to the temperature
gradient spectra using high signal to noise levels (Dillion & Caldwell, 1980). The last
20
method uses an algorithm to fit the Batchelor spectrum to the measured spectrum
(Ruddick et al., 2000). The Self Contained Autonomous Microstructure Profiler
(SCAMP, used in this experiment) processing software uses the last method to estimate
the rate of dissipation which is used in this study. The dissipation is estimated by fitting
the Batchelor spectrum and noise spectrum to the observed temperature gradient. The
model noise spectrum filters out noise occurring from the thermistor and the processing
circuitry with a 6-pole low-pass filter (Ruddick et al., 2000). Using the following equation
in conjunction with the measured values of χT, kB becomes the only free variable.
(1-23)
The dissipation, ε is then solved from the Batchelor wavenumber, Equation 1-24.
(1-24)
The algorithm seeks the best kB within a range of 9 x 10-11 m2s-3 to 1.5 x 10-5 m2s-3
(Steinbuck et al., 2009).
Using this background knowledge, field observations of hydrographic structure and
tidal flows were investigated and compared to the results of Meyers (2007)’s Estuarine
Coastal Ocean Circulation Model and the results of Valle-Levinson (2008)’s analytical
solution. The methods used for collecting and processing the data will be discussed in
the next chapter.
21
CHAPTER 2 METHODS
The chapter will be presented by a brief overview of the Hillsborough Bay study
area. The techniques of collecting the desired data will be explained, followed by the
description and methodology behind the instruments used in these field observations.
This chapter will conclude with a description of how the data are separated, outlined
and processed.
Study Area
Tampa Bay, located on the west-central coast of Florida, is a drowned riverbed
estuary (Morrison et al., 2006). As Florida’s largest open water estuary, Tampa Bay has
an area of approximately 1030 km2, a shallow mean depth of 4 m, and a drainage area
of 1,930 km2. The Bay is subdivided into four sections: Old Tampa Bay, Hillsborough
Bay, McKay Bay, and New Tampa Bay. Tampa Bay’s watershed reaches from the
Hillsborough River and extends to the Gulf of Mexico. Over 100 small tributaries
contribute to the Bay’s freshwater sources. Shipping channels have been dredged to 14
m and reach from the mouth of the bay through the lower and middle Tampa Bay. From
there the channels are directed toward Old Tampa Bay and Hillsborough Bay.
This investigation was conducted along a transect across Hillsborough Bay which
has a surface area of 96 km2 (Morrison et al., 2006). Being the most industrialized of all
four Bay segments, the cross-sectional bathymetry is characterized by two shoals
separated by a 14 m deep channel. The channel is located biased toward the left
(North-West) shoal, looking into the bay, which is markedly smaller than the right. The
Bay is governed by a mixed (diurnal and semi-diurnal) tide which is often characterized
22
by unequal high and low tides and a maximum spring range of 1 m. The vertical water
column is partially to well-mixed (Morrison et al., 2006).
Data Collection
Current velocity, temperature and conductivity measurements were collected over
one semidiurnal tidal period across Hillsborough Bay on February 24, 2009. The
transect line was 4.5 km in length and contained five vertical hydrographic stations, four
located over the shoals and one in the channel (Figure 2-1). Sampling lasted
approximately 11.35 hours and yielded a total of 12 transect repetitions, 6 of which
included hydrographic transects. Current velocity measurements are necessary to
determine the exchange flow, while temperature and electrical conductivity
measurements are needed to investigate the frictional effects. An Acoustic Doppler
Current Profiler (ADCP) and a Self Contained Autonomous Microstructure Profiler
(SCAMP) were the two instruments utilized in collecting the data.
The RD Instruments Workhorse ADCP used in this investigation measures
profiles of currents by transmitting pings of sound at a constant frequency into the
water. The sound waves returning to the instrument from particles moving away from
the instrument have a lower frequency than those returning from particles moving
toward it. The difference between frequencies is known as the Doppler Shift, and is
used to calculate the velocity of the particle and subsequently water surrounding it. The
1200 kHz ADCP was positioned on a small catamaran and towed off the starboard side
of the boat. The boat traveled at a speed of 1.5 to 2 m/s. The beam range was from 1.7
m to 14.7 m and each ping was recorded at .5 m bins. The ping rate was 2 Hz with a
beam angle of 20°. Currents were measured in North- South and East- West
components. WinRiver software was used to collect the data obtained from the
23
instrument which incorporated navigational data collected from a Garmin Global
positioning system (GPS).
The Self Contained Autonomous Microstructure Profiler (SCAMP) is a small,
lightweight device that measures small scale values and fluctuations of temperature and
electrical conductivity. Developed by Precision Measurement Engineering (PME), the
SCAMP samples at a rate of 100 Hz and can be deployed either ascending or
descending, depending on the area of interest. For this particular investigation, the
bottom of the water column was of interest and the descending mode was used. This
instrument was utilized to investigate the turbulence occurring in the water column and
to relate that to friction. The instrument was weighted and released directly downward at
a rate of 10 cm/s until it reached the bottom. Data from casts were recorded internally
and uploaded onto a computer. The software supplied allowed for calibration, data
acquisition and shows a graphical display of the previous cast’s parameters such as
velocity, temperature and salinity profiles. MATLAB is used for analysis in which salinity
and density can be computed and turbulent kinetic energy dissipation can be derived.
Data Processing
To further explore tidal variability, the ADCP data were converted to ASCII files and
loaded into MATLAB for analysis. The raw data were arranged into a large matrix,
where velocities were corrected by taking into account the ship’s velocity (Joyce, 1989).
Finally, the origin was defined to separate the large data set into transect repetitions
and the data were interpolated onto a regular grid. Time was either measured or
converted to Greenwich Mean Time (GMT). The process for calculating the residual
exchange flow pattern in order for it to be compared with the numerical model and
theoretical results is discussed next.
24
Tidal Variability
The E-W and N-S current velocities were rotated into along and across estuary
components. To find the principal axis of maximum variance, N-S velocities were plotted
along the y axis, and the E-W velocities were plotted along the x axis. A trend line was
determined, and the angle between this line and the x axis was computed. This angle
was needed to rotate the flows in order to achieve the appropriate along and across
estuary components. A grid of current measurements for the cross-section looking into
the estuary was created for each transect, resulting in 29 rows and 179 columns, with
vertical spacing of .5 m and a horizontal spacing of 25 m. Using the current velocities,
flow contours were created for depth versus distance across for each transect in the
along and across components. A mean bathymetry was calculated and plotted onto
each of these contours, masking the lower 10% to account for error from the ADCP’s
side lobe effects. The grid cells of the five hydrographic stations were found using the
latitude and longitude coordinates. Contours of along and across estuary flow were also
plotted with depth versus time for each of the five hydrographic stations. Using a least
squares technique, the data were fitted to a periodic function with a semidiurnal (12.42
hr) harmonic. The amplitude and phase (necessary to investigate frictional influences
from the bottom) as well as the residual exchange flow were obtained from this fit.
These contours were plotted over bathymetry for the along and across components.
After determining the tidally averaged flow patterns, it was necessary to compute
the theoretical exchange flow patterns to compare with the observed exchange flow.
The model described by Valle-Levinson (2008) was used to obtain theoretical along and
across estuary flows using the observed bathymetry and various values of vertical eddy
viscosity, Az. The Kelvin number used in the analysis was .49. Three values of Az were
25
used (1e-04, 10e-04, and 20e-04 m3/s). These three values were chosen to represent low,
moderate, and high frictional influences. These calculations were plotted against
bathymetry and used to compare with the observed residual flows to determine the
influence of frictional effects.
To investigate the frictional effects on the hydrographic variables, the SCAMP
processing software was used to extract profiles of temperature, salinity, and density for
each drop, which resulted in 6 casts per station. The data were interpolated onto a
uniform grid and temperature, salinity, and density contours were created for depth
versus distance across with the bathymetry plotted on top. This was completed for all
transects.
To study the friction term of the momentum balance, the turbulent kinetic energy
must be examined. To calculate the turbulent kinetic energy dissipation, the profiles
must first be separated into segments before the TKE dissipation can be estimated.
Several methods are currently being used to divide the profile into segments of SCAMP
data, which are each individually fitted to the Batchelor spectrum as discussed in
Chapter 1. Supplied with the SCAMP processing software was the option to use either
an adaptive method or a stationary segment method. For this investigation, the rate of
turbulent kinetic energy dissipation was processed using the stationary segmentation
method. Dissipation rates were calculated using 128 and 256 scans per segment. The
dissipation estimates for each drop along with the associated mean depths were
extracted for every cast. The interpolated dissipation contours were plotted for depth
versus distance across for all the transect repetitions. In addition, dissipation time series
26
contours were created for each of the five stations for the duration of the sampling
period using 128 and 256 scans per segment.
Given that stratification is known to suppress turbulence, the areas of high
stratification are of interest. In order to investigate the variations in stratification, the
potential energy anomaly, Equation 2-1, was utilized. This is a measurement of the
stratification of a whole water column and is representative of the potential energy
deficit in the water column (departure of the water’s column center of mass from mid-
depth) due to stratification. The mean density, ρm, was calculated for each column
(McDowell & O’Conner, 1977, pp. 48-51) and then the potential energy anomaly, Φ
(Simpson et al, 1990).
(2-1) Values of Φ were then plotted over the bathymetry for each of the hydrographic transect
repetitions. Potential energy anomaly contours were also generated for time versus
distance across the estuary, to observe the temporal stratification variation.
To look at the influences of velocity gradients and density gradients from an
energy standpoint, the Richardson number was utilized (Equation 2-2).
(2-2)
The Richardson number is a dimensionless ratio that determines the importance of
mechanical energy and buoyancy effects in the water column. It is the ratio of buoyancy
production and shear production. When Ri is small (< .25), velocity shears are
considered significant enough to overcome the stratifying effects of density. This
concept will be compared to temporal variations of TKE dissipation to see if there is any
27
correlation of buoyancy and shear production with dissipation. The next section
describes the time averaged distribution of the hydrographic variables, which was used
to investigate the temporal influences of friction on these parameters.
Subtidal Structure
In order to study the temporal influences of friction on hydrography, the
temperature, salinity, density, and TKE dissipation were averaged over time. From the
previously calculated temperature, salinity, density and dissipation data, tidally
averaged distributions were computed and plotted for the cross-section sampled.
28
Figure 2-1. Map of Hillsborough Bay Estuary, showing the transect line and five hydrographic stations.
29
CHAPTER 3 RESULTS
The results of this investigation are presented in terms of tidal flow variability,
residual flow and it’s comparison to the Ekman- Kelvin solution, hydrographic variables,
and TKE dissipation sections. Within the tidal variability section, the tidal flow phase and
amplitude and residual exchange flow are calculated. The tidal phase and amplitude are
used to investigate the frictional influences on the flow from the bottom. The observed
exchange flow is used to compare to the numerical model and semi-analytical solution
results, which was subsequently calculated. The results of the semi-analytical solution
are shown in the Ekman-Kelvin parameter space. These results are used to compare to
the observed exchange flow, to make inferences on the whether the pattern is being
influenced by low, moderate, or high frictional conditions. In order to examine the
frictional influences on hydrography, the temperature, salinity, density, and potential
energy anomaly are shown as transect repetitions and time averaged contours in the
hydrographic variables section. Given that the most appropriate way to represent friction
is through turbulence, the following section presents the results for the turbulent kinetic
energy distributions. These results were calculated using 128 and 256 scans and are
shown as transect repetitions and time averaged contours. In order to examine the
influences of velocity and density gradients on TKE dissipation, the Richardson number
was used. The time series contours of TKE dissipation using 128 and 256 scans for
Station 2 were compared to the Richardson number contours to see if any correlations
exist between them. After examining the frictional influences from an energy
perspective, the subsequent section explores the frictional effects on the momentum
30
balance. The friction and Coriolis terms from the momentum equation were plotted over
bathymetry, in order to determine the dominating force in the momentum balance.
Tidal Variability
The along-channel tidal velocities varied markedly each of the 12 transect
repetitions and ranged from -30 to 50 cm/s (Figures 3-1 and 3-2). The across-estuary
tidal current velocities showed positive and negative values (Figures 3-4 and 3-5). The
positive currents were representative of across-estuary currents traveling to the left
(looking into the estuary) of the cross-section (North-West), and the negative values
indicated current traveling to the right (South-East) and these current velocities ranged
from -30 to 20 cm/s. The initial conditions began with strongly positive along estuary
flow, flood tide, in the bottom of the channel and weak (~0 cm/s) velocities over the
shoals and at surface waters of the channel. Negative across estuary currents were in
the channel and positive values in the right shoal. The along-estuary flow progressively
strengthened across the entire cross-section, where it was strongest throughout the
entire water column of the channel and was weaker over the shoals. Negative across
estuary flow increased as the flood waters increased, with peak values near the surface
and decreasing positive flow along the right shoal. The isotachs of constant flow velocity
followed the bathymetry over the shoals indicating bottom friction effects. The along-
estuary current velocities eventually decreased and the flow became weak in the
channel and close to zero over the shoals. Negative across-estuary velocities
decreased as the flood waters decreased, with most velocities nearly zero except over
the surface waters of the channel. The current velocities became negative first over the
shoals and remained positive in the channel, before eventually becoming negative,
indicating ebb tide. Ebb tide developed everywhere except in the lower half of channel,
31
where the current was nearly zero. Across-estuary velocities eventually became positive
over the right shoal, during ebb. The sampling concluded with strongly ebbing
(negative) along estuary currents near the surface, weakening with depth until they
reached positive values near the bottom. The greatest positive across-estuary current
was in the upper surface waters of the channel and the left shoal, where the velocity
decreased with depth.
The along-estuary tidal current amplitude ranged from 0 to 30 cm/s, and depicted
the greatest amplitude near the surface over the channel and left shoal (looking into the
estuary), where it weakened with depth. The lowest amplitude was located along the
right shoal, which also decreased with depth. The isopleths of the amplitude contours
followed the bathymetry. The phase for along channel flow was measured in radians
and ranged from -1.4 to 0.4. The smallest phase for the along channel component was
present in the far right shoal, located along the bottom as well as the right wall of the
channel. The largest phase is in the surface waters above of the channel (Figure 3-5).
This indicated that semidiurnal tidal currents changed earlier over shoals, relative to the
channel, and near the bottom, relative to the surface. The across-estuary tidal current
amplitude ranged from 0 to 16 cm/s and was greatest at the surface waters over the
channel and left shoal where it decreased with depth. Weaker amplitudes were over the
right shoal. The tidal current phase ranged from 3.14 to -3.14 rad (Figure 3-6). The
greatest values were over the left shoal and channel and the smallest values were mid-
distance across the cross-section as well as along the bottom of the channel. The depth
averaged along-estuary tidal flow (cm/s) for time versus distance across was calculated
to show the transition between flood and ebb tide across the transect (Figure 3-7). This
32
showed the strongest flows in the channel for both flood and ebb tides. The transition
between flood and ebb took place between the hours of 20 and 21, with the shoals
leading the channel. The next section describes the observed exchange flow results,
which was needed to compare with the numerical model results.
Exchange Flow
The observed along-channel residual exchange flow ranged from -5 to 25 cm/s
and was strongly positive in the channel and left shoal, where it increased with depth
(Figure 3-8). Negative flow existed on the far right shoal, the surface waters of the
channel and adjacent portion of the right shoal. The isotachs followed the bathymetry
over the right shoal, indicating frictional influences from the bottom. The across-channel
residual flow, which ranged from -5 to 6 cm/s, showed positive values near the surface
over the left shoal and far right shoal. Negative and weak (~0 cm/s) flow values were
mid-depth of the channel and shoals. Given that the observed exchange flow has been
calculated, the theoretical exchange was used to find indications of frictional influences
that are causing the pattern.
Ekman- Kelvin Solution
The results of the model for the along-estuary component mean flow showed that
under low friction, the isotachs were horizontal and a vertically sheared pattern
developed. This pattern featured inflow at depth of the channel, and outflow at the
surface (Figure 3-9). Under moderate friction, a combined horizontal and vertical
sheared exchange flow was observed. This pattern showed inflow at depth in the
channel, and outflow at the surface as well as over the shoals. Under high friction,
horizontally sheared exchange flow was observed. The frictional influences allows for
outflow to occur over the shoals, while net volume inflow intrudes in the channel. The
33
across-estuary component of residual flow for the low frictional condition showed
negative flow (South-East) along the surface and positive (North-West) flow beneath it
(Figure 3-10). For the moderate frictional conditions, the solution showed positive
(North-West) flow throughout the water column of the channel and negative (South-
East) flow over the shoals. For the high frictional conditions, the flow was negative
(South-East) in the channel and positive (North-West) over the shoals. Provided that the
theoretical solution indicated a highly frictional condition causing this exchange, patterns
from frictional influences on hydrography were used to verify this condition.
Hydrographic Variables
The hydrographic variables were examined to investigate the frictional influences
from the bottom. Temperature over the sampling period ranged from 16 to 18°C (Figure
3-11). The survey began with the lowest temperatures located on the left shoal and
generally increased from left to right. Temperature was characterized by sharp
gradients along the shoals and upper waters of the channel. Progressively, the
temperature over the right shoal developed a trend where the highest values were
located near the surface, decreasing with depth and marked by horizontal isotherms.
The channel was distinguished by sharp temperature gradients. This trend grew with an
expanding thermocline that eventually reached across the entire cross-section. As the
sampling concluded, the thermocline was marked by crowded isotherms in the first few
meters of water. Below the thermocline, the isotherms transitioned vertically, indicating
a uniform temperature water column. The temperature was much cooler with the
minimum temperature values located along the bottom of the left shoal and channel.
Salinity over the sampling period ranged from 30 to 32 psu (Figure 3-12). Salinity
was low in the surface waters of the left shoal, and increased from left to right across
34
the estuary, marked by sharp salinity gradients. The salinity increased with depth,
separated by layers of horizontal isohalines, indicating a stratified water column, with
the highest values in the shipping channel. As time progressed, the cross-section
showed low values of salinity located everywhere except in the shipping channel, where
the salinity increased with depth. Eventually, this high salinity area in the channel began
to increase encompassing the surface waters over the channel and the initial portion of
the adjacent shoals. The salinity increases with depth in the channel and sharp salinity
gradients appeared on the left and right side of the channel. Eventually the right side of
the cross-section showed a halocline with the lowest values of salinity located along the
surface, where it increased with depth and were separated by crowded horizontal
isohalines, indicating stratified conditions. The left side of the cross-section had vertical
isohalines and decreased from left to right. The survey concluded with a salinity
gradient along the entire cross-section, where the salinity distribution was increasing
with depth.
The density anomaly over the sampling period ranged from 22 to 32 kg/m3 (Figure
3-13). This density structure initially showed the lowest values over the upper left shoal,
where it gradually increased from left to right, marked by sharp density gradients. The
highest values were found in the channel, increasing with depth. As the tide progressed,
density across the transect transitioned to low values of density everywhere except in
the channel. Eventually, the low density water shifted to the right shoal, and higher
values were found along the channel and left shoal, which increased with depth. This
area of high density broadened to encompass part of the adjacent right shoal. The
sampling concluded with the entire cross section showing a density distribution that
35
increased with depth, separated by horizontal isopycnals which indicated stratified
conditions. As seen, the water density structure followed the salinity structure closely.
Time-averaged temperature showed maximum temperatures along the surface
that decreased with depth to minimum values located in the channel and along the
bottom of the shoals. Horizontal isotherms were present across the entire sampling
transect distance. Time-averaged salinity contours showed the lowest values along the
surface and in the far right shoal. The salinity distribution increased with depth to the
maximum values located in the channel. Horizontally aligned isohaline were everywhere
with the exception of the far right shoal, where the isohaline transitioned vertically,
indicating a mixed water column. The time averaged density distribution showed the
lowest values along the surface and far right shoal. The density increased with depth,
reaching maximum values in the channel. The density distribution was characterized by
horizontal isopycnals everywhere except the far right shoal, where vertically oriented
isopycnals were present (Figure 3-14).
To find out where the stratification was the greatest across the transect, the
potential energy anomaly was used. Peak values of potential energy anomaly were in
the channel for all six hydrographic transect repetitions (Figure 3-15). This makes sense
because this is the area of highest stratification. The first transect decreased linearly
over the right shoal and ranged from 1 to 5.5 Jm-3. The second transect had a range
from 2 to 4 Jm-3, and also decreased linearly over the right shoal before reaching a
minimum on the right side of the mid-shoal spike in the bathymetry. From there the
potential energy anomaly slightly increased. The same trend was observed for the third
and fourth transects with a notably smaller range of 0 to 2 Jm-3. The fifth and six
36
transects peaked in the channel and decreased linearly over the right shoal, ranging
from 0 to 4 Jm-3. The potential energy anomaly time series contours for time versus
distance across (Figure 3-16), showed highest values in the channel, between 14 to 17
hrs and 21 to 22 hrs. The mean potential energy anomaly ranged from 0 to 3.5 Jm-3 and
showed the highest values in the channel (Figure 3-17). Turbulent kinetic energy
dissipation is one of the most appropriate ways to look at friction directly and these
results were investigated next.
TKE Dissipation
Turbulent kinetic energy dissipation distribution ranged from 10 -8 to 10 -4 m2s-3
over the sampling period (Figure 3-18). The first transect repetition (using the 128 scans
per segment processing method) displayed the highest dissipation values along the left
shoal, near the bottom of the bathymetry of the far right shoal, and the bottom of the
channel. Lower values are mid-distance across the transect line. The second transect
repetition showed the highest dissipation in the left shoal and shipping channel.
Generally, the left side of the transect showed higher values than those of the right. The
third transect showed maximum values located over the left shoal and bottom of the
channel. The left side of the cross-section showed higher dissipation than the right. The
fourth transect showed the highest values along the bathymetry of the right shoal and in
the channel. The surface waters had the lowest dissipation. The fifth transect showed
the highest dissipation in the channel and left shoal, again decreasing from left to right.
The final repetition, showed the highest dissipation on the left shoal and mid-depth of
the channel.
37
Generally being very similar to the 128 scans per segment method, the 256 scans
per segment showed the highest values of dissipation in the channel or near the
bathymetry (Figure 3-19). The only exception is the fourth transect, which showed high
dissipation near the surface of the left shoal and channel.
The mean turbulent dissipation shows very little variation between the 128 and
256 scans per segment (Figure 3-20). The highest values were along the bottom of the
left shoal, mid-depth of the channel and along the bottom of the right shoal. The lowest
values were in the waters above the high dissipation values along the right shoal
(Figure 3-21).
To examine the role of velocity and density gradients on dissipation, the
Richardson number was calculated. The Richardson number time series contours
ranged from 0 to 2.5 (Figure 3-22). This concept was utilized to see if a correlation
existed between time series contours of Richardson number and TKE dissipation. Along
the bottom, the Richardson number was consistently low, while the TKE dissipation was
high. Other than this trend, there was no distinct correlation between these contours.
To investigate friction from the momentum balance, the friction and Coriolis terms
were plotted over bathymetry (Figure 3-23). The results showed that friction dominates
the flow over Coriolis, with friction being one order of magnitude higher than Coriolis.
38
Distance Across (km)
Depth
(m
)
(a) Transect 1
1 2 3 4-14
-12
-10
-8
-6
-4
-2
Distance Across (km)
Depth
(m
)
(b) Transect 2
1 2 3 4-14
-12
-10
-8
-6
-4
-2
Distance Across (km)
Depth
(m
)
(c) Transect 3
1 2 3 4-14
-12
-10
-8
-6
-4
-2
Distance Across (km)
Depth
(m
)
(d) Transect 4
1 2 3 4-14
-12
-10
-8
-6
-4
-2
Distance Across (km)
Depth
(m
)
(e) Transect 5
1 2 3 4-14
-12
-10
-8
-6
-4
-2
-20
-10
0
10
20
30
40
50
Distance Across (km)
Depth
(m
)
(f) Transect 6
1 2 3 4-14
-12
-10
-8
-6
-4
-2
Figure 3-1. Along Estuary Tidal Flow (cm/s) for February 24. A) Transect 1. B) Transect 2. C) Transect 3. D) Transect 4. E) Transect 5. F) Transect 6.
39
Distance Across (km)
Dep
th (m
)
(a) Transect 7
1 2 3 4-14
-12
-10
-8
-6
-4
-2
Distance Across (km)
Dep
th (m
)
(b) Transect 8
1 2 3 4-14
-12
-10
-8
-6
-4
-2
Distance Across (km)
Dep
th (m
)
(c) Transect 9
1 2 3 4-14
-12
-10
-8
-6
-4
-2
Distance Across (km)
Dep
th (m
)
(d) Transect 10
1 2 3 4-14
-12
-10
-8
-6
-4
-2
Distance Across (km)
Dep
th (m
)
(e) Transect 11
1 2 3 4-14
-12
-10
-8
-6
-4
-2
-20
-10
0
10
20
30
40
50
Distance Across (km)
Dep
th (m
)
(f) Transect 12
1 2 3 4-14
-12
-10
-8
-6
-4
-2
Figure 3-2. Along Estuary Tidal Flow for February 24. A). Transect 7. B) Transect 8. C) Transect 9. D) Transect 10. E) Transect 11. F) Transect 12.
40
Distance Across (km)
Depth
(m
)
(a) Transect 7
1 2 3 4-14
-12
-10
-8
-6
-4
-2
Distance Across (km)
Depth
(m
)
(b) Transect 8
1 2 3 4-14
-12
-10
-8
-6
-4
-2
Distance Across (km)
Depth
(m
)
(c) Transect 9
1 2 3 4-14
-12
-10
-8
-6
-4
-2
Distance Across (km)
Depth
(m
)
(d) Transect 10
1 2 3 4-14
-12
-10
-8
-6
-4
-2
Distance Across (km)
Depth
(m
)
(e) Transect 11
1 2 3 4-14
-12
-10
-8
-6
-4
-2
-25
-20
-15
-10
-5
0
5
10
15
20
Distance Across (km)
Depth
(m
)
(f) Transect 12
1 2 3 4-14
-12
-10
-8
-6
-4
-2
Figure 3-4. Across Estuary Tidal Flow for February 24. A) Transect 1. B) Transect 2. C) Transect 3. D) Transect 4. E) Transect 5. F) Transect 6.
41
Distance Across (km)
Dep
th (
m)
(a) Transect 7
1 2 3 4-14
-12
-10
-8
-6
-4
-2
Distance Across (km)
Dep
th (
m)
(b) Transect 8
1 2 3 4-14
-12
-10
-8
-6
-4
-2
Distance Across (km)
Dep
th (
m)
(c) Transect 9
1 2 3 4-14
-12
-10
-8
-6
-4
-2
Distance Across (km)
Dep
th (
m)
(d) Transect 10
1 2 3 4-14
-12
-10
-8
-6
-4
-2
Distance Across (km)
Dep
th (
m)
(e) Transect 11
1 2 3 4-14
-12
-10
-8
-6
-4
-2
-25
-20
-15
-10
-5
0
5
10
15
20
Distance Across (km)
Dep
th (
m)
(f) Transect 12
1 2 3 4-14
-12
-10
-8
-6
-4
-2
Figure 3-5. Across Estuary Tidal Flow for February 24. A). Transect 7. B) Transect 8. C) Transect 9. D) Transect 10. E) Transect 11. F) Transect 12.
42
Distance Across (km)
Depth
(m
)
Tidal Current Amplitude (cm/s) for Along Channel Flow
0.5 1 1.5 2 2.5 3 3.5 4-14
-12
-10
-8
-6
-4
-2
5
10
15
20
25
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
Distance Across (km)
Depth
(m
)
Tidal Current Phase (radians) for Along Channel Flow
0.5 1 1.5 2 2.5 3 3.5 4-14
-12
-10
-8
-6
-4
-2
Figure 3-6. Tidal Current Amplitude (cm/s) and Phase (radians) for Along Channel Flow in February 24, as calculated from the least squares fit to the semi-diurnal tide.
.
43
Distance Across (km)
Depth
(m
)
Tidal Current Amplitude (cm/s) for Across Channel Flow
0.5 1 1.5 2 2.5 3 3.5 4-14
-12
-10
-8
-6
-4
-2
2
4
6
8
10
12
14
-3
-2
-1
0
1
2
3
Distance Across (km)
Depth
(m
)
Tidal Current Phase (radians) for Across Channel Flow
0.5 1 1.5 2 2.5 3 3.5 4-14
-12
-10
-8
-6
-4
-2
Figure 3-7. Tidal Current Amplitude (cm/s) and Phase (radians) for Across Channel Flow in February 24, as calculated from the least squares fit to the semi-diurnal tide.
44
Distance Across (km)
Tim
e (h
ours
)
0.5 1 1.5 2 2.5 3 3.5 414
15
16
17
18
19
20
21
22
23
24
-5
0
5
10
15
20
25
30
Figure 3-8. Depth Averaged Along Estuary Tidal Flow (cm/s) for Time versus Distance Across for February 24, 2009.
45
Distance Across (km)
Depth
(m
)
Mean Along Channel Flow (cm/s)
0.5 1 1.5 2 2.5 3 3.5 4-14
-12
-10
-8
-6
-4
-2
-5
0
5
10
15
20
-4
-3
-2
-1
0
1
2
3
4
5
Distance Across (km)
Depth
(m
)
Mean Across Channel Flow (cm/s)
0.5 1 1.5 2 2.5 3 3.5 4-14
-12
-10
-8
-6
-4
-2
Figure 3-9. Residual Along and Across Channel Flow (cm/s) for February 24, as calculated using least squares fit to semi-diurnal tidal cycle.
46
Distance Across (km)
Dep
th (m
)Along Channel Flow in terms of Ekman and Kelvin numbers (K=.49 Az=1E-04)
0.5 1 1.5 2 2.5 3 3.5 4-14
-12
-10
-8
-6
-4
-2
-0.2
0
0.2
0.4
0.6
0.8
Distance Across (km)
Dep
th (m
)
Along Channel Flow in terms of Ekman and Kelvin numbers (K=.49 Az=10E-04)
0.5 1 1.5 2 2.5 3 3.5 4-14
-12
-10
-8
-6
-4
-2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Distance Across (km)
Dep
th (m
)
Along Channel Flow in terms of Ekman and Kelvin numbers (K=.49 Az=25E-04)
0.5 1 1.5 2 2.5 3 3.5 4-14
-12
-10
-8
-6
-4
-2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 3-10. Results from Ekman Kelvin Model for Along Estuary Residual Flow using low, middle, and high Ekman numbers.
47
Distance Across (km)
Dep
th (m
)
Across Channel Flow in terms of Ekman and Kelvin numbers (K=.49 Az=1E-04)
0.5 1 1.5 2 2.5 3 3.5 4-14
-12
-10
-8
-6
-4
-2
-1
-0.8
-0.6
-0.4
-0.2
0
Distance Across (km)
Dep
th (m
)
Across Channel Flow in terms of Ekman and Kelvin numbers (K=.49 Az=10E-04)
0.5 1 1.5 2 2.5 3 3.5 4-14
-12
-10
-8
-6
-4
-2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
Distance Across (km)
Dep
th (m
)
Across Channel Flow in terms of Ekman and Kelvin numbers (K=.49 Az=20E-04)
0.5 1 1.5 2 2.5 3 3.5 4-14
-12
-10
-8
-6
-4
-2
-1
-0.5
0
0.5
Figure 3-11. Results from Ekman Kelvin Model for Across Estuary Residual Flow using low, middle, and high Ekman numbers.
48
Distance Across (km)
Depth
(m
)
(a) Transect 1
0.5 1 1.5 2 2.5 3
0
2
4
6
8
10
12
14
Distance Across (km)
Depth
(m
)
(b) Transect 2
0.5 1 1.5 2 2.5 3
0
2
4
6
8
10
12
14
Distance Across (km)
Depth
(m
)
(c) Transect 3
0.5 1 1.5 2 2.5 3
0
2
4
6
8
10
12
14
Distance Across (km)
Depth
(m
)
(d) Transect 4
0.5 1 1.5 2 2.5 3
0
2
4
6
8
10
12
14
Distance Across (km)
Depth
(m
)
(e) Transect 5
0.5 1 1.5 2 2.5 3
0
2
4
6
8
10
12
14 16
16.2
16.4
16.6
16.8
17
17.2
17.4
17.6
17.8
18
Distance Across (km)
Depth
(m
)
(f) Transect 6
0.5 1 1.5 2 2.5 3
0
2
4
6
8
10
12
14
Figure 3-12. Temperature (Celsius) for February 24. A) Transect 1. B) Transect 2. C) Transect 3. D) Transect 4. E) Transect 5. F) Transect 6. The “x” symbols represent the hydrographic stations.
49
Distance Across (km)
Depth
(m
)
(a) Transect 1
0.5 1 1.5 2 2.5 3
0
2
4
6
8
10
12
14
Distance Across (km)
Depth
(m
)
(b) Transect 2
0.5 1 1.5 2 2.5 3
0
2
4
6
8
10
12
14
Distance Across (km)
Depth
(m
)
(c) Transect 3
0.5 1 1.5 2 2.5 3
0
2
4
6
8
10
12
14
Distance Across (km)
Depth
(m
)
(d) Transect 4
0.5 1 1.5 2 2.5 3
0
2
4
6
8
10
12
14
Distance Across (km)
Depth
(m
)
(e) Transect 5
0.5 1 1.5 2 2.5 3
0
2
4
6
8
10
12
14 30
30.2
30.4
30.6
30.8
31
31.2
31.4
31.6
31.8
32
Distance Across (km)
Depth
(m
)
(f) Transect 6
0.5 1 1.5 2 2.5 3
0
2
4
6
8
10
12
14
Figure 3-13. Salinity (psu) for February 24. A) Transect 1. B) Transect 2. C) Transect 3. D) Transect 4. E) Transect 5. F) Transect 6. The “x” symbols represent the hydrographic stations.
50
Distance Across (km)
Depth
(m
)
(a) Transect 1
0.5 1 1.5 2 2.5 3
0
2
4
6
8
10
12
14
Distance Across (km)
Depth
(m
)
(b) Transect 2
0.5 1 1.5 2 2.5 3
0
2
4
6
8
10
12
14
Distance Across (km)
Depth
(m
)
(c) Transect 3
0.5 1 1.5 2 2.5 3
0
2
4
6
8
10
12
14
Distance Across (km)
Depth
(m
)
(d) Transect 4
0.5 1 1.5 2 2.5 3
0
2
4
6
8
10
12
14
Distance Across (km)
Depth
(m
)
(e) Transect 5
0.5 1 1.5 2 2.5 3
0
2
4
6
8
10
12
14 22
23
24
25
26
27
28
29
30
31
32
Distance Across (km)
Depth
(m
)
(f) Transect 6
0.5 1 1.5 2 2.5 3
0
2
4
6
8
10
12
14
Figure 3-14. Density Anomaly (kg/m3) for February 24. A) Transect 1. B) Transect 2. C) Transect 3. D) Transect 4. E) Transect 5. F) Transect 6. The “x” symbols represent the hydrographic stations.
51
Distance Across (km)
Depth
(m
)(a) Temperature (Celsius)
0.5 1 1.5 2 2.5 3
0
5
10
16
16.5
17
17.5
18
Distance Across (km)
Depth
(m
)
(a) Salinity (psu)
0.5 1 1.5 2 2.5 3
0
5
10
30
30.5
31
31.5
32
22
24
26
28
30
32
y g p y
Distance Across (km)
Depth
(m
)
(a) Density Anomaly (kg/m3)
0.5 1 1.5 2 2.5 3
0
5
10
Figure 3-15. Mean Temperature (Celsius), Salinity (psu), and Density Anomaly (kg/m3) Contours for February 24. The “x” symbol represents the five hydrographic stations.
52
0.5 1 1.5 2 2.5 3 3.50
2
4
6
Distance Across (km)
(PH
I J/m
3)
(a) Transect 1
0.5 1 1.5 2 2.5 3 3.50
2
4
Distance Across (km)
(PH
I J/m
3)
(b) Transect 2
0.5 1 1.5 2 2.5 3 3.50
1
2
Distance Across (km)
(PH
I J/m
3)
(c) Transect 3
0.5 1 1.5 2 2.5 3 3.50
1
2
Distance Across (km)
(PH
I J/m
3)
(d) Transect 4
0.5 1 1.5 2 2.5 3 3.50
2
4
Distance Across (km)
(PH
I J/m
3)
(e) Transect 5
0.5 1 1.5 2 2.5 3 3.50
2
4
Distance Across (km)
(PH
I J/m
3)
(f) Transect 6
0.5 1 1.5 2 2.5 3 3.5-15
-10
-5
0
Distance Across (km)
Depth
(m
)
(g) Bathymetry
0.5 1 1.5 2 2.5 3 3.5-15
-10
-5
0
Distance Across (km)
Depth
(m
)(h) Bathymetry
0.5 1 1.5 2 2.5 3 3.5-15
-10
-5
0
Distance Across (km)
Depth
(m
)
(i) Bathymetry
Figure 3-16. Potential Energy Anomaly (J/m3). A) Transect 1. B) Transect 2. C) Transect 3. D) Transect 4. E) Transect 5. F) Transect 6. G-I) Bathymetry.
53
Distance Across (km)
Tim
e(hr
)
0.5 1 1.5 2 2.5 3 3.5
14
15
16
17
18
19
20
21
22
23
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Figure 3-17. Potential Energy (J/m3) Contours for Time versus Distance Across for February 24.
54
0.5 1 1.5 2 2.5 3 3.50
1
2
3
4
Distance Across (km)
PH
I (J/
m3)
gy y
0.5 1 1.5 2 2.5 3 3.5-15
-10
-5
0Bathymetry
Distance Across (km)
Dep
th (m
)
Figure 3-18. Mean Potential Energy Anomaly (J/m3) and Bathymetry for February 24.
55
Distance Across (km)
Dep
th (m
)
(a) Transect 1
0.5 1 1.5 2 2.5 3
-12
-10
-8
-6
-4
-2
Distance Across (km)
Dep
th (m
)
(b) Transect 2
0.5 1 1.5 2 2.5 3
-12
-10
-8
-6
-4
-2
Distance Across (km)
Dep
th (m
)
(c) Transect 3
0.5 1 1.5 2 2.5 3
-12
-10
-8
-6
-4
-2
Distance Across (km)
Dep
th (m
)
(d) Transect 4
0.5 1 1.5 2 2.5 3
-12
-10
-8
-6
-4
-2
Distance Across (km)
Dep
th (m
)
(e) Transect 5
0.5 1 1.5 2 2.5 3
-12
-10
-8
-6
-4
-2
-8
-7.5
-7
-6.5
-6
-5.5
-5
-4.5
-4
Distance Across (km)
Dep
th (m
)
(f) Transect 6
0.5 1 1.5 2 2.5 3
-12
-10
-8
-6
-4
-2
Figure 3-19. Turbulent Kinetic Energy Dissipation (m2/s3) using 128 scans for February 24. A) Transect 1. B) Transect 2. C) Transect 3. D) Transect 4. E) Transect 5. F) Transect 6. The “x” symbols represent the stations.
56
Distance Across (km)
Dep
th (m
)
(a) Transect 1
0.5 1 1.5 2 2.5 3
-12
-10
-8
-6
-4
-2
Distance Across (km)
Dep
th (m
)
(b) Transect 2
0.5 1 1.5 2 2.5 3
-12
-10
-8
-6
-4
-2
Distance Across (km)
Dep
th (m
)
(c) Transect 3
0.5 1 1.5 2 2.5 3
-12
-10
-8
-6
-4
-2
Distance Across (km)
Dep
th (m
)
(d) Transect 4
0.5 1 1.5 2 2.5 3
-12
-10
-8
-6
-4
-2
Distance Across (km)
Dep
th (m
)
(e) Transect 5
0.5 1 1.5 2 2.5 3
-12
-10
-8
-6
-4
-2
-8
-7.5
-7
-6.5
-6
-5.5
-5
-4.5
-4
Distance Across (km)
Dep
th (m
)
(f) Transect 6
0.5 1 1.5 2 2.5 3
-12
-10
-8
-6
-4
-2
Figure 3-20. Turbulent Kinetic Energy Dissipation using 256 scans for February 24. A) Transect 1. B) Transect 2. C) Transect 3. D) Transect 4. E) Transect 5. F) Transect 6. The “x” symbols represent the hydrographic stations.
57
Distance Across (km)
Dep
th (m
)
Mean TKE Dissipation 128 Scans (m2/s3)
0.5 1 1.5 2 2.5 3
-12
-10
-8
-6
-4
-2
-8
-7.5
-7
-6.5
-6
-5.5
-5
-4.5
-4
Distance Across (km)
Dep
th (m
)
Mean TKE Dissipation 256 Scans (m2/s3)
0.5 1 1.5 2 2.5 3
-12
-10
-8
-6
-4
-2
-8
-7.5
-7
-6.5
-6
-5.5
-5
-4.5
-4
Figure 3-21. Mean Turbulent Kinetic Energy Dissipation using 128 and 256 scans for February 24. The “x” symbol represents the five hydrographic stations.
58
Time (hr)
Dep
th (m
)(a)Richardson Number
14 16 18 20 22-14-12-10-8-6-4-2
0
0.5
1
1.5
2
2.5
Time (hr)
Dep
th (m
)
(b) TKE Dissipation using 128 scans
14 16 18 20 22-14-12-10-8-6-4-2
-8
-7
-6
-5
-4
Time (hr)
Dep
th (m
)
(c) TKE Dissipation using 256 scans
14 16 18 20 22-14-12-10-8-6-4-2
Figure 3-22. Time series contours for Station 2. A) Richardson Number. B) TKE Dissipation using 128 Scans. C) TKE Dissipation using 256 Scans.
59
0 20 40 60 80 100 120 140 160 1800
1
2x 10
-5
(cd
u2)/
H
Friction Term
0 20 40 60 80 100 120 140 160 180-5
0
5x 10
-6
fv
Coriolis Term
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-15
-10
-5
0
Distance Across (km)
Dep
th (
m)
Bathymetry
Figure 3-23. Comparison of Friction and Coriolis Momentum Balance Terms.
60
CHAPTER 4 DISCUSSION
The purpose of this analysis was to compare the observed exchange flow pattern
from Hillsborough Bay with the results Meyer’s (2007) numerical circulation model and
Valle Levinson’s (2008) semi-analytical solution. It was also necessary to find
observational evidence to support the high frictional theoretical condition that is causing
this exchange pattern that is not often observed.
To get an understanding of the water entering and exiting the estuary, the tidal
flows were investigated. The observed along-estuary current velocities began in flood
tide, with the strongest flows in the channel. Slack water, where the tide transitions
from flood to ebb, occurred between the hours of 20 and 21. The ebb tide also showed
the strongest velocities in the channel. The across-estuary component showed
negative (South-East) currents during ebb waters and positive currents during flood.
The strongest currents were at the surface of the left shoal.
The exchange flow pattern, computed using a least squares fit technique, was
found to compare with the results of the numerical model and the theoretical solution.
The along-estuary residual exchange flow showed horizontally sheared distribution, with
full net volume inflow in the channel and weak outflow over the shoals. This horizontally
sheared distribution is among few observations of this exchange flow pattern. This
pattern is consistent with Meyer’s (2007) model results for Hillsborough Bay and is most
consistent with the results of Valle-Levinson’s (2008) semi-analytical solution using a
high Ekman number. The high Ekman number indicated high frictional influences.
The tidal current phase and amplitude were examined to get an understanding of
the frictional influences on the flow. The along-estuary amplitude was strongest in the
61
surface waters of the channel and weakest over the shoals and at depth in the channel.
These weaker flows are due to frictional influences from bottom drag affecting the entire
water column over the shoals. Due to depth of the channel, frictional influences do not
affect the entire water column, allowing for larger currents to take place. The isotachs
mimicked the bathymetry, which also indicated the frictional effect of the bottom. The
phase distribution showed the lowest values were at the surface and in the channel.
The highest phase values were near the bottom and at depth of the channel. The
currents at depth and near the bathymetry of the shoals lead the currents at the surface.
Given that friction is causing weaker flows in these areas, the reversing tide is more
capable of overcoming momentum. These results suggested high frictional influences
occurring over the shoals.
Now, to verify that there were high frictional influences taking place, the
hydrographic variables were examined. The temperature increased throughout the day
and consistently decreased with depth. This is a result of diurnal effects from the sun
heating the surface layers of the cross-section. The salinity and density findings were
very similar, suggesting that salinity governs when calculating the density for this
particular survey. Both salinity and density distribution increased with depth, with the
highest values in the channel. This observation is consistent with the concept that
denser water follows the path of least resistance. The contour lines were horizontally
oriented in the channel, signifying stratified conditions. Over the shoals, the contour
lines were more vertically oriented, which indicated a more mixed condition than in the
channel. This pattern is due to frictional influences from the bottom which affected the
entire water column. Velocity shears from bottom drag created turbulence, which
62
resulted in mixing. Throughout the sampling, the potential energy anomaly was
consistently highest in the channel, showing that the greatest stratification existed here.
The potential energy anomaly contours showed that with time, the greatest stratification
occurred in the beginning and end of the sampling duration.
Given that the tidal phase and amplitude and hydrography indicated highly
frictional conditions, the friction was investigated using the results for the turbulent
kinetic energy dissipation. The 128 and 256 scans per segment turbulent dissipation
distribution regularly showed the highest values and variation existed in the channel.
This is where velocities were the strongest and stratification was the greatest. Peak
dissipation values were also near the bathymetry of the shoals, due to velocity gradients
caused by bottom drag. These peak values were considered high (>10-5 m2s-3) when
compared to other estuaries (Luketina & Imberger, 2001). These high values indicated
that there are high frictional influences taking place compared to other estuaries.
Next, friction was investigated using the concept of energy. Given that, neglecting
transported TKE, turbulent kinetic energy is a balance of shear production, buoyancy
production, and dissipation, the Richardson number was studied. The Richardson
number contours were compared to TKE dissipation time series for Station 2 to see if
there is any correlation between TKE dissipation created by velocity shears or density
gradients. The Richardson number time series contours showed low (< .25) values near
the bottom, where TKE dissipation was the highest, which was expected because of the
high frictional influences previously observed. Other than this trend, there was no
distinct correlation between these three figures. This observation is valid because the
turbulent kinetic energy is a balance of shear production, buoyancy production and
63
dissipation, not just shear and buoyancy production. The highest Richardson number
values occurred mid-duration of sampling.
Provided that the frictional influences have been investigated using the concept of
energy, friction was then analyzed using the momentum balance. The friction and
Coriolis terms were plotted over bathymetry. The frictional term was one order of
magnitude higher than the Coriolis term, indicating that friction is dominating the flow
over Coriolis.
The observed exchange flow pattern compares favorably with the results of the
numerical model and the theoretical solution. Not only does the observed exchange flow
pattern indicate highly frictional conditions, but the tidal phase and amplitude,
hydrography, TKE dissipation and momentum balance suggest high frictional
influences, which supports this claim.
64
CHAPTER 5 CONCLUSION
The main findings of this study showed that the observed residual exchange flow
in Hillsborough Bay compared favorably with both numerical and theoretical results.
Showing a horizontally sheared pattern, it was characterized by net volume inflow in the
channel, and outflow over the left shoal. The other observational evidence showed that
highly frictional influences are evident in this estuary and further reinforced that high
frictional conditions were present. Stronger amplitudes were in the channel and weaker
values were over the shoals, due to frictional influences weakening the flows over the
shoals. The phase distribution showed that the currents over the shoals and at depth in
the channel lead the currents at the surface in the channel. The momentum of the
weaker flow is overcome before the momentum of the water in the channel, attributable
to frictional influences that slow down these flows over the shoals. The depth of the
water column in the channel allowed for stratification to develop, while stratification was
rather weak over the shoals. This is a sign of frictional effects from the bottom affecting
the entire water column over the shoals (creating turbulence that causes mixing). High
frictional influences yield high turbulent kinetic energy dissipation near the bottom when
compared to other estuaries. In conclusion, despite sustaining weak tidal currents, the
Hillsborough Bay Estuary still exhibited significant frictional influences. This highly
frictional condition resulted in a laterally sheared exchange flow that was dominated by
friction and is among few observed examples of this type of exchange flows.
65
LIST OF REFERENCES
Batchelor, G.K. (1959) Small-scale Variation of Convected Quantities like Temperature in Turbulent Fluid, J. Fluid Mechanics, 5, 113-133. Dillion, T.M., & Caldwell, D.R. (1980) The Batchelor Spectrum and Dissipation in the Upper Ocean. J. Geophys. Res., 85, 1910-1916. Hughes, W.F., & Brighton, A.J. (1999) Schaum’s Outline of Theory and Problems of Fluid Dynamics. New York: McGraw-Hill Companies, Inc. Lewis, R. (1997) Dispersion in Estuaries and Coastal Waters. New York: John Wiley & Sons Ltd. Luketina, D.A., & Imberger, J. (2001), Determining Turbulent Kinetic Energy Dissipation from Batchelor Curve Fitting. J. Atmos. Ocean Tech., 18, 100-113. Joyce, T. M., (1989) In situ “Calibration” of Ship-Board ADCPs. J. Atmos. Oceanic Tech. 6, 169- 172. McDowell, D.M., & O’Conner, B.A. (1977) Hydraulic Behavior of Estuaries. New York: John Wiley & Sons, Inc. Meyers S.D., Luther M.E., Wilson M., Havens H., Linville A., et al. (2007) A Numerical
Simulation of Residual Circulation in Tampa Bay. Part I: Low-frequency Temporal Variations. Estuaries and Coasts: Vol. 30, No. 4 pp. 679–697
Morisson, G., Sherwood, E.T., Boler, R., & Barron, J. (2006), Variations in Water Clarity and Chlorophyll a in Tampa Bay, Florida, in Response to Annual Rainfall, 1985- 2004. Estuaries and Coasts, 29, 6A, 926-931. Pielke, R.A. (2002) Mesoscale Meteorological Modeling. San Diego: Academic Press. Pritchard, D.W. (1956) The Dynamic Structure of a Coastal Plain Estuary. Journal of Marine Research 13, 133-144. Ruddick,B., Anis, A., & Thompson, K. (2000) Maximum Likelihood Spectral Fitting: The Batchelor Spectrum. J. Atmos. Ocean. Tech., 17, 1541-1555. Simpson, J.H., Brown, J., Matthews, J. & Allen, G. (1990) Tidal straining, density currents, and stirring in the control of estuarine stratification. Estuaries, 13, 125– 132. Soga, C.L.M., & Rehma, C.R. (2004), Dissipation of Turbulent Kinetic Energy near a Bubble Plumn. Journal of Hydraulic Engineering, 130, 441-119.
66
Steinbuck, J.V., Stacey, M.T., & Monismith, S.G. (2009) An Evaluation of Xt Estimation Techniques: Implications for Batchelor Fitting and ε. American Meteorological Society. Valle-Levinson, A. (2008), Density-driven Exchange Flow in Terms of the Kelvin and Ekman Numbers, J. Geophys. Res., 113. Wong, K.C. (1994), On the Nature of Transverse Variability in a Coastal Plain Estuary. Journal of Gepphysical Research. 14, 209-222.
67
BIOGRAPHICAL SKETCH
Kim began her educational career at the University of Central Florida, enrolled in
the Industrial Engineering Program. After several years, she decided to change her
major to civil engineering and move to Jacksonville. Finishing the final two years of
school at the University of North Florida, she received her bachelor’s degree in May
2008. After graduating, Kim moved to Gainesville to begin graduate school at the
University of Florida in the Coastal and Oceanographic Engineering Program.