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Internal tidal mixing in the Bay of Biscay
Deep-SeaResearch,Vol. 35, No. 5, pp. 691-709, 1988.
0198-0149/88$3.00+ 0.00 © 1988PergamonPressplc.
Printedin GreatBritain.
Internal tidal mixing in the Bay of Biscay A. L. NEW* (Received 18 May 1987; in revised form 8 September 1987; accepted 27 November 1987) Abstract--The continental shelf break in the Bay of Biscay often corresponds to a region of high phytoplankton abundance and cool water near the surface. It has been conjectured that these phenomena are due to mixing created by internal tidal motions that result from the interaction of the surface tide with bottom topography. This process is investigated in the present paper by employing a linear numerical model that describes the topographic generation of the internal tides. After demonstrating that the model is in good agreement with existing observations, it is found that there are typically two thermocline regions near the shelf break in which the Richardson number falls below the critical value of ] at some stage in the tidal cycle, indicating an unstable flow. By considering linear stability theory, mixing seems likely in these two regions during the summer months and at spring tides but not at neaps. The conclusion is consistent with available sea surface temperature data.
1. I N T R O D U C T I O N
THE continental shelf break in the Bay of Biscay is often found to be a region of cool surface water (Fig. 1) and high phytoplankton abundance, as described by PINGREEet al. (1986, to be referred to as PMN in the following). Both these phenomena, occurring also in other parts of the world's oceans, have been conjectured to be manifestations of physical mixing caused by internal motions of semidiurnal period (internal tides) that result from the interaction of the surface tide with shelf break topography (SASDSTROM and ELLIOT1",1984; BRISCOE, 1984; HOLLIGAS et al., 1985; PMN). This mixing would result in an upward transport of cool water, rich in nutrients, from below the thermocline which, upon reaching the well-illuminated surface layers, would provide a favourable environment for the growth of plant cells as well as the observed decrease in SST. In a survey of the area shown in Fig. i with a towed cycling CTD (the SeaSoar), PMN found internal tidal depressions of the thermocline travelling away from the shelf break in both the deep ocean and the shallow shelf waters. A typical example of the data collected in this manner is shown in Fig. 2, and the long wavelength (55 kin) tidal signal is dearly evident. [Similar observations have also been presented by PINGREEand MARDELL (1985) in which moored thermistor arrays indeed show that this signal has a semidiurnal period.] Figure 2 is complicated, however, by relatively high frequency internal waves (with apparent periods close to 10 min), which seem to appear with largest amplitudes in the tidal troughs. These short waves produce a roughening of the sea surface (THOMPSON and GASPAROVIC,1986) that is often visible as a series of long-crested (many km) "rips" with a spacing of the order of 1 km. These rips are also detectable from space using satellite imagery, as shown in Fig. 3a. The complex nature of the surface signature takes * Institute of Oceanographic Sciences, Brook Road, Wormley, Godalming, Surrey GU8 5UB, U.K. 691
692
A.L. NEW 8
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/ Io /
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21
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:~pOOm-46
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Fig. 1. Chartof the Bay of Biscay, from PMN, showing the main survey area (rectangle), mean sea surface temperature (SST) contours (*C), positions of internal tidal troughs 21-23, and the location of mooring 080.
o8oo
i
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100-
^ 21
^ 22
^ 23
Fig. 2. Example of SeaSoar tow on 19 September 1985 recorded by PMN approximately 50-150 km oceanward of the shelf break. Isopycnals are plotted at densities of 1026, 1026.5 and 1027 kg m-3, and tidal troughs 21-23 are indicated.
the form o f packets of the short internal waves travelling away from the shelf break (typically 3-6 waves in each packet), the envelope of all such contiguous packets indicating the approximate position of a tidal depression of the thermocline (two such pulses visible on shelf, o n e in the deep ocean). Figure 3b is an interpretation of this S A R image and shows, in relation to the bathymetry, the main wave packets, which appear to be radiating from the topographic ridges b e t w e e n canyons.
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Fig. 3. (a) Synthetic aperture radar image of the Bay of Biscay on 20 August 1978, showing internal wave packets near the shelf break. Courtesy Deutsche Forschungs- und Versuchsanstalt fiir Luft- and Raumfahrt (DFVLR). (b) Interpretation of (a), showing the main wave packets in relation to the bathymetry and possible sites for their generation (O).
Internal tidal mixing in the Bay of Biscay
695
Two alternative types of linear numerical model for the generation of internal tides by two-dimensional topography have now been widely applied in the literature. The first of these is the ray theory of BARNES(1974), which is applicable to arbitrary topography, but only to certain forms of (continuous) density profile. In particular, difficulties are encountered whenever a thermocline is present, producing a rapidly varying stratification. However, a simplified version of the theory has been developed by BAINES (1982) that can accomodate a step-like interface over a uniformly stratified lower layer, and has been compared with laboratory experiments by BAINESand FANG (1985). Alternatively, modal models, applicable to the case of a linear shelf slope but valid for arbitrary stratification, are now also available. The theory has been developed by PRINSENBERGet al. (1974) and PmNSENaERG and RATrRAY (1975), and compared with oceanographic observations by DEwrrr et al. (1986). The principal aim of the present paper is to investigate whether numerical models such as these, applied to the Bay of Biscay, predict thermocline mixing near the shelf break as discussed above. Since it was considered more important to accurately model the density structure than the precise form of the shelf break profile, a modal theory similar to that of PRINSENBERGand RATIRAY(1975) was employed. An important consideration, however, is to ascertain to what extent the theory is applicable to the real oceanic situation. In particular, PMN found that the summer thermocline over the deep ocean in the Bay of Biscay was unexpectedly dominated by internal tides of modes 3 or 4, or some combination of both. The present model strongly supports dominance by a mode 3 tide, and is favourably compared not only with the observations of PMN, but also with moored current meter data. Further, the possibility of internal tidal mixing is positively demonstrated, and seasonal variations, as well as those occurring over the spring-neap cycle, are discussed. 2. T O P O G R A P H I C G E N E R A T I O N M O D E L
It is easy to show that a stratified fluid has possible two-dimensional internal modes of oscillation of the form
rl(x, z, t) = acb(z)e i(k~-°O
(1)
for the vertical displacement of a fluid particle with mean position x, z and at time t, in which qb satisfies 0 ~ + k2
¢(0) = ¢ ( - h ) = 0,
dp = 0
(2a)
(2b)
where N 2 = -g/p0(0) • Opohgz for a background density field of p0(z), f = 1.075 x 10-4 s-1 is the Coriolis frequency, and z = 0, -h correspond to the mean positions of the surface and horizontal bed, respectively. This equation was solved by standard numerical techniques (e.g. BELL, 1971), for which the z-increment was taken to be 1 m in the upper 150 m of the water column, and 5 m elsewhere. Two typical N(z) profiles will be used in the present work, corresponding to summer and winter conditions. For the upper 150 m, the N-values for the summer were
696
A . L . NEW
calculated from a SeaSoar tow recorded in September 1985 and averaged over three complete internal tidal wavelengths, while for winter, N was set equal to 2.4 x 10-3 s-1 to represent relatively well-mixed conditions, as shown in Fig. 4a. Below this level, N(z) was modelled from data collected by PINGm~Eand MORPasoN (1973) as
N(z) =
2x(z + 150)~
-],-150 >t z/> -1200 m
(3a)
2.4 x 1003exp(0.65103 x 1003(z + 1200)), -1200 t> z >t -2400 m
(3b)
1.1 x 10°3
(3c)
2.4 x 1003 + 0.4 x 10-3sin \
l~lJ
,-2400 ~> z ~> -4400 m
which allows for the permanent pycnocline at about 900 m, with gradual decay below. The present numerical model is based on that of PRINSENBERGand RArrKAV (1975), and will be described only briefly. For the two-dimensional shelf break topography as shown in Fig. 4b and oscillations of semidiurnal frequency (6 = 1.405 x 10-4 s-l), we
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20
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1 "~"
[
100
10
/ ./"/"
I "/ | ./
b
2~60m 1
~x
2
4400 m
Fig. 4. (a) Brnnt-Vaisala frequency (N) profiles for typical summer (solid line) and winter (broken line) stratifications, and from data recorded at mooring 080 (chained line). (b) Idealized topography for the numerical model, showing the shelf and oceanic regions, 1 and 2, respectively, connected by a slope of gradient a = 0.09. The origin of co-ordinates, in the mean level of the surface, is as indicated.
Internal tidal rni~g in the Bay of Biscay
697
suppose that the vertical displacements in regions 1 and 2 are given, respectively, by r l l = [ a°ldp~e-ik~ + l~n~l ale-/k~ dP~] e-/at n2 =
(a~ e -ik~ + b N ilc~x) , 2 + ~
a2m eik2~ dp2
(4a) e..i6t.
(4b)
rnzl
The first term in equation (4b) represents an incident deep water surface tide in region 2 of amplitude ~ , whereas b 2 is the amplitude of its reflected component and a~ in equation (4a) is the amplitude of the component transmitted over the shelf. Since these barotropic tides are essentially hydrostatic, we deduce
(5a) (5b) To these surface tides are added sums of internal tides travelling away from the shelf break, so that (~l(z), k~, a~) and (tp2(z), k2,,,, a2m)correspond to the n th and m th internal tides calculated for the shelf and ocean depths, hi and h2, repectively. The only essential difference between this model and that of PR~SENaERC and RAa'rRAY (1975) is the allowance for a travelling surface tide on the shelf rather than a standing tide, since this was considered to be more appropriate for the Bay of Biscay (e.g. SCmVIDEltSIO,1979, 1981). The amplitudes of the surface tides in the two regions are related by a 2 = a~(k~ + ~ ) / 2 k 1
(6a)
b2
(6b)
1 1 - k~o)12k~ = ao(ko
from continuity of surface elevation and mass flux, so that only ao~ is an externally specified parameter (taken as a01 = 0.75, 2.00 m for neap and spring tides, respectively). Proceeding with analysis similar to that described by PRINSErOERC and RATa'RAY (1975), we find, in order to satisfy the watching conditions on the sloping boundary x = -z/a between the two regions, that the modal amplitudes satisfy N
M
alnF~ = 4 S ~ + ~,a2m Q,~, 2 n=l M
~, a2 p f i m = 4 ~ m=l
r=l,N
(7a)
j= I,M,
(7b)
m=l N
+ ~,a~Qln, n=l
where the coefficients are given in the Appendix. These relations were solved on a computer with M = 60 and N = 9. Since the amplitudes of the higher modes appeared to be successively less reliable, and also because the numerical solutions typically contained spurious oscillations on approximately the length scale of the highest modes retained (see Fig. 2 of PmNSEI,mEROet al., 1974), it was found desirable to apply an ad hoc filter to mode numbers rn = 30--60 such that the amplitude of the m th mode was
698
A.L. NEW
multiplied by 1-(m-30)/30. A similar filter was applied to modes n = 3-9 for the shelf motion. 3.
GENERAL
F O R M OF S O L U T I O N
To reveal the general form of the solution, and also as a test of the numerical procedure as a whole (including the filters), Fig. 5 exhibits the nature of the displacement field, which can be written as rl(x, z, t) = Re(fi(x,
z)e -iat) =
Re(lfild°e-i~
c o s ( 6 t - 0),
=
(8)
and shows vertical profiles of Ifi(x, z) l, the maximum elevation of a fluid particle from its mean position, and the phase, 0(x, z), converted to degrees, at which this is attained. We see that large vertical oscillations are possible at certain depths (e.g. in excess of 200 m in amplitude at 2.5 km depth and x = 50 km), and that, generally speaking, the phases vary little in the vertical except at these depths. (Note that since each of the internal tidal amplitudes is directly proportional to ao~, it is unnecessary to describe results for any tidal state other than spring tides.) The regions of large oscillations result from beams of energy (described by BAINES, 1982) which emanate from the point of the shelf break and travel along ray paths that have the characteristic slope
dz
--=
+
dx
-
( o -f
(9)
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200
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' ~ / "
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'\\ \.\ l\'\\\\,\i]\j!\\, \
\
" i \ iI /
I\\
/
i~
Fig. 5. Profiles of amplitudes I~1 (solid fines) and phases 0 (broken lines) of the vertical displacement at spring tides and for the winter stratification.The vertical lines coincidewith the zeroes of I~1 and phases of 0°, while the scales for these two quantities are shown above the profiles at 125 km. The chained lines represent the rays emanatingfrom the shelf break.
Internal tidal mixing in the Bay of Biscay
699
and reflect from both the surface and the horizontal bed. These rays (two in the deep ocean region, one over the shelf) are also illustrated in the figure, and theoretically should coincide with the positions of largest Ifi 1. The good agreement is taken as a validation of the numercial model as a whole. The corresponding situation for the summer stratification reveals that neither the maximum amplitudes nor the phases along the rays are subject to much seasonal variation, but that the vertical spacing between the beams in the deep ocean is now much increased. The agreement between the positions of the primary maxima of [rl I and the rays is comparable to that shown in Fig. 5, and this is taken as an indication that the seasonal thermocline is adequately represented. The horizontal velocity fields, which may also be easily calculated and are discussed in detail by NEW (1987), reveal a beam-like structure similar to that already described for the displacements. At spring tides, for instance, velocities of the order of 0.5 m s-1 in excess of those due to the surface tides are achieved along the rays, both by the acrossslope and along-slope components, but are lower elsewhere. These currents should not be neglected in the design and operation of offshore structures. 4. T H E R M O C L I N E MOTION
The internal tides in the Bay of Biscay are quite possibly the most energetic found anywhere in the world (BAINES, 1982). Using results from the present model, calculations of/~, the depth-integrated, phase-averaged energy, were performed for each mode, and for both winter and summer stratifications. In both cases, and for both shelf and ocean, mode 1 was the most energetic, with a general decrease for the higher modes. For example,/~ was equal to 8590, 6690, 6010, 400, and 2670 J m - 2 for the first five ocean modes, for the summer stratification and spring tides, with similar values for the winter stratification. However, it does not follow that mode 1 should necessarily give the largest contribution to the vertical displacement at all depths. For instance, decomposing the thermocline displacement (at a depth of 50 m, say) into its constituent modal components at spring tides reveals that, although the summer thermocline over the shelf is dominated by mode 1 (amplitudes of 9.7, 6.2, 3.1, 2.4 and 7.3 m for the first five modes, respectively), that over the deep ocean is dominated by mode 3 (amplitudes of 1.8, 4.3, 13.0, 2.5 and 3.8 m for the first five modes), in agreement with the observations of PMN. (For the winter stratification this predominance by mode 3 disappears.) To illustrate this more clearly, Fig. 6 shows instantaneous thermocline displacements at six equally spaced intervals through the tidal cycle. At time t = 0 h (Fig. 6a), the thermocline above the shelf break has attained its maximum positive elevation. As the flooding tide then begins to decrease, the thermocline in this region sinks until the time of maximum ebb (Fig. 6d), at which point two troughs appear to be forming. The offshelf trough then begins to move offshelf (Fig. 6d-f) and, from about the beginning of the succeeding tidal cycle (Fig. 6a) has attained an almost uniform speed and results in a nearly sinusoidal displacement of the thermocline with the wavelength characteristic of mode 3 (about 55 km). Similarly, the trough that forms shelfwards of the shelf break travels onto the shelf and has a wavelength of about 35 km, corresponding to mode 1. (The less regular nature of the thermocline motion over the shelf is due to frequent oscillations of the ray between the surface and the bottom.)
700
A.L. NEW
0
.~
I
-50 [
I
i
i
m~ 100-j
0
.
|
I
i
i
50
i
. . . . . . . .
I
km
i
,
i
100
i
1
. . . . . . . .
--A
A
A
e
Fig. 6. Numerically predicted displacements, for the summer stratification and spring tides, of the isopy~al with a mean depth of 50 m at times of (a) 0 h, maximum onshelf flow, (b) 2.1 h, (c) 4.1 h, (d) 6.2 h, (e) 8.3 h and (f) 11).3 h. The positions of the tidal troughs, estimated by eye, are indicated, and the shelf break is at x = 2.9 kin.
-50-
fJ /X ,aX / /0
0
0
tX / .s
x
~ /
o\ x
10
\ \ x
km
o OQ
2'0
h
'
\
o\
x\ \ o
~
o\,, o~ \ 5O
\
NO X,,, ,,0
x
0
\ \ OX
100-
Fig. 7. Comparison of internal tidal trough positions as observed by PMN (o) and predicted by the numerical model (x). The horizontal axis represents computational time, and the broken lines represent a mode 1 internal tide travelfing onto the shelf and a mode 3 into the ocean.
Internal tidal mixingin the Bay of Biscay
701
The positions of these troughs have been plotted in Fig. 7 and compared with the observed positions of the tidal thermocline depressions presented by PMN, taken from SeaSoar tows as in Fig. 2. To enable this comparison to be made, the time of maximum ebb was taken as 2 h after predicted high water at Plymouth (see PMN for full details). The positions of the two data sets are in remarkably good agreement, and further, are closely fitted by two straight lines representing the propagation of a mode 3 tide into the ocean and a mode 1 onto the shelf, both lines emanating from the shelf break at the time of maximum ebb. This provides a simple rule for predicting the positions of both the tidal thermocline depressions and the packets of higher frequency internal waves with which they seem to be associated. 5.
COMPARISON
WITH OBSERVATIONS
In addition to the above comparison, the numerical model has also been tested against the laboratory results of BAn~_s and FANO (1985). In their apparatus, a barotropic "tide" was produced in a uniformly stratified fluid (N = 0.97 s-1) by a 45 ° piston situated at one end of the wave tank. This interacted with representative shelf break topography to produce internal oscillations, and some of their results are summarized in Fig. 8. Superimposed is the flow field predicted by the present numerical model at 0.46 m from the shelf break along the lower "oceanic" ray, a position used for some of the laboratory results (theoretically, the solution should be independent of this co-ordinate). Overall, there is good agreement, except for regions near the rays (at ~ = 0 and 9.7 cm) which
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Fig. 8. Horizontalvelocity(u) comparisonof the present numericalmodel (with o = 0.248 s-1, f -- 0 s"-1,a -- 0.38 and a~ = 1.214 x 10-3 m, solidlines) with the laboratoryresults (e) and raytracing model (broken lines) of BAINESand FANG(1985) at (a) maximumonshelfflow, (b) high water, (c) maximumebb and (d) low water. Usbis the maximumbarotropic velocityat the shelf break, and ~ the perpendicular distance from the lower ray in the oceanic region.
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A . L . NEW
occupy some 10% of the fluid volume. These regions contain significant contributions from the higher modes in the model, which in the laboratory experiment may be quickly damped out by the action of viscosity, as discussed by BAINESand FANC (1985). Figure 8 also shows the linear ray tracing theory of BAINES(1982) which, for this case of constant N(z), should theoretically be equivalent to the present modal model, provided a sufficient number of modes is chosen. Overall, a close agreement between the two theories is attained, the slight oscillations in the modal model being expected to decrease for an increase in the number of modes. This comparison furnishes a further test of the present model, and in particular indicates that a sufficient number of modes has been taken to adequately describe the narrow beams, and also that the applied smoothing does not distort the solution to any great extent. A comparison has also been made with oceanographic data collected in the Bay of Biscay during August 1984 by R. D. PIN~R~E (personal communication). A mooring, with current meters at depths of 50, 100, 500 and 1000 m was positioned at 47°23'N, 6°40'W in a total depth of 1500 m (site 080 in Fig. 1). The across-slope currents at these depths were averaged over eight successive tidal periods near spring tides, and are plotted in Fig. 9. The numerical predictions, for the Brunt-Vaisala profile as shown in Fig. 4a, a~ = 2.00 m, and at x = 16.5 km, are in remarkably good agreement with the data. The largest differences occur at 50 m depth and could possibly result from nonlinear effects in the thermocline, but these are not large and overall the comparison is encouraging. In particular, the near 180° phase difference between the motions at 50 m and 1000 m is well represented by the model.
-0"5 0
OoI
f"
0.5 -i
I
Fig. 9. Comparison of across-slope velocities at mooring 080 (positive values offshelf) at depths of (a) 50 m, (b) 100 m, (c) 500 m and (d) 1000 m. The solid lines are the observations, the broken lines the numerical predictions.
703
Internal tidal mixing in the Bay of Biscay 6. I N T E R N A L T I D A L M I X I N G
The above comparisons of numerically predicted and observed velocities, both in the laboratory and the ocean, give us confidence to investigate variations in the Richardson number
where (u, v) are the horizontal components of velocity. Figure 10 shows contours of the minimum values (R~ in) achieved during the tidal cycle for a summer stratification and at spring tides. The contours generally follow the internal ray paths emanating from the point of the shelf break, and regions of g rain < ¼ Occur primarily where the rays cross the seasonal thermocline at a depth of about 50 m, with an additional region centred around 900 m, corresponding to the permanent pycnocline. (Note also that part of the energy in the upper oceanic ray is apparently reflected from near the base of the thermocline, at about 130 m, producing regions of R~ in < 10, as indicated by the additional rays. This behaviour might be expected theoretically.) A steady, stratified shear flow is generally thought to become unstable to small perturbations, always present in any physical flow, whenever the value of Ri is less than (MILES, 1961), eventually resulting in the formation of Kelvin-Helmholtz billows and mixing (THORPE, 1971, 1973). If the flow contains no horizontal boundaries, and the steady background density and horizontal velocity fields can be modelled by either hyperbolic tangent or error function profiles with the same vertical length scale d, then -2,o
_
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,
~
-,o
o
,
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~
kr
,
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3,0
,
__,o, x\\
~ix "4
\
1/ \\\\ '\',,\\
\\ \\ , \% \',", I,4
O/
\
X\,, \\
Fig. 10. Contours (at values of 10, 1 and ¼) of the minimum Richardson numbers, R mi~, achieved during the tidal cycle, for spring tides and the summer stratification. The broken lines represent the rays and values of y at particular positions along these paths are indicated.
704
A . L . NEW
the most unstable disturbance may, according to linear stability theory, have amplitude, wavenumber, and growth rate given by a(t) = ao e kct
(lla)
k = 0.441d
(llb)
kci = 0.8N(¼ - Ri)IR~.
(llc)
These equations, in which Ri is taken as the minimum value occurring in the flow (at the centre of the pycnocline, and N at the same level), are essentially a simple fit to data presented by THORPE (1971) and HAZEL (1972). Although our tidally varying flow is unsteady, we may nevertheless expect growth of similar instabilities during any period for which Ri < ¼, at least if the tidal flow varies on a timescale longer than that of the instabilities (the quasi-steady approximation of THORPE, 1971), which seems to be the case here. However, horizontal boundaries are present near the thermocline (i.e. the surface), the density and velocity profiles in the numerical model are not of the required simple forms (although a reasonable fit may be obtained for a certain depth range near the thermodine), and the length scales of these two quantities are not identical (a factor of about three different). Nevertheless, equation (11c) may give the growth rate of the most unstable disturbance correct at least to an order of magnitude. For positions near the ray paths, this relationship was integrated over the time interval for which subcritical values of R i (i.e. R i < ¼) were maintained. This enabled the ratio of the final (af) to the initial (ao) amplitudes of the disturbance to be calculated. The estimates of the "growth factors", defined by y = loglo(af/ao),
(12)
are also indicated in Fig. 10. The growth rates in equation (11c) are strictly only applicable for small amplitude disturbances. When a sufficiently large amplitude has been attained, we may expect overturning and mixing to occur as already described. If we were able to identify this "breaking amplitude", and also estimate the likely initial disturbance level in the real world, this would enable a critical value of T = Tc, say, to be proposed, such that if T I> To, mixing might be expected to occur. For example, if the breaking amplitude is of the order of 10 m, a reasonable physical value, and the initial disturbance has an amplitude of 1 mm, this would give Tc = 4. However, since all such estimates of T~ are somewhat speculative, and also since linear stability theory will cease to apply during the final stages of the growth and may then overestimate the growth rate, it would perhaps be safer to suggest a value of Tc = 5 or 6 for the onset of mixing. Returning to Fig. 10, we see immediately that the deep region of subcritical Ri values is not likely to cause significant mixing. However, the region caused by the passage of the shelf beam through the thermodine has typical values of ), = 8, so that mixing seems at least to be a possibility. The oceanic thermocline region, on the other hand, achieves even larger values, typically T = 20, so that a somewhat more energetic mixing region might be expected here. Values of T were also calculated for the winter stratification and spring tides, and these indicated the possibility of a weak mixing region on the shelf only, with typical values of y = 5. For neap tides, no mixing was predicted by the model for either stratification, since Ri was found to be nowhere subcritical.
705
Internal tidal mixing in the Bay of Biscay -50 I
I
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i
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0 I
I
i
i
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i
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kin
i
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50 I
I
i
I
o
I
°:t I
18T
,,']~
~
6
~
~::::::::~:::
I
~
~
\
Fig. 11. (a) Sea surface temperature (SST) traces across the shelf break in the Bay of Biscay at spring (solid line) and neap (broken line) tides during September 1985. (b) Caption as for (a). (c) Positions of the mixing regions as predicted by the numerical model for spring tides and the
summer stratification. Figure l l a and b show records of the sea surface temperature obtained with a ship's thermosalinograph (at a mean depth of about 3 m) during September 1985 (R. D. PINGREE, personal communication). The data were obtained by steaming directly across the shelf break in the survey area indicated in Fig. 1. Figure l l a illustrates two such runs with approximately the same temperature in the deep ocean, one at neap, and one at spring tides. The SST in both cases is seen to decrease (indicative of internal mixing) at about 20-30 km offshelf, and to remain more or less constant over the shelf. Figure l l b presents a further example of the same phenomenon, but now also shows a slight rise in SST further than about 10 km onto the shelf. In both these cases, the temperature drop is more pronounced at spring than at neap tides, in qualitative agreement with the numerical model, while the temperature decrease evident at neap tides could plausibly result from the mixing on the previous spring tides. Figure l l c then shows the predicted mixing regions, as described above and in Fig. 10. The mixing occurring in these regions might be expected to become smeared-out as a result of tidal advection and shear over the shelf and near the shelf break, and consequently to result in a reasonably uniform re#on of decreased SST extending up to perhaps 25-30 km offshelf, in good agreement with the observations. 7. DISCUSSION AND SUMMARY An important aspect of the present paper is an attempt to ascertain the reliability of linear generation models in describing the tidal response in physical situations, in
706
A . L . NEW
particular in the Bay of Biscay. In spite of the two-dimensionality of the present model, the comparison both with the observed positions of the tidal troughs in PMN, and the mooring in 1500 m water depth is encouraging, so that other results from the model may also be expected to reflect the physical situation. For instance, it is found that vertical oscillations in excess of 400 m in total excursion may occur in the deep ocean. The resulting alteration to instantaneous density profiles may have a significant effect upon the propagation of sound through the ocean, and deserves further investigation. In addition, horizontal currents with typical maximum values of the order of 0.5 m s-1 are associated with the internal tides, both in the deep ocean and on the shelf. These currents are in addition to those due to the surface tides, and should be allowed for in the design and operation of offshore structures. The only other attempt to compare a similar numerical model with oceanographic data of which we are aware is that of DEWrrr et al. (1986), using data from the JASIN experiment in the Rockall Channel. The model used by these authors is that of PRINSENBERGand RArrRAV (1975), in which a total of 30 oceanic modes was chosen to ensure the accuracy of the first 10 modes. The numerical results presented in their paper have been reconstructed using only these 10 modes (L. M. oEWrrr, personal communication), effectively applying a filter with a sharp cut-off, so that, since a seasonal thermocline is present in their observational data, some of the finer scale motion in the upper layers of the ocean may not be adequately resolved. In particular, the minimum Richardson numbers achieved with their model appear to be approximately Ri -- 2, attained when an internal ray crosses the thermocline. The present model was run with the same number of modes, and a similar result was indeed achieved, indicating that DEWrrr et al. (1986) may also have found significantly smaller values of Ri had more modes been used. The main aim of the present paper is an investigation of the possibility of internal tidal mixing near the shelf break. For a summer stratification and spring tides there are two thermocline regions in which the Richardson number falls below }, and the factors, y, relating to the amount by which an initially small amplitude disturbance might be expected to grow, were calculated. Although the relevance of the absolute values of y is perhaps open to question, their relative magnitudes may be useful in inferring the likelihood of mixing in different regions. The predicted mixing regions in the model extend to 25-30 km offshelf during the summer, and are most intense at spring tides. Both these results are consistent with the available data. In addition, the model (in the absence of dissipation) would predict a further mixing region where the ray, initially propagating downwards from the point of the shelf break and then reflected by the ocean floor, again crosses the thermocline, at about 150--160 km offshelf. This seems to be consistent with the results of HOLLIOAN et al. (1985), who observed temperature inversions and patches of turbulent mixing near the surface at about this distance from the shelf break. The interaction of the surface tide with the sea bed over the shelf might be expected to produce considerable turbulence, but this would be to some extent prevented from reaching the surface by the seasonal thermocline, and would presumably produce progressively more mixing further onto the shelf where the tidal flows are larger (see PINGREE et al., 1983). Consequently, it is difficult to see how this mechanism would produce a region of cool surface water primarily above the shelf break. Another possible mechanism for the production of the cool surface water is coastal
Internal tidal mixing in the Bay of Biscay
707
upwelling. It is thought that this would require a steady wind blowing parallel with the shelf break for a period of at least a week or so (YosHIDA, 1955). However, the dominant wind direction in the Bay of Biscay is from the west-southwest (ISEr~R and HASSE, 1985), nearly perpendicular to the shelf break, and the region of cool water seems to exist throughout much of the summer season (Satmer Bulletin, Nos 10-13, 1984), when the wind speeds are reasonably low (<2 m s-1 on average, ISEMER and HASSE, 1985). Further, the SST traces in Fig. 11 appear to be correlated with the spring-neap cycle in the way that the present numerical model would suggest. Consequently, it seems that upwelling is unlikely to be solely responsible for the cool water region. The physical situation, however, is also complicated by nonlinear phenomena which, although they are not describable by the present linear model, could contribute significantly to the mixing and should at least be mentioned. For instance, nonlinear wavewave interactions (as well as viscous dissipation) in the real world might act to considerably reduce the amplitudes of the higher mode internal tides (ScI-IOTr, 1977), and consequently might also affect the amounts of mixing produced. Further, the higher frequency internal waves (Figs 2 and 3) quite probably result from the nonlinear steepening and subsequent dispersion of the internal tidal depressions of the thermocline, much as in the formation of an undular bore. These internal waves would distort the velocity field due to the internal tides, and could either locally enhance or diminish the degree of mixing, depending upon whether the local Richardson number was decreased or increased. In addition, turbulent internal hydraulic jumps, another nonlinear phenomenon, could occur if the surface tidal currents were sufficiently strong that an internal Froude number became supercritical (e.g. BAIr,~S, 1984), and could result in significant mixing near the shelf break. However, the present numerical model indicates that the flow can only achieve a supercritical Froude number for the higher modes (modes 2 and higher on the shelf, modes 7 and higher in the ocean), which are of relatively small amplitude, so that large amounts of mixing would not be expected. Further, no such effect seems to have been reported in the Bay of Biscay, and no definite evidence of any hydraulic jump has been observed in echo-sounding recordings across the shelf break (R. D. PInOREE, personal communication). In any case, these nonlinear effects are beyond the scope of the present paper, in which we have confined our attention to a relatively simple, linear numerical model; this seems to describe the internal motions of semidiurnal frequency remarkably well (refer to Figs 7 and 9). We have also demonstrated that mixing of the thermocline by these internal tides is a distinct possibility, especially during the summer and at spring tides. This mechanism may consequently account for the observed region of high phytoplankton abundance and cool surface water which aligns approximately with the shelf break, although nonlinear effects could also be of significance. A field experiment is currently being planned to investigate the internal tidal mixing mechanism more closely. Acknowledgemena'--This work has been carried out with the support of the Procurement Executive, Ministry of Defence. The author is indebted to Dr. R. D. Pingree for many helpful discussions on the subject matter of the paper, and to Dr. J. C. Scott for bringing the satellite imagery to his attention. REFERENCES BAINESP. G. (1974) The generation of internal tides over steep continental slopes. Philosophical Transactions of the Royal Society, A277, 27-58.
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A . L . NEW
BAU~-S P. G. (1982) On internal tide generation models. Deep-SeaResearch, 29, 307-338. BAINESP. G. (1984) A unified description of two-layer flow over topography. Journal of Fluid Mechanics, 146, 127-167. BAINES P. G. and X.-H. FANO (1985) Internal tide generation at a continental shelffslope junction: a comparison between theory and laboratory experiment. Dynamics of Atmospheres and Oceans, 9, 297-314. BELL T. H. (1971) Numerical calculation of dispersion relations for internal gravity waves. Naval Research Laboratories, Washington, Report No. 7294. BRlSCOE M. G. (1984) Tides, solitons and nutrients. Nature, 312, 15. DEWrrr L. M., M. D. LEVINE, C. A. PAULSONand M. V. BURT (1986) Semidiurnal internal tide in JASIN: observations and simulation. Journal of Geophysical Research, 91, 2581-2592. HAZEL P. (1972) Numerical studies of the stability of inviscid stratified shear flows. Journal of Fluid Mechanics, 51, 39-61. HOLLIGAN P. M., R. D. PINGREE and G. T. MARDELL (1985) Oceanic sohtons, nutrient pulses and phytoplankton growth. Nature, 314, 348-350. ISEMERH.-J. and L. HASSE(1985) The Bunker climate atlas of the North Atlantic ocean. Vol. 1: observations. Springer-Verlag, New York, 218 pp. MILES J. W. (1961) On the stability of heterogeneous shear flows. Journal of Fluid Mechanics, 10, 496-508. NEW A. L. (1987) Internal tidal currents in the Bay of Biscay. In: Advances in underwatertechnology, ocean science and offshore engineering. Vol. 12: Modelling the offshore environment. Graham and Trotman Ltd., London, pp. 279-293. P~OREE R. D. and G. K. MORRISON(1973) The relationship between stability and source waters for a section in the Northeast Atlantic. Journal of Physical Oceanography, 3, 280-285. PINGREE R. D. and G. T. IV[ARDELL(1985) Solitary internal waves in the Celtic Sea. Progress in Oceanography, 14, 431-441. PINGREE R. D., D. K. GP,n~nTHSand G. T. MARDELL(1983) The structure of the internal tide at the Celtic Sea shelf break. Journal of the Marine BiologicalAssociation of the United Kingdom, 64, 99-113. PINt3REER. D., G. T. MARDELLand A. L. NEW (1986) Propagation of internal tides from the upper slopes of the Bay of Biscay. Nature, 321, 154-158. PRI~SENaERGS. J. and M. RATrRAV(1975) Effects of continental slope and variable Bnmt-Vaisala frequency on the coastal generation of internal tides. Deep-SeaResearch, 22, 251-263. PRINSENBERO S. J., W. L. WILMOTand M. RATrE~Y (1974) Generation and dissipation of coastal internal tides. Deep-SeaResearch, 21,263-281. SANDSTaOMH. and J. A. ELUOTr (1984) Internal tide and solitons on the Scotian shelf: a nutrient pump at work. Journal of Geophysical Research, 89, 6415--6426. Satmer (1984) Monthly bulletin. Edited by: Le centre de meteorologie spatiale, 22302 Lannion Cedex, B.P. 147, France. ScHorr F. (1977) On the energetics of baroclinic tides in the North Atlantic. Annales de G~ophysique, 33, 41-62. SCHWIDERSKIE. W. (1979) Global ocean tides, Part II: the semidiurnal principal lunar tide (Mz), atlas of tidal charts and maps. Naval Surface Weapons Center, Dahlgren, Virginia, U.S.A. Report No. NSWC TR 79--414. SCHWIDERSKIE. W. (1981) Global ocean tides, Part III: the semidiurnal principal solar tide ($2), atlas of tidal charts and maps. Naval Surface Weapons Center, Dahlgren, Virginia, U.S.A. Report No. NSWC TR 81-122. THOMPSON O. R. and R. F. GASPAROVIC(1986) Intensity modulation in S.A.R. images of internal waves. Nature, 320, 345--348. THORPE S. A. (1971) Experiments on the instability of stratified shear flows: miscible fluids. Journal of Fluid Mechanics, 46, 299-319. THORPE S. A. (1973) Experiments on instability and turbulence in a stratified shear flow. Journal of Fluid Mechanics, 61,731-751. YOSHIDAK. (1955) Coastal upwelling off the California coast. Records of Oceanographic Works in Japan, 2, 1-13. APPENDIX
Following methods similar to those described by PRINSENBERG and RATrRAY (1975), the c~effidents in equation (7) are given by P~=
fo--h~ (N _ g
dz,
, r *
,r=n
(A1)
Internal tidal mixing in the Bay of Biscay
f O
02=
(N 2 -
d)
l .L2 a z ~^..i(k2m+k ~)zla~~,,~,m
709
(A2)
-hi =
f4, (N2 -- 02) e-ikl"z/adPl (COSk2oz/Ot+ ~01 sin/C'oZ/a) ,~ -
[ (o (N ~ _ o2) P~cm= ] J-hz- ~m ei(F'-F')z/a*2*2mdz, [ g/ik2m
Q}" = -
f (o
, j :/: m
(A3)
(A4)
,j = m
O (N 2 _ 0-2) ei(#+k~)z/ad~2d~1 dz ik~
(A5)
--hi
-h2
=
(cos k~la + i sin k~zla)dp~} dz
"/~"
~01cos keoz/ct + i sin k20z/ct eiF'z/a~b2 dz
(N z_ o2) ei(~+k~)z/a~b2dp1 dz. J-~l ikl
(A6)
0198-0149/88$3.00+ 0.00 © 1988PergamonPressplc.
Printedin GreatBritain.
Internal tidal mixing in the Bay of Biscay A. L. NEW* (Received 18 May 1987; in revised form 8 September 1987; accepted 27 November 1987) Abstract--The continental shelf break in the Bay of Biscay often corresponds to a region of high phytoplankton abundance and cool water near the surface. It has been conjectured that these phenomena are due to mixing created by internal tidal motions that result from the interaction of the surface tide with bottom topography. This process is investigated in the present paper by employing a linear numerical model that describes the topographic generation of the internal tides. After demonstrating that the model is in good agreement with existing observations, it is found that there are typically two thermocline regions near the shelf break in which the Richardson number falls below the critical value of ] at some stage in the tidal cycle, indicating an unstable flow. By considering linear stability theory, mixing seems likely in these two regions during the summer months and at spring tides but not at neaps. The conclusion is consistent with available sea surface temperature data.
1. I N T R O D U C T I O N
THE continental shelf break in the Bay of Biscay is often found to be a region of cool surface water (Fig. 1) and high phytoplankton abundance, as described by PINGREEet al. (1986, to be referred to as PMN in the following). Both these phenomena, occurring also in other parts of the world's oceans, have been conjectured to be manifestations of physical mixing caused by internal motions of semidiurnal period (internal tides) that result from the interaction of the surface tide with shelf break topography (SASDSTROM and ELLIOT1",1984; BRISCOE, 1984; HOLLIGAS et al., 1985; PMN). This mixing would result in an upward transport of cool water, rich in nutrients, from below the thermocline which, upon reaching the well-illuminated surface layers, would provide a favourable environment for the growth of plant cells as well as the observed decrease in SST. In a survey of the area shown in Fig. i with a towed cycling CTD (the SeaSoar), PMN found internal tidal depressions of the thermocline travelling away from the shelf break in both the deep ocean and the shallow shelf waters. A typical example of the data collected in this manner is shown in Fig. 2, and the long wavelength (55 kin) tidal signal is dearly evident. [Similar observations have also been presented by PINGREEand MARDELL (1985) in which moored thermistor arrays indeed show that this signal has a semidiurnal period.] Figure 2 is complicated, however, by relatively high frequency internal waves (with apparent periods close to 10 min), which seem to appear with largest amplitudes in the tidal troughs. These short waves produce a roughening of the sea surface (THOMPSON and GASPAROVIC,1986) that is often visible as a series of long-crested (many km) "rips" with a spacing of the order of 1 km. These rips are also detectable from space using satellite imagery, as shown in Fig. 3a. The complex nature of the surface signature takes * Institute of Oceanographic Sciences, Brook Road, Wormley, Godalming, Surrey GU8 5UB, U.K. 691
692
A.L. NEW 8
°W
I
6
I
I
I
,,
,/. /
/ Io /
/
/k.._--~
22
/
'~- '-~\ls',-.
/
,'
t
~
/
21
/
/
/ -
i
/
/
/
/
:~pOOm-46
/
/
/
t..
f
I
I
I
Fig. 1. Chartof the Bay of Biscay, from PMN, showing the main survey area (rectangle), mean sea surface temperature (SST) contours (*C), positions of internal tidal troughs 21-23, and the location of mooring 080.
o8oo
i
lOOO I
I
12oo h I
I
m.
100-
^ 21
^ 22
^ 23
Fig. 2. Example of SeaSoar tow on 19 September 1985 recorded by PMN approximately 50-150 km oceanward of the shelf break. Isopycnals are plotted at densities of 1026, 1026.5 and 1027 kg m-3, and tidal troughs 21-23 are indicated.
the form o f packets of the short internal waves travelling away from the shelf break (typically 3-6 waves in each packet), the envelope of all such contiguous packets indicating the approximate position of a tidal depression of the thermocline (two such pulses visible on shelf, o n e in the deep ocean). Figure 3b is an interpretation of this S A R image and shows, in relation to the bathymetry, the main wave packets, which appear to be radiating from the topographic ridges b e t w e e n canyons.
(a)
6
"W Fig. 3a
5
(b) 47
,.
II
,Io,, ,
~11
~x oN
ir~
I /I
I
1
'
"
r
j ~
""
\I I
~
\ 1
//
I~'I ...'~
I
If~
/
/
'
~ 200m
~
/ / //
~_~
I I
I f
~
-10
"~ /
0m
I
~,~'2000m I
I I r
~
I
-"
~.
.
.
.
.
~
3000m
\t
..... t
, I \
/
I
400Om
. 1 ',"
6
°W
46
5
Fig. 3. (a) Synthetic aperture radar image of the Bay of Biscay on 20 August 1978, showing internal wave packets near the shelf break. Courtesy Deutsche Forschungs- und Versuchsanstalt fiir Luft- and Raumfahrt (DFVLR). (b) Interpretation of (a), showing the main wave packets in relation to the bathymetry and possible sites for their generation (O).
Internal tidal mixing in the Bay of Biscay
695
Two alternative types of linear numerical model for the generation of internal tides by two-dimensional topography have now been widely applied in the literature. The first of these is the ray theory of BARNES(1974), which is applicable to arbitrary topography, but only to certain forms of (continuous) density profile. In particular, difficulties are encountered whenever a thermocline is present, producing a rapidly varying stratification. However, a simplified version of the theory has been developed by BAINES (1982) that can accomodate a step-like interface over a uniformly stratified lower layer, and has been compared with laboratory experiments by BAINESand FANG (1985). Alternatively, modal models, applicable to the case of a linear shelf slope but valid for arbitrary stratification, are now also available. The theory has been developed by PRINSENBERGet al. (1974) and PmNSENaERG and RATrRAY (1975), and compared with oceanographic observations by DEwrrr et al. (1986). The principal aim of the present paper is to investigate whether numerical models such as these, applied to the Bay of Biscay, predict thermocline mixing near the shelf break as discussed above. Since it was considered more important to accurately model the density structure than the precise form of the shelf break profile, a modal theory similar to that of PRINSENBERGand RATIRAY(1975) was employed. An important consideration, however, is to ascertain to what extent the theory is applicable to the real oceanic situation. In particular, PMN found that the summer thermocline over the deep ocean in the Bay of Biscay was unexpectedly dominated by internal tides of modes 3 or 4, or some combination of both. The present model strongly supports dominance by a mode 3 tide, and is favourably compared not only with the observations of PMN, but also with moored current meter data. Further, the possibility of internal tidal mixing is positively demonstrated, and seasonal variations, as well as those occurring over the spring-neap cycle, are discussed. 2. T O P O G R A P H I C G E N E R A T I O N M O D E L
It is easy to show that a stratified fluid has possible two-dimensional internal modes of oscillation of the form
rl(x, z, t) = acb(z)e i(k~-°O
(1)
for the vertical displacement of a fluid particle with mean position x, z and at time t, in which qb satisfies 0 ~ + k2
¢(0) = ¢ ( - h ) = 0,
dp = 0
(2a)
(2b)
where N 2 = -g/p0(0) • Opohgz for a background density field of p0(z), f = 1.075 x 10-4 s-1 is the Coriolis frequency, and z = 0, -h correspond to the mean positions of the surface and horizontal bed, respectively. This equation was solved by standard numerical techniques (e.g. BELL, 1971), for which the z-increment was taken to be 1 m in the upper 150 m of the water column, and 5 m elsewhere. Two typical N(z) profiles will be used in the present work, corresponding to summer and winter conditions. For the upper 150 m, the N-values for the summer were
696
A . L . NEW
calculated from a SeaSoar tow recorded in September 1985 and averaged over three complete internal tidal wavelengths, while for winter, N was set equal to 2.4 x 10-3 s-1 to represent relatively well-mixed conditions, as shown in Fig. 4a. Below this level, N(z) was modelled from data collected by PINGm~Eand MORPasoN (1973) as
N(z) =
2x(z + 150)~
-],-150 >t z/> -1200 m
(3a)
2.4 x 1003exp(0.65103 x 1003(z + 1200)), -1200 t> z >t -2400 m
(3b)
1.1 x 10°3
(3c)
2.4 x 1003 + 0.4 x 10-3sin \
l~lJ
,-2400 ~> z ~> -4400 m
which allows for the permanent pycnocline at about 900 m, with gradual decay below. The present numerical model is based on that of PRINSENBERGand RArrKAV (1975), and will be described only briefly. For the two-dimensional shelf break topography as shown in Fig. 4b and oscillations of semidiurnal frequency (6 = 1.405 x 10-4 s-l), we
0 I
I
I
N xl03 s"1
20
I
I
I
1 "~"
[
100
10
/ ./"/"
I "/ | ./
b
2~60m 1
~x
2
4400 m
Fig. 4. (a) Brnnt-Vaisala frequency (N) profiles for typical summer (solid line) and winter (broken line) stratifications, and from data recorded at mooring 080 (chained line). (b) Idealized topography for the numerical model, showing the shelf and oceanic regions, 1 and 2, respectively, connected by a slope of gradient a = 0.09. The origin of co-ordinates, in the mean level of the surface, is as indicated.
Internal tidal rni~g in the Bay of Biscay
697
suppose that the vertical displacements in regions 1 and 2 are given, respectively, by r l l = [ a°ldp~e-ik~ + l~n~l ale-/k~ dP~] e-/at n2 =
(a~ e -ik~ + b N ilc~x) , 2 + ~
a2m eik2~ dp2
(4a) e..i6t.
(4b)
rnzl
The first term in equation (4b) represents an incident deep water surface tide in region 2 of amplitude ~ , whereas b 2 is the amplitude of its reflected component and a~ in equation (4a) is the amplitude of the component transmitted over the shelf. Since these barotropic tides are essentially hydrostatic, we deduce
(5a) (5b) To these surface tides are added sums of internal tides travelling away from the shelf break, so that (~l(z), k~, a~) and (tp2(z), k2,,,, a2m)correspond to the n th and m th internal tides calculated for the shelf and ocean depths, hi and h2, repectively. The only essential difference between this model and that of PR~SENaERC and RAa'rRAY (1975) is the allowance for a travelling surface tide on the shelf rather than a standing tide, since this was considered to be more appropriate for the Bay of Biscay (e.g. SCmVIDEltSIO,1979, 1981). The amplitudes of the surface tides in the two regions are related by a 2 = a~(k~ + ~ ) / 2 k 1
(6a)
b2
(6b)
1 1 - k~o)12k~ = ao(ko
from continuity of surface elevation and mass flux, so that only ao~ is an externally specified parameter (taken as a01 = 0.75, 2.00 m for neap and spring tides, respectively). Proceeding with analysis similar to that described by PRINSErOERC and RATa'RAY (1975), we find, in order to satisfy the watching conditions on the sloping boundary x = -z/a between the two regions, that the modal amplitudes satisfy N
M
alnF~ = 4 S ~ + ~,a2m Q,~, 2 n=l M
~, a2 p f i m = 4 ~ m=l
r=l,N
(7a)
j= I,M,
(7b)
m=l N
+ ~,a~Qln, n=l
where the coefficients are given in the Appendix. These relations were solved on a computer with M = 60 and N = 9. Since the amplitudes of the higher modes appeared to be successively less reliable, and also because the numerical solutions typically contained spurious oscillations on approximately the length scale of the highest modes retained (see Fig. 2 of PmNSEI,mEROet al., 1974), it was found desirable to apply an ad hoc filter to mode numbers rn = 30--60 such that the amplitude of the m th mode was
698
A.L. NEW
multiplied by 1-(m-30)/30. A similar filter was applied to modes n = 3-9 for the shelf motion. 3.
GENERAL
F O R M OF S O L U T I O N
To reveal the general form of the solution, and also as a test of the numerical procedure as a whole (including the filters), Fig. 5 exhibits the nature of the displacement field, which can be written as rl(x, z, t) = Re(fi(x,
z)e -iat) =
Re(lfild°e-i~
c o s ( 6 t - 0),
=
(8)
and shows vertical profiles of Ifi(x, z) l, the maximum elevation of a fluid particle from its mean position, and the phase, 0(x, z), converted to degrees, at which this is attained. We see that large vertical oscillations are possible at certain depths (e.g. in excess of 200 m in amplitude at 2.5 km depth and x = 50 km), and that, generally speaking, the phases vary little in the vertical except at these depths. (Note that since each of the internal tidal amplitudes is directly proportional to ao~, it is unnecessary to describe results for any tidal state other than spring tides.) The regions of large oscillations result from beams of energy (described by BAINES, 1982) which emanate from the point of the shelf break and travel along ray paths that have the characteristic slope
dz
--=
+
dx
-
( o -f
(9)
-
rn
o
I
o
I
0
km
50
lOO
"x
" i
i
\.\
J
,/
I
I
3 6 0°
I
I
i
I
'\
200
I
\
' ~ / "
\
'\\ \.\ l\'\\\\,\i]\j!\\, \
\
" i \ iI /
I\\
/
i~
Fig. 5. Profiles of amplitudes I~1 (solid fines) and phases 0 (broken lines) of the vertical displacement at spring tides and for the winter stratification.The vertical lines coincidewith the zeroes of I~1 and phases of 0°, while the scales for these two quantities are shown above the profiles at 125 km. The chained lines represent the rays emanatingfrom the shelf break.
Internal tidal mixing in the Bay of Biscay
699
and reflect from both the surface and the horizontal bed. These rays (two in the deep ocean region, one over the shelf) are also illustrated in the figure, and theoretically should coincide with the positions of largest Ifi 1. The good agreement is taken as a validation of the numercial model as a whole. The corresponding situation for the summer stratification reveals that neither the maximum amplitudes nor the phases along the rays are subject to much seasonal variation, but that the vertical spacing between the beams in the deep ocean is now much increased. The agreement between the positions of the primary maxima of [rl I and the rays is comparable to that shown in Fig. 5, and this is taken as an indication that the seasonal thermocline is adequately represented. The horizontal velocity fields, which may also be easily calculated and are discussed in detail by NEW (1987), reveal a beam-like structure similar to that already described for the displacements. At spring tides, for instance, velocities of the order of 0.5 m s-1 in excess of those due to the surface tides are achieved along the rays, both by the acrossslope and along-slope components, but are lower elsewhere. These currents should not be neglected in the design and operation of offshore structures. 4. T H E R M O C L I N E MOTION
The internal tides in the Bay of Biscay are quite possibly the most energetic found anywhere in the world (BAINES, 1982). Using results from the present model, calculations of/~, the depth-integrated, phase-averaged energy, were performed for each mode, and for both winter and summer stratifications. In both cases, and for both shelf and ocean, mode 1 was the most energetic, with a general decrease for the higher modes. For example,/~ was equal to 8590, 6690, 6010, 400, and 2670 J m - 2 for the first five ocean modes, for the summer stratification and spring tides, with similar values for the winter stratification. However, it does not follow that mode 1 should necessarily give the largest contribution to the vertical displacement at all depths. For instance, decomposing the thermocline displacement (at a depth of 50 m, say) into its constituent modal components at spring tides reveals that, although the summer thermocline over the shelf is dominated by mode 1 (amplitudes of 9.7, 6.2, 3.1, 2.4 and 7.3 m for the first five modes, respectively), that over the deep ocean is dominated by mode 3 (amplitudes of 1.8, 4.3, 13.0, 2.5 and 3.8 m for the first five modes), in agreement with the observations of PMN. (For the winter stratification this predominance by mode 3 disappears.) To illustrate this more clearly, Fig. 6 shows instantaneous thermocline displacements at six equally spaced intervals through the tidal cycle. At time t = 0 h (Fig. 6a), the thermocline above the shelf break has attained its maximum positive elevation. As the flooding tide then begins to decrease, the thermocline in this region sinks until the time of maximum ebb (Fig. 6d), at which point two troughs appear to be forming. The offshelf trough then begins to move offshelf (Fig. 6d-f) and, from about the beginning of the succeeding tidal cycle (Fig. 6a) has attained an almost uniform speed and results in a nearly sinusoidal displacement of the thermocline with the wavelength characteristic of mode 3 (about 55 km). Similarly, the trough that forms shelfwards of the shelf break travels onto the shelf and has a wavelength of about 35 km, corresponding to mode 1. (The less regular nature of the thermocline motion over the shelf is due to frequent oscillations of the ray between the surface and the bottom.)
700
A.L. NEW
0
.~
I
-50 [
I
i
i
m~ 100-j
0
.
|
I
i
i
50
i
. . . . . . . .
I
km
i
,
i
100
i
1
. . . . . . . .
--A
A
A
e
Fig. 6. Numerically predicted displacements, for the summer stratification and spring tides, of the isopy~al with a mean depth of 50 m at times of (a) 0 h, maximum onshelf flow, (b) 2.1 h, (c) 4.1 h, (d) 6.2 h, (e) 8.3 h and (f) 11).3 h. The positions of the tidal troughs, estimated by eye, are indicated, and the shelf break is at x = 2.9 kin.
-50-
fJ /X ,aX / /0
0
0
tX / .s
x
~ /
o\ x
10
\ \ x
km
o OQ
2'0
h
'
\
o\
x\ \ o
~
o\,, o~ \ 5O
\
NO X,,, ,,0
x
0
\ \ OX
100-
Fig. 7. Comparison of internal tidal trough positions as observed by PMN (o) and predicted by the numerical model (x). The horizontal axis represents computational time, and the broken lines represent a mode 1 internal tide travelfing onto the shelf and a mode 3 into the ocean.
Internal tidal mixingin the Bay of Biscay
701
The positions of these troughs have been plotted in Fig. 7 and compared with the observed positions of the tidal thermocline depressions presented by PMN, taken from SeaSoar tows as in Fig. 2. To enable this comparison to be made, the time of maximum ebb was taken as 2 h after predicted high water at Plymouth (see PMN for full details). The positions of the two data sets are in remarkably good agreement, and further, are closely fitted by two straight lines representing the propagation of a mode 3 tide into the ocean and a mode 1 onto the shelf, both lines emanating from the shelf break at the time of maximum ebb. This provides a simple rule for predicting the positions of both the tidal thermocline depressions and the packets of higher frequency internal waves with which they seem to be associated. 5.
COMPARISON
WITH OBSERVATIONS
In addition to the above comparison, the numerical model has also been tested against the laboratory results of BAn~_s and FANO (1985). In their apparatus, a barotropic "tide" was produced in a uniformly stratified fluid (N = 0.97 s-1) by a 45 ° piston situated at one end of the wave tank. This interacted with representative shelf break topography to produce internal oscillations, and some of their results are summarized in Fig. 8. Superimposed is the flow field predicted by the present numerical model at 0.46 m from the shelf break along the lower "oceanic" ray, a position used for some of the laboratory results (theoretically, the solution should be independent of this co-ordinate). Overall, there is good agreement, except for regions near the rays (at ~ = 0 and 9.7 cm) which
a ......
....
t2° -
;-
",'"
iii:ii '
_'_"~.~o
0
- :"i .Y
0i3
0i6
~
~
d
I
!
I
Fig. 8. Horizontalvelocity(u) comparisonof the present numericalmodel (with o = 0.248 s-1, f -- 0 s"-1,a -- 0.38 and a~ = 1.214 x 10-3 m, solidlines) with the laboratoryresults (e) and raytracing model (broken lines) of BAINESand FANG(1985) at (a) maximumonshelfflow, (b) high water, (c) maximumebb and (d) low water. Usbis the maximumbarotropic velocityat the shelf break, and ~ the perpendicular distance from the lower ray in the oceanic region.
702
A . L . NEW
occupy some 10% of the fluid volume. These regions contain significant contributions from the higher modes in the model, which in the laboratory experiment may be quickly damped out by the action of viscosity, as discussed by BAINESand FANC (1985). Figure 8 also shows the linear ray tracing theory of BAINES(1982) which, for this case of constant N(z), should theoretically be equivalent to the present modal model, provided a sufficient number of modes is chosen. Overall, a close agreement between the two theories is attained, the slight oscillations in the modal model being expected to decrease for an increase in the number of modes. This comparison furnishes a further test of the present model, and in particular indicates that a sufficient number of modes has been taken to adequately describe the narrow beams, and also that the applied smoothing does not distort the solution to any great extent. A comparison has also been made with oceanographic data collected in the Bay of Biscay during August 1984 by R. D. PIN~R~E (personal communication). A mooring, with current meters at depths of 50, 100, 500 and 1000 m was positioned at 47°23'N, 6°40'W in a total depth of 1500 m (site 080 in Fig. 1). The across-slope currents at these depths were averaged over eight successive tidal periods near spring tides, and are plotted in Fig. 9. The numerical predictions, for the Brunt-Vaisala profile as shown in Fig. 4a, a~ = 2.00 m, and at x = 16.5 km, are in remarkably good agreement with the data. The largest differences occur at 50 m depth and could possibly result from nonlinear effects in the thermocline, but these are not large and overall the comparison is encouraging. In particular, the near 180° phase difference between the motions at 50 m and 1000 m is well represented by the model.
-0"5 0
OoI
f"
0.5 -i
I
Fig. 9. Comparison of across-slope velocities at mooring 080 (positive values offshelf) at depths of (a) 50 m, (b) 100 m, (c) 500 m and (d) 1000 m. The solid lines are the observations, the broken lines the numerical predictions.
703
Internal tidal mixing in the Bay of Biscay 6. I N T E R N A L T I D A L M I X I N G
The above comparisons of numerically predicted and observed velocities, both in the laboratory and the ocean, give us confidence to investigate variations in the Richardson number
where (u, v) are the horizontal components of velocity. Figure 10 shows contours of the minimum values (R~ in) achieved during the tidal cycle for a summer stratification and at spring tides. The contours generally follow the internal ray paths emanating from the point of the shelf break, and regions of g rain < ¼ Occur primarily where the rays cross the seasonal thermocline at a depth of about 50 m, with an additional region centred around 900 m, corresponding to the permanent pycnocline. (Note also that part of the energy in the upper oceanic ray is apparently reflected from near the base of the thermocline, at about 130 m, producing regions of R~ in < 10, as indicated by the additional rays. This behaviour might be expected theoretically.) A steady, stratified shear flow is generally thought to become unstable to small perturbations, always present in any physical flow, whenever the value of Ri is less than (MILES, 1961), eventually resulting in the formation of Kelvin-Helmholtz billows and mixing (THORPE, 1971, 1973). If the flow contains no horizontal boundaries, and the steady background density and horizontal velocity fields can be modelled by either hyperbolic tangent or error function profiles with the same vertical length scale d, then -2,o
_
,
,
~
-,o
o
,
?
,
~
kr
,
2o
,
3,0
,
__,o, x\\
~ix "4
\
1/ \\\\ '\',,\\
\\ \\ , \% \',", I,4
O/
\
X\,, \\
Fig. 10. Contours (at values of 10, 1 and ¼) of the minimum Richardson numbers, R mi~, achieved during the tidal cycle, for spring tides and the summer stratification. The broken lines represent the rays and values of y at particular positions along these paths are indicated.
704
A . L . NEW
the most unstable disturbance may, according to linear stability theory, have amplitude, wavenumber, and growth rate given by a(t) = ao e kct
(lla)
k = 0.441d
(llb)
kci = 0.8N(¼ - Ri)IR~.
(llc)
These equations, in which Ri is taken as the minimum value occurring in the flow (at the centre of the pycnocline, and N at the same level), are essentially a simple fit to data presented by THORPE (1971) and HAZEL (1972). Although our tidally varying flow is unsteady, we may nevertheless expect growth of similar instabilities during any period for which Ri < ¼, at least if the tidal flow varies on a timescale longer than that of the instabilities (the quasi-steady approximation of THORPE, 1971), which seems to be the case here. However, horizontal boundaries are present near the thermocline (i.e. the surface), the density and velocity profiles in the numerical model are not of the required simple forms (although a reasonable fit may be obtained for a certain depth range near the thermodine), and the length scales of these two quantities are not identical (a factor of about three different). Nevertheless, equation (11c) may give the growth rate of the most unstable disturbance correct at least to an order of magnitude. For positions near the ray paths, this relationship was integrated over the time interval for which subcritical values of R i (i.e. R i < ¼) were maintained. This enabled the ratio of the final (af) to the initial (ao) amplitudes of the disturbance to be calculated. The estimates of the "growth factors", defined by y = loglo(af/ao),
(12)
are also indicated in Fig. 10. The growth rates in equation (11c) are strictly only applicable for small amplitude disturbances. When a sufficiently large amplitude has been attained, we may expect overturning and mixing to occur as already described. If we were able to identify this "breaking amplitude", and also estimate the likely initial disturbance level in the real world, this would enable a critical value of T = Tc, say, to be proposed, such that if T I> To, mixing might be expected to occur. For example, if the breaking amplitude is of the order of 10 m, a reasonable physical value, and the initial disturbance has an amplitude of 1 mm, this would give Tc = 4. However, since all such estimates of T~ are somewhat speculative, and also since linear stability theory will cease to apply during the final stages of the growth and may then overestimate the growth rate, it would perhaps be safer to suggest a value of Tc = 5 or 6 for the onset of mixing. Returning to Fig. 10, we see immediately that the deep region of subcritical Ri values is not likely to cause significant mixing. However, the region caused by the passage of the shelf beam through the thermodine has typical values of ), = 8, so that mixing seems at least to be a possibility. The oceanic thermocline region, on the other hand, achieves even larger values, typically T = 20, so that a somewhat more energetic mixing region might be expected here. Values of T were also calculated for the winter stratification and spring tides, and these indicated the possibility of a weak mixing region on the shelf only, with typical values of y = 5. For neap tides, no mixing was predicted by the model for either stratification, since Ri was found to be nowhere subcritical.
705
Internal tidal mixing in the Bay of Biscay -50 I
I
I
i
I
0 I
I
i
i
I
I
i
I
'
'
kin
i
I
50 I
I
i
I
o
I
°:t I
18T
,,']~
~
6
~
~::::::::~:::
I
~
~
\
Fig. 11. (a) Sea surface temperature (SST) traces across the shelf break in the Bay of Biscay at spring (solid line) and neap (broken line) tides during September 1985. (b) Caption as for (a). (c) Positions of the mixing regions as predicted by the numerical model for spring tides and the
summer stratification. Figure l l a and b show records of the sea surface temperature obtained with a ship's thermosalinograph (at a mean depth of about 3 m) during September 1985 (R. D. PINGREE, personal communication). The data were obtained by steaming directly across the shelf break in the survey area indicated in Fig. 1. Figure l l a illustrates two such runs with approximately the same temperature in the deep ocean, one at neap, and one at spring tides. The SST in both cases is seen to decrease (indicative of internal mixing) at about 20-30 km offshelf, and to remain more or less constant over the shelf. Figure l l b presents a further example of the same phenomenon, but now also shows a slight rise in SST further than about 10 km onto the shelf. In both these cases, the temperature drop is more pronounced at spring than at neap tides, in qualitative agreement with the numerical model, while the temperature decrease evident at neap tides could plausibly result from the mixing on the previous spring tides. Figure l l c then shows the predicted mixing regions, as described above and in Fig. 10. The mixing occurring in these regions might be expected to become smeared-out as a result of tidal advection and shear over the shelf and near the shelf break, and consequently to result in a reasonably uniform re#on of decreased SST extending up to perhaps 25-30 km offshelf, in good agreement with the observations. 7. DISCUSSION AND SUMMARY An important aspect of the present paper is an attempt to ascertain the reliability of linear generation models in describing the tidal response in physical situations, in
706
A . L . NEW
particular in the Bay of Biscay. In spite of the two-dimensionality of the present model, the comparison both with the observed positions of the tidal troughs in PMN, and the mooring in 1500 m water depth is encouraging, so that other results from the model may also be expected to reflect the physical situation. For instance, it is found that vertical oscillations in excess of 400 m in total excursion may occur in the deep ocean. The resulting alteration to instantaneous density profiles may have a significant effect upon the propagation of sound through the ocean, and deserves further investigation. In addition, horizontal currents with typical maximum values of the order of 0.5 m s-1 are associated with the internal tides, both in the deep ocean and on the shelf. These currents are in addition to those due to the surface tides, and should be allowed for in the design and operation of offshore structures. The only other attempt to compare a similar numerical model with oceanographic data of which we are aware is that of DEWrrr et al. (1986), using data from the JASIN experiment in the Rockall Channel. The model used by these authors is that of PRINSENBERGand RArrRAV (1975), in which a total of 30 oceanic modes was chosen to ensure the accuracy of the first 10 modes. The numerical results presented in their paper have been reconstructed using only these 10 modes (L. M. oEWrrr, personal communication), effectively applying a filter with a sharp cut-off, so that, since a seasonal thermocline is present in their observational data, some of the finer scale motion in the upper layers of the ocean may not be adequately resolved. In particular, the minimum Richardson numbers achieved with their model appear to be approximately Ri -- 2, attained when an internal ray crosses the thermocline. The present model was run with the same number of modes, and a similar result was indeed achieved, indicating that DEWrrr et al. (1986) may also have found significantly smaller values of Ri had more modes been used. The main aim of the present paper is an investigation of the possibility of internal tidal mixing near the shelf break. For a summer stratification and spring tides there are two thermocline regions in which the Richardson number falls below }, and the factors, y, relating to the amount by which an initially small amplitude disturbance might be expected to grow, were calculated. Although the relevance of the absolute values of y is perhaps open to question, their relative magnitudes may be useful in inferring the likelihood of mixing in different regions. The predicted mixing regions in the model extend to 25-30 km offshelf during the summer, and are most intense at spring tides. Both these results are consistent with the available data. In addition, the model (in the absence of dissipation) would predict a further mixing region where the ray, initially propagating downwards from the point of the shelf break and then reflected by the ocean floor, again crosses the thermocline, at about 150--160 km offshelf. This seems to be consistent with the results of HOLLIOAN et al. (1985), who observed temperature inversions and patches of turbulent mixing near the surface at about this distance from the shelf break. The interaction of the surface tide with the sea bed over the shelf might be expected to produce considerable turbulence, but this would be to some extent prevented from reaching the surface by the seasonal thermocline, and would presumably produce progressively more mixing further onto the shelf where the tidal flows are larger (see PINGREE et al., 1983). Consequently, it is difficult to see how this mechanism would produce a region of cool surface water primarily above the shelf break. Another possible mechanism for the production of the cool surface water is coastal
Internal tidal mixing in the Bay of Biscay
707
upwelling. It is thought that this would require a steady wind blowing parallel with the shelf break for a period of at least a week or so (YosHIDA, 1955). However, the dominant wind direction in the Bay of Biscay is from the west-southwest (ISEr~R and HASSE, 1985), nearly perpendicular to the shelf break, and the region of cool water seems to exist throughout much of the summer season (Satmer Bulletin, Nos 10-13, 1984), when the wind speeds are reasonably low (<2 m s-1 on average, ISEMER and HASSE, 1985). Further, the SST traces in Fig. 11 appear to be correlated with the spring-neap cycle in the way that the present numerical model would suggest. Consequently, it seems that upwelling is unlikely to be solely responsible for the cool water region. The physical situation, however, is also complicated by nonlinear phenomena which, although they are not describable by the present linear model, could contribute significantly to the mixing and should at least be mentioned. For instance, nonlinear wavewave interactions (as well as viscous dissipation) in the real world might act to considerably reduce the amplitudes of the higher mode internal tides (ScI-IOTr, 1977), and consequently might also affect the amounts of mixing produced. Further, the higher frequency internal waves (Figs 2 and 3) quite probably result from the nonlinear steepening and subsequent dispersion of the internal tidal depressions of the thermocline, much as in the formation of an undular bore. These internal waves would distort the velocity field due to the internal tides, and could either locally enhance or diminish the degree of mixing, depending upon whether the local Richardson number was decreased or increased. In addition, turbulent internal hydraulic jumps, another nonlinear phenomenon, could occur if the surface tidal currents were sufficiently strong that an internal Froude number became supercritical (e.g. BAIr,~S, 1984), and could result in significant mixing near the shelf break. However, the present numerical model indicates that the flow can only achieve a supercritical Froude number for the higher modes (modes 2 and higher on the shelf, modes 7 and higher in the ocean), which are of relatively small amplitude, so that large amounts of mixing would not be expected. Further, no such effect seems to have been reported in the Bay of Biscay, and no definite evidence of any hydraulic jump has been observed in echo-sounding recordings across the shelf break (R. D. PInOREE, personal communication). In any case, these nonlinear effects are beyond the scope of the present paper, in which we have confined our attention to a relatively simple, linear numerical model; this seems to describe the internal motions of semidiurnal frequency remarkably well (refer to Figs 7 and 9). We have also demonstrated that mixing of the thermocline by these internal tides is a distinct possibility, especially during the summer and at spring tides. This mechanism may consequently account for the observed region of high phytoplankton abundance and cool surface water which aligns approximately with the shelf break, although nonlinear effects could also be of significance. A field experiment is currently being planned to investigate the internal tidal mixing mechanism more closely. Acknowledgemena'--This work has been carried out with the support of the Procurement Executive, Ministry of Defence. The author is indebted to Dr. R. D. Pingree for many helpful discussions on the subject matter of the paper, and to Dr. J. C. Scott for bringing the satellite imagery to his attention. REFERENCES BAINESP. G. (1974) The generation of internal tides over steep continental slopes. Philosophical Transactions of the Royal Society, A277, 27-58.
708
A . L . NEW
BAU~-S P. G. (1982) On internal tide generation models. Deep-SeaResearch, 29, 307-338. BAINESP. G. (1984) A unified description of two-layer flow over topography. Journal of Fluid Mechanics, 146, 127-167. BAINES P. G. and X.-H. FANO (1985) Internal tide generation at a continental shelffslope junction: a comparison between theory and laboratory experiment. Dynamics of Atmospheres and Oceans, 9, 297-314. BELL T. H. (1971) Numerical calculation of dispersion relations for internal gravity waves. Naval Research Laboratories, Washington, Report No. 7294. BRlSCOE M. G. (1984) Tides, solitons and nutrients. Nature, 312, 15. DEWrrr L. M., M. D. LEVINE, C. A. PAULSONand M. V. BURT (1986) Semidiurnal internal tide in JASIN: observations and simulation. Journal of Geophysical Research, 91, 2581-2592. HAZEL P. (1972) Numerical studies of the stability of inviscid stratified shear flows. Journal of Fluid Mechanics, 51, 39-61. HOLLIGAN P. M., R. D. PINGREE and G. T. MARDELL (1985) Oceanic sohtons, nutrient pulses and phytoplankton growth. Nature, 314, 348-350. ISEMERH.-J. and L. HASSE(1985) The Bunker climate atlas of the North Atlantic ocean. Vol. 1: observations. Springer-Verlag, New York, 218 pp. MILES J. W. (1961) On the stability of heterogeneous shear flows. Journal of Fluid Mechanics, 10, 496-508. NEW A. L. (1987) Internal tidal currents in the Bay of Biscay. In: Advances in underwatertechnology, ocean science and offshore engineering. Vol. 12: Modelling the offshore environment. Graham and Trotman Ltd., London, pp. 279-293. P~OREE R. D. and G. K. MORRISON(1973) The relationship between stability and source waters for a section in the Northeast Atlantic. Journal of Physical Oceanography, 3, 280-285. PINGREE R. D. and G. T. IV[ARDELL(1985) Solitary internal waves in the Celtic Sea. Progress in Oceanography, 14, 431-441. PINGREE R. D., D. K. GP,n~nTHSand G. T. MARDELL(1983) The structure of the internal tide at the Celtic Sea shelf break. Journal of the Marine BiologicalAssociation of the United Kingdom, 64, 99-113. PINt3REER. D., G. T. MARDELLand A. L. NEW (1986) Propagation of internal tides from the upper slopes of the Bay of Biscay. Nature, 321, 154-158. PRI~SENaERGS. J. and M. RATrRAV(1975) Effects of continental slope and variable Bnmt-Vaisala frequency on the coastal generation of internal tides. Deep-SeaResearch, 22, 251-263. PRINSENBERO S. J., W. L. WILMOTand M. RATrE~Y (1974) Generation and dissipation of coastal internal tides. Deep-SeaResearch, 21,263-281. SANDSTaOMH. and J. A. ELUOTr (1984) Internal tide and solitons on the Scotian shelf: a nutrient pump at work. Journal of Geophysical Research, 89, 6415--6426. Satmer (1984) Monthly bulletin. Edited by: Le centre de meteorologie spatiale, 22302 Lannion Cedex, B.P. 147, France. ScHorr F. (1977) On the energetics of baroclinic tides in the North Atlantic. Annales de G~ophysique, 33, 41-62. SCHWIDERSKIE. W. (1979) Global ocean tides, Part II: the semidiurnal principal lunar tide (Mz), atlas of tidal charts and maps. Naval Surface Weapons Center, Dahlgren, Virginia, U.S.A. Report No. NSWC TR 79--414. SCHWIDERSKIE. W. (1981) Global ocean tides, Part III: the semidiurnal principal solar tide ($2), atlas of tidal charts and maps. Naval Surface Weapons Center, Dahlgren, Virginia, U.S.A. Report No. NSWC TR 81-122. THOMPSON O. R. and R. F. GASPAROVIC(1986) Intensity modulation in S.A.R. images of internal waves. Nature, 320, 345--348. THORPE S. A. (1971) Experiments on the instability of stratified shear flows: miscible fluids. Journal of Fluid Mechanics, 46, 299-319. THORPE S. A. (1973) Experiments on instability and turbulence in a stratified shear flow. Journal of Fluid Mechanics, 61,731-751. YOSHIDAK. (1955) Coastal upwelling off the California coast. Records of Oceanographic Works in Japan, 2, 1-13. APPENDIX
Following methods similar to those described by PRINSENBERG and RATrRAY (1975), the c~effidents in equation (7) are given by P~=
fo--h~ (N _ g
dz,
, r *
,r=n
(A1)
Internal tidal mixing in the Bay of Biscay
f O
02=
(N 2 -
d)
l .L2 a z ~^..i(k2m+k ~)zla~~,,~,m
709
(A2)
-hi =
f4, (N2 -- 02) e-ikl"z/adPl (COSk2oz/Ot+ ~01 sin/C'oZ/a) ,~ -
[ (o (N ~ _ o2) P~cm= ] J-hz- ~m ei(F'-F')z/a*2*2mdz, [ g/ik2m
Q}" = -
f (o
, j :/: m
(A3)
(A4)
,j = m
O (N 2 _ 0-2) ei(#+k~)z/ad~2d~1 dz ik~
(A5)
--hi
-h2
=
(cos k~la + i sin k~zla)dp~} dz
"/~"
~01cos keoz/ct + i sin k20z/ct eiF'z/a~b2 dz
(N z_ o2) ei(~+k~)z/a~b2dp1 dz. J-~l ikl
(A6)