Oscillations of frontal currents

ContinentalShelfResearch,Vol.4, No. 6, pp. 699 to 707, 1985. Printedin GreatBritain.

0278-4343/85S3.00+ 0.00 O 1985PergamonPressLtd.

Oscmations of frontal currents T. A. MCCLIMANS* and L. L~NSETn~" (Received 25 June 1984; in revisedform 12 October 1984; accepted 12 December 1984) Abstract--The problem of interpreting Eulerian current measurements in sharp fronts with current shears much larger than the planetary vorticity is discuss~l for the case when there are tidal and/or inertial oscillations present. For a situation taken from the Norwegian Coastal Current, long-frontal oscillations in speed exceeded cross-frontal oscillations by a factor of about 4 as a result of the frontal shear field moving back and forth past the moored current meter. It is shown how these large amplifications can be used to estimate the cross-frontal shear. The example highlights the problem of computing energy density spectra in frontal regions.

INTRODUCTION

THE Norwegian Coastal Current contains the northern outflow of much of the fresh water runoff from Northern Europe. Although often depicted as a stationary residual flow with seasonal variations, recent observations from satellites (JOnANNESSEN and MORK, 1979; MCCLIMANS and NI~EN, 1982), moored current meters (LONSE~ et al., 1982), and laboratory simulations (VINGERet al., 198 l) show large meanders in which the front between the fresher coastal water and the more saline Atlantic water wanders from 20 to 100 km offshore. The satellite thermal image in Fig. 1 gives an example of the types of meanders encountered in this area. The lighter shades indicate cooler water. The meanders move northwards about l0 to 40 km day -~ and the local duration of these events is often several days. Long-frontal velocities within the coastal current may be more than 3 kn. An example of the large currents recorded at 2 m depth at the mooring marked with an x in Fig. 1 is shown in Fig. 2. The duration of the event is on the order of 2 to 3 days, judging from the presence of the colder coastal water at this large offshore distance. The general impression of the current development in Fig. 2 is an initial northeast (long-frontal) flow followed by a northern and a northwest (long-frontal) flow as would be experienced by a fixed observer situated in the outer region of a northward propagating meander. A semi-diurnal oscillation of the speed in the main flow direction is quite apparent. It is our goal to explain these large oscillations and describe how they may be used to estimate cross-frontal current gradients. The method can distinguish wind-induced oscillations from the total signal, allowing for improved estimates of air-sea interactions for these baroclinic flows. PERKINS (1976) and WELLER (1982), among others, have considered the effect of largescale vorticity on the characteristics of Eulerian currents. To our knowledge, no one has addressed the problem of measuring currents in a frontal region with large shears in the presence of tidal and inertial oscillations, although POLLARD and MILLARD 0970) were *Norwe~an Hydrotechnical Laboratory, Trondbeim, Norway. ~"Oceanographic Company of Norway, Trondbeim, Norway. 699

700

T . A . MCCLIMANS and L. L~NSETH

apparently aware of it. Using an example from the Norwegian Coastal Current, the present work shows that this effect can produce large amplitude pseudo-signals and thereby give a fictitious contribution to spectral energy density. P A R A D O X OF T A I L W I N D S OVER B A R O C L I N I C C O A S T A L C U R R E N T S

In an attempt to obtain wind factors for wind-induced currents off the west coast of Norway (LONSETHel al., 1982) there were observed several situations of anomalous behavior, i.e., negative wind factors. McCLIMANS and EIDNES (1983) explained in three paradoxes the possibilities of such anomalous results. One of them, relevant to the present work, states that the speed of a baroclinic coastal current increases when a sustained tail wind stops. The explanation follows. The distribution of the longshore velocity of a baroclinic coastal current is depicted for a downwelling situation (Fig. 3), i.e., a sustained tail wind. The lateral (offshore) component of the momentum equation is

dv/dt + f u + g~;/~y = 0

(1)

in which u is the vertically integrated coastal current speed in the longshore (x) direction, v is the vertically integrated coastal current speed in the offshore (y) direction, f is the Coriolis parameter, g is the accleration of gravity, ~ is the surface elevation, and t is time. When the tail wind abates, the stored potential energy will cause a lateral offshore spread of the buoyant coastal water as shown by the dashed line in Fig. 3. In the absence of friction, the acceleration of the longshore velocity u of a particle can be expressed

du/dt-fv = 0

(2)

after the wind stops. Since v > 0 when the tail wind stops, then du/dt > 0. With little damping, the ensuing motion should be characterized by inertial oscillations and it is tempting to conclude that the oscillations seen in Fig. 2 are such oscillations. It will be shown, however, that the lateral shear of the frontal velocity field plays a dominant role in the magnitude of the oscillations as observed by a moored current meter. All abrupt changes in wind will cause inertial oscillations (POLLARD and IVIILLARD, 1970). When the shear layer is narrow compared to the region being forced by the wind, it can be assumed that inertial oscillations will be superimposed on the basic flow such that the inertial velocities uz and vt are functions of time only (PERKINS, 1976). For the purpose of discussion, then, the solution for the undamped inertial motion will be taken as

ut(t) ----~ sin (2nt/Tl),

vi(t) = v~ cos (2nt/Tt).

(3)

Here T~ is the inertial period, which is close to the semidiurnal, lunar tidal period in the area of interest, and ~ is the (offshore) amplitude of the inertial oscillation. E U L E R I A N C U R R E N T O S C I L L A T I O N S IN T H E F R O N T A L R E G I O N

Of particular interest here is the frontal region for which 9u/gy < 0. Expanding (2) to an Eulerian frame, as recorded by a moored current meter, gives

~u/~t = f v - u ~u/~x - v ~u/~y.

(4)

/ili~ ?? ?i~i I / ~// / ! : i i I ~ 1 % 1

Fig. 1.

...... ...... . . . ..... ~

/ ........

Satellite thermal image of a coastal current whirl near the current meter mooring (x). NOAA-6 image from 13 April 1981 courtesy Troms¢ Telemetry Station.

Oscillationsof frontalcurrents

;-.-.

703

T

0 50 100 cmls SEN.E i , J

w

8 *C

6 v

4

-~--, "f

2 Fig. 2.

8

4 5 6 MARCH 1981

7

8

The time history of current vectors and temperature at 2 m depth during the passage of a coastal current meander c a . 70 km offshore.

Due to the large negative gradient in the frontal region, the last term in (4) acts as an amplifier for long-frontal components of tidal and inertial signals measured at a point. In the absence of meanders, au/ax = 0 and the frontal gradient au/ay can be computed from the field observations by au/ay = f - (l/v) au/at.

(5)

To obtain the cross-frontal motions, it is necessary to take into account the time changes in the frontal direction. This is done by constructing the 25 h running average velocity, assuming that the time scale of frontal direction changes is on the order of days. This is equivalent to setting uau/ax ,~ vau/ay in (4) and taking the x axis along the front. The stick diagram in Fig. 2 implies a slowly changing direction. y,

L '

..

I

"-.... " " - . . .

WIND ~-

.....

~ " ~

Y,,-.

FRONTAL REGION -

~u/(3YI:'=" f

..............

>×

COAST Fig. 3. Assumed lateral distribution of longshore currents (u) before ( ) and after ( - - - - - ) a sustained tail wind ceases to blow over a baroclinic coastal current (northern hemisphere).

704

T . A . MCCLIMANS a n d L. L~NSETH

Tidal and inertial oscillations are obtained by subtracting the 25 h running average from the observed values. The time series of this (deviation) current is then presented as a hodograph. Long-frontal elongations on this hodograph would reveal possible contributions of the amplification term in (4). The amplification of the long-frontal tidal and inertial currents in Eulerian measurements may be expressed in terms of an amplification factor a = ff'/(~)

(6)

in which ~ is the observed amplitude of the long-frontal velocity oscillations and u r $ " u l is the amplitude of the long-frontal velocity oscillations of the tidal motion (computed from the long-term harmonic analysis of current records) plus the long-frontal inertial motions obtained by applying (3) to the cross-frontal component (which is obtained by subtracting the cross-frontal component of the tidal motion from the observed cross-frontal motion), i.e., u i = lYI .

Knowing vl, it is possible to isolate inertial oscillations in rather short records from a single mooring in a frontal region. With simultaneous wind observations, it is then possible to obtain a wind factor defined as the ratio of the wind-induced surface current to the wind speed measured at 10 m height. With a linear negative gradient in the frontal region as depicted in Fig. 3, v and ~ u / ~ t are in phase and the velocity shear in the frontal region can be estimated by 3u/3y = f - 2~u'/vT,

(7)

in which ~ is the amplitude of the cross-frontal oscillations and T is the period of the oscillation. In cases where the inertial oscillations dominate over the tidal current (~>>~) or the tidal ellipse is circular (~r = ~r), (7) can be approximated 3 u / 3 y = f - 2rra/T.

(8)

AN EXAMPLE FROM THE NORWEGIAN COASTAL CURRENT Both the instantaneous (10 rain average) and the 25 h running average of the north and east components of the current of Fig. 2 are shown in Fig. 4. The hodograph for the (deviation) current is given in Fig. 5. It shows a large elongation in the long-frontal direction. The phases of these oscillatory motions reveal that the maximum northeasterly (deviation) currents (positive long-front) occur when the front has advanced to the northwest (positive cross-fron0. The computed tidal hodograph is shown in Fig. 6. A comparison of the cross-frontal components shows more action in the data than can be explained from tides alone, implying wind or other forcing. The important difference, however, is in the long-frontal direction. Here the observed amplitudes are an order of magnitude larger than the tidal components. Wind forcing can hardly explain these values. The difference between the observed cross-frontal velocity and the computed cross-frontal tidal component is shown in Fig. 7 together with the wind speed and direction. Clearly, the oscillations here can be interpreted as wind-induced inertial oscillations, knowing the circumstances so well. The arguments leading to the solution (3) imply that the long-frontal component of an inertial oscillation has the same amplitude as the cross-frontal component, shifted in phase by something in excess of 3 h. Adding this to the long-frontal tidal oscillation of Fig. 6 shows that the amplification factor for this situation is a = 5.4 on the average.

Oscillations of frontal currents

705

100

cm S

~....~ 50

'" " t

NORTH

w

o

0

, ---~.,-%

v

-5¢

100, cm

s

50

r~.4: I .

EAST

.~.

o .~ -50

/%.."'I'" V".,I ~"\ ~-... , , ~ / t~...; ~ U~L!U v

2

3

4

5

6

7

Fig. 4. North and east components of the 10 min average ( ( - - - ) currents shown in Fig. 2.

8 MARCH 1981 ) and 25 h running average

With a m a x i m u m operating angle o f 20 ° from the vertical, the 300 m taut mooring used for the measurements o f Fig. 2 sways about the anchor with a radius o f 100 m. The current direction varies +45 o during the oscillations, producing a lateral motion o f + 70 m. In comparison, the tidal excursion is about 1 km. Thus, the observed lateral motion is a bit less than the actual motion. This applies also to the longitudinal motions. The amplification factors computed with the present procedure are therefore larger than those obtained from a truly Eulerian measurement. F r o m (7) the cross-frontal velocity gradient is estimated to be a u / a y = 1.3 x 10 -4 s -l - 5.9 x 10 -4 s -1 = - 4 . 6 x 10 -4 s -l _~ - 1 kn k m -1

which is about 4 f . The m a x i m u m shear within this region was p r o b a b l y higher.

cm/sSOl

!

/ ~ / ~ y212

Fig. 5. Hodograph for the deviation current at 2 m depth during the passage of the meander of Fig. 2. Numbers denote hour: ( ~ ) 4 March; ( - - - ) 5 March.

706

T.A. MCCLIMANSand L. LONSETH

10.

cm/s NORTH 6

5-

I

-10

-5

18

' /~,~' ~ 0L/?, 21/ 24

~

5 cm/s EAST

10

t ~'~

\ \N \ \

-5"

II I

12

\ql 12

-10 Fig. 6.

The hodograph for the local tidal current computed from long-term measurements. (See also Fig. 5.)

Figure 7 reveals an oscillation of amplitude 7 cm s -~ induced by the 12 to 15 m s -1 wind on 4 March. The large amplitude at the end of the series indicated that the frontal rotation was too rapid for the 25 h running average to be a meaningful fdter. In this case u a u / a x is not negligible. The impulse of the wind produced a current of 7 cm s -~ at 2 m depth within a few hours. The magnitude of the current at 2 m depth for the ca. 10 h impulse of ca. 12 m s -I wind measured at 80 m height corresponds to a wind factor of 0.7% for the reduced I0 m wind of 10 m s -l, implying that the 2 m depth gives a low value for a wind-induced surface current over a stratified sea (I.~NSETH et al., 1982). MCCLIMANS and EIDNES (1983) explain some of the circumstances which can lead to reduced wind factors in baroclinic coastal currents. POLLARD and MILLARD(1970) discuss others.

DISCUSSION

Single point (Eulerian) current moorings in baroclinic coastal currents reveal a variety of phenomena including densimetric flows, tides, and inertial oscillations. An example from Norwegian coastal waters shows a long-frontal amplification of the tidal and inertial motions, an artifact of the Eulerian measurements. The method described here gives an estimate of this

707

Oscillations of frontal currents

l o (a)

0

~

=

:--

-1(

01

-

!l 'c' NJ

I

F..o,o ~ ' ~ ~ _ ~ = ~ - -

3 AR

4 AR

'

SMAR

~ " CROSS FRONTAL

6MAR

Fig. 7. A comparison of (a) the difference between the cross-frontal components of the deviation current (Fig. 5) and the tidal current (Fig. 6) with (b) wind speed and (c) wind direction observed at 80 m altitude.

amplification using hodographs for the observed (deviation) currents and instantaneous tidal currents derived from long-term measurements. The shear amplification makes it difficult to assess energy spectra in frontal regions. It is possible, however, to extract information on cross-frontal velocity gradients. Acknowledgements--We wish to thank A/S Norske Shell for permission to use the field observations and anonymous referees for valuable suggestions for improving the presentation. This work has been supported by the Fund of License Fees. REFERENCES JOHANNESSEN O. M. and M. MORK (1979) Remote sensing experiment in the Norwegian coastal waters, Samarbeidsprosjektet den Norske kyststram. Report 3/79, Geophysical Institute, University of Bergen. LONSETH L., S. HAVER, J. P. MATHISEN and S. TRYGGESTAD (1982) Environmental conditions at Block 31/2: analysis of METOCEAN data from June 1980 to June 1981. OTTER Report STF88 F82009 to A/S Norske Shell (Proprietary). McCLIMANS T. A. and J. H. NILSEN (1982) Whirls in the Norwegian Coastal Current. In: Coastal oceanography, H. G. GADE, A. EDWARDS and H. SVENDSEN,editors, Plenum, New York, pp. 311-320. McCLIMANS T. A. and G. EIDNES (1983) Three paradoxes of tail winds over baroclinic coastal currents. Ocean Modelling, 50, 12- ! 3. PERKINS H. (1976) Observed effect of an eddy on inertial oscillations. Deep-SeaResearch, 23, 1037-1042. POLLARD R. T. and R. C. MILLARD, Jr. (1970) Comparison between observed and simulated wind-generated inertial oscillations. Deep-SeaResearch, 17, 813-821. VINGER /~., T. A. MCCLIMANS and S. TRYGGESTAD (t981) Laboratory observations of instabilities in a straight coastal current. In: The Norwegian coastal current, Vol. 2, R. S.4~TRE and M. I~ORK, editors, University of Bergen, pp. 553-582. WELLER R. A. (1982) The relation of near-inertial motions observed in the mixed layer during the JASIN (1978) experiment to the local wind stress and to the quasi-geostrophic flow field. Journal of Physical Oceanography, 12, 1122-1135.