Get documents about "QUANTIFYING VERTICAL FLUXES FROM TURBULENCE IN THE OCEAN
Document Sample
FEATURE
QUANTIFYING VERTICAL FLUXES FROM TURBULENCE
IN THE OCEAN
By J.N. Moum
TURBULENCE REPRESENTS the smallest numerical models, in which physical pro- tions to scales of a few millimeters or of
scales of the oceanic flowfield, ranging cesses on scales smaller than can be re- velocity fluctuations to scales of a few
from the order of 1 mm diffusive scales solved by the m o d e l - - t y p i c a l l y on the centimeters. Another estimate, based on
to eddy scales on the order of 1 m in the order of I00 m vertically and l0 km hor- a mixing length analogy, uses informa-
thermocline, to 10 m in the wind-mixed izontally-are simply parameterized. The tion about the energy-containing eddies
upper ocean, and to 100 m in the most parameterized physics includes turbu- of the turbulence (Moum, 1996a), which
energetic tidal mixing or convectively lence. in the main thermocline requires resolu-
mixing flows. Although models of the The only direct measure of the vertical tion of velocity fluctuations at ~10 cm
general circulation cannot possibly re- fluxes due to turbulence comes from ob- scale. It should be emphasized that these
solve the scales of turbulence, its effects servations of the temporal changes in ver- microstructure flux estimates are n o t flux
must be accounted for. This is because tical distributions of scalar variables. For measurements. The assumptions leading
turbulence is the primary agent for irre- example, the change in the center of mass to these flux estimates are fraught with
versibly mixing mass, heat, nutrients and of a stratified fluid exposed only to turbu- uncertainty.
other scalar properties important to the lence-enhanced mixing represents the In the Reynolds-averaged equations,
ocean's stratification, to its lifeforms, and system's gain in potential energy. The the vertical flux of a scalar C is the aver-
to its ability to rid itself of pollutants. By fact that other processes move density age value of the correlation between the
stretching material surfaces, turbulence surfaces vertically in the ocean (and yet vertical velocity fluctuation and the scalar
acts to enhance concentration gradients do not result in irreversible energy trans- fluctuation, <w'C'>. This is referred to as
so that molecular diffusion can proceed fer), makes this measurement difficult. the eddy-correlation flux. It has been
rapidly enough to play a role in the large- What we specifically want to know is the measured by the atmospheric boundary
scale, irreversible redistribution of prop- scalar flux across density surfaces, that is layer community since the 1950s, and is
erties. the diapycnal flux (I use the term vertical now measured using commercially avail-
Because the ocean is stably stratified, flux here as a general approximation). A able instrumentation. Such measurements
the largest property gradients are in the means to determine the scalar flux across produced classical results in the 1960s,
vertical direction. The biggest effect of density surfaces has been devised by for example, in the Kansas experiment
turbulence is to enhance vertical fluxes of Ledwell e t al. (1993), who injected dye (Haugen et al., 1971). Only in the 1990s
properties. It is critical that we quantify on a density surface in the thermocline have we seen equivalent types of eddy-
these vertical fluxes. This has been at- and observed its spreading over periods correlation measurements in the ocean
tempted from indirect inferences based of a year or so. Although a measure of made by a few groups using homemade
on the large-scale flowfield, from mea- the end result of the turbulence is ob- instrumentation. This lag in oceanic flux
surements of the end effects of turbu- tained, this method does not directly measurements is due to the difficulties of
lence-enhanced diffusion and from direct measure the properties of the turbulence making such measurements in the ocean
observations of properties of the turbu- itself, nor does it tell us about the specific (including platform motion, internal wave
lence itself. processes involved. contamination and small signal levels)
Vertical fluxes may be inferred from Indirect estimates of vertical fluxes and to the fact that the smaller scales of
the larger scale circulation's need to come from direct measurements of some oceanic turbulence dictate finer measure-
make ends meet (Garrett, 1994). Large- properties of the turbulence (or in ocean- merit resolution.
scale budget equations consistently re- ographer's parlance, microstructure).
quire more turbulent mixing in the main These include flux estimates based on Complexity of Small-Scale Flowfields
thermocline and below than has been ob- measurement of the temperature variance Although it will be beyond our abili-
served (Munk, 1966). So do large-scale dissipation rate, X (Osborn and Cox, ties for the foreseeable future to image
1972), estimates based on the turbulent the flowfield on all of the scales of mo-
kinetic energy dissipation rate, e (Os- tion, in time and three spatial dimensions,
J.N. Mourn. College of Oceanic and Atmo-
born, 1980), and conservation equation we can gain an appreciation for the rich-
spheric Sciences, Oregon State University. Corval- balances. The dissipation estimates re- ness of the small-scale flowfield from nu-
lis, OR 97331, USA. quire resolution of temperature fluctua- merical simulations, even though numeri-
OCEANOGRAPHY'VoI. 10, No. 3"1997 111
cal simulations are limited by finite com- w Im s -1) 0.08
puter memory in the range of scales they 0
I
0.04
can resolve. Fully resolved simulations, ~-~ 20
0.02
termed direct numerical simulations "~ 40 0.00
(DNS), require resolution of the smallest
-0.02
scales of the flowfield. This puts an upper
~ 80 -0.04
limit on the size of scales that can be re-
-0.08
solved. Current computing capabilities 100
permit full resolution of only - 1 0 - c m
thick patches of turbulence representative T (°c)
29.0
o
of the smallest turbulent events in the
i
,-.,, 20 28.0
ocean thermocline (Moum, 1996a). An
alternative is large eddy simulation "~" 40 27.0
(LES), designed to resolve the energy- 80
containing eddies of the flow, but also to
~ 80
parameterize the smallest scales of the
100
turbulence that actually accomplish the
irreversible mixing. loe,,Je] (W ke - t )
A sample LES intended to represent 0
the response of the upper ocean in the
--. 20
western equatorial Pacific to a westerly
"--" 40
wind burst indicates the range of scales
excited (E.D. Skyllingstad, W.D. 80
Smyth, J.N. Moum and H. Wijesekera,
unpublished). Shown is a two-dimen-
sional slice at one point in time of a
0o
100
0 40 80 120 160 200 240 280 3:~0 ~ou
--0
three-dimensional, temporally varying Zonal Distance (m)
flowfield. The domain is 384 m × 384
m horizontally and 96 m vertically, with Fig. I: Depth-zonal section of vertical velocity, w, temperature, T, and log e, from a
a uniform grid spacing of 1.5 m and time large eddy simulation of the upper ocean's response to a westerly wind burst in the
step of 1.2 s. western equatorial Pacific. Initial conditions and surface forcing were derived from ob-
The subsample of simulated data in servations taken during the Coupled Ocean Atmosphere Response Experiment
Figure 1 shows vertical velocity, w, tem- (COARE), from E.D. Skyllingstad, W.D. Smyth, J.N. Moum and H. Wijesekera (unpub-
perature, T, and e. The length scales evi- lished).
dent in w range from a few meters to the
depth of the boundary layer. In the
ocean, the best in situ observation we layers (especially in the atmosphere) events simultaneously taking place at the
could make of this field would either where the energy-containing scales are two closely spaced locations. Note that the
trace a single vertical, or perhaps hori- of the order of the boundary layer thick- spacing between the two ships is not that
zontal line. Such observations are blind nCSS. much greater than the horizontal extent of
to the flowfield 1 m to either side. For Another view of the complexity of the the simulation shown in Figure 1.
operational reasons, it is not possible to small-scale flowfield comes from observa-
repeat vertical profiles to 100 m depth tions. During the Tropical Instability
until at best five min, or 250 time steps Wave Experiment in fall 1991, two ships
later. Looking closely at the section of T, maintained station within several kilome- • APtJUW ,,. A M P ~ o • " lJ
•" ~ ~ '~- ;o~,
one can see what appears to be an over- ters of each other for a period of 3.5 days
turning wave 20 m into the section at 70 (Mourn et al., 1995). Approximately 1,000
m depth. This feature is suggestive of a turbulence profiles were made and c(z)
Kelvin-Helmholtz billow, the form of in- computed for each. Time series of 5-m o 8~ o ]
stability associated with stratified shear vertical averages centered at 50 m depth, 10-10
325.5 326 326,5 327 327,5 328 32B5
J
329
flow. The billow appears at the base of well below the mixed layer at that time, year~iay 1991
the mixed layer, where the fluid is suffi- indicate that the long-term trends, domi-
ciently stratified that the energy-contain- nated by daily variations, are duplicated in Fig. 2: Time series of e at 50 m depth
ing scales of the turbulence are not well records from each data set (Fig. 2). Mean made by two different groups from verti-
resolved by the LES. A fundamental lim- profiles over the 3.5-day period agree cal profilers on two separate ships lo-
itation of LES is its inability to resolve within confidence limits, but daily aver- cated within a few km of each other.
the smaller energy-containing scales in ages do not. The lack of agreement on OSU, Oregon State University using
stratified flows. This limitation has re- daily time scales results from variations of their profiler Chameleon; APL/UW, Ap-
stricted the use of LES in strongly strati- several orders of magnitude in E and sev- plied Physics Lab~University of Washing-
fied flows, although it has been widely eral hours in duration that can be seen in ton using their profiler AMP. From
used in studies of convecting boundary Figure 2. This indicates very different Moum et al. (1995).
1 12 OCEANOGRAPHY'VoI. 10, NO. 3°1997
What Do Large-Scale Flow Modelers from microstructure data obtained in the able estimate of net flux requires averag-
Want from Us? thermocline over the past 20 years. It also ing many down- and counter-gradient in-
As indicated by the two examples dis- agrees with microstructure estimates of stantaneous values.
cussed above, insufficient resolution of K v made during the NATRE experiment In a stratified fluid, we usually find
the space and time variability of the itself, but is about 10 times smaller than turbulent patches much greater in hori-
small-scale flowfield will continue to that inferred from large-scale budgets and zontal extent than vertical. This means
haunt our interpretations of measure- numerical models. Such a small value of that a longer record is obtained from a
ments for the foreseeable future. How K, strongly suggests that most mixing in horizontal pass through a patch than from
can we make a reasonable contribution to the upper part of the thermocline does a vertical pass. We therefore expect to
the problem confronted by large-scale not happen in situ but occurs while fluid obtain more degrees of freedom and
flow modelers? Modelers require an esti- is in contact with the surface, after which smaller confidence limits in the estimates
mate of fluxes of properties across their it is stirred along sloping density surfaces made from data obtained in each patch
grid cells due to processes occuring at (Garrett, 1993). from horizontal tows. However, in the
scales smaller than their grid size. These What we learn from tracer release ex- thermocline, where turbulence occurs in-
fluxes are usually parameterized by the periments is the average rate of mixing termittently, fewer patches per unit length
product of a turbulent diffusion coeffi- between two endpoints in time over a re- of record will be sampled by horizontal
cient and a property gradient. In many in- gion many times larger than individual tows. More important, it is the vertical
stances, large-scale modelers can do mixing events. What we do not learn is divergence of the flux that we really want
nothing about the physics at scales how this mixing was achieved, nor even to know, so there is good reason to try to
smaller than their model's grid size other how it evolved between the two end- make sense of eddy-correlation measure-
than to assign a value to a turbulent dif- points. Consequently, we can only guess ments obtained from vertical profiles.
fusion coefficient. This value may be var- at the processes responsible for the mix- An example of a vertical profile of
ied from grid cell to grid cell and in time, ing. In light of atmospheric and oceanic eddy-correlation flux measurement
according to some specified parameteri- variability on decadal and longer time comes from about the same depth as the
zation. But it remains an oversimplifica- scales, we require a better understanding horizontal tow shown in Figure 3, but
tion of the real physics that will include, of the physics responsible for the mixing 1,000 nautical miles off northern Califor-
as well as the three-dimensional turbu- so that we can have better predictive ca- nia (Fig. 4, from Mourn, 1996b). Tem-
lence, Langmuir circulations near the sur- pabilities. A first attempt at this, at least perature and vertical velocity fluctuations
face, internal gravity waves propagating in the thermocline, is the internal wave are about the same magnitude as those
throughout the volume, surface waves scaling first proposed by Gregg (1989) shown in Figure 3. The length of record
breaking at the free surface, and shear in- and since revised by others. through the patch is ~10 times smaller,
stability in the interior of the fluid, to however, so confidence limits are consid-
name a few of the physical processes we Eddy-Correlation Measurements erably larger.
recognize. In the past few years, several groups Typical signal levels obtained from
It seems likely that current choices of have found ways to measure eddy-corre- aircraft flying through the atmospheric
subgrid scale parameterization range lation fluxes from horizontal tows (Ya- boundary layer over the ocean are on the
from bad to worse. A reasonable goal for mazaki and Osborn, 1993: Fleury and order of 1°K temperature fluctuation and
the investigators of small-scale ocean Lueck, 1994; Gargett and Mourn, 1995). 1 ms' vertical velocity fluctuation
physics is to gain an understanding that is These measurements are akin to those ob- (Friehe, 1986). The comparatively
at least sufficient to avoid the worst tained by flying through the atmospheric smaller signals in the ocean thermocline
choices. boundary layer with a rack of probes highlight one of the difficulties in making
mounted on the nose of an aircraft. Fig- these measurements.
Examples of Flux Measurements ure 3 shows an example of such a record
Examples of flux measurements using derived from a vehicle towed behind a
very different methods give us quite dif- ship at speeds of 0.7-1.5 m s ' at 4 0 0 m
ferent insights. depth east of Barbados (Fleury and
00~
Lueck, 1994). Temperature fluctuations OO8
Tracer Releases are on the order of 0.01°K, and vertical 016 ~.. I , Cgl
0 5 In 15 20 25 3C 3S 40 45 SO 55 60 55 so 75
A direct measure of the vertical (really velocity fluctuations are several m m s '.
diapycnal) flux is obtained by observing Their product is the instantaneous eddy-
the spreading of a purposely introduced correlation heat flux, w ' 0 ' . Along this
artificial tracer. This has been done in tow, instantaneous values of w ' 0 ' may
several locations by Jim Ledwell and have either sign, positive values repre-
0 5 10 15 2O 25 30 35 4O 45 5O 55 an ,~b ~c 75
coworkers and has provided a very im- senting restratification of displaced fluid
portant result in the North Atlantic Tracer parcels, and negative values representing
Release Experiment (NATRE; Ledwell et down-gradient transport, that is, either Fig. 3: Nearly horicontal space series o[
al., 1993). The rate of vertical spreading downward transport of warm fluid or up- fluctuations of temperature, ~'. vertical
can be interpreted in terms of an eddy ward transport of cool fluid. Down-gradi- velocity, w ', attd their product w 'H'. The
diffusion coefficient, Kv. The estimate of ent transport increases system potential series qf R' attd w ' have been highpass
K, from NATRE o f - 1 0 ~ m e s ' at 300 m energy by moving mass upward. This filtered at 0.5 cpm. From Fleurv and
depth appears to confirm estimates of K v must be the end result of mixing. A reli- Lueck (1994).
OCEANOGRAPHY'Vo[.10, No. 3"1997 II 3
e~
o['c 1
e ~ 6s6
w (m~s]
-0006 O000 0006 range of scales excited by turbulence in rectly proportional to the computational
tidal channels ranges from the order of m e m o r y requirement for simulations.
ae~ 100 m to diffusive scales of sub-millime- Present and 1987 capabilities are noted at
ter s i z e - - a range of over 5 orders of the bottom right of Figure 5, and the
354
magnitude. The large range in scales shaded region denotes present capabilities
g makes such a flowfield impossible to rep- in e - N space. Although flowfields can-
resent with current computing power. not be represented, and only a single tur-
as~
However, in weaker, more stratified bulent patch can be simulated, it is now
4; flows, such as the weakest cases typical possible to observe the complete life
a~
of the deep ocean, where L o - 0.1 m and cycle, growth through decay, of turbu-
diffusive scales for T are approximately lence representative of the weakest cases
a~
-ool oo[¢J~l OOl .o.ool oooo o o m 0.001 m, it becomes feasible to represent observed in the abyss and thermocline.
~o" ,,,;o ['c nu4
turbulent patches with Pr = 7. Although One way these simulations are being used
typically turbulence simulations are run is to test the representativeness of our
Fig. 4: Vertical profiles o f O (instanta-
with Pr = 1, which is correct for air, there limited ability to sample the smallest
neous, solid; reordered, dashed), temper-
is reason to believe that physical misin- scales within the context of a fully re-
ature fluctuation, R' (the difference be- terpretations can result if the correct Pr is solved three-dimensional temporally
tween the two d curves), w ; and w ' ~ .
not used for seawater (W. Smyth, per- varying turbulence.
From Mourn (1996b). At the present rate of increase of com-
sonal communication). At present, it is
not possible to consider the small salinity puting power, extrapolated from the two
Prospects for Understanding scales, because Sc is so large. data points in Figure 5, it is clear that we
Small-Scale Flowfields The ratio of energy-containing length should not expect to see representative
One way to summarize what types of scale to diffusive length scale is simulations of all the scales of turbulent
small-scale flowfields have been sampled (dvNe)l'C The cube of this quantity is di- patches even in the thermocline, within
in the ocean and what range of scales is our careers. Probably the way we are
involved in the turbulence is a plot of log going to learn more about the small-scale
e versus log N (Fig. 5). The scales of the
. . . . = . . . .
flowfields is to obtain better and more
turbulence in a stratified fluid range from Strmt of Gibraltar
hydrauhc ~ump comprehensive measurements, especially
lO-4 tidal channel (Wesson&Gregg '94}
the energy-containing scale, set by buoy- LGargett&Moum 'ctSi d,ffuslv,, ~calos those including the role of the initial in-
ancy forces in stratified turbulence as Lo ~,~ ~. -- velocity 0 6 mm
lemperature 0 2 mm
stability in creating the turbulence.
~ sahmlV 0 02 m~r
= (tiN b*'-" (dashed lines rising to the right
lO 4 ~: L
in Fig. 5) to the diffusive scale (vD'-/e) ''~ % wtnd mixed layers
(Lombardo&Gregg '89)
[
i Acknowledgements
convectlvely upper equatorlaJ ~ 1997
(horizontal dashed lines in Fig. 5). Here, mlxedtayers
[Shay&Gregg '86, shelf bottom
thermOclme
IMoum etal '89~
g}
~/1~ h, 18
Doug Caldwell, Ann Gargett, Rolf
Anls&Moum'941 boundary laver t
N is the b u o y a n c y frequency, v is the (Dewey oral'88) ' ~ 1987 Lueck, Jonathan Nash and Bill Smyth all
polar pycn~hne [ ~/l~ hi~) 3
kinematic viscosity and D is replaced by o 10 4 over topography
(W~jesekera etal'93)
mare thermochne
contributed thoughtful comments on early
either v for velocity, or by the molecular (Gregg '89, Moum '96) [
drafts of this paper. Thanks to Roll Lueck
near abyssal slopes letupetalure 2 mm
diffusivity for temperature, D T, or salin- (Tool~ oral '94) ~. salinity 02 mm
for providing a copy of Figure 3 and to
ity, D s. This means that scalars such as 10 - m abyssal ocean
DNS
Ann Gargett for a well-organized 1997
(Toole etal '94)
temperature and salinity vary on length TOS session in Seattle.
scales smaller than the smallest length 10 -4 10 ~3 10 -2 10 -1
scales of velocity variations by a factor log N[s-1] References
L,~qo~,,:/L,~, = v ~ . In seawater, the Anis, A. and J.N. Mourn, 1994: Prescriptions fl)r
Prandtl number, Pr = v/D r = 7 and Fig. 5: Log-log plot of turbulent kinetic heat flux and entrainment rate in the upper
ocean during convection. J. Phys. Oceanogr.,
Schmidt number Sc = Ds/D t = 100. The energy dissipation rate, ~, t,ersus buoy-
24. 2142-2155.
smallest scales in temperature are ancy frequency, N. The range covers Dewey, R.K., P.H. Leblond and W.R. Crawford,
times smaller than those in velocity, most o f the o b s e r v e d cases front the 1988: The turbulent bottom boundary layer
whereas the smallest scales in salinity are ocean, representative examples o f which and its influence on local dynamics over the
another ~ g c times smaller still. The ex- are noted. The energy-containing scale, continental shelf. Dvn. Ann. Oceans, 12.
143 172.
amples shown in Figure 5 are intended to L, = (e/N') "e, is represented by dashed
Fleury, M. and R.G. Lueck, 1994: Direct heat flux
be representative rather than inclusive. In lines sloping up to the right and is plot- estimates using a towed vehicle. J. Phys.
some of the observational examples, the ted at decade intervals. The small, disffi~- Oceanogr., 24, 80 I-818.
energy-containing eddies may be affected sire scales, (vD'-/e)~C are represented by Friehe. C.A., 1986: Fine-scale measurements of ve-
by proximity to boundaries as well as by horizontal d a s h e d lines. The dif/usive locity, temperature and humidity in the atmo-
spheric boundary layer. In: Probing the At-
stratification. scales f o r velocity, temperature, and mospheric Bounds O" Layer, D.H. Lenschow.
We can see the range of turbulence salinio'for • = 10" nz: s ~'and e = 10 ~'nz: ed. Amer. Met. See., Boston, 29-38.
flowfields that have been observed, from s-' are noted at right. At any point on Gargett, A.E. and J.N. Mourn, 1995: Mixing effi-
the very energetic special cases at the top this curve, one can infer the range o f ciencies in turbulent tidal fronts: results
of Figure 5 to the more general situations scales in a stratified turbulent flow. The from direct and indirect measurements of
density flux. J. Phys. Oceanogr., 25, 2583
typical of the upper ocean, thermocline shaded region represents current capa- 2608.
and abyssal ocean. Unstratified, convec- bilities f o r computing fully resoh,ed, Pr Garrett, C.J.R., 1993: A stirring tale of mixing. Na-
tively mixed layers are off to the left. The = 7 turbulence. ture, 364, 670-671.
114 OCEANOGRAPHY'Vo1. 10, No. 3°1997
Garrett, C.J.R., 1994: Dispersion and mixing in the Mourn. J.N., 1996a: Energy-containing scales of Osborn, T.R. and C.S. Cox, 1972: Oceanic fine
ocean. In: Oceatt Processes in Climate Dy- turbulence in the ocean thermocline. J. Geo- structure. Ge¢q~h3's. Fluid Dyn,. 3, 321-
namics: Global and Mediterranean Exam- phys. Res., 101. 14,095-14,109. 345.
pies, P. Malanotte-Rizzoli and A.R. Robin- Moum, J.N., 1996b: Efficiency of mixing in the Shay, T.J. and M.C. Gregg, 1986: Convectivel)-
son, eds. Kluwer, Amsterdam, 61-77. main thermocline..k Geophys. Res., 101, driven turbulent mixing in the upper ocean.
Gregg, M.C., 1989: Scaling turbulent dissipation in 12,057- l 2,069. J. Phys. Oceanogr., 16. 1777-1798.
the thermocline. J. Geophys. Res., 94, 9686 Mourn, J.N., D.R. Caldwell and C.A. Paulson. Toole, J.M., K.L. Polzin and R.W. Schmitt, 1994:
9698. 1989: Mixing in the equatorial surface layer Estimates of diapycnal mixing in the abyssal
Haugen, D.A., J.C. Kaimal and E.F. Bradley, 197l: and thermocline. J. Geophys. Res., 94, ocean. &'ience. 264, I 120-1123.
An experimental study of Reynolds stress 2005-2021. Wesson, J.C. and M.C. Gregg, 1994: Mixing in Ca-
and heat flux in the atmospheric boundary Mourn, J.N., M.C. Gregg, R.C. Lien and M.E. Cam marinal Sill in the Strait of Gibraltar. J.
layer. Q. J. Roy. Met. Soc., 9Z 168 180. 1995: Comparison of turbulence kinetic en- Geophys. Res., 99, 9847-9878.
Ledwell, J.R., A.J. Watson and C.S. Law, 1993: Ev- ergy dissipation rate estimates from two Wijesekera. H., L. Padman, T. Dillon, M. Levine, C.
idence for slow mixing across the pycno- ocean microstructure profilers. J. Atmos. Paulson and R. Pinkel, 1993: The application
cline from an open ocean tracer release ex- Oceanic Technol.. 12, 345-366. of internal wave dissipation models to u re-
periment. Nature. 364, 701-703. Munk, W.H., 1966: Abyssal recipes. Deep-Sea Res.. gion of strong mixing. J. Phys. OceanoA,r.,
Lombardo, C.P. and M.C. Gregg, 1989: Similarity 13. 707-730. 23. 269-286.
scaling of viscous and thermal dissipation in Osborn, T.R., 1980: Estimates of the local rate of Yamazaki, H. and T.R. Osborn, 1993: Direct esti-
a convecting surface boundary layer. J. Geo- vertical diffusion from dissipation measure- mation of heat flux in the seasonal thermo-
phys. Res., 94, 6273-6284. ments. J. Phys. Oceanogr., 10, 83-89. cline. ,1. Phys. Oceanogr., 23. 503-516, ~]
OCE4NOGRAPHY*VoI. 10. No. 3*1997 1 15
