Tidal dynamics in the northern Adriatic Sea › ~cushman › papers › 2000-JGR-Tidal Dynamics… · Tidal dynamics in the northern Adriatic Sea Viado Mala•.i•. National Institute

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 105, NO. Cll, PAGES 26,265-26,280, NOVEMBER 15, 2000

Tidal dynamics in the northern Adriatic Sea Viado Mala•.i•.

National Institute of Biology, Marine Station Piran, Piran, Slovenia

Dino Viezzoli

Osservatorio Geofisico Sperimentale, Villa Opicina, Italy

Benoit Cushman-Roisin

Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire

Abstract. Tides in the northern Adriatic Sea are investigated using two distinct numerical models. First, a two-dimensional (2-D) finite difference model is implemented with very high horizontal resolution (556 m) to simulate the northern Adriatic. After calibration of open boundary conditions the model gives very satisfactory results: The averaged vectorial difference between observed and simulated elevations is <1.3 cm for each of the seven

major tidal constituents. Next, a 3-D finite element model is applied to the entire sea in order to provide a better simulation of the tidal currents in the vicinity of the open boundary of the first model. Results show that the northern Adriatic behaves like a narrow rotating channel in which the instantaneous sea surface elevation (SSE) contours are aligned with the depth-averaged velocity vectors and in which the SSE is always higher to the right of the local current. These features emphasize the rotational character that tides can exhibit in a relatively small basin. Wave fitting to the current elevation structure shows that semidiurnal tidal constituents are well represented with a system of two frictionless Kelvin waves (incident and reflected). In contrast, the diurnal constituents are best described as a topographic wave propagating across, not along, the basin, from the Croatian coast to the Italian shore. Despite this obvious disparity the semidiurnal and diurnal tides can be understood as distinct members of a single family of linear waves, which exist under the combined actions of gravity and topography.

1. Introduction

Early studies of the tides in the Adriatic Sea (Figure 1) began in the nineteenth century (as reported by Defant [1961]), and it has long been known that only seven tidal constituents, four semidiurnal and three diurnal, make a significant contri- bution to the sea surface elevation (SSE). Defant [1961] showed that except within straits the Mediterranean tides reach their highest amplitude in the northern Adriatic Sea. Generally, Mediterranean tides are weak, with surface elevations not exceeding 1 rn [Tsimplis et al., 1995]. The tide in the northern Adriatic is of a mixed type, with the semidiurnal component M 2 and diurnal component K• having comparable amplitudes [Polli, 1959].

Taylor [1921] proposed a theory according to which tides in a rectangular gulf (semiclosed channel) are combinations of incident and reflected Kelvin and Poincar• waves superim- posed in such a way that the normal velocity vanishes along all sides, including the end of the channel. A feature of the solution is the possible existence of one or several amphidromic points inside the gulf. For the Adriatic Sea, there is an amphidromic point approximately two thirds up the basin for each semidiurnal constituent and none for the diurnal constituents

[Polli, 1959]. In studying the problem of the attenuation of the Adriatic tidal wave at the head of the basin (the coastline

Copyright 2000 by the American Geophysical Union.

Paper number 2000JC900123. 0148-0227/00/2000JC900123509.00

extending from Venice to Trieste), Hendershott and Speranza [1971] showed how partial reflection causes a displacement of the M 2 amphidromic point from the channel axis toward the western (Italian) coast. Later, the Taylor approach was again applied to the northern Adriatic by Mosetti [1986], who then successfully compared M 2 current amplitudes and phases to observations. Thus at least the M 2 tide in the northern Adriatic can be understood in terms of Kelvin and Poincar• waves. The

same cannot be said of the other tidal constituents.

Early numerical models of tides in the northern Adriatic Sea were limited by driving the model with only one or two constituents (M 2 and K• [McHugh, 1974] and M 2 [Cavallini, 1985]). Cavallini [1985] further investigated the elliptic motion produced by the M 2 tide and the effect of different boundary conditions along the open boundary.

The purpose of the models presented here is to simulate accurately the tidal motions in the northern Adriatic, with special emphasis on the Gulf of Trieste and the area leading to it. First, the two-dimensional Tidal Residual Intertidal Mudflat (TRIM) model of Cheng et al. [1993] is selected because of its suitability to this type of study; it was shown to be successful in simulating tidal and residual currents in San Francisco Bay [Cheng et al., 1993]. The paper reviews briefly the model for- mulation and the procedure for the calibration of the open boundary conditions. It then follows with model results and comparison of surface elevations with observations. Next, a three-dimensional finite element model is employed to obtain better tidal velocity profiles along the open boundary of the

26,265

26,266 MALA•II2 ET AL.: TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA

' I

12

46

' I ' I

13 14

' I ' I ' I ' I ' I

15 16 17 18 19

Longitude (o)

TRIESTE

46-

45-

PESARO

43-

14 18 19

Figure 1. Position of the model domain within the Adriatic Sea. Numbers along the axes are degrees of longitude and latitude.

first model in order to interpret the dynamical nature of the dominant tides. The paper finally discusses the physical nature of the tidal waves by matching analytical solutions to the numerical results.

2. Two-Dimensional Numerical Model

2.1. Model Geometry

The model domain (Figure 2) comprises an area extending northward from a straight open boundary line (x axis) connecting Pesaro in Italy to Kamenjak at the southern tip of the Istria Peninsula in Croatia. This line is 124 km long. The tidal stations nearest to the domain corners are Pesaro and Pula.

Figure 2 also shows the bathymetry. The overall picture is that of a depth increasing almost linearly with distance from the Italian coast (left) for •30 km, beyond which the bottom is nearly flat. On the Croatian side the topography exhibits steep jumps between trenches and submarine ridges and even is- lands, all within a few kilometers from the coast. The rugged topography in the vicinity of the Croatian coast complicates tidal modeling since enhanced local variations in bottom friction, wave reflection, and wave refraction affect the amplitude and arrival time of the wave in the Gulf of Trieste.

The major difference between the modeled area in the northern Adriatic Sea and that in San Francisco Bay, to which the TRIM model was first applied [Cheng et al., 1993], is the length of the open boundary. While the geometry of San Fran- cisco Bay is a complicated set of bays (San Pablo Bay, Central Bay, and South Bay), it is connected to the ocean only through a narrow strait, the Golden Gate. In contrast, the northern

Adriatic Sea is relatively compact, with a single appendix, the Gulf of Trieste, at its northeasternmost end and a wide open- ing on its southern side.

More than 6100 depth soundings and over 3100 coastal positions were taken from maritime charts and interpolated using the Kriging procedure [Davis, 1986]. From these the model topography was generated on a staggered finite difference grid with a spatial resolution of 0.3 nautical miles (about 556 m). Such resolution is deemed sufficient for the study of local sea surface elevation (SSE) and Eulerian depth-averaged velocities as it will be verified below from the velocity fields near capes and inside the Gulf of Trieste.

2.2. Model Equations

Tidal dynamics are numerically simulated with the two- dimensional TRIM model [Cheng et al., 1993], which is semi- implicit and therefore unconditionally stable. It solves the depth-integrated continuity equation

Or/ O[(r/+ D)u] O[(r/ + D)v] oD- = - ox - oy '

where (u, v) is the vertically averaged velocity, r/is the SSE above the mean level, and D(x, y) is the resting depth, together with the vertically averaged momentum equations

du Or/ g(r/ + D) Op rx ø - rx d•-- fv = -g Ox 2p0 Ox + vI-IAU + Po(r/ + D)

(2) d v Or/ g(r/+ D) Op ry ø -- Ty dt + fu = -g Oy 290 Oy + u/_/Au + P0(r/ + D) '

(3)

where d/dt - O/Ot + uO/Ox + vO/Oy is the Lagrangian derivative, f = 1.04 x 10 -4 S -1 is the Coriolis parameter, p is the vertically averaged density, Po is a reference density, uu is a horizontal eddy viscosity, (rx ø, ry ø) is the surface wind stress, and (rx, ry) is the frictional bottom stress. Density variations, horizontal diffusion of momentum, and wind stress are set to zero in our tidal analysis.

The bottom stress is taken as a nonlinear function of the

depth-averaged velocity according to the classical quadratic bottom drag law:

rx: poCou Su 2 + u 2 ry: poCoux/u 2 + u 2. (4) Because the drag coefficient depends on the water depth, we take

6.13 x 10 -3

= (2 - e ' (5) where D is given in meters. This gives a value of 2.55 x 10 -3 at 2 m and of 1.53 X 10 -3 at >12 m. The preceding expression for the drag coefficient is in accordance with the parameter- ization of the bottom friction developed for the San Francisco Bay by Cheng et al. [1993], who opted for a variation of the drag coefficient with depth instead of holding the latter constant. The rationale behind (5) is the Ch6zy coefficient of hydraulics [Cheng et al., 1993].

The system (1)-(5), which is solved for the unknowns u, v, and r/, is nonlinear through advection and bottom friction terms. In our application these nonlinearities couple the vari- ous tidal constituents and generate residual currents.

MALA•I• ET AL.: TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA 26,267

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J•IALAMOCCO

o 50 i i i i i i

VENEZIA LIDO ] N

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PULA

I•M•N•AKI • • , •

_

Figure 2. Sea ports and isobaths in the area retained by the 2-D model. Tidal data are available from Pesaro, Porto Corsini, Malamocco, Venezia-Lido, Trieste, Rovinj, and Pula. The axes are in model units, with one unit equal to 556 m (=0.3 nautical miles). The open boundary of the model, extending from Pesaro to Kamenjak, is the x axis.

The semi-implicit staggered grid method of Casulli [1990] is applied to the system (1)-(3), reducing them to a pentadiago- nal system of linear equations for the SSE values on the grid (for details, see Cheng et al. [1993]). The matrix of the system is positive definite and can be solved very efficiently. Although the code is unconditionally stable, a time step of 900 s is chosen for accuracy.

2.3. Model Calibration and Open Boundary Conditions

The length of the tidal record in the port of Trieste excqeds 100 years: Observations of the water level began in 1859, and monthly and annual mean sea level values have been published since 1905 [Godin and Trotti, 1975]. Hourly data since 1939 are also available [Stravisi and Ferraro, 1986]. Because this is one of the longest SSE records in the Mediterranean Sea, the constants of the tidal constituents are precisely known for this port of the Adriatic Sea [Crisciani et al., 1995]. Next in order of availability are the tidal records of Rovinj and Venezia-Lido. The first step in the calibration of the model is to seek a match between model results and observations at Trieste, Rovinj, and Venezia-Lido [Hydrographic Institute of the Republic of Croatia (HIRC), 1994; Istituto Idrografico della Marina (IIM), 1994].

For this task we determine the SSE values to be prescribed along the open boundary line connecting Pesaro to Kamenjak for each separate tidal constituent, starting with M 2. The tidal constants (amplitude and phase) for each tidal constituent along the open boundary are taken as quadratic and cubic polynomials:

amplitude = H = a0 + a• • + a2 (6)

x phase = # = b0 + b• • + b2 + b3 , (7)

where x is the distance from Pesaro (x = 0 at Pesaro and x = L at Kamenjak, L being the length of the open boundary line). Initially, the coefficients a o to b 3 are fitted to the charts of Polli [1959]. The model is then run from rest until all transients have fallen below the level of numerical noise (about 9 days). Cal- culated SSE values at the M 2 frequency in Trieste, Rovinj, and Venezia-Lido are then compared to the observations. Because the phase discrepancy is found to be larger than the amplitude discrepancy, the two end slopes ( !7' (0) at Pesaro and 17' (1) at Kamenjak) of the cubic polynomial for the phase profile are taken as adjustable parameters. The distribution of the amplitude and phase discrepancies between calculated and observed values as those parameters are varied is then examined. For this, amplitudes and phases are interpreted as vectors (or complex numbers), and the vectorial (complex) difference between observations and calculations is taken. Figure 3 displays the absolute value of this difference as the two tuning parameters are varied.

The plot reveals that the error between observed and calculated values reaches a minimum for a certain set of values of

17'(0) and 17'(1). These values are then adopted for the boundary conditions in the remaining simulations. There is no

26,268 MALA•I(• ET AL.: TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA

Figure 3. Distribution of the difference between M 2 tidal observations and 2-D calculations at three stations (Trieste, Rovinj, and Venezia-Lido) as a function of the gradient of the phase lag at both ends of the open boundary line, #' (0) and 9'(1). Note the minimum near 9'(0) = 4 ø and 9'(1) : -28 ø '

need to adjust the quadratic polynomial for the amplitude profile along the boundary. Finally, the entire procedure is repeated for the six other tidal constituents. Table 1 lists the optimized coefficients obtained for both amplitude and phase profiles along the open boundary line expressed as (6) and (7).

3. Two-Dimensional Model Results

After calibration the model is spun up for 31 days and then run for 190 days (i.e., slightly more than 6 months). The starting time is December 1, 1996, so that the actual simulation begins on January 1, 1997. The Rayleigh criterion for the separation of the S2 and K 2 frequencies from the simulated record demands a time series of 182.6 days (the so-called synodic period [Pugh, 1987]. So, the duration of our simulation (190 days) is sufficient. The results are sampled hourly, and the tidal constituents at five ports in the northern Adriatic (Porto Corsini, Mallamocco, Venezia-Lido, Trieste, and Rovinj, see Figure 2) are extracted from these hourly time SSE series.

Table 2 compares the amplitudes and phase lags of the model results with the observed values. These values were

taken from Polli [1959], Trotti [1969], Mosetti and Manca [1972], Godin and Trotti [1975], Mosetti [1987], and Ferraro and Maselli [1995] as well as from official reports for the port of Rovinj [HIRC, 1994] and for the port of Venezia-Lido [IIM, 1994]. It follows from Table 2 that while the model amplitudes differ from their respective observed values by <1 cm for the majority of ports and constituents, there are a few outliers (K2 in Venezia-Lido, Kl in Malamocco, and M 2 in Porto Corsini). These errors, nonetheless, fall below 2.2 cm. The majority of phase differences between model and observations is well below 10 ø, while the worst results are obtained for Venezia-Lido (10.8 ø error for K2 and up to 21.9 ø for Pl).

The performance of the 2-D model is summarized on Table 3. For each tidal constituent the average amplitude difference is <1 cm, the average vectorial difference is <1.3 cm, and the

average phase lag difference is <7.2 ø . When all seven tidal constituents in all five ports are considered together (35 values), the average amplitude difference is 0.5 cm, the average vectorial difference is 0.8 cm, and the average phase lag difference is 4.4 ø. We conclude that the 2-D model was success-

fully calibrated and that it provides reliable values, allowing us to consider the distribution of tidal elevation and velocity inside the northern Adriatic.

The distributions of SSE amplitude and phase lag of the principal semidiurnal (M2) and diurnal (K•) constituents over the model domain are shown in Figures 4 and 5. The amplitude of each constituent increases northward and then northeast-

ward from the forced open boundary to the Gulf of Trieste. In other words, the amplitude rises over decreasing depth, as one might have expected from the principle of wave action conser- vation.

For each constituent the phase lag generally increases west- ward from the Croatian coast (right-hand side of Figures 4 and 5) to the Italian shore (left-hand side of Figures 4 and 5). While the M 2 cotidal lines diverge, the K• cotidal lines tend to be more parallel; this tendency is related to the fact that the M 2 tide has an amphidromic point somewhere south of the domain (where all cotidal lines gather into a single point), while the K• tide does not [Polli, 1959]. In the northeastern corner of the domain the cotidal lines of both M 2 and K• tides bend into the Gulf of Trieste, where they diverge slightly. This is expected since the flow must be parallel to the coastline, the semiminor axis of the velocity ellipses must be small, and the cotidal lines must conserve their angle with respect to the coastline [Pugh, 1987, p. 439]. The remaining semidiurnal (K2, N2, and S2) and diurnal (P• and O•) constituents are substantially weaker but reveal SSE amplitude and phase lag distributions similar to those of the M 2 and K• constituents, respectively.

There exists a peculiar K• amplitude minimum between Pe- saro and Porto Corsini in the southwestern corner of the do-

main (see Figure 2 for the geographical location of these two ports). This minimum locally distorts the otherwise gradual distribution of the K• amplitude. Because there is no hint of such local minimum in the observations, we conclude that its existence is an artifact of the model, most likely attributable to an imperfect open boundary condition. The same problem was also noted by McHugh [1974, Figure 7] for the same tidal constituent in the same region of the same model domain. This consistency in the location of a K• amplitude minimum and the fact that both models rely on the same type of boundary condition lead us to conjecture that the prescribed SSE along the

Table 1. Coefficients of the Quadratic and Cubic Polynomials, (6)-(7), Fitted by the Calibration Procedure to Prescribe the Elevation Amplitudes and Phases Along the Open Boundary •'

H, cm g, deg

a. a• a 2 b o bl b2 b3

M 2 12.79 -9.2 9.4 311.0 4.0 -136.0 80 K 2 1.81 -1.4 1.6 313.0 -10.0 -104.0 66 N 2 2.20 - 1.0 0.9 305.0 - 5.0 -67.0 33 S2 6.83 -5.4 5.9 313.0 2.5 -110.5 62 Kl 15.4 -1.2 0.5 84.0 -41.4 66.1 -40 Pl 5.1 -0.3 0.1 84.0 -37.1 51.7 -29 O1 4.2 0.7 -0.3 69.0 -7.2 -4.8 3

aThere is a set of coefficients for every tidal constituent.


Table 2. Comparison Between Observations and Model Results of Elevation Amplitudes H and Phases g at Five Tidal Stations in the Northern Adriatic a

H o, H m ' H o _ H m ' gO, gm, gO _ gm, d, d/H ø, Site Constituent cm cm cm deg deg deg cm %

Rovinj M 2 19.3 19.3 0.0 270.0 270.6 0.6 0.2 1.0 K2 3.0 2.9 -0.1 277.0 274.8 -2.2 0.2 5.5 N 2 3.5 3.2 -0.3 266.0 273.4 7.4 0.5 15.5 S2 11.2 10.7 -0.5 277.0 278.1 1.1 0.6 5.0 K• 16.1 16.0 -0.1 71.0 70.4 -0.6 0.2 1.4 P• 5.3 5.3 0.0 71.0 71.0 0.0 0.1 0.9 O1 4.9 4.8 -0.1 56.0 61.2 5.2 0.4 9.1

Trieste M 2 26.7 26.6 -0.1 277.5 278.8 1.3 0.6 2.3 K 2 4.3 4.0 -0.3 286.1 283.0 -3.1 0.3 8.0 N 2 4.5 4.3 -0.2 274.9 280.9 6.0 0.5 11.0 S2 16.0 15.0 - 1.0 286.1 286.5 0.4 1.0 6.3 K• 18.2 17.3 -0.9 71.1 73.1 2.0 1.1 5.9 P1 6.0 5.7 -0.3 71.1 73.7 2.6 0.4 6.9 O• 5.4 5.2 -0.2 61.1 63.6 2.5 0.3 6.0

Venezia-Lido M 2 23.4 23.6 0.2 288.0 287.7 -0.3 0.2 0.9 K2 5.3 3.5 - 1.8 281.0 291.8 10.8 1.9 36.6 N 2 3.8 3.9 0.1 299.0 289.3 -9.7 0.6 17.1 S2 13.8 13.1 -0.7 293.0 295.4 2.4 0.9 6.3 K1 16.0 16.8 0.8 79.0 77.4 -1.6 0.9 5.7 P• 4.3 5.5 1.2 56.0 77.9 21.9 2.2 51.2 O• 5.2 5.1 -0.1 70.0 67.7 -2.3 0.3 4.9

Malamocco M 2 23.5 23.3 -0.2 296.0 288.7 - 7.3 3.0 12.8 K 2 4.0 3.5 -0.5 299.0 292.8 -6.2 0.6 16.1 N 2 4.1 3.8 -0.3 295.0 290.3 -4.7 0.4 10.6 S2 14.0 13.0 -1.0 305.0 296.5 -8.5 2.3 16.1 K• 18.3 16.7 -1.6 82.0 77.9 -4.1 2.0 10.9 P• 5.8 5.5 -0.3 70.0 78.4 8.4 0.9 15.2 O• 5.3 5.0 -0.3 65.0 68.2 3.2 0.4 7.3

Porto Corsini M 2 15.6 17.6 2.0 303.0 300.1 -2.9 2.2 14.0 K 2 2.5 2.5 0.0 310.0 303.9 -6.1 0.3 10.8 N 2 3.1 2.9 -0.2 295.0 299.6 3.6 0.3 9.7 S2 9.2 9.4 0.2 310.0 306.9 -3.1 0.6 6.0 K• 15.9 15.3 -0.6 81.0 81.9 0.9 0.7 4.2 P• 5.3 5.0 -0.3 81.0 81.9 0.9 0.3 5.9 O• 5.0 4.7 -0.3 67.0 72.1 5.1 0.5 10.8

aSuperscripts o and m refer to observed and model values, respectively. The quantity d is the vectorial difference.

open boundary probably causes an artificial reflection of the diurnal tidal wave back into the domain. Thus the nature of the

open boundary condition needs reconsideration, at least for the diurnal frequencies. A possible remedy is the optimization

Table 3. Average and Standard Deviation of the Absolute Difference Between Observed and 2-D Model Values of

Amplitude AH, Vectorial Difference Ad, and Phase Lag A# for All Seven Tidal Constituents at Five Stations

ZIH, cm zid, cm zig, deg

M 2 0.5 1.2 2.5 STD 0.8 1.1 2.6

K• 0.5 0.7 5.7 STD 0.6 0.7 3.0

N 2 0.2 0.5 6.5 STD 0.1 0.1 1.9

S2 0.7 1.1 3.1 STD 0.3 0.6 2.9

K1 0.8 1.0 1.8 STD 0.5 0.6 1.2

P1 0.4 0.8 6.8 STD 0.4 0.8 8.1

O1 0.2 0.4 3.7 STD 0.1 0.1 1.2 All 0.5 0.8 4.3 STD 0.5 0.7 4.1

of the open boundary condition by minimizing an objective functional [Shulman and Lewis, 1995], but this falls beyond the scope of this paper.

The rotary coefficient Ca = +- (1 - e) of the ellipses drawn by the M 2 and K 1 velocity vectors over their respective cycles was calculated at every fifth grid point. Here e is the ellipse eccentricity [Pugh, 1987], and a positive sign is assigned for clockwise rotation. Thus, for pure rectilinear motion, e = 1 and Ca = 0, while for pure counterclockwise rotation, e = 0 and Ca = -1. This definition agrees with that of Gonella [1972]. Over the northern Adriatic the M 2 tidal current rotates counterclockwise (Figure 6), with the sense of rotation being reversed locally along the eastern coast, in bays and around capes, especially inside the Gulf of Trieste. For this gulf our results match almost perfectly the observations reported by Mosetti and Purga [1990], thus leading support to our calculations. In the central part of the northern Adriatic the M 2 tidal ellipses are elongated and aligned with the channel axis. Along this channel axis, from Venice to the middle of the open boundary, our along-channel variation of Ca for the M 2 constituent are consistent with the variation of e = 1 - Icl derived analytically by Mosetti [1986] and numerically by Cav- allini [1985]. The K 1 ellipses, too, are strongly aligned with the channel axis-in the central part of the basin. A noteworthy feature is the reversal of the sense of rotation along a line

26,270 MALA•I• ET AL.: TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA

240-

220 -

200-

180-

160 -

t40- 120-

tO0-

80- x j I "•'--t. .".. '>4., •',,

60-

40- • •[-••13 '•"••••••<•.•:,•

20. 0 ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I

Figure 4. M 2 tidal elevations (solid and short-dashed lines, in cm) and cotidal lines (long-dashed and dotted lines, in degrees) according to the 2-D model. •s numbers are grid indices.

stretching from west to east at midbasin, with clockwise rotation to the south and counterclockwise rotation to the north.

Figure 7 shows the instantaneous flow field and elevation distribution at the time of maximum rate of elevation increase

in the port of Trieste, with all seven tidal constituents included. This time is nearly (within an hour) the time of maximum inflow in Trieste. Note the rightward intensification of the currents and surface elevation along the Croatian/Slovenian coast, as well as the local intensification of the velocity in the vicinity of the Po River mouth (Cape Maestra, see Figure 2) and around the cape marking the entrance of the Gulf of Trieste. There currents exceed 20 cm s -•. Note also that the

surface elevation is greater on the shoreward side of the currents, as one would expect from a coastal Kelvin wave.

Figure 8 shows how the 2-D model performs over time in the port of Trieste by comparing the surface elevation time series (Figure 8a) and frequency spectrum (Figure 8b) with all seven tidal constituents included in the model. The agreement is found to be excellent. (Minor peaks contained in the model spectrum have no correspondent in the observed spectrum because the energy of those peaks is so low that it falls within the instrumental error and was neglected in the data analysis.)

In every fifth cell of the domain the time series of the

240-

220 -

200-

180-

160 -

t40-

120-

tOO-

80-

60-

40-

20-

0

Figure 5. K• tidal elevations (solid and short-dashed lines, in cm) and cotidal lines (long-dashed and dotted lines, in degrees) according to the 2-D model. Axis numbers are grid indices.


N

240

220

180

160

140

120

100

Figure 6. Rotary coefficient C/• of current ellipses of (top) M 2 and (bottom) K• tidal constituents. The coefficient is positive for clockwise rotation, zero for rectilinear oscillatory motion, and negative for counterclockwise rotation.

depth-averaged speed (4560 hourly values) was Fourier trans- formed, and the very low frequency energy was extracted (Fig- ure 9). Considering this low-frequency component to be the tidally rectified flow, we find that the tidally rectified currents in the northern Adriatic are quite weak, being <1 cm s -• almost everywhere, except near the coast, where their magnitudes reach 3 cm s-•. These higher values appear to be related

to the irregularities of the bottom topography and coastline configuration offshore of Venice (see Figure 2). Like sharp corners along the coastline, sharp submarine irregularities, too, are responsible for large velocity gradients, which create tidal residuals. (The larger values in the southwestern corner are suspect because of their relation to the problem with the open boundary condition in that area.)


(a)

Time (hours)

97/04/22 18:00

[[] selection

•> predicted ß model N

(b)

170

160

130

cm/s

97/04/22 18:00

cm/s ...... \ "",•"• I//• 50 • '"' 'x '• '•a, "•'•"*•" ' /

Time (h)

N

120

290 300 310 320 330 340 350 360

Figure 7. Tidal currents (arrows) and surface elevations (solid contours) of all seven tidal constituents combined at the time marking halfway between the lowest and next highest elevation in Trieste (i.e., approximately flood time): (a) entire basin and (b) enlarged view of the Gulf of Trieste.

MALA(2I• ET AL.: TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA 26,273

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-o.6 / ....................... I ....................... i ....................... i,, ,¾ ................... i 2640 2664 2688 2712 2736

Time (hours)

b 10"

i LJ,., .,I , -' ' ""-: ""'" r., _

i i i i I i i i i i i i i i i i i j i i i i i i i i i i i i i I

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10"

10"-

10'" -

10"

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

Frequency (cpd)

Figure 8. Comparison of the surface elevation between 2-D model results and observations in the port of Trieste: (a) sample of the 6 month time series (open circles are observations, heavy dots are model results, and the thin line is the difference) and (b) frequency spectrum of the 'entire 6 month record (thick line is observations, dots are model results, and thin line is the spectrum of the difference).

4. Check on the Open Boundary Conditions As a final check on the model results, we compare the

optimized open boundary conditions (derived in section 2.3) with the numerical predictions of a larger model. This model [Lynch et al., 1996; Naimie, 1996] is three-dimensional, has a finite element mesh, and uses a turbulence closure scheme [MeNor and Yamada, 1982]. It is applied to the entire Adriatic Sea, with a resolution varying from 16 to 2 km. Along the Pesaro-Kamenjak line the model has 50 triangular elements, with a side length of 2 km near each coast, 4 km farther offshore, and 8 km in the central part of the channel (Figure 10). The amplitudes and phases of the surface elevation and velocity are easily obtained from the proximate nodal values using the linear basic functions used inside every finite element.

Figure 11 compares the amplitudes and phases of the sea surface elevations of the seven major tidal constituents computed by the two models. The agreement is satisfactory and therefore validating the open boundary conditions of our first model. There are, nonetheless, some differences. The finite element model predicts slightly lower values for the M2 and S2 tidal amplitudes and smoother phase profiles for the diurnal constituents. These differences are not surprising since the 3-D model has coatset resolution than the 2-D model (>-2 km versus 556 m).

While we think that the surface elevations produced by the 2-D model are superior to those of the 3-D model (because of much finer horizontal resolution), we also believe that the 3-D depth-averaged velocity predictions along the Pesaro- Kamenjak line are more reliable. Indeed, in the 3-D model this

line lies well within the integration domain, whereas it is the open boundary of the 2-D model, and the nature of the open boundary condition does not allow for optimization of the velocity.

5. Interpretation and Discussion In order to gain additional insight into the nature of the tides

in the northern Adriatic, we now interpret the dynamics of the M2 and K• tides. (The other semidiurnal and diurnal constituents do not require separate interpretations, for their structures are very similar to those of the M2 and K• tides, respectively.) The relatively simple structures of the amplitude and phase profiles of the surface elevation and depth-averaged velocities across the basin at the Pesaro-Kamenjak line (hence- forth P-K line) suggests a straightforward explanation, such as the superposition of a few linear waves. Although the M2 tide has been explained as the superposition of a pair of incident and reflected Kelvin waves [Hendershott and Speranza, 1971; Mosetti, 1986], no dynamical interpretation has yet been proposed for the K• tide. Here we shall not only clarify the dynamics of both tides but also show that the semidiurnal and

diurnal tides are two manifestations of a single family of waves, which exist under the combined actions of gravity and topography. In the semidiurnal case, gravity dominates, and the M2 tide takes on aspects of a set of Kelvin waves propagating along the basin, while topography dominates in the diurnal case, and the K• tide resembles a continental shelf wave propagating across the basin.


Residual current (cm/s)

24O

220

200

180

160

140

N

Figure 9. Magnitude of the depth-averaged velocity (in cm s -•) in the very low frequency band (<0.8570 cpd).

5.1. Topography-Gravity Waves

The following mathematical developments are not meant as a theory but rather as a set of arguments presented to provide a certain intuition about the dynamical nature of some wave motions. We then infer that these wave motions are the mech- anisms behind the diurnal and semidiurnal tides in the Adriatic

type exp(-itot). Elimination of v between (8) and (10) then yields a single equation for the y structure of

oy z>

Sea. Consider the linear, barotropic, frictionless equations of motion on an f plane, over a sloping bottom, and in the ab- sence of velocity along isobaths:

-- + (D,) = 0, ot

-fu= -# Ox'

Ot - -g Oy ' where the water depth D (y) varies in only one direction, which is meant to capture the general shoaling of the Adriatic along its main axis from the South Adriatic Pit to the Venice-Trieste

coastline. Thus the x axis points across the basin, and the y axis points along the basin. As we consider flow fields deprived of cross-basin velocity (u = 0), we ignore the effect of lateral boundaries. Because our interest lies in forced oscillatory motions at specified frequencies, we take the time dependency of the surface elevation •/and cross-isobath velocity v to be of the

(8)

(9)

Figure 10. Superposition of the Pesaro-Kamenjak open boundary line of the northern Adriatic model on the local triangulation of the 3-D finite element model.

MALA•I• ET AL.' TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA 26,275

15 360

N2, K2

øo 0:2 0:4

340

320'

300

280

M2 26O

1 0 0.2 0.4 0.6 0.8 1

16 11o

12

o10

8

K1

6 P1,01

4' i i i 0 0.2 0.4 0.6 0.8

x/L

lOO

90

• 80 • K1

70. P1 K1, P1 60

5O 0 0.2 0.4 0.6 0.8 1

•L

Figure 11. Profiles of the (left) surface elevation amplitude and (right) phase of the (top) semidiurnal and (bottom) diurnal constituents along the Pesaro-Kamenjak line. Thick lines with larger symbols are the optimized open boundary conditions derived for the 2-D model, and thin lines with smaller symbols are the results of the 3-D model covering the entire sea.

If we assume that the topography varies slowly over y (admit- tedly not the case for the Adriatic Sea but, nonetheless, a fruitful assumption to elucidate some dynamics), then a solution of the form exp[rr(y)] with rr being a slow function of its variable y can be sought. We find

0.) 2 Do "2 q- D'rr' + = 0, (12)

where rr' stands for drr/dy and can be considered as the in- verse of an e-folding length in the y direction or a wavenumber if it happens to be imaginary. Likewise, D' is dD/dy, the topographic slope. The assumption of a slowly varying function has permitted us to ignore a term containing the second derivative of rr. The solution is

D ' /D '2 00 2 rr' = + (13) 5-b - ¾ gO'

We note that this expression always contains a real part -D'/2D, which implies a growth of the amplitude toward shallow water. Integration of this component over y yields growth that is inversely proportional to the square root of the depth, i.e., a factor 7 over a depth change from 1000 to 20 m. Amplification of the tidal elevation is indeed noted in the Adriatic for all tides, including semidiurnal and diurnal constituents. The remaining part of rr', however, may be either real or imaginary, leading to additional amplitude growth (or attenuation) or to wave behavior in the cross-isobath direction, respectively.

To illustrate this possible dichotomy, let us take a constant Coriolis parameterf = 1.03 x 10 -4 S -1 (characteristic of the Adriatic) and a parabolic topography D(y) = Do(1 - ay) 2 with values D O - 571 m and a - 1/935 km (obtained by least squares fitting to the cross-basin depth average as a function of

the distance along the main axis of the entire sea, Figure 12). Then the expression under the square root becomes

a 2 -- (1 - ay) 2'

where the coefficient a 2 - ro2/gDo is equal to -2.38 x 10 -12 m -2 for the M 2 tide (to -- 1.41 10 -4 S -1) and equal to + 1.95 x 10 -13 m -2 for the K 1 tide (to = 7.29 10 -s s-l). Thus we see a reversal in sign between semidiurnal and diurnal tides, imply- ing that the semidiurnal tides have an oscillatory (and therefore propagating) character, while the diurnal tides only have a gradual amplitude variation from deep to shallow. Returning to the terms that make the quantity under the square root, with the first term depending on the bottom slope and the second term depending on surface variability (via gravity), we conclude that the semidiurnal tides are essentially surface gravity waves with a topographic distortion, while the diurnal tides are topographic waves modified by surface variability. In the limit of no bottom slope (a - 0) the semidiurnal tide is a pure surface Kelvin wave, while in the limit of the rigid lid approximation (g -• oo) the diurnal tide is a pure topographic wave.

Equation (10), which gives the across-isobath velocity,

ig oil

ro Oy (14) i g o"

reveals that the propagating component of r/(with the imaginary part of rr') in a semidiurnal tide has a component v that is in phase (both are real or both are imaginary), while the nonpropagating r/(with rr' real) of a diurnal tide has an ac- companying v that is in exact quadrature (one is real while the

26,276 MALA0•I0• ET AL.' TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA

0 I I I I I I I

-lOO

-200

-3oo

-400

-5OO

Do -600

-700

0 100 200 300 400 500 600 700 800

Figure 12. Parabolic fit to the depth profile of the Adriatic Sea along its main axis, from Otranto Strait to the northwestern shoreline. For each position along the main axis the depth value is the average depth across the basin, from the southwest shore to the northeast coast.

other is imaginary). Then inserting the value of v from (9), we get

( i#rr' ) Orl (15) f •-rl =# Ox'

Physically, the structure in the x direction has the opposite character of that in the y direction: When one is propagating, the other is not. Thus, for semidiurnal tides the gravitational component is propagating in y and trapped in x, while for the diurnal tides the wave propagates in x but is attenuated in y.

We now turn to the numerical results and explore the extent to which these may conform to the preceding remarks. We choose to perform the analysis on the finite element results only because of the superiority of its depth-averaged velocity predictions along the Pesaro-Kamenjak section (P-K line). These are displayed on Figure 13. The amplitudes of the ve-

--1

locity component parallel to the P-K line are below 1.5 cm s for all constituents, indicating that the tidal flow in the across- channel direction is weak. Therefore we can limit ourselves to

explaining the tidal flow in the along-channel direction only. A noticeable feature of the along-channel velocity profiles seen on Figure 13 is that the amplitudes are significantly higher on the right (eastern) side. This left-right asymmetry may be attributable to the difference in bottom topography between both sides (see Figure 2) or to a pair of waves, with a stronger incident wave coming from the south along Croatia and a weaker reflected wave returning from the north along Italy.

5.2. Semidiurnal Tides

The profiles of phase differences between computed elevation and velocity are very similar for all semidiurnal compo-

nents (Figure 13, top right), being about 90 ø in the center and varying antisymmetrically on both sides. As Appendix A shows, this is revealing of an incident-reflected standing wave pattern.

This leads us to investigate to which extent a simple set of two, incident and reflected, Kelvin waves can explain the structure of the semidiurnal tides. Approximating the northern Ad- riatic Sea from the Pesaro-Kamenjak line inward as a rectangular gulf with uniform depth and considering all 222 depths at model nodes along the P-K line, we estimate the mean width L = 137 km, the average depthD = 46.4 m, and the Coriolis parameter f = 1.03 x 10 -4 S -1, which yield a radius of deformation R -- 207 km and an aspect ratio R/L = 1.51. According to (A2a) the square of the elevation amplitude can be expressed as

H2(x) = C] + C2 e2'c/R + C3e -z•/R, (16)

which is linear in its coefficients. A least squares fit between the preceding expression and the data (Figure 11, top left) yields estimates of the coefficients C• to C3. Then the incoming and outgoing wave amplitudes, A o and A1, can be calculated, as can the phase 2ky + ok, from

cos (2/cy + = C1

The results are reported in Table 4 for each semidiurnal constituent. These show that the incoming wave has for each constituent a slightly higher amplitude than the outgoing wave (,4• > .40). We can then use these estimates to reconstruct

MALA•I(2 ET AL.: TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA 26,277

14o

12o •

100

80

60 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

: 140

12o

lOO

8o

60 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

x/L x/L

Figure 13. Structure of the depth-averaged tidal velocities along the Pesaro-Kamenjak (P-K) line: (top) semidiurnal tides, (bottom) diurnal tides, (left) amplitudes, and (right) phase differences with surface elevations. Thick lines are along-channel velocity components; thin lines are across-channel velocity components (same symbols as for Figure 11).

the structures of surface elevation, normal velocity, and phase difference along the P-K line and compare them to the numerical results (Figure 14). The common features indicate that the semidiurnal tides in the northern Adriatic are primarily the result of a superposition of an incoming Kelvin wave with its partial reflection. The good fit of elevation amplitudes is a direct result of the least squares fit, but the velocity amplitude and phase difference provide an independent check. Consid- ering that the northern Adriatic is not close to being a rectangular basin with a flat bottom and that the bottom slope ought to affect the wave properties as noted in (13), the agreement is better than what could have been expected. This indicates that the semidiurnal tides in the northern Adriatic primarily consist in an incoming Kelvin wave progressing along the eastern coast, turning with the coastline (as a set of Poincard waves) and returning in an attenuated form along the Italian (western) coast. Such behavior also explains the amphidromic point observed farther south [Polli, 1959] as the location where both

Table 4. Amplitude of Incoming Wave (A •), Amplitude of Outgoing Wave (Ao), and Phase Difference 2ky + d) Determined From a Least Squares Fit of the Pesaro- Kamenjak Results to the Two-Kelvin-Wave Theory a

A1, Ao, 2ky + ½, Constituent cm cm deg X 2

M 2 11.5 10.8 - 110.9 1.280 K 2 2.6 2.4 - 118.1 0.003 N 2 2.0 1.9 -- 107.3 0.001 S2 7.0 6.5 - 117.5 0.136

aTh e )(2 values express the "goodness of the fit" according to 1, 2 statistics. Note that the outgoing wave is systematically weaker than the incoming wave.

incident and reflected waves have equal and opposite phases. This explanation, which is not new, confirms the previous con- jectures of Hendershott and Speranza [1971] and Mosetti [1986].

5.3. Diurnal Tides

When the same procedure is applied to diurnal tides, it fails because no pair of exponential curves can be fitted to both ends without creating an unrealistic negative value in the middle. The conclusion must be that diurnal tides do not consist of

Kelvin waves. The preceding arguments, indeed, show that we should expect not a gravity-type wave but a topographic wave propagating across the basin (although the distance is rather short!) with an amplitude amplification from deep to shallow.

For the fitted parabolic profile (Figure 12) the amplification coefficient rr' takes the form

+a - x/a 2- w2/gDo or' - 1 - ay (18)

(The sign in front of the square root was chosen to yield the smallest absolute value, corresponding to the wave with the least amplification from south to north, i.e., the wave with the least energy.) Integration over the direction of the main axis of the Adriatic yields

f0 y ') or(y) = cr' (y dy'

= 1- 1 gD0a 2 In 1-ay ' from which we deduce the amplification/attenuation factor

(1) •-¾•-"'2/gøøa2 exp [o-(y)] = 1 - ay . (19)

26,278 MALA(2I• ET AL.: TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA

12

10 {

8

o 6=

4

S2

N2, K2 .............................

0 0.5 0 0.5 1 0 0.5 1

Figure 14. Comparison of the (left) surface elevation amplitude, (middle) normal velocity amplitude, and (right) phase difference between the numerical results (thin lines) and the fit of the two-Kelvin-wave theory (solid lines).

For the values quoted above (f = 1.03 x 10-4/S, ro -- 7.29 x 10-S/s, D O = 571 m, a = 1/935 km, and L = 137 km), this factor equals 3.1 over a distance of 800 km (the length of the entire Adriatic) and 1.1 over a distance of 144 km (the length of the portion retained for this model). These values are only slightly smaller than the observed north-south amplification factors observed for the K• tide ([Polli, 1959] and Figure 5).

We can further examine whether the topography-wave ar- gument predicts a phase shift across the basin that agrees with the observed value. Seeking a solution of the type exp[- ikx] (with k real positive to correspond to a phase that decreases from left to right, from Italy to the opposite shore), (15) yields

frr' f a - x/a 2 - ro2/gD o k .... . (20) ro ro 1 - ay

The phase shift across the basin, i.e., over the distance L, is

fL a - x/a • - •o•/gDo kL = . (21) ro 1 - ay

For the values quoted above the predicted phase drop from Italy to Croatia at y - 640 km (about the location of the Pesaro-Kamenjak line) is 0.39 rad = 22 ø. In comparison, Fig- ure 11 reveals a phase difference of 15ø-20 ø, in the same direction. Appendix B provides a more precise comparison by using a theoretical framework slightly more rigorous than the basic arguments proposed at the beginning of this section and also by comparing the along-basin velocity magnitudes. The conclusion remains the same: The Kz tide and all other diurnal tides can be explained as topographic waves progressing from the northeast to the southwest, from the Croatian coast to the Italian shore.

6. Conclusions

Until this study, Adriatic Sea tides had not been simulated by means of nonlinear numerical models, except in the context of a tidal analysis of the entire Mediterranean Sea [Tsimplis et al., 1995], which made no specific mention of the particular structure of the tides in the northern Adriatic. The objectives of the present study were the accurate 2-D simulation and dynamical analysis of the tides within the subdomain sur- rounded by five ports (Rovinj, Trieste, Venezia-Lido, Malam- occo, and Porto Corsini) and extending slightly to the south (line joining Pesaro to the southern tip of the Istrian Peninsula).

The model's open boundary conditions were calibrated so as to obtain an optimum fit with the known tidal elevations in the five ports. The simulation results were found successful, for

they yielded arms error of all surface elevations in the five ports smaller than 1 cm (0.5 _+ 0.5 cm). Similar results were obtained for the vectorial difference between modeled and observed complex values (combinations of amplitudes and phases): 0.8 _+ 0.7 cm. The phase errors generally fell below 5 ø (4.4 ø _+ 4.2ø).

The model shows that the surface elevation is always higher on the right side of the flow, indicating that both the northern Adriatic and the Gulf of Trieste behave like narrow channels [Gill, 1982], in which the velocity component along the channel is significantly stronger than the cross-channel velocity and is subject to the Coriolis force. At times of high inflow/outflow the isolines of surface elevation are nearly aligned with the depth-averaged velocity. As such, the Gulf of Trieste may be considered as a miniature of the northern Adriatic Sea.

The surface elevations and along-channel velocities across the open boundary (line from Pesaro to Kamenjak) of the present 2-D model compare well with the similar quantities obtained with a larger and 3-D model. The analysis of these cross-channel profiles then led to the following three results: (1) the M 2 and other semidiurnal tides can be understood as having been formed by a standing set of incident and reflected Kelvin waves, (2) the northward amplification of these Kelvin waves is caused by the shoaling bottom, and (3) the Kz and other diurnal tides can be understood as topographic waves propagating across the basin with the shallow water on their right, namely, from the Croatian coast to the Italian shore, and subject to attenuation from shallow to deep. While conclusion 1 is a confirmation of earlier propositions [Hendershott and Speranza, 1971; Mosetti, 1986], conclusions 2 and 3 are new. In particular, no dynamical interpretation of the diurnal tides in the Adriatic had been proposed prior to this study.

Appendix A: Kelvin Waves in a Flat Bottom Channel

Consider two oppositely traveling barotropic Kelvin waves in a channel extending along the y axis, of constant width L and of uniform depth D. The surface elevation •l(x, y, t) and longitudinal velocity v(x, y, t) can be written as

•q = A oe -x/R cos (ky + rot) + A ,e (x-L)/R cos (ky - rot + rk)

g -x/R u = •-• [-Aoe cos (ky + rot)

+ A he (x-L)/• cos (ky - rot + rk) ],

(Ala)

(Alb)

MALA•I• ET AL.' TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA 26,279

where R = V'#D/f = to/fk is the external radius of deformation, # is the gravitational acceleration, f is the (constant) Coriolis parameter, k is the longitudinal wavenumber, and to is the angular frequency. One wave reaches its largest amplitude (.40) along the coast x -- 0, while the other reaches its largest amplitude (A•) along the opposite wall x = L. There is a phase difference (k between the two waves. If we combine the two waves into a single oscillatory field beating at the frequency to, namely, r• = H(x, y) cos [tot - (kr•(x, y)] and v = V(x, y) cos [tot - qbr.(x, y)], we obtain

H(x,y)

= x/Ao2e -2x/R + A•2e 2(x-*)/R + 2AoA,e

g

V(x, y) = fR

cos (2ty + q,),

(A2a)

ß x/A•e -zv'• + A•2e 2(x-L)/'•- 2AoA•e -•/'• cos (2ky + (A2b)

2A 0,4 le -•/• sin (2ky + 4)) tan ((kn - (k•,) = Ao2e -zvR - A •2e2(X-6)/e ' (A2c)

Note that if the incoming and reflected waves have the same amplitude (.40 = .4 •), the phase difference is

sin ( 2ky + 4)) tan ((k, - (kv) = sinh [ (L - 2x)/R]' (A3)

which is equal to _+90 ø at the middle of the channel (x = L/2) and varies antisymmetrically on both sides.

Appendix B: Topographic Waves in a Shoaling Channel

The linear barotropic equations governing waves in a channel of variable depth can be written as

Ou Ot fv =-# Ox' (Bla)

Ov

OZ- + fu = -# Oy ' (Bib) O rl Ou 0

0•- + D •xx + •yy (Dr) = 0, (Blc) where x is directed across the channel (0 <- x <- L), y is directed along the channel, f is the (constant) Coriolis parameter, # is the gravitational acceleration, and D(y) is the resting depth, which we take as a function of y only. Had the surface elevation term Or•/Ot been ignored, the set of equations would have been that governing continental shelf waves [Gill, 1982, p. 409]. In other words, we are considering here topographic waves modified by the gravitational influence of the SSE.

If we seek solutions having a given frequency to, as in the tidal problem, and having a wave expression in x, namely, [r•, u, v](x, y, t) = [H, U, V](y) exp [-i(kx + tot)], the cross-channel ampli.tudes H(y), U(y), and V(y) must satisfy

-f gH' + togkH v = ?_ , (B2)

i ( togH' - f gkH) V = f2_ to2 , (B3)

(DH') (k2D+fkD + f2-to2) .... H: 0 (B4) to g '

where a prime indicates a derivative with respect to y. Equation (B4) is difficult to solve exactly for a depth profile

D(y) given a priori, even as a simple analytical function. Thus, instead of constructing a D(y) topography profile and solving for H(y), let us anticipate a solution H(y) that has realistic features and seek the D(y) profile to which it corresponds. Then, if that topography has realistic features, we accept the solution.

Observations [Polli, 1959] as well as our present simulations (Figure 5) reveal (1) that the amplitude of the K• tide increases smoothly and gradually along the basin and (2) that the cross- basin velocity is very weak. Let us then adopt H(y) = .4 exp (sy), where s (>0) is an e-folding length scale and u = 0. According to (B2), there is a wavenumber k that guarantees no cross-basin flow:

fs k = --. (BS)

Equation (B4) becomes

6O 2 sD' + s2D +- = 0, (B6)

g

which is satisfied if D(y) is of the form

6O 2 D (y) = D •e -sy 2, (B7)

gs

where D• and s are two adjustable parameters. If we set y - 0 at the P-K line, where the cross-basin average depth is 46.4 m, we have D• - 46.4 m + to2/gs2. Then, if we impose a zero depth at the Venice-Trieste shoreline, which is 140 km away, we obtain an equation for the constant s:

46.4 m + e-(140 km)s __ (B8) gs 2'

The solution with the lowest absolute value (yielding the least energetic wave) is s = 1.87 x 10 -6 m-•, and the topographic profile compatible with H(y) = .4 exp (sy) that best fits the actual bottom topography of the northern Adriatic is

D(y) = (201.6 m)exp [-(1.87 10-6/m)y] - (155.2 m).

(B9)

Over the 140 km of basin length the e-folding scale s yields a wave amplification of exp (sy) = 1.30, i.e., corresponding to a surface elevation amplitude increase of 30% from the P-K line to the northwestern shoreline. In comparison, the numerical model (Figure 5) revealed an increase from 14.5 to 17.5 cm, which is a 21% increase. Considering the radical assump- tions of the theoretical model, we find reasonable agreement.

The cross-channel wavenumber k given by (B5) is found to be approximately equal to 2.6 x 10 -6 m -•, which yields a phase change kL from Pesaro to Kamenjak (L = 137 km) of about 21 ø . This value agrees with the phase differences determined from the numerical simulations and shown in Figure 11 (lower right).

Turning now to the along-basin velocity, we derive from (B3)

26,280 MALA0•I(2 ET AL.: TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA

that the y structure of the v velocity component is related to the amplitude profile H(y) by

V(y) = -i --H(y). (B10)

Along the P-K line, where the K• amplitude H is 14.5 cm, the theory predicts a K• velocity magnitude of 3.6 cm s -•, which agrees quite well with the values obtained by the numerical simulations (Figure 13, lower left, top curve). Furthermore, the presence of the -i factor in the expression for V indicates that the along-basin velocity lags the sea surface elevation by 90 ø , which is precisely the value noted in the numerical simulations (Figure 13, lower right).

In conclusion, the preceding theory is validated by favorable comparisons with numerical simulation results (as well as observations), and since the theory reduces to the classical continental shelf wave theory in the limit of no gravitational effects (rigid lid approximation obtained for # • •), the K• tide of the northern Adriatic is a topographic wave modified by gravitational effects. Note that the wave is evanescent in the down-

channel direction and propagates in the cross-channel direction, from the Croatian coast to the Italian shore.

Acknowledgments. Malafiifi was supported by the Ministry of Sci- ence and Technology of Slovenia through grant Z1-7045-0105. The Osservatorio Geofisico Sperimentale in Trieste (Italy) supported Viez- zoli. Malafiifi and Cushman-Roisin also acknowledge the support of the U.S. Office of Naval Research, through grant N00014-93-7-0391 to Dartmouth College. All three authors are indebted to Christopher E. Naimie of Dartmouth College for having performed the tidal calculations with the finite element model (to be published elsewhere) and extracted the values along the Pesaro-Kamenjak section for the purpose of the present study. The Abdus Salam International Centre for Theoretical Physics supported the participation of the authors at the International Workshop on the Oceanography of the Adriatic Sea, where fruitful discussions took place.

References

Casulli, V., Semi-implicit finite-difference methods for the two- dimensional shallow water equations, J. Comput. Phys., 86, 56-74, 1990.

Cavallini, F., A three-dimensional numerical model of tidal circulation in the northern Adriatic Sea, Boll. Oceanol. Teor. Appl., III, 205-218, 1985.

Cheng, R. T., V. Casulli, and J. W. Gartner, Tidal, Residual and Intertidal Mudflat (TRIM) model and its applications to San Fran- cisco Bay, California, Coastal Shelf Sci., 36, 235-280, 1993.

Crisciani, F., S. Ferraro, and B. Patti, Do tidal harmonic constants depend on mean sea level? An investigation for the Gulf of Trieste, Nuovo Cimento Soc. Ital. Fis. C., 18, 15-18, 1995.

Davis, J. C., Statistics and Data Analysis in Geology, 2nd ed., 654 pp. John Wiley, New York, 1986.

Defant, A., Physical Oceanography, vol. 2, 729 pp., Pergamon, Tarry- town, N.Y., 1961.

Ferraro, S., and M. Maselli, Golfo di Trieste, previsioni di marea per il 1996, Nova Thalassia, 12, suppl., 47 pp., 1995.

Gill, A. E., Atmosphere-Ocean Dynamics, 662 pp., Academic, San Di- ego, Calif., 1982.

Godin, G., and L. Trotti, Trieste water levels 1952-1971: A study of the tide, mean level and seiche activity, Misc. Spec. Publ. Fish. Mar. Serv. Can., 28, 24 pp., 1975.

Gonella, J., A rotary-component method for analysing meteorological and oceanographic vector time series, Deep Sea Res., 19, 833-846, 1972.

Hendershott, M. C., and A. Speranza, Co-oscillating tides in long, narrow bays: The Taylor problem revisited, Deep Sea Res., 18, 959- 980, 1971.

Hydrographic Institute of the Republic of Croatia (HIRC), Tide Ta- bles, Adriatic Sea-East Coast, 110 pp., Split, Croatia, 1994.

Istituto Idrografico della Marina, (IIM), Tavole di Marea, Mediterra- neo-Mar Rosso e delle Correnti di Marea, Venezia-Stretto di Messina, 96 pp., Genova, Italy, 1994.

Lynch, D. R., J. T. C. Ip, C. E. Naimie, and F. E. Werner, Compre- hensive coastal circulation model with application to the Gulf of Maine, Cont. Shelf Res., 16, 875-906, 1996.

McHugh, G. F., A numerical model of two tidal components in the northern Adriatic Sea, Boll. Geofis. Teor. Appl.,Xvi, 322-331, 1974.

Mellor, G. L., and T. Yamada, Development of a turbulence closure model for geophysical fluid problems, Rev. Geophys., 20, 851-875, 1982.

Mosetti, F., Distribuzione delle maree nei mari italiani, Boll. Oceanol. Teor. Appl., V, 65-72, 1987.

Mosetti, F., and B. Manca, Le maree dell'Adriatico: Calcoli di nuove costanti armoniche per alcuni porti, in Studi in onore de Giuseppina Aliverti, pp. 163-177, Ist. Univ. Nav. di Napoli, Ist. di Meteorol. e Oceanogr., Naples, Italy, 1972.

Mosetti, F., and N. Purga, Courants c6tiers de diff6rente origine dans un petit golfe (Golfe de Trieste), Boll. Oceanol. Teor. Appl., VIII, 51-62, 1990.

Mosetti, R., Determination of the current structure of the M 2 tidal component in the northern Adriatic by applying the rotary analysis to the Taylor problem, Boll. Oceanol. Teor. Appl.,/V, 165-172, 1986.

Naimie, C. E., Georges Bank residual circulation during weak and strong stratification periods: Prognostic numerical model results, J. Geophys. Res., 101, 6469-6486, 1996.

Polli, S., La Propagazione delle Maree nell'Adriatico, paper presented at IX Convegno Dell' Associazione Geofisica Italiana, Rome, 1959.

Pugh, D. T., Tides, Surges and Mean Sea-Level, 472 pp., John Wiley, New York, 1987.

Shulman, I., and J. K. Lewis, Optimization approach to the treatment of open-boundary conditions, J. Phys. Oceanogr., 25, 1006-1011, 1995.

Stravisi, F., and S. Ferraro, Monthly and annual mean sea levels at Trieste 1890-1984, Boll. Oceanol. Teor. Appl., IV, 97-103, 1986.

Taylor, G. I., Tidal oscillations in gulfs and rectangular basins, Proc. London Math. Soc., 20, 193-204, 1921.

Trotti, L., Nuove costanti armoniche per i porti Adriatici e Ionici, Atti Accad. Ligure Sci.-Lett., 26, 43-56, 1969.

Tsimplis, M. N., R. Proctor, and R. A. Flather, A two-dimensional tidal model for the Mediterranean Sea, J. Geophys. Res., 100, 16,223- 16,239, 1995.

B. Cushman-Roisin, Thayer School of Engineering, Dartmouth Col- lege, Hanover, NH 03755-8000. (Benoit. [email protected])

V. Malafiifi, National Institute of Biology, Marine Station Piran, Fornace 41, Piran 6330, Slovenia.

D. Viezzoli, Osservatorio Geofisico Sperimentale, Borgo Grotta Gi- gante 42/c, Sgonico 34010, Trieste, Italy.

(Received February 22, 1999; revised December 8, 1999; accepted April 5, 2000.)

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