A FREQUENCY DOMAIN FINITE ELEMENT MODEL
FOR TIDAL CIRCULATION
by
Joannes J. Westerink, Keith D. Stolzenbach,
and Jerome J. Conner
Energy Laboratory Report No. MIT-EL 85-006
January 1985
A FREQUENCY DOMAIN FINITE ELEMENT MODEL
FOR TIDAL CIRCULATION
by
Joannes J. Westerink
Keith D. Stolzenbach
Jerome J. Conner
Energy Laboratory
and
R. M. Parsons Laboratory for
Water Resources and Hydrodynamics
Department of Civil Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
Sponsored by
Northeast Utilities Service Company
and
New England Power Service Company
under the
MIT Energy Laboratory Electric Utility Program
and by
The Sea Grant Office of NOAA
U.S. Department of Commerce
Energy Laboratory Report No. MIT-EL 85-006
January 1985
ABSTRACT
A highly efficient finite element model has been developed for the
numerical prediction of depth average circulation within small scale
embayments which are often characterized by irregular boundaries and
bottom topography.
Traditional finite element models use time-stepping and have been
plagued with requirements for high eddy viscosity coefficients and small
time steps necessary to insure numerical stability, making application
to small bays infeasible. These problems are overcome by operating in
the frequency domain, an intrinsically more natural solution procedure
for a highly periodic process such as tidal forcing. In order to handle
non-linearities, an iterative scheme which updates non-linearities as
right hand side force loadings must be implemented.
Pioneering efforts with the harmonic approach have had shortcomings
in either not modeling all physically relevant terms and/or in not
gearing towards application to small scale regions. Small embayments
are often quite shallow and have rapidly varying depth, making the
nonlinear terms in the governing hydrodynamic equations much more
significant. This requires that more frequencies be used in order to
resolve the tide and account for the greater nonlinear coupling due to
bottom friction, convective acceleration and finite amplitude effects.
In order to make the process of handling this wide range of frequencies
manageable, a hybrid frequency-time domain approach is applied. The
iterative scheme revolves around a highly efficient linear core code
which can handle a wide range of frequencies. Furthermore, instead of
Fourier expanding the nonlinear terms, an efficient least squares error
minimization algorithm is used for the discrete spectral analysis of the
iteratively updated psuedo-force time history generated by the nonlinearities.
With this highly efficient scheme it is now possible to efficiently
study both short period and long term residual circulation within small
scale embayments.
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ACKNOWLEDGMENTS
This report is part of a research program to develop more efficient
and accurate ciculation and dispersion models for coastal waters.
The
report describes a two-dimensional non-linear frequency domain model
(named TEA-NL) for the analysis of circulation in tidal embayments.
The
work is an extension of a linear model (TEA) described in
Westerink, J. J., Connor, J. J., Stolzenbach K. D., Adams,
E. E., and Baptista, A. M., "TEA: A Linear Frequency Domain
Finite Element Model for Tidal Embayment Analysis," Energy
Laboratory Report No. MIT-EL 84-012, February 1984
Also developed as part of this reserch program is a two-dimensional
transport model (ELA) which combines Eulerian and Lagrangian techniques
and is described in
Baptista, A. M., Adams, E. E., and Stolzenbach, K. D.,
"Eulerian-Lagrangian Analysis of Pollutant Transport in Shallow
Water," Energy Laboratory Report No. MIT-EL 84-008, June 1984
(Also published as Technical Report No. 296, R. M. Parsons
Laboratory for Water Resources and Hydrodynamics, M.I.T.)
Support for this research was provided in part by the Sea Grant
Office of NOAA, Department of Commerce, Washington, D.C., and in part by
Northeast Utilities Service Company and New England Power Company
through the M.I.T. Energy Laboratory Electric Utility Program.
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TABLE OF CONTENTS
Page
..........
ABSTRACT ...................
ACKNOWLEDGMENTS
...................
TABLE OF CONTENTS
.......
4
......
5
. ..................
LIST OF TABLES ...
LIST OF FIGURES
3
. ..
....
. ..
....
..
. . . ..
..
...................
.....
INTRODUCTION ...................
2.
DESCRIPTION OF TIDES IN SHALLOW EMBAYMENTS . .........
LINEAR CORE MODEL
5.
NONLINEAR MODEL
7.
..
....................
.
.
. . ..
.
. .
REFERENCES ....
. ......
.
.
.
....
.......
............
........
-5-
.
.
........
79
97
104
124
170
193
Description of Bight of Abaco and Its Tides . .......
.
..
....
Overtide Computations for the Bight of Abaco
Compound Tide Computations for the Bight of Abaco .....
.....
Discussion ...................
CONCLUSIONS
79
104
.......
...................
43
47
56
61
.. ......
...
24
43
....
...................
.............
24
29
Harmonic Analysis of Non Linear Pseudo Forcings
Iterative Convergence . ..................
APPLICATION
6.1
6.2
6.3
6.4
. .........
Weighted Residual Formulation .
.......
......
.
............
Finite Element Method Formulation.
. . . . . . . . . . . . . .
Frequency Domain Formulation
4.
5.1
5.2
.
..
Harmonic Tidal Components in Estuaries
NUMERICAL FORMULATION
13
..............
Governing Equations
2.2
3.1
3.2
3.3
6.
. ....
2.1
6
8
.......
1.
3.
. . .
.
202
205
LIST OF TABLES
Page
22
Table 1.1
Wavelengths of a 12.4 Tide in Various Water Depths ..
Table 2.1
Astronomical Tides of Importance . .........
.
30
Table 2.2
Major Overtides. . ..................
.
31
Table 2.3
Major Compound Tides . ................
Table 2.4a
Response-Forcing Table for Overtides as Generated by
31
Finite Amplitude Term at Cycle No. 2 of Iteration. .
.
35
.
35
Response-Forcing Table for Compound Tides as Generated
by Finite Amplitude Term at Cycle No. 1 of Iteration .
37
Response-Forcing Table for Compound Tides as Generated
by Finite Amplitude Term at Cycle No. 2 of Iteration .
37
Table 2.6
Tides of Interest (High Freq. End) . .........
40
Table 4.1
Sizes and Ranks of Various Matrices. . ........ .
64
Table 4.2
Variation of Convergence with cs
69
Table 4.3
Comparison of Analytical and Numerical Elevations and
Velocities for Example Channel Case at Various
Locations
Table 2.4b
Response-Forcing Table for Overtides as Generated by
Finite Amplitude Term at Cycle No. 3 of Iteration. .
Table 2.5a
Table 2.5b
Table 5.1
Table 5.2a
Table 5.2b
.
.
. .
.
. . .
. .
.
(a) Linearized Friction Factor X = 0.0000. . ..... .
74
(b) Linearized Friction Factor X = 0.0010. . ..... .
75
(c) Linearized Friction Factor X = 0.0100. . ..... .
76
LSQ Analysis Results Showing Effects of Variation of
Number of Frequencies and Time Sampling Points;
Example Simulating Overtide Type Frequencies . ....
89
LSQ Analysis Results Showing Effects of Variation of
Number of Frequencies and Time Sampling Points;
Example Simulating Closely Spaced Compound Tide
.
. . . . . . .
Frequencies. . ............
92
LSQ Analysis Results Showing Effects of Variation of
Number of Frequencies and Time Sampling Points;
Example Simulating Closely Spaced Compound Tide
.
Frequencies . . . . . . . . . . . . . . . . . . . .
93
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Page
Table 6.1
Table 6.2
Table 6.3a
Table 6.3b
Table 6.4
Table 6.5a
Table 6.5b
Reflection and Transmission Coefficients for a Long
Wave Passing Over a Step from Depth h i to Depth h
2
for Various Depth Ratios. . . . . . . . . . . . . . .
. 107
Summary of Measured Astronomical Tides Along the
.. . ................
Open Ocean Boundary .. . ...
* 111
Measurement Error for Each Frequency in Terms of
Proportional Variance, Vm . . ..........
J
* 122
Measurement Error for Each Frequency in Terms of
..........
Proportional Standard Deviation, S
J
* 123
Values for Friction Factor cf in Terms of Depth and
Bottom Roughness (from Wang and Connor, 1975) . .
* 136
Overtide Computation Errors Expressed as Error
Between Measurements and TEA Predictions in Terms of
. . ............. .
Proportional Variance, VP .. . .
J
* 165
Overtide Computation Errors Expressed as Error
Between Measurements and TEA Predictions in Terms of
Proportional Standard Deviation, SP .
J
.
.......
* 166
Table 6.6
Tides of Possible Interest for M 2 and N 2 Interaction. . 171
Table 6.7a
Compound Tide Computation Errors Expressed as Error
Between Measurements and TEA Predictions in Terms of
Proportional Variance, V. . . . . . . .............
Table 6.7b
Compound Tide Computation Errors Expressed as Error
Between Measurements and TEA Predictions in Terms of
.........
Proportional Standard Deviation, SP
i
-7-
* 190
* 191
LIST OF FIGURES
Page
Definition Sketch Showing Typical Elevation Prescribed
.
and flux prescribed (Q ) Boundaries . . . . . .
(FT)
28
Figure 2.2
Schematic of Major Astronomical and Shallow Water Tides
41
Figure 3.1
Schematic of Iterative Non Linear Scheme. . ...... .
60
Figure 4.1
Definition Sketch of Depth Varying Channel Which
Illustrates Convergence Problems of Iterative Linear
.............
. ..
Scheme. . ...........
68
Finite Element Grid Discretization for Closed Ended
......... . .
Channel Example Case. . ........
73
Linear Equation Generated by Least Squares Analysis
Procedure . . . . . . . . . . . . . . . . . . . . . . .
83
Figure 2.1
Figure 4.2
Figure 5.1
Figure 5.2
Effects of Variation in Frequency and Time Sampling
.
Rates for Typical Overtide Frequency Distribution .
Figure 5.3
91
.. .
105
Geography of Bight of Abaco, Bahamas. . ......
Figure 6.2
Bathymetry of the Bight of Abaco, Bahamas . ......
Figure 6.3
Field Data for M 2 Astronomical Constituent (after
Filloux & Snyder, 1979)
(a) Amplitude in centimeters. . ...
109
. . . . . . . .
.
112
. . . .
.
113
...... .
114
(b) Phase lag in radians. . .........
Field Data for M 4 Overtide Constituent (after Filloux
& Snyder, 1979)
(a) Amplitude in centimeters. . ...... . .
(b) Phase lag in radians. . ......
Figure 6.5
88
Effects of Variation in Frequency and Time Sampling
Rates for Typical Compound Tide Frequency Distribution
(maximum period is T = 12.4 hours and maximum synodic
. . . . . . . . . . .
period is TS = 27 days) . ....
Figure 6.1
Figure 6.4
.
. . . . . . .
.
115
Field Data for M 6 Overtide Constituent (after Filloux
& Snyder, 1979)
(a) Amplitude in centimeters. . ..
. . . . . . . . .
.
116
(b) Phase lag in radians. . ....
. . . . . . . . .
.
117
-8-
Figure 6.6
Figure 6.7
Figure 6.8
Figure 6.9
Field Data for N 2 Astronomical Constituent (after
Filloux & Snyder, 1979)
(a) Amplitude in centimeters . . . . . . . . . . . . .
118
(b) Phase lag in radians . . . . ..
119
.............. .
Finite Element Grid Discretization for Bight of Abaco,
Bahamas. . . . . . . . . . . . . . . . . . . . . . . ..
Results of TEA with Full Non Linear Friction Effects
(cf = 0.003) for M 2 Astronomical Constituent
(a) Amplitude in centimeters . . . . . . . . . . . . .
127
(b) Phase lag in radians . . . . . . . . . . . . . . .
128
Results of TEA with Full Non Linear Friction Effects
(cf = 0.006) for M 2 Astronomical Constituent
(a) Amplitude in centimeters . . .
...
. . . . . ....
(b) Phase lag in radians . . . . . . . . . . . . . .
Figure 6.10
125
129
.
130
. . . . .
131
(b) Phase lag in radians . . . . . . . . . . . . . . .
132
Results of TEA with Full Non Linear Friction Effects
(cf = 0.009) for M 2 Astronomical Constituent
(a) Amplitude in centimeters . . . . . .
. .
Figure 6.11a Trajectory along which M 2 Elevation Amplitudes are
Compared for Varying Friction Factor in Figure 6.11b
133
Figure 6.11b Comparison of M 2 Elevation Amplitudes for Varying
Friction Factor, cf, along Trajectory S . . . . . . .
134
Figure 6.12
Figure 6.13
Results of TEA with Full Non Linear Friction Effects
......
(cf = 0.009) for Steady State Constituent...
Results of TEA with Full Non Linear Friction Effects
(cf = 0.009) for M 4 Overtide Constituent
(a) Amplitude in centimeters ..
. ...
............
(b) Phase lag in radians . . . . ................
Figure 6.14
137
138
139
Results of TEA with Full Non Linear Friction Effects
(cf = 0.009) for M 6 Overtide Constituent
(a) Amplitude in centimeters
(b) Phase lag in radians
-9-
.
............. .
................
140
141
Figure 6.15
Results of TEA with Full Non Linear Friction Effects
(cf = 0.009) and Finite Amplitude Effects for M 2
Astronomical Constituent
(a) Amplitude in centimeters . . . . . . . . . . . . .
142
. . . . . . . . . . . .
143
Results of TEA with Full Non Linear Friction Effects
(c = 0.009) and Finite Amplitude Effects for
Steady State Constituent . . . . . . . . . . . . . . .
144
(b) Phase lag in radians . . .
Figure 6.16
Figure 6.17
Results of TEA with Full Non Linear Friction Effects
(cf = 0.009) and Finite Amplitude Effects for
M 4 Constituent
(a) Amplitude in centimeters . . . . . . . . . . . . .
145
. . . . . . . .
146
(b) Phase lag in radians . . .
Figure 6.18
. . . .
Results of TEA with Full Non Linear Friction Effects
(c = 0.009) and Finite Amplitude Effects for
M 6 Constituent
(a) Amplitude in centimeters . . . . . . . . . . . . .
147
. . . . . . . . .
148
(b) Phase lag in radians . . . . . .
Figure 6.19
Figure 6.20
Figure 6.21
Figure 6.22
Results of TEA with Full Non Linear Friction Effects
(cf = 0.009), Finite Amplitude Effects and Convective
Acceleration Effects for M 2 Astronomical Constituent
(a) Amplitude in centimeters . . . . . . . . . . . . .
150
(b) Phase lag in radians . . . . . . . . . . . . . . .
151
Results of TEA with Full Non Linear Friction Effects
(cf = 0.009), Finite Amplitude Effects and Convective
Acceleration Effects for Steady State Constituent. . .
152
Results of TEA with Full Non Linear Friction Effects
(cf = 0.009), Finite Amplitude Effects and Convective
Acceleration Effects for M 4 Overtide Constituent
(a) Amplitude in centimeters . . . . . . . . . . . . .
153
(b) Phase lag in radians . . . . . . . . . . . . . . .
154
Results of TEA with Full Non Linear Friction Effects
(cf = 0.009), Finite Amplitude Effects and Convective
Acceleration Effects for M 6 Overtide Constituent
(a) Amplitude in centimeters . . . . . . . . . . . . .
155
(b) Phase lag in radians . . . . . . . . . . . . . . .
-10-
156
Figure 6.23a Velocity Results of TEA with Full Non Linear Friction
Effects (cf = 0.009), Finite Amplitude Effects and
Convective Acceleration Effects for Steady State
Component . . . . . . . . . ..................... .
158
Figure 6.23b Velocity Results of TEA with Full Non Linear Friction
Effects (cf = 0.009), Finite Amplitude Effects and
Convective Acceleration Effects for M 2 Component at
Time of Maximum Ebb for the M 2 Component Relative to
the Ocean Boundary. . . . . . . . . . . . . . . . . .
159
Figure 6.23c Velocity Results of TEA with Full Non Linear Friction
Effects (cf = 0.009), Finite Amplitude Effects and
Convective Acceleration Effects for M 4 Component at
Time of Maximum Ebb for the M 2 Component Relative to
the Ocean Boundary. . . . . . . . ................. .
160
Figure 6.23d Velocity Results of TEA with Full Non Linear Friction
Effects (c = 0.009), Finite Amplitude Effects and
Convective Acceleration Effects for M 6 Component at
Time of Maximum Ebb for the M 2 Component Relative to
the Ocean Boundary. . . . . . . . .................
161
Figure 6.24a Continuity Equation Pseudo Forcing Vector Ratios Due
to M 2 - N 2 Interaction. . . . . . . . . . . . . . . .
173
Figure 6.24b Momentum Equation Pseudo Forcing Vector Ratios Due to
M 2 - N 2 Interaction . . . . . . . . . . . .
.......
174
Figure 6.25
Results of TEA with M 2 -N 2 Interaction and with Full
Non Linear Friction (cf = 0.009) and Finite Amplitude
Effects for M 2 Astronomical Constituent
(a) Amplitude in centimeters. .
. . . . ............. .
(b) Phase lag in radians. . . . . . . . .
Figure 6.26
. . .
. . . . ............. .
(b) Phase lag in radians. . . . . . . . . . .
Figure 6.28
177
Results of TEA with M 2 -N 2 Interaction and with Full
Non Linear Friction (cf = 0.009) and Finite Amplitude
Effects for N2 Astronomical Constituent
(a) Amplitude in centimeters. .
Figure 6.27
.....
176
......
178
179
Results of TEA with M 2 -N2 Interaction and with Full
Non Linear Friction (cf = 0.009) and Finite Amplitude
Effects for Steady State Constituent.. . . . . . . ...
180
Results of TEA with M 2 -N2 Interaction and with Full
Non Linear Friction (cf = 0.009) and Finite Amplitude
Effects for MN Compound Constituent . . . . . . . . . .
181
Figure 6.29
Figure 6.30
Figure 6.31
Results of TEA with M2-N 2 Interaction and with Full
Non Linear Friction (cf = 0.009) and Finite Amplitude
Effects for MN 4 Compound Constituent
(a) Amplitude in centimeters . . . . . . . . . . . . .
182
(b) Phase lag in radians . . . . . . . . . . . . . ..
183
Results of TEA with M 2 -N 2 Interaction and with Full
Non Linear Friction (cf = 0.009) and Finite Amplitude
Effects for M 4 Overtide Constituent
(a) Amplitude in centimeters . . . . . . . . . . . . .
184
(b) Phase lag in radians . . . . . . . . . . . . . . .
185
Results of TEA with M 2 -N 2 Interaction and with Full
Non Linear Friction (cf = 0.009) and Finite Amplitude
Effects for 2MN 6 Compound Constituent
(a) Amplitude in centimeters . . . . .
(b) Phase lag in radians . . .
Figure 6.32
. . . . ....
186
. . . . . . . . . . . .
187
Results of TEA with M 2 -N 2 Interaction and with Full
Non Linear Friction (cf = 0.009) and Finite Amplitude
Effects for M 6 Overtide Constituent
(a) Amplitude in centimeters . . . ..
(b) Phase lag in radians . . . . . .
-12-
............
. . .
. . ....
.
188
189
CHAPTER 1.
INTRODUCTION
Recent years have seen the development of numerous coastal
circulation models which apply the finite element method.
The principal
advantage of finite element methods over the more traditional finite
difference methods is the greater versatility allowed in grid
discretization which is especially important for small scale coastal
embayments.
This feature permits the convenient fitting of the often
irregular boundaries and allows refinement of the grid in such critical
areas as high flow and bottom depth gradient regions and/or the narrow
mouths connecting these embayments to the open ocean.
These circulation models share as a general starting point the well
established shallow water equations which are derived by depth averaging
the conservation of mass and momentum equations with the application of
the hydrostatic and Boussinesq assumptions. Therefore the equations that
are applied are based on first principles and require empirical support
only for the turbulent exchanges and surface and bottom stresses.
Major
differences between these models lie in features such as the type of
localized expansions used to resolve the spatial dependence of the
variables and more importantly the way in which they discretize the time
dependence.
Traditionally time marching schemes have been applied which
are either explicit or implicit.
Early time domain models [Grotkop, 1973; Taylor and Davis, 1975;
Wang and Connor, 1975; Kawahara, 1978; King et al.,
1974] have had
severe problems relating to the economy and accuracy of the schemes
developed.
Economic constraints stem both from the large amount of
numerical manipulation required for the schemes and a maximum allowable
-13-
time step required for the accuracy and/or stability of the
computation.
For explicit schemes a Courant stability constraint
necessitates the maximum time step to decrease along with element size,
making it especially infeasible to apply these models to small scale
geometries/elements.
Furthermore, these early models have been plagued
with accuracy problems which relate to short wave length artificial
oscillations in elevation and velocity produced by the finite
discretization of the domain [Gray, 1980; Sani, et al., 1980].
These
models either require artificially high eddy viscosity to damp out this
short wave length noise or the numerical scheme is such that it is
inherently overdamped.
The accuracy problem, however, arises from the
fact that not only the numerical noise is damped, but the longer
physical waves being simulated are also damped.
This aspect of
overdamping has drawn strong criticism as to the ability of these models
to adequately simulate the physical problem described by the shallow
water equations.
Efforts to overcome these shortcomings in accuracy and
efficiency have been numerous and have had varying degrees of success.
Alternatives which have been investigated include different time
integration schemes [Gray and Lynch, 1977, 1979; Niemeyer, 1979], mass
lumping schemes [Kawahara, 1982] and the examination of the effects of
mixed interpolation of elevation and velocity
[Walters and Cheng,
1980;
Walters and Carey, 1983; Platzman, 1981; Williams and Zienkiewicz,
1981].
Certain investigators paid special attention to the numerically
troublesome convective terms by either applying a Petrov-Galerkin
weighting scheme (equivalent to upwinding) [Nakazawa, et al., 1980] or
by using the method of characteristics in conjunction with finite
elements [Benque, et al.,
1981].
One of the more promising schemes
-14-
developed is
the use of a wave type equation in
conjunction with the
fundamental momentum equation as the basis of the finite element
formulation [Lynch and Gray,
Although some of these alternative
1979].
schemes have been successful at eliminating short wave length noise
without damping the longer physical waves and are more efficient than
earlier models, all of the above methods still have maximum allowable
time steps making them economically unattractive for either long term
simulations and/or small scale embayments.
A very attractive alternative
to time stepping schemes which has
recently been employed is the use of harmonic analysis in conjunction
with finite elements.
Because of the periodic nature of the tidal
phenomenon, the harmonic method is an intrinsically more natural
solution procedure and was one of the traditional methods for analysis
before the advent of finite difference and finite element methods
[Dronkers, 1964].
There are no time stepping limitations since this
procedure generates a set of quasi-steady (or time independent)
equations.
In addition, truncation errors and the associated stability
problems caused by time stepping are precluded.
Furthermore,
eliminating the time dependence from the governing equations reduces
them from equations of the difficult and time consuming hyperbolic type
to that of the elliptic type which are much more readily solved by
finite element methods.
The harmonic method also offers the potential
of economically performing realistic long term simulations in tidal
embayments (e.g., up to 30 days) and calculating the associated residual
circulation.
A possible drawback of the harmonic method is the increased number
of frequencies which would be required to model a non-periodic
-15-
phenomenon such as wind driven circulation.
However some winds (steady
winds and sea breezes) are periodic and furthermore any wind spectrum
can readily be harmonically decomposed.
Time domain schemes become
economically more attractive when the number of frequencies required to
adequately represent the wind spectrum becomes excessive.
A more
significant difficulty which arises in implementing this frequency
domain technique is that non linear terms generate additional responses
at frequencies other than the base forcing frequency.
Strategies to handle this harmonic coupling produced by the
non linear terms (finite amplitude, convective acceleration and bottom
friction terms) have consisted of either iterative procedures [Pearson
and Winter, 1977; Kawahara, 1978] or some type of perturbation analysis
[Askar and Cakmak, 1978; Kawahara et al.,
1978; Le Provost et al.,
1981].
1977; Le Provost and Poncet,
The iterative scheme applied by Pearson
generates a finite spectral series representing the pseudo-forcings due
A
to all the non linear components of the shallow water equations.
linear solution is then used to evaluate the elevation response due to
each pseudo forcing component (at pre-determined frequencies).
Pseudo
forcings are then updated using the updated responses in elevation and
velocity until convergence in the elevation is achieved.
The full non
linear model has only been tested on geometries with a relatively small
number of elements while a simplified linear version has been applied to
a large grid of a deep bay [Jamart and Winter,
1980].
Kawahara [1977], on the other hand, applied a perturbation analysis
which allows grouping of terms in the expanded (by a power series)
shallow water equations in order to generate several sets of linearized
equations of varying order.
In a later paper, Kawahara [1978] applied
-16-
the periodic Galerkin method for the time dependence which produces a
coupled system of non linear simultaneous equations which are then
solved iteratively.
An important limitation with Kawahara's schemes is
that both bottom friction and Coriolis are omitted from the governing
equations while keeping the much less important eddy viscosity terms.
A further weak point of both Pearson's and Kawahara's work is that
they have used a Fourier series expansion in terms of integer values of
a base frequency (the M2 tide) to represent the variables.
This
precludes the possibility of investigating the interaction between the
majority of the tidal components (see Table 2.1 for major tidal
components).
Therefore, such effects as monthly (spring/neap) variation
(caused to a large extent by beating effects of closely packed tidal
components) can not be looked at and only a major base tide (e.g., M2)
and its harmonics may be studied.
Le Provost [1981] applies an expansion which considers the
interaction of the major closely spaced forcing components (M2 , S2 , N2,
K1 ). However, the perturbation analysis and the quasi-linearization for
bottom friction which are used only account for the non linear coupling
between the major forcing component (M2 ) and its first harmonic (M4).
The remaining astronomic constituents of the tide generating potential
(S2 , N2 , KI) are treated linearly.
The resulting computer code has been
applied to the English Channel and Le Provost states that the method is
constrained in its application to very shallow waterbodies.
Finally, Lynch [1981] has developed a linear harmonic model based
on the same re-arrangement of the fundamental equations as an earlier
time stepping scheme [Lynch and Gray, 1979] which was found to minimize
short wave length noise while retaining the computational accuracy of
-17-
the longer physical waves.
The equations which he uses as the starting
point of his finite element scheme are the wave equation
(found by
substituting the momentum equation into the continuity equation) and the
momentum equation.
He found that when using similar interpolation
orders for elevation and velocity, this scheme will give accurate
solutions without noise.
As previously mentioned the harmonic method naturally lends itself
to the calculation of long term variations and residual circulations.
Residual currents may be thought of as a complicated combination of
steady and slowly varying currents due to the effects of non linear
interactions of both the main and various other astonomic tidal
components [Ianniello, 1977].
Due to infeasibility of running long term
simulation with time stepping programs, investigators have commonly
computed residual circulations either by time averaging the governing
equations [Walters and Cheng, 1980; Bonnefille, 1978; Tee, 1981] or by
time averaging the results produced by a time domain model over one or
several tidal cycles.
The former technique places the inability to
perform long term computations in a set of additional unknown time
averaged terms (tidal-stress terms).
relating to both expense and accuracy.
The latter technique has problems
In order to capture the effects
of the variations in tidal forcing (which cause the non linear
interaction between tides and produce residual/long term circulations)
with sufficient accuracy, the simulation would have to be run over
extended periods of time.
It would not only be expensive to run a time
stepping model for long periods of time but round off errors would also
propagate through the solution.
Hence it is doubtful that these
previous efforts are able to model the effects that drive residual
-18-
circulation since they are not able to capture the actual physics of the
residual currents.
As is discussed in Chapter 2, the major variation in the currents
of coastal embayments generated by astronomic tidal forcings are
well described by considering a one month period.
Responses then occur
at these astronomical forcing frequencies and their associated higher
harmonics (and certain lower frequencies due to the closeness of certain
of the components).
Hence we have a limited number of frequencies which
need to be considered.
Therefore it now becomes even clearer that applying the harmonic
method is extremely well suited to assess low period fluctuations and
residual circulation due to tidal forcing components and their non
linear interaction.
It
is
not only computationally convenient to do so
but also allows the calculation to be done in a manner which is based on
the same first principles with which we presently perform short term
circulation computations and furthermore allows all the significant
effects to be caught.
In summarizing the many advantages of the harmonic method, we note
that:
(i)
no time stepping constraints due to small element sizes are
required;
(ii)
it is well suited for the highly periodic tidal computations
in estuaries;
(iii) the results may be stored in a much more economical and
convenient form for applications with a transport model;
(iv)
it allows computations of long term residual currents; and
(v)
there are no cold start problems which time domain approaches
often have.
-19-
In spite of its many unique features,
being exploited to its full potential.
the harmonic method is
far from
Pioneering efforts which have
applied this technique have had shortcomings in not modeling all
physically relevant terms, by not allowing for full interaction between
the major tidal components and in not gearing towards application of the
method to residual circulation computation and to small scale
geometries.
Even though application of the method to small scale
estuaries increases the computational effort due to the often shallower
depths which increases the significance of non linearities, it is
nonetheless in these small scale regions that the effort associated with
time domain approaches becomes entirely insurmountable due to the
exessively small time steps needed.
We conclude that there is a definite need for the development of
improved strategies for computing tidally induced circulation within
coastal embayments.
The present research addresses this issue with the
development of a general harmonic finite element model which allows the
in-depth study of the many complex non linear interactions which occur
in shallow waterbodies.
This includes not only the investigation of the
coupling occurring between a given astronomical tide and the harmonics
it generates through the non linear terms in the governing equations but
also the complicated interactions between the various astronomical tides
Among the nonlinear harmonic responses
and their associated harmonics.
to be investigated are steady and other very long period residual
circulations which are generated.
A direct iteration scheme will be used to handle the non linear
terms in
the governing equations.
However,
inherent to any solution
scheme which iteratively updates the non linearities as right hand side
-20-
the limitation that the relative magnitude of the right hand
loadings is
side non linear terms must be small compared to the left hand side
linear terms (Ketter and Prawel,
1969).
The implication of this is
the non linear solution must be a perturbed linear solution.
that
The
significance of the non linear friction term far exceeds that of the
other non linear terms for the case of tidal estuaries.
In order to
minimize the importance of non linear friction as a right hand side
term, a close approximation of the linear part of the friction term (a
major part of the friction term is
linear as is
shown in
be included on the left hand side of the equations.
Chapter 2) will
This greatly
enhances iterative stability and allows computations to be performed for
very shallow estuaries.
Furthermore there may be theoretical limitations for iterative
schemes relating to the relative size (with respect to wavelength) of
the estuary.
Lamb (1932) shows that when solving for the case of an
open ended canal, the solution obtained by treating the finite amplitude
term by successive approximation will be unstable if 2n ()
small, where x = the size of the canal and X
=
wavelength.
(. ) is not
Even though
the same difficulty does not necessarily occur for the case of a closed
ended canal,
the criterion may be viewed as being indicative of
potential instabilities.
Table 1.1 shows the wavelengths associated
with a 12.4 hour tide in various water depths without considering bottom
friction.
Since it
is the intent to apply the method to estuaries and
bays and not to coastal seas, the stability criterion will be small and
this potential instability will not be a limitation of the method.
In the following chapters the solution strategies applied are
discussed in detail.
Chapter 2 describes the governing equations used
-21-
Table 1.1
Wavelengths of a 12.4 Tide in Various Water Depths
x
h
(m)
(km)
2
200
10
442
100
1400
-22-
and furthermore describes important tides and the types of interactions
that may occur.
In Chapter 3 the governing partial differential
equations are discretized such that numerous sets of linear algebraic
equations are generated.
These sets of equations are in the frequency
domain and are coupled through the non linearities.
Chapter 4 examines
various strategies to optimally solve each of these linear sets of
equations and Chapter 5 discusses the details of the implementation of
the fully non linear scheme.
In Chapter 6, the program TEA-NL (Non
Linear Tidal Embayment Analysis), developed with the methods described,
is applied to an example case.
-23-
DESCRIPTION OF TIDES IN SHALLOW EMBAYMENTS
CHAPTER 2.
2.1
Governing Equations
The equations which are used to describe
tidal wave propagation may
be readily developed by depth averaging the Navier-Stokes and continuity
equations with the assumptions of hydrostatic pressure distribution,
constant density fluid, constant pressure at the air-water interface and
negligible momentum dispersion (or eddy viscosity).
(Dronkers, 1964):
equations are
n9t + [u(h + n)],x + [v(h + n)] y
- fv + gn
gt ,x
u
v
,t
+ fu -
+ g~
,y
The resulting
(2.1)
0
+ vu ) = 0
C /p(h + r) + Tb/p(h + n) + (uu
,y
,x
x
x
(2.2a)
) = 0
(2.2b)
y
/p(h + n) + Tb/p(h + n) + (uv
y
,x
+ vv
,y
where:
u(x,y,t)
v(x,y,t)
components of water velocity in x and y directions,
respectively
r(x,y,t) =
surface elevation relative to mean sea level (MSL)
t
=
time
h
=
depth to MSL
g
=
acceleration due to gravity
p
=
water density
f
=
the Coriolis factor
= the applied surface stresses
x,
x y
b
T ,'
b
= the bottom stresses
-24-
.
I~-----------
Bottom stresses are quantified as [Daily and Harleman, 1966]:
b
S/0
2 1/2
c f (u
xb
x /0
(2.3a)
2 +v2) 1/2
+v)
v
= C
(2.3b)
where
Cf = friction factor =
1/8 fDW
Darcy-We isbach
g/c2
Chezy
2
ng
(2.4)
Manning
h
The fully non linear friction terms may be approximated by linearized
friction terms as follows:
blin /p=
[h/(h+n)] b/P
b
[h/(h+n)] b/
y
=
b,lin
ln/p
y
=
X
(2.5a)
XV
(2.5b)
where
X
=
cfU
=
U
=
representative flow velocity
linearized friction coefficient
(2.6)
For tidal flow with only one frequency present and the linearization
being performed on an equivalent work over a tidal cycle basis, X has
the following form [Ippen, 1966]:
8
max
3n
(2.7)
f
where
Umax
max
representative maximum velocity during a cycle
-25-
Although the linear friction term does not characterize the fully non
linear one in spreading energy to other frequencies, it can approximate
the magnitude of the actual non linear friction term quite reasonably.
Linearized friction is helpful in the fully non linear scheme as an
iterative stabilizer.
Wind stresses may be approximated by the empirical formulas [Van
Dorn, 1953; Wu, 1969]:
S=
x
s
y
where:
U10
air
D
U2 cos 0
w
10
(2.8a)
p ir
air
U2 sin e
w
10
(2.8b)
D
air
=
= wind speed
pair = density of air
cD
= wind drag coefficient
S
w
= wind approach angle
The applied wind stress term in the governing equations may be
simplified by assuming that finite amplitude effects will be unimportant
and approximating 1/(h+n) as 1/h, a very reasonable assumption when
considering the empirical nature of the formula used to represent this
term and the inherent limitations of a depth averaged model in
simulating wind-driven circulation.
Finally, the Coriolis parameter is calculated as:
f =
(2.9)
20 sin6
where 4 = degrees latitude and 0 = radian frequency of the Earth.
The boundary conditions associated with the governing equations are
elevation prescribed and normal flux (or velocity) prescribed conditions
which are respectively expressed as:
-26-
=
n (x,y,t)
on P
(2.10)
Qn(x,y,t) =
Qn(x,y,t)
on TQ
(2.11)
(x,y,t)
and
where
P
=
elevation prescribed boundary
r
=
flux prescribed boundary
As is shown in Figure 2.1, elevation prescribed boundaries are usually
associated with open ocean boundaries and flux prescribed boundaries are
usually associated with land (including rivers) boundaries.
Since the governing equations (2.1 and 2.2) are based on first
principles (with the exception of surface stresses), their ability to
model the circulation in estuaries depends on whether the assumptions
made in their derivation are satisfied.
The depth averaging of the
equations precludes the accurate modeling of strongly stratified
estuaries and/or wind induced circulation in deep waterbodies.
The
constant density assumption does not allow density driven currents to be
simulated and the hydrostatic pressure assumption rules out the modeling
of short or intermediate length waves.
The importance of eddy viscosity
has been deemed as negligible in most estuaries [Dronkers, 1964] and
hence only empirical support is required for the surface stress terms.
Since we have confidence that Eqs. 2.1 and 2.2 accurately describe tidal
circulation in well mixed estuaries, the task at hand is to implement a
numerical scheme which allows these complicated partial differential
equations to be solved accurately for arbitrarily shaped estuaries
over extended periods, such that the effects of the variations
-27-
River
Inflow
Q
TIDAL
EMBAYMENT
r
OCEAN
Figure 2.1
Definition Sketch Showing Typical Elevation Prescribed (P )
and flux prescribed (rQ) Boundaries
-28-
in
tidal forcing can be included.
As we have seen,
the harmonic method
in conjunction with finite elements is excellently suited for this
purpose.
2.2
Harmonic Tidal Components in Estuaries
Prior to proposing a general harmonic finite element scheme which
allows the treatment of the more cumbersome non linear terms in the
shallow water equations, it will be beneficial to discuss in more detail
the nature of the non linear responses which arise in shallow estuaries.
Tides are the result of complex gravitational interactions between
the moon, sun and oceans.
Tides in the open ocean may be described by
the superpositioning of a series of harmonic components which are
unaffected by the non linear terms in the governing equations.
In
shallow water, however, non linear effects such as the finite amplitude
of the tide (compared to the overall depth),
bottom friction and
convective acceleration become significant, and as a consequence the
astronomical tides present (see Table 2.1) generate shallow water
tides which cannot be ignored [Dronkers, 1964].
Shallow water tides may be classified as either overtides or
compound tides.
Overtides correspond to the generation of a tidal
response through the non linearities by one astronomical component.
The
frequencies associated with the overtides are exact multiples of the
frequencies of the astronomical tidal components which generate them
(see Table 2.2 for overtides of importance).
Compound tides are the
result of the non linear interaction between two astronomical
constituents, and their associated frequencies correspond to sums and
differences of frequencies of the astronomical tide (see Table 2.3 for
-29-
Table
2.1
Astronomical Tides of Importance
Tide
-1
f(hr - )
Mm
MSf
Mf
0.00131
0.00151
0.00305
763 = 32 days
661 = 27.5 days
328 = 13.6 days
M2
0.08054
12 hr 25 min
S2
0.08333
12 hr 0 min
46
N2
0.079051
12 hr 39 min
19
K2
0.083449
11 hr 59 min
13
L2
0.082080
12 hr 11 min
4
K1
0.04178
23 hr 56 min
58
01
0.04198
23 hr 49 min
41
P1
0.04158
24 hr 03 min
19
T(hr)
P /P
(%)
1M
LOW PERIOD
1.2
9.1
17.2
SEMI-DIURNAL
100
DIURNAL
-30-
_~I~_
Table 2.2
Major Overtides
f(hr- 1 )
Tide
Freq. Comp.
M4
2wM2
0.161074
6.21
M6
3
wM2
0.241611
4.14
M8
4n2
0.322148
3.10
S4
20
0.166667
6.00
S6
302
0.250000
4.00
Table 2.3
Tide
Major Compound Tides
Freq. Comp.
MS4
T(hr)
M2 + wS2
f(hr-1)
T(hr)
0.163870
6.10
2MS 6
2M2 +
2
0.244407
4.09
2SM 6
2
+
+S2
2
0.247204
4.05
2MS 2
2
-
WS2
0.077740
12.86
wM2 + wN2
0.159588
6.27
+
N2
0.240125
4.16
-
2
ON2
0.082022
12.19
MN
2MN6
2MN
2MN 2
2w
2
2o2
. 2.
..
-31-
compound tides of importance).
Residual circulation corresponds to both
steady state (zero frequency) circulation which results from an overtide
type interaction or very low period compound tides produced by
frequencies of two major closely spaced tides.
Let us examine how the non linearities interact with the
astronomical forcing tide to produce overtides and compound tides.
We
shall do this by following an iterative type procedure which is
indicative of the overall non linear iterative numerical scheme that
The non linearities are treated as right hand
has been implemented.
side force loadings of sets of equations produced by harmonically
separating the (time domain)
governing equations.
Details concerning
this harmonic decomposition of the governing equations are given in
Chapter 3.
For the present analysis,
we shall simply state that a non
linear forcing at a given frequency will generate a response at
additional frequencies.
In general the variables describing tidal motion may be expressed
as a harmonic series of the form (for a one-dimensional tidal wave):
where:
(x,y,t)
=
_ J(x,y) cos(wjt + * )
(2.12)
u(x,y,t)
=
uj(x,y) cos(w t +
(2.13)
j)
wj
=
frequency of the jth harmonic component
'n
=
amplitude of elevation of the jth harmonic
uj
=
amplitude of velocity of the jth harmonic
j
=
phase shift for elevation of the jth harmonic
*j
=
phase shift for velocity of the jth harmonic
By definition these series representations are exact for the case of
linear deep water tides where only astronomical species exist.
-32-
They are
~W~CIX~_
EDIIYLil~
-LU~-~--~.
-~I~
II
approximate
UI*E*?6~(1LYL
for the case of non linear shallow water tides and the
accuracy with which they represent the tides depends on how many terms
in
the series are considered.
The general type of terms produced by substituting our series
representations into the non linear finite amplitude term may be studied
by only considering a two term series.
(un),x
f {[u
[;
cos(wt +
cos(wit +
This yields:
,) + um cos(&mt +
) + r m cos(Wmt +
,m)1
m)
9x
(2.14)
This may readily be expanded to:
S(u),x
(Uln),x cos(W t +
I)
cos(W t + c9)
+ (uIm),x cos( Lt + * ) cos( mt +
m)
+
)
Vm ) cos(Wmt +
m)
+ (Um1
+ (um
We note that Eq.
,x cos(wmt + Jm) cos(t
) ,x cos(mt +
(2.15)
2.15 contains both terms which involve only one
frequency and mixed terms which involve two frequencies.
The former
describes the effect of finite amplitude produced by a single tidal
component while the latter describes the interaction of two tidal
components.
The mixed term is the representative term in Eq. 2.15 and
may be further expanded through a trigonometric identity as follows:
(U1m),x cos(Wt +
=2
(Ufm),x
)
) co(Wmt
cos[(
- ~,)t + ('
-
m)
+ Wm)t + (0 + rm ) ]l
+ cos[w
+
I COE
it
-33-
(2.16)
We conclude that the non linear finite amplitude term may be represented
by a harmonic expansion with associated frequencies equal to the sums
and differences of all the possible combinations of frequencies present
in the expansion for responses.
We now apply the previous expansions to examine how overtides are
generated through the finite amplitude term.
We start out our iterative
investigation by assuming that the non linearities are non-existent and
that only one astronomical species of frequency wl exists in our
estuary.
If
we treat the finite amplitude
term as a forcing term in
the
continuity equation, we may deduce from Eqs. 2.14 - 2.16 that a tide at
frequency wl produces a forcing at both steady state (or zero
frequency) and at 2wl.
Both these forcings in turn generate responses
at their associated frequencies.
Hence after one cycle of our iteration
we have responses at frequencies 0,
wl and 2wl.
Each of these
harmonic responses will again generate two forcings equal to the sum and
difference of the associated frequency.
forcings at 0, 201 and 4wl.
At Cycle 2 this generates
Furthermore the various harmonic
repsonses will interact to generate mixed forcings at frequencies equal
to the sums and differences of the mixed frequencies.
generates forcings at wl, 201 and 3wI.
For Cycle 2 this
Table 2.4a summarizes the
response-forcing interaction which are produced at Cycle 2 by the finite
amplitude term when only one astronomical tidal forcing frequency is
present.
Table 2.4b shows the interaction occuring at Cycle 3 and the
process is repeated until a sufficient number of frequencies has been
found.
These integer multiples of the base frequency wl then are the
overtides associated with the astronomical tide at frequency wl.
For
the tidal problem, the energy transferred to each successive harmonic
-34-
__IIILILU l__ylLI_1Ls~.~llY--..C^~___
Table 2.4a
Response-Forcing Table for Overtides as Generated
Finite Amplitude Term at Cycle No. 2 of Iteration
Response
Frequency
211
0
0
1
2w1
2w1
w1
3w
0
l
4w1
0
1
Table 2.4b
__
Response-Forcing Table for Overtides as Generated by
Finite Amplitude Term at Cycle No. 3 of Iteration
Response
Frequency
0
1
0
w1
20
1
2"1
3w1
2wl
3w1
41
0
.
1
4w 1
0
1
3
2w1
15w
.1
6w1
301
0
-35-
I
L -1 11^11.
will decrease substantially.
Hence only a limited number of overtide
frequencies need be considered in order to accurately model the non
linear interactions occurring.
In fact, for the finite amplitude term,
the most significant harmonics are the zero frequency and the 2w~,
although an entire series of overtides (3wl,
4wl, etc.) exist.
We
note that each of these response-forcing tables is indicative of the
amount of energy being transferred to the various harmonics, in that the
more cycles it takes for a non linear frequency to show up, the less
important it is.
The convective acceleration terms will produce the same overtide
frequencies as the finite amplitude terms, even though the energy will
be distributed somewhat differently to each of the various harmonics due
to there being different phase shifts for the terms.
When two or more astronomical forcing tides exist we can again
deduce from our previous expansion that the finite amplitude and
convective terms will produce not only the harmonic overtides associated
with each individual astronomical tide, but also tidal species with
frequencies equal to the sums and differences of combinations of the
astronomical frequencies present and their associated compound tides and
overtides.
Table 2.5a and b illustrate the frequencies present at the
first and second cycle.
We note that the cycle at which a frequency
shows up is indicative of its relative importance.
Furthermore we note
that all frequencies which appear at a particular cycle are not
necessarily equally important as can be seen by examining the individual
expansions.
The non linear friction term differs somewhat from the finite
amplitude and convective terms in that all frequencies generated by the
-36-
LILI___LL1IILU__Y___LI~-XXI
Table 2.5a
Response-Forcing Table for Compound Tides as Generated by
Finite Amplitude Term at Cycle No. 1 of Iteration
Table 2.5b
Response-Forcing Table for Compound Tides as Generated by
Finite Amplitude Term at Cycle No. 2 of Iteration
-37-
non linearity are present at the very first cycle.
Subsequent cycles
only update the magnitude of the forcings and the associated responses.
Let us examine the bottom friction term for a one dimensional case.
b
7
x
P
h
h
S-q,cf u u
(2.17)
This equation may be approximated by Taylor series expanding the finite
depth ratio term to get:
b
h
h+
x
u u
cfpUnU
(2.18)
cf1 uI
Assuming only one astronomical tide exists we have at the first cycle of
the iteration a response of
=
u
(2.19)
u cos(w,t)
Substituting into the dominating term of our approximation for friction
and performing a Fourier expansion we find that the non linear forcing
will be:
2
cffulu
-
cos 5
cfu (0.8488 cos olt + 0.1698 cos 3w1 t - 0.0242
1t
+ ... )
(2.20)
Hence there are forcing terms at all the odd harmonics of the base
frequency wl.
We note that the major forcing component remains at the
astronomical frequency, whereas both the finite amplitude and convective
acceleration terms distributed forcing to frequencies other than that of
the responses that were generating the forcing.
Hence after the first
cycle we expect responses at frequencies wl, 3w1, 5w1,
-38-
etc.
It is
quite simple to demonstrate that at the second and subsequent cycles,
when all
these odd harmonics are included in
the expansion for velocity,
the term cfulul will still only produce forcings at the same odd
The precise coefficients of the
harmonics as the first cycle.
approximating Fourier series will be case dependent.
However since the
major forcing is at the base frequency, the major reponse generated
should be at the same base frequency.
For this case of dominant
response frequency remaining the same, the Fourier series expansion
representing all the interacting frequencies should be quite similar
(for the first few loading terms at least) to the very first expansion
equation (2.20).
It is also the case for compound tides that when one
frequency component dominates, the Fourier series approximating cfu
uI
will be similar to the one generated if only the dominant terms were
considered.
Finally we note that the (n/h)cflulu term will generate
even harmonics (including zero frequency) in the same way that cf uIu
generates odd harmonics.
It
is
obvious that the forcings and hence
responses due to this second order term are of smaller magnitude than
that of the main term.
Table 2.6 shows a summary of the astronomical, overtide and
compound tidal constituents (at the high frequency end) which could be
important in shallow estuaries.
Table 2.6 in graphical form.
Figure 2.2 shows the information from
It is emphasized that these constituents
are case dependent and are affected by such factors as the relative
importance of the various astronomical tides for the bay under
consideration and/or
important.
which non linear terms in the equations are most
We note that all these frequencies modulate over a period of
up to 208 days or possibly longer if other components are considered.
-39-
Table 2.6
f
T
Af
T
=
=
=
=
Tides of Interest (High Freq.
End)
frequency of tide
period of tide
freq. increment between consecutive tides
1/Af = synodic period (length of record required to distinquish
between tidal constituents)
-40-
~L-LI-L-^LL
.--I~Xr
S
-~~L-1_.ICI__
~_
6
H,
0I
I
S4 **2SM6
"*.* -*
.*..__
...-
*
MN 4 * ****
* *
**
*
K2
2MN 2 & L2 .......
M2
N2 2MS
2
U1
p,
MSf
|I I
isl
So'
C
-41-
--
----
steady---
.-
,--(
I
M4
II
I
It
is
clear that it
is
not possible to include in
a general sense
all of these tidal components with either a perturbation analysis (in
which the modeler determines a priori which terms and interactions are
to be investigated) or a scheme which uses a Fourier expansion in terms
of integer values of a base frequency (allowing only overtides of one
major frequency to be studied).
The most general and computationally convenient scheme which allows
investigation of any number of these closely spaced tidal components is
an iterative scheme in which the non linear interactions are treated as
force loadings which are somehow harmonically analyzed.
chapters discuss in detail the numerical
The following
techniques applied in
the
iterative frequency domain scheme used to develop the numerical code.
-42-
_
CHAPTER 3.
3.1
~__I~
NUMERICAL FORMULATION
Weighted Residual Formulation
The spatial dependence of our governing equations will be resolved
through the use of the finite element method.
In order to apply the
finite element method a weighted residual formulation [Connor and
Brebbia, 1976] must be established which shall be used as the basis of
our finite element formulation.
As is shown in Chapter 4, in order to
avoid matrix rank problems the algebraic matrix equations produced with
our finite element scheme must be solved by substituting the finite
element momentum equation into the finite element continuity equation.
The resulting formulation is similar to that of Lynch [1980] with the
exception that the finite element discretization is
continuity and momentum equations are merged.
applied before the
These manipulations,
however, dictate that the continuity equation be used in order to
establish the symmetrical weak weighted residual form.
It may readily
be shown that formulating the fundamental weak form in this fashion
leads to elevation prescribed boundary conditions,
j*, being essential
and normal flux prescribed boundary conditions, Qn*, being natural.
Specifically applying Galerkin's method to establish the fundamental
weak form, the error in the continuity equation is weighted by the
variation in elevation, 6r, and is integrated over the interior domain,
Q.
Furthermore the natural boundary error must be accounted for by
weighting it with 8n and integrating it over the natural boundary
SN .
It is required that the combined integrated and properly
-43-
__~TI_*_____*CC3_1___*_A~-.
weighted interior and natural boundary errors vanish and the following
expression results:
ff
+ Q* l i
18n dO + f T-Q
+ [v(h+-)]
+ [u(h+n)]
ft)
d
= 0
(3.1)
0
theorem in order to eliminate the derivatives on the
Applying Gauss'
flux terms yields:
ff {,
9
+
8r - u(h+n)(6r)
f
fu(h+n)a
r
where anx,
nx
,x
- v(h+)(p)
+ v(h+j)a
1
ny
I
do
6n dV +
(-Q
r
n
+ Qn ) 5
n
dP = 0
(3.2)
any are the direction cosines on the boundary.
However, the normal flux may be expressed as:
Q
n =
anx Q + any Qy
nx x
ny
(3.3)
y
where
Q
=
u(h+)
(3.4a)
Q
=
v(h+)
(3.4b)
Using the previous relationships for flux simplifies Eq.
ff In I
- u(h+n)(8n)
+ f Qn 6
r
dr +
3.2 to:
I dO
- v(h+r)(8 )
y
f (-Qn+
r.,
n)'
dP = 0
Furthermore the entire boundary, r, is divided into an essential
boundary, r E , and a natural boundary r N .
-44-
This then leads to:
(3.5)
I___- .ii._
I___WIIPI__ _e
II1L_1LIIII____Yli
} dQ
- v(h+n)(8n)
ff {rn 8n - u(h+n)(8r)
+ f QnSndr + f Q 8dr r
nr
r
E
N
Qn 6dr+
o
( Q&n 6dr-
(3.6)
N
N
The natural boundary integrals of normal flux cancel and the essential
boundary integral of normal flux vanishes due to the selection of
elevation as an essential boundary condition.
This implies that
exactly satisfied on PE and therefore the variation, An,
"* is
is by
With these simplifications, Eq. 3.6 leads to
definition zero on r E .
the symmetrical weighted residual weak form:
ff
{n
- v(h+n)( n) IdQ + f Qn
tn - u(h+r)(~i)
dTr - 0
(3.7)
N
We now proceed with the weighted residual method by establishing
the weighted residual form of the momentum equations.
In order to allow
for the possibility of establishing symmetrical final system matrices
each of the momentum equations is multiplied through by depth, h. The
weighted residual forms for this modified form of the momentum equations
is then obtained by weighting the associated errors with residual
velocities and integrating over the interior domain, resulting in:
ffhu
+ gh
x
+ h(uu
ff
fhv
+ gh,
+ h(uv
- fhv -
,x
+ vu
,y
s
2 2)1/2
u
x/p + (h/h+n) cf(u2+
)I 6u dQ
=
- fhu - T/o + (h/h+n) cf(u2+
,x
+ vv
)1 6v dQ
,y
=
-45-
(3.8a)
0
0
2 )/2
(3.8b)
The weighted residual equations which will be the basis of the
finite element formulation have now been established.
The final
weighted residual equations include first order spatial derivatives of
all the variables, n, u, and v.
continuity requireumen
Hence the associated functional
imposed on these variables is that they be
continuous over the domain.
Furthermore there are first order
derivatives taken of the weighting function 6T, again requiring
continuity over the domain, while no derivatives are taken of the
weighting functions 6u and 6v, requiring only that they be finite over
the domain.
Since the non-linearities will be treated in an iterative manner
Eqs. (3.7) and (3.8) are re-arranged such that non linear terms appear
on the right hand side where they can be conveniently updated as pseudo
force loadings to a linear problem with each iteration.
All boundary
and non-variable loadings are also placed on the right-hand sides.
Finally, in order to enhance iterative stabilty, a linearized friction
term is included on both sides of the momentum equation.
These
modifications result in the following equations:
ff
{n
&r - uh(6n)
fQ
} dQ =
- vh(6)
6n dr + ff {un(6)
+ vyn()
}
dQ
(3.9)
N
ff
{ hu
+ ghj
,t
- fhv + Xu
,x
ff{Ts/p + (X - (h/h+)cf(u
x
} 6u dQ =
+
v2)/2u - h(uu
-46-
,x
- vu
)}6u dQ
,y
(3.10a)
ff
{ hvt
0
x
3.2
+ ghy
- fhu + Xv } 6v dQ
-
,y
f,9x
Finite Element Method Formulation
In order to generate a system of algebraic equations from integral
equations, the finite element method is applied to the final form of the
This involves dividing the global domain,
weighted residual equations.
Q, into element subdomains, Qe, and representing the variables within
each element by polynomial expansions.
Contributions from all elements
are summed and inter-element functional continuity requirements are
taken into account in order to generate a global system of equations.
To satisfy the minimum functional continuity requirements on the
variables linear or higher order expansions must be used for the finite
For the development of this method it was felt
element approximations.
that the simplicity of linear expansions outweighed the improved
accuracy achieved (for the same number of total nodes) of higher order
elements.
Therefore the simplest possible element, the linear triangle,
was selected and the variables are expressed within each element as:
S=
tl1nI1
+
2"2
+4
2
u2
++
u
=
1u
1
+
v
=
( 1V 1
+
2 v2
3 13
=
(n
u3
=
u(n)
3v 3
=
3
+
where:
-47-
(n)
()v
(3.11)
(3.12a)
(3.12b)
2
(n)
1
(3.13)
are the nodal elevation values for element n.
v1
U1
u( n
)
and
u2
v(n)
u3
v2
(3.14a,b)
v3
are the nodal velocity values in each coordinate direction for element
n.
Finally,
--
(3.15)
2 62]
[ l
-
are the normalized element coordinates for each element.
The same
linear expansions are used for the weighting functions which is
expressed as:
(3.16)
)(n)
6
=
4
6u
=
,
6 u(n)
(3.17a)
6v
=
,
6v
( n)
(3.17b)
6
Furthermore, linear element expansions were selected for mean water
level depths h.
In this manner elevations, velocities and depths are
-48-
all defined at the nodes such that inter-element fluxes are both
Hence:
continuous and cleary defined.
=
h
4
(3.18)
h(n)
Finally, for reasons discussed in Chapter 5 it is desirable to have
nodal values for both the linearized friction factor X and our friction
factor cf, again requiring linear expansions:
and
X
=
cf
=
X( n)
(3.19)
(n)
(3.20)
_
Substituting the finite element expansions into the final weighted
residual form of the continuity equation (Eq. 3.7) for each element
sub-domain, Pe, and summing over all elements within the domain, Q,
leads to:
T (n) 8(n)
Sel
-
,t
h
4 u
- -,x
--
(n )
(n)
(n)
n)
(n)
(n)
(n)
v
4,
d, h
dQ
e
6 (n)d r +(n)
*
JQ n
_
.
rT
N
e
n
u
dr + ff
(n)
4
4r
(n) (n)
(n)
Sn..
+4)v
4)9
4
-,x-
-
-
-
(n)
,9
(
lpdQl
e
(3.21)
Re-arranging somewhat gives:
T
S6
T
[(n[
)
9e
el
=(-fQ4
TN e
T
T
dQ)n(n)
0f
4 h(n)4 dQ)u(n)
d )v(n)
X1
e
e
T
(n) (n)
T
+4 4)
tx 4) I 4u
(
dr) +
(n)
T 6 h(n)
(n)
T
e
-49-
(n)
(n)
}dQ]
4 v
(3.22)
U(n)
Letting
E
L
u(n)
(3.23)
v(n)
6u(n)
,
6U(n)
and
(3.24)
Eq.
3.22
may
expressed
(n)be
therefore
as:
Eq. 3.22 may therefore be expressed as:
(n)
)
el
(n) -D (n) U (n)
Sp(n),lin + p(n) ,nl
-
el
In)
(3.25)
where the element sub-matrices are defined as:
S
(3.26)
T
d
fT
4 h
e
-E
_ h (n )
ff j
_ -,y
SdQ
d
(3.27)
e
P(n),lin
f 4T Q dr
T)
rN
(3.28)
n
Ne
P(n),nl
ff (4
-)
-G -
(n)
4u-
(n)
-
T
+4S
-,y
6 r
(n) 4)vv(n))dQ
)dQ
(3.29)
e
into global matrices leads to the
Loading the element sub-matrices
following global equations:
6
T
[M
, - D= -U
=T) - t
-
-P
lin
nl1
+ P ]
T)
-T
-50-
(3.30)
where
'n
=
global elevation vector (1 elevation per node)
U
=
global velocity vector (2 velocities per node)
=
global continuity equation coefficient matrix
=
global derivative matrix
=
global load vector for flux prescribed boundaries
=
global load vector for finite amplitude effects
11
D
lin
pnl
-71
leads to a final set of
Allowing for an arbitrary variation in 68q
non linear algebraic equations which minimize the error incurred in the
continuity equation due to the finite representation of the spatial
variables:
M r,t - DU
lin + Pnn1
- Plin
=
(3.31)
Substituting the finite element expansion into the weighted
residual forms of the momentum equation (Eqs. 3.10a,b) for each element
and summing over all elements within the domain, Q,
sub-domain, Qe,
leads to:
(n)
_(n)
[ff{sh
elQ
X
u9 +
u
$u
(n)
(n)
h(n)
(
(n)
(n)
h(n)(n)
(
+ 0 h
_fd
4 h n) v
_9
dQ
--
e
s
ff {9
ff{
Q
4 6u(n)} d Q + ff
Q
h
.
Sh)
4u
-
(n)
4, u
-,x-
(n)
I{A(n) 46 u(n)ldQ
fric-u
(n)
h
+ -..
for the x-momentum equation and
-51-
..v
(n)d
,
u(n)
,y-
}Su
--
(n)
dQ ]
(3.32a)
Sf
S[ff{$ h
el 9
e
f L
(n)
v(n)
v
v
+ 4
n )
(n)
(n)
(n)+ g
(n)
_6v(n)
-f h ()un+
_
(
n
(n)
) (n
dO
s
f {-~.
p
_ 6v(n)}do + ff
{
A-v(n)}dQ
)
fric-v
e
e
(h
- ff
Q
e
(n)
()
v
6 u(
(n)
for the y-momentum equation.
h
+
()
(n)
v(n)
v
(n)
},v
(n )
dQ
(3.32b)
The right-hand side load terms
representing the difference between linear and non linear friction are
now denoted as:
(n)
fric-u
A
v2
cf(u 2
c (u 2+
f
h
h+
=
(n)
fric-v
frn)==[ch+n
c fv(u
2
v )
1/2
2)1/2
+
u1
(3.33a)
v
(3.33b)
and have not been expanded in terms of the finite element approximations
used since they require a numerical integration scheme.
Re-arranging Eqs. 3-32a,b leads to:
,T
Su(n)[
-U -
-
el _
h(n) i , dQ)u(n) + (fT,
-, t
-±-
Qe
e
h(n)_ dQ)v(n) + g(f4T
T
+(_-ff
X(n)_ dQ)u(n)
-
Sp
h(n)
-
,x
dQ)n(n)
e
e
s
d)
STTd)
f
x(n) + (ffrA()
(ff±T T-=dQ)
e
SP
-( ffT
d
fric-u
e
h(n)_ u(n)
(n)
T
e
e
-52-
h(n)
(n)}do)]
(3.34a)
___IIILYIYY__JYI__~_L-I~~~IWICiYIZI
and
6 v(n)
el
t-
Tfr
9
h(n)f
)
h (n)4 dQ)v (n
-f.T
dC)v(n) + (
-
-,t9--
e
+(ffT
h(n)
(n)
g(ffT
+
+ g(ffh
d()u(
e
(ff T
e
(+
+(J
e
p
e
-(ff{
P) dP) l(n)
T
h(n)
(n4)
cfric-v
(n)S,x-
T dQ)
-
(n)
T
h(n) -..- -
,y - (n)}dQ)
(3.34b)
e
Adding together expressions 3.34a and 3.34b (which does not change these
equations due to the arbitraryness of 6u(n) and 6v(n)) and recalling
expressions 3.23, 3.24 and 3.27 we have:
e
_(
n)
n)
[M(n) u(n) +
(n)
U(
+!
+ g D(n)
i
(n)
=
el
+ p(n) ,nl
S fric
_(n),nl
(3.35)
-cony
where the element sub-matrices are defined as:
0
effT Sh(n)4) dQ
M(n)
--U
e
f
0
II
fT
M(n)
=F
T
h(n)
(3.36)
Qe
(n)X dQ
ff -)
-53-
(3.37)
h(n)X dQ
_fffT
dQ
e
(n)
C
h(n)
jf±T
(3.38)
)
C
dQ
Qe
S
x dQ
ff
e
.(n)
Sff
e
T
(n) ,nl
ff -4T
-A-fric
(3.39)
s
- d
- dQ
p
(n)
(3.40)
dQ
n)
fric-v
e
u(n)+
h(n) 4 u (n)
ff(4T
- - --,x-
T
h(n)
v(
n )
u(n))dQ
ii
)do
e
p(n),nl
-conv
(3.41)
( T
T
(
n
4)u (n)4) v (n)+ 44h (n)4 v
h (n)
y v
)d
v (n))dQ
,y -
, x-
Again loading element sub-matrices into global matrices leads to the
following set of global equations:
+ M U + M U + g DT
6UT [MU
-U
,t
-F
-W
-C
-54-
nl
A-fri
c
nl
cony
(3.42)
where
_
=
global momentum equation mass matrix
M
=
global linearized frictional distribution matrix
M
=
global Coriolis matrix
P
=
global wind stress loading vector
nl
P -
=
global load vector containing difference between
linearized friction and full nonlinear friction
nl
P
-cony
=
global load vector containing convective
acceleration effects (non linear)
-C
Allowing for arbitrary variation in 6U results in the following final
set of non linear algebraic equations which minimize the error incurred
in the momentum equation due to the finite representation of the
spatial variables:
M
-U
Eqs.
U
+ M
,t
T
U + M U + g D
-C
-F
=
P + P
W
-
A-fric
- P
(3.43)
cony
3.31 and 3.43 are still differentially time dependent which will be
resolved by applying the harmonic analysis procedure presented in the
next section.
With the exception of the non-linear frictional
difference load vector,
(n),nl , all the previous element matrices
P-f
*-d-fric
and vectors may be readily developed in an analytical fashion.
The
procedure is very straight-forward and the resulting element matrices
and vectors are presented in Appendix A.
The matrices M
matrix, MC,
,
M and M are all symmetric while the Coriolis
is skew symmetric.
In the development of the prescribed
flux load vector (calculated only on natural boundaries TN) a linear
varying normal flux is assumed.
For the wind stress load vector PM
wind shear stress is considered constant over each element.
-55-
Since it
is
not possible to analytically evaluate
the integrals in
the friction difference load vector, a numerical integration procedure
is applied.
Allowing for the fact that velocity, elevation and depth
all vary linearly, at least a cubic integration formula is required for
the non-linear friction contribution.
3.3
Frequency Domain Formulation
The finite element technique has been used to resolve the spatial
dependence in the governing equations and has thus reduced the non
linear partial differential form of the governing equations to the
following set of non linear differentially time dependent algebraic
equations:
- DU
M
-n - t
M U
+M
tUF
tC
lin
=
nl
(3.44)
Pin +
-71
--n
U + M U + gD
-
P
(3.45)
+ P
_FU
where
nl
-U
(3.46)
-A-fric
--conv
Both the variables r and U and the loadings P lin
and P
-U
are time dependent vectors.
Pn
P
In addition all terms are
linear with the exception of the right hand side pseudo forcing terms
pnland Pnl which contain non linear combinations of the variables
-U
n
velocity and elevation.
The differential time dependence in Eqs. 3.44 and 3.45 will be
resolved by a scheme which reduces them to sets of harmonic equations
-56-
which are coupled through the non linear terms.
The non linear coupling
will be treated by an iterative updating scheme which shall be discussed
The reduction of Eqs. 3.44 and 3.45 from the time
in more detail later.
domain to the frequency domain assumes that the responses of the system
in elevation and velocity may be expressed as a harmonic series of the
form:
N
(t)
Re
(3.47)
e
=
J=1
Nf
=
U(t)
i
t
(3.48)
)
Re{ 7 U. e
j=1 -
It is noted that the complex quantities denoted by
magnitude and a phase shift.
^
include both a
Furthermore it is assumed that both the
linear and the non linear load vectors may be represented by similar
harmonic series:
=
P(t)
Re{
Nf
A
Y
P
io
t
(3.49)
e
j=1
Substituting the harmonic series representations of the variables into
Eqs. 3.44 and 3.45 and taking appropriate time derivatives leads to:
Nf
A
M ( Y iw
= J= 1=
N
f
iW t
e
j=1 -rj
N
ijt
lin
- y P
Nf ^
) - D( 7 Ue
J=1
e
+
i jt
)
iWtj
f ,n
y
P
J=1 -J
-57-
=
e
(3.50)
and
Nf
A
i
Nf
iWJ t
Nf
N
N
N
.e
+ gDT(
e
PW
j)
pUnl e
J+
(3.51)
j
j=1
=1
= j=1
)
ijt
nl
f
it
it
f
t
J=l
j=1
J=1
ij
e
(
!
)+
e
U
7
+
e
U
( 7Ji
t
These equations may be re-arranged as:
Nf
j=1
(iw M
-
fA
-
A
A
T
-J
-iUj_ +
=-J
F
A
^
U + gD
(3.52)
0
ei
P
-
(1w !U U + M U + M
i& t
An
- DUj + P
j
j
Ain
A
A
-P
iWj t
nl
- WJ
-P
-U
)e
= 0
(3.53)
Due to the orthogonality properties of sinusoidal functions (and hence
complex exponentials), each of the expressions within the brackets must
equal zero.
This leads to Nf sets of time independent linear equation
of the following type:
A
iW
A
Anlin
Pi
- =D -U.
M
=
A
ij 5i!M
-
A
+
U +M!
+
U +M!
4nl
(3.54)
P
-i
A
U
^
T A
+
+ g D
=
W +
^nl
P
(3.55)
Note that the natural flux prescribed boundary conditions are
Alin
The essential boundary
incorporated in the load vectors Pj .
condition may also be expressed as a harmonic series and hence
Nf A
j
j=1
1
iwj t
e
.
ij
e
rNj
N
-58-
t
(3.56)
which leads to a set of boundary conditions associated with each of the
set of equations (3.54-3.55) of the form:
l
N
=
(3.57)
'n
The iterative solution strategy starts out with the assumption that
the non linear loadings are zero.
Each of these Nf sets of equations
are then solved for the boundary loadings imposed.
Time histories may
then be generated for velocity and elevation with Eqs. 3.47 and 3.48.
This in turn allows time histories of the non linear psuedo loading
vectors Pnl(t) and Pnl(t) to be produced.
n
--U
As was assumed earlier,
these time domain pseudo forcings may now be approximated as harmonic
series.
Hence the total non linear loadings for continuity and momentum
are distributed to all or some of the Nf sets of frequency domain
equations.
Now each of these Nf sets of equations
the entire procedure is
is
solved again and
repeated until convergence is reached.
schematic of the iterative scheme is
shown in Figure 3.1.
A
Strategies of
how to optimally harmonically decompose the nonlinear load vectors and
details such as the number of frequencies required for the harmonic
series representation used will be discussed in
Chapter 5.
Each of the Nf sets of equations are linear at each cycle of the
iteration although they are coupled through the non linear loadings.
As
is clear from Figure 3.1 solving each set of linear equations is the
heart of our overall non linear solution scheme.
In the next chapter we
shall examine methods to solve these linear sets of equations in an
optimal manner.
-59-
BOUNDARY CONDITIONS
n
AND Q
FOR ALL FREQUENCIES wj
j
= 1,M
I:
,-
LINEAR SOLUTION AT GIVEN Wj
A
lin
^*
SP l
An
^*
,(Q )+
RESULTS IN U
SELECT NEXT
AND 'j
I
CHECK CONVERGENCE
GENERATE TIME DOMAIN RESPONSE
HISTORY BY SUPERPOSITIONING
OF SOLUTIONS Uj AND nj
U(t) AND n(t)
GENERATE TIME HISTORY LOADING
Pn(t) USING TIME
HISTORY RESPONSE
HARMONIC ANALYSIS OF Pnl(t)
TO GENERATE
FREQ. DOMAIN LOADING P
J = 1,M
SELECT SET OF w j's
FOR
WHICH LINEAR CORE IS RUN
Figure 3.1.
Schematic of Iterative Non Linear Scheme
-60-
CHAPTER 4.
LINEAR CORE MODEL
In Chapter 3 the finite element method was applied in order to
resolve the spatial dependence of the governing partial differential
equations and reduce them to a set of non linear algebraic equations in
space with the differential time dependence left unresolved.
The
assumption of harmonic forcings on the system and harmonically
decomposible pseudo forcings due to the non linearities allowed the
elimination of the time dependence from this set of equations and thus
produced numerous sets of equations in the frequency domain.
Furthermore the concurrent assumption of an iterative type solution
scheme which updates the non linear pseudo forcings yielded sets of
linear algebraic equations of the following type for each required
frequency:
A
A
is M r - D U
A
=
(4.1)
P
iw M U + M U + M U + g D
n
P
(4.2)
A
The continuity loading vector P
can include both contributions from a
flux loading at frequency w and from the component at frequency W of a
harmonically decomposed finite amplitude pseudo forcing.
Similarly the
momentum loading vector PU can include contributions from a harmonic
type wind loading at frequency w and the components of the non linear
pseudo loadings at frequency w due to convective acceleration and the
difference between a linearized friction and the full non linear
-61-
friction.
As was noted in
Chapter 3 these equations form the core of
our fully non linear scheme.
As was discussed in Chapter 2, numerous frequencies are needed to
accurately simulate shallow water tides with a frequency domain
approach.
The required frequencies represent either the astronomical
tides of importance which are present or their associated non linear
Hence the linear core equations need to be
over and compound tides.
solved for each of these frequencies.
model is
iterative,
In addition, the fully non linear
which means that all (or at least most) of these
frequencies need to be solved for at each cycle of the iteration until
convergence is reached.
Therefore it is important that the linear core
solution strategy be not only accurate and free of spurious oscillations
but also very efficient.
The method selected to solve Eqs. 4.1 and 4.2 should also take into
account the wide range of frequencies that may be required, from the
zero frequency and low frequency astronomical tides and residuals
generated up to the very high frequency harmonics.
Finally, the method
applied should take into account the physical characteristics of
typical tidal embayments and the nature of the tidal flows within these
Let us now examine some of the various possible ways in
embayments.
which Eqs. 4.1 and 4.2 may be solved.
One possible solution method is
to substitute for elevation into
the momentum equation and obtain a final equation with velocity as the
basic variable.
Solving for elevation with the continuity equation
yields:
A
S=
1
i
-1
A
w MT-n(P
A
+ =D U)
-62-
(4.3)
Substituting Eq. 4.3 into Eq. 4.2 and re-arranging leads to:
2
(w M U + iM -- + iM+
T-1
g DM
=
D) U-=
U
_
DT
gD
-1
M--
(4.4)
P
-
Hence Eq. 4.4 is solved for velocity U which may then be backsubstituted into Eq. 4.3 in order to obtain elevation.
This strategy,
however, fails due to a rank problem with the total left-hand side
system matrix generated in Eq.
the momentum mass matrix,
4.4.
Due to the w2 factor multiplying
, and the w factors multiplying the
linearized friction and Coriolis matrices, M and MC, in Eq. 4.4, the
contribution of the gDTM
matrices.
D matrix dominates the sum of these
For higher harmonics, the effect of
&2M
;;U
could disappear
entirely due to the round off accuracy of the computer.
Table 4.1 shows
that the rank of the g DTM1 D matrix is N, making the rank of the total
=
=T) =
system matrix also N. Clearly we can not solve for 2N velocities with a
system of 2N equations of rank N. This then indicates that the
appropriate manner in which to solve Eqs. 2.1 and 2.2 is to substitute
for velocity in the continuity equation which produces a wave-like
equation with n as the basic variable.
Let us now go into some of the various ways that Eqs.
2.1 and 2.2
can be solved if velocity is substituted into the continuity equation.
The first technique examined involves solving for velocity with the
momentum equation such that only the mass matrix M
need be inverted.
Note that if the mass lumping procedures are applied, the matrix M
= U
simplifies to a diagonal matrix which makes the required inversion
extremely economical.
More details on the mechanics and implications of
lumping procedures are discussed later in this chapter.
Eq. 4.2 for U as indicated:
-63-
Hence solving
Sizes and Ranks of Various Matrices
N = Number of Node Points
Table 4.1
Size
Matrix
U
M
'
M- 1
D M-
1
D
n M- 1 nT
-64-
Rank
2N x 2N
2N
NxN
N
2N x 2N
N
N xN
N
^
-
1
-1
iw
U
- M
U
A
P
- (M +M
-U
T^
A
) U - g DT1
(4.5)
=C
The above procedure bars zero frequency cases from being solved due to
the 11w term.
This problem may readily be corrected by adding the
U to both sides of Eq. 4.2, where cs is an arbitrary non-zero
term c
constant.
Therefore Eq. 4.2 is now solved for U as:
^
U
TU
(i
1
1
-1
+ cs
t-U
A
- (M + M - c
-C
-F
T
M )U - g D
U
A
(4.6)
such that the zero frequency case is not excluded from investigation.
Substituting for U using Eq. 4.6 into the continuity equation (4.1) and
re-arranging produces:
(0
2
- iwc)M
-1
- gD
-DM
-1
DT
A
P - (
(F
-U
=
+- M
_-C
A
+ cs)P
-cn
(4.7)
-1 T
gD MU D , and the additional
Now both the dominating component,
component,
A
1
- (i,
_)UI
)M , of the left hand size matrix are of rank N.
(w2 - iwc s---n
Hence the rank of the left-hand side matrix is sufficient to solve for
N unknown elevations.
However,
since Eq.
4.7 contains the variable U on the right hand
side of the equation an iterative scheme is required to solve Eqs. 4.6
and 4.7.
This linear type iteration is distinct from the iteration
scheme discussed in previous chapters which updated the non linear terms
in the governing equations.
However both types of iterations could
proceed simultaneously.
Placing the very small (w
hand side,
-
iWC )M term in Eq. 4.7 on the right-
leads to the following attractive iterative linear core
-65-
solution scheme:
gDM
-1
T
D
^k+l
2
(
+DM
=U
1
Ak+l
- (M +
F +-C
P
-U
-1
^
-TP
+ cs
-c
M )
(4.8)
sU U
T
k
JU- (F
(iW+ c )
-U-(
^k
n + (i
- ic)
c
- -gD
Cs M )U U
+
^k+1
n
The superscript k refers to the cycle of the iteration.
iterative solution procedure is
^1
convergence
is
(4.9)
The linear
started by initially evaluating the
^o
o
right-hand side of Eq. 4.8 using r = 0 and U
then evaluating U
1
0, solving for
^1
I
,
using Eq. 4.9 and continuing the iteration until
achieved.
As previously mentioned this linear iterative scheme has some very
desirable features.
First of all,
the system matrix,
gD
-1 T
1D , that
needs to be solved is both real and symmetric thus saving in
computational expense and storage.
Furthermore, the system matrix does
not have frequency, w, embedded in it which eliminates the need to
re-set and re-solve a system matrix for each of the many frequencies
required in the fully non linear scheme.
The advantages of this linear iterative scheme,
however,
by the severe iterative stability problems which can occur,
leading to divergence or extremely slow convergence.
are offset
either
The convergence
criterion for this linear core model is:
Ronv
^k
^k+1
k
^k^k
^ -- ^k-
=
c
2 2
gh
(1
+
)gh
+
-66-
(c s --- h
f)(a + c s
(4.10)
(4.10)
where Rconv
=
local convergence criterion and I = element dimension.
The actual convergence rate may be checked by examining
log(l/Rconv) which indicates
improving every cycle.
how many decimal places the solution is
For example,
Rconv equals 0.1 when the
solution is improving by one decimal place every cycle of the
iteration.
The smaller Rconv the more rapid the convergence while for
Rconv > 1 the iteration diverges.
It is stressed that Rconv is a
local convergence criterion and must be satisfied everywhere within the
domain.
Typical iterative stability problems can be demonstrated by
examining a zero frequency flow in a depth varying channel with a
constant linear friction factor and no Coriolis effects (see Figure
4.1).
For this case the convergence criterion reduces to:
R
cony
1= -.
(4.11)
he
Note that convergence is dependent on both the selection of a value for
c s and the location in the channel.
Table 4.2 shows values of Rconv
for various selections of cs at different locations in the channel in
terms of v, the ratio of maximum to minimum depth in the channel.
Only
by setting c s = X/hmin will convergence be guaranteed everywhere in
the channel regardless of the depth variation.
For example, for a depth
variation of y = 10, the maximum value of Rconv obtained at any point
in the channel by using c s = X/hmin, cs = X/havg and cs
X/hmax are respectively 0.9,
4.5 and 9.
=
Even though the optimal
selection of c s = X/hmin produces a stable convergent scheme, it
very slow due to the value of Rconv being close to 1.
-67-
is
M.S.L.
min
max
h
avg
h
depth ratio = y -
max
h.
min
Figure 4.1
Definition Sketch of Depth Varying Channel
Which Illustrates Convergence Problems of
Iterative Linear Scheme
-68-
> 1
-
Table 4.2
Variation of Convergence with c s
R
cony
c
shallow end
average depth
deep end
hx0
min
2
avg
1
hmax
I1
1+1
Y
-69-
0
The convergence criterion shows that for cases in which the depth
X, vary substantially within the domain,
or the linear friction factor,
Even though
stability problems will arise with this particular scheme.
for some cases it
convergence
is
possible to adjust the global factor c s
is
guaranLeed everywhere within the domain,
it
is
such that
a
cumbersome procedure to find this value and will result in at best very
It is clear from examining either Table 4.2 or
slow convergence rates.
Eq. 4.11, that optimal convergence (i.e. Rconv = 0) is achieved by
selecting local values of c s equal to X/h.
is that instead of adding c
M
What this implies
U using a global value for c s
to the right and left hand sides of Eq. 4.2, local (or element) matrices
S(n) (n) which use the local optimal value for cs should be used to
establish a global matrix which is added to the left and right hand
sides of Eq. 4.2.
However, it is noted that this optimal global
matrix is exactly M.
What this indicates is that the frictional
distribution matrix should be kept on the left-hand side when solving
Eq. 2.2 for U.
Based on examination of values of Rconv and running computer
simulations which apply the linear iterative scheme discussed above for
a variety of embayments,
this iterative scheme proved impractical for
use as the linear core solver.
other schemes
to solve Eq.
Furthermore, examination of various
2.1 and 2.2 led to the conclusion that a
direct one pass non-iterative solution technique for the linear core was
optimal.
As we shall see, this optimal method yields a final system
matrix which is complex and non-symmetric.
Furthermore, it has
frequency embedded into it requiring that at each cycle of the non
linear iteration the system matrix must be re-set and re-solved.
-70-
However, the overall amount of computational effort is substantially
less for the one pass linear solution when considering the number of
iterations required for the other linear solution methods.
This direct
linear scheme was implemented as the linear core solver for the overall
non linear code.
For the direct solution scheme, the momentum equation is now solved
for U as follows:
A
A
U
-
-1
^
T
M
(P - g D
- TOT -U
A
(4.12)
)
where:
TO
=
(4.13)
MF +)
(iwM+
Substituting for U into our continuity equation produces:
(i
^ -1 IT
D
M + g D M
=
=
TOT
)
A
r
-
P
-n
^ -1
P
+ DM
(4.14)
= :TOT -U
As previously mentioned the total left hand system matrix in Eq. 4.14 is
complex, non-symmetric (since the Coriolis matrix M
MTOT ) and has frequency embedded into it.
is contained in
Due to storage limitations
it is preferable to re-set and resolve the system matrix for each
frequency at each cycle of the iteration rather than storing the matrix
produced for each frequency at the first cycle and using it in
subsequent cycles of the iteration.
Finally in order to make this technique viable, the lumping
procedure has been applied not only to the mass matrix MU, but also to
the linearized frictional distribution matrix M and the Coriolis
M.
matrix, ;;c
M
This now allows the required inversion of ;O
-71-
to be
performed economically.
also lumped.
For the sake of consistency the matrix M
was
The lumping for the symmetric mass and friction matrices
involves combining all terms on a row onto the diagonal and for the skew
symmetric Coriolis matrix combining rows onto off-diagonal terms (in
order to retain the skew symmetric natural of the matrix).
TOT will be tri-diagonal.
Hence matrix
The lumping procedure in effect amounts to
a slight re-distribution of mass and the linear friction and Coriolis
Model results have been shown to be
forcings between neighboring nodes.
quite insensitive to these lumpings.
The linear core model has been verified against the analytical
solution for a tidal wave entering a rectangular channel closed at
one end both with and without bottom friction damping (Ippen, 1966).
The example channel used for this simulation was 25 km long and 4 km
wide and had a depth of 10 m.
shown in Figure 4.2.
The grid representing this channel is
A constant element size of 1 km was used yielding
5 nodes across the channel width.
A 12.4 hour forcing tide of 1 m in
amplitude at the open (ocean) end was used.
The linear core model
was run for a no bottom friction case, a lightly damped case (with a
linearized friction factor, X, equal to 0.001 m/sec) and a heavily
damped case (X = 0.01 m/sec).
Results of the linear core model for all three cases are shown
and compared to the corresponding exact analytical solution (at
various locations) in Table 4.3.
Agreement between analytical values
and numerical predictions is excellent for both elevation amplitude
and phase.
For the undamped and lightly damped cases, the linear
core model slightly overpredicts elevation amplitudes by an average
-72-
1
4
i~~.tLL
/V
A\-....
't
I . '
~t
i
I V
'1/VV 1/1/~.\~;\;
~-~r
Ir
I
~
C-
t.S
VV
\~~V\~
~~.'"'
-- ,
-
1----
;n
L//VI
-~------,----
i
LI IVik
1
x (kin)
Figure 4.2
Finite Element Grid Discretization for Closed Ended Channel Example Case
'
Table 4.3
Comparison of Analytical and Numerical Elevations and Velocities
for Example Channel Case at Various Locations
(a) Linearized Friction Factor X = 0.0000
Elevations
Numerical
Analytical
x
Amplitude
Phase
Amplitude
(m)
(m)
(rad)
(m)
Phase
(rad)
0
1.00000
0.00000
1.00000
0.00000
6000
1.02796
0.00000
1.02800
0.00000
13000
1.05113
0.00000
1.05119
0.00000
19000
1.06273
0.00000
1.06277
0.00000
25000
1.06661
0.00000
1.06657
0.00000
Velocities
Numerical
Analytical
x
Amplitude
(m/sec)
Phase
(rad)
Amplitude
(m/sec)
Phase
(rad)
0.36747
0.00000
0.36847
1.57080
6000
0.28178
0.00000
0.28408
1.57080
13000
0.18015
0.00000
0.18181
1.57080
19000
0.08997
0.00000
0.09260
1.57080
25000
0.00000
0.00103
1.57080
(m)
0
-74-
Table 4.3
Comparison of Analytical and Numerical Elevations and Velocities
for Example Channel Case at Various Locations
(b) Linearized Friction Factor X = 0.0010
Elevations
Analytical
Numerical
x
Amplitude
Phase
(m)
(m)
(rad)
Amplitude
(m)
Phase
(rad)
1.00000
0.00000
1.00000
0.00000
6000
1.02744
-0.02026
1.02748
-0.02029
13000
1.05039
-0.03637
1.05044
-0.03641
19000
1.06194
-0.04422
1.06198
-0.04425
25000
1.06581
-0.04680
1.06578
-0.04678
0
Velocities
Numerical
Analytical
x
Amplitude
(m/sec)
Phase
(rad)
Amplitude
(m/sec)
Phase
(rad)
0.36721
1.53906
0.36821
1.53905
6000
0.28158
1.53267
0.28388
1.53274
13000
0.17915
1.52744
0.18167
1.52749
19000
0.08990
1.52485
0.09253
1.52485
25000
0.00000
0.00103
1.52589
(m)
0
-75-
Table 4.3
Comparison of Analytical and Numerical Elevations and Velocities
for Example Channel Case at Various Locations
(c) Linearized Friction Factor X = 0.0100
Elevations
Numerical
Analytical
x
Phase
(rad)
Amplitude
(m)
Phase
(rad)
Amplitude
1.00000
0.00000
1.00000
0.00000
6000
0.98152
-0.18880
0.98144
-0.18904
13000
0.98420
-0.34690
0.98409
-0.34718
19000
0.99167
-0.42508
0.99152
-0.42532
25000
0.99507
-0.45097
0.99483
-0.45070
(m)
0
(m)
Velocities
Numerical
Analytical
x
(m)
0
Amplitude
(m/sec)
Phase
(rad)
Amplitude
(m/sec)
Phase
(rad)
0.34439
1.27029
0.34530
1.27017
1.20652
0.26540
1.20734
6000
0.26328
13000
0.16729
1.15433
0.16963
1.15483
19000
0.08393
1.12844
0.08637
1.12843
25000
0.00000
0.00096
1.13888
,
-76-
of about 0.00005 m which equals approximately 0.005 percent of the
amplitude at each point.
For the heavily damped case, the numerical
predictions for elevation amplitude are slightly below exact values.
The predictions for velocity amplitude and phase are also excellent
although errors are somewhat larger for velocity than for elevation.
We note that this is consistent with the fact that velocities are
The linear core model consistently
computed as derivatives of elevation.
overpredicts velocity amplitudes by about 0.002 m/sec.
Although the
absolute error for velocity is about the same throughout the channel,
the relative error (defined as a percentage of the exact velocity
amplitude at a point) becomes large at the closed end of the channel
We note that this slight
due to velocities decreasing to zero.
overprediction corresponds to a small amount of leakage through the
closed end of the channel.
This is due to the fact that normal flux
is treated as a natural boundary condition and will only be satisfied
exactly (no leakage) in the limit as the grid is refined.
However, in
terms of the velocity at the entrance, the error between exact and
Furthermore we note that the
predicted velocities is less than 1%.
finite element method is an error minimization procedure.
The errors
depend on the degree of spatial discretization and solutions will
improve with increased grid refinement.
The numerically predicted values shown in Table 4.3 are values
for the channel centerline.
Node to node oscillations across the width
of the channel were extremely small.
The character of the maximum
node to node oscillation remained about the same regardless of the
amount of damping.
For elevation, node to node oscillations for this
case were somewhat less than the discrepancy which existed between the
-77-
exact and numerical solution (typical maximum difference in elevation
amplitude across the channel was 0.00002 m which is about 0.002% of
the amplitude at each point).
For velocity the node to node
oscillations were slightly greater than the discrepancy between the
exact and numerical solution (typical maximum difference in elevation
amplitude across the channel was 0.004 m/sec).
For general two-
dimensional flows these node to node oscillations will increase
somewhat.
However it is estimated that even under severe depth and
geometry changes the elevation amplitude oscillations will typically
remain less than 1% with corresponding velocity amplitude oscillations
increasing to several percent.
We conclude that the linear core model accurately simulates the
linearized governing equations at very low node to node oscillation
levels.
Now that an accurate linear core model has been developed,
we are ready to proceed with the more complex task of incorporating
the non linear terms in the governing equations into our computations.
-78-
NON LINEAR MODEL
CHAPTER 5.
The iterative solution technique used in
the development of the
fully non linear model was described in Section 3.3.
Chapter 4 examined
strategies for the optimal solution of each of the linear sets of
equations produced with this iterative scheme.
Attention will now be
focused on the details regarding the implementation of the fully non
linear scheme.
Of primary importance is the selection of a harmonic
analysis procedure for the pseudo loadings generated by the non linear
terms in the governing equations.
In addition, issues such as iterative
stability and convergence rates will be addressed in this chapter.
5.1
Harmonic Analysis of Non Linear Pseudo Forcings
The selection of a harmonic analysis procedure for the non linear
pseudo forcing vectors is of vital importance for the efficiency,
accuracy and generality of the fully non linear model.
The efficiency
of the model is strongly influenced by the number of time history points
required for the harmonic analysis procedure, since the procedure must
be applied at every node in the grid at each iteration.
The accuracy
and generality of the harmonic analysis relate to the type and the
detail of harmonic information extracted from a given time history
record.
A variety of standard Fourier harmonic analysis procedures can be
applied to convert time history loadings to frequency domain loadings.
All standard Fourier procedures operate with integer multiples of some
base frequency.
Therefore, Fourier analysis is quite satisfactory when
-79-
However, as
examining one major astronomical tide and its overtides.
was noted in Chapter 2, tidal harmonics are not limited to frequencies
which are integer multiples of some base frequency.
Tidal energy exists
throughout a wide range of frequencies which may be extremely closely
spaced and are, in general, irregularly distributed.
Hence, in order to
obtain sufficient frequency resolution when Fourier analyzing the non
linear pseudo forcing time histories, an extremely small base frequency
step is required.
Associated with this very small frequency step is a
very large total number of frequencies being processed, most of which
In Fourier analysis procedures, time
have no associated tidal energy.
history record lengths and the total number of time sampling points
increase
inversely with respect to the frequency step.
Hence,
the finer
the desired frequency resolution, the larger the number of time history
data points which need to be generated.
Even application of the very
efficient Fast Fourier Transform algorithm would be impractical due to
the excessive amount of numerical operations required to obtain the
frequency resolution needed to separate important tidal components
[Oppenheim and Schafer, 1975; Newland, 1980].
A very attractive alternative
to standard Fourier analysis
procedures is the least squares harmonic analysis method.
This method
consists simply of a common least squares error minimization procedure
which uses a harmonic series as the fitting function.
This harmonic
series only contains frequencies which are known to exist in the time
history record.
The method is able to extract extremely closely and
irregularly spaced frequency information, yet it only requires a number
of time history sampling points equal to twice the number of frequencies
contained in the time history record.
-80-
The almost infinite frequency
resolution and the extremely low number of required time history
sampling points make the least squares method the optimal choice for
Furthermore,
the analysis of tidal records.
since there are no set
requirements for record length and time sampling intervals, this method
is ideally suited for the analysis of field data [Munk and Hasselman,
1964; Filloux and Snyder, 1979a; Speer, 1984].
The method is even
better qualified for the analysis of analytically generated harmonic
response and non-linear pseudo forcing histories since these signals are
guaranteed to contain only the exact predictable harmonics associated
with a given set of astronomical forcing frequencies (i.e., there is no
energy due to non-tidal forcings).
In order to harmonically decompose a time history record with
values f(ti) at time sampling points ti; i
frequency content wj; J = 1,M,
-
1,N and with known
the following harmonic series is
used
for the least squares procedure:
M
g(t)
=
faj cosw t + b sinj
j=1
(5.1)
ti
where aj, bj are the unknown harmonics coefficients.
The squared
error between the sampling points and the fitting function is:
N
E
If(ti)
=
- g(ti )
2
(5.2)
Hence:
N
E
=
S r7
i=1
2
M
(a cos
t
+ b sinwt
j=1
-81-
i )
- f(t
(5.3)
The error minimization is
accomplished by setting equal to zero the
partial derivatives of E with respect to each of the coefficients aj
and bj.
Hence:
N
M
(a coswjLi + b sinw t)
[
=
E
-_
f(ti)]cosw t = 0
j=1,N
(5.4a)
t = 0
j=1,N
(5.4b)
i=1 J=1
j
M
N
[
7 (a cosW ti
T
=
8
j
i=1 J=1
+ b sinw ti)
- f(t
)]sinj
These equations lead to a complete set of 2M simultaneous linear
equations:
LSQ a
=
(5.5)
SLS Q
where:
is
the least squares
a
is
the vector of unknown coefficients
-LSQ
is
S
-LSQ
(LSQ)
matrix
M
the LSQ signal vector
Equation 5.5 is shown in expanded form in Figure 5.1.
Steady state components in the signal being analyzed simply
correspond to a frequency equal to zero in the harmonic analysis series
(Eq. 5.1).
When setting up the least squares matrix, the zero frequency
component will generate a row and a column of zeros.
These should
obviously be eliminated when generating the LSQ matrix.
This then
results in a 2M-1 x 2M-1 matrix when zero frequency is included as an
analysis frequency.
The harmonic least squares method does not have the matrix ill
conditioning problems often associated with the least squares procedure
-82-
i
i
cos 2 w t
sinw t
Ycosltisinl t1 i
t
t
Tsinwtcosolt
YcoswMtisinwl t
sinwMtisin lti
Icosw
Tcos m cosw
Isin2 wlti
i
i
i
rf(ti)coslti
i
Yf(ti)sinw lt
2
sinblticos $t
t
i
t
Ycosw tisinMt
i
Ysinwltsin NMt
i
Icoswlticos
Figure 5.1
Tcos
I
i
...
,sinMt cosw Mt i
f( t
Tsin2 Nti
i
ff(ti)sinwMti
ti
YsinwMticoswMti
i
i
Linear equation generated by least squares analysis procedure
)coswMti
applied with polynomial fitting functions.
As is seen by examining
Figure 5.1, the LSQ matrix is diagonally dominant due to the squaring of
the diagonal summation terms, while terms in the matrix are still of the
same order of magnitude due to the nature of the sine and cosine
functions.
The LSQ matrix, MLSQ, need only be generated once in order to
analyze any of a number of time history signals with the same frequency
content.
Hence,
SQ is set and LDLT decomposed only once before
the cycling begins in the overall non linear iterative scheme.
Upon
each cycle of the iteration the right hand side of Eq. 5.5, SLSQ
which contains the information on the actual time history signal being
harmonically analyzed, is re-set.
The vector a, which contains the
harmonic coefficients being sought, may then be solved for by a forward
and backward substitution procedure.
Each pair of coefficients ai,
bi, contained in vector a for every analysis frequency, are readily
converted to the complex form required for the frequency domain pseudo
loading vectors.
Setting up the signal vector, SLSQ, involves approximately 8MN
operations and solving the vector a requires roughly 4M2 operations
since the matrix ~SQ is already in decomposed form.
These operations
must be performed for every nodal point at every cycle of the
iteration.
In addition, the nodal time history values, f(ti), must be
generated at every node for each of the N sampling points.
The effort
required for this is dependent on the node to element ratio of the grid
and on which non linear terms are included in the overall non linear
analysis.
-84-
~1M~___
___I__
For a time history signal for which the entire frequency content is
both known and used in
the harmonic fitting series,
the number of time
sampling points required to find the precise signal equals twice the
number of frequencies in
the signal.
Together with the orthogonality
properties of sinusoidal functions, this requirement insures that all
the equations in the system of equations (5.5) will be linearly
independent.
The fact that the reproduction of the signal is precise
may be inferred by examining Figure 5.1 which shows that if f(ti) is
substituted by the original harmonic generating signal,
becomes an identity.
this equation
We note that although there is no noise in the
signal, there is still an inherent round-off accuracy for the elements
in
the LSQ matrix and signal vector which places requirements on the
time sampling procedures.
Let us now examine some of the sampling
criteria which allow the LSQ method to be optimally applied.
In order to avoid duplicity of information in
the elements of the
LSQ system of equations, time sampling points should be contained within
the overall period of the signal being sampled.
However, to allow
maximum dissimilarity in each equation, sampling points should be spread
throughout the period of modulation of the signal.
modulation of a signal is
The period of
obtained by examining its frequency content
and selecting the maximum value of all periods or synodic periods of the
harmonic components contained in the signal.
The synodic period
describes the period of beating of two closely spaced frequencies and is
calculated as:
T
S
2
1
(5.6)
2
-85-
where wl,
w2 = adjacent frequencies in radians/sec.
It is
especially important to sample throughout the modulation period of the
signal when analyzing records with extremely closely spaced frequencies
in order to avoid round-off accuracy problems which could lead to
singularity of the matrix M SQ .
Hence, when sampling a signal which
contains only two very closely spaced frequencies, the four sampling
points should be spread over the synodic period which will be much
greater than the periods of the individual components.
Unlike field
sampling procedures, the numerical generation of time sampling points is
only affected by the number of points generated and is unaffected by the
period of time over which they are generated.
If
the procedures
discussed are adhered to, harmonic signals generated with a typical
tidal frequency content (e.g.
frequencies of Table 2.6) can be exactly
(to within machine accuracy) recuperated while using only twice as many
time sampling points as frequencies contained in the signal.
The pseudo forcing signals generated by the non linear terms are
such that energy is transferred indefinitely to over and compound tidal
frequencies.
The amount or order of the energy transfer to each
frequency is roughly described by the various levels of response-forcing
tables in Chapter 2.
The number of frequencies appearing, as each
subseqent table is generated, increases exponentially.
However, the
amount of energy transferred to new frequencies becomes increasingly
insignificant as they appear.
Since it is numerically impossible (and
meaningless) to consider all the frequencies that energy is spread to,
the harmonic series representing the pseudo forcing must be truncated,
which establishes an order of accuracy for the harmonic analysis.
-86-
This
_
~_~__~4~~
accuracy should be compared with the order of spatial accuracy being
achieved by the finite element method.
Truncating the series has the effect that no interaction is
between the truncated harmonics and those being considered.
This will
therefore affect the response at the harmonic being analyzed.
if
allowed
However,
we have consistently selected as analysis frequencies all those which
correspond to a harmonic non linear pseudo forcing above some given
threshold, the effect of there being no feedback from the frequencies
not considered into the analysis frequencies introduces no more error
into the overall computation than the neglect of these frequencies in
the first
place.
We shall now consider the effects the truncations have on the LSQ
analysis procedure.
Let us examine some results of numerical
experiments which are illustrative of the behavior of the LSQ procedure
when fewer frequencies are present in the LSQ analysis series than are
present in the signal being analyzed.
Figure 5.2a shows an input signal with seven equally spaced
harmonics with the same input amplitude at each harmonic.
When
analyzing this input signal with all seven frequencies present in the
manner previously prescribed (i.e.,
14 time sampling points spread over
the maximum modulation), we find that the input signal is exactly
recuperated (to accuracy of 14 digits) as is shown in Case 1 of Table
5.1.
However, if only four sampling frequencies are included in the
analysis series, and 8 time sampling points are used (while still
sampling over the entire modulation period), severe aliasing occurs at
the higher sampling frequencies.
Increasing the number of time sampling
points by only 2 corrects the aliasing problem and accurate amplitudes
-87-
2.0
1.0.
0
-111111
2
3
1 F quency
Frequency
4w 1
5W
6w i
(a) Input Signal
2.0.
1.0
0
WF
2eI
3w
Frequency
(b) Results for LSQ Analysis with 4 Frequencies
and 8 Time Sampling Points
1.0-
I~-I~
G
0
W
3w1
2wI
Frequency
(c) Results for LSQ Analysis with 4 Frequencies
and 10 Time Sampling Points
Figure 5.2
Effects of Variation in Frequency and Time Sampling Rates
for Typical Overtide Frequency Distribution
-88-
I
Table 5.1
Analysis
Case Frequencies
I
X
.
LSQ Analysis Results Showing Effects of Variation of Number of Frequencies and
Time Sampling Points; Example Simulating Overtide Type Frequencies
Resulting Coefficients for Frequencies
N*
At**
(hrs)
1
2
3
4
5
6
1
1 - 7
14
5.57
1.000000
0.000000
1.000000
0.000000
1.000000
0.000000
1.000000
0.000000
1.000000
0.000000
1.000000
0.000000
2
1 - 4
8
9.75
1.000980
0.000000
1.002718
-0.000161
2.001712
0.005177
1.998599
0.004993
-
-
3
1 - 4
10
7.80
1.000048
0.000000
1.000252
-0.000410
1.000820
-0.000818
1.002468
-0.001220
-
-
4
1 - 4
14
5.57
0.999265
0.000000
0.998617
-0.000639
0.998885
-0.001276
0.999371
-0.001911
-
-
5
1 - 4
56
1.395
1.003836
0.000000
1.007666
0.000410
1.007645
0.000821
1.007611
0.001232
-
-
6
1 - 4
224
0.3482 0.999219
0.000000
0.998439
-0.000023
0.998439
-0.000046
0.998440
-0.000070
* N = number of time history sampling points
**At = time step between consecutive time history sampling points
7
1.000000
0.000000
-
and phases are recuperated from the signal for all the frequencies
However,
included in the sampling series.
increasing the number of time
sampling points further does not increase the accuracy obtained
whatsoever.
5.1.
These effects are summarized in both Figure 5.2 and Table
The extent of aliasing depends on how many frequencies are
neglected and their associated energy level (i.e., neglecting higher
harmonics with small associated amplitudes produces less aliasing).
Finally, we note that in the example considered, which is representative
of an overtide frequency distribution, there was an increase in economy
in neglecting higher harmonics in the LSQ analysis since both the number
of frequencies and the number of time sampling points decreased.
Care
must be taken that responses are not calculated for frequencies which
have aliased pseudo forcing amplitudes since this can lead to
instabilities in the overall iterative process.
Let us now consider an input signal which includes closely spaced
frequencies and hence is more representative of compound tidal
frequencies.
Figure 5.3a shows an input signal with four groups of
clustered frequencies.
Again, if we sample all 10 frequencies present
and adhere to the prescribed sampling procedures (20 time sampling
points distributed over the full modulation), we can exactly recuperate
the input signal, as is shown in Case 1 of Table 5.2a.
However, if only
the first seven frequencies are used in the LSQ analysis series (and the
last cluster is neglected), the number of time sampling points must be
increased by more than 4 times over what it was when all ten frequencies
were present.
Table 5.2a shows that severe errors occur (e.g., more
than 1000 fold increases in amplitude) if the time sampling point
density is not sufficiently high.
This type of error leads to
-90-
-LLsrr
~.~u~'-----CI-?~--~-?iliryL"1~-
0!0
0.5
.5
10
Frequency (hr- 1 )
2 0
(a) Input Signal
2.0
1.01
0.0
0.5
1.5
1.0
-1
Frequency (hr )
2.0
(b) Results for LSQ Analysis with 7 Frequencies
(one entire cluster neglected) and 80 Time
Sampling Points
2.0.
1.0
0.0
0.5
1.5
1.0
-1
Frequency (hr )
2.0
(c) Results for LSQ Analysis with 9 Frequencies
(one frequency neglected from within cluster)
and 160 Time Sampling Points
Figure 5.3
Effects of Variation in Frequency and Time Sampling Rates
for Typical Compound Tide Frequency Distribution (maximum
period is T = 12.4 hours and maximum synodic period is
TS = 27 days)
-91-
Table 5.2a
Analysis
Case Frequencies
LSQ Analysis Results Showing Effects of Variation of Number of Frequencies and
Time Sampling Points; Example Simulating Closely Spaced Compound Tide Frequencies
Resulting Coefficients for Frequencies
N*
At**
(hrs)
1
2
3
4
1.0000
0.0000
1.0000
0.0000
5
1
1 - 10
20
38.4
1.0000
0.0000
2
1 - 7
14
54.9
3.5800 1.6209
0.0000 -0.0109
0.9163 1.9452
0.1232 -0.3419
3
1 - 7
16
48.0
1.0543
0.0000
3.0844 3644.
0.0855 -137.
4
1 - 7
40
19.2
1.0589 3.9483 -0.5780
0.0000 -2.0861
1.4183
5
1 - 7
80
6
1 - 7
160
1.0000
0.0000
1626.
-94.
1.0000
0.0000
6
1.0000
0.0000
7
1.0000
0.0000
1.3661 1.9080 -1.3900
0.0769 -0.2909 -1.0710
0.5364 -1623.
-0.3611 89.
-361.
1383.
8
9
10
1.0000
0.0000
1.0000
0.0000
1.0000
0.0000
-
-
-
-
-
-
0.8362
0.5070
0.9887
0.0312
1.0152
0.0746
1.0601
0.1003
-
-
-
9.6
1.0271 1.0903 1.0743 1.0735
0.0000 -0.0547 -0.0425 -0.0632
0.9905
0.0158
1.0097
0.0606
1.0490
0.0922
-
-
-
4.8
1.0120
0.0000
0.9925
0.0176
0.9813
0.0603
1.0056
0.1110
-
-
-
1.0207
0.0072
1.0193
0.0068
1.0320
0.0004
* N = number of time history sampling points
**At = time step between consecutive time history sampling points
I,I4
.
I
I P
I
#
0
Table 5.2b
Analysis
Case Frequencies
LSQ Analysis Results Showing Effects of Variation of Number of Frequencies and
Time Sampling Points; Example Simulating Closely Spaced Compound Tide Frequencies
N*
At**
(hrs)
1
2
Resulting Coefficients for Frequencies
'
3
4
5
6
7
8
0.1001
0.0092
0.0996
0.0068
-
0.7510
0.2290
1.0618
1.0762
0.0157 -0.0147
-
0.9977
0.0006
0.9961
0.0042
-
1.0644 1.0548
0.0219 -0.0059
-
1.0046
0.0239
0.5231
0.3497
-
1.1610 0.8487 0.9925
0.3483 -0.2594 -0.0128
1.0498
0.0529
1.0562
0.0231
-
0.9954
0.0121
0.9989
0.0252
1.0128
0.0374
1.0499
0.0535
1.0572
0.0239
-
0.9982
0.0059
0.9987
0.0150
1.0062
0.0228
1-6, 8-10
18
48.0
0.1908
0.0000
8
1-6, 8-10
40
19.2
0.9809 1.4910
0.0000 -0.3460
9
1-6,
8-10
42
18.3
1.9852
0.0000
10
1-6, 8-10
80
9.6
1.0028 1.0035 1.0037 1.0044
0.0000 -0.0157 -0.0149 -0.0048
11
1-6, 8-10
82
9.4
0.9973 1.0561 0.8064
0.0000 -0.0213 -0.3497
12
1-6, 8-10
160
4.8
1.0039 1.0053 1.0050 0.9992
0.0000 -0.0126 -0.0118 -0.0063
13
1-6, 8-10
162
4.7
1.0039 1.0061 1.0058 0.9994
0.0000 -0.0133 -0.0124 -0.0073
1.0468
0.0843
10
0.9891 1.0170
0.0448 -0.1295
7
0.1101
0.1948
9
0.9872
0.0623
1.0015 0.9923
0.0066 -0.0471
1.1660 1.7550
0.2095 -0.3962
* N = number of time history sampling points
**At = time step between consecutive time history sampling points
0.1064 0.1601
-0.0155 -0.1720
1.0134
0.0216
0.0259
0.1286
1.0146
0.0235
1.0020
0.0293
0.9993 0.9661
-0.0036 -0.0886
1.0019
0.0499
1.0122
0.0305
1.0241
0.0356
exponential growth of response in
the full non linear iterative scheme
and causes complete iterative instability in a matter of several
cycles.
Again the accuracy of the LSQ analysis doesn't increase with
further increases in
the number of time sampling points beyond a certain
number.
Comparing the computational economy of using all the signal
frequencies versus using less sampling frequencies and more time
sampling points, shows that for this case it is far better to include
all the signal frequencies.
When using all 10 signal frequencies and 20
time points, 2000 operations are required, while when using only 7
frequencies and the 80 time sampling points, 4676 operations are
required.
These numbers are only indicative of the computational effort
required for the actual LSQ analysis per nodal point (i.e.,
the
conversion of a time history signal to a set of amplitudes in the
frequency domain for the pseudo forcing at a node) and do not include
the effort required to generate the time history sampling points.
This
effort could very well be even greater than the increased effort
associated with the LSQ analysis.
In general, signals with very closely spaced frequencies are much
more sensitive to exclusion of frequencies in the LSQ analysis series
than signals with a widely spaced frequency content.
The net increase
in the number of required time sampling points is very case specific and
depends on the closeness of frequencies in the signal, the number of
dropped frequencies and their associated magnitude.
Often we may want
to increase the number of sampling frequencies, even though they have
magnitude which fall outside of the range of interest, simply in order
to decrease the number of time sampling points.
-94-
There will be an
-UI~~-IYYIYIOIF~-Lq~
L~--
optimal balance between the number of sampling frequencies and the
number of time sampling points for every case.
We note that storage
requirements are also effected by this balance between number of time
sampling points and number of sampling frequencies and should be taken
into consideration.
Finally the effect of excluding only one frequency contained in a
given frequency cluster from the analysis series is shown in Figure 5.3c
and Table 5.2b.
We note that frequencies within the cluster from which
the frequency is excluded are most sensitive to the exclusion.
not illustrated in Table 5.2b,
can be shown that if
it
Although
the requirement
of spreading time sampling points throughout the modulation period is
not met, energy from the missing frequency will be lumped to the other
frequencies in the cluster in a constructive or destructive manner,
depending on where in the modulation period the time sampling points
lie.
Table 5.2b also illustrates a convenient methodology for
determining whether or not a certain number of time sampling points is
sufficient for the number of frequencies and characteristics of the
signal being analyzed.
By simply adding two time sampling points and
accordingly adjusting the sampling time step (such that all time points
are evenly spaced over the modulation period) a totally different set of
time sampling points is generated.
If the results of the LSQ analysis
procedure are the same, then the number of time sampling points is
sufficient.
As has become apparent in the preceding paragraphs, two categories
of frequencies may be defined.
The first category consists of those
frequencies which have sufficiently large associated forcing/response
-95-
amplitudes that they should be included in
the full non linear analysis
in order for the computation to be consistent to a given order of
accuracy.
At these full non linear analysis frequencies, responses are
calculated (and hence the linear core model is run) allowing full
interaction between all frequencies in this category.
The second
category of frequencies consists of frequencies which do not have
significant enough levels of forcing/response amplitudes to be included
as full non linear analysis frequencies, but are used to allow the LSQ
procedure to more efficiently extract accurate information at the full
non linear analysis frequencies.
These frequencies do not affect the
interaction occuring between the full non linear analysis frequencies.
As seen earlier, the number of time sampling points required is
very case dependent.
Furthermore a re-analysis of the signal with a
slightly higher number of time sampling points allows us to assess if
the time sampling density is sufficiently high for the LSQ procedure to
accurately analyze the signal.
This allowed the convenient
implementation of an automatic time step selection feature in the
computer code TEA-NL (Non Linear Tidal Embayment Analysis), which
chooses the minimum time sampling rate to achieve a specified level of
accuracy for the non linear frequency domain pseudo forcing amplitudes
at each of the LSQ analysis frequencies.
This time step selection
process need only be applied at a few representative nodes in the grid
at each cycle.
If
the number of time sampling points required exceeds a
user specified maximum number (the number at which it is more efficient
to include more frequencies for the LSQ analysis), TEA-NL stops
execution and allows the user to input additional LSQ frequencies
thereby premitting the LSQ analysis to be performed efficiently.
-96-
The number of frequencies generated for a typical tidal problem
with several significant astronomical forcing frequencies is quite
large.
Although the tabular method of Chapter 2 gives a rough idea of
the importance of frequencies, we won't a priori know precisely which
frequencies have sufficiently large forcing/response amplitudes to be
included in the overall non linear analysis.
TEA-NL has been set up to
select those frequencies which should be included as full non linear
analysis frequencies based on a user specified non linear pseudo forcing
threshold.
The threshold is a percentage of the maximum spatially
averaged non linear forcings of all the frequencies.
This automatic
selection procedure of full non linear analysis frequencies allows the
convenient and economical use of TEA-NL while ensuring that the order of
accuracy of the harmonic analysis (and hence the non linear interaction)
is consistent providing the user does not neglect to input certain
important frequencies.
Frequencies with spatially average forcing
amplitudes less than the specified percentage of the maximum will be
excluded as full non linear analysis frequencies and are used only as
LSQ analysis frequencies.
This percentage is taken as a fraction of the
expected smallest quantity of interest.
5.2
Iterative Convergence
The iterative stability of every scheme which solves a set of non
linear algebraic equations through direct iteration is dependent on the
magnitude of the right hand side term being smaller than the magnitude
of the linear terms on the left hand side of the equations.
Therefore
the success of the fully non linear scheme used in TEA-NL is dependent
on the non linearities being of sufficiently small magnitude with
-97-
respect to the linear terms so that the non linear solution is
perturbed linear solution.
only a
For most tidal embayments, the non linear
terms in the governing equations do not dominate the linear terms.
However, the magnitude of the non linear friction term may become quite
substantial in shallow embayments with rapid velocities.
Fortunately,
as is indicated by Eq. 2.20, for the case of a single astronomical
forcing tide, the major portion of the harmonically decomposed friction
term is distributed to the main astronomical forcing frequency itself.
It
can also be shown that for the case of several astronomical forcing
tides,
the dominant tidal frequency (usually M2 ) will still have the
largest pseudo forcing contribution from the non linear friction term.
Furthermore the magnitude of the coefficient of the forcing at the
dominant frequency will be close to the case where only the dominant
astronomical forcing exists.
We note that this dominant harmonic
friction term may be approximated as a linearized friction term and
incorporated with the other linear terms on the left hand side of the
equations.
This led to the use of the linearized friction term on both
sides of the momentum equations in Chapter 3.
Hence the iteration now
occurs about a right hand side loading term which equals the difference
between a linearized friction and the fully quadratic friction term.
the linearized friction factor is properly estimated, this can reduce
the right hand side loadings by an order of magnitude.
For the fully
non linear scheme this increases the rate of convergence substantially
and in cases where friction dominates, makes an otherwise divergent
iteration converge.
We note that the value of the non linear friction coefficient,
cf,
is dependent on the bottom surface, whereas the linearized
-98-
If
friction coefficient (see Eq.
2.6) is a property of both the non linear
friction coefficient and the local flow.
The effectiveness of including
linear friction in decreasing the magnitude of the right hand side
loading term is dependent on how closely the linearized friction term
approximates the loading term of the harmonic expansion for non linear
friction (in Eq. 2.20).
The most convenient scheme to obtain a good
local estimate for linear friction X is to update the user prescribed
value at the beginning of the second cycle of iteration using nodal
values of cf and the results of the first iteration for the nodal
magnitude of velocity for the dominant frequency.
Hence, program TEA-NL
only requires that a reasonable global linearized friction factor (based
on some global cf and global estimate for velocity) be specified in
order to start the iterative process.
The updated nodal values for X
obtained in the second iteration are not only helpful in speeding
convergence rates, but also allow an improved fully linear solution (if
TEA-NL is run in the linear mode) due to the much improved local values
for X.
Finally we note that iterative convergence rates may be improved by
including the computation of finite amplitude and convective
The
acceleration effects only beyond the second cycle of iteration.
reason for this is that friction is usually the dominant term and once
the repsonses in elevation and velocity have adjusted for it, the
effects of the other terms are more accurately assessed.
if
For example,
the user specifies a linearized friction factor which is too low, the
elevation amplitude in the first cycle would be overestimated.
This in
turn results in the over-adjustment for finite amplitude effects in the
second cycle, while the elevation amplitudes being calculated in the
-99-
second cycle would have adjusted better to the actual non-linear
friction.
However, including finite amplitude effects only beyond the
third cycle of iteration does not allow this type of overcompensation.
So far we have seen how convergence rates could be improved.
However, the question that remains is at what point we can consider the
solution to have converged.
TEA-NL is
Obviously the computational effort of
directly related to the total number of iterative cycles that
need to be run.
When determining the degree of accuracy which the
iteration process should achieve we should consider the following
points:
(i)
There is an order of accuracy associated with the finite
element method which was used for the spatial discretization
of the governing differential equations.
The accuracy depends
on the grid size and the types of gradients (relative to the
grid size) for both the flow field and the depth variation.
Furthermore, the linear core solution exhibits a certain
degree of node to node oscillation in
for elevation and velocity.
the solution calculated
This oscillation may be typically
quantified to the order of several percent of the magnitude of
the solution at a given node.
The oscillation is somewhat
greater for velocity than for elevation.
The accuracy of the
computation is not improved by carrying the iterative accuracy
beyond the estimated percentage of the node to node
oscillation.
This then indicates that achieving a relative
accuracy of several percent at each frequency is sufficient
for iterative convergence.
-100-
(ii) We note that non linear pseudo forcings are generated with
elevations and velocities which contain a certain amount of
node to node oscillation.
Hence we expect some deterioration
in the solution achieved at each of the various levels of
frequencies described in Chapter 2 (i.e., more node to node
oscillation).
The degree of deterioration depends on the
magnitude of oscillation relative to the overall magnitude of
the forcing (signal to noise ratio).
Furthermore, it depends
on which of the non linear terms are included in the analysis
and their relative importance.
For example,
the finite
amplitude forcing term is due to a gradient of the product of
elevation and velocity.
If the gradients in elevation and
velocity are smaller than the relative node to node
oscillation (which depends on the magnitude of the terms), we
would expect a meaningless set of harmonic pseudo forcings to
be generated.
In general,
the forcing signal to noise ratio
will be such that it will allow the meaningful calculation of
significant response frequencies.
For the finite amplitude
and convective acceleration pseudo forcing terms, this level
of deterioration increases as energy is cascaded down to the
various levels of frequencies.
For the friction pseudo
forcings this signal/noise effect is less pronounced due to
the fact that the forcing term does not include any
derivatives and furthermore energy is cascaded from the major
astronomical forcing levels to all the compound and overtide
frequencies at once.
We conclude that this noise in the
-101-
pseudo forcings must be considered when determining the
convergence achievable at each of the frequencies.
(iii) There is an order of accuracy associated with the truncation
of the harmonic series used for the time discretization of the
governing equations.
Hence the frequencies taken into
consideration are effected to a certain degree by the lack of
interaction with the missing harmonics.
As previously
mentioned the overall accuracy of the computation by not
considering this interaction is no worse than that caused by
not considering these terms in the first place.
However,
performing a calculation of a given frequency beyond the
estimated percent of the missing non linear interaction would
be meaningless.
(iv) Bottom friction and bottom topography can only be described to
within a certain degree of accuracy.
Therefore the iterative
accuracy being sought should also take into account the
variability in response associated with the uncertainty in
parameters.
All these points should be taken into consideration when
determining the level of accuracy which the iteration process should
achieve.
This level is case dependent and also varies for each of the
frequencies for which the calculations are being performed.
Program TEA-NL allows the determination of the level of accuracy
achieved and the rate of convergence by calculating the following
-102-
~'-L-~'mrrrrrYr~-lir~jYBrr~
a~rr~.
-rrc~
parameters at each cycle and for each freuqency under consideration:
Dn = maximum global difference between magnitude of elevation
calculated at consecutive cycles
DU = maximum global difference between magnitude of velocity
calculated at consecutive cycles
Dr = average global difference between magnitude of elevation
calculated at consecutive cycles
DU = average global difference between magnitude of velocity
calculated at consecutive cycles
n
average global value of elevation amplitude
U
= average global value of velocity ampltiude
RP
= convergence rate for elevation amplitude
RU
-
L
= relative convergence level for elevation
convergence rate for velocity amplitude
LU = relative convergence level for velocity
The relative convergence levels are defined as:
D
L
(5.7a)
DU
LU
(5.7b)
--
U
It is these values that should be used to determine convergence in
TEA-NL.
Typically they will be on the order of several percent.
We
shall discuss values for these parameters in more detail in Chapter
Finally, we note that R,
6.
and RU are not only indicative of the
improvement in accuracy with each cycle of iteration, but may also be
indicative of the level of signal to noise in the pseudo forcings.
-103-
CHAPTER 6.
APPLICATION
Program TEA-NL is a very flexible computer code which allows the
general calculation of non linear tides.
ability to compute compound tides.
TEA-NL is unique in its
As an example application, TEA-NL
will be used to investigate tidal circulation within the Bight of Abaco
in the Bahamas.
This is an ideal application not only because
significant non linear tides are generated, but also because extensive
field data collection and analysis have been performed for this shallow
semi-enclosed basin (Filloux and Snyder, 1979).
Furthermore, the basin
is such that it allows simple boundary conditions to be applied for the
non linear tides.
6.1
Description of Bight of Abaco and Its Tides
The Bight of Abaco (Figure 6.1) is a shallow embayment with land
boundaries consisting of the Island of Abaco along the southern and
eastern parts of the embayment and the islands of Little Abaco and Grand
Bahama along the northern parts.
Although the northwestern part of the
embayment does have a number of shallow connections to the open ocean,
data taken by Filloux and Snyder (1979) showed that these openings were
relatively opaque to the tides and could therefore be treated as land
boundaries.
They reached this conclusion by comparing the amplitudes
of major astronomical tides at several locations lying within the bight
along the northwestern boundary to those at a nearby location in the
open ocean.
-104-
--- EICILI^*
-- ~-*r--rrs(ul
--I- ~*axmrrr^--
Figure 6.1
Geography of Bight of Abaco, Bahamas.
represents 200 m contour.
-105-
~-lrr
(lre~~
~ ---- l-~(~
Dotted line
The western edge of the bight is connected to the open ocean and is
characterized by an extremely sharp discontinuity in depth.
The 200 meter
depth contour lies between 1 to 3 kilometers from the 5 meter contour and
the 1000 meter contour lies between 3 to 15 kilometers from the 5 meter
contour.
Ultimately depths drop to between 1500 and 2000 meters.
Hence
depths on the ocean side of the boundary drop by a factor of between
200 and 800.
reflected.
As a wave passes over a step of this size, it is largely
This may be shown by considering the reflection and
transmission coefficients of an incoming long wave passing over a step
from depth h1 to depth h 2 which are expressed as (Ippen, 1966):
/h I /h
K
r
=
K
=
t
-1- 1
(6.1a)
/h 2 + 1
1 2
/h/h
1
(6.1b)
+ 1
2
Values for various depth ratios h1 /h2 are shown in Table 6.1.
These
equations, which were derived for a vertical step, will apply to the
Bight of Abaco case since even the shortest possible overtide wavelength
(M6 at a depth of 2 m has a wavelength of 66 km) is many times greater
than the distance over which the most substantial portion of the depth
drop occurs.
For the Bight of Abaco h /h2 has a range of between 0.005
and 0.001 and it is seen that the reflected wave has an amplitude
between 0.87 and 0.94 of the incoming wave and is reflected out of phase
with respect to the incoming wave.
The transmitted wave has an amplitude
between 0.13 and 0.06 of the incoming wave and is in phase with the
-106-
_ _____1_LI1LCIIYlliI~~i -I
Ue~-~I~
-I~__~IXI
-(~~
Table 6.1
Reflection and Transmission Coefficients for a Long Wave
Passing Over a Step from Depth h1 to Depth h2 for
Various Depth Ratios
h
1
h
Kh
r
K
t
0.1
-0.52
0.48
0.01
-0.82
0.18
0.005
-0.87
0.13
0.001
-0.94
0.06
-1.00
0.00
0
-107-
-^X
incoming wave.
When considering the lateral expansion that occurs, the
amount of reflected energy increases and the transmitted wave becomes
even smaller.
Hence it is a reasonable approximation in this case to
assume that the wave is totally reflected.
This allows the very
convenient treatment of the boundary conditions for the non linear tides.
Since depths on the open ocean side of the boundary are very deep, there
will be essentially no non linear tides generated there.
The non linear
tides generated within the shallow bight, however, will be reflected
back into the bight due to this severe depth discontinuity.
As a result
no non linear tidal species will exist in the open ocean and their
amplitudes should be specified as zero along the ocean boundary.
Figure 6.2 shows the bathymetry within the bight.
In the region
along the open ocean boundary, depths vary between 2 and 5 meters.
This
region actually forms a sill since depths increase again in the interior
of the bight.
In the northern half of the bight, a 7 - 8 meter
depression is the dominant feature.
Depths become very shallow along
the northern edge.
Bottom characteristics also vary somewhat within the bight.
The
sill region has a bottom surface characterized by numerous sand waves
with heights between 1 and 3 meters.
represented in the depth distribution.
These sand bores are not
The northern depression region
contains muddy mounds with a relief of 10 cm and horizontal scale of
several meters.
The relatively flat southern portion of the bight has a
bottom surface consisting of a thin sediment cover over rock, punctuated
in patches by sea fans and corals (Filloux and Snyder, 1979; Snyder,
Sidjabat and Filloux, 1979).
-108-
Figure 6.2
Bathymetry of the Bight of Abaco,
Depth Contour in Meters.
-109-
Bahamas.
Filloux and Snyder (1979) have run a series of three field
experiments, each lasting approximately one month, which measured
elevation at 15 locations within the bight.
At each location bottom
mounted tide gauges with a sensitivity of 1 cm collected a time history
of bottom pressure.
The bandpass characteristics of the data are such
that steady motion, surface wave motion and turbulence are excluded from
the data records collected.
These time history records were then
harmonically decomposed using a least squares analysis procedure which
uses 5 astronomical frequencies (01, K 1 , N 2 , M2, S2) and two overtide
frequencies (M4 and M 6 ) for the analysis series.
A time point sampling
rate of 4 data points per hour was used from the available recorded
40 readings per hour (Filloux, private communication).
Atmospheric
pressure records were also collected and harmonically analyzed in the
same manner as the tidal records.
This allowed the bottom pressure
amplitudes to be adjusted to reflect only water pressure variation.
Hence surface elevation amplitude and phase data were obtained for the
M 2 , N 2 , S2, 01 and K1 astronomical tides and the M 4 and M 6 overtides
at 25 points throughout the bight.
The M 2 tide is the major component
having an amplitude of 40 cm along the open ocean boundary.
The
amplitudes and phases along the open ocean boundary of the five
astronomical components are summarized in Table 6.2.
The amplitudes
and phase lags for surface elevation, obtained at the various measurement
points in the bight for the M 2 , M 4 , M 6 and N2 tides are shown in
Figures 6.3 through 6.6.
These figures only reflect data which Filloux
and Snyder (1979) considered to yield consistent and stable estimates.
-110-
eru~irarrari-pr---,~--uLIrruu,~,~yiyaF1
Table 6.2
Summary of Measured Astronomical Tides
Along the Open Ocean Boundary
Tide
Amplitude
(cm)
Phase Lag
(radians relative
to M 2 )
K1
9.
3.5
01
7.
3.7
N2
10.
5.9
M2
40.
0.0
S2
6.
-111-
0.9
__~~~~~_~V
+39.2
Measured Elevation
Amplitude
+ 21.E
419.6
16.4
+17.2
37.4
+39.0
17.5
18.5
+29.2
+14.6
15.6
13.5
16.9
+18.3
15. 3
+14 .3
1I.
26
9
+38.8
+
12.2
+ 39.4
\.15
.5
15.9
14.5
27.9+
+12.7
15.8+
14.1
16.7
+ 16.7
15.9
+16.6
+16.3
+18.1
17.617.9
17.1 1 .1
Figure 6.3
19.1
19.2+
18.8
Field Data for M2 Astronomical
Constituent (after Filloux & Snyder, 1979)
(a) Amplitude in centimeters
-112-
*L(LI*_IIY~
_1_I1IILI__II~
+
-0.07
+0.56
+0.87
Measured Elevation
Phase Lag
0.96
+0.89
0.00
+0.00
1.15
+0.96
0.87+
0.86
1.29
1.33
0.82
1.32
+1.10
+0.17
0.35+
+
0.19
-0.02
40.94
+ 0.07
1.50
1l.52
1.52
0.17+
+1.70
1.94+
1.83
+1.68
1.80
+ 1.85
+ 1.73
1.92
88
Figure 6.3
2.01
1.95+
1.921.76
Field Data for M2 Astronomical
Constituent (after Filloux & Snyder, 1979)
(b) Phase lag in radians (relative to the
M2 tide)
-113-
-_I~--~-
I
M
+0.2
0F
Measured Elevation
Amplitude
+0.8
+2.1
1.2
1.4
+0.2
0.2
,1.2
+1.2
+
1
+0.3
0.9
0.8
1.2
1.1
0.8
+0.7
0.6+
40.2 +0.3
+0.2
0.3+
0.9 +
0.5
0.5+
0 .2+
0.5
0.3 +
0.5
+0.5
0.3
0.5
+ 0.7
+ 0.6
0.5
0.5 +
7 0.6 0.9+
Figure 6.4
.0+
0.7
0.9
Field Data for M4 Overtide Constituent
(after Filloux & Snyder, 1979)
(a)
Amplitude in centimeters
-114-
, I,
-r*--a~
Y1-Ya~ul---rrrcP;~~iIIYTPll)lli-
+1.57
Measured Elevation
Phase Lag
+ 0.33
+ 1.22
1.26
1.33
A+
1.33
1.06
+;
+6.02
0.38+
1.74
1. 06
+2.39
+0.38
+1
i.59
+1.52
44.92
+ 1.95
+ 1.
5.65+
+ 3.56 4.05+
4.05+
3.91
2.22
2.44
2.88
+2.91
+2.73
2.72
2.60
+2.56
2.88 2.90+
Figure 6.4
2.60+
2.95
3.04
Field Data for M4 Overtide Constituent
(after Filloux & Snyder, 1979)
(b)
Phase lag in radians
M2 tide)
-115-
(relative to the
+0.3
r
Measured Elevation
Amplitude
+0.5
+0.1
0.2
0.3
0.2
0.2
0.8
1.6
+1.3
0
0.8
0.7
0.8
+1.0
+0.3
0.6 +
+
0.4
+0.2
+0.7
+0.2
+0.7
0.7
0.7
0.3+
+0.3
0.4+
0.4
0.2
+0.4
0.4
+ 0.3
+0.6
0.5
0.3 + 0.4
0.5
0.5+
Figure 6.5
07
0.5+
0.6
Field Data for M Overtide Constituent
(after Filloux &6 Snyder, 1979)
(a) Amplitude in centimeters
-116-
~II-----"II~-I~L~L~"
-T~ yl
~~_
+3.80
""
Measured Elevation
Phase Lag
+1.71
+0.23
4.40
4.69
+ 5.67
4.63
+4.40
4.82
4
3.79
+ 3.16
3.7
3.63 3.32+
5.10
+5.15
+
2.37
+5.25
+3.18
44.28
4.92
+5.95
2.34+
4.97
+4.76
4.94
3.91+
1.74
1.90
5.58
5.29
5.67
+0.45
+5.64
0.38
6. 25 0.10
6. 21
'
0 .28
0.70+
1.17
0,45
Figure 6.5
Field Data for M6 Overtide Constituent
(after Filloux & Snyder, 1979)
(b) Phase lag in radians (relative to the
M 2 tide)
-117-
_
+9.7
Measured Elevation
Amplitude
-,o
+ 3.7
+3.2
3.4
+10.3
9.5
4.2
4.1
+7.6
2.7
3.8
3.4
4.3
4.0
+ +7.0
+10.3
3.3
2.6
+ 2.9
2t 9
+10.0
7.7+
2.2
+ 3.8
3.3
+2.0
3.6+
3.4
2.2
2.7
3.5
+2.3
+2.4
2.7
2.1
1.7
+4.3
4.0
3.6
3.2
+
2.3
Figure 6.6
Field Data for N2 Astronomical
Constituent (after Filloux & Snyder,
(a)
Amplitude in centimeters
-118-
1979)
~~-X
aar~rs-- I.aar~
i)iWII~~~PIYILc--arx~
+0
'
+
5.78
Measured Elevation
Phase Lag
'.19
o061
+0.61
0.52
0.70
+0.79
5.85
+6.02
4+
6.00
6.14
+
0.87
+~1.13
0. 61
+1. 10
+6.005.85
+0.68
-.0.00
6.02
0.93
+1.03
+
1.33
0.98
+
1.41
1.41
+
1.95
0.91
+ 1.20
1.69
+ 1.22
41.06
1.31
0.98 1.48
1.43+
1.82
Figure 6.6
1.50+
1.85
+1.50
Field Data for N2 Astronomical
Constituent (after Filloux & Snyder, 1979)
(b) Phase lag in radians (relative to the
M2 tide)
-119-
Filloux and Snyder (1979) predicted the theoretical statistical
error associated with their data analysis procedure.
Their actual
errors, determined by the variability in results at a point from
experiment to experiment, generally substantially exceeded the computed
theoretical statistical error.
They suggest that this may be the result
of not having included more harmonic components in their least squares
analysis procedure.
As we shall see later in this chapter, not including
more harmonic components could very well have led to increased errors
associated with certain of the harmonics which Filloux and Snyder were
able to extract from the time history records.
The variability in the analyzed measurement data may be quantified
by calculating the proportional variance for each harmonic.
The
proportional variance is computed by summing the square of the difference
between the elevation amplitude value obtained from experiment to
experiment at a particular location and the average value at that location
and then dividing by the sum of the squares of all the measured values.
For the computation of proportional variance of measurement data only
locations with more than one data point are used.
The proportional
variance of measurement data for each harmonic j is expressed as:
'
K
L
Vm=
£=
K
IKm
nj (x,
k)
-
1
Z
2
k
n. (x'
k)
(6.2)
k k=1
k=l
L
SK
£
Snj(x
2
, k)
£=l k=1
where
m
n.
=
measured elevation amplitude component for jth harmonic
xk
measurement location within the bight
=
-120-
L
=
total number of measurement locations with multiple
data points
K
=
total number of measurement data points at location Z
The reduced data presented by Filloux and Snyder (1979) was used to
compute the proportional variances for each of the seven harmonics
used in their data analysis.
These proportional measurement variances,
Vm, were computed for the bight as a whole and are presented in Table
6.3a.
Furthermore proportional measurement variances are presented
for the three distinct regions (the sill region, the northern bight
and the eastern bight) into which the bight may be divided.
Each of
these three regions contain an equal number of multiple data point
measurement locations.
The proportional variances are generally
equal or somewhat lower in the sill region and eastern part of the
bight (when compared to values for the entire bight) and are generally
substantially greater in the northern part of the bight.
A somewhat
easier way to interpret these proportional variances is to examine the
proportional standard deviation which is equal to the square root of
the proportional variance:
Sm
j
=
i
(6.3)
j
The proportional standard deviation may be viewed as the standard
deviation in terms of a fraction of a global representative measure of
amplitude.
Table 6.3b presents the associated proportional standard
deviations for measurement data.
It is noted that the M 2 , 01 and K1
tides (K1 measurement errors are consistent with M 2 and 01 in the
northern and eastern bight and K1 is therefore included) all show very
-121-
Table 6.3a
Measurement Error for Each Frequency in
Terms of Proportional Variance, Vm
J
m
V
Tide
Entire
Sill
Bight
Region
Northern
Bight
K1
0.0063
0.0015
0.0154
0.0006
01
0.0008
0.0007
0.0013
0.0005
N2
0.0171
0.0055
0.0450
0.0126
M2
0.0009
0.0012
0.0007
0.0013
S2
0.0205
0.0134
0.0530
0.0009
M4
0.0105
0.0134
0.0285
0.0014
M6
0.0144
0.0054
0.0294
0.0120
-122-
Eastern
Bight
~YI~
Table 6.3b
Measurement Error for Each Frequency in
Terms of Proportional Standard Deviation,
m
S
Tide
Entire
Sill
Northern
Bight
Eastern
Bight
Bight
Region
K1
0.08
0.04
0.12
0.02
01
0.03
0.03
0.04
0.02
N2
0.13
0.07
0.21
0.11
M2
0.03
0.03
0.03
0.04
S2
0.14
0.12
0.23
0.03
M4
0.10
0.12
0.17
0.04
M6
0.12
0.07
0.17
0.11
-123-
modest measurement errors while the N2, S2, M 4 and M 6 tides all exhibit
somewhat higher errors.
There is no correlation between average
amplitude in a region and the error incurred in the reduced measurements.
As previously mentioned, we will later show that the N
2 , S2 , M 4 and M 6
tides display greater measurement error due to the existence of significant
(relative to each of these tides) closely spaced compound tides which
were not used in the data reduction procedure.
In the following two sections we shall examine the circulation
patterns predicted by TEA-NL for the bight.
First, only the M 2
astronomical tide and the steady, M4 and M 6 overtides it generates
will be considered.
Then the much more complex interaction of the
closely spaced M 2 and N 2 astronomical tides and their associated
compound and overtides will be investigated.
6.2
Overtide Computations for the Bight of Abaco
TEA-NL requires both a description of the geometry of the
embayment, in the form of a finite element grid, and a set of boundary
conditions for each of the various harmonic components being
considered.
The finite element grid discretization for the Bight of
Abaco is shown in Figure 6.7.
The grid has been refined in the area
surrounding the small island which lies along the ocean boundary and
also in the region adjacent to the Island of Grand Bahama.
In this section we shall examine the circulation resulting from the
main astronomical (boundary forcing) component and its most significant
overtides.
Along the ocean boundary the M 2 amplitude of 40 cm measured
by Filloux and Snyder (1979) is used as the elevation prescribed boundary
-124-
~---~~--L-ieu
---l-----
Figure 6.7
Finite Element Grid Discretization for
Bight of Abaco, Bahamas.
-125-
condition.
Furthermore for the steady, M 4 and M 6 overtides, amplitudes
of zero are prescribed along the ocean boundary due to its reflective
nature.
Land boundaries are all specified with zero normal flux for
all frequencies.
In order to determine the sensitivity of variations in bottom
friction, three sets of overtide runs were performed with TEA-NL
corresponding to non linear friction factors of 0.003, 0.006 and 0.009.
Results for both elevation amplitude and phase of the M2 component are
shown in Figures 6.8 through 6.10.
The effects of the increase in
friction factor cf are illustrated in Figure 6.11b
which shows M 2
elevation amplitudes along the trajectory defined in Figure 6.11a.
This trajectory passes through three representative areas in the bight;
the sill region, the central bight and the northern bight.
The increases
in bottom friction correspond to decreases in elevation amplitude and
increases in phase shifts although the overall pattern of the distributions
remain similar.
The sill region, which corresponds to a very shallow
region which has rapid flow, is a high gradient region and shows the
most substantial reduction (damping) in elevation amplitude and the
largest increases in phase lag.
in the
Elevation amplitudes are the smallest
center part of the basin and increase again somewhat towards the
eastern and northern land boundaries.
The increase in amplitude is the
most significant along the northern edge of the bight where depths decrease
very rapidly (from 8 m to 1 m).
We note that for a non linear friction
coefficient of cf = 0.009 there is excellent agreement between
the M 2
elevations predicted by TEA-NL (Figure 6.10) and the measurements by
Filloux and Snyder (1979) (Figure 6.3).
-126-
This value for cf is consistent
Irr~.
-L--~-------cPm~i~
r~.*~.~g(.
-u~~
32-
528
Figure 6.8
Results of TEA with Full Non Linear
Friction Effects (cf = 0.003) for M2
Astronomical Constituent
(a)
Amplitude in centimeters.
-127-
O.N
0.o(
\
run D2
PHASE
1.0 2
1.5
Figure 6.8
Results of TEA with Full Non Linear
Friction Effects (cf = 0.003) for M2
Astronomical Constituent
(b)
Phase lag in
tide
-128-
radians (relative to M2
run el
AMPLITUDE
2'/
32
Figure 6.9
Results of TEA with Full Non Linear
Friction Effects (cf
-
0.006) for M 2
Astronomical Constituent
(a)
Amplitude in centimeters.
-129-
0.0
10
PHASE
1...
1,5
Figure 6.9
Results of TEA with Full Non Linear
Friction Effects (cf = 0.006) for M2
Astronomical Constituent
(b)
Phase lag in radians
tide
-130-
(relative to M2
20
2q
Figure 6.10
2
Results of TEA with Full Non Linear
Friction Effects (cf = 0.009) for M2
Astronomical Constituent
(a)
Amplitude in
-131-
centimeters.
__;II_~Y
_XI~_;~
1.1
5,43
Figure 6.10
Results of TEA with Full Non Linear
Friction Effects (cf - 0.009) for M2
Astronomical Constituent
(b)
Phase lag in
tide)
-132-
radians (relative to M2
Figure 6.11a
Trajectory along which M Elevation
Amplitudes are Compared for Varying
Friction Factor in Figure 6.11b
-133-
i
50
_
Sill
Region
'II
I
.
Northern
Depression
Central
Bight
45
40
35
nMI
(cm)
30
25
0.003
c
20 -
cf = 0.006
15
Cf = 0.009
10
5
0"
I
I
I
I
I
I
"
I
I
I
S
Figure 6.11b
Comparison of M 2 Elevation Amplitudes for Varying Friction
Factor, cf, along Trajectory S
with what would be expected for a water depth of 2 - 7 m with large
bedforms (dunes of 1 m) as is shown in Table 6.4.
Figures 6.12 through 6.14 show overtide (steady, M 4 and M 6 )
elevation amplitudes and phases corresponding to the M 2 forcing tide
with a friction factor cf = 0.009.
These computations did not include
the effects of finite amplitude in the continuity equation or convective
acceleration in the momentum equation and hence the overtide responses
shown are generated by the non linear friction term.
The most
significant overtide is the M 6 tide (Figure 6.14) since as was discussed
in Chapter 2, the harmonically decomposed friction pseudo forcing term
is distributed mainly to the M 2 and M 6 frequencies.
Of secondary
importance is the finite amplitude effect in the bottom friction term
which generates pseudo forcings/responses at even harmonics.
This is
reflected in the weaker responses at the steady and M 4 overtides (as is
shown in Figures 6.12 and 6.13).
We note that TEA-NL requires the
specification of a zero reference at some point in the domain with
respect to which all elevations are computed.
For this case a point
along the ocean boundary was used for convenience.
Figures 6.15 through 6.18 illustrate the effects of including in
the computation both the finite amplitude term from the continuity
equation and the friction term (again with a friction factor of
cf = 0.009).
As was shown in Chapter 2, the most significant pseudo
forcings due to the finite amplitude term in the continuity equation
are distributed to the steady and M 4 overtides and are a result of the
responses at M 2 .
This is reflected in the substantial increase in
response for the steady and M 4 overtides (compared to computations
-135-
________L_
(l___jllIC--~-~L-~-~
-~I~I
Table 6.4
Bottom
Values for Friction Factor cf in Terms of Depth and Bottom
Roughness (from Wang and Connor, 1975)
H
H
[m]
1
2
5
10
20
30
40
50
100
roughness
k[] [sec
k [m]
Stones
0.07
0.025 0.0061 0.0049 0.0036 0.0028 0.0023 0.0020 0.0018 0.0017 0.0013
Small
rocks
0.20
0.030 0.0088 0.0070 0.0052 0.0041 0.0033 0.0028 0.0026 0.0024 0.0019
7unes
0.50
1.10
0.035
0.0095 0.0070 0.0056 0.004410.003910.0035 0.0033 0.0026
0.040
0.0092 0.0073 0.0058 0.0051 0.0046 0.0043 0.0034
"'
-136-
Figure 6.12
Results of TEA with Full Non Linear
Friction Effects (cf = 0.009) for Steady
State Constituent.
Amplitude in
centimeters.
-137-
I~L
__LLI1___~____1~1_~_C--II
Y~U.LII
~tl -^-.~I_
0.4
run e2
AMPLITUDE
0.T
0-
OO.LL
0. L
Figure 6.13
Results of TEA with Full Non Linear
Friction Effects (cf - 0.009) for M4
Overtide Constituent
(a)
Amplitude in
-138-
centimeters.
i.
o
M4
run e2
/5
PHASE
3.O
Figure 6.13
Results of TEA with Full Non Linear
Friction Effects (cf = 0.009) for M4
Overtide Constituent
(b)
Phase lag in radians (relative to M 2
tide)
-139-
-~
-L-*~ II~I~L~i~~^---~~--I1I~-~-IIY..P~---
Figure 6.14
Results of TEA with Full Non Linear
Friction Effects (cf = 0.009) for M6
Overtide Constituent
(a)
Amplitude
in centimeters.
-140-
Figure 6.14
Results of TEA with Full Non Linear
Friction Effects (cf - 0.009) for M6
Overtide Constituent
(b)
Phase lag in radians (relative
tide)
-141-
to M2
I
I
20'
Figure 6.15
Results of TEA with Full Non Linear
Friction Effects (cf = 0.009) and Finite
Amplitude Effects for M 2 Astronomical
Constituent
(a)
Amplitude in centimeters.
-142-
0.7L
I.A
Ii
1.8
Figure 6.15
Results of TEA with Full Non Linear
Friction Effects (cf = 0.009) and Finite
Amplitude Effects for M2 Astronomical
Constituent
(b)
Phase lag in radians
tide)
-143-
(relative to M2
~I=-IIOL~I~-Y..
~------~
0.0
STEADY
run e3
AMPLITUDE
.2.5
2,25
12.o
2.5
2.25
2,0
2.5
Figure 6.16
Results of TEA with Full Non Linear
Friction Effects (cf = 0.009) and Finite
Amplitude Effects for Steady State
Amplitude in centimeters.
Constituent.
-144-
Figure 6.17
Results of TEA with Full Non Linear
Friction Effects (cf = 0.009) and Finite
Amplitude Effects for M4 Overtide
Constituent
(a)
Amplitude in centimeters.
-145-
..-..
mlux.-r~---^--r*llra~-*IY"eY"Lnar~~.--
run e3
PHASE
1.5
2.5
Figure 6.17
Results of TEA with Full Non Linear
Friction Effects (cf = 0.009) and Finite
Amplitude Effects for M4 Overtide
Constituent
(b)
Phase lag in radians
tide)
-146-
(relative to M2
Figure 6.18
Results of TEA with Full Non Linear
Friction Effects (cf = 0.009) and Finite
Amplitude Effects for M6 Overtide
Constituent
(a)
Amplitude in
-147-
centimeters.
/
Figure 6.18
Results of TEA with Full Non Linear
Friction Effects (cf a 0.009) and Finite
Amplitude Effects for M6 Overtide
Constituent
(b)
Phase lag in
tide)
-148-
radians (relative to M2
III__IUY__~___I*__CI__LII___,
with friction effects alone) as is shown in Figures 6.16 and 6.17.
Contributions of the finite amplitude pseudo forcing to other
frequencies (M2, M6, etc.) are of secondary importance since these
are generated by either overtide responses (which are much smaller than
the M2 response) or through the interaction of an overtide response
with the main M2 tide.
This is unlike the friction term for which
all overtides may be directly generated by the astronomical tides.
The limited effect of the finite amplitude term on frequencies other
than steady and M 4 is demonstrated in Figures 6.15 and 6.18 which
show that responses at M2 and M 6 remain essentially unchanged
(compared to computation with friction effects alone).
The changes resulting from including convective acceleration
in the computation in addition to friction and finite amplitude
may be seen by examining Figures 6.19 through 6.22.
As expected
responses at M 2 and M 6 remain essentially unchanged while there are
very slight changes for the steady state and M4 elevation amplitude
distributions.
All three overtides calculated have responses of approximately
equal importance.
In general, patterns of elevation amplitude vary
somewhat for these overtides.
in the sill region.
However they all exhibit high gradients
High gradients for both elevation and flux are
also prominent for the M2 astronomical tide in this region.
The high
gradients of the main tide, together with higher M2 velocities and
elevations and (furthermore) shallower depths, result in much greater
non linear pseudo forcings in this region relative to the rest of the
-149-
Figure 6.19
Results of TEA with Full Non Linear
Friction Effects (cf = 0.009), Finite
Amplitude Effects and Convective
Acceleration Effects for M2 Astronomical
Constituent
(a)
Amplitude in centimeters.
-150-
~--(irrrrr~-~i-i 3 -li L-yCyrsUlpl
lrrYI*~C'_r--_-
__
0.6
_-O,L)
" o.
I~1e
Figure 6.19
1
Results of TEA with Full Non Linear
Friction Effects (cf = 0.009), Finite
Amplitude Effects and Convective
Acceleration Effects for M 2 Astronomical
Constituent
(b) Phase lag in radians (relative to M 2
tide)
-151-
s~l-~_qii~L~LBgirCLirY-n~-~
--.~^.-~X
STEADY
0.0
run e4
AMPLITUDE
2.7 5
Z.5
2.25
Figure 6.20
Results of TEA with Full Non Linear
Friction Effects (cf = 0.009), Finite
Amplitude Effects and Convective
Acceleration Effects for Steady State
Constituent.
Amplitude in centimeters.
-152-
---~'*siPII-YLXlr~i
0.,
Figure 6.21
LI.0
- -
'
Results of TEA with Full Non Linear
Friction Effects (cf = 0.009), Finite
Amplitude Effects and Convective
Acceleration Effects for M4 Overtide
Constituent
(a)
Amplitude in
-153-
centimeters.
_ _
run e4
PHASE
3.0
Figure 6.21
Results of TEA with Full Non Linear
Friction Effects (cf = 0.009), Finite
Amplitude Effects and Convective
Acceleration Effects for M4 Overtide
Constituent
(b)
Phase lag in
tide)
-154-
radians
(relative to M2
_III____YII__LYiUI~_CIIP.I.XI~1__X^~.
1.0 o.5
Run e4
AMPLITUDE
2.5
,.5
2.5
2.0
I.0
I.
Figure 6.22
~
c
Results of TEA with Full Non Linear
Friction Effects (cf = 0.009), Finite
Amplitude Effects and Convective
Acceleration Effects for M6 Overtide
Constituent
(a)
Amplitude in
-155-
centimeters.
3.0
Run e4
PHASE
4.5
4-S
so
6.0
Figure 6.22
Results of TEA with Full Non Linear
Friction Effects (cf = 0.009), Finite
Amplitude Effects and Convective
Acceleration Effects for M6 Overtide
Constituent
(b) Phase lag in radians (relative to M2
tide)
-156-
bight.
The sill region is therefore responsible for a substantial
portion of the generation of the overtides.
We note that for the steady state component, the high elevation
gradients in the sill region correspond to a seaward flushing current
which is shown in Figure 6.23a.
This steady residual current, which is
the result of the finite amplitude term in the continuity equation,
exists only in the sill region since the gradients in elevation are
small in the rest of the bight.
It should be stressed that the steady
residual current computed and shown in Figure 6.23a is a residual
velocity current and is equivalent to the time averaged velocity
expressed as:
uR
=
u(t)
=
(6.4)
uw= 0
This steady Eulerian residual velocity is associated with the net drift
of a particle traveling with the velocity of the fluid.
It is distinct
from the time averaged flux which gives the following residual flux
current:
QR
=
u(t)(h + n(t))
=
uRh + u(t)n(t)
(6.5)
We note that mass is not conserved when considering the steady velocity
currents by themselves since, as was noted in Chapter 3, the harmonic
continuity equation for the steady component is coupled with harmonic
continuity equations for other frequency components.
For the residual
flux currents, however, mass is conserved for each individual frequency
constituent.
This is due to the fact that the continuity equation in
terms of flux is a linear differential equation and therefore leads to
-157-
Figure 6.23a
Velocity Results of TEA with Full Non Linear
Friction Effects (cf = 0.009), Finite
Amplitude Effects and Convective Acceleration
Effects for Steady State Component.
-158-
Figure 6.23b
Velocity Results of TEA with Full Non Linear
Friction Effects (c = 0.009), Finite
Amplitude Effects and Convective Acceleration
Effects for M 2 Component at Time of Maximum
Ebb for the M2 Component Relative to the
Ocean Boundary
-159-
Figure 6.23c
Velocity Results of TEA with Full Non Linear
Friction Effects (cf = 0.009), Finite
Amplitude Effects and Convective Acceleration
Effects for M4 Component at Time of Maximum
Ebb for the M 2 Component Relative to the
Ocean Boundary
-160-
Figure 6.23d
Velocity Results of TEA with Full Non Linear
Friction Effects (cf = 0.009), Finite
Amplitude Effects and Convective Acceleration
Effects for M 6 Component at Time of Maximum
Ebb for the M2 Component Relative to the
Ocean Boundary
-161-
uncoupled harmonic continuity equations for all the various frequencies.
Hence residual flux currents indicate net mass flushing patterns.
For
our particular case, while net steady velocity currents are quite
significant (maximum velocities in the bight for the steady residual
component are approximately 10% of the maximum velocities in the bight
for the main M 2 component at maximum ebb), the net steady flux currents
will be insignificant.
This is due to the fact that the residual velocity
current, uR, is generated by the time averaged finite amplitude term,
u(t)n(t).
Hence the two terms in Eq. 6.5 balance to yield no net residual
flux, QR.
However, in general, net steady flux currents can exist and
will depend on the type of ocean connections and forcings, the depth and
bottom friction factor distributions, as well as the geometry of the
embayment.
Figure 6.23b shows the predicted velocities at maximum ebb (relative
to the ocean boundary) for the main component (M2 ).
We note that the
scaling is greater by a factor of 10 relative to Figure 6.23a.
Figures
6.23c and d show the predicted velocities associated with the M 4 and M 6
overtide components at the time of maximum ebb for the M 2 component
(relative to the ocean boundary).
varies for these figures.
We note again that the velocity scaling
The actual total velocity is obtained by
adding all four components at any one time.
Finally we note that at some
locations along land boundaries, non-zero normal velocities exist.
These
are especially severe along the northern boundary where very sharp depth
gradients exist.
Hence the actual amount of flux leakage is limited due
to the shallow depths in these areas.
Furthermore these non-zero
velocities can be eliminated by refinement of the grid in any problem
-162-
areas since normal fluxes are natural boundary conditions and are thus
satisfied exactly in the limit.
So far we have seen that the overtides generated are all of about
equal magnitude.
The friction term is the most important in that it is
responsible for the responses of the main astronomical tide (M2 ) in
addition to the generation of the M 6 overtide.
The finite amplitude
term in the continuity equation generates for the most part the M 4 and
steady tides, while the convective acceleration term has little effect
compared to the other non linear terms.
Let us now compare the
overtides computed by TEA-NL, for a friction factor of 0.009 and with
all non linearities included, to those experimentally obtained by
Filloux and Snyder (1979).
As was previously noted, agreement between TEA-NL predictions and
measurements by Filloux and Snyder (1979) for both elevation amplitude
and phase of the M 2 tide was excellent.
Since the measurements by
Filloux and Snyder do not reflect steady state circulation, no
comparisons can be made.
For the M4 overtide, TEA-NL predictions
(Figure 6.21) and measurements (Figure 6.4) show good agreement.
However, when comparing TEA-NL results for the M6 tide (Figure 6.22)
with measurements (Figure 6.5), there is some discrepancy.
The
numerical predictions for the M 6 amplitude exceed measurements by a
factor of about 2.0.
Phase errors are less pronounced although
agreement is not as good as for the M 4 tide.
The variability between the TEA-NL overtide predictions and
measurements may be quantified by calculating the proportional variance
for each of the harmonics at which reduced measurement data are
-163-
This proportional variance is computed by summing the
available.
square of the difference between each of the experimental elevation
amplitude values available at each measurement location and the value
computed by TEA-NL at that location and then dividing by the sum of
the squares of all the measured values (Snyder, Sidjabat and Filloux,
1979).
Hence the proportional variance evaluating the error between
predictions and measurements for each harmonic j is expressed as:
L
K
E
In (x,, k) - n (x)
2
(6.6)
£=l k=l
Vp
L
K1
z
E
£=l k=l
where
measured elevation amplitude component for the jth
harmonic
n~
3
TEA-NL predicted elevation amplitude component for
the jth harmonic
L
=
total number of measurement locations
K
=
total number of measurement data points at location k
The proportional prediction variances, V3, are calculated for the
entire bight and the three sub-regions previously defined and are
presented in Table 6.5a.
It is noted that regional values for V
are
about the same or less in the northern and eastern bight compared to
values for the entire bight while in the sill region they are somewhat
higher.
These proportional prediction variances V
uncertainties in the reduced measurement data Vm.
include the
Hence if the average
measured value at each point were correct, the net error between predicted
-164-
Table 6.5a
Overtide Computation Errors Expressed as
Error Between Measurements and TEA Predictions
in Terms of Proportional Variance, V
j
P
Tide
Entire
Sill
Bight
Region
Northern
Bight
M2
0.0114
0.0142
0.0038
0.0099
M4
0.0907
0.1489
0.0387
0.0767
M6
0.9198
0.9774
1.2368
0.7892
-165-
Eastern
Bight
Table 6.5b
Overtide Computation Errors Expressed as
Error Between Measurements and TEA Predictions
in Terms of Proportional Standard Deviation,
sP
i
Sp
Tide
Entire
Bight
Sill
Region
Northern
Bight
Eastern
Bight
M2
0.11
0.12
0.06
0.10
M4
0.30
0.39
0.20
0.28
M6
0.96
0.99
1.11
0.89
-166-
and measured elevation amplitudes would be obtained by subtracting VM.
3
from V . Values for proportional prediction standard deviations are
3
defined by:
S
j
=
(6.7)
j
and are shown in Table 6.5b.
All values for the predicted M 6 tide
are greater than the measured amplitude.
Table 6.5b shows that this
amplitude excess is equal to approximately one.
M 6 values are too high by a factor of 2.0.
Hence the predicted
Snyder, Sidjabat and
Filloux (1979) had the same overprediction problems for the M 6 tide
with their numerical model.
The numerical model applied to the Bight of Abaco by Snyder,
Sidjabat and Filloux (1979) is a frequency domain model which uses
finite differences to resolve the spatial dependence of the governing
equations.
As for TEA-NL, the non linear harmonic coupling in their
model is handled with an iterative scheme which cycles through the
various sets of harmonic equations.
However, their model is based on an
analytical harmonic separation of the governing equations and uses a
number of approximate expansions for the various terms.
Furthermore,
their model only performs computations for 5 astronomical tides (K ,
1
01, N 2 , M 2 and S2 ) and two overtides (M4 and M 6 ) and does not
consider any compound type interactions.
Snyder, Sidjabat and Filloux
found that their optimal overall solution was obtained at a friction
factor, cf, equal to 0.007 and that this resulted in proportional
prediction variances of VM2 = 0.07, Vp = 0.33 and VM = 0.79. These
to
proportional standard
deviation
2prediction
values of
4
6
values of
deviation
standard
prediction
proportional
correspond to
-167-
S
M2
= 0.89.
= 0.56 and S
S
M6
M4
0.26
Hence TEA-NL predictions are
somewhat better than their predictions for the M 2 and M4 tides while
the error for the M 6 tide is about the same for both models.
Snyder, Sidjabat and Filloux (1979) were able to obtain better
agreement between their numerical model predictions and their field
measurements by deviating from the standard quadratic law by either
including a linear friction component or by allowing for significant
The first mechanism assumes bottom friction to be
nontidal currents.
composed of a linear and a quadratic part of the form:
b
T
p
=
c
fl
(6.8)
u + cf2 uIu
f2
For this two parameter friction law, the quadratic friction coefficient,
cf2 , is substantially reduced from that used for the fully quadratic
one parameter law.
The reasons for their improved agreement may be
readily explained as follows.
The friction forcing felt by the M 2
tidal component is about the same as the one parameter friction law if
the linear friction factor cfl is sufficiently large to compensate for
the reduction in the quadratic friction factor, cf2 (recall that the
largest portion of the quadratic friction term acted as a linear term at
the actual forcing frequency).
This then allows the response at M 2 to
be the same as that calculated when the one parameter fully quadratic
law was used.
The M 4 overtide response will be largely unaffected
since this is generated by the finite amplitude term.
Hence, since the
M 2 response remains the same, the finite amplitude forcing at M4
will remain the same.
The M 6 overtide response, however, will be
reduced depending directly on how much the quadratic coefficient cf 2
-168-
has been reduced from the one parameter law.
This is due to the fact
that the M 6 tide is now generated by the reduced quadratic term
cf21ulu (and the dominant M2 response velocities have remained the
same for both friction laws).
Snyder, Sidjabat and Filloux found that
with friction factors of cfl = 0.00086 m/sec and cf2 = 0.0033 they were
able to reduce the proportional prediction variance for the M 6 tide
to VP = 0.178 (SP = 0.42). The second mechanism by which they were
M6
M6
able to reduce the error for the M 6 tide was to include an rms nontidal
current of 0.28 m/sec.
This nontidal current is about equal to the
maximum tidal velocity in the bight.
The proportional prediction
= 0.231 (SP = 0.48) in
variance for the M6 tide was reduced to V'
M6
M6
this manner. However the deviations required from the standard
quadratic law for the first mechanism and the large nontidal current
required for the second mechanism in order to significantly improve
the results are not supported by the flow conditions which exist in
the bight.
As this point we note that Filloux and Snyder (1979) and Snyder,
Sidjabat and Filloux (1979) did not consider steady state in either
their analysis of the experimental data or in their numerical model.
However, as was seen from TEA-NL results, the steady state overtide
response was of the same order of magnitude as other velocities
considered making it inconsistent to not consider this
steady term.
Furthermore, Snyder, Sidjabat and Filloux (1979) did not consider any
compound tides generated through the non linear interactions between
the various astronomical tides.
In the next section we shall establish
the importance of compound tides in the bight by examining the
interaction of the M2 and N 2 tides.
-169-
Compound Tide Computations for the Bight of Abaco
6.3
In this section we shall study the compound tidal interactions in
the bight.
In particular, we are interested in examining the
significance of compound tides with frequencies in the neighborhood of
and M 6 tides.
the M
From Table 6.2 we note that the N 2 tide is
roughly of the same magnitude as the diurnal constituents K 1 and
01 .
However, the importance of the compound tides generated by these
diurnal constituents which lie close to the M 4 and M 6 tides is
much less than the compound tides generated by semi-diurnal
constituents.
Furthermore, Table 6.2 shows that the N2 forcing tide
is roughly twice as large as the S2 tide.
Therefore looking at the
compound tides generated by the M 2 and N 2 tides will give us a good
understanding of the compound tidal interaction.
In addition it will
allow us to assess the importance of compound tides in the vicinity of
the M 4 and M 6 overtides.
The frequencies most likely to be of importance for the M 2 - N 2
interaction are readily obtained by using the response-forcing tables
discussed in Chapter 2.
The frequencies produced with this technique
after the second cycle of the procedure are listed in Table 6.6.
We
note that these compound tides separate into five frequency clusters.
The first cluster consists of the steady zero frequency response and two
long period (28 days and 14 days) residual tidal components.
The
remaining response clusters are grouped around 12, 6, 4 and 3 hours.
The synodic period for these frequencies is 28 days.
The frequencies listed in Table 6.6 were used in the application of
TEA-NL.
A time sampling rate of 132 points (spread over 28 days) was
-170-
I--L--L*nin~~_~
Table 6.6
Tides of Possible Interest for M2 and N2 Interaction
-1
)
(rad/sec
_________I___
28.05
0.0
0
2.5927x10-6
MN
2MN
(days)
,________
____________
steady
~T
-T
Freq.
Freq. Comp.*
Tide
~.;~~~a*--*l~l~l~i~P---~ --~n~m~~-l----- .~~x.,._
-6
5.1854x106
-4
1.3538x10-
2w1 -2w2
2
28.05 days
28.05
14.02 days
0.56
12.89 hrs
28.05
4
12.65 hr s
28.05
1.4056x10-4
12.42 hrs
28.05
1.4316x10-4
12.19 hrs
0.56
3NM4
2.7335x10-
6.38 hrs
28.05
N
2.7594x10-
6.33 hrs
28.05
-4
2.7853x10-4
6.27 hrs
28.05
2.8113x10
4
6.21 hrs
28.05
2.8372x10
4
6.15 hr s
28.05
wl-2w2
2NM 2
N2
1.3797x10
"'2
1
2
M2
2w2
2MN
2
4
MN4
1 2
3 2w
1-3w
2
M4
3MN 4
132
21+2
2w+2w2
1
N6
-4
4.1391x10 -
4.22 hrs
0.56
2NM 6
4.1650x10 -
4.19 hrs
28.05
4.1909x10 - 4
4.16 hrs
28.05
4.14 hrs
28.05
5.5189x10 -4
3.16 hrs
0.56
5.5447x10-4
3.15 hrs
28.05
5.5707x10-4
5.5966x10-
3.13 hrs
28.05
3.12 hrs
28.05
w
2MN
2
6
4.217x10 -
M6
2w 1 +2w 2
N8
3NM 8
1
2 +2 2
2MN 8
3MN 8
5.6225X10
M
8
=
=
-4
3.10 hrs
I
I
*wl
W2
4
WM
2
UN 2
-171-
a
required in order to obtain accurate harmonic analysis results up to and
including the 4 hour period cluster.
Friction and finite amplitude
effects were considered while convective acceleration was neglected in
the computations due to its limited importance.
The boundary conditions are specified such
that the M 2 and N 2
astronomical constituents have amplitudes and phases set equal to values
measured by Filloux and Snyder (1979).
For all overtide and compound
tides, zero elevation is specified along the ocean boundary.
Globally averaged harmonic pseudo forcing amplitude distributions
for the continuity and momentum equations are shown in Figures 6.24a,b.
These figures show the ratios of the harmonic forcing at each frequency
to the maximum harmonic forcing of all frequencies considered.
Furthermore these figures only reflect tides with a pseudo forcing
greater than 1% of the maximum harmonic continuity or momentum pseudo
forcing.
We note that the tides with the most prominent forcings
correspond to frequencies in the first two rows of Table 2.5b.
Figure 6.24a shows the finite amplitude pseudo forcings being
distributed mainly to the steady cluster and the M 4 cluster.
The
forcings are distributed to the cluster in a similar way as for the
overtide case.
However, now not only are certain overtides generated
but compound interactions of relative importance also exist.
Besides
the steady pseudo forcing, a 28 day period finite amplitude pseudo
forcing exists.
now significant.
Furthermore both an M 4 and MN 4 pseudo forcing are
The N 2 overtides themselves are not of importance.
Figure 6.24b shows the friction pseudo forcing being distributed
mainly to the astronomical frequencies themselves.
-172-
Besides the M
2
-YIYL6~-~I"--"-~-CYrrr-~~I~I~
P
4
()
.10
.0908 .07.06-
N2
M6
N
2MN
12MN 2
.04
.03.02 -
.01
0.00
1.00
2.00
3.00
4.00
-4
(rad/sec)
w x 10
Figure 6.24a
Continuity Equation Pseudo Forcing Vector Ratios
Due to M 2 - N 2 Interaction
-173-
P
M2.10
2MN2
2
.08
.07 .06
STEADY
.05.04
M
6
H
M4
MN
MN4
2N
6
.03.02 -
.010.00
1.00
2.00
3.00
4.00
x 10- 4 (rad/sec)
Figure 6.24b
Momentum Equation Pseudo Forcing Vector Ratios
Due to M 2 - N 2 Interaction
-174-
5.00
~"----~~--L-"L~~~L-~^-l*l
ll~-at~--TY r~-L~~---i~r~a~*~il_
4~,,
overtides, MN, 2MN 2 , MN4 and 2MN 6 compound tides are now of
significance.
We note that for both the continuity equation and
momentum equation pseudo loadings, the compound pseudo forcings shown
are typically 40-50% of the magnitude of the adjacent M 2 overtides.
Hence to be consistent in the order of approximation of the analysis we
must take these compound tides into consideration.
Figures 6.25 through 6.32 show the most significant tides
associated with the non linear interaction of the M 2 and N 2 tides.
The steady state (Figure 6.27), M 2 (Figure 6.25) and M4 (Figure
6.30) constituents are essentially the same as for the M 2 overtide
computations.
The M 6 constituent (Figure 6.32) does show some
reduction in amplitudes but predicted values still substantially
exceed measured values.
The N 2 astronomical constituent (Figure
6.26) shows very good agreement with measurements (Figure 6.6).
The variability between the TEA-NL compound tide predictions
and measurements are again quantified by calculating the proportional
prediction variance, VP.
3
Values for V
J
are shown in Table 6.7a.
Proportional prediction variances V? are again about the same or
J
less in both the northern and eastern bight compared to values for the
entire bight with the exception of the values for V
in the northern bight.
for the N 2 tide
However as was noted from Table 6.3a, the
proportional measurement variances, Vm, in the northern bight were
in general substantially greater than values for Vm for the bight as
a whole.
This was especially true for the N 2 tide.
Hence the net
error between TEA-NL predictions and measurements will be substantially
reduced.
We conclude that in general the numerically predicted
-175-
_
__
20'
Figure 6.25
Results of TEA with M2 -N 2 Interaction
and with Full Non Linear Friction (cf 0.009) and Finite Amplitude Effects for
M2 Astronomical Constituent
(a)
Amplitude in centimeters
-176-
r~~~-I"-~a-C
-r
-~nu~rr~ ~9~
0.O
I.2-
.5
Figure 6.25
0'(0
Results of TEA with M2-N2 Interaction
and with Full Non Linear Friction (cf -
0.009) and Finite Amplitude Effects for
M2 Astronomical Constituent.
(b) Phase lag in radians (relative to M2
tide)
-177-
Figure 6.26
Results of TEA with M2 -N 2 Interaction
and with Full Non Linear Friction (cf =
0.009) and Finite Amplitude Effects for
N2 Astronomical Constituent.
(a)
Amplitude in
-178-
centimeters
---~I
Figure 6.26
Results of TEA with M2 -N 2 Interaction
and with Full Non Linear Friction (cf 0.009) and Finite Amplitude Effects for
N2 Astronomical Constituent.
(b)
Phase lag in
tide)
-179-
radians (relative to M2
Figure 6.27
Results of TEA with M2 -N 2 Interaction
and with Full Non Linear Friction (cf =
0.009) and Finite Amplitude Effects for
Steady State Constituent.
Amplitude in
centimeters.
-180-
Figure 6.28
Results of TEA with M2 -N 2 Interaction
and with Full Non Linear Friction (Cf 0.009) and Finite Amplitude Effects for MN
Compound Constituent
Amplitude in centimeters
-181-
MN4
run h3
0,
AMPLITUDE
0.2
0.L
Figure 6.29
Results of TEA with M2 -N 2 Interaction
and with Full Non Linear Friction (cf =
0.009) and Finite Amplitude Effects for
MN4 Compound Constituent
(a)
Amplitude in
-182-
centimeters.
rrl--~i~i-P.-~ly
Clli~iY~-~L~
C1~yl~
s
Figure 6.29
Results of TEA with M2 -N 2 Interaction
and with Full Non Linear Friction (cf =
0.009) and Finite Amplitude Effects for
MN4 Compound Constituent
(b) Phase lag in radians (relative to M
2
tide)
-183-
Figure 6.30
Results of TEA with M2 -N 2 Interaction
and with Full Non Linear Friction (cf =
0.009) and Finite Amplitude Effects for
M4 Overtide Constituent
(a)
Amplitude in
-184-
centimeters.
ILCILLL~mltl IIII111
Figure 6.30
Results of TEA with M2 -N 2 Interaction
and with Full Non Linear Friction (cf =
0.009) and Finite Amplitude Effects for
M4 Overtide Constituent
(b)
Phase lag in radians (relative to M2
tide)
-185-
~
00.
0*6
6.6
Figure 6.31
Results of TEA with M2 -N 2 Interaction
and with Full Non Linear Friction (cf =
0.009) and Finite Amplitude Effects for
2MN 6 Compound Constituent.
(a) Amplitude in centimeters
-186-
~
Figure 6.31
.I-.IXII-UIL ~I~CI*^
L
I~---%~IIItl~._
Results of TEA with M2-N 2 Interaction
and with Full Non Linear Friction (cf =
0.009) and Finite Amplitude Effects for
21iN4
6 Compound Constituent.
(b)
Phase lag in radians (relative to H2
tide)
-187-
~.1II___
run h3
0.5,
AMPLITUDE
1.0
Figure 6.32
Results of TEA with M2 -N 2 Interaction
and with Full Non Linear Friction (cf =
0.009) and Finite Amplitude Effects for
M6 Overtide Constituent
(a)
Amplitude in
-188-
centimeters.
_I_____LPL__ILLI____~I ~1IY-~-~.-..l.^~
Figure 6.32
Results of TEA with M2-N2 Interaction
and with Full Non Linear Friction (cf 0.009) and Finite Amplitude Effects for
M6 Overtide Constituent
(b)
Phase lag in
tide)
-189-
radians (relative to M2
I_
Table
6
.7a
Compound Tide Computation Errors Expressed
as Error Between Measurements and TEA
Predictions in Terms of Proportional
Variance, V
3
P
Vj
Tide
Entire
Sill
Northern
Eastern
Bight
Region
Bight
Bight
N2
0.0253
0.0187
0.0680
0.0305
M2
0.0114
0.0138
0.0066
0.0077
M4
0.0799
0.1800
0.0384
0.0304
M6
0.4693
0.6011
0.5356
0.3964
-190-
Table 6.7b
Compound Tide Computation Errors Expressed
as Error Between Measurements and TEA
Predictions in Terms of Proportional
Standard Deviation, SI
3
-191-
distributions are better for the northern bight than for the bight as
a whole.
For the sill region proportional prediction variances, VP ,
for the M 4 and M 6 overtides exceed values for the bight as a whole.
Values for the proportional prediction standard deviation are shown
in Table 6.7b.
Hence agreement between predictions and measurements for both
the M 2 and N 2 tides overall is excellent (Snyder, Sidjabat and Filloux
(1979) obtained a proportional prediction variance for the N2 tide
of V~ = 0.02).
N2
Agreement is good for the M
tide and has improved
for the M 6 tide when compared to the overtide computations.
For the
M 2 tide about 75% of the locations have predicted amplitude values
which exceed the average measured values at a location while for the
N 2 tide the fraction is only 60%.
For the M 4 tide only about 50%
of the locations have overpredicted amplitudes while 25% of the
locations have predicted values equal to the average of the measured
values.
Finally for the M 6 tide all locations have overpredicted
amplitudes with the exception of locations actually on the ocean
boundary.
As may be deduced from Table 6.7b, the overprediction
factor for the M 6 tide has been reduced to about 1.7 for these
compound tide computations.
As would be expected from our examination of pseudo forcing
values, there are now also significant compound responses.
There is
a monthly varying MN compound tide (Figure 6.28), a MN 4 compound tide
(Figure 6.29) adjacent to the M 4 and a 2MN 6 compount tide (Figure 6.31)
adjacent to the M 6 .
Although these compound tides are somewhat smaller
than their adjacent overtides (by a factor of approximately 2 to 3),
-192-
.
they are important to the dynamics of the bight.
................. iJ ---
i---
i
Furthermore, it is
noted that patterns for both the elevation amplitude and phase shift
distributions of adjacent compound tides and overtides are very
similar.
6.4
Discussion
In the previous section it was ascertained that agreement between
the reduced experimental data and the TEA-NL numerical predictions which
included the full non linear interaction of the M 2 and N 2 astronomical
tides was excellent for the astronomical tides themselves and good for
the M 4 overtide.
However M 6 predictions exceeded measured values by
a factor of about 1.7.
In this section we shall explore some of the
various possibilities that might explain and/or improve the discrepancy
which exists between measurements and predictions for the M 6 overtide.
Let us first determine what effect neglecting compound tides has
had on the measurement data reduction procedure used by Filloux and
Snyder (1979).
As was discussed in Chapter 5, the least squares
harmonic analysis procedure is much more sensitive to the neglect of
frequencies of relative importance within a cluster than when a
frequency is dropped outside of a cluster.
Hence the procedure may have
trouble resolving a tide if a closely spaced adjacent tide exists and
is not included as an analysis tide.
The associated error in the results
will depend on both the relative significance of the two tides and the
time point sampling density.
However the error introduced into the
reduced measurement data under consideration should be about the same
for both the M 4 and M 6 tides since as the results of the compound tide
-193-
-
computations showed, the MN 4 and 2MN 6 compound tides are of about the
same relative importance with respect to their adjacent M and
M6
4
overtides.
Furthermore the spacing between the MN 4 and M 4 tides and
between the 2MN 6 and M6 tides is equal (Ts - 28 days).
confirms that reduced measurement
the M 4 and M 6 tides.
Table 6.3
errors are about the same for both
Hence we conclude that data errors for the M
6
tide are not responsible for the large discrepancy that exists between
predictions and measurements for that tide.
However the grouping of measurement data errors discussed in
Section 6.1 may be readily explained by examining the importance of
closely spaced compound tides which were neglected in the data reduction.
Recall that the K1 , 01 and M 2 tides generally had very low measurement
errors (S
~ 0.03) while the N2 , S2, M 4 and M 6 tides all had larger
errors (Sm - 0.10-0.14).
The tides with higher measurement errors all
have relatively important compound tides in their vicinity which were
neglected in the measurement data analysis.
As was seen in Figures
6.24a and b, the 2MN 2 tide is of relative importance with respect to
the N 2 tide and will also certainly be important with respect to the
S2 tide.
Furthermore as was previously mentioned, the MN and 2MN
4
6
tides are proportionally significant with respect to the M and
M6
4
tides.
However the tides with lower measurement errors do not have
significant closely spaced compound tides.
While the 2MN 2 tide is
significant with respect to the N 2 tide, it is not significant with
respect to the much larger M 2 tide.
The 01 and K1 tides will have
no important compound tides located in their vicinity even when all
five major astronomical tides are included in the computations.
-194-
I__IYL_____lill*I_~ll~LlI~-~
We note from Table 2.6 that the K1 and 01 tides are extremely closely
spaced (Ts - 208 days). However the error associated with the data
reduction procedure is still very low.
Hence we conclude that the
proportional measurement variances for the N 2 , S2, M 4 and M 6 tides
can be reduced to the levels achieved for the 01, K1 and M 2 tides
if select compound tides (2MN2 , MN 4 , 2MN and most likely the 2MS
6
2,
MS4 and 2MS 6 tides) are included in the least squares analysis procedure
used to reduce the measured elevation time history records to harmonic
amplitudes.
The time sampling rate could be kept about the same as that
used by Filloux and Snyder (1979) which may be deduced from the fact
that the K1 and 01 reduced measurements showed very low error.
Thus far we have seen that experimental data error levels can
In
not be responsible for the poor fit of the predicted M 6 tide.
fact the measurement error levels calculated in Section 6.1 are in
general quite modest compared to TEA-NL prediction-measurement error
levels.
Therefore let us now examine some possible ways in which the
fit for the M 6 tide could be improved.
The first issue to be examined is the correctness of the
boundary conditions which were applied with TEA-NL.
A good indication
which justifies treating the shallow connections to the open ocean in
the northwestern part of the bight as land boundaries is the low
prediction-measurement errors in the northern bight.
In fact as was
seen in the previous section the net prediction-measurement errors
are in general substantially less in the northern part of the
bight when compared to those for the bight as a whole.
This confirms
Filloux and Snyders' (1979) conclusion regarding this boundary.
-195-
_._._
~i ii.llLI-LI
-XLII
L^
Let us now assess whether the treatment of the ocean boundary
along the western edge of the bight as being totally reflective for
overtides is justifiable.
In the previous section we noted that
prediction-measurement error levels for the M and especially
the M 6
4
tides were substantially greater than those for the M 2 and N2 tides.
In addition, regional error levels for both the M 4 and M 6 tides were
the highest in the sill region.
We also note that although measured
values along the ocean boundary are small for both the M and
M6
4
tides, they certainly are not negligibly small compared with overall
bight values for these tides as would be the case for a totally
reflective boundary.
These facts possibly indicate that either the
assumption of the boundary being totally reflective is not entirely
correct and/or the location of the reflective boundary is incorrect.
As was seen in Section 6.1, the actual reflection coefficient
was only about 0.90.
Hence a certain amount of leakage of overtide
energy into the open ocean does occur.
In fact if the ocean boundary
were totally reflective as was assumed, then no astronomical tides
would be allowed to enter the bight either.
The question is whether
the somewhat inflated reflection coefficient of 1.0 which was used
contributes significantly to overpredicted overtides.
tide almost all predicted values are too high.
For the M 6
For the M 4 tide only
50% of the comparison locations were overpredicted with another 25%
of the comparison locations having equal predicted and measured
values.
However it may be shown that the overall contribution to
the proportional variance for the M 4 tide from overpredicted points
far outweighs that from underpredicted points not only due to there
-196-
being more overpredicted points but also due to the fact that the
prediction-measurement differences were on the average much greater
for the overpredicted points.
This indicates that both M4 and M 6
computations suffer mainly from overprediction which is consistent
with a reflection coefficient which is too high.
Hence accounting
for the correct degree of reflection will definitely improve
prediction-measurement error levels for both the M4 and M 6 overtides.
However since an approximately equal influence of reflection
coefficients would be expected on both the M4 and M 6 tides there
will still remain a substantial discrepancy between the M and M
4
6
error levels.
The other possible problem with the open ocean boundary condition
applied for the overtides in the computations could be the location
of the reflection boundary.
The grid used (Figure 6.7) has the ocean
boundary located at the beginning of the sharp depth drop as if a vertical
step were located there.
However it would seem more logical to place
this boundary somewhere in the middle of the range of the most substantial
depth drop in order to better simulate a would be reflective boundary.
We recall that the 1000 meter contour was between 3 and 15 kilometers
from the present boundary.
Hence placing the reflective boundary
halfway between the 5 and 1000 meter contours would put it between
1 to 7 kilometers (depending on where along the boundary) away from
the boundary used in Figure 6.7.
This adjustment distance can be
significant when compared to the overall scale of the sill region.
We note that defining an ocean boundary in the manner just described
would create difficulties in this case since there are no astronomical
-197-
tide measurements at that location.
We conclude that allouing for some overtide transmission out of
the bight and accounting for the fact that the reflection does not
totally occur at the upper edge of the depth drop will contribute
towards improving the M 4 and M 6 distributions but are not the
dominating physical mechanisms which explain the much larger
prediction-measurement errors of the M 6 overtide.
Let us now assess whether the use of one equal value for friction
factor, cf, for the entire bight contributed significantly to the
large error level of the M 6 tide.
The use of spatially dependent
friction factor values is certainly physically well motivated due to
the significant variation in bottom surface characteristics within
the bight.
The sill region has a substantially greater bottom
roughness (dunes of 1 to 3 m in a depth of 2 - 5 m) than other areas
in the bight and, as Table 6.4 shows, a value of cf greater or equal
to 0.009 would be expected in the sill region.
Table 6.4 also
indicates that a value of cf = 0.009 is somewhat too high in other
regions of the bight.
Hence in the sill region cf could be greater
or equal to the value used in the computations while in the remainder
of the bight a lower value should be used.
If the value of cf used
in the computations were significantly below the actual value for the
sill region, then the higher than actual values in other parts of the
bight might compensate for this.
However this hypothesis which favors
the use of localized friction factors is not supported by the error
distributions for the M 2 and N2 tides.
The M 2 tide was dominantly
overpredicted due to both the number of overpredicted points and the
-198-
Illlll__l~i
L~--
*~*~llb~
fact that the average overpredicted differences far exceed underpredicted differences.
For the N 2 tide, overprediction only slightly
dominated underprediction.
As has been previously discussed, the sill region is the most
important region in terms of both the effect of friction on the
main tides and the generation of non linear tides.
That the most
significant impact of friction on all distributions is in the sill
region is demonstrated by the high gradients in elevation amplitude
and phase which exist there.
Furthermore since the non linearities
are the most significant in the sill region (due to high elevation
amplitudes, high velocities and shallow depths relative to other
parts of the bight), the non linear overtides and compound tides
are largely generated there.
Hence a spatially varying friction
factor will not drastically effect any of the computed distributions
if values for cf in the sill region are kept the same.
The limited
impact of spatially varying friction factor is confirmed by the
findings of Snyder, Sidjabat and Filloux (1979) who performed a
limited number of computations to check for the sensitivity of
this effect.
In Section 6.2 the effects of variations in global friction
factor were checked with intervals of cf equal to 0.003.
Given the
general dominance of the overpredictions it is likely that more
refined increases in cf beyond the value of 0.009 will have some
effect in reducing general error levels.
As was state earlier an
increase of the value for cf in the sill region would be justifiable
due to the bottom roughness there.
Furthermore values for cf in the
-199-
remainder of the bight could most likely be reduced to physically
more realistic values without effecting the tidal distributions in
the bight.
An increase in cf in the sill region not only causes
decreases in the amplitudes of the main tide distribution but also
generally decreases the M 6 amplitudes.
This is due to the fact that
for this case the effect of the reduction in M 2 velocities due to
increased friction proportionally outweighs the actual increases in
the friction factor itself.
and hence a reduced M 6 tide.
This results in a reduced M 6 pseudo forcing
However extrapolating the effects of
the reduction in amplitudes of the M and M tides for
changes in cf
2
6
of 0.003, we conclude that this fine tuning process will not have a
major impact in reducing M 6 error levels.
Finally let us consider the effects of only including
the M 2
and N2 astronomical tides and neglecting the 01, K 1 and S2 tides in
the computations performed.
As stated in Section 6.3, after the M
2
and N 2 tides, the S2 will probably be the most influential to the
M6 .
A very significant improvement in M 6 errors was achieved by
including the N 2 tide in the computation (recall VP
0.9198 to 0.4693 and Sp
M6
dropped from
dropped from 0.959 to 0.685).
However as
was seen from Table 6.2, the S2 tide is only about half as large as
the N2 tide.
Assuming that the improvement in the M 6 solution due
to considering the S2 tide (in addition to the M 2 and N 2 ) is
proportionally (to the amplitude of the tide) the same as that
achieved when the N2 tide was included, the error for the M 6 could
be brought down to about VP = 0.24 (Sp = 0.50). However this
M6
M6
assumes that the processes are linear, which of course they are not.
-200-
-~I-r_--rr~unr~r*nluBhrl--dOYIPYYt;P~__ - ~-iiu-.ru~,y---
Hence any actual improvement could be more or less than this.
We
note that we do not expect any modification to the M4 tide due to
the inclusion of the S2 tide, in the same way that the M 4 remained
essentially unchanged when the N 2 tide was added.
We have examined a number of possibilities for their effectiveness
in improving the fit between the predicted and measured overtides.
No single mechanism seems capable of reducing overtide error levels
to those of the astronomical tides.
However a combination of these
mechanisms may achieve significantly better overtide fits.
The
largest reduction in M 6 error will most likely be brought about by
the inclusion of the S 2 astronomical tide in addition to the M 2 and
N 2 tides in the computation.
As was seen this could very well lead
to reducing M 6 error levels close to those presently achieved for
the M4 tide.
Improved treatment of the main ocean boundary would
bring about improvements in fit for both the M4 and M 6 tides.
Finally, the fine tuning of the friction coefficient, cf, could
produce minor improvements for the error levels of all tides.
-201-
CHAPTER 7.
CONCLUSIONS
A computer model, TEA-NL, which computes tidally driven circulation
in coastal embayments, has been developed.
The finite element method
was used to resolve the spatial dependence in the governing equations
while a hybrid time domain - frequency domain approach was used to
resolve the time dependence.
With this approach the non linear terms
are iteratively updated in the time domain to produce time histories
which are then harmonically decomposed with the least squares method.
The least squares method was extremely well suited for this purpose
since it allows the resolution of very closely spaced and narrowly
banded energy in an extremely efficient manner.
With harmonic forcings
on the system and the harmonically decomposed non linear terms
(pseudo-forcings), the governing equations separated into sets of
linear equations in the frequency domain.
This led to the development
of a linear core solver which solved each set of linear equations
at a given frequency in an extremely efficient manner.
The linear
core solver yields accurate solutions (not overdamped) while it
shows very low spurious oscillations.
TEA-NL allows the general investigation of the effects of the
non linear interaction between tidal components in shallow estuaries.
Hence not only can overtides be computed but compound tides can also
be assessed.
The importance of compound tides was seen in the
application of TEA-NL to the Bight of Abaco where certain of the
compound tides generated through the interaction
of the M 2 and N 2
astronomical tides had responses equal to about 50% of the corresponding
adjacent M 2 overtides.
We note that these compound tides can be
-202-
a
;__)~ I __X~
~_~
especially important in the assessment of long period residual
fluctuations which exist in addition to any steady state residual
currents.
There are several aspects of TEA-NL which could be improved.
The first aspect concerns the fact that at present the TEA-NL user
must specify all frequencies of possible interest.
However use of
an FFT at several selected locations (typifying various parts) in
the embayment would allow the identification of all important
frequencies.
These frequencies could then be used for the much
more economical least squares harmonic analysis procedure for all
points in the embayment.
This would make the model more user
friendly and also ensure that no important frequencies are neglected.
Furthermore this will simplify the simulation of complex wind
histories.
A further aspect which would improve TEA-NL would be the use
of higher order finite elements because of their increased accuracy
per number of total nodes and the convenience of the larger elements
with which the embayment may be discretized.
We note that the size
of the largest element in the domain must reflect the size of the
smallest wavelength present (e.g. Mg) (so that wave shape can be
adequately represented).
In very shallow embayments wavelengths
decrease while the importance of higher harmonics increases, possibly
requiring a very fine grid.
The higher the order of the element,
however, the larger the minimum element size which can be used.
Furthermore higher order elements would be more convenient to
accomodate rapid changes in geometry and depth.
-203-
___ _i 1_~~1
_I__
j/*~~_
Finally the treatment of flux boundary conditions could be
improved.
TEA-NL presently treats them as natural which, unless
boundaries are sufficiently refined, could lead to leakage.
The
specific refinement of boundary areas however is often inconvenient.
Hence TEA-NL would be improved by re-formulating such that fluxes
were treated as essential boundary conditions and as such were
more strictly enforced (i.e., no errors were allowed regardless
of element sizes along the boundary).
Despite these minor inconveniences in usage, TEA-NL is an
effective model for simulating both short term (1 day) and long
term (1 month and more) tidally driven circulations in embayments.
Its most important feature is that it allows an accurate assessment
of compound tides which include long term periodically fluctuating
residual circulations.
p
-204-
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