Oscillations of the inertia period on the Adriatic Sea shelf

Conthwatal Shell" Research. Vol. 7. No. 6. pp. 577-598, 1987. 11278-4343/87 $3.00 + 0.00 Printed in Great Britain. Pergamon Jourmds Ltd.

Oscillations of the inertia period on the Adriat ic Sea she l f

MIRKO ORLIC*

(Received 1 August 1985; in revised form 2 April 1986; accepted 30 April 1986)

Abstract--Wind, current and hydrographic data, taken during three summer seasons ( 1979, 1980 and 1983) on the Adriatic Sea shelf, have been analysed for evidence of the inertia-period oscillations. The data originated from four stations: one close to the lateral boundary, one at a typical mid-basin location, and two close to the longitudinal boundary of the Adriatic Sea.

The inertia-period oscillations occurred in episodes lasting for a few days. Vertically, the oscillations displayed a simple structure: the clockwise current-vector rotations were opposed in phase across the thermocline. The partition of energy between two layers depended on the thermocline depth. Horizontally, the inertia-period currents accounted for about 10% of the total current variance at stations close to the longitudinal boundary, and for 20--30% at the stations farther offshore. The oscillations in the current field were accompanied by temperature variations. The complexity of the phenomenon could well be explained by the internal mode and a few horizontal modes of the two-layer sea contained in a rotating rectangular channel.

The two-layer fluid model was also found to be capable of introducing the non-adiabatic problem in an illuminative way. The typical Adriatic Sea wind stress (0.25 N m 2) caused in the model inertia-period currents of 5-1(I cm s ~, and pycnocline displacements of -1 m--in fair agreement with the observations. The linearized bottom friction damped the oscillations with the realistic decay time (1-2 days).

1. INTRODUCTION

TttE deep-sea inertia-period motions have been intensively observed and modelled during the past two decades. The progress is well documented in two review papers, by WEBSTER (1968) and Fu (1981). Webster has presented considerable empirical evidence on the inertia-period oscillations, and pointed to the transient nature of the phenomenon, its thin vertical extent, and its possible occurrence anywhere in the oceans. Fu has analysed newly acquired data, and interpreted the results both in terms of local sources and of turning point effects on remotely generated internal waves.

The inertia-period oscillations in land-locked basins differ from the oceanic oscillations. In lakes and inland seas the local generation and damping seem to be the dominant mechanisms determining the transient character of the inertia-period motions. More- over, due to the presence of solid boundaries and the interference of multiply reflected internal waves, it appears that a suitable description of the inertia-period oscillations can be given in terms of normal modes. The first normal-mode solutions were published by DEFANT (1940, 1952) and PROUDMAN (1953). The.se authors have shown that transverse standing oscillations may occur in rotating rectangular channels. The baroclinic mode of these oscillations is most often characterized by a period which differs only slightly from the inertia period. A similar close-to-the-inertia period was found to be connected with

* Geophysical Institute, Faculty of Science, University of Zagreb, P.O. Box 224, 41001 Zagreb, Yugoslavia.

577

578 M. ORUC"

the internal Poincar6-type modes of both rectangular (KRAuSS, 1966; RAO, 1977) and circular (CSANADY, 1967) basins. An interesting feature of these modes is a clockwise rotation of the phase around the basin perimeter.

Simultaneously with the theoretical research, empirical investigations have been performed in various lakes and inland seas all over the world. The inertia-period motions had been detected in the Baltic Sea as early as the thirties (WEBSTER, 1968), and more recently they were investigated by TOMCZAK (1969). The North American Great Lakes have received a great deal of attention (Lake Michigan: MORTIMER, 1963; VERBER, 1964, 1965, 1966; MALONE, 1968; Lake Superior: SMITH, 1972; Lake Ontario: BLANTON, 1974, 1975; PICKEVr and RICttARDS, 1975; MARMORINO, 1978). Although the majority of authors analysed the summer situations, the inertia-period oscillations were also found in winter, in the presence of "reverse thermocline". Studies have also been conducted in the Black Sea (YAMPOL'SKIY, 1961; FOMIN and SAWN, 1973), the Japan Sea (NAN-NITI et al., 1966), the Mediterranean Sea (GONELLA et al., 1969; PERKINS, 1970; MILLOT and CREPON, 1981 ), the North Sea (ScHoTr, 1971), and the inertia-period motions have been detected even in the small Lake Geneva (BAUER et al., 1981). The common feature of the inertia-period oscillations in land-locked basins is their coincidence with the stratified conditions. The oscillations are usually manifested by considerable pycnocline movements and current- vector rotations that dominate the central parts of the basins. The rotations in the current field change their phase from one side of the pycnocline to the other.

The first observations of the inertia-period motions in the Adriatic Sea seem to be the ones originating from a station close to the Yugoslav coast: ZAmC (1977) noticed a maximum of energy--coinciding with the local inertia-period--in the sea-surface current spectrum, whereas GAeIC" (1980) analysed the same data set in the time domain. The inertia-period motions have subsequently been observed along the Italian coast (AccER- BONI et al., 1979; MICttELATO, 1983). Finally, the investigations have been extended to the open Adriatic by ACCERBONI etal . (1981) and by GAClC" and VUC'AK (1982). These authors have shown that significant energy maxima appear in the current spectra at the inertia period, that the episodes of the inertia-period oscillations coincide with the transients in the wind field above the Adriatic Sea, and that the phase difference of the current oscillations is approximately 180 ° across the thermocline.

The purpose of this paper is two-fold. On the one hand, the empirical investigations of the Adriatic Sea inertia-period motions will be expanded (Section 2) by considering a new data set and performing analyses in both time (band-pass filtering) and frequency (rotary spectral analysis, rotary systems analysis) domains. On the other hand, simple conceptual models will be used for the interpretation of the empirically determined characteristics of the inertia-period oscillations in the Adriatic Sea. In Section 3 the free, undamped oscillations will be described first, and then the sources and sinks of energy will be explored on the basis of a two-layer fluid model. The results will be summarized in Section 4.

2. D A T A D E S C R I P T I O N A N D A N A L Y S I S

We had at our disposal currents measured at four stations in the Northern Adriatic (Fig. 1). The bottom depths, the depths of the current meters and the measurement intervals are given in Table 1. At Stas A and B, Alexaev BPV-2 current meters were suspended from an oil-drilling platform, whereas at Stas C1 and C2 Aanderaa RCM-4

Inertia-period oscillations 579

Fig. 1.

. ~ ' ~ , - , ~ "~ T RIESTE

:"- . . - 50 - - . . . . C',_.

PESARO~?~ PESARO~?~ ~... ",!

' \-.i',, ~o ,'o ~ ~ ,~o

KM

N

l "%

",~ ~ ZADAR

, / .#

i /

. / / ~

/ t /

I %.

" ~ u ' J I . . . . . . . . . . . . . , "~,. ' ,~,. _ , 1

• 7- . . . . . . . . . . ".-..._.

Topography of the Adriatic Sea shelf and positions of sampling stations (A, B, CI and C2).

Table 1. Description of the current-vector time series

Station/bottom Immersion of depth (m) current meter (m) Measurement interval (number of data)

A/28 3 1 June-26 July 1979 (1344) 15 1 June-31 July 1979 (1464) 25 28 June-31 July 1979 (816)

B/61 3 26 July-28 August 1980 (816) 30 26 July-28 August 1980 (816) 58 26 July-28 August 1980 (816)

C1/28 8 17 August-16 October 1983 (1464) 23 17 August-4 October 1983 (1176)

C2/35 8 17 August-9 October 1983 (1296) 30 17 August-25 October 1983 (1680)

580 M. ORUe

current meters were set on subsurface moorings. The sampling interval was 15 min at the first pair of stations, and 10 min at the latter pair. From the original values the hourly means were calculated, and these were used as input data for our analyses.

Besides the current-meter measurements, wind and hydrographic data were also of interest for the present study. At Stas A and B an anemograph was located 35 m above the sea surface. For Stas C1 and C2 wind data from the nearby Pula Airport (63 + 10 m) were used. The hydrographic data set included bathythermograph profiles taken three times a day at Stas A and B, and a single temperature time series originating from the lower level of Sta. C2.

The main drawback of our data set is the lack of synopticity: only the measurements at Stas C1 and C2 were performed simultaneously. However, the spatial distribution of stations is believed to be advantageous. The Sta. A data described the conditions close to the lateral boundary of the Adriatic Sea, Sta. B is placed at a typical mid-basin location, whereas the measurements from Stas C1 and C2 illustrate the processes close to the longitudinal boundary of the Adriatic Sea area.

As already mentioned, data were analysed in both time and frequency domains. To extract the inertia contents of the current-vector time series, a band-pass filter was

used (see TOMCZAK, 1969). The filter was applied primarily to eliminate the tidal signal which obscures the inertia events. The cut-off frequencies were 0.042 and 0.074 cph, the slope of the filter characteristics was 0.008 cph -~, and the length of the band-pass weight function equalled 49. The filter weights were computed according to DAvis (1971). The effect of band-pass filtering was tested on a few synthetic time series, as suggested by KUNDU (1976), and it was shown that the filtering process spreads the inertia event by a day, that it cuts down the current magnitude, but does not alter the phase. From the band-passed currents the kinetic energy per unit mass was calculated (BLACKFORD, 1978; MARMORINO, 1978). The time series of the energy were plotted, after the high frequency oscillations--connected with departures of the current-vector tips from the circular path--had been smoothed.

In the frequency domain, rotary spectral analysis was performed. The method had been developed by MOOERS (1970), Perkins (1970) and GONELLA (1972); Perkins' approach was adopted here. All the rotary spectra were computed with six degrees of freedom. The two vector time series were compared using rotary systems analysis, whose main components are the squared coherence and the phase factor spectra, proposed by MOOERS (1970) and GONELLA (1972), and the gain factor spectrum defined by GODIN (1978). Let it be added that the limiting value of the squared coherence was computed at the 80% significance level.

Station A, 1 June-31 July 1979

There appear to have been six inertia events (Fig. 2), centered around 6 and 15 June, 5, 17, 20 and 24 July, and coinciding with the wind pulses. During the periods characterized by low-speed winds (8-12, 22-28 June, 8-12 July) there were no inertia- period episodes in the sea. However, not all high-speed winds were followed by oscillations in the sea. We shall show later that--besides high wind speed--a sudden change of wind is necessary to generate the inertia-period oscillations in the Adriatic Sea.

Figure 2 shows that the ratio of kinetic energy per unit mass at 3 m to that at 15 or 25 m was quite variable. This is particularly well illustrated by the last three events: although insignificant around 16 July, the energies at 15 and 25 m rose to considerable portions of


JUNE 1979 JULY 1979 5 10 15 20 25 30 5 10 15 20 25

I I I I I I I I I I I

WIND,DAILY MEAN K.E./MASS

0

0"

Fig. 2.

INERTIA PERIOD OSCILLATIONS, K.E./MASS: 3 rn . . . . 15rn

Kinetic energy per unit mass for daily-mean wind and inertia-period oscillations at 3, 15 and 25 m, Sta. A, 1 June-31 July 1979.

the "surface" energy by 23 July. The bathythermograph profiles for the same period (Fig. 3) show that the persistent winds brought about deepening of the thermocline and cooling (heating) of the surface (bottom) layer, most probably through the mixing processes (SPINEL and IMBERGER, 1980). The change in the depths of the surface and bottom layers would affect the partition of energy between the two, as will be shown in Section 3. Let us point out that the thermocline oscillations are visible in Fig. 3. However, since three-times-a-day sampling does not permit adequate time-series analysis to be carried out, the inertia-period motions cannot be isolated from the tidal oscillations or even from the possibly aliased high-frequency oscillations.

The rotary spectral analysis for Sta. A shows peaks that occur at clockwise frequencies slightly above 1/16.8 cph, which is the local inertia frequency. The peaks at the corresponding anticlockwise frequencies are one or two orders of magnitude lower. The squared coherence is significantly high for both pairs of vector time series (Fig. 4). The phase shift across the thermocline is close to 180 °, whereas the inertia-period currents in the bottom layer appear to be almost in phase. The ratio of the radius of a circle along which the current vector rotates at 15 m to that at 3 m is close to 0.6. The similar ratio is obtained for 25-15 m radii, indicating suppression of the near-bottom currents.

Station B, 26 July-28 August 1980

The time series of kinetic energy per unit mass (Fig. 5) were dominated by the inertia event that occurred between 14 and 19 August. It was preceded by two wind pulses, each characterized by a wind shift from southeast to northeast direction. In particular, it seems that the sudden impositions of the northeast wind effectively generated the inertia-period oscillations in the Adriatic Sea. As it follows from the analysis of related synoptic situations (DEuTSCHER WETFERDIENST, 1980), an example of which is given in Fig. 6, these wind pulses were caused by cyclones travelling along the Adriatic and Tyrrhenian seas. As is well known, cyclones on these paths bring about the characteristic southeast to northeast wind shifts above the Adriatic Sea, and therefore may provide the generating mechanism for the inertia-period motions in the basin.

582 M. 0 P,.t+I¢

~ 10 r

20 121

21 h tL

14 h

14 16 18 07 h , ,

14 16 18 20 22 0 i ~ i i q

30

TEMPERATURE [*C]

16 18 20 22 24 i i i I r

20 22 2/* i i i

24

16 JULY 1979

Fig. 3.

10 E ...¢. i-- a_ 20 LU O

30

TEMPERATURE [*C] 21 h

14 16 18 20 22 24 1/, h + ~ , , ,

14 16 18 20 22 24 87 h i , ; , , ,

14 16 18 20 22 24 w i i i i r

0 7 h ~ ~ k Y 1979

Vertical profiles of temperature measured three times a day tit Sta. A, on 16 and 23 July 1979.

Figure 7 shows that the wind episodes also caused changes in the temperature of the surface and bottom layers, and that the thermocline deepened between 14 and 19 August. On the first of these dates kinetic energy per unit mass at 30 m represented only a small fraction of the "surface" energy (Fig. 5). After 19 August the two were comparable, which indicates that the partition of the energy between the two layers is strongly controlled by their depths. For Sta. B we have again only three bathythermograph samples per day, so that the thermocline oscillations, visible in Fig. 7, have to remain unexplained.

The rotary spectra (Fig. 8) have significant peaks at clockwise frequencies slightly greater than the local inertia frequency, i.e. 1/17.1 cph, and no peaks at corresponding anticlockwise frequencies. By comparing this result with the analogous conclusion for Sta. A, we can see that the shifts of the energy maxima along the frequency axis parallel the variations in the inertia period with latitude. This raises the question of how the beta effect would modify the normal modes of land-locked basins--a problem which certainly deserves attention. Figure 8 also illustrates the lessening of the inertia energy with depth.


Fig. 4.

1

a ~ 08

" ' ~ 0.6 rv <n- " ~ UJ 0.t., O ' I - ( D O

0.2

0

360°[-

18oo~_

r O_ 0 °t- 'I o 0.6

~ 0•6 < LL

0.~ z < 0.2

0

Z~,...- A ~zxL i

. . . . . , 0 % . . . . .

E]

1 I

I

~'o" °'°, ,~" • C]~ D

I I

I

I I II I -0.070 -0.065 -0060 -0.055

FREQUENCY [cph]

o cz 3 - " 1 5 m zx zx 1 5 - , " 2 5 m

Results of the rotary systems analysis for two current pairs of Sta. A. Dashed line is at the local inertia frequency (1/16.8 cph).

JULY 1980 AUGUST 1960 30 5 10 15 20 25

I I I I I I

N W

S

E N

15

01 10

E s

0

WIND DIRECTION

I I WIND SPEED

200r K.E./MASS : - - 3 m / . . . . . 30m (M

• . . . . . . . . . 58m

s° I 0

Fig. 5. Wind direction, wind speed and band-passed kinetic energy per unit mass at 3, 30 and 58 m, Sta. B, 26 July-28 August 1980. Arrows indicate the northeast wind pulses that seemingly

caused the great inertia episode in the sea.

584 M. ORt~I( ~

Fig. 6. Successive synoptic charts, for (a) 15, (b) 16, (c) 17 and (d) 18 August 1980, 12 h GMT. The cyclonic disturbance travelled along the Italian peninsula and brought about the southeast to

northeast wind shift above the Adriatic Sea (compare with Fig. 5).

The same follows f rom the Sta. B rotary systems analysis (results not shown) , a l though the squared coherence is barely significant for the 30-58 m current -vector pair. The phases differ by 180 ° across the thermocl ine , and they do not differ inside the bo t tom layer.

Stations C1 and C2, 17 August-25 October 1983

There were four inert ia-period events at Sta. C1 that had their counte rpar t s at Sta. C2, and one episode which was clearly visible at the first station and almost unnot iceable at


TEMPERATURE [°C3

21hl2 1/. 16 18 20 22 2/, 1-4 h- , , ~ r , i

07 h 12, 1/,, 16f 18, 20~ 22, 2/,, 12 1/, 16 18 20 22 2/.

0 I I I I i I I ] I

10

2O

~ 30

o ~0

0

50 1/, AUGUST 1980

6O

Fig. 7.

21 h TEMPERATURE [ 'C] 1/h 12 14 16 18 20 22 24

0"I h 12 1/,, 16, 18 20, 22~ 26

12 14 16 18 20 22 2l. o

1C

2( -/

~ 3o

Q 40

50 19 AUGUST 1980

G0

A s in Fig. 3 excep t fo r Sta . B, 14 a n d 19 A u g u s t 1980.

the second station (Fig. 9). Of interest is the fact that some of the inertia-period episodes seem to propagate from Sta. C1 to Sta. C2 (see particularly the event of 2-3 September); qualitatively, this would conform to the clockwise rotation of the internal Poincar6-type modes in land-locked basins. However, the phase speeds encountered ( -1-2 km h -l) are much smaller than the apparent propagation velocity of the mentioned modes at the shore (-10-20 km h-l), and they are similar to the internal wave speeds for the Adriatic Sea. The fact that the strongest wind pulses correlate better with the currents at Sta. C2 than with the sea motions at Sta. C1 further complicates matters. Since the wind data originate from the Pula Airport, which is closer to Sta. C2, this could suggest that the propagation of wind disturbances is of some importance.

586 M. ORLIC

1

E ~9

r ~

uJ 161

Fig. 8,

_18° %

i ! \ ° l 0

/ [ / /

o - - o 3 m o - - o 30 rn a - - a 58 m

I -0.05

I i / ~ I I I -0.07 -0.06 0.05 0.06 0.07

161 m z m 7 0

3

162

FREOUENCY [cph]

Rotary spectra of the 3, 30 and 58 m currents at Sta. B. Dashed line is at the local inertia frequency (1/17.1 cph).

For the summer of 1983 we have at our disposal the temperature time series measured at the lower level of Sta. C2. The data for the inertia-period event of 10-11 September are shown in Fig. 10. They are characterized by an increase in temperature, which lasted throughout the above-mentioned time interval, and by a series of oscillations. The general increase in temperature could be attributed to the mixing processes mentioned previously in the interpretation of the bathythermograph profiles, the downwelling effects due to the southeast wind pulse and/or the internal Kelvin waves propagating along the Yugoslav coast (the offshore distance of Sta. C2 was 8 km, i.e. comparable to the internal radius of deformation). However, here the temperature oscillations are of main interest. Therefore, the data were band-pass filtered using the same filter as was described for the currents. The band-passed temperatures are also shown in Fig. 10. Although the inertia event is spread due to the filtering process, it clearly coincides with the inertia-period oscillations in the current field, which indicates that such oscillations in the Adriatic Sea are horizontally divergent.

The rotary spectra (Fig. 11) show the peaks that occur at clockwise frequencies smaller than the inertia frequency. However, one may question the validity of these results, since the elevation of peaks above the surrounding noise is barely significant, the data being taken close to the longitudinal boundary of the Adriatic Sea. The comparison of the "surface" inertia-period oscillations at Stas C1 and C2 shows, with a high squared coherence, a 90 ° phase lead and a gain factor of about 2.5. Therefore, the radius of the circle along which the current vector rotates is greater offshore than close to the coast. The phase factor suggests clockwise or offshore propagation of the phase with a speed ( -2 km h -l) similar to the phase speeds determined in the time domain.


AUGUST 1983 SEPTEMBER 1983

20 25 30 5 10 15 20 I I I I I 1 I

WIND DIRECTION N

o

N 2,31i K.E., MAss : - - s m .. . . . m

0 ~

120

9O

60 E u 30

C2, K .E . /MASS: - - 8m

. . . . . 30m

Fig. 9. From top tO bottom: the Pula Airport wind direction and wind speed, the Sta. CI band- passed kinetic energy per unit mass at 8 and 23 m, the Sta. C2 band-passed kinetic energy per unit

mass at 8 and 30 m; 17 August-20 September 1983. Arrows indicate major wind pulses.

Summary of the empirical evidence

The current data from the Northern Adriatic contain oscillations at the periods close to the local inertia periods. Vertically, the oscillations display a simple structure: the clockwise current-vector rotations are opposed in phase across the thermocline. The partition of energy between the two layers seemingly depends on the thermocline depth. Horizontally, the oscillations are characterized by energies that are lower along the longitudinal boundary of the Adriatic Sea than farther offshore.

The variance of the band-passed currents is compared to the total variance in Table 2. The inertia-period currents account for about 10% of the total current variance at stations close to the longitudinal boundary. They account for more variance (20-30%) at the upper levels of the mid-basin station and the station close to the lateral boundary of

588 M. ORLlC

Fig. 10.

18

17

16

o

LU ne

r Y k lJ O.. ~r I L l I---

0.t,

t o

t w i i i

to 0 a LI.I itl

U_ -0.t

j

I I I I I 8 9 10 11 12 1

SEPTEMBER 1983

The Sta. C2 temperatures measured at 30 m (top) and band-passed values (bottom), 8/12 September 1983.

Fig. 11.

10 -1 tt}

E u

$ o~

LU 11~2

o•••• / /

i . . ~ . 8m/C1 - , ~ A 23 m / C 1 o ~ o 8m/C2

t 8 a ~ , ~ 30m/C2

0°/o

I L I I / / I I I -0.065 -0.060 -0055 0.055 0.060 0.065

FREQUENCY [cph]

As in Fig. 8 except for the Stas C1 (8, 23 m) and C2 (8, 30 m), and the local inertia frequency (1/17.0 cph).


Table 2. The variance o f northeast and northwest current components

Station

Band-passed Total variance variance (cm 2 s -2) (cm 2 s 2) . Ratio (%)

Depth (m) NE NW NE NW NE NW

A 3 19.6 17.9 78.8 72.2 24.9 24.8 15 6.9 8.3 26.0 33.6 26.5 24.7 25 3.1 3.3 22.8 20.6 13.6 16.0

3 36.3 3(I.3 103.1 118.9 35.2 25.5 30 6.2 12.7 16.2 53.6 38.3 23.7 58 0.7 2.5 5.8 29.9 12.1 8.4

C1 8 3.2 3.5 42.8 54.6 7.5 6.4 23 0.7 0.9 5.2 4.8 13.5 18.7

C2 8 12.8 13.7 198.5 157.7 6.4 8.7 30 8.1 7.5 102.8 73.2 7.9 10.2

the Adriatic Sea. However, the near-bottom inertia-period motions are strongly sup- pressed at these two locations: the ratio of their variance to the total current variance is - 10%.

The inertia-period oscillations are caused by changes of rather strong winds, and it seems that the suddenly imposed northeast wind provides a particularly efficient generating mechanism. The current-vector rotations are accompanied by variations of temperature, indicating that the Adriatic Sea inertia-period motions cannot be modelled as simple, horizontally non-divergent inertia oscillations.

3. T W O - L A Y E R FLU ID MODEL

The basic conceptual model, which reproduces some of the salient features of the inertia-period oscillations in land-locked basins, approximates a lake or an inland sea by a two-layer fluid contained in a rectangular channel. It is assumed that the motions are hydrostatic, that there is no frictional transfer of momentum across the interface, and that the non-linear and lateral-friction terms may be omitted from the governing equations. The equations of motion and continuity for the bottom layer and for the baroclinic mode of oscillations then read (see, e.g. CSANADY, 1971):

Ou' h 0~' ~x h ~x' - - - f v ' = - g e - - Ot h + h' Ox (h + h')p h'(h + h') p'

Or' h 0~ ' T.y h Xy, - - + f u ' = - g e - - (1) Ot h + h' Oy (h + h')p h'(h + h') P'

+ + - - = 0 , \ Ox Oy / Ot

where u' and v' are vertically averaged baroclinic velocity components in the bottom layer, and ~' is the pycnocline elevation/depression due to the baroclinic mode of motions. With f the Coriolis parameter is denoted, g is the acceleration due to gravity, p

590 M. ORLIC

(p') and h (h') are the density and the depth of the surface (bottom) layer, whereas ~ is the proportionate density defect of the surface layer, i.e. e = (p' - p)/p' (usually a small quantity); zx and ry are the surface stresses, ~x' and Zy, are the bottom stresses.

The vertically averaged baroclinic velocity components in the surface layer (u and v) are given by

V ---- - - - -

h !

/A r

h

h !

V r

h

(2)

The sea-surface elevation ~ may be neglected for the internal mode considered here. The baroclinic mode of free oscillations has been considered because it is characterized

by the inertia period. The barotropic mode is of a much shorter period, and has therefore been neglected. Likewise, the barotropic component of forced motions has not been taken into account, the baroclinic component being the only one relevant for triggering the inertia-period oscillations.

If the coordinate system is orientated in such a way that the x axis lies across the channel, the y axis being parallel to the shores, (1) simplifies to

Ou' h 0~' Zx h z~, - - - f v ' = - g ~ - - Ot h + h' Ox (h + h')p h'(h + h') p'

Ov' zy h zy, - - + f u ' = - - - - (3) Ot (h + h')p h'(h + h') P'

Ou' a¢' h ' - - + - - = 0 ,

Ox Ot

where the longshore gradients are ignored. The boundary conditions at the shores are that the velocity normal to the shore vanishes, i.e. u'(x = 0, L) = 0, where L is the channel width.

The adiabatic problem

If we neglect the surface and bottom stresses in (3), the resulting equations of motion and continuity, together with the side-wall boundary conditions, are satisfied by the standing Sverdrup waves:

h'Kn

fin - - - An cos K.x c o s ( ~ t - a .)

u" = A. sin K.x sin (~nt - an) (4)

f v" = - - An sin •nX COS(O,t -- a ,) ,

(3" n


where n ~

Kn - - L

2 2 2 2 ~,, = f + K,,c (5)

hh' C2 = ge

h + h '

A,, and a,, are constants that depend on the initial conditions, and n is the nodal index (n = 1, 2, 3 . . . ) . The above solution is illustrated in Fig. 12 for the specific case of uninodal oscillations. Some of its elements were published by DEFANT (1940, 1952), and the whole solution was given by PROUDMAN (1953).

Let us now turn our attention to the Adriatic Sea shelf. The quantities chosen to represent the two-layer rectangular-channel approximation of this basin are listed in Table 3. From these quantities and the above theoretical results it may easily be verified that the angular frequencies of the first few horizontal modes are only slightly higher than the inertia angular frequency (a~ = 1.005 x 10-4 S -1, 62 = 1.020 x 10 -4 s -1,

~3 = 1.044 × 10 -4 s-l). This agrees with the empirical results for the open Adriatic.

t= t 0

t = t o + 2-~t

t= t o +

31~ t = t o *

"~/////////////////////////////////,~

~I/////////////////////////////~~ ~ ~ "/" ×

~///////////////////////////////~

K Fig. 12. One cycle of the transverse internal seiche in the rotating rectangular channel (n = 1).

592 M. ORLIC

Table 3. The parameters describing the Adriatic Sea, modelled as a two-layer sea in rectangular channel

Parameter Value

Width (L) 160 × 103 m Surface layer depth (h) 20 m Bottom layer depth (h') 40 m Density defect (~) 2 × 10 3 Acceleration due to gravity (g) 9.81 m s Wave speed (c) 0.51 m s i Coriolis parameter (f) 11) 4 S I

Equalities (2) point to the current oscillations whose phases differ by 180 ° across the pycnocline--again in accordance with the empirical results. The same equalities confirm our earlier conclusion that the partition of energy between the two layers depends on the pycnocline depth. Solution (4) reproduces the empirically determined onshore contrac- tion of the current vectors. These vectors follow circular paths (~n-'~ f ) and their rotations are accompanied by pycnocline motions, which suggests an explanation of the inertia-period temperature oscillations measured in the Adriatic Sea. To summarize, we can say that standing Sverdrup wave conceptual model may be taken to account for some prominent characteristics of the inertia-period oscillations in the Adriatic Sea.

With the aim of defining the limitations of the two-layer fluid model, the motion of a continuously stratified fluid in the rectangular channel has been investigated. The vertical and horizontal problems have been solved sep/lrately, the former numerically for an observed density distribution, and the latter analytically (FJELDSTAD, 1958). It has been found that the continuously stratified fluid model reproduces the variations of currents with depth slightly more realistically. Moreover, this model also simulates the density oscillations at each point in the fluid interior. Otherwise, the two layers appear quite acceptable, since the first baroclinic mode obviously dominates over all other vertical modes in the Adriatic Sea.

The w i n d f o r c i n g

To elucidate the generation of the inertia-period oscillations in the Adriatic Sea, let us analyse the simplest case of a horizontally uniform wind stress. From our empirical considerations if follows that the inertia episodes are effectively generated by the suddenly imposed northeast wind; therefore, we put

~x = Z,~y = 0, t > 0 .

The bottom stresses are neglected, and the initial conditions are

¢ ' = u' = v' = 0, t ~ < 0 .

The solution of (3) may then be found by the Laplace transform method. It reads (CSANADV, 1973, with some corrections):

I n e r t i a - p e r i o d osc i l l a t ions 593

h' z { ! cosh [(f/c)(x - L)] - cosh (f/c)x h + h' pf 2 sinh (f/c)L

COS KIX COS O ' l l + ~ COS K3X COS 133I + . . . L ~3

"c ( ! [ f s i n l ~ l x s i n ~ l t + f sinK3xsinG3t + . . . ]} (6) U t - -

p(h + h')f 3~13

V ~ - - p(h + h')f

sinh [([/c)(x - L)] - sinh (flc)x sinh (f/c)L

,2 ]} sin KIX cos (ylt + 3or 2 sin K~ cos ~3 t + . . . .

The above expressions contain time-independent components, which represent the directly forced motions. The remaining terms constitute the standing Sverdrup waves generated by the sudden imposition of the wind. Only odd horizontal modes are excited by a spatially uniform wind stress.

For a typical Adriatic Sea wind stress (~ = 0.25 Nm-2), approximate value of density (p = 103 kg m -3) and the model data of Table 3, the zero-time modal distribution of the pycnocline displacements and the longshore velocities is given in Table 4. Both the pycnocline and the current amplitudes appear to be of a correct order of magnitude, when compared with our earlier empirical results. Of interest is the fact that the kinetic energy is concentrated in the first two or three horizontal modes, whereas the potential energy is distributed more widely. The data available for the Adriatic Sea do not permit this result to be verified. However, SCHWAB (1977) has presented temperature and current spectra that reveal such an energy distribution in Lake Ontario.

One of the referees raised the question of the inertia-period oscillations being caused by air pressure. There are two reasons why synoptic air-pressure disturbances are not

Table 4. 7he pycnocline displacements from the equilibrium and the longshore velocities at time t = 0

N o d e n ~',(x = 0) (m) v;,(x = L/2) (m s l)

1 - 0 . 4 1 - 0 . 0 5 2 3 - 0 . 3 8 + 0 . 0 1 6 5 - 0 . 3 3 - 0 . 0 0 8 7 - O. 28 + 0 .005 9 - 0 . 2 3 - 0 . 0 0 3

9

Z - 1.63 -0 . (142 n = l

F o r c e d s o l u t i o n + 3 . 2 6 + 0 . 0 4 2

594 M. ORLIC"

expected to excite the oscillations in the Adriatic Sea. On the one hand, the dimensions of synoptic atmospheric disturbances are much greater than the dimensions of the Adriatic Sea shelf and, consequently, only small differences of air pressure occur across the Sea (e.g. Fig. 6). On the other hand, air pressure varies gradually in time, with typical periods far above the inertia period (KARABEG and ORLIC, 1982), which excludes the possibility of resonant excitation. Probably mesoscale air-pressure disturbances may prove to be of some importance, but it is difficult to detect them above the Adriatic Sea due to the sparsity of observations in both space and time.

The influence o f bottom friction

With the aim of analysing the damping of the inertia-period oscillations, the situation after the cessation of wind will be modelled. The influence of wind will therefore be ignored, whereas the bottom friction is assumed to follow the linear law

t

Zx = kp 'u ' , z' y = kp 'v ' ,

where k is the coefficient of bottom friction. It is supposed that the baroclinic contribution to the bottom-layer velocity is much greater than the barotropic contribution. This is supported by the results of the rotary spectral analysis for the Adriatic Sea: whereas all the spectra show maxima at the inertia period (which is characteristic for the baroclinic mode of transverse standing oscillations), there is no peak at the periods of 3-5 h (characterizing the barotropic mode of the mentioned oscillations). Such a disparity may easily be understood if we take into account that gravity is reduced for the baroclinic mode, which implies interface displacements larger than sea-surface displacements due to the barotropic mode. Moreover, the baroclinic currents of the same direction are confined to either of the two (shallow) layers, whereas the barotropic currents occupy the whole water column. Consequently, the former currents surpass the latter in magnitude, despite the baroclinic mode period being much larger than the barotropic mode period.

Putting for brevity

kh 6 - h'(h + h')

and assuming f 2 -> ~ c 2, we obtain (ORLI~', 1984):

4" "~ ~ A,,e -~' cos K,,x cos ~,,t - a,, - arc tan

u" = A~ e -~' sin ~,,x sin(~,,t - a,,) (7)

v;, ~ A,, e -~' sin K~ cos(~,,t - a,,).

The influence of bottom friction is manifested in the amplitude ratio between the pycnocline oscillations and the oscillations in the current field, as well as in the phase difference between these oscillations. The damping enters through the exponential term, which depends on the coefficient of bottom friction and on the depths of the surface and bottom layers.

Solution (7), along with our empirical results, allows the coefficient of bottom friction to be computed for the Adriatic Sea. The exponential curves were fitted to the


Table 5. The results o f the exponential curve fitting to the time series o f kinetic energy per unit mass, measured at Sta. B

Absolute Date Depth correlation Number of degrees k ( m s ~)

(1980) ( m ) coefficient of freedom ~(h i) ( × 10 ~)

311-31 July 3 0.96 34 0.046 1.53

9-10 August 3 0.98 29 /I.021 0.70

17-19 August 3 0.99 40 0.0211 0.67 30 0.98 26 0.019 11.63

19-22 August 3 0.99 44 0.018 I).60 30 0.99 27 11.019 0.63

23-24 August 3 0.99 22 0.028 0.93 30 0.97 31 0.1121 0.70

"descending" segments of the time series of kinetic energy per unit mass. The data taken at Sta. B were chosen for the computation, because the stratification at this station appears to agree well with the two layers of Table 3. Moreover, the computation was restricted to the inertia episodes whose peaks exceed 30 cm 2 s -2. The results given in Table 5 show that the coefficients of bottom friction fall between 0.6 and 1.5 x 10 -3 m s -t. This corresponds to a decay time of 1-2 days--a rather low value, due to the pycnocline depth being a considerable fraction of the total depth. The coefficients of bottom friction, obtained here, are smaller than the values (k = 1.7-10.5 x 10 -3 m s -~) needed to explain the seiche decay in the Gulf of Trieste (CALoI, 1938). However, GODIN and TRo~111975) give the damping coefficients for the longitudinal surface seiches of the Adriatic Sea, which--with the mean depth of 231 m--imply k = 0.9-1.4 × 10 -3 m s -~, in fair agreement with the present results. ORLI~ et al. (1986) also calculate similar values (k = 1.1 - 1.2 x 10 -3 m s -t) from the simultaneous measurements of winds and currents in the Northern Adriatic, supposing balance between the surface and bottom stresses.

4. C O N C L U S I O N S

Several characteristics of the inertia-period motions in the Adriatic Sea are success- fully reproduced by the models presented in this paper. The vertical structure of the inertia-period oscillations is well described by the internal mode of the two-layer fluid model: the current-vector rotations are opposed in phase across the pycnocline whereas the partition of energy between the two layers depends on the pycnocline depth. Horizontally, the suppression of the inertia-period oscillations towards the longitudinal boundary of the Adriatic Sea is adequately explained by the structure of transverse seiches in the rotating rectangular channel. The continuously stratified fluid model is needed for simulating the inertia-period oscillations of temperature, which--according to our observations--accompany the variations in the current field. On the other hand, the two-layer fluid model introduces the non-adiabatic problem in an illuminative way. The wind suddenly imposed on the two-layer sea brings about motions which agree in order of magnitude with the observed inertia-period oscillations. Moreover, the linearized bottom friction damps the oscillations of the same two-layer sea with the realistic decay time (1-2 days).

596 M. ORLIC

Although the experimental and theoretical results agree in a number of points, some empirical findings cannot be explained by the models presented here. However, they suggest the lines along which the modelling efforts should proceed. Frequencies smaller than the inertia frequency appear close to the northeast Adriatic coast, possibly due to a Doppler effect. The shifts of the inertia peaks with latitude are apparent in the data, which leads to the investigation of land-locked basins in a beta plane. The near-bottom suppression of the inertia-period motions is very well documented for the Adriatic Sea; probably, the no-slip bottom condition is the cause. Finally, the propagation of phase is evident from the data, and it should be modelled either through the influence of the travelling atmospheric disturbances or as an effect of the lateral boundary of the Adriatic Sea.

On the other side, the available Adriatic Sea data are not complete enough to exhaust the present, albeit rather simple, models. Various measurements are needed for testing the models further. Both the horizontal and the vertical modal structure of the inertia- period oscillations should be investigated, using extended arrays of moored current meters as well as towed and dropped instruments. Moreover, for answering the question of how the kinetic and potential energies are distributed over the horizontal modes, simultaneous time series of temperature and currents are necessary.

Acknowledgements--The Stas A and B data were provided by Z. Vu~ak (Hydrographic Institute of the Yugoslav Navy, Split), the measurements at Stas CI and C2 were carried out in co-operation with the colleagues from the Center for Marine Research (Rovinj and Zagreb), whereas the wind data from the Pula Airport were provided by the Hydrometeorological Office of the SR Croatia (Zagreb). I am indebted to M. Kuzmi¢5 for his help in the preparation of this manuscript, especially for the time he spent on discussions. I would also like to thank the anonymous referees, who offered valuable critiques of the preliminary manuscript. This work was supported by the Self-managing Community of Interest for Scientific Research of the SR Croatia, under Contract No. 43/0119.

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Oscillations of the inertia period on the Adriatic Sea shelf

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