JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 105, NO. C6, PAGES 14,215–14,222, JUNE 15, 2000
Internal hydraulics and mixing in a highly stratified estuary
Robert J. Chant
Institute for Marine and Coastal Sciences, Rutgers University, New Brunswick, New Jersey
Robert E. Wilson
Marine Science Research Center, State University of New York at Stony Brook
Abstract. Shipboard acoustic Doppler current profiler and conductivity-temperaturedepth data obtained during highly stratified conditions in the Hudson River estuary along
a section of variable width and breadth are presented. The observations emphasize tidal
period asymmetries in the vertical structure of current and salinity. However, these
asymmetries exhibit significant along-channel structure which is determined by channel
morphology. During the ebb the flow is linearly sheared, and steep halocline slopes in the
vicinity of channel contractions are maintained by momentum advection. A minimum in
vertical shear across the pycnocline occurs in channel contractions. During food the
pycnocline sharpens and flattens with a middepth velocity maximum embedded in the
pycnocline which separates a stratified surface layer from a bottom mixed layer. The
along-channel structure in vertical shear is consistent with a lateral vorticity equation.
Estimates of Richardson numbers suggest that vertical mixing across the pycnocline is
enhanced downstream of channel contractions.
1.
Introduction
Early work describing density-driven circulation in estuaries
focused on tidal residual fields. The seminal observational
work of Pritchard [1954, 1956] suggested that a mean baroclinic
pressure gradient is balanced by a vertical stress divergence,
with advection of momentum playing a smaller yet potentially
significant role. A balance between baroclinicity and vertical
stresses was assumed in the theoretical work of Hansen and
Rattray [1966], whose work produced widely cited estuarine
circulation diagrams and an estuarine classification scheme.
Hansen and Rattray [1996] suggest that estuaries be classified
by two nondimensional parameters, a stratification parameter
and a circulation parameter, both of which, in theory, could be
obtained observationally. Despite the simplicity of this scheme,
actual application of the theory was complicated by difficulties
in obtaining representative tidal mean quantities [Dyer, 1997].
Although both Pritchard [1956] and Hansen and Rattray
[1966] recognized that tidal residual stresses were largely associated with tidal motion, the tidal motion itself was treated
implicitly. More recent work has explicitly discussed the interaction between tides and stratification [Jay and Smith, 1990a, b;
Geyer and Smith, 1987; Geyer and Farmer, 1987]. Based on field
observations Geyer and Farmer [1987] discuss the tidal period
advance and breakdown of a bottom salt front. They characterize the flow in terms of a two-layer Froude number and
conclude that the salt wedge is destroyed during the ebb tide
when the flow becomes supercritical and vertical shears, augmented by bottom stress, increase until stratification is broken
down by vertical mixing.
Jay and Smith [1990a, b] describe the interaction between
stratification and tidal forcing based on a two-layer analytical
model with an interface of finite thickness. They suggest that
Copyright 2000 by the American Geophysical Union.
Paper number 2000JC900049.
0148-0227/00/2000JC900049$09.00
tidally mean motion arises from a flood to ebb asymmetry in
vertical mixing which occurs because of tidal period variations
in stratification. They emphasize the importance of tidal period
internal motion in characterizing the estuary and use an internal Froude number to define the stability of the salt wedge.
However, in contrast to Geyer and Farmer [1989], Jay and Smith
[1990a] conclude that the salt wedge can be destroyed on the
flood if the Froude number exceeds unity. Jay and Smith
[1990a, b] neglect bathymetric effects, which they justify by
scaling arguments.
However, long ago, Stommel and Farmer [1952] recognized
that topographic effects can control halocline structure and
exchange in estuaries. Stommel and Farmer’s [1952] work was
extended by Armi and Farmer [1986] and Farmer and Armi
[1986] to develop a two-layer internal hydraulic theory which
includes effects of channel morphology and barotropic forcing.
Gan and Ingram [1992] used a similar two-layer model to
describe intratidal variability in halocline structure in the
Manitounuk Sound of the southern Hudson Bay, and related
the adjustment of this structure to vertical mixing associated
with internal bores and solitary waves. Partch and Smith [1978]
discuss the importance of internal hydraulic jumps on mixing in
the Duarmish estuary in Washington State, and conclude that
these jumps are associated with channel morphology.
In this paper we describe the effect of channel morphology
on density and velocity structure in a highly stratified estuary
and discuss the implications this structure has on vertical mixing. This description is based on data collected in a 1993 field
program conducted in the Hudson River estuary (Figure 1).
Observations were made along a highly stratified 10 km reach
with pronounced variations in channel cross-sectional area
(Figure 2). The amplitude of the M2, N2, S2, K1, and O1 tidal
constituents at the Battery (Figure 1) are 66, 14, 13, 10, and 5
cm. The survey extended from neap to spring tidal conditions.
The Hudson has an annual mean discharge of 550 m3/s, with
the highest monthly mean discharge of 1000 m3/s occurring
during the month of April.
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2.
Figure 1. Map of survey area. Dots show locations of conductivity-temperature-depth (CTD) stations. Along-channel
sections were run along these transects. Cross-channel sections
were across the channel at stations 5, 6, 7, and 8. Battery is
denoted with B.
In section 2 of this paper we describe the field program. In
section 3 the data are used to describe the spatiotemporal
structure of instantaneous velocity and density fields. In section
4 we present estimates of terms in the along-channel momentum balance. Richardson number estimates are made in section 5. In section 6 a discussion places these observations in
context with a two-layer model and a lateral vorticity equation.
Concluding comments are made in section 7.
Field Program
Between April 29 and May 3, 1993, an acoustic Doppler
current profiler (ADCP) and conductivity-temperature-depth
(CTD) survey was conducted in a 10 km reach of the Hudson
River estuary (Figure 1). The survey included along channel
sections on April 29 and May 3 with cross-channel sections run
the intermediate days. This reach is characterized by a large
fractional change in channel cross section (Figure 2). River
discharge of approximately 2000 m3/s produced strong stratification throughout the survey, though increasing tidal currents
during the survey tended to weaken stratification. Winds were
calm during the survey. Instrumentation included a 1200 kHz
narrowband RD Instruments ADCP and an Applied Microsystems CTD with a sampling rate of 5 Hz. The ADCP had a
right-angle head and was towed abeam on a catamaran, placing
the instrument near the surface. Current measurements were
made at 1 m intervals, with the uppermost bin centered at
1.8 m and the deepest reliable bin at 85% water depth. Additionally, we used a 200 kHz Furino depth sounder to resolve
details in halocline structure. Position was recorded from a
Global Positioning System (GPS).
The survey was designed to provide data suitable for time
series analysis at a fixed point while providing quasi-synoptic
sections. Maximum tow speed of the catamaran was approximately 6 knots (11.1 km/h) through the water. Along-channel
surveys consisted of running repeated transects approximately
every 1.5 hours over a 12 hour period. Each transect consisted of
10 CTD casts at selected stations (Figure 1). Stations are separated by approximately 1 km and were selected to resolve halocline variability associated with variations in channel morphology.
ADCP data were processed and averaged into 1 min ensembles,
which typically resulted in 200 m ADCP footprints. Details of the
lateral surveys are given by Chant and Wilson [1997].
Channel constrictions are located at the George Washington
Bridge at station 7 (km 18 on Figure 2) and between stations
3 and 4 (km 12). A channel expansion is located between
stations 5 and 6 (km 16). The channel tends to deepen in
contractions and shoal in expansions. South of the survey region the channel tends to increase in width and cross section.
However, the increase is not monotonic. Local minimums in
channel cross section occur 1 and 6 km north of the Battery
(Figure 2).
Figure 2. Channel cross-sectional area. Region surveyed is indicated by thick line.
CHANT AND WILSON: HYDRAULICS AND MIXING IN A STRATIFIED ESTUARY
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Figure 3. Vertical profiles of velocity (solid line) and density (dashed line) from station 5 and station 7 near
maximum ebb and maximum flood.
3.
3.1.
Observations
Vertical Structure
Vertical profiles of along-channel current and density during
maximum ebb and maximum flood in a channel constriction
(station 7) and a channel expansion (station 5) are presented in
Figure 3. During ebb in the expansion, currents and stratification are linear with depth. Currents 2.8 m below the surface are
ebbing at approximately 140 cm/s while at 20 m, currents are
ebbing at 15 cm/s. Surface and bottom densities are 2 ␴ t and 16
␴ t , respectively. Vertical structure differs markedly during the
flood (Figure 3, top right plot). Currents have a middepth
maximum at approximately 10 m below the surface where
currents are flooding at approximately 70 cm/s. Vertical shears
are strongest above the middepth maximum. Vertical shears
above the middepth velocity maximum column are commensurable to those during the ebb. Vertical stratification during
the flood is characterized by a two-layer system. Density varies
by 12 ␴ t units across 5 m thick halocline which separates a
weakly stratified surface layer from a bottom mixed layer.
In the channel constriction, vertical shear during the ebb is
considerably weaker than in the channel expansion. This reduction is associated with increased ebbing velocities at depth
while surface currents, despite variations in channel width,
remain fairly constant. Stratification during the ebb in the
contraction is bottom intensified. During the flood, velocity in
the constriction has a middepth maximum, approximately at
the same depth as at station 5. In the bottom mixed layer,
vertical shears in the contraction and expansion are of similar
magnitude. However, in the stratified fluid above the middepth
velocity maximum, vertical shear in the contraction is substantially weaker than in the channel expansion.
3.2.
Along-Channel Density Structure
Along-channel sections of ␴ t are presented in Figure 4 during maximum ebb and maximum flood on April 29. During
maximum ebb the halocline exhibits large-amplitude excursions over the reach. The two regions of depressed halocline
are associated both with contractions and with deepening of
the main channel. Associated with these excursions are
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Figure 4. Density from CTD: (top) the first, (middle) the fourth and (bottom) the fifth transects. Maximum
ebb occurred during the first transect. Maximum flood occurred between the fourth and fifth transects.
changes in halocline thickness; the halocline thickens as it
descends in the water column and sharpens as it rises. The
halocline remains intact throughout this reach with very little
variation in surface to bottom salinity over the 10 km stretch.
The lower two plots in Figure 4 present concurrent transects
taken around the time of maximum flood. Flood is characterized by a rapid growth of the bottom mixed layer. Neph and
Geyer [1996] suggest that this thickening is primarily due to
overstraining whereby the horizontal density gradient is overturned by the vertical shear during flood. Halocline slopes tend
to weaken during the flood.
3.3.
Along-Channel Velocity Structure
A contour of instantaneous along-channel velocity during
maximum ebb and around maximum flood on April 29 is
shown in Figure 5. Although a tidally driven eddy develops
between stations 5 and 7 [Chant and Wilson, 1997], it is not
present in these two fields during this time. Isotachs during the
ebb exhibit similar behavior to the density field. Isotachs
spread in the channel constrictions and compress in channel
expansions. Despite the twofold change in channel width, surface currents exhibit only weak along-channel variability. In
contrast, currents at depth accelerate in channel contractions.
At slack water the strong pycnocline slopes internally adjust.
During flood, strong vertical shear is found above the halocline
south of the constriction at station 7 with very weak shear
north of this point. Bottom currents are strongly convergent in
the southern third of the domain during the flood. The reduction of shear in the vicinity of the contraction at section 7
suggests that the tidal period internal motion is blocked by the
channel contraction.
3.4.
Froude Number
The state of the flow can be defined by the two-layer Froude
number 公U 21 ( g⬘H 1 ) ⫹ U 22 ( g⬘ H 2 ) , where U 1 , U 2 , H 1 , and
H 2 are the upper and lower layer velocities and upper and
lower layer depths, respectively, and g⬘ is reduced gravity
( g⌬ ␳ / ␳ ). Estimates of the Froude number were made by
CHANT AND WILSON: HYDRAULICS AND MIXING IN A STRATIFIED ESTUARY
14,219
defining reduced gravity and layer velocities with the mean
quantities above and below the 7 ␴ t isopyncal. During maximum ebb, Froude numbers exceed 3 throughout the reach,
indicating that the flow is appreciably supercritical at this
phase of the tide (Froude number estimates during ebb remain
appreciably supercritical when chosing other reasonable isopycnals to define the interface). During the flood the two-layer
Froude number is just below unity, suggesting that the flow is
subcritical.
In summary, these observations emphasize tidal period
asymmetry in the vertical structure of current and density.
During the ebb the vertical structure is characterized by a
nearly linear velocity and density profile. In contrast, the velocity profile during the flood has a middepth maximum embedded in a sharp halocline. The halocline separates a bottom
mixed layer from a stratified surface layer. The vertical structure, however, exhibits significant along-channel variability.
The along-channel variability is most pronounced during the
ebb. In particular, vertical shears are weaker within channel
contractions. Reduced vertical shears in channel contractions
are associated with increased lower layer velocities while upper
layer flows remain fairly constant along the reach. Estimates of
the two-layer Froude number indicate that the flow is appreciably supercritical during the ebb and subcritical on the flood.
4.
Momentum Balance
Strongly convergent horizontal flow indicates that advection
contributes to the momentum balance. To assess this, we begin
with the momentum equation
ut ⫹ g␩x ⫹
g
␳
冕
␩
␳ x⭸ z ⫹ uu x ⫹ ␶ z ⫽ 0,
(1)
z
where t is time, x and z are the along-channel and vertical
directions, u is the along-channel current, ␩ is the surface
elevation, g is the acceleration due to gravity, and ␶ are the
Reynolds stresses. By neglecting stresses near the surface a
momentum balance at the surface is
u0 t ⫹ g ␩ x ⫹ u0u0 x ⫽ 0.
(2)
The barotropic pressure gradient is removed by subtracting 2
from 1 to form a shear equation
共u ⫺ u0兲 t ⫹
g
␳
冕
Figure 5. Along-channel velocity obtained from ADCP during the same transects presented in Figure 4.
lated to contributions by vertical stresses, particularly in the
bottom mixed layer, and by secondary circulation which is
active in this channel [Chant and Wilson, 1997].
␩
␳ x⭸ z ⫹ 共uu x ⫺ u0u0 x兲 ⫹ ␶ z ⫽ 0.
(3)
z
The second and third (uu x ⫺ u0u0 x ) terms represent a density-driven shear term and an advectively driven shear term,
respectively. These two terms are estimated between each station during maximum ebb and around maximum flood on April
29, and the result is plotted in Figure 6. Since this estimate is
made near the time of maximum current, the time-dependent
term should be small. The sign of the pressure gradient has
been switched to facilitate comparison. During the ebb both
the baroclinic and advective shear term are enhanced relative
to the flood. More importantly, the sign and magnitude of
these two terms tend to be the same, particularly in the vicinity
of the channel contractions. Although obvious discrepancies
are evident, Figure 6 indicates that during the ebb there is a
tendency for baroclinically driven shears to be balanced by
advective shears. Discrepancies in the terms are probably re-
5.
Richardson Number Estimates
Geyer and Smith [1987] suggest that halocline thickness in a
highly stratified estuary during a supercritical ebb tends to
remain close to an equilibrium value whereby the flow is marginally stable with active but intermittent vertical mixing. Slight
increases in shear trigger instabilities which vertically mix both
momentum and salt, thus reestablishing stable conditions. Our
observations suggest that vertical shear exhibits an alongchannel structure which is related to channel morphology. To
assess if the observed longitudinal structure in vertical shear
influences mixing, we compare estimates of Richardson numbers in the contraction at section 7 to those in the channel
expansion at sections 5 and 6.
Since we are interested in the development in shear instabilities within the halocline, Richardson number estimates are
restricted to locations where stratification exceeds a specified
criterion. Since stratification weakened over the survey, the
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Figure 6. Estimates for the second and third terms in equation (3). The solid line is the baroclinic term,
while the dashed line is the advective term. Horizontal ticks are at increments of 0.01 cm2/s.
criteria decreased from 0.5 ␴ t /m on April 29 to 0.25 ␴ t /m on
May 3. These values were chosen to maximize the number of
estimates without being influenced by the near-zero Richardson numbers in the bottom mixed layer.
Histograms of Richardson numbers for sections 5–7 are
plotted in Figure 7. Richardson numbers tend to be smaller in
the channel expansion at sections 5 and 6 than in the contraction at section 7. On one hand, it may appear counterintuitive
to have reduced mixing within channel constrictions. However,
data presented here indicate that vertical shears are strongest
downstream of channel contractions, and subsequently, we expect to find reduced Richardson numbers downstream of channel contractions.
6.
6.1.
Discussion
Two-Layer Model
Certain aspects of the spatial and temporal density and velocity structure can be understood in light of a relation for an
invicid two-layer flow in rectangular channel of variable width
and depth.
共u 2 ⫺ u 1兲 t ⫽ g⬘共1 ⫺ G 2兲
⭸h
⭸H 1
1 ⭸B 2
⫺ g⬘F 22
⫺
共u ⫺ u 22兲,
⭸x
⭸x B ⭸x 1
(4)
where u 1 , u 2 , H 1 , and H 2 are upper and lower layer velocities
and thicknesses, h is the height of the bottom above a refer-
ence level, B is channel width, G 2 is the two-layer Froude
number, and t and x are time and distance, which are positive
up estuary. Neglecting time dependency, (4) can be rewritten
to express halocline slope as a function of channel morphology
and the flow state
⭸H 1
⫽
⭸x
F 22
⭸h
1 ⭸B 2
⫹
共u ⫺ u 22兲
⭸ x g⬘B ⭸ x 1
.
共1 ⫺ G 2兲
(5)
R. E. Wilson et al. (Influence of channel morphology on halocline behavior and trapping of fine-particles in a stratified
estuary, submitted to Journal of Physical Oceanography, 1999)
demonstrated that halocline slopes along this reach during the
ebb are consistent with this two-layer model and that flow
along this reach is appreciable supercritical during the ebb.
The bathymetric terms, B x and H x , tend to be of the same sign
because the channel deepens in contractions and shoals in
expansions. Thus the first two terms are of the same sign when
surface layer velocities exceed lower layer velocities, as is the
case during the ebb. Therefore effects of a deepening and
contracting channel act in concert to thicken the surface layer
during the supercritical ebb. Likewise, effects of a shoaling and
expanding channel act together to thin the surface layer. In
contrast, during the flood, lower layer velocities exceed those
in the upper layer, and the first two terms in (6) are opposed.
In summary, width and depth variations in this channel act
CHANT AND WILSON: HYDRAULICS AND MIXING IN A STRATIFIED ESTUARY
14,221
Figure 7. Richardson number estimates made in the pycnocline at sections 5, 6, and 7.
together to produce steep halocline slopes during the ebb. In
contrast, during flood these bathymetric effects are opposed.
Reduced shears during the supercritical ebb in the contraction are also elucidated by the two-layer model. During the ebb
the upper layer largely determines the two-layer Froude number and can be viewed as a single active layer. This causes
surface ebb velocities to remain fairly constant along this
reach, despite the factor of 2 change in channel width, because
supercritical flow draws from its kinetic energy to squeeze
through a contraction, in contrast to subcritical flow, which
draws from its potential energy [Turner, 1973]. If the lower
layer were motionless, surface flows would actually decrease
within contractions, which itself would decrease shears in the
contraction. However, the lower layer is in motion, and lower
layer flows must increase because both the contracting channel
and thinning bottom layer reduce cross-sectional area in the
lower layer. Consequently, in the constriction the shear between the layers decreases during the supercritical ebb.
6.2.
Shear Tendency During the Ebb
The tendency for vertical shears to weaken in a contraction
during a supercritical ebb can be demonstrated more directly
by a lateral vorticity, or vertical shear, equation. This equation
is obtained by first writing a steady, nonrotating, invicid momentum equation
u
⭸u
⭸u
1 ⭸P
⫹w
⫽⫺
⭸x
⭸z
␳ ⭸x
(6)
and a laterally averaged continuity equation
⭸
⭸
共uB兲 ⫹
共wB兲 ⫽ 0,
⭸x
⭸z
(7)
where P is pressure, composed of both a barotropic and a
baroclinic component, B( x,z) is the channel width, and u and
w are the horizontal and vertical currents. Taking the vertical
gradient of 6, assuming linear vertical shear, and substituting in
(7) yields a shear tendency equation
u
⭸
⭸x
冉 冊
⭸u
⭸z
⫽
⭸u
⭸z
冉
u
⭸B
⭸B
⫹w
⭸x
⭸z
冊
⫹
1 ⭸␳
␳ ⭸x
(8)
or, equivalently, a lateral vorticity equation for an invicid nonrotating flow though a channel of variable width. The first term
on the right-hand side represents vortex stretching/squashing
while the second term represents baroclinically driven shears.
The term on the left-hand side represents the horizontal advection of shear.
As a supercritical ebb enters a contraction the halocline
descends in the water column, causing the baroclinic pressure
gradient to change sign from the estuarine-scale along-channel
baroclinic pressure gradient. This tends to reduce shear as the
flow enters the contraction. Similarly, the vortex-stretching
term tends to reduce vertical shears as the flow enters a contraction because both the horizontal flow and downward vertical motion squash the fluid into regions of reduced channel
breadth. Consequently, both terms on the right-hand side act
in concert to reduce shears in the contraction. In this invicid
example the shear tendency is balanced by the horizontal advection of vertical shear which requires shears to weaken moving into the contraction.
In contrast, downstream of the contraction the halocline
rises, and both the baroclinic term and stretching terms change
sign. Here the sign of the baroclinic term is the same as the
estuarine-wide baroclinicity. Consequently, both of the terms
on the right-hand side act to increase the estuarine shear
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downstream of the contraction. This is balanced by advection
which requires shears to increase moving away from the contraction. More generally, (8) shows that for supercritical continuously stratified sheared flow, vertical shears are reduced in
a channel contraction. Subsequently, we expect to find enhanced vertical shears and vertical mixing downstream of channel contractions.
7.
Conclusions
Observations from the Hudson River estuary emphasize a
tidal period asymmetry in the vertical structure of density and
current velocity. During the flood a sharp, relatively flat halocline separates a bottom mixed layer from a stratified surface
layer. Embedded in this sharp halocline is a middepth velocity
maximum. In contrast, during the ebb both velocity and density
have a linear vertical structure. The vertical structure, however,
exhibits significant along-channel variability. Steep halocline
slopes occur during the ebb, with the halocline depressed
where the channel deepens and elevated where the channel
shoals and expands. This behavior is consistent with a two-layer
model which emphasizes that the halocline is more responsive
to channel morphology during the ebb because effects of channel deepening and contracting act in concert to produce steep
halocline slopes. In contrast, both observations and the twolayer model indicate that the halocline is less responsive to
variations in channel morphology during the flood.
A shear tendency equation explains the observed reduction
in vertical shears in channel contractions during the ebb. Surface currents exhibit weak along-channel variability, despite a
factor of 2 change in channel width. In contrast, currents at
depth accelerate in channel contractions and weaken in channel expansions.
Observations indicate that the advection of shear contributes significantly to an instantaneous momentum balance. Although we do not estimate tidally mean terms, it is likely that
the tidally mean terms reflect the ebb conditions, since halocline slopes and streamwise velocity gradients are more pronounced on the ebb. These 1–2 km scale fluctuating baroclinic
pressure gradients are superimposed on a larger-scale but
weaker estuarine scale baroclinic pressure gradient. Although
the tidally mean momentum balance in the vicinity of these
small-scale morphological features is probably imparted by
advection, the estuarine-wide balance may still be between
baroclinicity and mixing. However, these small-scale advective
features indirectly influence the estuarine-wide momentum
balance by driving localized regions of enhanced mixing. Consequently, we suggest that estimates of tidally averaged momentum balances in highly stratified estuaries are dependent
on the length scale over which an estimate is made.
Acknowledgments. R.E.W. acknowledges support from the National Science Foundation (OCE8917692) and from the Hudson River
Foundation (01291A and GF0192). R.J.C. acknowledges partial support from New York State Sea Grant Institute. We also acknowledge
John Brubaker, who collected and processed the ADCP.
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R. J. Chant, Institute for Marine and Coastal Sciences, Rutgers
University, 71 Dudley Road, New Brunswick, NJ 08901. (chant@
imcs.rutgers.edu)
R. E. Wilson, Marine Science Research Center, State University of
New York at Stony Brook, Stony Brook, NY 11769.
(Received February 17, 1999; revised October 20, 1999;
accepted February 7, 2000.)