DeepSea Research, Vol.29, No. 3A, pp.285 to 306, 1982'Priilted in Great Britain.

0198-0149/82/030285-22 $03.00/0Pergamon Press Ltd.

Tidal currents in the northwest African upwelling region

J. M. HurnNANcE* and P. G. BruNrsf

(Received 27 October 1980; in rgvised.form l0 July l98l ; accepted 29 July l98l)

Abstract-Tidai currents are analysed fuom 2'l instruments on five moorings having 28 days incommon during "Auftrieb '75-Upwelling

'75". The diurnal currents are largely incoherent,probably owing to proximity ofthe local inertial frequency. The barotropic semidiurnal currentsare modelled b! a Kelvin wive propagating northwaids with an estimated M2 tidal energy flux of2xl}rL W across 23"N. The baroclinic semi-diurnal currents are much less coherent and arestrongest at the moorings in 50Gm water depth on the steep continental slope, where they.aredominated by intermittent near-surface and bottom-intensified motion, These bottom-intensifiedcurrents are mostly aligned with the shelf and can only sometimes be modelled as a corlrbination ofRHnIrs' (1970, Geophysical Fluid Dynamics, l, 273-302\ bottom-trapped waves. The bottomcurrents are also much larger than the 0 (2 cm s-1) estimate for internal tides generated by theonshore component ofthe barotropic current according to colventional models. We suggest thatthe strong currents are caused by the stronger longshore component of the barotropic currentinteracting with longshore topographic irregularities in the shelf and slope. Variability in thebasocliniclide is mainly attributed to changing stratification in the upper 150 m. The Mo harmonicoccurs intermittently at 50G'm depth on the near-critical continental slope.

INTRODUCTION

THe "Auftrieb '75-Upwelling '75" experiment took place off northwest Africa from late

January to early March 1975 (Bnocr,rMANN, HucHes and ToMczAK,1977). Numerous CTDcasts and five successful current meter moorings along three cross-shelf sections {, B, and C(Fig. 1) enable study of water movements over periods of days or weeks (Torrrcznr andHucHrs, 1980) and of tidal currents in three dimensions.

The tidal currents are intefesting owing to (a) their amplitude, which is comparable with

that of the longer-period motions; (b) proximity of the diurnal and inertial frequencies;(c) the contribltion of northward energy fluxes to the large North Atlantic semi-diurnaltides; and (d) ihe presence of significant internal tides, already studied in this area by HoRNand MeNcre (1976) and GonnoN (1979). Since the length of continental shelf and slopefrom 21" to 26'N was'chosen because of its minimal longshore topographic variations, theobservations provide a favourable opportunity for testing present conceptual models oftidal currents; these models alrnost invariably'assume uniform conditions alongshore.

The plan of the paper is as follows. After a discussion of the data, including spectralanalysis that leads us to specialize on the semi-diurnal tides, we consider a Kelvin-wavemodel for the barotropic currents. The model assists extrapolation oceanwards from theimmediate vicinity of the observations, so that an estimate can be made of the totalnorthwards tidal energy flux across 23"N. We then consider the large baroclinic currents at

* Institute of Oceanographic Sciences, Bidston Observatory, Birkenhead, Merse,yside L43 7RA, E{tgland.

t CSIRO Division ofatmospheric Physics, P.O. Box 77, Mordialloc, Victoria 3195, Australia.Crown Copyright 1982.

t2285

286 J. M. HurHN,c,Ncn and P. G. BAINES

g1') at-,

Fig. l. Plan of current meter moorings. Lupine (40Gm water depth; see GoRDoN, 1979) and thearea 'HM'of measurements discussed by Honu and MerucxB (1976) arc also shown.

82 and C2 on the steep continental slope in terms of presently known models for (a) freewaves propagating over a shelf or slope and (b) internal tides generated over the shelf andslope by the onshore barotropic component of the tidal current. Purely empirical analysisinto orthogonal modes is also performed, providing a measure of the efficacy of theconceptual models. In the final sections we discuss various aspects of the interpretation ofthe data, including the first harmonic (Mo), and the conclusi6ns are summarized.

DATA

Table I shows the positions of the 27 cuqrent meters (from five moorings) yielding datafor a common period of 28 days. The time series were reduced by separate diurnal andsemi-diurnal band-pass filters to 336 2-hourly readings commencing at 1200,3 February1975. With regard to the diurnal currents, three aspects of the observations indicate thatthey contain extra inertial contributions, viz. (a) the diurnal currents were found to have acoherence of only I5/, relative to the local surface tide deduced from tide tables(Anurn*rv, 1979), whereas for the semi-diurnal currents the comparable figure was 53/o;(b) the r.m.s. diurnal current speeds were more than half those of the semi-diurnal currents,greatly exceeding expectations based on diurnal tidal elevations, which are only l0/, of thesemi-diurnal elevations; and (c) the sum of section B current spectra (Fig. 2) shows thediurnal tides barely rising above the broadband inertial motion. These aspects woulddegrade any analysis for the diurnal tides, which has not been pursued further. HonN andMnrNcrr (1976) found a similar merger of diurnal currents into a broad inertial peak, even'though their measurements at 19' to 20'N were further from 'critical latitudes' (e.g., 27.6'N,where the inertial frequency matches the O, diurnal tidal constituent). We note thepossibilities at critical latitudes for internal tide generation by topography (HeNoERsHorr,

Tidal currents in the northwest African upwelling region 287

Table 1. Current meter moorin{ data

Mooring Water depth Instrument depths

A3B1B2B3C2

25'37.3',N l6'38.9'W22"44.t'N 17' 3.dW22"47.6'N t7"t2.gw22'54.0N r7"45.1'W2l'22.8',N 17"40,1',W

3026't4

5152015507

99,228,1 105,437,750139 ,49 ,59 , @,69 ,7275, 165,290,365,5057 5, 165, 290, 365, 505, 8156'.1 , t57 ,282, 357 , 497*

fNo temperature time series.Short velocity time series.

o l z

Fig,2. Sum of all section B current spectra divided by 17 : number of meters. Units, cm2 s-2 ineach band (width l/30 cycle day-r).

19?3) or by coupling with the barotropic tide via the horizontal components of the Coriolisforce (Mlles ,1974). The resulting baroclinic currents may be comparable with barotropicdiurnal currents (which will not be substantially affected), and incoherent away from theimmediate area of generation.

By contrast, the semi-diurnal tides are prominent in Fig. 2, and we follow Honu andMencxe (1976) and GoRDoN (1979) in concentrating on them. Figure 3 represents thehorizontal currents at each meter by a tidal current ellipse, the mean over28 days : 4 weeks. A measure of variability is

where i indexes weekly averages and j the two component directions. The u,, are complex

current amplitudes relative to the barotropic spring-neap cycle modelled below (so thatonly variability additional to the spring-neap cycle is measured) and an overbar denotes an

average over the four weeks. Variability within weeks is usually somewhat less, owing to

the narrow semi-diurnal spectral peak (Fig. 2), which is well covered by the semi-diurnalband-pass filter of width.0.4 cycles day-1, approximately. The values of 52 (Fig. 3) showthe greatest variability to be at the slope moorings 82 and C2 in 50Gm water depth. Unlikeobservations offOregon (ToncruusoN and Htcrev, 1979), there is little correlation betweenl/S2 and onshore-offshore ellipse alignment. the latter being prominent only at Bl as part

of the barotropic Kelvin-wave form (see below).

4 2 1 2S ' = i ; f l u r r - a t l ' f \ l u 1 l ' ,

i = l j = l I j = r( 1 )

288 J. M. HurHNeNcE and F, G. Bernns

S * i * l s/t /' rt$ltg

+I

+ 1 1o

6

s $ s s E\ . \ \ \ \\ \ \ \ \

I P e e eo o o 6 6

t q $ = s .q | \'-+- --+-

+

$* B. C5*5 s-{

+

6Jo

o +

tr

fr/ ,/ F-{-

8 t BN 6

t l

g o & ; B-,--t-:- -u-T_ ---u-

8 B{ o

$ 3d o

, t ( Ftro 9'-r'

+, r

+ + l E

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+ k '\t-i- "u+

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o

Tidal currents in the northwest African upwelling region

BAROTROPIC MODEL

Model description

We consider the semi-diurnal band-passed data and take the barotropic current at eachmooring to be the mean of the 3, 5, or 6 current records from the various depths. The non-zero internal tide contribution to such simple mooring averages is expected to be removedin subsequent averaging; we seek a coherent motion between the five.moorings over aperiod offour weeks, and internal tides are usually incoherent over such large distances andtimes (WuNscn, 1975). Such mooring averages are also supported by analysis intoempirical orthogonal modes (e.g., KuNou, AILrN and SuIru, 1975), which.shows that mostof the coherent energy is retained : the first mode of the mooring-averaged currentS includes90% of the energy contained in the first mode based on all the individual currents. Themooring averages include 64.7/, of the total data variance.

The model for the barotropic current comprises (a) a ldirect tide'and (b) a Kelvin wave.Respectively, (a) and (b) are a particular integral and a complementary solution of thelinearized inviscid shallow-water (long-wave) equations forced by the tidal potential. Theamplitude and phase of the free Kelvin wave are chosen to give the bestleast-squares fit tothe data. This procedure follows Munx, SNoocnms and WurusH (1970), except that hereno Poincard waves ate included, because their pattern of currents is almostindistinguishable from a Kelvin-wave pattern over the limited extent of the observations.

In calculating the direct tide and the Kelvin-wave form for the mean frequency Mr, astraight shelf was assumed, but separate calculations were made for the actual offshoredepth profiles along each of the three sections A, B, and C, using the numerical methoddescribed by CnrowelI., CurcHrN and LoNcurr-HtccrNs (L972). The boundary conditionsfor both (a) and (b) are: no flow through the coast and decay far offshore. The arbitraryKelvin-wave amplitudes were'linked between the sections A, B, and C to conserve thelongshore energy flux, and the phase differences corresponded to the calculatedpropagation speed or wavelength (7700, 67-50, and 7260 km at A, B, and C). Hence, thedirect tide, together with any overall amplitude and phase of the calculated Kelvin wave,satisfies all the conditions for barotropib motion against a straight coast and shelf profile.Baroclinic motion generattid. by interaction between the barotropic current andtopography (e.g., Bnwrs, 1973) is considered later.

52 (only) is added in the same proportion as in the equilibrium tide, so that the model,'stime dependence is simplified to include only the mean semi-diurnal frequency M,modulated by a spring-neap cycle. A lag T (the 'age' of the tide) in the Kelvin-wavemodulation is also allowed and chosen for the best least-squares fit to the data. By using theKelvin-wave structure to link all the records in the least-squares fitting process for T, wemay estimate the age despite its inaccessibility from individual current records, each havingincoherent contributions giving almost random ages. This would also be achieved byjointly analysing the barotropic currents into empirical orthogonal functions.

Resalts: currents

The modbl currents, after fitting over the whole 28 days, are shown in Fig. 3 and accountfor 60.4/,of the barotropic current variance. The direct tide is at most 0.3 cm s-r for M2,so that most. of the current (and surface elevation) is contributed by the Kelvin wave.Confidence limits of 95/" determined by the F-test for linear regression are approximately+12% in amplitude and *7o in phase, given the model.

289

290 J. M. HtmrNlNce and P. G. BelNes

The given spring-neap cycle appears to be followed. Separate fits for each ofthe 14 2-daysegments of record account for 63/. of the barotropic current variance, an insigrrificantimprovement over the 28-day fit assuming the cycle. An age T of 36 + 15 h (gs%confidencelimits) is found, consistent with observed values averaging about 40 h for tidal elevations inthe area.

The model currents match the observations in alignment, amplitude, and phase quitewell overall. The figure 60.4/. may be compared with the maximum 77.3/, of thebarotropic current variance attributable to any one spatial form and time dependence (thefirst empirical mode). [Both figures are sample-based, incorporating artificial coherence,and overestimate any predictive skill of the respective models (Devls, t976).f However,there are local differences between the Kelvin-wave model and the first empirical moderepresenting observations. The model currents are mis-aligned (10' to 50') at 43 and 81,early at B1, and too small (by 20 to 4O/.) and late at B3 (Fig.'3). Two likely causes for thesedifferences are non-parallel shelf topography and coherent baroclinic contributions to themooring-averaged current. Moorings A3,82, 83, and C2 all show coherent variations ofthe currents with depth (Fig.3); sampling only the upper levels at A3 bnd 83 probablycauses the mooring average to exceed the barotropic current by a baroclinic contribution.Less probably, the particularly large currents at the deep mooring 83 might result fromsubstantial (but uncharted!) topography; convex features may enhance currents (e.g.,HursNeucr,1974). The observed alignment of the barotropic current at 81 cannot besimulated by a model with parallel depth contours; it is probably a result of localtopography, of limited extent [0 (10 km) say] so that 82 currents are not similarlydeflected.

The currents atB2andc2 confirm those at Lupine (water depth 400 m, Fig. 1 ; Gonnon,r979t.

Surface elevation

The model's surface elevation is 1..08 m, phase lag 335' for M, at the coast inshore ofsection B, and is due almost entirely'to the Kelvin wave rather than the direct tide. Muchadditional information regarding the surface elevation is available for comparison. Manyco-tidal charts have been derived both empirically (e.g., Drernror, L944\ and numericallyfrom first principles (Acclo and Pernms, 1978). Both methods show a combination of(a) an almost plane progressive wave arriving from the South Atlantic and (b) a Kelvin-likewave propagating cyclonically along the North Atlantic coasts around an amphidromenear 50"N, 39'W. Figure 4 for M, has been sketched using published harmonic constantsin the Aournalrv Trne Tesr,Es (1979), cnnrwrrcur, zerLER and HeuoN (1979), 29-dayanalyses of four equatorial Atlantic records obtained by I.O.S. in November to December1978, and a year's analysis of a Brasilian record on Fernando de Noronha (D. E.Cnnrwnrcnr, personal communication). S, is similar to M, with an amplitude factor :0.4and a phase lag:49' .

Surface elevations in the model are about 20' earlier than observed, and are larger with alonger wavelength (each x 1.8 approximately) than the local estimates 0.6 m, 4000 kmfrom Fig. 4. These amplitude and wavelength factors are consistent, cancelling in theireffect on the currents by the longshore momentum equation

Tidal currents in the northwest African upwelling region 291

Fig. 4. Sketch of M2 tidal amplitudes (----) in centimeters and phases (-).

wherever the onshore velocity component is small relative to ouf f, i.e., everywhere exceptBl (/is the Coriolis parameter, o the M2 frequency). However, the factor being 1.8 and notI dces suggest the presence of a Poincar6 wave in addition to the Kelvin wave modelled.Accepting in particular the observed 400Gkm wavelength alongshore, the dispersionrelation (e.g., Muxx et al.,1970) in the.deep water further offshore (supposed uniform at2500 m) gives an offshore decay scale of 750 km, considerably less than the figure 2300 kmfor the Kelvin wave alone. We emphasize that these estimates depend on tidal elevationmeasurements over a region much more extensive than the current meter sections A to C.

Energy tluxAlthough the current measurements are confined to the coast, we can extrapolate

offshore by means of the model to estimate the longshore energy flux

(ClnrwntcHr, EDDEN, SpeNcen and Vessre, 1980; (" is the equilibrium tide). Allowing forthe reduced elevations and offshore scale discussed above, the flux across 23"N is estimatedas 2 x 10r r W. The major uncertainty is the offshore decay scale; 750 km is probably alower limit and if increased the flux estimate increases nearly proportionally.

For comparison, GonooN (1979) found 7x10f Wm-l along the 4OG'm contour atLupine. Scaled up to 4x10s Wm-l in the deeper water (2500m) offshore, where thelongshore current is similar, and extending over the energy flux decay length |(750 km)offshore, a similar total flux is found.

oo ln

292 J. M. [IursNeNcB and P. G. BlrNres

At 38"N, CnnrwnrcHr et al. (1980) found a northward flux of 2.46xl0rr W estimatedfrom observations, and R. A. Ft-nrnen (personal communication) founC a value5.5 x 1011 p from a numerical model of northeast Atlantic tides (with observed elevationsas open-bogndary input). The discrepancy between these estimates has to be taken as ameasure of their uncertainty. Between 23" and 38'N, there are no obvious energy sinks, butsome work,

101 I w,

is done against tidal forces. The balance is completed by a substantial influx of energy fromthe west between 23" and 38'N, as suggested by the amphidromic system of North Atlanticco.tidal charts; the influx between 23" and 38'N equals

work against tidal forces

*northward flux across 38"N-northward flux across 23"N

_ J ( t+S.S-2)x 1011 W:4 .5 x 1011 W (max imum)

L ( 1 + 2 . 5 - 3 ) x 1 0 1 1 W : 0 . 5 x 1 0 1 1 W ( m i n i m u m ) .

The observed phas,e delay in the Bl currents suggests a small shoreward energy fluxcomponent pSh(((-(")u) of about 4t2 kW m-1, or a total of 2 x 10e W if maintainedover the whole shelf between sections A and C. These figures are negligible compared eitherwith the longshore flux or with global tidal energy dissipation 0(3 x 1012 W) (e.g., Accenand Pexerus,1978).

ANALYSIS OF BAROCLIN IC MOTION

Distribution

.Both the variability estimates and the mean tidal current ellipses in Fig. 3 suggest thatthe currents at A3, 81, and 83 are nearly barotropic. Their mooring averages account for,respectively,74,94,and82\of the semi-diurnal variance, contrasting with only 34lat82and C2. All these barotropic percentages are larger than, but distributed like, thoseobserved in 19" to 20'N by HonN and MmNcre (I976), who found a baroclinic energy peakat 100Gm water depth. Our finding of greater baroclinic energy at the slope mciorings 82and C2 is also to be expected in general terms from models of internal tide generation atcontinental slopes (Benes, 1982).

82 and C2 also contribute 68/. of all the recorded temperature variance with only nineout of 26 records. Only 4l of all the temperature variance was attributable to up-slope-down-slope motion of the barotropic current. Nevertheless, both the currents and thetemperature variations according to the barotropic model were subtracted from theoriginal (semi-diurnal-filtered) time series. The 'baroclinic residual', described hereafter,remains.

Fig. 5. 'Baroclinic residual'currents at (a)82 and (b) C2. Abscissa is days from 0000, 3 Februaryt1975. Vertical scale is -30 to +30 cm s-1 on all plots. r, onshore; u, longshore north.

- oo [

Tidal currents in the northwest Afrfu:an upwelling region

v

v

294 J. M. HuurNrwcB and P. G, Bert.rBs

67m

157 m

?82m

Tidal currents in the northwest African upwelling region 295

The currents atBz tndC2 (Fig. 5) are stronger near the sea surface and sea floor' as was

found by HonN and MBncre (1976). Root-mean-square values are 9.0, 4.5, 5.7,7.0, and

11.4cm s-l on average between B2 and C2, working from the top current meter

downwards. Separate surface and bottom-intensified motions are suggested'

Empirical modes

The currents (u, onshore; u,longshore) and temperature (T) series atB2 or C2 canbe

jointly analysed into empirical orthogonal functions:

'baroclinic residuals'^ i o,r",rr' (2)

where the leading'organized'contributions brdr, ir6r," ' in turn optimize (forleast-

square error) the ipproximation for N : !,2, . .., NM (e.g., Kurou et al.,1975). The bo are

N^M-vectors,whereNMisthetotalnumberof ser ies, 15atB2(u,u,T at5meters)and 14

tt C2. Temperature is measured in different units from velocity and might accidentally

dominate (oi not influence at all) the fitting process according to the choice of scale; we give

temperatuie equal weight by scaling to average the same variance numerically as the mean-var ibnceof a l l the u induser iesa i thes ta t ion .Twomodes( i 'e . ,N :2 \a re min ima l fo r

representing phase differences between u, u, and T, and include a majority of the variance

at82 and C2. If we further impose 6rG): dStldt,i 'e','barocl inic residual ' :brdr(t)+b rdQrldt, (3)

then b, and b, are more readily interpreted as in-phase and quadrature contributions to

baroclinic motion oscillating as {t(t).At 82, the first two (of iS;.-piti"ul modes account for 63.8/. of the variance; the

constrained form (3) accounts for 56.3/.and is illustrated in Fig. 6. Figure 7 illustrate_s-the

form (3) atC2,where the respective figures for variance accounted for are 68'5 and 55'5%'

nt Uoih moorings, the tempeiature is approximately in phase throughout the depth, with a

mid-depth -u*irnu*, whireas the velocity has one phase revetsal and a near'bottom

maximum. This structure suggests the lowest baroclinic mode'

Comparing the longshore and onshore pressure gradients calculated from the curr-ents

via the inviscid momentum equations

0pl0y:-p(0ul0t+fu) f+t0pl0x: - p(1ul1t- fu),

a larger gradient (or shorter length scale) alongshore than onshore is suggested where u

"*.r"-dr, 1b."uur" o > f),i.e.,near the bottom; at all levels uandu are comparable, so that

ipl1x and' 0pl0y are comparable (Figs 6, 7). As a topographically imposed onshore length

scale of l0 km is expect;d, this suggests that a similarly' short scale prevails alongshore'

With viscosity included the observed u:u ratio is still incompatible with 0l0y:0. At the

bottom current meters (only), 10 m above the sea floor, viscous boundary layer effects may

increase the longshore component u relative to u, but not to the extent observed atB.2if

0l0y:0. In thelhicker 'criiical'boundary layer considered by Gonoon (1980) (assuming

0l0y = 0, u and u rnay both be increased but remain in the relation lul : lfulol < lul'

Internal tide generation

All present models assume a straight continental shelf so that internal tides are forced by

296 J. M, HursN,{qcr and P. G. Benns

v Amplirudr (f)0

Depih

m

500lSOo Ptose (X) 360.

T Anlplitude (+)

Iig. 6' Earoclinic empirical mode at 82. Phases are leads (only relative phases are significant).Connecting lines are merely visual aids and there is.no basis for interpolating between data leveis

( + , x ) .

u A(plitud€

l8O' Ptrose (X)

O 20 C T AmDlitud€ (+)

Fig. 7. Baroclinic empirical mode at C2. Conventions as Fig. 6,

the onshore-offshore barotropic component of the current. Then the internal tide can bedetermined, if the stratification, barotropic tide, and shelf profile are all known. We use themodels of B,qJl.res (1973, 1981) for 'flat' topography a < c and'steep' topography a > c,respectively, where fl is the sea-floor slope and

c : (o2 _ TzyL l(N2 _ oz\t

is the characteristic (or raylslope: N, f,and o are the Brunt-Viiistilfl, inertial, and tidalfrequarcies.

l8O' Phe (x) 360.

v Arplilu& (+)

Tidal currents in the northwest African upwetling region

For section B, Fig. 8 shows vertical profiles of o, from CTD casts within 5 km of 82 taken

at various times while the currents were recorded, and the assumed stratification for

modelling. The slope at 82 is 'steep': 0.057: d,> c*0.039;BetNns'(1981) model was

used. Thi model slbpe is uniform and adjoins a level shelf (Fig. 9); internal motion and

characteristics are reflected at the underside of a surface mixed layer of depth 40 m.

For section C, Fig. 10 shows oe from CTD casts within 10 km of C2 early in the cruise,

and the assumed stratification for modelling. The continental slope is 'steep' between 100

and22Um depth approximately, where we use BAINn's'(1981) model, and possibly also

below 1050 -. We neglect the latter as a source of internal tides at C2 in view of its distance

(35 km) and the absence of CTD data at these depths. Between 220 and 105Gm depth the

ilope is 'flat' (Fig. 11), for example at C2O.O4:= fl < c:0.042 and a is less elsewhere; here

we use B,upes' (fgZt) model. Calculations were made with and without a 3Gm surface

mixed layer.

Fig. 8. osnear 82. Numbers indicate date in days after 3 February 1975 (when current meterrecords began). ----'.6e for M2 characteristics'

297

m

800

fig.9. Depth profile, model profile -- _11T:# meter positions +, and M2 characteristics

298 J. M. HurnNeNcr and P. G. BArNes

Fig. 10. o, near C2. Conventions as Fig. 8.

80 60 40

Fig. 11. Depth profile, current meter positions +, and M2 characteristics _ '_ near C2.

The Kelvin wave model gives the onshore-offshore barotropic current component (U,say) that forces the internal tidal currents, which may be large near (i.e., within 10 m) thetwo characteristics originating at the shelf break (Figs 9, 1l), 0(u) between the twocharacteristics, and small elseyhere. Neither at 82 nor C2 are any current meters closeenough to a characterislic for the internal tidal currents to exceed about 2 and 3 cm s-r.resPectively [i.e., 0(U)]. However, the currents at C2 are considerably.modified by thesurface mixed layer (Fig. 12); the'beam'between characteristics from the shelf break isdisplaced if reflected at 30 m rather than at the surface.

These calculated internal tides are weak compared with the observed currents,particularly at the bottom meters at 82 and C2, and satisfy lul:lulflo, contrary toobservation.

Tidal currents in the northwest African upwelling region 299

2 cm/t

5OO m

Fig. 12, Calculated onshore M, baroclinic current at C2. Coinponents - and ---- are,reipectively, in phase with and lag-by 90" the onshore barotropic currint (1.6 cm.s-

I at C2). (a) Nomixed layer. (b) 3Gm mixed layer.

Bottom-traqPed waves

We subtract the calculated internal tide from the 'baroclinic residual' and regard the

remainder as unforced motion to be modelled by free baroclinic waves. (We argue belowthat this motion is probably due to the large longshore barotropic motion interacting'withunknown longshore topographic variations.)

The comilete set of free baroclinic waves above the inertial frequency in a continentalshelf context awaits systematic study. In the present case, an energy balance for thebaroclinic motion of the rough form input rate - radiational loss : energy density x'radiation coefficient'suggests a small 'radiation coeffibient', as the energy density is large

300 J. M. HursNlxcr and P. G. Blrr.rns

compared with the forced internal tide, whilst the (unidentified) energy input mayreasonably be supposed to be no more than drives the forced internal tide. Hence, we seekfairly well-trapped waves, which suggests short length scales; unless a wave is confinedabove the slope, a long wavelength alongshore implies substantial radiational losses abovethe inertial frequbncy (HurHNnNce ,1978). Bottom-trapped waves (RHwes, 1970) satisfy thetrapping requirement (and indeed are the limiting waveform for short longshorewavelengths at sub-inertial frequencies). The solutions for waves progressing up or down awedge (Wunscu, 1969) also show bottom intensification but radiate energy rapidly. Noother 'prototype' solutions appear to be known at present.

Hence, we test whether bottom-trapped waves can model the observed 'baroclinic

residual'. Such waves are possible at the M, frequency o if o O(0 0) propagating wave was also considered at 82. Fitting was by leastsquares to the M, complex amplitudes of the 'baroclinic residual' current, for eachindividual week of the 4-week record. Only the bottom three current meters were included.The small energy near 160 m has little elTect on a fit by bottom-trapped waves and isexcluded as being too near the surface for the bottom-trapped wave form to apply. Themeters near 70 m are also excluded on this score and on account of their extra motion to bediscussed later. Significance offits was aqsessed by the F-test on residual variance, based onthree degrees of freedom for each wave fitted (K and two for complex V) and 12 degrees offreedom in the data (two each for the complex amplitudes of two velocity components atthree meters).

At 82, a significant fit by bottom-trapped waves was found only in the most energeticweeks 2 and 3 (Fig. 5a), when two waves gave a significantly better fit than one. Values of Zand K for the two waveforms (5) are given in Table 2.The data variance attributable tobottom-trapped waves in weeks 2 and 3 amounts to 307. of the 82 total (barotropic and

lablel, 92bottom,'tappeilwaves.su!fixesl,2ilenotewaves.progressingilown-slopeandup-slope,respectively,0t: 42I - -02. -lrlKl : 2.29 : lzlKz in (5). lgsy"aconfidenci limirs-. phases rilative to modei Keiuin-ware

Week

K r ' m - rK 2 , m - tVr , cm s- rVr , cm s- l

\ variance accounted for

0.0015 + 50%0.0163+3vA

-7.73-4.67i+13%8.40+5.72itr9%

98.3

0.002r t 100%0.0016+ 100%

-8.16-2.69i+30%8.6'l +1.26i+28%

93.9

Tidal currents in the northwest African upwelling region

baroclinic) over the whole 4 weeks. ilhe associated energy flux (averaged over the 4 weeks) is

nearly 500Wm-1: 308Wm-l onshore and 177 Wm-l offshore (the energy flux is

proportional to lVlKl2 and is onshore if phase propagation is down-slope and vice versa)''Eviiently, the onshore flux is reflected or dissipated before reaching 81, but the current

meters at 83 offshore are too high to give firm evidence about fluxes there. The whole flux

at B2 represent a loss from the barotropic tide, but is insignificant compared with

barotropic energy fluxes. [For comparison with GonnoN's (1979) figure of 300 W m-r at

Lupine in 40Gm water depth, the total baroclinic energy at 82 propagating seawards at the-

rp"ia of thd first vertical internal mode (assuming a flat bottom) would be a flux of

1SOO W m-1, an overestimate because (a) the first vertical mode is fastest and (b) the

bottom is not flat; e.g., WuNscn's (1969) solutions propagate energy slower over a slope.]

The vertical velocity w atB2 was inferred from the temperature oscillations. Its phase

relative to the 'baroclinic residual' onshore current u fluctuated from week to week

(Table 3), but the 4-week means show a and w roughly in phase. At mid-depth, w/u was

approximately equal to the characteristic slope, as for a plane internal wave propagating

o* or offshore. Near the sea floor, wlu approximated the bottom slope a. However, these

4-week average results apply only to the individual week 2, when the bottom-trappedwaves fitted best. Then, w and uwere in phase within 15' at the bottom three meters and

wfu: u within lO/,,indicating motion parallel to the sea floor'AtC2,the phase of w relative to z fluctuated widely, particularly at the top meter. 1'.i.282

and 357 m, w and uwere in phase within 11' and wfu : u* 0.04 within 2O/"duringweek 1(only),.suggesting motion parallel to the sea floor then. Unfortunately, the bottom metergave no temperature record from which to infer w.

The calculated bottom-trapped waves at C2 have 0.= 0 because o: Nc, so that the

model currents (5) are aligned onshore-offshore and cannot match the observedpredominantly longshore currents. The attempted fit by bottom-trapped waves is never

statistically significant.

Discussion of baroclinic models

Available dynamical models (specifically, bottom-trapped waves) give a generally poor

representation of the baroclinic motion over the slope at 82 and C2, despite the simple

empirical first vertical mode.ihe 82, week 3 (only), combination of nearly equal up-slope and down-slope

propagating bottom-trapped waves gives net phase and energy propagation mainly along

lhe slope. As bottom-trapped waves at the M, frequency o require o

392 J, M. HurnNercp and P. G. Bewrs

(HurHNeNce,1978).Its possibility above the inertial frequency merits investigation, despitethe inevitability of some radiational energy loss.

In the alternative description in terms of rays, one likely factor contributing to thebottom-intensified currents is simply the superposition of incident and reflectedlnternalwaves (or ''beams'), which are nearly in phase close to the bottom (as in simple modes).

By (4), the greater longshore (u) than onshore (u) currents,imply shorter longihore lengthscales than the onshore-offshore scale, which is at most the.tbpographic scale 0(l0k;).Hence, internal-tide models neglecting longshore variations (as they all do) may beinadequate. Longshore shOlf irregularities potentially generate internal tides from thelongshore barotropic current component, which is stronger than the onshore componentforcing present models.

DISCUSSION OF INTERNAL T IDE VARIAB IL ITY

The records from 67 (75) m depth at 82 (C2) app€ar most energetic in the first l| to 2weeks, when stratification was strong at this depth (Figs 8, l0). BerNBs'(l9gl) analysisimplies the generation of wave motion on the shallow thermocline; the motion propug"i""seawards and subsequently leaks downwards on characteristics. Such motion is polaiizedanticyclonically with lulul: flo, is strongly dependent on the stratification neaithe shelfbreak, and is absent when there is no distinct thermocline. A detailed comparison betweenthe model and data at this depth is not feasible, but qualitatively they agree quite well forthe first I to 2 weeks.

Themixed layer i tsd l f ex tendsdownto 67 (75)mforpar to f the t ime,as ind ica tedbyflat-topped temperature records (Bnocrvaun et a\.,1977). A mixed liyer of varying deptirreflects characteristics into different paths, affecting the internal tide generated niar ttreshelf brbak as illustrated by Fig. 12. Near-surface internal-waVe reflection may also beaffected by a current in the rnixed layer (Srenr.r , l977lby a factor of two for a 20 cm s- 1current in a 4Gm mixed layer at 82, for example.

Stratification changes allect characteristics and o, to a similar extent (both are a firstintegral of N2). Figures 8 and 10 indicate that these changes are large in the top 150 m;otherwise modifications to Figs 9 and 1l are slight. Such stratification changis in thegenerating region over the shelfbreak, or near the observing point, are suflficient to explainthe current variability at the upper level. If the large bottom currents at 82 and C2 arcgenerated near the shelf break (we suggest by the large longshore barotropic currentsadjusting to longshore topographic variations), then variability in these bottom-intensifiedmotions may also be attributed to the changing dpperJevel stratification.

In addition to changes in the surface mixed layer and thermocline (i.e., upper 150 m),there are several other possible sources of internal tide intermittency, or of departure frompredictions by linear theory using characteristics. Scattering by microstructure is expectedto be small over the short distance from the generation regions to 82 ( < l0 km) ind C2(20 km, or about I day at the speed of the lowest internal mode). lnteraction with otherinternal waves is probably also small over only one oscillatory cycle. [WuNscn (1976)found rapid spatial changes in internil wave energy generated (?) near a seamount, within10 km for waves of 5-h period.

-However, the changes were not obviously due to lonJinear

interactiorls and did not imply interactions with the tidal currents, to which the internalwave energy was not related.] Mean current shear distorts internal tides (Moonns, 1975),

Tidal currents in the northwest African upwelling region

but only slightly in our context, because the horizontal shear is much less than/and thevertical shear causes isopycnal slopes much smaller than the slope of characteristics.

Mean currents (u) can advect the internal tide. Onshore currents are typically 5 cm s- I'-

or less and probably not generally important, but longshore flows exceed 20 cm s-r at

times. Hence, the internal tide, propagating at only 0 (15 cm s - 1) in its (fastest) first vertical

mode form near the shelf edge, may be advected alongshore faster than it can propagate

offshore. Then the particular depth profile and characteristics of section B or C (say) no

longer apply. Furthermore, if the internal tide wavenumber k has a component parallel to(u), then the"intrinsic'frequency o-(u.k) may differ from the IV.lz frequency o, altering

the spatial structure of the internal tide. Estimates of k would require measurements with a

longshore spacing of 0 (1 km).Fluctuating 'mean' currents correspondingly cause intermittency, and also induce

changes in the perceived internal tide frequency (of zero crossings, say) in the records.

Figure 5 shows only slight variations, usually within the range of periods 12.4+ t [r, so that

conditions seem to be quasi-steady. Both perceived frequency changes and intermittency(from the various sources mentioned) broaden the tidal peaks in the enqrgy spectrum(Fig. 2). The total effect is rather modetate for semi-diurnal tides.

BOTTOM BOUNDARY LAYER

Any boundary layer effects recorded by the lowest meter (10 m above the bottom) at B2

andC2are masked by other vertical structures. However, at 81, the bottom meter was only

2 m off the sea floor, and Fig. 3 indicates a slight trend towards cyclonic polarization and aphase advance. This trend may be attributed to the boundary layers for the anticyclonicind cyclonically polari2ed components of the oscillatory tidal current having different

thicknesses (MuNr et at.,1970). The cyclonic layer is thinner, so that there is a range at the

top of the anticyclonic layer where the anticyclonically polarized component has begun to

dJcrease (to zero at the sea floor) brit the cyclonic component is still undiminished. [If a

uniform eddy viscosity v is assumed, then the two boundary layer depths are

6*:l2vl@+f)lr, where o is the tidal frequency and/is the Coriolis parameter; the

observations give v ( 10-3 m2 s- I in the bottom 10 m at 81, corresponding to d- 5 5 m

for M, and 6 - S 11 m for the diurnal tides (WrlrHenlv, Bruusncx and Btnp; 1980).1 At the

end oi the cruise, turbid water appeared to occupy the bottom 3 m at 81, beneath clear

water (KuIIENBERG, 1978). This layering may have suppressed bottom turbulence, which isprobably generated by the low frequency and semi-diurnal currents (being strongest).

M" HARMONIC

Onshore-offshore tidal excursions of order 1 km take fluid particles through substantialchanges of depth near the shelf edge. Non-linear effects may both influence internal tidegeneration and generate higher harmonic.s, notably at the Ma frequency (twice Ma) near4 cycles day-t.

Ma currents and temperature fluctuations were observed at Lupine (Fig. 1), especiallywhen the baroclinic tide was strongest, but the currents were consistently related to the

semi-diurnal tide only near the bottom (GonooN, 1979). The bottom currents were

strongest, surface currents were nearly as large, and temperature fluctuations were greatest

at mid-depth.

303

304 J. M. HursNlNcB and P. G. BarNnS

AtB2, Mn appears in the onshore-offshore velocity component at the bottom (Fig. 13)when the total semi-diurnal currents are strongest, and in the bottom temperature record inthe appr6priate phase relation; low temperatues lag the onshore current. In contrast withLupine, however, there is a minimum in Ma temperature fluctuations at mid-depth, and thenear-surface currents contain only l5l of the energy in the bottom currents.

Mo is small at C2 despite even stronger semi-diurnal currents at the end of the recordthan at 82, and despite being quite close to Lupine. (The C2 and l-upine records are fromdillerent times, however.) c2 ditfers in having an expected minimum period ZnlNafor Rsnres' (1970) bottom-trapped waves of 14 h, compared with 7 h both at Lupine(N : 1.8 cycles h-1, bottom slope a : 0.076) and on theiteepest slope just 4 km inshorefrom 82. Equivalently, the Mn characteristic slope is only. just greater than a at B2 andLupine but twice as steep at C2. WuNscs's (1969) progressive solutions in a wedgetherefore suggest that any Mn currents at 82 and Lupine should be concentrated near thebottom.

We conclude that the Mn harmonic is associated with large onshore-offshore semi-diurnal currents and a near-characteristic bottom slope for Ma. The latter conditiondepends on the stratification, makingfor even more variable Mo generation than impliedby changing semi-diurnal currents.

t l

cm/so -

-60

60

cm /s

o

Jonuor! Feuuory Mcch

Fig. 13. Hourly onshore (u) and longshore (u) current and temperature (T) at the lowest 82 andC2 meters.

B2

60cmy's

Tidal currents in the northwest African upwelling region

C$NCLUSIONS

The diurnal currents were largely incoherent with the tidal elevations and have been

mostly attributed to a broad inertial peak in the energy spectrum.Thi spatial and temporal coherence of the barotropic semi-diurnal tide permits'its

evaluation from the five current meter moorings, In particular, analysing the moorings

simultaneously gives a good estimate of the age of the tide (36 + 15 h) and confirms the

spririg-neap cycle. The mooring-averaged currents are generally consist€nt with a Kelvin-

wave model. Extrapolation offshore, using the Kelvin-wave model with allowance for the

observed tidal elevations, permits an estimate of the northward tidal energy flux at 23'N of

2 t o 3 x 1 0 1 r w .Significant baroclinic currents occur at the two moorings 82 and C2 on the continental

slopJ at 500 m. These currents are intermittent, stronger near the surface with a

(onshore) > u (longshore) and, at different times, stronger near the bottom with u > a. The

near-surface motion may be attributed to generation near the shelf break on a shallow

thermocline, as the motion and thermocline co-exist. The currents in the deeper water are

stronger than expected from two-dimensional theory and are probably due to the large

longshore barotropic currents interacting with unknown longshore topographic variations.

Foiall of these motions we attribute the intermittency to the varying density stratification

in the upper 150 m.The majority of the baroclinic variancQ is represented empirically by a first vertical mode

form. However, dynamical models for this form over a significant sea-floor slope are

lacking. At 82, but not C2, RunrEs'(1970) bottom-trapped waves model the near-bottom

motion when its amplitude is large; at both sites large amplitudes near the bottom may be

partly due to the superposition of incident and reflected waves.- The Ma harmonic has (relatively) large amplitude currents at 82 (and Lupine) where the

ray slope approximately equals the bottom slope; but not at C2 where the ray slope is twice

the bottom slope.A better understanding of the currents appears to call for (a) models of internal tide

generation through the longshore barotropic currents adjusting to an irregular shelf and

[b1 u rrror" systematic and extensive knorivledge of internal wave forms over a continental

shelf and slope.

Acknowledgemenrs-We are very grateful to Dr P. HuoHes for making the 'Upyetling ']!'aat-a

lrry_ty available.

ftt. *ot[-*"r partially funded 5y Natural Environment Research Council Grants GR3/1818 (J.M'H.) and

GR.l12623 (P.G.B.).

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