Deep-Sea Research, 1972, Vol. 19, pp. 289 to 296. Pergamon Press. Printed in Great Britain.

Inertial oscillations in the Mediterranean*

HENRY PERKINS?

(Received 20 August 1970; in revised form 30 November 1971; accepted November 1971)

Abstract--Measurements of ocean currents were made in the Western Mediterranean Sea at five depths for two months during early 1969. The data are of particular interest in that they show a dominant and persistent presence of inertial oscillations; that is, of circularly polarized currents having periods of a half pendulum day. The oscillations are found to have frequency very nearly 3 70 higher than the local inertial frequency and vertical dependence simpler than that heretofore reported. This latter feature is held due to properties of the local stratification. It is also found to be consistent in certain respects with the motion having been generated by near-surface processes.

INTRODUCTION

INERTIAL oscillations are defined for the purpose of this investigation as horizontal ocean currents in which the current vector rotates in a nearly circular, clockwise sense (counter-clockwise in the southern hemisphere) with period of about a half pendulum day. It is now generally believed that these are commonly, if not universally, found in the deep oceans of the world away from equatorial latitudes.

Earlier observations revealed that they are superimposed on a broad spectrum of other processes but are usually quite noticeable in the data and often are the most energetic constituent present, dominating even the mean current. However, the motions are transient, persisting for several days at most; indeed, the persistence is often for only a few cycles. The spatial structure of these currents is complex, having typical scales of 100 m in the vertical and, for the few cases known, 10 lan in the horizontal (WLmSTER, 1968).

DESCRIPTION OF SITE AND MOORING

From JaI1ua~ 22 to March 12, 1969, currents were measured in the Mediterranean Sea at 38 ° 01'N 5 ° 00'E, about 120 km north of the Algerian coast during Atlantis I I Cruise 49 (Mooring 289). The single mooring involved contained five instruments, modified versions of Geodyne model 850 current meters, located at 500 m depth intervals between 200 and 2200 m.

Interest in this region derived in part from its particularly simple geometry, consisting of a nearly flat bottom and a long east-west coastline to the south (Fig. 1). Also, the absence of sitmificant tides in the Mediterranean simplifies identification and measurement of inertial oscillations. Finally, historical hydrographic data show the

*Contribution No. 2546 from the Woods Hole Oceanographic Institution. Based on a thesis submitted in partial fulfilment of the requirements for the Ph.D. degree, Massachusetts Institute of Technology and Woods Hole Oceanographic Institution.

tPresent address: School of Marine and Atmospheric Sciences, University of Miami, Miami, Florida.

289

290 HENRY PERKINS

Fial.

3 o 4 " 5" ~- 7" 8 o

S

Bathymetric chart of the observing region after cdaaxt BC 3916 of the U.S. Oceanographic Otto. Contours are in fathoms (1 fathom = 1.8288 m).

Brunt-Viiis~l~i frequency to have similar vertical dependence throughout this area at the time of year when the measurements were made, and to be the same as that found when the mooring was set (Fig. 2). The region to the north of that given in Fig. 1, however, has local areas of near-neutral stability and intense vertical mixing (VOORmS and WENS, 1970; ANATI and STOMUEL, 1970). At the observing site, stability of the deep water is sufficiently low to challenge the precision of hydrographic measurements.

DESCRIPTION OF MEASUREMENTS

Mean currents at all depths were found to be nearly identical to that at 1700 m (Fig. 3) until about Feb. 10, averaging some 2 kin/day roughly southward. After that date, the four deepest instruments continue to have comparable mean properties, while that at 200 m indicated an accelerating mean current towards the north, the speed of which eventually reached about 5 kin/day.

The influence of inertial period motion is evident in all the data series (Fig. 4). Spectral analysis revealed the expected dominant peak near inertial frequency and little else except some harmonics probably introduced by the instrumentation. No tidal components were detected. The spectral energy at inertial frequency can be decomposed into two circularly polarized constituents (Moovms, 1970), one rotating clockwise and the other anti-clockwise. These are given respectively by

(Pu,~ + Pvv - - 2Q,,,~)/4 and

(P~u -q- P,,v + 2Quv)/4

where Pu,, and Pvv are the autospectra of the east and north components respectively

Inertial oscillations in the Mediterranean 291

FREQUENCY (CPH) 0 2 & 6

1000.

,=,

2000.

Fig, 2. Brunt-Vaisalii or stability frequency versus depth in the observing region at the time when the mooring was set. Depths of the five current meters are indicated by crosses.

2895 N IOIKM 31"NS*E I 1700M

O.L

2g-

<

c

FEHUAIIY

-lS

Fig. 3. An example of one of the data series as a traditional progressive vector or virtual displacement diagram. The evident oscillations have period close to inertial, 19.43 hr at the

observing site.

292 HENRY PEaaInS

DEPTHS IN METERS

% ~:~ ~,~ ~ .i / ~ ~=

m ~

---;:_+ ( ' ? --) ~ ~._~

/~+ '+-~ ...+ +., . g:

oooooooooo ooooooooooo~

Inertial oscillations in the Mediterranean 293

and Quv the quadrature spectrum between them. The oscillations are found to be almost completely circularly polarized in the clockwise sense (Table 1).

The strength of the inertial oscillations in this data permits their frequency to be resolved somewhat better than is available through conventional spectral analysis. The technique selected for doing this consisted of breaking each data record into overlapping pieces of two inertial periods in length and subjecting each piece to a harmonic analysis at inertial frequency for the energy constituent having circular clockwise polarization. Small departures from inertial frequency cause a slow drift in phase from one section of the data series to another, a fact confirmed by testing the procedure on synthetic signals having much higher noise levels than the data.

Table 1. Kinetic energy density in (cm/sec)S/cph at inertial frequency decomposed into counterclockwise and clockwise constituents.

Depth (m) Counterclockwise Clockwise

200 12 2167 700 4 434

1200 3 164 1700 3 603 2200 9 106

Application of this method indicates a clear systematic departure of the data towards periods shorter than inertial (Fig. 5). Particularly noteworthy is the first part of the data at the fourth (1700 m) level, which is exceptionally sinusoidal in appear- ance. The rate of phase change there is remarkably uniform and corresponds to a frequency 3 % above inertial. (In comparison, long-term drift of the internal instrument clocks was less than 0" 1%). A similar trend is apparent in the other four curves during

AMPLITUDE (cml,,ec)

,°°I.

0,0 t •

PHASE (DEGREES)

o,\

° I

180 23 312 I0 20 Z8 10 23 312 10 20 28 tO JANUARY FEBRUARY MARCH JANUARY FEBRUARY MARCH

Fig. 5. Harmonics at inertial frequency of successive short sections of the five series of current measurements are presented here as ampfitude and phase. The inset shows percentage increase

of observed over local inertial frequency as a function of slope of the phase curves.

294 HENRY PERKINS

about the first 3 weeks of the data series. Also of interest during this period are the phase relations between the various data series, particularly the reversal between the second (700 m) and fourth (1700 In) series. Those portions of data for which the phase estimates seem least stable correspond in general to intervals where amplitudes are smallest.

Coherence estimates do not seem a particularly useful characterization for signals of this type, but we note that each consecutive pair of data series shows coherence at the 95 ~ confidence level for frequencies near inertial and for no others. The single exception is for the 1700 and 1200 m pair where untimely phase changes between the two signals sharply reduce the computed coherence.

DISCUSSION

Mean or quasi-steady currents can produce significant effects on osciUations of the type considered here. A straightforward Doppler shift of 3 ~ near inertial frequency with a 2 k i n / d a y mean current, corresponding to the first portion of the observing interval, indicates a wavelength of order 100 kin. However, the behavior of inertial oscillations in meridional mean currents is not known. Nevertheless, steady zonal currents with shear in the vertical or horizontal can modify the limits of the frequency band where internal waves exist (MAGAARD, 1968) and vertical shear may transfer energy into the inertial frequency range (FgANKmNOUL, 1970). The observed mean vertical shear was significantly different from zero only between the two uppermost instruments and there only after about Feb. 10, which may account for the change in the character of the inertial oscillations at that time (Fig. 3). In what follows, all of these effects are neglected.

An exact, numerical evaluation of the eigensolutions for the governing equations in the irregularly shaped Western Mediterranean basin does not appear within reach. But by making suitable geometric approximations, some insight into their structure can be gained. Supposing the water depth to be constant at 2800 m leads to a simple, separate equation for the vertical dependence of the vertical component of velocity W for periodic motions (MUNK and PmLLn'S, 1968). It is given by

d 2 W / ~ ~ ÷ k2(z) W = O,

where k(z ) : 7 ( N l ( z ) - - 'r2 D")~/2a O .

Boundary conditions on W at the free surface and bottom are respectively

d W / d z - - g T * W / ( 2 a D ) ~' : 0

and W = 0 .

Here z is the upward-directed vertical co-ordinate, N the Brunt-V~listUli frequency, a the radius of the Earth, ~9 the Earth's rotational rate, g the acceleration of gravity, 7 a dimensionless eigenvalue to be determined as part of the solution, and q the wave frequency divided by t2. The expression for k, the local vertical wave-number, differs from that of Munk and Phillips in that it depends on a, which arises from retaining time derivatives in the vertical momentum equation. Once W ( z ) is known, the vertical structure Z (z) of the horizontal currents is found from

z (z) = 4al,, ~,~ d Wlaz,

Inertial oscillations in the Mediterranean 295

where a D is taken to be 1.03 times f ( f : 2 9 times the sine of the latitude) in the present case.

From representative solutions to the above equations (Fig. 6) it is evident that the great variation in N(z), especially its low value below a few hundred meters depth,

~r= 1028 181i4 3078 39~16

i,i o o o o

Fig. 6. Horizontal currents of the first four vertical modes at inertial frequency at the observing site together with the corresponding eigenvalues y. The modes are referred to as Nos. 1-4 in order from left to right. Not shown is mode 0, the barotropic mode, which is nearly independent of depth and corresponds to y ~ 5. It can be dismissed on the grounds that its horizontal scale

is too large for the Mediterranean basin.

has a strong influence on the modal structure. The very small values of N, and therefore of k, which prevail at these depths require slower current variation with depth than would typical oceanic stratification. This effect may account for the apparently simple vertical structure of the present observations.

Generation of inertial oscillations by meteorological processes is now established (C'~PON, 1969; POLLARD and MILLARD, 1970; GONELLA, CREVON and MADELAIN, 1969), the mechanisms believe to be responsible acting only near the sea surface. The resulting oscillations may, under suitable circumstances, be expanded into vertical normal modes (HOLLAN, 1969; POLLARD, 1970). I f the initial excitation is confined to the upper few tens of meters, the resulting modal expansion has signifi- cant contributions from a large number of modes when N corresponds to the usual oceanic case. But the stratification considered here leads to modes with suitable near- surface structure so that relatively few low order modes suffice for such an expansion. Indeed, the first 4 modes of Fig. 6 have zero crossings near 50 m, the depth of the sur- face mixed layer. Again the effect would be to simplify the vertical structure every- where, particularly in the deep water. Although the present data could in principle be expanded into vertical modes, yielding amplitudes for at most 5 of them, the very sparse vertical sampling together with uncertainties in the stability make such an expansion unduly speculative.

Known separable solutions for the horizontal dependence of inertial period currents consists of sine waves in the east-west direction and Airy functions in the

296 HENRY PlmrdNs

north-south direction (MUNK and PHILLIPS, 1968). By suitable adjustment of the zeroes of these functions, the solutions can be adapted to basins having meridional and zonal boundaries. A zonal boundary at 37°N and a wave frequency near f , for example, lead to horizontal scales smaller than the dimensions of the Mediterranean except for the barotropic mode, which is therefore excluded. Conversely, prescribing the horizontal scales by further inclusion of meridional barriers separated by some 600 km yields a discrete spectrum of eigenfrequeneies. These have been tabulated for the 5 lowest modes in each of the 3 space dimensions, a total of 125 frequencies. They were found to have periods scattered quasi-uniformly primarily between 16.0 and 19.5 hours, easily including the range of observations. The corresponding frequency spread is consistent with the observed signal persistence of around two weeks.

Thus it seems that the apparently simple vertical dependence of the observed inertial oscillations is related to the special density stratification of the Mediterranean. While the data is not adequate to determine the prevailing modal distribution the hypothesis of surface generation predicts relatively lower modes than would occur under typical ocean stratification. This hypothesis is not inconsistent with the hori- zontal scales of the Mediterranean nor with the vertical dependence, frequency and persistence of the observed motion.

Acknowledgements--The author is greatly indebted to Dr. FE~as W~rER for his help throughout the study. Assistance from many members of the moored current meter group at the Woods Hole Oceanographic Institution is also gratefully acknowledged.

This work was supported by the National Science Foundation through grant GA10208 and by the Office of Naval Research through contract No. 0014-66 CO241. Portions of the manuscript were prepared while the author was a guest investigator at the Institut f~r Meereskunde in Kiel, Germany.

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CREPES M. (1969) Hydrodynamique marine en r~gime impulsionnel. Cah.oc~anogr., 21, (9), 863-877.

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VOORmS A. D. and D. C. WEnB (1970) Large vertical currents observed in a winter sinking region of the Northwest Mediterranean. Cab. oedanogr., 22, (6), 571-580.

W ~ s r ~ F. (1968) Observations of inertial-period motions in the deep sea. Rev. Geophys., 6, (4), 473-490.