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
OsservatorioGeofisicoSperimentale,Villa Opicina, Italy
Benoit
Cushman-Roisin
Thayer Schoolof Engineering,Dartmouth College,Hanover, New Hampshire
Abstract. Tides in the northernAdriatic Sea are investigatedusingtwo distinctnumerical
models.First, a two-dimensional
(2-D) finite differencemodel is implementedwith very
high horizontalresolution(556 m) to simulatethe northernAdriatic. After calibrationof
open boundaryconditionsthe model givesvery satisfactory
results: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 elementmodelis appliedto the entire seain
order to providea better simulationof the tidal currentsin the vicinityof the open
boundaryof the first model.Resultsshowthat the northernAdriatic behaveslike a
narrowrotatingchannelin whichthe instantaneous
seasurfaceelevation(SSE) contours
are alignedwith the depth-averaged
velocityvectorsand in whichthe SSE is alwayshigher
to the right of the local current.Thesefeaturesemphasizethe rotationalcharacterthat
tides can exhibit in a relativelysmallbasin.Wave fitting to the current elevationstructure
showsthat semidiurnaltidal constituentsare well representedwith a systemof two
frictionlessKelvinwaves(incidentand reflected).In contrast,the diurnal constituents
are
bestdescribedas a topographicwavepropagatingacross,not along,the basin,from the
Croatian coastto the Italian shore.Despite this obviousdisparitythe semidiurnaland
diurnal tides can be understoodas distinctmembersof a singlefamily of linear waves,
which existunder the combinedactionsof gravityand topography.
extendingfrom Venice to Trieste),Hendershott
and Speranza
[1971]showedhow partial reflectioncausesa displacementof
Early studiesof the tides in the Adriatic Sea (Figure 1) the M 2 amphidromicpoint from the channelaxistoward the
beganin the nineteenthcentury(asreportedbyDefant[1961]), western(Italian) coast.Later, the Taylor approachwas again
and it haslong been knownthat only seventidal constituents, appliedto the northernAdriatic by Mosetti[1986],who then
four semidiurnaland three diurnal, make a significantcontri- successfully
comparedM 2 current amplitudesand phasesto
bution to the sea surface elevation (SSE). Defant [1961] observations.Thus at leastthe 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 highestamplitude in the northern Adriatic Sea. same cannot be said of the other tidal constituents.
Generally,Mediterraneantides are weak, with surfaceelevaEarly numericalmodelsof tidesin the northernAdriatic Sea
tionsnot exceeding1 rn [Tsimpliset al., 1995].The tide in the were limited by drivingthe model with only one or two connorthern Adriatic is of a mixed type, with the semidiurnal stituents (M 2 and K• [McHugh, 1974] and M 2 [Cavallini,
componentM 2 and diurnalcomponentK• havingcomparable 1985]).Cavallini[1985]furtherinvestigated
the ellipticmotion
amplitudes[Polli, 1959].
producedby the M 2 tide and the effect of differentboundary
Taylor[1921]proposeda theoryaccordingto whichtidesin conditionsalongthe open boundary.
a rectangulargulf (semiclosedchannel)are combinationsof
The purposeof the modelspresentedhere is to simulate
incident and reflected Kelvin and Poincar• waves superim- accuratelythe tidal motions in the northern Adriatic, with
posedin sucha waythat the normalvelocityvanishesalongall specialemphasison the Gulf of Triesteandthe arealeadingto
sides,includingthe end of the channel.A feature of the solu- it. First, the two-dimensionalTidal Residual Intertidal Mudflat
tion is the possibleexistenceof one or severalamphidromic (TRIM) modelof Chenget al. [1993]is selectedbecauseof its
points inside the gulf. For the Adriatic Sea, there is an am- suitabilityto thistypeof study;it wasshownto be successful
in
phidromicpointapproximately
twothirdsup the basinfor each simulatingtidal and residual currentsin San FranciscoBay
1.
Introduction
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 surfaceelevationswith observations.Next, a
three-dimensional
finite elementmodel is employedto obtain
Paper number 2000JC900123.
0148-0227/00/2000JC900123509.00
better tidal velocityprofilesalong the open boundaryof the
26,265
26,266
'
MALA•II2 ET AL.: TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA
I
'
12
I
13
'
I
'
14
I
15
'
I
'
16
I
'
17
I
18
'
I
19
Longitude (o)
46
46-
TRIESTE
45-
PESARO
Adriatic Sea is relativelycompact,with a singleappendix,the
Gulf of Trieste, at its northeasternmost
end and a wide opening on its southernside.
More than 6100 depth soundingsand over 3100 coastal
positionswere taken from maritime chartsand interpolated
using the Kriging procedure[Davis, 1986]. From these the
model topographywas generatedon a staggeredfinite differencegrid with a spatialresolutionof 0.3 nauticalmiles(about
556 m). Suchresolutionis deemedsufficientfor the studyof
localseasurfaceelevation(SSE)andEuleriandepth-averaged
velocitiesas it will be verified below from the velocity fields
near capesand insidethe Gulf of Trieste.
43-
2.2.
Model Equations
Tidal dynamicsare numerically simulated with the twodimensionalTRIM model [Chenget al., 1993],which is semiimplicit and therefore unconditionallystable. It solvesthe
depth-integratedcontinuityequation
Or/
O[(r/+ D)u]
oD=-
ox
O[(r/ + D)v]
-
oy
'
where (u, v) is the verticallyaveragedvelocity,r/is the SSE
abovethe mean level, and D(x, y) is the restingdepth, togetherwith the verticallyaveragedmomentumequations
14
18
19
du
Or/ g(r/ + D) Op
d•--fv= -g Ox
rx
ø - rx
2p0 Ox+ vI-IAU
+ Po(r/
+D)
Figure 1. Position of the model domain within the Adriatic
Sea. Numbers along the axes are degreesof longitudeand
latitude.
(2)
dv
Or/ g(r/+ D) Op
dt+fu= -g Oy
ry
ø -- Ty
290 Oy+ u/_/Au
+ P0(r/
+D)'
(3)
first model in order to interpret the dynamicalnature of the
dominanttides.The paper finallydiscusses
the physicalnature
of the tidal wavesby matchinganalyticalsolutionsto the numerical
results.
where d/dt -
O/Ot + uO/Ox + vO/Oy is the Lagrangian
derivative,
f = 1.04 x 10-4 S-1 is the Coriolisparameter,
p
is the verticallyaverageddensity,Pois a referencedensity,uu
is a horizontal
eddyviscosity,
(rx
ø, ry
ø) is the surface
wind
stress,and (rx, ry) is the frictionalbottomstress.Density
2.
Two-Dimensional
Numerical
Model
variations, horizontal diffusion of momentum, and wind stress
are set to zero in our tidal analysis.
2.1. Model Geometry
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:
nectingPesaroin Italy to Kamenjakat the southerntip of the
Istria Peninsulain Croatia. This line is 124 km long.The tidal
rx: poCou
Su2 + u2 ry: poCoux/u
2 + u2. (4)
stations nearest to the domain
corners are Pesaro and Pula.
Figure 2 alsoshowsthe bathymetry.The overallpictureis that
of a depth increasingalmost linearly with distancefrom the
Italian coast (left) for •30 km, beyondwhich the bottom is
nearlyflat. On the Croatiansidethe topographyexhibitssteep
jumps between trenchesand submarineridges and even islands,all within a few kilometersfrom the coast.The rugged
topographyin the vicinity of the Croatian coastcomplicates
tidal modeling sinceenhancedlocal variationsin bottom friction, wavereflection,andwaverefractionaffectthe amplitude
Becausethe drag coefficientdependson the water depth,we
take
6.13 x 10 -3
= (2- e
'
(5)
whereD is givenin meters.Thisgivesa valueof 2.55 x 10-3
at 2 m andof 1.53X 10-3 at >12 m.Thepreceding
expression
for the drag coefficientis in accordancewith the parameterization of the bottomfrictiondevelopedfor the San Francisco
and arrival time of the wave in the Gulf of Trieste.
Bayby Chenget al. [1993],whooptedfor 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 Ch6zycoefficientof hydraulics
the TRIM model was first applied [Chenget al., 1993], is the [Chenget al., 1993].
lengthof the openboundary.While the geometryof SanFranThe system(1)-(5), whichis solvedfor the unknownsu, v,
ciscoBay is a complicatedset of bays(San PabloBay, Central and r/, is nonlinear through advectionand bottom friction
Bay, and SouthBay), it is connectedto the oceanonlythrough terms.In our applicationthesenonlinearitiescouplethe varia narrow strait, the Golden Gate. In contrast, the northern
ous tidal constituentsand generateresidualcurrents.
MALA•I•
ET AL.: TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA
26,267
o
i
J•IALAMOCCO
50
i
i
i
i
i
VENEZIA
LIDO ]
N
240
220
2OO
180
160
PULA
I•M•N•AKI •
• , •
_
Figure 2. Seaportsandisobaths
in the arearetainedbythe 2-D model.Tidal dataare availablefrom Pesaro,
Porto Corsini,Malamocco,Venezia-Lido,Trieste,Rovinj, and Pula.The axesare in modelunits,with one unit
equalto 556 m (=0.3 nauticalmiles).The openboundaryof the model,extendingfrom Pesaroto Kamenjak,
is the x axis.
The semi-implicitstaggeredgrid methodof Casulli[1990]is
appliedto the system(1)-(3), reducingthem to a pentadiagonal systemof linear equationsfor the SSE valueson the grid
(for details,seeChenget al. [1993]).The matrixof the system
is positivedefiniteandcanbe solvedveryefficiently.Although
the codeis unconditionallystable,a time stepof 900 s is chosen
for accuracy.
2.3.
Model Calibration and Open Boundary Conditions
amplitude
=H
phase
=#
(6)
= a0+ a•• + a2
x
= b0+ b•• + b2
+ b3
,
(7)
wherex is the distancefrom Pesaro(x = 0 at Pesaroandx =
L at Kamenjak,L beingthe lengthof the openboundaryline).
Initially, the coefficientsa o to b3 are fitted to the chartsof Polli
[1959].The modelis thenrun from restuntil all transientshave
fallen belowthe level of numericalnoise(about 9 days).CalculatedSSEvaluesat the M 2 frequencyin Trieste, Rovinj, and
Venezia-Lidoare then comparedto the observations.
Because
the phasediscrepancy
is foundto be largerthan the amplitude
discrepancy,
the two end slopes( !7'(0) at Pesaroand17'(1) at
Kamenjak)of the cubicpolynomialfor the phaseprofile are
taken as adjustableparameters.The distributionof the amplitude andphasediscrepancies
betweencalculatedand observed
valuesas thoseparametersare varied is then examined.For
this,amplitudesandphasesare interpretedasvectors(or complexnumbers),andthe vectorial(complex)differencebetween
observationsand calculationsis taken. Figure 3 displaysthe
absolutevalue of this differenceas the two tuningparameters
The lengthof the tidal recordin the port of Trieste excqeds
100 years:Observationsof the water level beganin 1859, and
monthlyandannualmeansealevelvalueshavebeenpublished
since1905[Godinand Trotti,1975].Hourly data since1939are
alsoavailable[StravisiandFerraro,1986].Becausethisis one of
the longestSSE recordsin the MediterraneanSea, the constantsof the tidal constituents
are preciselyknownfor thisport
of the Adriatic Sea [Criscianiet al., 1995]. Next in order of
availabilityare the tidal recordsof Rovinj and Venezia-Lido.
The first stepin the calibrationof the modelis to seeka match
betweenmodel resultsand observationsat Trieste, Rovinj, and
Venezia-Lido[Hydrographic
Instituteof theRepublicof Croatia
(HIRC), 1994;IstitutoIdrografico
dellaMarina (IIM), 1994].
For this task we determinethe SSE valuesto be prescribed
alongthe openboundaryline connectingPesaroto Kamenjak are varied.
The plot revealsthat the error betweenobservedand calcufor eachseparatetidal constituent,startingwith M 2. The tidal
constants(amplitudeand phase) for each tidal constituent lated values reaches a minimum for a certain set of values of
along the open boundaryare taken as quadraticand cubic 17'(0) and 17'(1). These values are then adopted for the
boundaryconditionsin the remainingsimulations.There is no
polynomials:
26,268
MALA•I(• ET AL.: TIDAL DYNAMICSIN THE NORTHERNADRIATIC SEA
averagephase lag difference is <7.2 ø. When all seventidal
constituentsin all five ports are consideredtogether(35 values),the averageamplitudedifferenceis 0.5 cm, the average
vectorialdifferenceis 0.8 cm, and the averagephaselag difference
is 4.4 ø. We conclude
that the 2-D model was success-
fully calibratedand that it providesreliable values,allowingus
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•) constituents
over
the model domainare 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 amplituderisesover decreasingdepth,as one
might have expectedfrom the principleof wave actionconservation.
Figure 3. Distribution of the difference between M 2 tidal
observationsand 2-D calculationsat three stations(Trieste,
Rovinj,and Venezia-Lido)asa functionof the gradientof the
phaselag at both endsof the openboundaryline, #' (0) and
9'(1). Note the minimumnear 9'(0) = 4ø and 9'(1) :
-28
ø'
need to adjust the quadratic polynomialfor the amplitude
profile along the boundary.Finally, the entire procedureis
repeatedfor the six other tidal constituents.
Table 1 liststhe
optimizedcoefficientsobtainedfor both amplitudeand phase
profilesalongthe openboundaryline expressed
as(6) and(7).
3.
Two-Dimensional
Model
Results
saro and Porto
After calibrationthe model is spunup for 31 daysand then
run for 190 days(i.e., slightlymore 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
separationof the S2 and K2 frequenciesfrom the simulated
record demandsa time series of 182.6 days (the so-called
synodicperiod [Pugh,1987].So,the durationof our simulation
(190 days)is sufficient.The resultsare sampledhourly,andthe
tidal constituentsat five portsin the northernAdriatic (Porto
Corsini, Mallamocco, Venezia-Lido, Trieste, and Rovinj, see
Figure 2) are extractedfrom thesehourlytime SSE series.
Table 2 comparesthe amplitudesand phase lags of the
model
results with
the observed
values.
These
For eachconstituentthe phaselag generallyincreaseswestwardfrom the Croatiancoast(right-handsideof Figures4 and
5) to the Italian shore (left-handside of Figures4 and 5).
While the M 2 cotidal lines diverge,the K• cotidallinestend to
be more parallel;thistendencyis relatedto the fact that the M 2
tide hasan amphidromicpoint somewheresouthof the domain
(whereall cotidallinesgatherinto a singlepoint),whilethe K•
tide doesnot [Polli, 1959]. In the northeasterncorner of the
domainthe cotidal lines of both M 2 and K• tidesbend into the
Gulf of Trieste, where they divergeslightly.This is expected
sincethe flow must be parallel to the coastline,the semiminor
axisof the velocityellipsesmustbe small,and the cotidallines
mustconservetheir anglewith respectto the coastline[Pugh,
1987,p. 439].The remainingsemidiurnal(K2, N2, and S2) and
diurnal(P• and O•) constituentsare substantially
weaker but
reveal SSE amplitude and phase lag distributionssimilar to
those of the M 2 and K• constituents,respectively.
There existsa peculiarK• amplitudeminimumbetweenPeCorsini
in the southwestern
corner of the do-
main (seeFigure 2 for the geographicallocationof thesetwo
ports). This minimum locally distortsthe otherwisegradual
distributionof the K• amplitude. Becausethere is no hint of
such local minimum in the observations, we conclude that its
existenceis an artifact of the model, mostlikely attributableto
an imperfectopenboundarycondition.The sameproblemwas
also noted by McHugh [1974, Figure 7] for the same tidal
constituentin the sameregionof the samemodeldomain.This
consistency
in the locationof a K• amplitudeminimumand the
fact that both modelsrely on the sametype of boundarycondition lead us to conjecturethat the prescribedSSE along the
values were
taken from Polli [1959], Trotti [1969], Mosetti and Manca
[1972],Godinand Trotti[1975],Mosetti[1987],andFerraroand
Maselli [1995] as well as from official reportsfor the port of
Rovinj [HIRC, 1994] and for the port of Venezia-Lido [IIM,
1994].It followsfrom Table 2 that while the model amplitudes
differ from their respectiveobservedvaluesby <1 cm for the
majorityof portsandconstituents,
thereare 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
phasedifferencesbetweenmodel and observations
is well below 10ø, while the worst results are obtained for Venezia-Lido
(10.8ø error for K2 andup to 21.9øfor Pl).
The performanceof the 2-D modelis summarizedon Table
3. For each tidal constituentthe averageamplitude difference
is <1 cm, the averagevectorialdifferenceis <1.3 cm, and the
Table 1. Coefficientsof the Quadraticand CubicPolynomials,
(6)-(7), Fitted by the CalibrationProcedureto Prescribethe
Elevation Amplitudes and PhasesAlong the Open
Boundary•'
H, cm
M2
K2
N2
S2
Kl
Pl
O1
g, deg
a.
a•
a2
bo
bl
b2
b3
12.79
1.81
2.20
6.83
15.4
5.1
4.2
-9.2
-1.4
- 1.0
-5.4
-1.2
-0.3
0.7
9.4
1.6
0.9
5.9
0.5
0.1
-0.3
311.0
313.0
305.0
313.0
84.0
84.0
69.0
4.0
-10.0
- 5.0
2.5
-41.4
-37.1
-7.2
-136.0
-104.0
-67.0
-110.5
66.1
51.7
-4.8
80
66
33
62
-40
-29
3
aThereis a set of coefficientsfor everytidal constituent.
MALA•I•
ET AL.: TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA
26,269
Table 2. ComparisonBetweenObservationsand Model Resultsof ElevationAmplitudes
H and Phasesg at Five Tidal Stationsin the Northern Adriatica
Site
Rovinj
Trieste
Venezia-Lido
Malamocco
Porto Corsini
Constituent
H o,
cm
H m'
cm
H o _ H m'
cm
gO,
deg
gm,
deg
gO_ gm,
deg
d,
cm
d/H ø,
%
M2
K2
N2
S2
K•
P•
O1
M2
K2
N2
S2
K•
P1
O•
M2
K2
N2
S2
K1
P•
O•
M2
K2
N2
S2
K•
P•
O•
M2
K2
N2
S2
K•
P•
O•
19.3
3.0
3.5
11.2
16.1
5.3
4.9
26.7
4.3
4.5
16.0
18.2
6.0
5.4
23.4
5.3
3.8
13.8
16.0
4.3
5.2
23.5
4.0
4.1
14.0
18.3
5.8
5.3
15.6
2.5
3.1
9.2
15.9
5.3
5.0
19.3
2.9
3.2
10.7
16.0
5.3
4.8
26.6
4.0
4.3
15.0
17.3
5.7
5.2
23.6
3.5
3.9
13.1
16.8
5.5
5.1
23.3
3.5
3.8
13.0
16.7
5.5
5.0
17.6
2.5
2.9
9.4
15.3
5.0
4.7
0.0
-0.1
-0.3
-0.5
-0.1
0.0
-0.1
-0.1
-0.3
-0.2
- 1.0
-0.9
-0.3
-0.2
0.2
- 1.8
0.1
-0.7
0.8
1.2
-0.1
-0.2
-0.5
-0.3
-1.0
-1.6
-0.3
-0.3
2.0
0.0
-0.2
0.2
-0.6
-0.3
-0.3
270.0
277.0
266.0
277.0
71.0
71.0
56.0
277.5
286.1
274.9
286.1
71.1
71.1
61.1
288.0
281.0
299.0
293.0
79.0
56.0
70.0
296.0
299.0
295.0
305.0
82.0
70.0
65.0
303.0
310.0
295.0
310.0
81.0
81.0
67.0
270.6
274.8
273.4
278.1
70.4
71.0
61.2
278.8
283.0
280.9
286.5
73.1
73.7
63.6
287.7
291.8
289.3
295.4
77.4
77.9
67.7
288.7
292.8
290.3
296.5
77.9
78.4
68.2
300.1
303.9
299.6
306.9
81.9
81.9
72.1
0.6
-2.2
7.4
1.1
-0.6
0.0
5.2
1.3
-3.1
6.0
0.4
2.0
2.6
2.5
-0.3
10.8
-9.7
2.4
-1.6
21.9
-2.3
- 7.3
-6.2
-4.7
-8.5
-4.1
8.4
3.2
-2.9
-6.1
3.6
-3.1
0.9
0.9
5.1
0.2
0.2
0.5
0.6
0.2
0.1
0.4
0.6
0.3
0.5
1.0
1.1
0.4
0.3
0.2
1.9
0.6
0.9
0.9
2.2
0.3
3.0
0.6
0.4
2.3
2.0
0.9
0.4
2.2
0.3
0.3
0.6
0.7
0.3
0.5
1.0
5.5
15.5
5.0
1.4
0.9
9.1
2.3
8.0
11.0
6.3
5.9
6.9
6.0
0.9
36.6
17.1
6.3
5.7
51.2
4.9
12.8
16.1
10.6
16.1
10.9
15.2
7.3
14.0
10.8
9.7
6.0
4.2
5.9
10.8
aSuperscripts
o andm refer to observedand modelvalues,respectively.
The quantityd is the vectorial
difference.
of the open boundaryconditionby minimizingan objective
functional[ShulmanandLewis,1995],but thisfallsbeyondthe
open boundarycondition needsreconsideration,at least for scopeof this paper.
the diurnalfrequencies.A possibleremedyis the optimization
The rotarycoefficientCa = +-(1 - e) of the ellipsesdrawn
by the M 2 and K 1 velocityvectorsover 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 DifferenceAd, and PhaseLag
and Ca = 0, while for pure counterclockwise
rotation,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 currentrotates
counterclockwise
(Figure 6), with the senseof rotationbeing
M2
0.5
1.2
2.5
reversedlocally along the easterncoast,in bays and around
STD
0.8
1.1
2.6
K•
0.5
0.7
5.7
capes,especiallyinsidethe Gulf of Trieste. For this gulf our
STD
0.6
0.7
3.0
resultsmatch almost perfectly the observationsreported by
N2
0.2
0.5
6.5
Mosettiand Purga[1990], thusleadingsupportto our calculaSTD
0.1
0.1
1.9
tions.In the centralpart of the northernAdriatic the M 2 tidal
S2
0.7
1.1
3.1
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 conopen boundaryprobablycausesan artificial reflectionof the
diurnal tidal wave back into the domain. Thus the nature of the
P1
0.4
0.8
STD
0.4
0.8
6.8
8.1
O1
0.2
0.4
3.7
STD
All
STD
0.1
0.5
0.5
0.1
0.8
0.7
1.2
4.3
4.1
stituentare consistent
with the variationof e = 1 - Icl
derivedanalyticallyby Mosetti[1986] and numericallyby Cavallini [1985].The K 1 ellipses,too, are stronglyalignedwith the
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
240220 200-
180160 -
t40120-
tO0-
80-
x
6040- •
j
•[-••13
I "•'--t.
."..
'>4., •',,
'•"••••••<•.•:,•
20.
0
' I ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I
Figure 4. M 2 tidalelevations
(solidandshort-dashed
lines,in cm)andcotidallines(long-dashed
anddotted
lines,in degrees)accordingto the 2-D model.•s
numbersare grid indices.
stretchingfrom west to east at midbasin,with clockwiserota-
surfaceelevationis greater on the shorewardsideof the currents, as one would expectfrom a coastalKelvin wave.
Figure 7 showsthe instantaneousflow field and elevation
Figure8 showshowthe 2-D modelperformsovertimein the
distribution at the time of maximum rate of elevation increase
port of Triesteby comparingthe surfaceelevationtime series
in the port of Trieste,with all seventidal constituents
included. (Figure8a) andfrequencyspectrum(Figure8b) 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 peakscontainedin the model
currentsand surfaceelevation along the Croatian/Slovenian spectrumhave no correspondent
in the observedspectrum
coast,aswell as the local intensification
of the velocityin the because the energy of those peaks is so low that it falls
vicinityof the Po River mouth (Cape Maestra,seeFigure 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
tion to the south and counterclockwise
rotation
to the north.
240220 200-
180160 t40120tOO806040200
Figure 5. K• tidal elevations
(solidandshort-dashed
lines,in cm) andcotidallines(long-dashed
anddotted
lines,in degrees)accordingto the 2-D model.Axis numbersare grid indices.
MALA•I• ET AL.:TIDAL DYNAMICSIN THE NORTHERNADRIATICSEA
26,271
N
240
220
180
160
140
120
100
Figure 6. Rotary coefficientC/• of currentellipsesof (top) M 2 and (bottom) K• tidal constituents.
The
coefficientis positivefor clockwise
rotation,zerofor rectilinearoscillatory
motion,andnegativefor counterclockwise
rotation.
depth-averaged
speed(4560 hourlyvalues)wasFourier transformed,andthe verylow frequencyenergywasextracted(Figure 9). Consideringthis low-frequencycomponentto be the
tidally rectifiedflow, we find that the tidally rectifiedcurrents
in the northernAdriatic are quite weak, being <1 cm s-•
almosteverywhere,exceptnear the coast,where their magni-
to the irregularitiesof the bottom topographyand coastline
configurationoffshoreof Venice (see Figure 2). Like sharp
cornersalongthe coastline,sharpsubmarineirregularities,
too,
are responsiblefor large velocitygradients,which create tidal
residuals.(The larger values in the southwesterncorner are
suspectbecauseof their relationto the problemwith the open
tudesreach3 cms-•. Thesehighervalues
appeartoberelated boundaryconditionin that area.)
26,272
MALA•I•
ET AL.: TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA
(a)
[[]
selection
•>
predicted
ß
model
N
Time (hours)
97/04/22 18:00
cm/s
(b)
97/04/22
18:00
N
Time (h)
170
cm/s
I//•
......
50
\ "",•"•
• '"''x'•'•a,"•'•"*•"' /
160
130
120
290
300
310
320
330
340
350
360
Figure 7. Tidal currents(arrows)and surfaceelevations(solidcontours)of all seventidal constituents
combinedat the time markinghalfwaybetweenthe lowestand next highestelevationin Trieste(i.e.,
approximately
floodtime): (a) entirebasinand (b) enlargedviewof the Gulf of Trieste.
MALA(2I• ET AL.: TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA
26,273
i
1
0.4
0.2
•
0o0
-o.6 / .......................
2640
b
I .......................
2664
i .......................
2688
i,, ,¾...................
2712
i
2736
Time (hours)
10"
10'2
10"
i LJ,.,
.,I
10"10'"-
-'
' ""-: ""'"
_
10"
iiiiIiiii
0.5
1.0
1.5
ii ii
iiiijiiii
2.0
2.5
3.0
iiii
3.5
ii
,
i
4.0
ii
4.5
5.0
5.5
r.,
I
6.0
Frequency(cpd)
Figure 8. Comparisonof the surfaceelevationbetween2-D model resultsand observations
in the port of
Trieste:(a) sampleof the 6 monthtime series(opencirclesare observations,
heavydotsare modelresults,and
the thin line is the difference)and (b) frequencyspectrumof the 'entire 6 month record (thick line is
observations,
dotsare modelresults,and thin line is the spectrumof the difference).
4.
Check on the Open Boundary Conditions
As a final check on the model results,we compare the
optimizedopenboundaryconditions(derivedin section2.3)
with the numericalpredictionsof 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
Sea, with a resolutionvaryingfrom 16 to 2 km. Along the
Pesaro-Kamenjakline 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 centralpart of the channel(Figure
10). The amplitudesand phasesof the surfaceelevationand
velocityare easilyobtainedfrom the proximatenodal values
usingthe linearbasicfunctionsusedinsideeveryfinite element.
Figure 11 comparesthe amplitudesand phasesof the sea
surfaceelevationsof the sevenmajor tidal constituentscomputed by the two models.The agreementis satisfactoryand
thereforevalidatingthe openboundaryconditionsof our first
line lies well within the integration domain, whereasit is the
open boundaryof the 2-D model, and the nature of the open
boundarycondition does not allow for optimization of the
velocity.
5.
Interpretation and Discussion
In order to gainadditionalinsightinto the natureof the tides
in the northernAdriatic,we now interpretthe dynamicsof the
M2 and K• tides. (The other semidiurnaland diurnal constituentsdo not require separateinterpretations,for their structuresare very similarto thoseof the M2 and K• tides,respectively.) The relativelysimplestructuresof the amplitudeand
phase profiles of the surfaceelevation and depth-averaged
velocitiesacrossthe basinat the Pesaro-Kamenjak
line (henceforth P-K line) suggests
a straightforward
explanation,suchas
the superposition
of a few linear waves.Althoughthe M2 tide
has been explainedas the superpositionof a pair of incident
model. There are, nonetheless, some differences. The finite and reflectedKelvin waves[Hendershott
and Speranza,1971;
elementmodelpredictsslightlylowervaluesfor the M2 and S2 Mosetti,1986], no dynamicalinterpretationhasyet been protidal amplitudesand smootherphaseprofilesfor the diurnal posedfor the K• tide. Here we shallnot only clarify the dyconstituents.
Thesedifferencesare not surprisingsincethe 3-D namics of both tides but also show that the semidiurnal and
model has coatset resolutionthan the 2-D model (>-2 km diurnaltidesare two manifestationsof a singlefamily of waves,
versus556 m).
which existunder the combinedactionsof gravityand topogWhile we think that the surfaceelevationsproducedby the raphy.In the semidiurnalcase,gravitydominates,and the M2
2-D model are superiorto thoseof the 3-D model (becauseof tide takeson aspectsof a setof Kelvinwavespropagatingalong
muchfiner horizontalresolution),we alsobelievethat the 3-D the basin,while topographydominatesin 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 modelthis 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. Magnitude
ofthedepth-averaged
velocity
(incms-•) intheverylowfrequency
band(<0.8570
cpd).
5.1. Topography-GravityWaves
typeexp(-itot). Elimination
of v between(8) and(10) then
The followingmathematical
developments
are not meantas yieldsa singleequationfor they structureof
a theorybutratherasa setof arguments
presented
to provide
a certainintuition about the dynamicalnature of somewave
oy z>
motions. We then infer that these wave motions are the mechanisms behind the diurnal and semidiurnal tides in the Adriatic
Sea.Considerthe linear,barotropic,
frictionless
equations
of
motionon an f plane, over a slopingbottom,and in the absenceof velocityalongisobaths:
--
ot
+
(D,) = 0,
-fu= -# Ox'
(8)
(9)
Ot- -g Oy'
wherethewaterdepthD (y) variesin onlyonedirection,which
is meantto capturethegeneralshoaling
of theAdriaticalong
its main axis from the South Adriatic Pit to the Venice-Trieste
coastline.
Thusthex axispointsacross
thebasin,andthey axis
pointsalongthe basin.As we considerflowfieldsdeprivedof
cross-basin
velocity(u = 0), we ignorethe effectof lateral
boundaries.
Because
our interestliesin forcedoscillatory
mo- Figure 10. Superposition
of the Pesaro-Kamenjak
open
tionsat specified
frequencies,
we takethe timedependency
of boundary line of the northern Adriatic model on the local
thesurfaceelevation•/and cross-isobath
velocityv to beof the triangulationof the 3-D finite element model.
MALA•I• ET AL.' TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA
15
26,275
360
340
320'
300
280
N2, K2
M2
øo
26O
0:2 0:4
1
0
0.6
0.8
1
lOO
K1
90
12
• 80
•
o10
K1
70.
8
4'0
0.4
11o
16
6
0.2
P1,01
K1,
P1
60
i
0.2
P1
i
0.4
0.6
i
5O
0
0.8
0.2
0.4
0.6
0.8
1
•L
x/L
Figure 11. Profilesof the (left) surfaceelevationamplitudeand (right) phaseof the (top) semidiurnaland
(bottom) diurnal constituentsalong the Pesaro-Kamenjakline. Thick lines with larger symbolsare the
optimizedopen boundaryconditionsderivedfor the 2-D model, and thin lineswith smallersymbolsare the
resultsof the 3-D model coveringthe entire sea.
If we assumethat the topographyvariesslowlyovery (admit- the distancealongthe 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 elucidatesomedynamics),then a solution of the form exp[rr(y)] with rr beinga slowfunctionof its
a 2 -(1 - ay)2'
variabley can be sought.We find
0.)2
Do"2 q-D'rr' +
= 0,
(12)
wherethecoefficient
a2 - ro2/gDo
isequalto -2.38 x 10-12
m-2 fortheM2tide(to-- 1.4110-4 S-1) andequalto + 1.95x
10-13m-2 fortheK1tide(to= 7.2910-s s-l). Thusweseea
where rr' standsfor drr/dy and can be consideredas the inverseof an e-foldinglengthin they directionor a wavenumber
if it happensto be imaginary.Likewise,D' is dD/dy, the
topographicslope.The assumption
of a slowlyvaryingfunction
has permitted us to ignore a term containingthe secondde-
reversalin signbetweensemidiurnaland diurnal tides,implying that the semidiurnaltides have an oscillatory(and therefore propagating)character,while the diurnaltidesonly have
a gradualamplitudevariationfrom deepto shallow.Returning
to the termsthat makethe quantityunderthe squareroot,with
rivative of rr. The solution is
the first term dependingon the bottom slopeand the second
term dependingon surfacevariability(via gravity),we conD' + /D'2 00
2
rr' =
(13) clude that the semidiurnaltides are essentiallysurfacegravity
5-b- ¾
gO'
waveswith a topographicdistortion,while the diurnaltidesare
We note that this expressionalwayscontainsa real part topographicwavesmodifiedby 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 approxgrowththat is inverselyproportionalto the squareroot of the imation(g -• oo)the diurnaltide is a pure topographicwave.
Equation(10), whichgivesthe across-isobath
velocity,
depth,i.e., a factor 7 over a depthchangefrom 1000to 20 m.
Amplification of the tidal elevationis indeed noted in the
ig oil
Adriatic for all tides, includingsemidiurnaland diurnal constituents.The remainingpart of rr', however,may be either
real or imaginary,leadingto additionalamplitudegrowth(or
i g o"
attenuation)or to wavebehaviorin the cross-isobath
direction,
respectively.
To illustratethis possibledichotomy,let us take a constant revealsthat the propagatingcomponentof r/(with the imagiCoriolis
parameterf= 1.03 x 10-4 S-1 (characteristic
of the nary part of rr') in a semidiurnaltide hasa componentv that
Adriatic)anda parabolic
topography
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 nonpropagating
r/(with rr' real) of a diurnaltide has an acv that is in exactquadrature(one is real while the
squaresfittingto the cross-basin
depthaverageasa functionof companying
roOy
(14)
26,276
MALA0•I0•ET AL.' TIDAL DYNAMICSIN THE NORTHERNADRIATIC SEA
0
I
I
I
I
I
I
I
100
200
300
400
500
600
700
-lOO
-200
-3oo
-400
-5OO
Do
-600
-700
0
800
Figure 12. Parabolicfit to the depthprofile of the Adriatic Sea alongits main axis,from Otranto Strait to
the northwesternshoreline.For eachpositionalongthe main axisthe depthvalueis the averagedepthacross
the basin, from the southwestshore to the northeast coast.
nents(Figure 13, top right), beingabout90øin the centerand
varyingantisymmetrically
on both sides.As AppendixA shows,
thisis revealingof an incident-reflected
standingwavepattern.
This leadsus to investigateto which extenta simpleset of
f
•-rl =# Ox'
two, incidentand reflected,Kelvin wavescan explainthe strucPhysically,the structure in the x direction has the opposite ture of the semidiurnaltides.Approximatingthe northernAdcharacterof that in the y direction:When one is propagating, riatic Sea from the Pesaro-Kamenjakline inward as a rectanthe other is not. Thus, for semidiurnaltidesthe gravitational gulargulfwith uniformdepthand consideringall 222 depthsat
componentis propagatingin y and trappedin x, while for the model nodesalongthe P-K line, we estimatethe meanwidth
diurnal tidesthe wave propagatesin x but is attenuatedin y. L = 137 km, the averagedepthD = 46.4 m, and the Coriolis
f = 1.03 x 10-4 S-1, whichyield a radiusof
We now turn to the numericalresultsand explorethe extent parameter
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 elevationamplitudecan
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
locitycomponentparallelto the P-K line are below1.5 cm s
which is linear in its coefficients.A least squaresfit between
for all constituents,indicatingthat the tidal flow in the across- the precedingexpressionand the data (Figure 11, top left)
otheris imaginary).Then insertingthe valueof v from (9), we
get
( i#rr'
) Orl
channel direction
is weak. Therefore
(15)
we can limit ourselves to
yieldsestimatesof the coefficientsC• to C3. Then the incomexplainingthe tidal flow in the along-channel
directiononly.A ing and outgoingwave amplitudes,A o and A1, can be calcunoticeablefeature of the along-channelvelocityprofilesseen lated, as can the phase2ky + ok,from
on Figure 13 is that the amplitudesare significantly
higheron
C1
the right (eastern)side.This left-right asymmetrymay be atcos (2/cy +
=
tributable to the difference in bottom topographybetween
both sides(seeFigure 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 resultsare reported in Table 4 for each semidiurnal
5.2.
Semidiurnal
Tides
constituent.These showthat the incomingwave hasfor each
The profilesof phasedifferencesbetweencomputedeleva- constituenta slightlyhigheramplitudethan the outgoingwave
tion and velocityare very similar for all semidiurnalcompo- (,4• > .40). We can then use theseestimatesto 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
0.2
0.4
0.6
0.8
1
0.6
0.8
1
140
:
12o
lOO
8o
60
0
0.2
0.4
0.6
0.8
1
0
x/L
x/L
Figure 13. Structureof the depth-averagedtidal velocitiesalong the Pesaro-Kamenjak(P-K) line: (top)
semidiurnaltides,(bottom)diurnaltides,(left) amplitudes,and (right) phasedifferences
with surfaceelevations.Thick linesare along-channel
velocitycomponents;
thin linesare across-channel
velocitycomponents
(samesymbolsas for Figure 11).
the structuresof surfaceelevation,normalvelocity,and phase
differencealongthe P-K line and comparethem to the numerical results(Figure 14). The commonfeaturesindicatethat the
semidiurnaltides in the northernAdriatic are primarily the
resultof a superposition
of an incomingKelvin wavewith its
partial reflection.The good fit of elevationamplitudesis a
directresultof the leastsquaresfit, but the velocityamplitude
and phasedifferenceprovidean independentcheck.Consideringthat the northernAdriatic is not closeto beinga rectangularbasinwith a flat bottomandthat the bottomslopeought
to affectthe wavepropertiesasnotedin (13), the agreementis
better than what couldhavebeen expected.This indicatesthat
the semidiurnaltidesin the northernAdriaticprimarilyconsist
in an incomingKelvin wave progressingalong the eastern
coast,turningwith the coastline(as a set of Poincardwaves)
andreturningin an attenuatedformalongthe Italian (western)
coast.Suchbehavioralsoexplainsthe amphidromicpoint observedfarther south [Polli, 1959] as the locationwhere both
Table 4. Amplitudeof IncomingWave (A •), Amplitudeof
OutgoingWave (Ao), and PhaseDifference2ky + d)
Determined From a Least SquaresFit of the PesaroKamenjak Resultsto the Two-Kelvin-WaveTheorya
Constituent
M2
K2
N2
S2
incidentand reflectedwaveshaveequal and oppositephases.
This explanation,whichis not new,confirmsthe previousconjecturesof Hendershott
and Speranza[1971]andMosetti[1986].
5.3.
Diurnal
Tides
When the sameprocedureis appliedto diurnaltides,it fails
becauseno pair of exponentialcurvescan be fitted to both
endswithout creatingan unrealisticnegativevalue in the middle. The conclusion must be that diurnal tides do not consist of
Kelvinwaves.The precedingarguments,indeed,showthat we
shouldexpectnot a gravity-typewavebut a topographicwave
propagatingacrossthe basin(althoughthe distanceis rather
short!)with an amplitudeamplificationfrom deepto shallow.
For the fitted parabolicprofile(Figure 12) the amplification
coefficient
rr' takes the form
or'-
Ao,
2ky + ½,
cm
cm
deg
X2
11.5
2.6
2.0
7.0
10.8
2.4
1.9
6.5
- 110.9
- 118.1
-- 107.3
- 117.5
1.280
0.003
0.001
0.136
aThe )(2valuesexpress
the "goodness
of the fit" according
to 1,2
statistics.Note that the outgoingwaveis systematically
weakerthan the
incomingwave.
1- ay
(18)
(The signin front of the squareroot was chosento yield the
smallestabsolutevalue, corresponding
to the wave with the
leastamplificationfrom southto north, i.e., the wavewith the
leastenergy.)Integrationoverthe directionof the main axisof
the Adriatic yields
or(y) =
A1,
+a - x/a2- w2/gDo
f0
y ')
cr'(y
= 1-
dy'
1 gD0a
2 In 1-ay '
from whichwe deducethe amplification/attenuation
factor
(1) •-¾•-"'2/gøøa2
.
(19)
exp
[o-(y)]
= 1- ay
26,278
MALA(2I•ETAL.:TIDALDYNAMICS
IN THENORTHERN
ADRIATICSEA
12
10{
8
o
S2
6=
4
N2, K2
.............................
0
0.5
0
0.5
1
0
0.5
1
Figure
14. Comparison
ofthe(left)surface
elevation
amplitude,
(middle)
normal
velocity
amplitude,
and
(right)
phase
difference
between
thenumerical
results
(thinlines)
andthefitofthetwo-Kelvin-wave
theory
(solidlines).
For thevaluesquotedabove(f = 1.03 x 10-4/S,ro-- 7.29x
they yielded arms error of all surfaceelevationsin the five
10-S/s,
DO= 571m,a = 1/935km,andL = 137km),this portssmallerthan1 cm(0.5 _+0.5 cm).Similarresultswere
factorequals
3.1overa distance
of 800km(thelengthof the obtained for the vectorial difference between modeled and
entireAdriatic)
and1.1overa distance
of 144km(thelength observed
complex
values(combinations
of amplitudes
and
of theportion
retained
forthismodel).
These
values
areonly phases):
0.8_+0.7cm.Thephaseerrorsgenerally
fellbelow5ø
slightly
smaller
thanthe observed
north-south
amplification(4.4ø _+4.2ø).
factors
observed
fortheK• tide([Polli,1959]andFigure
5).
Themodelshows
thatthesurface
elevation
isalways
higher
We canfurtherexamine
whetherthetopography-wave
ar- ontherightsideof theflow,indicating
thatboththenorthern
gument
predicts
a phaseshiftacross
thebasinthatagrees
with Adriatic and the Gulf of Trieste behavelike narrowchannels
theobserved
value.Seeking
a solution
of thetypeexp[-ikx] [Gill,1982],
inwhich
thevelocity
component
along
thechannel
(withk realpositive
to correspond
to a phasethatdecreasesis significantly
stronger
thanthecross-channel
velocity
andis
fromlefttoright,fromItalytotheopposite
shore),
(15)yields subject
to the Coriolisforce.At timesof highinflow/outflow
the isolines
of surface
elevation
are nearlyaligned
withthe
frr' f a - x/a2- ro2/gD
o
velocity.
Assuch,
theGulfof Trieste
maybe
k ....
.
(20) depth-averaged
ro
ro
1 - ay
consideredas a miniature of the northern Adriatic Sea.
The surfaceelevations
andalong-channel
velocities
across
Thephaseshiftacross
thebasin,i.e.,overthedistance
L, is
fL a - x/a• - •o•/gDo
kL = ro
.
1 - ay
the openboundary
(linefromPesaro
to Kamenjak)
of the
present2-D modelcompare
wellwiththe similarquantities
(21) obtained
witha largerand3-D model.The analysis
of these
cross-channel
profiles
thenledto thefollowing
threeresults:
For thevaluesquotedabovethepredicted
phasedropfrom (1) the M 2 and other semidiurnaltidescanbe understoodas
Italy to Croatiaat y - 640 km (aboutthe locationof the havingbeenformedbya standing
setof incidentandreflected
Pesaro-Kamenjak
line)is0.39rad= 22ø.In comparison,
Fig- Kelvinwaves,(2) thenorthward
amplification
of theseKelvin
ure 11 revealsa phasedifferenceof 15ø-20
ø,in the samediwavesis caused
bytheshoaling
bottom,and(3) theKz and
rection.
Appendix
B provides
a moreprecise
comparison
by otherdiurnaltidescanbe understood
astopographic
waves
usinga theoretical
framework
slightly
morerigorous
thanthe propagatingacrossthe basinwith the shallowwater on their
basicarguments
proposed
at thebeginning
of thissection
and
right,namely,
fromtheCroatian
coastto theItalianshore,
and
alsoby comparing
the along-basin
velocitymagnitudes.
The subject
to attenuation
fromshallow
to deep.Whileconclusion
conclusion
remainsthesame:TheKz tideandall otherdiurnal
1 is a confirmation
of earlierpropositions
[Hendershott
and
tidescanbeexplained
astopographic
waves
progressing
from Speranza,
1971;Mosetti,1986],conclusions
2 and3 arenew.In
the northeastto the southwest,
from the Croatiancoastto the
Italian
6.
shore.
Conclusions
particular,no dynamical
interpretation
of the diurnaltidesin
theAdriatichadbeenproposed
priorto thisstudy.
Appendix A: Kelvin Waves
Until this study,Adriatic Sea tideshad not been simulated in a Flat Bottom Channel
bymeans
ofnonlinear
numerical
models,
except
in thecontext Consider
twooppositely
traveling
barotropic
Kelvinwaves
in
of a tidalanalysis
oftheentireMediterranean
Sea[Tsimplis
et a channelextending
alongthey axis,of constant
widthL and
al., 1995],whichmadeno specific
mention
of theparticular of uniform
depthD. Thesurface
elevation
•l(x, y, t) and
structure
of thetidesin thenorthern
Adriatic.Theobjectiveslongitudinal
velocityv(x, y, t) canbe writtenas
of the presentstudywere the accurate2-D simulationand
dynamicalanalysisof the tides within the subdomainsur-
•q= Aoe-x/R
cos(ky+ rot)+ A ,e(x-L)/R
cos(ky- rot+ rk)
roundedby fiveports(Rovinj,Trieste,Venezia-Lido,Malamocco,andPortoCorsini)andextending
slightlyto the south
(Ala)
(linejoining
Pesaro
tothesouthern
tipoftheIstrian
Peninsula).
Themodel'sopenboundary
conditions
werecalibrated
soas
to obtainan optimumfit with the knowntidal elevations
in the
fiveports.The simulation
resultswerefoundsuccessful,
for
g
-x/R
u=•-•[-Aoe cos
(ky+ rot)
+ A he(x-L)/•
cos(ky- rot+ rk)],
(Alb)
MALA•I• ET AL.' TIDAL DYNAMICS IN THE NORTHERNADRIATIC SEA
26,279
whereR = V'#D/f = to/fkis the externalradiusof 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 otherreachesits largest
to
g
'
amplitude(A•) along the oppositewall x = L. There is a
phasedifference(kbetweenthe two waves.If we combinethe where a prime indicatesa derivativewith respectto y.
Equation(B4) is difficultto solveexactlyfor a depthprofile
two waves into a single oscillatoryfield beating at the frequencyto,namely,r• = H(x, y) cos[tot - (kr•(x, y)] and v = D(y) givena priori, evenasa simpleanalyticalfunction.Thus,
insteadof constructing
a D(y) topographyprofile and solving
V(x, y) cos [tot - qbr.(x,y)], we obtain
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.
(DH')(k2D+fkD
+f2-to2)
(A2a)
g
V(x,y)=fR
ßx/A•e
-zv'•+ A•2e
2(x-L)/'•2AoA•e
-•/'•cos(2ky+
Observations[Polli, 1959] aswell as our presentsimulations
(Figure5) reveal(1) that the amplitudeof the K• tide increases
smoothlyand graduallyalongthe basinand (2) that the crossbasinvelocityis veryweak. Let us then adoptH(y) = .4 exp
(sy), where s (>0) is an e-foldinglengthscaleand u = 0.
Accordingto (B2), thereis a wavenumberk that guarantees
no
cross-basin
flow:
(A2b)
fs
k = --.
2A 0,4le-•/• sin (2ky + 4))
tan((kn
- (k•,)
= Ao2e
-zvR
- A•2e2(X-6)/e
'
Equation(B4) becomes
Note that if the incomingand reflectedwaveshave the same
amplitude(.40 = .4 •), the phasedifferenceis
sin ( 2ky + 4))
tan((k,- (kv)
= sinh
[(L- 2x)/R]'
(BS)
(A2c)
6O
2
sD' + s2D+- g = 0,
(A3)
whichis equalto _+90
øat the middleof the channel(x = L/2)
and varies antisymmetricallyon both sides.
(B6)
whichis satisfiedif D(y) is of the form
6O
2
D (y) = D •e-sy
gs
2,
(B7)
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
in a Shoaling Channel
46.4m, we haveD• - 46.4 m + to2/gs2.
Then,if we impose
The linear barotropicequationsgoverningwavesin a channel of variable depth can be written as
46.4 m +
Ou
Ot fv =-# Ox'
(Bla)
e-(140
km)s
__gs2'
(B8)
The solutionwith the lowestabsolutevalue (yieldingthe least
energetic
wave)iss = 1.87 x 10-6 m-•, andthetopographic
Ov
compatiblewith H(y) = .4 exp (sy) that bestfits the
(Bib) profile
actualbottom topographyof the northernAdriatic is
OZ+fu= -# Oy'
Orl
a zero depth at the Venice-Trieste shoreline,which is 140 km
away,we obtain an equation for the constants:
Ou
0
0•-+D •xx+ •yy(Dr)= 0,
(Blc)
D(y) = (201.6 m)exp [-(1.87 10-6/m)y]- (155.2 m).
(B9)
where x is directedacrossthe channel(0 <- x <- L), y is
directedalongthe channel,f is the (constant)Coriolisparameter, # is the gravitationalacceleration,andD(y) is the resting
depth,whichwe take as a functionof y only. Had the surface
elevationterm Or•/Otbeen ignored,the set of equationswould
havebeenthat governingcontinentalshelfwaves[Gill, 1982,p.
409]. In other words, we are consideringhere topographic
wavesmodifiedby the gravitationalinfluenceof the SSE.
If we seek solutionshaving a given frequencyto, as in the
tidal problem,and havinga wave expressionin 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) mustsatisfy
-f gH' + togkH
v=
?_
,
Over the 140 km of basinlengththe e-foldingscales yields
a waveamplificationof exp(sy) = 1.30, i.e., corresponding
to
a surfaceelevationamplitude increaseof 30% from the P-K
line to the northwesternshoreline.In comparison,the numerical model (Figure 5) revealedan increasefrom 14.5 to 17.5
cm, which is a 21% increase.Consideringthe radical assumptions of the theoreticalmodel, we find reasonableagreement.
The cross-channel
wavenumberk givenby (B5) is found to
be approximately
equalto 2.6 x 10-6 m-•, whichyieldsa
phasechangekL from Pesaroto Kamenjak(L = 137 km) of
about 21ø. This value agreeswith the phasedifferencesdetermined from the numericalsimulationsand shownin Figure 11
(lower right).
(B2) Turningnowto the along-basinvelocity,we derivefrom (B3)
26,280
MALA0•I(2ET AL.: TIDAL DYNAMICS IN THE NORTHERN ADRIATIC SEA
that they structureof the v velocitycomponentis relatedto the
amplitudeprofileH(y) by
V(y) = -i --H(y).
(B10)
Along the P-K line, where the K• amplitudeH is 14.5 cm, the
theorypredicts
a K• velocity
magnitude
of 3.6cms-•, which
Godin, G., and L. Trotti, Triestewater levels1952-1971:A studyof the
tide, mean level and seicheactivity,Misc.Spec.Publ.Fish.Mar. Serv.
Can., 28, 24 pp., 1975.
Gonella,J., A rotary-componentmethodfor analysingmeteorological
and oceanographicvector time series,Deep Sea Res., 19, 833-846,
1972.
Hendershott, M. C., and A. Speranza,Co-oscillatingtides in long,
narrowbays:The Taylor problemrevisited,Deep SeaRes.,18, 959980, 1971.
agreesquite well with the valuesobtainedby the numerical
simulations(Figure13,lowerleft, top curve).Furthermore,the
presenceof the -i factorin the expression
for V indicatesthat
the along-basinvelocitylags the sea surfaceelevationby 90ø,
whichis preciselythe valuenotedin the numericalsimulations
(Figure 13, lowerright).
In conclusion,
the precedingtheoryis validatedby favorable
comparisons
with numericalsimulationresults(as well as observations),and sincethe theoryreducesto the classicalcontinentalshelfwavetheoryin the limit of no gravitationaleffects
(rigid lid approximation
obtainedfor # • •), the K• tide of
the northernAdriatic is a topographicwavemodifiedby grav-
HydrographicInstituteof the Republicof Croatia (HIRC), Tide Tables,Adriatic Sea-EastCoast,110 pp., Split, Croatia, 1994.
IstitutoIdrograficodellaMarina, (IIM), Tavoledi Marea, Mediterra-
itational
Mosetti, F., and B. Manca, Le maree dell'Adriatico: Calcoli di nuove
effects. Note that the wave is evanescent in the down-
channel direction and propagatesin the cross-channeldirection, from the Croatian coast to the Italian shore.
Acknowledgments.Malafiifiwas supportedby the Ministry of Science and Technologyof Sloveniathrough grant Z1-7045-0105. The
Osservatorio
GeofisicoSperimentale
in Trieste(Italy) supportedViezzoli. Malafiifiand Cushman-Roisinalsoacknowledge
the supportof the
U.S. Office of Naval Research,through grant N00014-93-7-0391 to
Dartmouth College.All three authorsare indebtedto ChristopherE.
Naimie of Dartmouth Collegefor havingperformedthe tidal calculationswith the finite elementmodel(to be publishedelsewhere)and
extractedthe valuesalong the Pesaro-Kamenjaksectionfor the purposeof the presentstudy.The Abdus SalamInternationalCentre for
Theoretical Physicssupportedthe participationof the authorsat the
International Workshopon the Oceanographyof the Adriatic Sea,
where fruitful discussions
took place.
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, Comprehensivecoastalcirculationmodel with applicationto the Gulf of
Maine, Cont. Shelf Res., 16, 875-906, 1996.
McHugh, G. F., A numericalmodel of two tidal componentsin the
northernAdriatic Sea,Boll. Geofis.Teor.Appl.,Xvi, 322-331, 1974.
Mellor, G. L., and T. Yamada, Developmentof a turbulenceclosure
model for geophysicalfluid problems,Rev. Geophys.,20, 851-875,
1982.
Mosetti, F., Distribuzione delle maree nei mari italiani, Boll. Oceanol.
Teor.Appl., V, 65-72, 1987.
costantiarmonicheper alcuniporti, in Studiin onorede Giuseppina
Aliverti, pp. 163-177, Ist. Univ. Nav. di Napoli, Ist. di Meteorol. e
Oceanogr.,Naples, Italy, 1972.
Mosetti, F., and N. Purga,Courantsc6tiersde diff6renteoriginedans
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
componentin the northernAdriaticby applyingthe rotary analysis
to the Taylor problem,Boll. Oceanol.Teor.Appl.,/V, 165-172, 1986.
Naimie, C. E., GeorgesBank residualcirculationduring weak and
strongstratificationperiods:Prognosticnumericalmodel results,J.
Geophys.Res., 101, 6469-6486, 1996.
Polli, S., La Propagazionedelle Maree nell'Adriatico,paper presented
at IX ConvegnoDell' AssociazioneGeofisicaItaliana, Rome, 1959.
Pugh,D. T., Tides,Surgesand Mean Sea-Level,472 pp., John Wiley,
New York, 1987.
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of open-boundaryconditions,J. Phys. Oceanogr.,25, 1006-1011,
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(ReceivedFebruary22, 1999;revisedDecember8, 1999;
acceptedApril 5, 2000.)