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Tidal propagation in Ria de Aveiro Lagoon, Portugal
Phys. Chem. Earth (B), Vol. 25, No. 4, pp. 369-374, 2000 0 2000 Elsevier Science Ltd. All rights resewed 1464-1909/00/$ - see front matter
Pergamon
PII: S1464-1909(00)00028-9
Tidal Propagation in Ria de Aveiro Lagoon, Portugal J. M. Dias’, J. F. Lopes’ and I. Dekeyser’ ‘C.Z.C.M. - Dept. Fisica, Universdade de Aveiro, 3810-193 Aveiro, Portugal *C.O.M., Universite de la Mediterranee, 13288 Marseille Cedex 09, France Received 23 April 1999; accepted 12 January 2000
Abstract. Ria de Aveiro is a shallow vertically homogeneous lagoon, located in the Northwest of Portugal, connected to the Atlantic through a very narrow artificial channel, and supplied with freshwater by two major rivers, Antua and Vouga. The present study describes the application of a twodimensional depth integrated mathematical model, to predict tide-induced water level and depth mean current for the entire lagoon. The model computes time series of those variables as well as their instantaneous distribution at different times of the tidal cycle. Properties of tides and tidal currents in Ria de Aveiro are characterized and discussed. Special emphasis is given to the study of the hydrodynamic behaviour of the lagoon in extreme conditions of astronomical tide forcing, considering spring and neap tide conditions. Tides propagate from the mouth of Ria de Aveiro and are present in the entire lagoon. The tidal amplitude decreases with the distance from the mouth while the phase lag in the high and low water (which is different) increases. The tidal range increases for the spring tide at the far end of the channels correspond to a local increase of the high tide level, which result in a fortnightly variation of the mean levels at those places. (Q2000 Elsevier Science Ltd. All rights reserved
1
Introduction
Ria de Aveiro is a shallow coastal lagoon situated on the Northwest Atlantic coast of Portugal (40’38’N, 8’44’W), separated from the sea by a sand bar. It has a very irregular and complex geometry (Fig.l), characterised by narrow channels and by the existence of significant intertidal zones, namely mud flats and salt marshes. It is connected with the Atlantic through an artificial channel and exchanges most part of its water with the ocean by tidal input across this narrow entrance, 1.3 km long, 350 m wide and 20 m deep. The lagoon has a maximum width and length of 10 and 45 km, respectively, and in a spring tide covers an area of 83 km2 at high tide, reduced to 66 km2 at low tide. The average depth of Correspondence
to: J. M. Dias
the lagoon is about 1 m, except in navigation channels where dredging operations are frequently carried out. The channel connecting the lagoon with the ocean has a depth of about 20 m and the other navigation channels are about 7 m deep. Two rivers, the Vouga and Antua, discharging into the East side of the lagoon contribute to a major source of fresh water. The average flows for the Vouga river and Antua river are around 29 m3/s and 2 m3/s, respectively. The total mean river discharge during a tidal cycle into the lagoon is about 1.8~10~ m3 (Moreira et al., 1993) while the tidal prism at the mouth in a spring tide with a tidal range of 2.48 m is about 70x lo6 m3 (Vicente, 1985). The tidal prism in each one of the main channels relative to its value at the mouth is about 38% for SJacinto channel, 26% for Espinheiro channel, 10% for Mira channel and 8% for Ilhavo channel (Silva, 1994). The tides at the mouth of the lagoon are predominantly semi-diurnal, with a mean tidal range of about 2.0 m. The minimum tidal range is 0.6 m (neap tides), and the maximum tidal range is about 3.2 m (spring tides), corresponding to a maximum and a minimum water level of 3.5 and 0.3 m, respectively. According to theses values Ria de Aveiro is a mesotidal lagoon (Davies, 1964). It has been observed that the difference between surface and bottom salinity and temperature values is very low (Dias et al., 1999). It can, therefore, be postulated that the Ria de Aveiro is a very well mixed lagoon (Pritchard, 1967). In this work will be studied the tidal propagation in Ria de Aveiro lagoon. To attain this purpose, and considering the lagoon characteristics, a two-dimensional depth integrated time-dependent numerical model will be used to predict tide induced water level and depth mean velocity in the lagoon.
2
The numerical
model
The hydrodynamic model used in this study is a classical two-dimensional vertically integrated model. In Cartesian coordinates, the system of depth-averaged shallow-water
370
J. M. Dias et aL: Tidal Propagationin Ria de Aveirolagoon, Portugal
Fig. 1. Geographic map of Ria de Aveiro.
Fig. 2. Numerical bathymetry, with the depth in m, relative to the hydrographic null (-2.0 m below mean sea level), with the locations of the stations used to calibrate the numerical model (a).
(Dronkers, 1969) can be derived from integrating the full three-dimensional governing equations between the sea-bed (z = -h) and the free-surface (z = <) to give: the 21 momentum equation
equations
2, P@+ Cl (1) the 22 momentum equation
(2) and the continuity equation X at+
a[@ + CVI + a[@ + C)Vl = o ax1 ax2
(3)
where U and V are the depth averaged velocity components
in the x1 and x2 directions, respectively, (x1, x2) are the Cartesian coordinates, C is the free-surface water elevation, h(zl , x2) is the water depth, u and u are the two components of the flow speed along the x1 (eastward) and 22 (northward) axis, respectively, t is the time, f is the Coriolis parameter, g is the gravitational acceleration, p is the water density
and Ah is the horizontal eddy viscosity. ~2”~and pi, are he of the shear stress at the bottom and are related to the frictional dissipation of momentum at the watersediment interface and are approximated by a quadratic drag law (Dronkers, 1969; Leendertse and Gritton, 1971) using the Manning-Chezy formulation. The system of equations (I)- (3) has been approached to a set of numerical finite difference equations and solved by the AD1 method (alternating direction implicit), using a spacestaggered grid. With appropriate boundary and initial conditions, this system of equations constitutes a well posed initial boundary value problem whose solution describes the depth-averaged circulation in a tidal basin. The simulation of the flow in a complex domain like Ria de Aveiro implies the use of the finest possible grid. For the hydrodynamic model, a rectangular computational grid with 160 cells in the-xl-direction (eastward) and 393 cells in the x2-direction (northward), was the finest possible (the cells measure 100x 100 m). With this grid the narrower channels are represented with its width exaggerated, but conserving its water volume. The numerical bathymetry used in this study (Fig.2) was defined, with the help of the Monte Carlo method to determine volume integrals for each cell (Dias et al., 1996), using data concerning depth and boundaries of the lagoon (obtained from a general survey carried out in 1987/88 by the Hydrographic Institute of Portuguese Navy).
magnitudes
371
J. M. Dias et al.: Tidal Propagation in Ria de Aveiro lagoon, Portugal
The numerical model implemented for the entire Ria de Aveiro was calibrated comparing spectral density and harmonic constants values of the tides generated by the model with the respective values of field data, for 20 different tidal stations (Fig.2) placed along the main channels of the lagoon (Dias et al., 1998). The amplitude and phase of the i’kfs, Ss, 01 and Ki constituents determined by harmonic analysis are shown in Figure 3. The final results reveal a good agreement between the model and the field data values, specially for the Ma constituent. The model results for Ilhavo channel are not so good as for the other channels, specially the phase values determined for the diurnal constituents. This phase lag occurs probably due to the difficulty in correctly representing a convergence zone with only 10 m wide (close to the town of Ilhavo), where the currents and the energy dissipation are very high, with the used grid. The semi-diurnal constituents are considerably higher than the diurnal ones and the lunar principal, Mz, is the main tidal constituent in all the stations.
3
Properties of tides and tidal currents
Fig. 4. a) Phase distribution (“) and b) amplitude distribution (m) for the M2 constituent.
To characterize tides and tidal currents of Ria de Aveiro lagoon, several numerical simulations have been carried out. A simulation was performed for 29 days with the results
saved at all the grid points. A least-squares harmonic analysis procedure (Foreman, 1977) was applied to the time series at each grid point to yield the spatial distribution of harmonic constants of the major tidal constituents. Other simulations were performed in order to study the hydrodynamic behaviour of the lagoon in extreme conditions of astronomical tide forcing. With this purpose the model was forced by the extreme spring and neap tide conditions measured at the lagoon mouth (tidal range of 3.2 and 0.6 m, respectively). For each one of the two cases the maximum and minimum water levels (not presented), the tidal range, the water level phase lag in the high and low water (presented only for the neap tide) and the mean water level were determined. 3.1
a
90.-
: 2 4 6 8 10 12 14 16 18 21
Station
2 4 6 8 10 12 14 16 18 20
Station
Fig. 3. Comparison between the amplitude and the phase of iI42. 5’2, 01 and K1 constituents computed for field data and simulated time series for 20 different tidal stations.
Amplitude
and phase distributions
of water level
Properties of four primary tidal constituents Mz, KI, 5%and Oi have been analyzed; the results for M2, which has most of the tidal energy and for this reason can be considered representative of the tide in Ria de Aveiro lagoon, are presented and discussed below. The tides are semi-diurnal with a small diurnal pattern. The model predicted phase distribution for M2 constituent is depicted in Fig.4a). The tides propagate into the lagoon and are altered by the channels geometry and bathymetry. There are rapid phase changes as the tides move to the North through S.Jacinto and Espinheiro channels, and at their far end the phase delay is around 130° (almost 4.5 hours). However, little phase shift is noticeable in Mira channel, and specially in Ilhavo channel. At the far end of Mira
312
J. M. Dias et al.: Tidal Propagation in Ria de Aveiro lagoon, Portugal
Fig. 5. Delay relative to the mouth of the local a) high and b) low water (minutes) in the neap tide.
channel there is a rapid phase change due to an increase in the friction in this zone. The tides propagate faster in the beginning of the channels, specially in Mira channel, but friction and reflection induce higher rate of changes in tidal phase at the far end of the channels, which are narrow and shallow areas. Orientation of the co-tidal lines (constant phase lines) is nearly perpendicular to the shoreline in all the lagoon. In Fig.5 is depicted the delay of the local high and low water for the neap tide, respectively, relatively to the mouth high and low water. The values are very close to the ones found for the M2 constituent (FigAa)),however the values for high and low water are slightly different, specially at the far end of the channels. The delay is higher in the low water situation due to the small velocity of the tide propagation when the water level is lower. According to the low depths of these areas it is reasonable to suppose that the tidal wave velocity at the far end of the channels will be dependent on the actual depth (free-surface water elevation plus water depth). This dependence of the tidal wave velocity on C explains why the crests travel faster than the troughs, and therefore why the delay is higher in the low water situation. The distribution of amplitude for Mz is depicted in FigAb). The amplitude is almost constant in the lagoon central area, rapidly decreasing toward the end of the channels, where its value is very low (specially at Mira channel). In Ilhavo channel this decreasing is not so important, and the M2 amplitude is still high at the end of the channel. In Fig.6 are depicted the tidal ranges for the extreme spring and neap tide conditions. Its general distribution is similar to that found for the A42 amplitude. The tidal range decreases
Fig. 6. Tidal ranges (m) for the extreme a) spring and b) neap tide conditions.
progressively along all the channels with the increase of the distance to the mouth in both tide conditions. However, the relative decrease in each place is higher in the spring tide than in the neap tide due to the lower tidal prism through the mouth of Ria de Aveiro in the neap tide. The tidal range in spring tide is much higher than in neap tide in the entire lagoon. At the far end of the channels this is essentially due to the much higher water levels that are reached at high tide in the spring tide. The mean water level during a tidal cycle was computed in spring and neap tide conditions, forcing the model with a wave including all the tidal constituents (Fig.7). During neap tide the mean water level is almost constant in the entire lagoon, but during spring tide it increases as the distance to the mouth grows, reaching a difference of about 0.5 m between the mouth and the far end of the main channels. The tidal range increase during spring tide at the far end of the channels corresponds to an increase of the high tide level. As the tide goes from neap to spring conditions the high water level increases from tidal cycle to tidal cycle, but the low water level remains constant. Therefore, the water volume reaching the far end of the channels during the flood is higher than the one leaving in the next ebb, leading to water accumulation in these areas. Consequently there is a mean water level increase during consecutive tidal cycles. AS result there is a fortnightly variation of the mean levels at those places. 3.2
Amplitude
and phase distribution
of tidal currents
Figure 8 shows the phase and amplitude distribution for M2 tidal current. Similar charts for K1, S2 and 01 have been
J. M. Dias et al.: Tidal Propagation in Ria de Aveiro lagoon, Portugal
Fig. 7. Mean water level (m) during a tidal cycle a) in spring and b) in neap tide conditions.
developed but are not shown. The phase of tidal current has a pattern similar to that found for the water level, with a phase delay towards the end of the channels. The minimum phase delay is again in the Ilhavo channel, and the highest values found at the far end of S.Jacinto channel reach about 100’ (about 3.5 hours). The smallest values computed for the phase of the tidal current weren’t found at the lagoon mouth as expected, but in small areas at the beginning of Ilhavo channel and specially at the beginning of Mira channel. This can be explained as follows. The propagation of the tide represents a balance between the inertia of the water mass, the pressure force due to the slope of the water surface and the retarding force of bottom friction. In this lagoon tidal elevations and velocities are usually not in phase; high water occurs before high slack tide and low water before low slack tide. This is because of the momentum of the flow, which causes the current to continue to flow against an opposing pressure gradient. The beginning of Ilhavo and Mira channels are very shallow areas comparing with the beginning of the other channels and with the lagoon mouth (Fig.2). In this situation the water column has less momentum, and the current direction changes when the water level begins to drop. Since the water level phase delay in these areas relatively to the mouth is very low (Fig.4) the current inversion occurs first in the shallow areas close to the lagoon mouth. The amplitude of the water depth and low 0.4 m/s in most rably in the narrow
the tidal current is clearly dependent on on the lagoon geometry. Its value is beof the grid points, but increases consideand deepest channels. The tidal current
313
Fig. 8. a) Phase (“) and b) amplitude (m/s) distribution for the Mz tidal current.
amplitude in the entrance channel can be higher than 1.2 m/s. At the beginning of Espinheiro and S.Jacinto channels the tidal current is also very intense, specially in the narrowest ZOnes of this last channel. In the intertidal area (negative values of water depth - Fig.2) the tidal current is very low. 3.3
Characterizing lagoon
tides and tidal currents in Ria de Aveiro
With the properties of tides and tidal currents determined from a calibrated numerical model the tidal propagation can be examined in detail. For an ideal standing wave the phase difference between the water level and the tidal current is 90°, whereas for a progressive wave this difference is 0”. In Ria de Aveiro the depth is very small and the water surface in most cases is large; the energy dissipation due to the bottom stress prevents the formation of standing waves, which result from the interference between the incident wave and the reflected waves inside the lagoon. Based on model results, the phase difference between the tides and the currents for the Ms constituent was computed and plotted in Fig.9. In most parts of Espinheiro and S.Jacinto channels the phase difference is in the range O-30”; thus characterizing the tide in these channels as a progressive wave is a fairly good description, As the tides propagate from the deep channels towards the shallow regions and due to the reflection from shore, the wave characteristics move from progressive waves to standing waves. At Ilhavo and Mira channels and at the far end of the other channels (including Laranjo Bay), the phase difference is between 30-50”
374
J. M. Dias et al.: Tidal Propagationin Ria de Aveirolagoon, Portugal
The phase of tidal currents in Ria de Aveiro has a pattern similar to the one found for the water level, with the phase delay increasing toward the end of the channels. The current inversion occurs first in the shallow areas close to the lagoon mouth because here the water column has a low momentum and the water level phase delay is very small. Tides generate strong currents in deep and narrow channels, but not in the intertidal area. The highest values are found at the beginning of Espinheiro and SJacinto channels, and specially in the entrance channel, where the tidal current amplitude can be higher than 1 m/s. In order to better characterize the processes occurring in this lagoon the numerical model will be used in the future to determine the tidal prism through the main channels in different tidal conditions, as well as the residual circulation induced by different forcing actions. Future research will also include the simulation of the thermal and saline horizontal structure under different runoff conditions. The authors thank JNICTKNRS and PRAXIS XXI, through project EICOS, for financial support granted to this work.
Acknowledgements.
Fig. 9.
Phase difference (“) between the tides and the
WI
for the MZ
and the tides are mixed progressive and standing waves. Consequently, the general characteristics of the tidal wave in Ria de Aveiro are those of a damped progressive wave: during the propagation the amplitude decreases and the phase lag increases.
4
Conclusions
The tides, which are semi-diurnal with a small diurnal pattern, propagate from the mouth of Ria de Aveiro and are present in the entire lagoon. At Ilhavo and Mira channels, at Laranjo Bay and at the far end of SJacinto channel the tide propagates as a mixed progressive and standing wave, while in the remaining areas it propagates as a progressive wave. Therefore, when the tidal wave moves from the deepest lagoon areas to the shallow ones it changes its characteristics from progressive to standing ones. The tidal amplitude decreases with the distance from the mouth while the phase lag in the high and low water (which is different) increases. The tidal range and the mean water level inside the lagoon are strongly affected by the tidal regime. As the tide goes from neap to spring conditions the tidal range increase at the far end of the channels corresponds to an increase of the high tide level, which results in a fortnightly variation of the mean levels at those places.
References Davies, J. H., A morphogenetic approach to world shorelines, Z. Geomorphol., R, 127-142, 1964. Dias, J. M., Lopes, J. F. and Dekeyser, I., Numerical modelling of tidal fluxes and passive pollutants concentration in Ria de Aveiro, Portugal. In: Coustal Environment - Environment Problems in Coastal Regions (Ferrante, A.J. and Brebbia, CA., Eds). Computational Mechanics Publications, Southampton, 285-294, 1996. Dias, J. M., Lopes, J. F., Dekeyser, I., An Exploratory Study of the Dynamics of Ria de Aveiro, Portugal. Hydrodynamics, (H.Kim, S.H.Lee and S.J.Lee, Eds.), 619-624, 1998. Dias, J. M., Lopes, J. F., Dekeyser. I., Hydrological characterization of Ria de Aveiro, Portugal, in early summer. Oceunologica Acta, 22,473485, 1999. Dronkers, J. J., Tidal computations in rivers and coastal waters. Journal of Hydraulics Division, ASCE, 95, No. HYl, 44-77, 1969. Foreman, H. G. G., Munual,for tidal heights analysis and predictions, Pac. Mar. Sci. Rep., 77-10, Inst. of Ocean Sciences, Victoria, Canada, 1977. Leendertse. J. J. and Gritton, E. C., A Water-Quality Simulation Modelfor Well-Mixed Estuaries und Coastal Seas, Vol.11.Computational Procedures, Rand Corporation, New York, 1971. Moreira, M. H., Queiroga, H., Machado, M. M., Cunha, M. R., Environmental gradients in a southern estuarine system: Ria de Aveiro, Portugal, Implication for soft bottom macrofauna colonization, Neth. J. Aquat. Ecol., 27 (2-4), 465482.
1993.
Pritchard, D. W., What is an estuary: physical viewpoint, in: Lauff, G.E. (Ed.), Estuaries, American Association for the Advancement of Science, Publication 83, Washington, 3-5, 1967. Silva, J. F., Circulu@ &I Jguo no Rio de Aveiro - contribuigio-para o estudo da qualidade ok~&a, PhD Thesis, Universidade de Aveiro, Portugal, 1994. Viceme, C. M., Caracterizqao hidriulica e aluvionar da Ria de Aveiro, Utiliza@o de modelos hidr&licos no estudo de problemas da Ria, in: Jorru&.~ & Ria de Aveiro, III, Ed&So da Chara Municipal de Aveiro, Aveiro, Portugal, 41-58, 1985.
Pergamon
PII: S1464-1909(00)00028-9
Tidal Propagation in Ria de Aveiro Lagoon, Portugal J. M. Dias’, J. F. Lopes’ and I. Dekeyser’ ‘C.Z.C.M. - Dept. Fisica, Universdade de Aveiro, 3810-193 Aveiro, Portugal *C.O.M., Universite de la Mediterranee, 13288 Marseille Cedex 09, France Received 23 April 1999; accepted 12 January 2000
Abstract. Ria de Aveiro is a shallow vertically homogeneous lagoon, located in the Northwest of Portugal, connected to the Atlantic through a very narrow artificial channel, and supplied with freshwater by two major rivers, Antua and Vouga. The present study describes the application of a twodimensional depth integrated mathematical model, to predict tide-induced water level and depth mean current for the entire lagoon. The model computes time series of those variables as well as their instantaneous distribution at different times of the tidal cycle. Properties of tides and tidal currents in Ria de Aveiro are characterized and discussed. Special emphasis is given to the study of the hydrodynamic behaviour of the lagoon in extreme conditions of astronomical tide forcing, considering spring and neap tide conditions. Tides propagate from the mouth of Ria de Aveiro and are present in the entire lagoon. The tidal amplitude decreases with the distance from the mouth while the phase lag in the high and low water (which is different) increases. The tidal range increases for the spring tide at the far end of the channels correspond to a local increase of the high tide level, which result in a fortnightly variation of the mean levels at those places. (Q2000 Elsevier Science Ltd. All rights reserved
1
Introduction
Ria de Aveiro is a shallow coastal lagoon situated on the Northwest Atlantic coast of Portugal (40’38’N, 8’44’W), separated from the sea by a sand bar. It has a very irregular and complex geometry (Fig.l), characterised by narrow channels and by the existence of significant intertidal zones, namely mud flats and salt marshes. It is connected with the Atlantic through an artificial channel and exchanges most part of its water with the ocean by tidal input across this narrow entrance, 1.3 km long, 350 m wide and 20 m deep. The lagoon has a maximum width and length of 10 and 45 km, respectively, and in a spring tide covers an area of 83 km2 at high tide, reduced to 66 km2 at low tide. The average depth of Correspondence
to: J. M. Dias
the lagoon is about 1 m, except in navigation channels where dredging operations are frequently carried out. The channel connecting the lagoon with the ocean has a depth of about 20 m and the other navigation channels are about 7 m deep. Two rivers, the Vouga and Antua, discharging into the East side of the lagoon contribute to a major source of fresh water. The average flows for the Vouga river and Antua river are around 29 m3/s and 2 m3/s, respectively. The total mean river discharge during a tidal cycle into the lagoon is about 1.8~10~ m3 (Moreira et al., 1993) while the tidal prism at the mouth in a spring tide with a tidal range of 2.48 m is about 70x lo6 m3 (Vicente, 1985). The tidal prism in each one of the main channels relative to its value at the mouth is about 38% for SJacinto channel, 26% for Espinheiro channel, 10% for Mira channel and 8% for Ilhavo channel (Silva, 1994). The tides at the mouth of the lagoon are predominantly semi-diurnal, with a mean tidal range of about 2.0 m. The minimum tidal range is 0.6 m (neap tides), and the maximum tidal range is about 3.2 m (spring tides), corresponding to a maximum and a minimum water level of 3.5 and 0.3 m, respectively. According to theses values Ria de Aveiro is a mesotidal lagoon (Davies, 1964). It has been observed that the difference between surface and bottom salinity and temperature values is very low (Dias et al., 1999). It can, therefore, be postulated that the Ria de Aveiro is a very well mixed lagoon (Pritchard, 1967). In this work will be studied the tidal propagation in Ria de Aveiro lagoon. To attain this purpose, and considering the lagoon characteristics, a two-dimensional depth integrated time-dependent numerical model will be used to predict tide induced water level and depth mean velocity in the lagoon.
2
The numerical
model
The hydrodynamic model used in this study is a classical two-dimensional vertically integrated model. In Cartesian coordinates, the system of depth-averaged shallow-water
370
J. M. Dias et aL: Tidal Propagationin Ria de Aveirolagoon, Portugal
Fig. 1. Geographic map of Ria de Aveiro.
Fig. 2. Numerical bathymetry, with the depth in m, relative to the hydrographic null (-2.0 m below mean sea level), with the locations of the stations used to calibrate the numerical model (a).
(Dronkers, 1969) can be derived from integrating the full three-dimensional governing equations between the sea-bed (z = -h) and the free-surface (z = <) to give: the 21 momentum equation
equations
2, P@+ Cl (1) the 22 momentum equation
(2) and the continuity equation X at+
a[@ + CVI + a[@ + C)Vl = o ax1 ax2
(3)
where U and V are the depth averaged velocity components
in the x1 and x2 directions, respectively, (x1, x2) are the Cartesian coordinates, C is the free-surface water elevation, h(zl , x2) is the water depth, u and u are the two components of the flow speed along the x1 (eastward) and 22 (northward) axis, respectively, t is the time, f is the Coriolis parameter, g is the gravitational acceleration, p is the water density
and Ah is the horizontal eddy viscosity. ~2”~and pi, are he of the shear stress at the bottom and are related to the frictional dissipation of momentum at the watersediment interface and are approximated by a quadratic drag law (Dronkers, 1969; Leendertse and Gritton, 1971) using the Manning-Chezy formulation. The system of equations (I)- (3) has been approached to a set of numerical finite difference equations and solved by the AD1 method (alternating direction implicit), using a spacestaggered grid. With appropriate boundary and initial conditions, this system of equations constitutes a well posed initial boundary value problem whose solution describes the depth-averaged circulation in a tidal basin. The simulation of the flow in a complex domain like Ria de Aveiro implies the use of the finest possible grid. For the hydrodynamic model, a rectangular computational grid with 160 cells in the-xl-direction (eastward) and 393 cells in the x2-direction (northward), was the finest possible (the cells measure 100x 100 m). With this grid the narrower channels are represented with its width exaggerated, but conserving its water volume. The numerical bathymetry used in this study (Fig.2) was defined, with the help of the Monte Carlo method to determine volume integrals for each cell (Dias et al., 1996), using data concerning depth and boundaries of the lagoon (obtained from a general survey carried out in 1987/88 by the Hydrographic Institute of Portuguese Navy).
magnitudes
371
J. M. Dias et al.: Tidal Propagation in Ria de Aveiro lagoon, Portugal
The numerical model implemented for the entire Ria de Aveiro was calibrated comparing spectral density and harmonic constants values of the tides generated by the model with the respective values of field data, for 20 different tidal stations (Fig.2) placed along the main channels of the lagoon (Dias et al., 1998). The amplitude and phase of the i’kfs, Ss, 01 and Ki constituents determined by harmonic analysis are shown in Figure 3. The final results reveal a good agreement between the model and the field data values, specially for the Ma constituent. The model results for Ilhavo channel are not so good as for the other channels, specially the phase values determined for the diurnal constituents. This phase lag occurs probably due to the difficulty in correctly representing a convergence zone with only 10 m wide (close to the town of Ilhavo), where the currents and the energy dissipation are very high, with the used grid. The semi-diurnal constituents are considerably higher than the diurnal ones and the lunar principal, Mz, is the main tidal constituent in all the stations.
3
Properties of tides and tidal currents
Fig. 4. a) Phase distribution (“) and b) amplitude distribution (m) for the M2 constituent.
To characterize tides and tidal currents of Ria de Aveiro lagoon, several numerical simulations have been carried out. A simulation was performed for 29 days with the results
saved at all the grid points. A least-squares harmonic analysis procedure (Foreman, 1977) was applied to the time series at each grid point to yield the spatial distribution of harmonic constants of the major tidal constituents. Other simulations were performed in order to study the hydrodynamic behaviour of the lagoon in extreme conditions of astronomical tide forcing. With this purpose the model was forced by the extreme spring and neap tide conditions measured at the lagoon mouth (tidal range of 3.2 and 0.6 m, respectively). For each one of the two cases the maximum and minimum water levels (not presented), the tidal range, the water level phase lag in the high and low water (presented only for the neap tide) and the mean water level were determined. 3.1
a
90.-
: 2 4 6 8 10 12 14 16 18 21
Station
2 4 6 8 10 12 14 16 18 20
Station
Fig. 3. Comparison between the amplitude and the phase of iI42. 5’2, 01 and K1 constituents computed for field data and simulated time series for 20 different tidal stations.
Amplitude
and phase distributions
of water level
Properties of four primary tidal constituents Mz, KI, 5%and Oi have been analyzed; the results for M2, which has most of the tidal energy and for this reason can be considered representative of the tide in Ria de Aveiro lagoon, are presented and discussed below. The tides are semi-diurnal with a small diurnal pattern. The model predicted phase distribution for M2 constituent is depicted in Fig.4a). The tides propagate into the lagoon and are altered by the channels geometry and bathymetry. There are rapid phase changes as the tides move to the North through S.Jacinto and Espinheiro channels, and at their far end the phase delay is around 130° (almost 4.5 hours). However, little phase shift is noticeable in Mira channel, and specially in Ilhavo channel. At the far end of Mira
312
J. M. Dias et al.: Tidal Propagation in Ria de Aveiro lagoon, Portugal
Fig. 5. Delay relative to the mouth of the local a) high and b) low water (minutes) in the neap tide.
channel there is a rapid phase change due to an increase in the friction in this zone. The tides propagate faster in the beginning of the channels, specially in Mira channel, but friction and reflection induce higher rate of changes in tidal phase at the far end of the channels, which are narrow and shallow areas. Orientation of the co-tidal lines (constant phase lines) is nearly perpendicular to the shoreline in all the lagoon. In Fig.5 is depicted the delay of the local high and low water for the neap tide, respectively, relatively to the mouth high and low water. The values are very close to the ones found for the M2 constituent (FigAa)),however the values for high and low water are slightly different, specially at the far end of the channels. The delay is higher in the low water situation due to the small velocity of the tide propagation when the water level is lower. According to the low depths of these areas it is reasonable to suppose that the tidal wave velocity at the far end of the channels will be dependent on the actual depth (free-surface water elevation plus water depth). This dependence of the tidal wave velocity on C explains why the crests travel faster than the troughs, and therefore why the delay is higher in the low water situation. The distribution of amplitude for Mz is depicted in FigAb). The amplitude is almost constant in the lagoon central area, rapidly decreasing toward the end of the channels, where its value is very low (specially at Mira channel). In Ilhavo channel this decreasing is not so important, and the M2 amplitude is still high at the end of the channel. In Fig.6 are depicted the tidal ranges for the extreme spring and neap tide conditions. Its general distribution is similar to that found for the A42 amplitude. The tidal range decreases
Fig. 6. Tidal ranges (m) for the extreme a) spring and b) neap tide conditions.
progressively along all the channels with the increase of the distance to the mouth in both tide conditions. However, the relative decrease in each place is higher in the spring tide than in the neap tide due to the lower tidal prism through the mouth of Ria de Aveiro in the neap tide. The tidal range in spring tide is much higher than in neap tide in the entire lagoon. At the far end of the channels this is essentially due to the much higher water levels that are reached at high tide in the spring tide. The mean water level during a tidal cycle was computed in spring and neap tide conditions, forcing the model with a wave including all the tidal constituents (Fig.7). During neap tide the mean water level is almost constant in the entire lagoon, but during spring tide it increases as the distance to the mouth grows, reaching a difference of about 0.5 m between the mouth and the far end of the main channels. The tidal range increase during spring tide at the far end of the channels corresponds to an increase of the high tide level. As the tide goes from neap to spring conditions the high water level increases from tidal cycle to tidal cycle, but the low water level remains constant. Therefore, the water volume reaching the far end of the channels during the flood is higher than the one leaving in the next ebb, leading to water accumulation in these areas. Consequently there is a mean water level increase during consecutive tidal cycles. AS result there is a fortnightly variation of the mean levels at those places. 3.2
Amplitude
and phase distribution
of tidal currents
Figure 8 shows the phase and amplitude distribution for M2 tidal current. Similar charts for K1, S2 and 01 have been
J. M. Dias et al.: Tidal Propagation in Ria de Aveiro lagoon, Portugal
Fig. 7. Mean water level (m) during a tidal cycle a) in spring and b) in neap tide conditions.
developed but are not shown. The phase of tidal current has a pattern similar to that found for the water level, with a phase delay towards the end of the channels. The minimum phase delay is again in the Ilhavo channel, and the highest values found at the far end of S.Jacinto channel reach about 100’ (about 3.5 hours). The smallest values computed for the phase of the tidal current weren’t found at the lagoon mouth as expected, but in small areas at the beginning of Ilhavo channel and specially at the beginning of Mira channel. This can be explained as follows. The propagation of the tide represents a balance between the inertia of the water mass, the pressure force due to the slope of the water surface and the retarding force of bottom friction. In this lagoon tidal elevations and velocities are usually not in phase; high water occurs before high slack tide and low water before low slack tide. This is because of the momentum of the flow, which causes the current to continue to flow against an opposing pressure gradient. The beginning of Ilhavo and Mira channels are very shallow areas comparing with the beginning of the other channels and with the lagoon mouth (Fig.2). In this situation the water column has less momentum, and the current direction changes when the water level begins to drop. Since the water level phase delay in these areas relatively to the mouth is very low (Fig.4) the current inversion occurs first in the shallow areas close to the lagoon mouth. The amplitude of the water depth and low 0.4 m/s in most rably in the narrow
the tidal current is clearly dependent on on the lagoon geometry. Its value is beof the grid points, but increases consideand deepest channels. The tidal current
313
Fig. 8. a) Phase (“) and b) amplitude (m/s) distribution for the Mz tidal current.
amplitude in the entrance channel can be higher than 1.2 m/s. At the beginning of Espinheiro and S.Jacinto channels the tidal current is also very intense, specially in the narrowest ZOnes of this last channel. In the intertidal area (negative values of water depth - Fig.2) the tidal current is very low. 3.3
Characterizing lagoon
tides and tidal currents in Ria de Aveiro
With the properties of tides and tidal currents determined from a calibrated numerical model the tidal propagation can be examined in detail. For an ideal standing wave the phase difference between the water level and the tidal current is 90°, whereas for a progressive wave this difference is 0”. In Ria de Aveiro the depth is very small and the water surface in most cases is large; the energy dissipation due to the bottom stress prevents the formation of standing waves, which result from the interference between the incident wave and the reflected waves inside the lagoon. Based on model results, the phase difference between the tides and the currents for the Ms constituent was computed and plotted in Fig.9. In most parts of Espinheiro and S.Jacinto channels the phase difference is in the range O-30”; thus characterizing the tide in these channels as a progressive wave is a fairly good description, As the tides propagate from the deep channels towards the shallow regions and due to the reflection from shore, the wave characteristics move from progressive waves to standing waves. At Ilhavo and Mira channels and at the far end of the other channels (including Laranjo Bay), the phase difference is between 30-50”
374
J. M. Dias et al.: Tidal Propagationin Ria de Aveirolagoon, Portugal
The phase of tidal currents in Ria de Aveiro has a pattern similar to the one found for the water level, with the phase delay increasing toward the end of the channels. The current inversion occurs first in the shallow areas close to the lagoon mouth because here the water column has a low momentum and the water level phase delay is very small. Tides generate strong currents in deep and narrow channels, but not in the intertidal area. The highest values are found at the beginning of Espinheiro and SJacinto channels, and specially in the entrance channel, where the tidal current amplitude can be higher than 1 m/s. In order to better characterize the processes occurring in this lagoon the numerical model will be used in the future to determine the tidal prism through the main channels in different tidal conditions, as well as the residual circulation induced by different forcing actions. Future research will also include the simulation of the thermal and saline horizontal structure under different runoff conditions. The authors thank JNICTKNRS and PRAXIS XXI, through project EICOS, for financial support granted to this work.
Acknowledgements.
Fig. 9.
Phase difference (“) between the tides and the
WI
for the MZ
and the tides are mixed progressive and standing waves. Consequently, the general characteristics of the tidal wave in Ria de Aveiro are those of a damped progressive wave: during the propagation the amplitude decreases and the phase lag increases.
4
Conclusions
The tides, which are semi-diurnal with a small diurnal pattern, propagate from the mouth of Ria de Aveiro and are present in the entire lagoon. At Ilhavo and Mira channels, at Laranjo Bay and at the far end of SJacinto channel the tide propagates as a mixed progressive and standing wave, while in the remaining areas it propagates as a progressive wave. Therefore, when the tidal wave moves from the deepest lagoon areas to the shallow ones it changes its characteristics from progressive to standing ones. The tidal amplitude decreases with the distance from the mouth while the phase lag in the high and low water (which is different) increases. The tidal range and the mean water level inside the lagoon are strongly affected by the tidal regime. As the tide goes from neap to spring conditions the tidal range increase at the far end of the channels corresponds to an increase of the high tide level, which results in a fortnightly variation of the mean levels at those places.
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