Tidal Dynamics in the Northern Adriatic Sea

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 OsservatorioGeofisico Sperimentale, Villa Opicina, Italy

Benoit Cushman-Roisin Thayer Schoolof Engineering,Dartmouth College,Hanover, New Hampshire

Abstract. Tides in the northern Adriatic Sea are investigatedusing two distinctnumerical models.First, a two-dimensional(2-D) finite differencemodel is implementedwith very high horizontalresolution (556 m) to simulatethe northernAdriatic. After calibrationof open boundaryconditions the model givesvery satisfactoryresults: The averagedvectorial difference between observed and simulated elevations is <1.3 cm for each of the seven major tidal constituents.Next, a 3-D finite elementmodel is appliedto the entire sea in order to providea better simulationof the tidal currentsin the vicinityof the open boundaryof the first model. Resultsshow that the northernAdriatic behaveslike a narrow rotatingchannel in whichthe instantaneoussea surfaceelevation (SSE) contours are alignedwith the depth-averagedvelocity vectors and in which the SSE is alwayshigher to the right of the local current.These featuresemphasize the rotationalcharacter that tides can exhibit in a relativelysmall basin.Wave fitting to the current elevationstructure showsthat semidiurnaltidal constituentsare well representedwith a systemof two frictionlessKelvin waves(incident and reflected).In contrast,the diurnal constituentsare best describedas a topographicwave propagatingacross, not along,the basin,from the Croatian coastto the Italian shore.Despite this obviousdisparity the semidiurnaland diurnal tides can be understoodas distinctmembers of a singlefamily of linear waves, which existunder the combinedactions of gravityand topography.

1. Introduction extendingfrom Venice to Trieste), Hendershottand Speranza [1971] showedhow partial reflectioncauses a displacementof Early studiesof the tides in the Adriatic Sea (Figure 1) the M 2 amphidromicpoint from the channelaxis toward the beganin the nineteenthcentury (as reported by Defant [1961]), western(Italian) coast.Later, the Taylor approachwas again and it has long been known that only seventidal constituents, applied to the northern Adriatic by Mosetti [1986], who then four semidiurnaland three diurnal, make a significantcontri- successfullycompared M 2 current amplitudesand phasesto bution to the sea surface elevation (SSE). Defant [1961] observations.Thus at least the M 2 tide in the northern Adriatic showed that except within straits the Mediterranean tides can be understood in terms of Kelvin and Poincar• waves. The reach their highest amplitude in the northern Adriatic Sea. same cannot be said of the other tidal constituents. Generally,Mediterranean tides are weak, with surfaceeleva- Early numericalmodels of tidesin the northernAdriatic Sea tions not exceeding1 rn [Tsimpliset al., 1995].The tide in the were limited by driving the model with only one or two con- northern Adriatic is of a mixed type, with the semidiurnal stituents (M 2 and K• [McHugh, 1974] and M 2 [Cavallini, componentM 2 and diurnal componentK• havingcomparable 1985]).Cavallini [1985] further investigatedthe ellipticmotion amplitudes[Polli, 1959]. producedby the M 2 tide and the effect of different boundary Taylor[1921] proposed a theory accordingto whichtides in conditionsalong the open boundary. a rectangulargulf (semiclosedchannel) are combinationsof The purposeof the modelspresented here is to simulate incident and reflected Kelvin and Poincar• waves superim- accuratelythe tidal motions in the northern Adriatic, with posedin sucha way that the normalvelocity vanishes along all specialemphasis on the Gulf of Trieste and the area leadingto sides,including the end of the channel.A feature of the solu- it. First, the two-dimensionalTidal Residual Intertidal Mudflat tion is the possibleexistence of one or severalamphidromic (TRIM) modelof Chenget al. [1993]is selectedbecause of its points inside the gulf. For the Adriatic Sea, there is an am- suitabilityto this type of study;it wasshown to be successfulin phidromicpoint approximatelytwo thirdsup the basinfor each simulatingtidal and residual currents in San FranciscoBay semidiurnal constituent and none for the diurnal constituents [Chenget al., 1993]. The paper reviewsbriefly the model for- [Polli,1959]. In studyingthe problemof the attenuationof the mulation and the procedurefor the calibrationof the open Adriatic tidal wave at the head of the basin (the coastline boundaryconditions. It then followswith model resultsand Copyright2000 by the American GeophysicalUnion. comparisonof surface elevationswith observations.Next, a Paper number 2000JC900123. three-dimensionalfinite elementmodel is employedto obtain 0148-0227/00/2000JC900123509.00 better tidal velocityprofiles along the open boundaryof the

26,265 26,266 MALA•II2 ET AL.: TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA

' I ' I ' I ' I ' I ' I ' I ' I Adriatic Sea is relativelycompact, with a singleappendix, the 12 13 14 15 16 17 18 19 Gulf of Trieste, at its northeasternmostend and a wide open- Longitude (o) ing on its southernside. 46 46- More than 6100 depth soundingsand over 3100 coastal TRIESTE positionswere taken from maritime charts and interpolated using the Kriging procedure [Davis, 1986]. From these the

45- model topographywas generatedon a staggeredfinite differ- encegrid with a spatialresolution of 0.3 nauticalmiles (about 556 m). Suchresolution is deemedsufficient for the studyof localsea surface elevation (SSE) and Euleriandepth-averaged PESARO velocitiesas it will be verified below from the velocity fields near capesand inside the Gulf of Trieste.

43- 2.2. Model Equations Tidal dynamicsare numerically simulated with the two- dimensionalTRIM model [Chenget al., 1993], which is semi- implicit and therefore unconditionallystable. It solvesthe depth-integratedcontinuity equation Or/ O[(r/+ D)u] O[(r/ + D)v] oD-= - ox - oy ' where (u, v) is the verticallyaveraged velocity, r/is the SSE abovethe mean level, and D(x, y) is the restingdepth, to- getherwith the verticallyaveraged momentum equations du Or/ g(r/ + D) Op rxø - rx 14 18 19 d•--fv = -g Ox 2p0 Ox+ vI-IAU+ Po(r/ + D) Figure 1. Position of the model domain within the Adriatic (2) Sea. Numbers along the axes are degreesof longitude and dv Or/ g(r/+ D) Op ryø -- Ty latitude. dt+ fu = -g Oy 290 Oy+ u/_/Au+ P0(r/ + D) ' (3) first model in order to interpret the dynamicalnature of the where d/dt - O/Ot + uO/Ox + vO/Oy is the Lagrangian dominanttides. The paper finally discussesthe physicalnature derivative,f = 1.04 x 10-4 S-1 is the Coriolisparameter, p of the tidal wavesby matchinganalytical solutions to the nu- is the verticallyaveraged density, Po is a referencedensity, uu merical results. is a horizontaleddy viscosity, (rxø, ryø) is the surfacewind stress,and (rx, ry) is the frictionalbottom stress. Density 2. Two-Dimensional Numerical Model variations, horizontal diffusion of momentum, and wind stress 2.1. Model Geometry are set to zero in our tidal analysis. The bottom stress is taken as a nonlinear function of the The model domain(Figure 2) comprisesan area extending depth-averagedvelocity accordingto the classicalquadratic northwardfrom a straightopen boundaryline (x axis) con- bottom drag law: nectingPesaro in Italy to Kamenjakat the southerntip of the Istria Peninsulain Croatia. This line is 124 km long. The tidal rx: poCouSu 2 + u2 ry: poCoux/u2 + u2. (4) stations nearest to the domain corners are Pesaro and Pula. Figure 2 alsoshows the bathymetry.The overallpicture is that Becausethe drag coefficientdepends on the water depth,we take of a depth increasingalmost linearly with distancefrom the Italian coast (left) for •30 km, beyondwhich the bottom is 6.13 x 10 -3 nearlyflat. On the Croatianside the topographyexhibits steep = (2- e ' (5) jumps between trenchesand submarineridges and even is- lands, all within a few kilometersfrom the coast.The rugged whereD is givenin meters.This gives a valueof 2.55 x 10-3 topographyin the vicinity of the Croatian coast complicates at 2 m andof 1.53X 10-3 at >12 m. Thepreceding expression tidal modeling sinceenhanced local variationsin bottom fric- for the drag coefficientis in accordancewith the parameter- tion, wavereflection, and wave refractionaffect the amplitude ization of the bottom friction developedfor the San Francisco and arrival time of the wave in the Gulf of Trieste. Bayby Chenget al. [1993],who optedfor a variationof the drag The major difference between the modeled area in the coefficientwith depth insteadof holding the latter constant. northernAdriatic Sea and that in San FranciscoBay, to which The rationalebehind (5) is the Ch6zycoefficient of hydraulics the TRIM model was first applied [Chenget al., 1993], is the [Chenget al., 1993]. lengthof the open boundary.While the geometryof San Fran- The system(1)-(5), whichis solvedfor the unknownsu, v, ciscoBay is a complicatedset of bays(San Pablo Bay, Central and r/, is nonlinear through advection and bottom friction Bay, and SouthBay), it is connectedto the oceanonly through terms. In our applicationthese nonlinearities couple the vari- a narrow strait, the Golden Gate. In contrast, the northern ous tidal constituentsand generateresidual currents. MALA•I• ET AL.: TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA 26,267

o 50 i i i i i i

J•IALAMOCCO VENEZIALIDO ] N 240

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_ Figure 2. Seaports and isobaths in the arearetained by the 2-D model.Tidal dataare availablefrom Pesaro, Porto Corsini,Malamocco, Venezia-Lido, Trieste, Rovinj, and Pula. The axesare in model units,with one unit equalto 556 m (=0.3 nauticalmiles). The openboundary of the model,extending from Pesaroto Kamenjak, is the x axis.

The semi-implicitstaggered grid methodof Casulli[1990] is appliedto the system(1)-(3), reducingthem to a pentadiago- amplitude= H = a0+ a•• + a2 (6) nal systemof linear equationsfor the SSE valueson the grid (for details,see Cheng et al. [1993]).The matrixof the system is positivedefinite and canbe solvedvery efficiently. Although phase= # = b0+ b••x + b2 + b3 , (7) the codeis unconditionallystable, a time stepof 900 s is chosen for accuracy. wherex is the distancefrom Pesaro(x = 0 at Pesaroand x = L at Kamenjak,L beingthe lengthof the openboundary line). 2.3. Model Calibration and Open Boundary Conditions Initially, the coefficientsa o to b 3 are fitted to the chartsof Polli The lengthof the tidal recordin the port of Trieste excqeds [1959].The modelis then run from restuntil all transientshave 100 years:Observations of the water level beganin 1859, and fallen belowthe level of numericalnoise (about 9 days).Cal- monthlyand annualmean sea level values have been published culatedSSE valuesat the M 2 frequencyin Trieste, Rovinj, and since1905 [Godinand Trotti,1975]. Hourly data since1939 are Venezia-Lidoare then comparedto the observations.Because alsoavailable [Stravisi and Ferraro, 1986]. Because this is one of the phasediscrepancy is found to be larger than the amplitude the longestSSE recordsin the MediterraneanSea, the con- discrepancy,the two end slopes( !7'(0) at Pesaroand 17' (1) at stantsof the tidal constituentsare preciselyknown for thisport Kamenjak) of the cubicpolynomial for the phaseprofile are of the Adriatic Sea [Criscianiet al., 1995]. Next in order of taken as adjustableparameters. The distributionof the ampli- availabilityare the tidal recordsof Rovinj and Venezia-Lido. tude and phasediscrepancies between calculated and observed The first stepin the calibrationof the modelis to seeka match valuesas those parametersare varied is then examined.For betweenmodel resultsand observationsat Trieste, Rovinj, and this,amplitudes and phases are interpretedas vectors (or com- Venezia-Lido[Hydrographic Institute of theRepublic of Croatia plexnumbers), and the vectorial(complex) difference between (HIRC), 1994;Istituto Idrografico della Marina (IIM), 1994]. observationsand calculationsis taken. Figure 3 displaysthe For this task we determine the SSE valuesto be prescribed absolutevalue of this differenceas the two tuning parameters alongthe openboundary line connectingPesaro to Kamenjak are varied. for eachseparate tidal constituent,starting with M 2. The tidal The plot revealsthat the error betweenobserved and calcu- constants(amplitude and phase) for each tidal constituent lated values reaches a minimum for a certain set of values of along the open boundary are taken as quadratic and cubic 17'(0) and 17'(1). These values are then adopted for the polynomials: boundaryconditions in the remainingsimulations. There is no 26,268 MALA•I(• ET AL.: TIDAL DYNAMICSIN THE NORTHERNADRIATIC SEA

averagephase lag difference is <7.2 ø. When all seven tidal constituentsin all five ports are consideredtogether (35 values), the averageamplitude difference is 0.5 cm, the average vectorial differenceis 0.8 cm, and the averagephase lag difference is 4.4 ø. We conclude that the 2-D model was success- fully calibratedand that it providesreliable values,allowing us to considerthe distribution of tidal elevation and velocity inside the northern Adriatic. The distributionsof SSE amplitude and phase lag of the principalsemidiurnal (M2) and diurnal(K•) constituentsover the model domain are shownin Figures4 and 5. The amplitude of each constituent increases northward and then northeast- ward from the forced open boundaryto the Gulf of Trieste. In otherwords, the amplituderises over decreasingdepth, as one might have expectedfrom the principle of wave actionconser- vation. For each constituentthe phaselag generallyincreases west- ward from the Croatiancoast (right-hand side of Figures4 and 5) to the Italian shore (left-hand side of Figures4 and 5). Figure 3. Distribution of the difference between M 2 tidal While the M 2 cotidal lines diverge,the K• cotidallines tend to observationsand 2-D calculationsat three stations(Trieste, be more parallel; this tendencyis related to the fact that the M 2 Rovinj, and Venezia-Lido)as a functionof the gradientof the tide hasan amphidromicpoint somewheresouth of the domain phaselag at both endsof the openboundary line, #' (0) and (whereall cotidallines gather into a singlepoint), while the K• 9'(1). Note the minimumnear 9'(0) = 4ø and 9'(1) : tide doesnot [Polli, 1959]. In the northeasterncorner of the -28 ø' domainthe cotidal lines of both M 2 and K• tides bend into the Gulf of Trieste, where they divergeslightly. This is expected sincethe flow must be parallel to the coastline,the semiminor need to adjust the quadratic polynomialfor the amplitude axisof the velocityellipses must be small,and the cotidallines profile along the boundary.Finally, the entire procedureis must conservetheir anglewith respectto the coastline[Pugh, repeatedfor the six other tidal constituents.Table 1 liststhe 1987,p. 439]. The remainingsemidiurnal (K2, N2, and S2) and optimizedcoefficients obtained for both amplitudeand phase diurnal (P• and O•) constituentsare substantiallyweaker but profilesalong the openboundary line expressedas (6) and (7). reveal SSE amplitude and phase lag distributionssimilar to those of the M 2 and K• constituents,respectively. There existsa peculiar K• amplitudeminimum betweenPe- 3. Two-Dimensional Model Results saro and Porto Corsini in the southwestern corner of the do- After calibrationthe model is spunup for 31 daysand then main (see Figure 2 for the geographicallocation of thesetwo run for 190 days(i.e., slightlymore than 6 months).The start- ports). This minimum locally distortsthe otherwisegradual ing time is December 1, 1996, so that the actual simulation distributionof the K• amplitude. Becausethere is no hint of begins on January 1, 1997. The Rayleigh criterion for the such local minimum in the observations, we conclude that its separationof the S2 and K 2 frequenciesfrom the simulated existenceis an artifact of the model, mostlikely attributableto record demands a time series of 182.6 days (the so-called an imperfectopen boundarycondition. The sameproblem was synodicperiod [Pugh,1987]. So, the durationof our simulation also noted by McHugh [1974, Figure 7] for the same tidal (190 days)is sufficient.The resultsare sampledhourly, and the constituentin the sameregion of the samemodel domain.This tidal constituentsat five portsin the northernAdriatic (Porto consistencyin the locationof a K• amplitudeminimum and the Corsini, Mallamocco, Venezia-Lido, Trieste, and Rovinj, see fact that both modelsrely on the same type of boundarycon- Figure 2) are extractedfrom thesehourly time SSE series. dition lead us to conjecturethat the prescribedSSE along the Table 2 comparesthe amplitudesand phase lags of the model results with the observed values. These values were taken from Polli [1959], Trotti [1969], Mosetti and Manca Table 1. Coefficientsof the Quadraticand CubicPolynomials, [1972],Godin and Trotti[1975], Mosetti [1987], and Ferraro and (6)-(7), Fitted by the CalibrationProcedure to Prescribethe Maselli [1995] as well as from official reports for the port of Elevation Amplitudes and PhasesAlong the Open Rovinj [HIRC, 1994] and for the port of Venezia-Lido [IIM, Boundary•' 1994].It followsfrom Table 2 that while the model amplitudes H, cm g, deg differ from their respectiveobserved values by <1 cm for the majorityof portsand constituents,there are a few outliers(K2 a. a• a2 bo bl b2 b3 in Venezia-Lido,Kl in Malamocco,and M 2 in Porto Corsini). M 2 12.79 -9.2 9.4 311.0 4.0 -136.0 80 These errors, nonetheless,fall below 2.2 cm. The majority of K 2 1.81 -1.4 1.6 313.0 -10.0 -104.0 66 phasedifferences between model and observationsis well be- N 2 2.20 - 1.0 0.9 305.0 - 5.0 -67.0 33 low 10ø, while the worst results are obtained for Venezia-Lido S2 6.83 -5.4 5.9 313.0 2.5 -110.5 62 (10.8ø error for K2 and up to 21.9ø for Pl). Kl 15.4 -1.2 0.5 84.0 -41.4 66.1 -40 The performanceof the 2-D modelis summarizedon Table 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 3. For each tidal constituentthe averageamplitude difference is <1 cm, the averagevectorial difference is <1.3 cm, and the aThereis a set of coefficientsfor everytidal constituent. MALA•I• ET AL.: TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA 26,269

Table 2. ComparisonBetween Observations and Model Resultsof ElevationAmplitudes H and Phasesg at Five Tidal Stationsin the Northern Adriatica 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 aSuperscriptso and m refer to observedand modelvalues, respectively. The quantityd is the vectorial difference. open boundaryprobably causes an artificial reflectionof the of the open boundarycondition by minimizingan objective diurnal tidal wave back into the domain. Thus the nature of the functional[Shulman and Lewis,1995], but thisfalls beyond the open boundarycondition needs reconsideration,at least for scopeof this paper. the diurnal frequencies.A possibleremedy is the optimization The rotarycoefficient Ca = +-(1 - e) of the ellipsesdrawn by the M 2 and K 1 velocityvectors over their respectivecycles was calculatedat every fifth grid point. Here e is the ellipse Table 3. Average and StandardDeviation of the Absolute eccentricity[Pugh, 1987], and a positive sign is assignedfor Difference Between Observed and 2-D Model Values of clockwiserotation. Thus, for pure rectilinear motion, e = 1 Amplitude AH, Vectorial Difference Ad, and PhaseLag and Ca = 0, while for pure counterclockwiserotation, e = 0 A# for All SevenTidal Constituentsat Five Stations and Ca = -1. This definition agreeswith that of Gonella ZIH, cm zid, cm zig, deg [1972].Over the northernAdriatic the M 2 tidal current rotates M 2 0.5 1.2 2.5 counterclockwise(Figure 6), with the senseof rotation being STD 0.8 1.1 2.6 reversedlocally along the eastern coast, in bays and around K• 0.5 0.7 5.7 capes,especially inside the Gulf of Trieste. For this gulf our STD 0.6 0.7 3.0 resultsmatch almost perfectly the observationsreported by N 2 0.2 0.5 6.5 STD 0.1 0.1 1.9 Mosettiand Purga [1990], thus leadingsupport to our calcula- S2 0.7 1.1 3.1 tions.In the central part of the northernAdriatic the M 2 tidal STD 0.3 0.6 2.9 ellipsesare elongatedand alignedwith the channelaxis. Along K1 0.8 1.0 1.8 this channel axis, from Venice to the middle of the open STD 0.5 0.6 1.2 boundary,our along-channelvariation of Ca for the M 2 con- P1 0.4 0.8 6.8 STD 0.4 0.8 8.1 stituentare consistentwith the variationof e = 1 - Icl O1 0.2 0.4 3.7 derivedanalytically by Mosetti[1986] and numericallyby Cav- STD 0.1 0.1 1.2 allini [1985].The K 1 ellipses,too, are stronglyaligned with the All 0.5 0.8 4.3 STD 0.5 0.7 4.1 channel axis-in the central part of the basin. A noteworthy feature is the reversal of the senseof rotation along a line 26,270 MALA•I• ET AL.: TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA

240-

220 -

200-

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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) andcotidal lines (long-dashed and dotted lines,in degrees)according to the 2-D model.•s numbersare grid indices.

stretchingfrom west to east at midbasin,with clockwiserota- surfaceelevation is greater on the shorewardside of the cur- tion to the south and counterclockwise rotation to the north. rents, as one would expectfrom a coastalKelvin wave. Figure 7 showsthe instantaneousflow field and elevation Figure8 showshow the 2-D modelperforms over time in the distribution at the time of maximum rate of elevation increase port of Trieste by comparingthe surfaceelevation time series in the port of Trieste,with all seventidal constituentsincluded. (Figure8a) andfrequency spectrum (Figure 8b) with all seven This time is nearly (within an hour) the time of maximum tidal constituentsincluded in the model. The agreementis inflow in Trieste. Note the rightward intensificationof the found to be excellent.(Minor peakscontained in the model currents and surface elevation along the Croatian/Slovenian spectrumhave no correspondentin the observedspectrum coast,as well as the local intensificationof the velocityin the because the energy of those peaks is so low that it falls vicinityof the Po River mouth (Cape Maestra,see Figure 2) within the instrumentalerror and was neglectedin the data and around the cape marking the entrance of the Gulf of analysis.) Trieste. There currents exceed 20 cm s-•. Note also that the In every fifth cell of the domain the time series of the

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t40-

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Figure 5. K• tidal elevations(solid and short-dashedlines, in cm) andcotidal lines (long-dashed and dotted lines,in degrees)according to the 2-D model.Axis numbersare grid indices. MALA•I• ET AL.:TIDAL DYNAMICSIN THE NORTHERNADRIATIC SEA 26,271

240

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Figure 6. Rotary coefficientC/• of current ellipsesof (top) M 2 and (bottom) K• tidal constituents.The coefficientis positivefor clockwiserotation, zero for rectilinearoscillatory motion, and negativefor counterclockwise rotation. depth-averagedspeed (4560 hourlyvalues) was Fourier trans- to the irregularitiesof the bottom topographyand coastline formed,and the verylow frequencyenergy was extracted (Fig- configurationoffshore of Venice (see Figure 2). Like sharp ure 9). Consideringthis low-frequencycomponent to be the cornersalong the coastline,sharp submarine irregularities, too, tidally rectified flow, we find that the tidally rectified currents are responsiblefor large velocity gradients,which create tidal in the northernAdriatic are quite weak, being <1 cm s-• residuals.(The larger values in the southwesterncorner are almosteverywhere, except near the coast,where their magni- suspectbecause of their relation to the problemwith the open tudesreach 3 cms-•. Thesehigher values appear to berelated boundarycondition in that area.) 26,272 MALA•I• ET AL.: TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA

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290 300 310 320 330 340 350 360 Figure 7. Tidal currents(arrows) and surfaceelevations (solid contours) of all seventidal constituents combinedat the time markinghalfway between the lowestand next highestelevation in Trieste(i.e., approximatelyflood time): (a) entirebasin and (b) enlargedview of the Gulf of Trieste. MALA(2I• ET AL.: TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA 26,273

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10'" - i LJ,.,.,I ,

_ 10" -' 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 Ir., 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. Comparisonof the surfaceelevation between 2-D model resultsand observationsin the port of Trieste:(a) sampleof the 6 monthtime series(open circles are observations,heavy dots are modelresults, and the thin line is the difference)and (b) frequencyspectrum of the 'entire 6 month record (thick line is observations,dots are modelresults, and thin line is the spectrumof the difference).

4. Check on the Open Boundary Conditions line lies well within the integration domain, whereas it is the open boundaryof the 2-D model, and the nature of the open As a final check on the model results, we compare the boundary condition does not allow for optimization of the optimizedopen boundaryconditions (derived in section2.3) velocity. with the numericalpredictions of a larger model. This model [Lynchet al., 1996;Naimie, 1996] is three-dimensional,has a finite element mesh, and uses a turbulence closure scheme [MeNorand Yamada,1982]. It is appliedto the entire Adriatic 5. Interpretation and Discussion Sea, with a resolutionvarying from 16 to 2 km. Along the In order to gain additionalinsight into the nature of the tides Pesaro-Kamenjakline the model has 50 triangular elements, in the northernAdriatic, we now interpretthe dynamicsof the with a side length of 2 km near each coast, 4 km farther M2 and K• tides. (The other semidiurnaland diurnal constit- offshore,and 8 km in the centralpart of the channel(Figure uentsdo not require separateinterpretations, for their struc- 10). The amplitudesand phasesof the surfaceelevation and tures are very similarto thoseof the M2 and K• tides,respec- velocityare easilyobtained from the proximatenodal values tively.) The relativelysimple structures of the amplitudeand usingthe linearbasic functions used inside every finite element. phase profiles of the surface elevation and depth-averaged Figure 11 comparesthe amplitudesand phasesof the sea velocitiesacross the basinat the Pesaro-Kamenjakline (hence- surfaceelevations of the sevenmajor tidal constituentscom- forth P-K line) suggestsa straightforwardexplanation, such as puted by the two models.The agreementis satisfactoryand the superpositionof a few linear waves.Although the M2 tide thereforevalidating the open boundaryconditions of our first has been explainedas the superpositionof a pair of incident model. There are, nonetheless, some differences. The finite and reflectedKelvin waves[Hendershott and Speranza,1971; elementmodel predictsslightly lower values for the M2 and S2 Mosetti,1986], no dynamicalinterpretation has yet been pro- tidal amplitudesand smootherphase profiles for the diurnal posedfor the K• tide. Here we shall not only clarify the dy- constituents.These differences are not surprisingsince the 3-D namics of both tides but also show that the semidiurnal and model has coatset resolutionthan the 2-D model (>-2 km diurnal tidesare two manifestationsof a singlefamily of waves, versus556 m). which existunder the combinedactions of gravityand topog- While we think that the surfaceelevations produced by the raphy.In the semidiurnalcase, gravity dominates, and the M2 2-D model are superiorto thoseof the 3-D model (becauseof tide takeson aspectsof a setof Kelvinwaves propagating along muchfiner horizontalresolution), we alsobelieve that the 3-D the basin,while topographydominates in the diurnalcase, and depth-averaged velocity predictions along the Pesaro- the K• tide resemblesa continentalshelf wave propagating Kamenjakline are more reliable.Indeed, in the 3-D model this across the basin. 26,274 MALA•I• ET AL.: TIDAL DYNAMICS IN THE NORTHERNADRIATIC SEA

Residual current (cm/s)

24O N

220

200

180

160

140

Figure9. Magnitudeofthe depth-averaged velocity (in cm s -•) in thevery low frequency band (<0.8570 cpd).

5.1. Topography-GravityWaves typeexp(-itot). Eliminationof v between(8) and(10) then The followingmathematical developments are not meantas yieldsa singleequation for they structureof a theorybut rather as a setof argumentspresented to provide a certain intuition about the dynamicalnature of somewave oy z> motions. We then infer that these wave motions are the mech- anisms behind the diurnal and semidiurnal tides in the Adriatic Sea.Consider the linear,barotropic, frictionless equations of motion on an f plane, over a slopingbottom, and in the ab- senceof velocityalong isobaths:

--ot + (D,) = 0, (8)

-fu= -# Ox' (9)

Ot- -g Oy' wherethe water depth D (y) variesin onlyone direction, which is meantto capturethe generalshoaling of theAdriatic along its main axis from the South Adriatic Pit to the Venice-Trieste coastline.Thus the x axispoints across the basin,and the y axis pointsalong the basin.As we considerflow fields deprived of cross-basinvelocity (u = 0), we ignorethe effect of lateral boundaries.Because our interestlies in forcedoscillatory mo- Figure 10. Superpositionof the Pesaro-Kamenjakopen tionsat specifiedfrequencies, we takethe timedependency of boundary line of the northern Adriatic model on the local thesurface elevation •/and cross-isobathvelocity v to be of the triangulationof the 3-D finite element model. MALA•I• ET AL.' TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA 26,275

15 360

340

320'

300

N2, K2 280

M2 26O øo 0:2 0:4 1 0 0.2 0.4 0.6 0.8 1

16 11o

K1 lOO

12 90 K1 o10 • 80• 8 70. P1 K1,P1

6 P1,01 60

4' i i i 5O 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 x/L •L Figure 11. Profilesof the (left) surfaceelevation amplitude and (right) phaseof the (top) semidiurnaland (bottom) diurnal constituentsalong the Pesaro-Kamenjakline. Thick lines with larger symbolsare the optimizedopen boundaryconditions derived for the 2-D model, and thin lineswith smallersymbols are the resultsof the 3-D model coveringthe entire sea.

If we assumethat the topographyvaries slowly over y (admit- the distancealong the main axisof the entire sea,Figure 12). tedly not the case for the Adriatic Sea but, nonetheless,a Then the expressionunder the squareroot becomes fruitful assumptionto elucidatesome dynamics), then a solution of the form exp[rr(y)] with rr beinga slowfunction of its a 2 -- variabley can be sought.We find (1 - ay)2' 0.)2 wherethe coefficient a 2 - ro2/gDois equalto -2.38 x 10-12 Do"2 q-D'rr' + = 0, (12) m-2 forthe M 2 tide(to -- 1.4110 -4 S-1) andequal to + 1.95x 10-13 m -2 for theK 1 tide(to = 7.2910 -s s-l). Thuswe seea where rr' standsfor drr/dy and can be consideredas the in- reversalin signbetween semidiurnal and diurnal tides, imply- verseof an e-foldinglength in they directionor a wavenumber ing that the semidiurnaltides have an oscillatory(and there- if it happensto be imaginary.Likewise, D' is dD/dy, the fore propagating)character, while the diurnaltides only have topographicslope. The assumptionof a slowlyvarying function a gradualamplitude variation from deep to shallow.Returning has permitted us to ignore a term containingthe secondde- to the termsthat make the quantityunder the squareroot, with rivative of rr. The solution is the first term dependingon the bottom slopeand the second term dependingon surfacevariability (via gravity),we con- rr' = D' + /D'2 002 (13) clude that the semidiurnaltides are essentiallysurface gravity 5-b- ¾ gO' waveswith a topographicdistortion, while the diurnaltides are We note that this expressionalways contains a real part topographicwaves modified by surfacevariability. In the limit -D'/2D, which implies a growth of the amplitude toward of no bottom slope (a - 0) the semidiurnaltide is a pure shallowwater. Integration of this componentover y yields surfaceKelvin wave,while in the limit of the rigid lid approx- growththat is inverselyproportional to the squareroot of the imation(g -• oo)the diurnaltide is a pure topographicwave. depth,i.e., a factor 7 over a depth changefrom 1000 to 20 m. Equation(10), which givesthe across-isobathvelocity, Amplification of the tidal elevation is indeed noted in the ig oil Adriatic for all tides, includingsemidiurnal and diurnal constituents.The remainingpart of rr', however,may be either roOy (14) real or imaginary,leading to additionalamplitude growth (or i g o" attenuation)or to wavebehavior in the cross-isobathdirection, respectively. To illustrate this possibledichotomy, let us take a constant revealsthat the propagatingcomponent of r/(with the imagi- Coriolisparameterf = 1.03 x 10-4 S-1 (characteristicof the nary part of rr') in a semidiurnaltide has a componentv that Adriatic)and a parabolictopography D(y) = Do(1 - ay)2 is in phase(both are real or both are imaginary),while the with valuesD O - 571 m anda - 1/935 km (obtainedby least nonpropagatingr/(with rr' real) of a diurnal tide has an ac- squaresfitting to the cross-basindepth average as a functionof companyingv that is in exactquadrature (one is real while the 26,276 MALA0•I0•ET AL.' TIDAL DYNAMICS IN THE NORTHERNADRIATIC 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. Parabolicfit to the depthprofile of the Adriatic Sea alongits main axis,from Otranto Strait to the northwesternshoreline. For eachposition along the main axisthe depthvalue is the averagedepth across the basin, from the southwestshore to the northeast coast. other is imaginary).Then insertingthe valueof v from (9), we nents(Figure 13, top right), beingabout 90 ø in the centerand get varyingantisymmetrically on both sides.As AppendixA shows, this is revealingof an incident-reflectedstanding wave pattern. This leadsus to investigateto which extent a simpleset of f ( i#rr'•-rl ) =# OrlOx' (15)two, incidentand reflected,Kelvin wavescan explainthe struc- Physically,the structure in the x direction has the opposite ture of the semidiurnaltides. Approximating the northernAd- characterof that in the y direction:When one is propagating, riatic Sea from the Pesaro-Kamenjakline inward as a rectan- the other is not. Thus, for semidiurnaltides the gravitational gulargulf with uniformdepth and consideringall 222 depthsat componentis propagatingin y and trappedin x, while for the model nodesalong the P-K line, we estimatethe meanwidth diurnal tides the wave propagatesin x but is attenuatedin y. L = 137 km, the averagedepthD = 46.4 m, and the Coriolis We now turn to the numericalresults and explorethe extent parameterf = 1.03 x 10-4 S-1, whichyield a radiusof to which these may conform to the precedingremarks. We deformationR -- 207 km and an aspectratio R/L = 1.51. chooseto perform the analysison the finite elementresults Accordingto (A2a) the squareof the elevationamplitude can only becauseof the superiorityof its depth-averagedvelocity be expressedas predictionsalong the Pesaro-Kamenjaksection (P-K line). H2(x) = C] + C2e2'c/R + C3e -z•/R, (16) These are displayedon Figure 13. The amplitudesof the ve- --1 locitycomponent parallel to the P-K line are below 1.5 cm s which is linear in its coefficients.A least squaresfit between for all constituents,indicating that the tidal flow in the across- the precedingexpression and the data (Figure 11, top left) channel direction is weak. Therefore we can limit ourselves to yields estimatesof the coefficientsC• to C3. Then the incom- explainingthe tidal flow in the along-channeldirection only. A ing and outgoingwave amplitudes,A o and A1, can be calcu- noticeablefeature of the along-channelvelocity profiles seen lated, as can the phase2ky + ok,from on Figure 13 is that the amplitudesare significantlyhigher on the right (eastern)side. This left-right asymmetrymay be at- C1 cos (2/cy + = tributable to the difference in bottom topographybetween 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 reflectedwave returningfrom the north alongItaly. The results are reported in Table 4 for each semidiurnal 5.2. Semidiurnal Tides constituent.These showthat the incomingwave has for each The profilesof phasedifferences between computed eleva- constituenta slightlyhigher amplitude than the outgoingwave tion and velocityare very similar for all semidiurnalcompo- (,4• > .40). We can then use theseestimates to reconstruct MALA•I(2 ET AL.: TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA 26,277

14o

12o•

100

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

140 : 12o lOO

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. Structureof the depth-averagedtidal velocitiesalong the Pesaro-Kamenjak(P-K) line: (top) semidiurnaltides, (bottom) diurnal tides, (left) amplitudes,and (right) phasedifferences with surfaceelevations.Thick linesare along-channelvelocity components; thin lines are across-channelvelocity components (samesymbols as for Figure 11). the structuresof surfaceelevation, normal velocity, and phase incidentand reflectedwaves have equal and oppositephases. differencealong the P-K line and comparethem to the numer- This explanation,which is not new, confirmsthe previouscon- ical results(Figure 14). The commonfeatures indicate that the jecturesof Hendershottand Speranza[1971] and Mosetti [1986]. semidiurnaltides in the northernAdriatic are primarily the result of a superpositionof an incomingKelvin wave with its 5.3. Diurnal Tides partial reflection.The good fit of elevationamplitudes is a When the sameprocedure is appliedto diurnaltides, it fails direct resultof the leastsquares fit, but the velocityamplitude becauseno pair of exponentialcurves can be fitted to both and phasedifference provide an independentcheck. Consid- endswithout creatingan unrealisticnegative value in the mid- ering that the northernAdriatic is not closeto being a rectan- dle. The conclusion must be that diurnal tides do not consist of gularbasin with a flat bottomand that the bottomslope ought Kelvin waves.The precedingarguments, indeed, show that we to affectthe waveproperties as noted in (13), the agreementis shouldexpect not a gravity-typewave but a topographicwave better than what couldhave been expected.This indicatesthat propagatingacross the basin (althoughthe distanceis rather the semidiurnaltides in the northernAdriatic primarily consist short!)with an amplitudeamplification from deep to shallow. in an incoming Kelvin wave progressingalong the eastern For the fitted parabolicprofile (Figure 12) the amplification coast,turning with the coastline(as a set of Poincardwaves) coefficient rr' takes the form andreturning in an attenuatedform alongthe Italian (western) coast.Such behavior also explains the amphidromicpoint ob- +a - x/a2- w2/gDo servedfarther south [Polli, 1959] as the locationwhere both or'- 1- ay (18) (The signin front of the squareroot was chosento yield the smallestabsolute value, correspondingto the wave with the Table 4. Amplitude of IncomingWave (A •), Amplitude of least amplificationfrom southto north, i.e., the wavewith the OutgoingWave (Ao), and PhaseDifference 2ky + d) leastenergy.) Integration over the directionof the main axisof Determined From a Least SquaresFit of the Pesaro- the Adriatic yields Kamenjak Results to the Two-Kelvin-WaveTheory a A1, Ao, 2ky + ½, or(y) = cr'(y dy' Constituent cm cm deg X2 f0y ') 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 = 1- 1 gD0a2 In 1-ay ' S2 7.0 6.5 - 117.5 0.136 from which we deducethe amplification/attenuationfactor aThe )(2values express the "goodnessof the fit" accordingto 1,2 statistics.Note that the outgoingwave is systematicallyweaker than the incomingwave. exp[o-(y)] = (1) 1- ay •-¾•-"'2/gøøa2. (19) 26,278 MALA(2I•ET AL.:TIDAL DYNAMICS IN THENORTHERN ADRIATIC SEA

10{

S2 o 6=

4 N2, K2

......

0 0.5 0 0.5 1 0 0.5 1 Figure14. Comparisonofthe (left) surface elevation amplitude, (middle) normal velocity amplitude, and (right)phase difference between thenumerical results (thin lines) and the fit of the two-Kelvin-wave theory (solidlines).

For thevalues quoted above (f = 1.03 x 10-4/S,ro -- 7.29x they yielded arms error of all surfaceelevations in the five 10-S/s,DO = 571m, a = 1/935km, and L = 137km), this portssmaller than 1 cm (0.5 _+0.5 cm).Similar results were factorequals 3.1 over a distanceof 800km (thelength of the obtained for the vectorial difference between modeled and entireAdriatic) and 1.1 over a distanceof 144km (the length observedcomplex values (combinations of amplitudesand of theportion retained for this model). These values are only phases):0.8 _+ 0.7 cm. The phase errors generally fell below 5ø slightlysmaller than the observednorth-south amplification (4.4 ø _+4.2ø). factorsobserved for the K• tide([Polli, 1959] and Figure 5). Themodel shows that the surface elevation isalways higher We canfurther examine whether the topography-wavear- on the rightside of theflow, indicating that both the northern gumentpredicts a phaseshift across the basin that agrees with Adriatic and the Gulf of Trieste behavelike narrow channels theobserved value. Seeking a solution of thetype exp[- ikx] [Gill,1982], in which the velocity component along the channel (withk realpositive to correspondto a phasethat decreases is significantlystronger than the cross-channel velocity and is fromleft to right,from Italy to theopposite shore), (15) yields subjectto the Coriolisforce. At timesof highinflow/outflow the isolinesof surfaceelevation are nearlyaligned with the frr' f a - x/a2 - ro2/gDo k .... . (20) depth-averagedvelocity. As such, the Gulf of Triestemay be ro ro 1 - ay consideredas a miniature of the northern Adriatic Sea. Thephase shift across the basin, i.e., over the distanceL, is The surfaceelevations and along-channelvelocities across the openboundary (line from Pesaro to Kamenjak)of the fL a - x/a• - •o•/gDo present2-D modelcompare well with the similarquantities kL = ro 1 - ay . (21) obtainedwith a largerand 3-D model.The analysisof these cross-channelprofiles then led to the followingthree results: For thevalues quoted above the predicted phase drop from (1) the M 2 and other semidiurnaltides can be understoodas Italy to Croatiaat y - 640 km (aboutthe locationof the havingbeen formed by a standingset of incidentand reflected Pesaro-Kamenjakline) is 0.39 rad = 22ø. In comparison,Fig- Kelvinwaves, (2) thenorthward amplification of theseKelvin ure 11 revealsa phasedifference of 15ø-20ø, in the samedi- wavesis causedby theshoaling bottom, and (3) theKz and rection.Appendix B providesa more precise comparison by otherdiurnal tides can be understoodas topographic waves usinga theoreticalframework slightly more rigorous than the propagatingacross the basin with the shallowwater on their basicarguments proposed at thebeginning of thissection and right,namely, from the Croatian coast to theItalian shore, and alsoby comparingthe along-basinvelocity magnitudes. The subjectto attenuationfrom shallow to deep.While conclusion conclusionremains the same:The Kz tideand all otherdiurnal 1 is a confirmationof earlierpropositions [Hendershott and tidescan be explained as topographic waves progressing from Speranza,1971; Mosetti, 1986], conclusions 2 and 3 arenew. In the northeastto the southwest,from the Croatiancoast to the Italian shore. particular,no dynamicalinterpretation of the diurnaltides in theAdriatic had been proposed prior to thisstudy.

6. Conclusions Appendix A: Kelvin Waves Until this study,Adriatic Sea tideshad not been simulated in a Flat Bottom Channel bymeans of nonlinearnumerical models, except in thecontext Considertwo oppositely traveling barotropic Kelvin waves in of a tidalanalysis of theentire Mediterranean Sea [Tsimplis et a channelextending along the y axis,of constantwidth L and al., 1995],which made no specificmention of theparticular of uniformdepth D. The surfaceelevation •l(x, y, t) and structureof thetides in thenorthern Adriatic. The objectives longitudinal velocity v(x, y, t) canbe writtenas of the presentstudy were the accurate2-D simulationand dynamicalanalysis of the tides within the subdomainsur- •q= A oe-x/R cos (ky + rot)+ A ,e(x-L)/R cos (ky - rot+ rk) roundedby fiveports (Rovinj, Trieste, Venezia-Lido, Malam- occo,and PortoCorsini) and extendingslightly to the south (Ala) (linejoining Pesaro tothe southern tipof the Istrian Peninsula). g -x/R The model'sopen boundary conditions were calibrated so as u= •-• [-Aoe cos(ky + rot) to obtainan optimumfit with the knowntidal elevationsin the fiveports. The simulationresults were found successful, for + A he(x-L)/• cos (ky - rot+ rk)], (Alb) MALA•I• ET AL.' TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA 26,279 whereR = V'#D/f = to/fk is the externalradius of deforma- i ( togH' - f gkH) tion, # is the gravitationalacceleration, f is the (constant) V= f2_to2 , (B3) Coriolisparameter, k is the longitudinalwavenumber, and tois the angularfrequency. One wave reachesits largestamplitude .... H: 0 (B4) (.40) alongthe coastx -- 0, while the other reachesits largest (DH')(k2D+fkD to +f2-to2) g ' amplitude (A•) along the oppositewall x = L. There is a phasedifference (k betweenthe two waves.If we combinethe where a prime indicatesa derivativewith respectto y. two waves into a single oscillatoryfield beating at the fre- Equation(B4) is difficultto solveexactly for a depthprofile quencyto, namely, r• = H(x, y) cos[tot - (kr•(x, y)] and v = D(y) givena priori, evenas a simpleanalytical function. Thus, V(x, y) cos [tot - qbr.(x,y)], we obtain insteadof constructinga D(y) topographyprofile and solving for H(y), let us anticipatea solutionH(y) that has realistic H(x,y) featuresand seek the D(y) profile to which it corresponds. Then, if that topographyhas realisticfeatures, we acceptthe = x/Ao2e-2x/R+ A•2e 2(x-*)/R + 2AoA,e cos (2ty + q,), solution. (A2a) Observations[Polli, 1959] aswell as our presentsimulations (Figure5) reveal(1) that the amplitudeof the K• tide increases g smoothlyand graduallyalong the basinand (2) that the cross- V(x,y) = fR basinvelocity is very weak. Let us then adoptH(y) = .4 exp (sy), where s (>0) is an e-foldinglength scaleand u = 0. ßx/A•e -zv'• + A•2e2(x-L)/'•- 2AoA•e -•/'• cos (2ky + Accordingto (B2), there is a wavenumberk that guaranteesno cross-basin flow: (A2b) fs 2A 0,4le -•/• sin (2ky + 4)) k = --. (BS) tan((kn - (k•,)= Ao2e-zvR- A •2e2(X-6)/e ' (A2c) Equation (B4) becomes Note that if the incomingand reflectedwaves have the same 6O2 amplitude(.40 = .4 •), the phasedifference is sD' + s2D+- = 0, (B6) g sin ( 2ky + 4)) tan((k, - (kv)= sinh[(L - 2x)/R]' (A3) whichis satisfiedif D(y) is of the form 6O2 whichis equalto _+90ø at the middleof the channel(x = L/2) D (y) = D •e-sy 2, (B7) and varies antisymmetricallyon both sides. gs where D• and s are two adjustableparameters. If we sety - 0 at the P-K line, where the cross-basinaverage depth is Appendix B: Topographic Waves 46.4m, we haveD• - 46.4 m + to2/gs2.Then, if we impose in a Shoaling Channel a zero depth at the Venice-Trieste shoreline,which is 140 km The linear barotropic equationsgoverning waves in a chan- away,we obtain an equation for the constants: nel of variable depth can be written as 46.4 m + e-(140km)s __ (B8) Ou gs2' Ot fv =-# Ox' (Bla) The solutionwith the lowestabsolute value (yieldingthe least Ov energeticwave) is s = 1.87 x 10-6 m-•, andthe topographic profile compatiblewith H(y) = .4 exp (sy) that bestfits the OZ-+fu = -# Oy' (Bib) actualbottom topographyof the northernAdriatic is Orl Ou 0 D(y) = (201.6 m)exp [-(1.87 10-6/m)y]- (155.2 m). 0•-+ D •xx+ •yy(Dr) = 0, (Blc) (B9) where x is directed acrossthe channel(0 <- x <- L), y is directedalong the channel,f is the (constant)Coriolis param- Over the 140 km of basin length the e-folding scales yields eter, # is the gravitationalacceleration, and D(y) is the resting a waveamplification of exp(sy) = 1.30, i.e., correspondingto depth, whichwe take as a functionof y only. Had the surface a surfaceelevation amplitude increaseof 30% from the P-K elevationterm Or•/Otbeen ignored,the set of equationswould line to the northwesternshoreline. In comparison,the numer- havebeen that governingcontinental shelf waves [Gill, 1982,p. ical model (Figure 5) revealedan increasefrom 14.5 to 17.5 409]. In other words, we are consideringhere topographic cm, which is a 21% increase.Considering the radical assump- wavesmodified by the gravitationalinfluence of the SSE. tions of the theoreticalmodel, we find reasonableagreement. If we seek solutionshaving a given frequency to, as in the The cross-channelwavenumber k givenby (B5) is found to tidal problem,and havinga wave expressionin x, namely, [r•, be approximatelyequal to 2.6 x 10-6 m-•, whichyields a u, v](x, y, t) = [H, U, V](y) exp [-i(kx + tot)], the phasechange kL from Pesaroto Kamenjak(L = 137 km) of cross-channelampli.tudes H(y), U(y), and V(y) mustsatisfy about 21ø . This value agreeswith the phasedifferences determined from the numericalsimulations and shownin Figure 11 -f gH' + togkH (lower right). v = ?_ , (B2) Turningnow to the along-basinvelocity, we derivefrom (B3) 26,280 MALA0•I(2ET AL.: TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA that they structureof the v velocitycomponent is relatedto the Godin, G., and L. Trotti, Triestewater levels1952-1971: A studyof the amplitudeprofile H(y) by tide, mean level and seicheactivity, Misc. Spec.Publ. Fish.Mar. Serv. Can., 28, 24 pp., 1975. Gonella, J., A rotary-componentmethod for analysingmeteorological V(y) = -i --H(y). (B10) and oceanographicvector time series,Deep Sea Res., 19, 833-846, 1972. Hendershott, M. C., and A. Speranza, Co-oscillatingtides in long, Along the P-K line, where the K• amplitudeH is 14.5 cm, the narrow bays:The Taylor problemrevisited, Deep SeaRes., 18, 959- theorypredicts a K• velocitymagnitude of 3.6 cm s-•, which 980, 1971. agreesquite well with the values obtainedby the numerical HydrographicInstitute of the Republicof Croatia (HIRC), Tide Ta- simulations(Figure 13, lowerleft, top curve).Furthermore, the bles,Adriatic Sea-EastCoast, 110 pp., Split, Croatia, 1994. IstitutoIdrografico della Marina, (IIM), Tavoledi Marea, Mediterra- presenceof the -i factorin the expressionfor V indicatesthat neo-Mar Rosso e delle Correnti di Marea, Venezia-Stretto di the along-basinvelocity lags the sea surfaceelevation by 90ø , Messina,96 pp., Genova, Italy, 1994. whichis preciselythe value notedin the numericalsimulations Lynch, D. R., J. T. C. Ip, C. E. Naimie, and F. E. Werner, Compre- (Figure 13, lower right). hensivecoastal circulation model with applicationto the Gulf of In conclusion,the precedingtheory is validatedby favorable Maine, Cont. Shelf Res., 16, 875-906, 1996. McHugh, G. F., A numericalmodel of two tidal componentsin the comparisonswith numericalsimulation results (as well as ob- northernAdriatic Sea,Boll. Geofis.Teor. Appl.,Xvi, 322-331, 1974. servations),and sincethe theoryreduces to the classicalcon- Mellor, G. L., and T. Yamada, Developmentof a turbulenceclosure tinental shelfwave theory in the limit of no gravitationaleffects model for geophysicalfluid problems,Rev. Geophys.,20, 851-875, (rigid lid approximationobtained for # • •), the K• tide of 1982. Mosetti, F., Distribuzione delle maree nei mari italiani, Boll. Oceanol. the northernAdriatic is a topographicwave modified by grav- Teor.Appl., V, 65-72, 1987. itational effects. Note that the wave is evanescent in the down- Mosetti, F., and B. Manca, Le maree dell'Adriatico: Calcoli di nuove channel direction and propagatesin the cross-channeldirec- costantiarmoniche per alcuniporti, in Studiin onorede Giuseppina tion, from the Croatian coast to the Italian shore. Aliverti, pp. 163-177, Ist. Univ. Nav. di Napoli, Ist. di Meteorol. e Oceanogr.,Naples, Italy, 1972. Mosetti, F., and N. Purga, Courantsc6tiers de diff6rente origine dans Acknowledgments.Malafiifi was supportedby the Ministry of Sci- un petit golfe (Golfe de Trieste),Boll. Oceanol.Teor. Appl., VIII, ence and Technologyof Sloveniathrough grant Z1-7045-0105. The 51-62, 1990. OsservatorioGeofisico Sperimentale in Trieste(Italy) supportedViez- Mosetti, R., Determination of the current structure of the M 2 tidal zoli. Malafiifiand Cushman-Roisinalso acknowledge the supportof the componentin the northernAdriatic by applyingthe rotary analysis U.S. Office of Naval Research, through grant N00014-93-7-0391 to to the Taylor problem,Boll. Oceanol.Teor. Appl.,/V, 165-172, 1986. Dartmouth College.All three authorsare indebtedto ChristopherE. Naimie, C. E., Georges Bank residual circulation during weak and Naimie of Dartmouth Collegefor havingperformed the tidal calcula- strongstratification periods: Prognostic numerical model results,J. tionswith the finite elementmodel (to be publishedelsewhere) and Geophys.Res., 101, 6469-6486, 1996. extractedthe valuesalong the Pesaro-Kamenjaksection for the pur- Polli, S., La Propagazionedelle Maree nell'Adriatico,paper presented poseof the presentstudy. The Abdus Salam International Centre for at IX ConvegnoDell' AssociazioneGeofisica Italiana, Rome, 1959. Theoretical Physicssupported the participationof the authorsat the Pugh, D. T., Tides,Surges and Mean Sea-Level,472 pp., John Wiley, International Workshop on the Oceanographyof the Adriatic Sea, New York, 1987. where fruitful discussionstook place. Shulman,I., and J. K. Lewis, Optimizationapproach to the treatment of open-boundaryconditions, J. Phys. Oceanogr.,25, 1006-1011, 1995. References Stravisi,F., and S. Ferraro, Monthly and annual mean sea levels at Trieste 1890-1984,Boll. Oceanol.Teor. Appl., IV, 97-103, 1986. Casulli, V., Semi-implicit finite-difference methods for the two- Taylor, G. I., Tidal oscillationsin gulfs and rectangularbasins, Proc. dimensionalshallow water equations,J. Comput.Phys., 86, 56-74, London Math. Soc., 20, 193-204, 1921. 1990. Trotti, L., Nuove costantiarmoniche per i porti Adriatici e Ionici,Atti Cavallini, F., A three-dimensionalnumerical model of tidal circulation Accad. Ligure Sci.-Lett.,26, 43-56, 1969. in the northernAdriatic Sea,Boll. Oceanol.Teor. Appl., III, 205-218, Tsimplis,M. N., R. Proctor,and R. A. Flather, A two-dimensionaltidal 1985. model for the Mediterranean Sea, J. Geophys.Res., 100, 16,223- Cheng, R. T., V. Casulli, and J. W. Gartner, Tidal, Residual and 16,239, 1995. Intertidal Mudflat (TRIM) model and its applicationsto San Fran- ciscoBay, California, CoastalShelf Sci., 36, 235-280, 1993. B. Cushman-Roisin,Thayer School of Engineering,Dartmouth Col- Crisciani, F., S. Ferraro, and B. Patti, Do tidal harmonic constants lege,Hanover, NH 03755-8000.(Benoit. [email protected]) dependon mean sealevel? An investigationfor the Gulf of Trieste, V. Malafiifi, National Institute of Biology, Marine Station Piran, Nuovo Cimento Soc. Ital. Fis. C., 18, 15-18, 1995. Fornace 41, Piran 6330, Slovenia. Davis, J. C., Statisticsand Data Analysisin Geology,2nd ed., 654 pp. D. Viezzoli, OsservatorioGeofisico Sperimentale, Borgo Grotta Gi- John Wiley, New York, 1986. gante 42/c, Sgonico34010, Trieste, Italy. Defant, A., PhysicalOceanography, vol. 2, 729 pp., Pergamon,Tarry- town, N.Y., 1961. Ferraro, S., and M. Maselli, Golfo di Trieste, previsionidi marea per il 1996,Nova Thalassia,12, suppl.,47 pp., 1995. Gill, A. E., Atmosphere-OceanDynamics, 662 pp., Academic,San Di- (ReceivedFebruary 22, 1999;revised December 8, 1999; ego, Calif., 1982. acceptedApril 5, 2000.)