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The vertical length scale of double-diffusive intrusions
Deep-Sea Research, Vol. 26A, pp. 903 to 913 t~ Pergamon Press Ltd 1979. Printed in Great Britain
0011-7471/79/0801-0903 $02.00/0
The vertical length scale of double-diffusive intrusions B. R. RUDDICK* a n d J. S. TURNER* (Received 23 October 1978: in revised /brm 19 March 1979," accepted 25 March 1979)
Abstract -A laboratory model has been developed to simulate the mixing at an oceanic front across which there are large T S anomalies but a small net horizontal density difference. Identical stable density distributions are established on the two sides of a central vertical barrier, using the analogue system of salt solution at one end of a tank and sugar solution at the other. When the barrier is removed, local density anomalies are produced by double-diffusive transports across the frontal region. These lead to the spontaneous formation of interleaving layers over the whole depth of the tank, with a scale directly proportional to the horizontal concentration differences across the front and inversely proportional to the vertical density gradient. This result is explained using a theoretical argument based on a comparison of the potential energy in the initial and final distributions and the mechanism of energy release by double-diffusive transports across the interfaces between alternate layers. The available data on the scale of intrusive layers near oceanic fronts are also broadly consistent with this energy argument.
1. I N T R O D U C T I O N
THERE ARE n o w m a n y o b s e r v a t i o n s in the ocean that d e m o n s t r a t e the i m p o r t a n c e of interleaving processes in d e t e r m i n i n g the rate of h o r i z o n t a l mixing between water masses. P a r t i c u l a r l y when there is a large c o n t r a s t in T - S p r o p e r t i e s in the h o r i z o n t a l b u t a small net density difference between the w a t e r masses, interleaving l a m i n a e a n d a s s o c i a t e d local t e m p e r a t u r e inversions are very c o m m o n . FEDOROV (1978) surveyed this field in some detail a n d p r e s e n t e d m u c h evidence to show that large intrusions (such as the M e d i t e r r a n e a n outflow into the Atlantic) b r e a k up into a n u m b e r of thinner layers t h a t then interleave with the s u r r o u n d i n g water. These effects are m o s t p r o m i n e n t near s h a r p fronts, which suggests that the interleaving is in some w a y driven by the h o r i z o n t a l g r a d i e n t s of p r o p e r t i e s at a rate related to the m a g n i t u d e of the gradients. The fact that interleaving layers are distinguished by b o t h t e m p e r a t u r e a n d salinity a n o m a l i e s (which m u s t be nearly c o m p e n s a t i n g to preserve static stability) i m m e d i a t e l y suggests that double-diffusive processes, i.e. those that d e p e n d essentially on the different m o l e c u l a r diffusivities of heat a n d salt, c o u l d be i m p o r t a n t for their f o r m a t i o n a n d p r o p a g a t i o n . M o s t of the detailed l a b o r a t o r y e x p e r i m e n t s on d o u b l e diffusion have c o n c e n t r a t e d on vertical t r a n s p o r t processes across interfaces between layers with different T S properties. TURNER a n d CHEN (1974) p o i n t e d out, however, t h a t these m a y n o t always be directly a p p l i c a b l e to the ocean, a n d they carried o u t a n u m b e r of e x p l o r a t o r y experiments that i n c l u d e d t w o - d i m e n s i o n a l effects. M o r e recently, TURNER (1978) explicitly studied the i n t r u s i o n of one fluid into a g r a d i e n t o f a n o t h e r a n d s u m m a r i z e d the features of ocean o b s e r v a t i o n s that can be e x p l a i n e d in terms of this a n d related processes. JOYCE, ZENK a n d TOOLE (1978) have now presented direct evidence (from m e a s u r e m e n t s
* Research School of Earth Sciences, Australian National University, P.O. Box 4, Canberra, Australia. 903
904
13. R. RUDDICK and J. S. TURNER
across the Antarctic Polar Front) of the importance of double-diffusion in driving the horizontal mixing of heat and salt across the front. The dynamics of this intrusive process are still far from understood, however, and there has as yet been no satisfactory quantitative explanation of even the most obvious property, namely, the vertical length scale of the individual layers. Various suggestions have been made about the mechanism that might determine this scale (see TURNER, 1978). It could be due to distortions by internal waves (the vertical wavelength of which sets the scale); to a process analogous to the 'side wall heating' mechanism described by THORPE, HUTT and SouLsav (1969) and CHEN, BRIGGSand WIRTZ (1971); or to some inherent instability that depends only on the presence of nearly compensating vertical property gradients, not on the specific mechanism that produced them (LINDEN, 1976). None of these explanations has yet been properly tested because of various inadequacies in all the two-dimensional laboratory work to date. The new experiments described below, however, have achieved a much greater degree of control over the important parameters and have produced clear-cut results that can be compared with a simple theory related specifically to the experimental geometry. Briefly, a vertical front has been set up by withdrawing a barrier separating regions containing sugar and salt solutions having the same vertical density distribution. After the initial wave disturbances and small scale turbulence have died away, a series of interleaving layers develops and gradually extends horizontally. In the present paper we will describe these observations qualitatively and then concentrate on the vertical length scale and its theoretical explanation. 2. LABORATORYEXPERIMENTS ON FRONTS It will be helpful first to sketch the steps that have led to the present experiment in order to emphasize the advantages of the technique adopted. Consider what happens when a source of fluid (say sugar solution) is injected into a tank containing a homogeneous solution of a second property (say salt), which has a different molecular diffusivity but the same density. TURNER and CHEN (1974) have shown that vigorous vertical convection ensues, with rapid mixing between the injected fluid and its surroundings. When a source of one fluid is released at its neutral buoyancy level into a gradient of the second, similar vertical convection is observed before the fluid spreads out at several levels (TURNER, 1978). This behaviour implies, however, that the properties of the fluid as it begins to intrude are not well known, because a rather uncontrolled mixing process has preceded the interleaving, which is our major concern. There have also been experiments that (to some extent at least) model more directly the effect of a discontinuity of properties across a vertical or inclined frontal surface. TURNER and CHEN (1974) and LINDEN and WEBER (1977) have shown how the introduction of an inclined boundary can produce a series of extending 'noses' in a tank containing opposing vertical gradients of two properties, which was previously quite stable. The disadvantage of this method is clearly the solid boundary, which inhibits horizontal motions. Horizontal differences have also been produced just by adding an excess of one component to the end of a previously stratified tank, using a nozzle moved up and down to produce rapid mixing over the whole depth. This can also be done with a central barrier in place, which is later removed to allow the different mixtures to communicate. It is difficult to avoid a substantial density mismatch and this inevitably produces intruding motions related (at least initially) to the overall density structure and not to the slower double-diffusive processes.
(a)
(b)
(c) Fig. 2(a)-(c).
[facing p. 904]
(d)
(e)
if) Fig. 2(d)-(f).
Fig. 3. (a) Another experiment in which the initial distortion of the front was small, showing the smooth increase in layer scale with increasing depth. (b) An experiment in which fibs across the front was constant in depth (see Fig. 4). The layer scale shows no systematic variation with depth.
Fig, 2, (Previo~ two pa~e:i.) Shadowgraph record O~alypical experiment, with a bottom density of 1.0" g c m -3. The scale is in centimetres, measured f r o ~ the bottom. (a) (t = 0 m 0 s) the barrier is wi .~drawn. (b) (t ~ m 55 s) the distorted fron t after the density anomalies have been adjusted by i r ~ r n a l wave motions. (c) (t = 8 m 28 s) interleaving mO~ta~ have formed, with alternating diffusive interfaces and fingering regions. (d) (t = 12 m 36 s) thelayers sprea~ horizontally and become more organized in appearance. (e), (f) (t = 25 m 25 s and 25 m 45 ~; th~-ell-developed interleaving layers. The motion of the dye streaks shows flow to the right just below the diffusive interfaces, and flow to the left just above them. The velocity of this flow is several times the advance speed of the layers. Note the increase in layer thickness with increasing del: and BAS.
The vertical length scale of double-diffusiveintrusions
905
In the present experiments, great care was taken to minimize these disturbances by producing density distributions on the two sides of the barrier as nearly identical as possible. With the 'dam' in place, the two ends of the tank are filled simultaneously using duplicate 'double bucket' devices to produce the same linear gradient of salt solution on one side and sugar on the other (with, of course, equal densities at the top and bottom). A schematic diagram of the apparatus is shown in Fig. 1. A small hole in the top of the dam just below the final depth allows an adjustment of water level if one side has been slightly overfilled. Before starting the experiment, the dam is lifted about 3 mm to allow the pressure at the bottom to be equalized. When the experiment proper is started, by raising the dam smoothly (at a typical rate of 2 mm s- 1), the behaviour is as shown in the sequence of photographs of Fig. 2. As the barrier is raised, small-scale mixing is generated locally and there are also larger-scale wave-like motions, generated by the remaining internal density anomalies [Fig. 2(a)]. These motions rapidly die away and we are left with a slightly distorted but essentially vertical front between fluids of different diffusivity [Fig. 2(b)]. It is important to emphasize from the start that the subsequent motion is insensitive to either the initial distortion or the small-scale mixing; different devices for mixing the interfacial region were added to the trailing edge of the dam, but the final state depended only on the property differences across this region and not on its width or detailed structure. As time goes on, the interleaving layers shown in Fig. 2(c)-(f) spontaneously form and extend. The interfaces above and below them are alternately 'finger' and 'diffusive' in character, and the motion is driven by local density anomalies produced by the release of potential energy due to the double-diffusive transports between layers. The extending layers have a small tilt to the horizontal, which can be attributed to the fact that the buoyancy flux through the finger interfaces is greater than that through the diffusive interfaces. Qualitatively, the motion in these layers can be explained in the same terms as the multiple
~4
MIXER t
. . . . .
REMOV~E
K -,ER
IL
._
_
//
T
S
t~
1,8m
0.3m
~'~
Fig. 1. Sketchof the experimentalapparatus, showingthe tank, the removablebarrier which divides the tank into two equal halves, and the two identical 'double bucket' filling systems, one for each half of the tank.
906
B . R . RUDDICK and J. S. TURNER
intrusions described by TURNER(1978), but there is the important difference that the initial conditions are now carefully controlled and measured. Both the size of the layers and their rate of horizontal extension increases systematically with depth, i.e. with the concentration anomaly across the front. Figure 3(a) shows another experiment, similar to the one just described but in which the density gradients on either side of the barrier were particularly well-matched. The initial distortions of the front were small, allowing the 'natural' scale to develop more readily. Although the systematic increase in the layer scale with depth is more regular and clearly visible, the scale of the developed intrusions is nearly the same in the two experiments just described. The initial distortion of the front in Fig. 2(b) has to a large extent been forgotten by the later stages [Fig. 2(e), (f)]. Another type of experiment, in which flAS across the front and dp/dz were independent of depth, is illustrated in Fig. 3(b). (The initial distributions of S and T corresponding to this case are sketched in Fig. 4.) The layer scale and rate of extension do not vary systematically with depth but now tend to be constant. The only results that will be considered quantitatively here are those for the vertical length scale of the layer. A series of experiments with different linear density gradients and top and bottom concentrations is summarized in Fig. 5. The observed layer scale H within each experimental run is, with some experimental scatter, directly proportional to the horizontal concentration difference across the front, and the various runs are brought together if H is inversely proportional to the vertical density gradient. (Note that dp/dz varies by a factor of 20 in these experiments.) This implies that, for experiments where the surface water is fresh, the layer scale is directly proportional to the depth below the free surface and is independent of the actual density at the bottom. More generally, if there is only one component on each side, with a higher concentration at the surface, the layer scale is proportional to the distance below the level at which the extrapolated linear concentration profile would fall to zero. The constant of proportionality and its relation to the mechanism of layer formation and extension will be considered in the following section.
~'S
,,T
~s
aT
f ~A S
BARRIER
,BAS
Fig. 4. Sketch of the initial profiles on either side of the barrier for the experiment in Fig. 3(b), in which [3AS was constant with depth. A basic density stratification (on both sides) is altered by the addition of sugar to both buckets of the left-hand filling system and a corresponding amount of salt to both buckets of the right-hand filling system. The extra sugar and salt increase the mean density on either side without altering the density gradient.
907
The vertical length scale of double-diffusive intrusions
•
~\'~ ,
B
/
N4
B//
A
I
~'r MIXED LAYERS
kx'~
I
I
20 d= /~AS /de P dz
GRADIENT
.o"
/
I
I0
Fig. 5.
,~j~
°/
d
LINEAR
,
30
(cm.)
Plot of the observed layer thickness, H versus
= flAS
/;o
dz
for the experiments to date. The data points A J are from the experiments described in Section 3, with dp/dz constant but flAS increasing in depth. Each data point represents one layer. The solid points (solid circle, square, triangle) are from experiments with flAS and dp/dz constant in depth. The layer thickness was also constant in depth, and the points represent an average over several layers. The experimental parameters are summarized below. The solid lines are described in the text. Data point 1 dp
A
-- - - x 103 (cm 1) P0 dz
flAS
B
0.33 1.1
C
D
E
F
G
H
I
J
1.5
1.9
2.0
3.0
3.5
3.7
5.6
7.5
Increases with depth
•
•
•
1.3
1.2
0.67
0.025 0.003 0.011
3. AN ENERGY A R G U M E N T FOR THE LAYER D E P T H
The argument developed below is based on a comparison of the potential energy in the initial distribution of properties with that corresponding to the final distribution. The latter is produced by the coupled transports through the quasi-horizontal interfaces between alternate layers, which are set up by incipient intrusive motions of an assumed form. By equating these energies, one arrives at a kind of finite amplitude stability criterion, which in fact includes a prediction of layer depth in terms of the given parameters of the system• In spirit, the theory is similar to the energy arguments used by TURNER (1968) to predict the mixed-layer depth produced when a stable salinity gradient is heated from below• Explicitly, suppose (as in our experiment) that there are identical linear gradients of two properties on each side of a barrier [Fig. 6(a)]. Denote by T the property with higher molecular diffusivity and by S that with the lower diffusivity, and let p = po(1 +flAS) = po(1 +~AT) be the density at the level z o on which we fix our attention,
908
B.R. RUDDICK and J. S. TURNER
a) Profiles before intrusion (Left)
(Right)
b) Profiles in centre immediately after intrusion I
.
,
'
.'I
|':.".:..
/9S(z)
cIT(z)
-~ O A S ~
-NaZ~T le
p(z)
oriltnal
gradient
c) Profiles after fingers have "run down" the intrusion BS(z)
GtT(z)
p(z)
oAT
-I
7, r~AT(l-n)
Fig. 6. Schematic diagram of the intrusion process. (al Profiles offlS(z) (left of barrier) and aT(z) (to the right of barrier) before intrusions occur. (b) Hypothetical profiles offlS(z), aT(z), and p(z) after an intrusion has caused a layer'of S to flow over a layer of T. (c) Hypothetical profiles of S, T, and p after the finger fluxes have 'run down' the S contrast between the upper and lower halves of the layer.
at a distance d below that at which the concentration difference is zero. The density gradient is thus 1/po(dp/dz ) = 7 = ~AS/d. When the barrier is removed, suppose that the motions that develop are such that locally a layer o f S, originally of depth H, intrudes on top of T to produce a layer of half that depth, as shown in Fig. 6(b). (The a r g u m e n t can be carried through using alternative assumptions a b o u t the intrusion process, which will be referred to below.) The situation shown is clearly unstable, but double-diffusive transports driven by the potential energy in the S distribution will rapidly cause it to run down towards a state where the S is uniformly distributed and the T (and therefore the net density) are hydrostatically stable. The 'final state' is shown in Fig. 6(c); it is implied here that the transport across the finger interface formed by the initial interleaving is so m u c h larger than that across the diffusive interfaces above and below that the latter can be ignored. It has also been assumed that over the whole period of rundown, the flux ratio of heat to salt is (in density terms)
BFs
= n = constant < 1.
(1)
The total changes in S and T and in net density shown on Fig. 6(c) have been plotted using (1). Using this notation, the initial potential energy, relative to the level z o, of the region
The vertical length scale of double-diffusiveintrusions
909
z = +_H/2 in which the density is p = po(1 + T z ) is Eo = - g P o
f
ill2
(1 +yz)z dz
, ) - HI2
(2)
= -gpoTH3/12.
The final potential energy of the stepped density distribution shown in Fig. 6(c) is E = t~gPo flAS(n + 1 )H 2 - l~gpoflAS(3 - n)H z = ~gpoflAS(n - 1 )H 2.
(3)
The potential energy remains unchanged or has decreased, so that it is possible to achieve the final state using only energy available internally if E o >~ E~, or - ~27H 3 >~ - ~(1 - n ) f l A S H 2 n ~< -~(1- n ) f l A S / 7 .
(4)
This has the same form as the laboratory experimental result, with a numerical constant that depends on the flux ratio n. A lower limit to the layer depth can be obtained as follows : The density in the upper half layer of the 'final' stepped density distribution shown in Fig. 6(c) is Pl
~--p0[1 + flAS(1 + n)/2].
(5)
The density of the lower half of the (assumed similar) pair above this is P2 = Po[ 1 + f l A S ( 3 - n ) / 2 - T H ] .
(6)
The condition that the interface between the two be statically stable leads to H > (flAS/7)(1 - n ) .
(7)
Thus H is bounded above and below: ( f l A S / 7 ) ( 1 - n ) < H < {(flAS/?)(1 - n ) .
(8)
Many variations on this argument are possible, and they lead to forms similar to (8) but with different (generally larger) numerical constants. Two examples will be described here. Suppose that the final density distribution is not stepped, as shown in Fig. 6(c), but remains linear (while redistributing the S component within the layer). This might be the case in the laboratory experiments, where fingers fill the entire region between diffusive interfaces. Then it is found that the two limits in (8) coalesce, giving H = 2(1 - n)flAS/7.
(9)
Thus allowing the fingers to fill the entire region only changes the numerical factor from 3/2 to 2. It can be argued that the intermediate state drawn in Fig. 6(b) is unrealistic in that it implies a temporary increase in potential energy. If instead we suppose that the gradient of T is just replaced over a depth H/2 by the S gradient (with no change in density gradient), then the run-down state is slightly different and (1 - n)flAS
(¼-- ¼n)7
< H <
(1 - n)flAS
( ~ - - ¼n)), '
(10)
910
B . R . RUDDICK and J. S. TURNER
If the basic stratification is double-diffusive, with some available potential energy in the vertical stratification, the predicted layer thickness is only weakly increased. This is because the S and T contrasts due to horizontal advection tend to be stronger than the contrasts already present in the stratification.
4. C O M P A R I S O N WITH O B S E R V A T I O N S
(a) Laboratory experiments Although the estimates (8-10) all give the same form for the predicted layer thickness, there is a considerable variation in the numerical factors according to the particular assumptions one makes in the arguments of Section 3. In what follows we use the experimental results to estimate the 'best' value of the numerical factor. GRIFFITHSand RUDDICK (1979) recently measured, more accurately than was previously possible, the ratio of buoyancy fluxes for sugar/salt fingers. They found that as the density anomaly ratio Rp = ctAT/flAS increases during a rundown experiment, n decreases rapidly and approaches a nearly constant value ofn = 0.88+0.01 for Ra > 1.3. We take this value for n rather than the earlier value of 0.92 measured by LAMBERTand DEMENKOW(1972). Note that the quantity 1 - n, which appears in equations (8)-(11 ), differs by 50% according to which estimate of n is used. In Fig. 5 are plotted the observed layer thicknesses, H, vs
d = flAS/ ,'
1 dp Po dz
for the experiments to date. The cross-frontal salinity difference for each layer, AS, was proportional to the depth below the free surface in experiments A to J. Each data point represents a single layer. The solid points (solid square, circle, triangle) are from experiments in which AS and dp/dz were constant with depth, as in Fig. 4. Each such point represents an average over several layers. Also plotted in Fig. 5 are the straight lines corresponding to equations (8) and (9) using n = 0.88. With the exception of experiment B, the data are contained by the three lines. (In experiment B, exceptionally strong distortions of the initial front pre-set the scale of interleaving. Although the scale decreased slowly with time, the initial structure persisted for the duration of the experiment. In the other experiments, the distortions were weak enough for their effects to diminish with time.) We take the central line, H = 3(l-n)d,
(11)
as the 'best fit' to the experimental data. TURNER (1978) described experiments in which an intrusion produced by a source of S in a gradient of T (or vice versa) broke up into a number of sloping layers, with a structure similar to the layers formed at a front. The observed layer scale in those experiments was considerably smaller than that predicted by equation (9). We can now attribute this to the intense mixing at the source, which diluted the incoming fluid with ambient fluid, causing the effective contrast flAS to be much smaller. However, the sense of the changes in layer scale with density gradient and source input rate described by TURNER is consistent with equation (9). THORPE et al. (1969) and CUEN et al. (1971) described experiments in which a salinity
The vertical length scale of double-diffusive intrusions
911
gradient was heated from the side, producing a series of layers with alternating diffusive and finger interfaces. Their analysis predicted a layer scale of ~ A T ' / 1 a_p, while experimentally /P
O2
the constant of proportionality was found to be 0.81 + 0.10. For heat/salt fingers,* equation (11) predicts a scale of 0.66eAT
Although the mechanism of layer formation is
apparently very different in the 'heated side wall' experiments and ours, it is worth remarking on the identical form and the close numerical agreement of the two results.
(b) Oceanic observations In his study of observations of oceanic finestructure, FEDOROV (1978, Fig. 6) described a front in the surface layer of the tropical Atlantic Ocean that exhibits several of the features found in the laboratory fronts. These properties, summarized by TURNER (1978), are directly attributable to the double-diffusive nature of the cross-frontal interleaving. They are (1) the presence of two or more temperature inversions, indicative of diffusive interfaces on either side of a salt finger interface ; (2) the slope of these interfaces upward away from the warm salty water mass ; and (3) changes of temperature and salinity along the layer in a manner consistent with the changes produced by the salt finger fluxes. Other fronts in which double diffusion may be the driving mechanism have been reported by JOVCE et al. (1978), POSMENTIERand HOUGHTON(1978), HORNE (1978), VOORHIS, WEaa and MILLARD (1976), and NACATA (1970). In Table 1 we compare the vertical scale predicted by equation (11 ) with the scale of interleaving actually observed. The computed and measured scales agree within a factor of two for all cases except that of Horne. [Horne's calculations for the front off N o v a Scotia indicate that the diffusive fluxes, neglected in the calculation leading to equation (11), dominate over the finger fluxes. Thus our simple estimate may not apply there.]
Table 1. A comparison between the vertical length scale of interleaving layers computed JJ'om equation (11) and the observed scale
Layer depth (m) Reference
FEDOROV(19781 JOVCEet al. (1978) POSMENTIER and
HOUGHTON(1978) HORNE(1978) VOORHISet al. (1976) NAGATA(1970)
* n = 0.56 (TURNER,1967).
AT
~
(C)
(°C)- 1 x 104
dp -p dz (m- 1t
2.9 0.95 1.7 1.7 1.55 2.0 1.43
1.6× 10-5 4.9 × 10-7 1.6×10 5 2.8×10 5 9.1 × 1 0 - 6 1.6×10 4 6.5 × 10 6
0.4 1.25 8 8 4 7 5
1
Computed
Observed
0.66d
H
4.8 160 56 32 45 5.8 73
10 100 28 23 10 11 120
912
B.R. RUDDICKand J. S. TURNER
Overall the agreement is good, considering that the laboratory data exhibit a scatter of a factor of two and were carried out using the sugar/salt system. Dimensional arguments suggest that
d = flAS/ is a natural vertical scale for intrusions (regardless of the driving mechanism), but our comparison with the observations seems to imply more than that. The multiplying term (1 - n ) in equation (11) changes by a factor of 4 from the laboratory value of 0.12 for sugar/salt to the 'oceanic' value of 0.44 for heat/salt. The good numerical agreement thus supports the mechanism we have proposed, whereby double-diffusive fluxes release the energy used for frontal interleaving.
5. CONCLUSIONS From the laboratory experiments modelling horizontal interleaving at a front, combined with a simple energy argument based on the double-diffusive driving mechanism, one can make the following estimate of the vertical length scale of cross-frontal intrusions :
H = ~(1-n)/~AS
(11)
1 dp p dz where n = density flux ratio, 0.56 for heat/salt fingers, 0.88 for sugar/salt fingers AS = the salinity contrast, measured across the front along an isopycnal (note that flAS = sAT)
1 dp - the vertical density gradient, and p dz 1 @ p ~S This estimate is probably accurate to about a factor of two. When extrapolated from laboratory scales (a few centimetres) to several examples of oceanic fronts (where the scale is tens or hundreds of metres), the observed interleaving scale matches that predicted by equation (i 1) to the expected accuracy. This agreement suggests that double-diffusive processes play a major role in mixing at oceanic fronts and that the layer scales are determined by the mechanism we have described. It has become apparent that observations near oceanic fronts should be made in enough detail to resolve the interleaving and that the interpretation of such data must take into account the double-diffusive nature of the process before one can properly estimate the various fluxes. In particular, the unrealistic concept of eddy diffusivities should be avoided. More laboratory experiments are in progress, with the aim of obtaining a better understanding of the horizontal fluxes of heat and salt associated with such interleaving systems of layers.
Acknowledgements--We are grateful to DEREKCORRIGANand Ross WYLDE-BROWNE for their assistance in building the apparatus and carrying out the experiments.
The vertical length scale of double-diffusive intrusions
913
REFERENCES CHEN C. F., D. G. BRIC,~Sand R. A. WmTZ (1971) Stability of thermal convection in a salinity gradient due to lateral heating. International Journal of Heat and Mass Transfer, 14, 57-65. FEDOROVK. N. (1978) The thermohalinefinestructure of the ocean. English translation, Pergamon Press, Oxford, 169 pp. GRIFFITHS R. and B. R. RUDDICK(1979) Accurate fluxes across a salt-sugar finger interface deduced from direct density measurements. Submitted to Journal of Fluid Mechanics. HORNE E. P. W. (1978) Interleaving at the subsurface front in the slope water off Nova Scotia. Journal of Geophysical Research, 83, 3659-3671. JOVCE T. M., W. ZENK and J. TOOLE (1978) The anatomy of the Antarctic Polar Front in the Drake Passage. Journal o( Geophysical Research, 83, 6093-6113. LAMBERTR. B. and J. W. DE~NKOW (1972) On vertical transport due to fingers in double diffusive convection. Journal of Fluid Mechanics. 54, 627 640. LINDEN P. F. (1976) The formation and destruction of fine-structure by double-diffusive processes. Deep-Sea Research, 23, 895-908. LINDEN P. F. and J. E. WEBER (1977) The formation of layers in a double-diffusive system with a sloping boundary. Journal of Fluid Mechanics, 81,757 773. NAGATA Y. (1970) Detailed temperature cross section of the cold-water belt along the northern edge of the Kuroshio. Journal of Marine Research, 28, 1-14. POSSirNTn~R E. S. and R. W. HOUGHTON (1978) Fine structure instabilities induced by double diffusion in the shelf/slope water front. Journal of Geophysical Research, 83, 5135-5138. THORPE S. A., P. K. HUTT and R. SOULSBY(1969) The effect of horizontal gradients on thermohaline convection. Journal of Fluid Mechanics, 38, 375-400. TURNER J. S. (1967) Salt fingers across a density interface. Deep-Sea Research, 14, 599-611. TURNER J. S. (1968) The behaviour of a stable salinity gradient heated from below. Journal of Fluid Mechanics, 33, 183-200. TURNER J. S. (1978) Double diffusive intrusions into a density gradient. Journal of Geophysical Research. 83, 2887 2901. TURNER, J. S. and C. F. CHEN (1974) Two-dimensional effects in double-diffusive convection. Journal qfFluid Mechanics. 63, 577-592. VOORmS, A. D., D. C. WEBB and R. C. MILLARD(1976) Current structure and mixing in the shelf/slope water front south of New England. Journal of Geophysical Research, 81, 3695-3708.
0011-7471/79/0801-0903 $02.00/0
The vertical length scale of double-diffusive intrusions B. R. RUDDICK* a n d J. S. TURNER* (Received 23 October 1978: in revised /brm 19 March 1979," accepted 25 March 1979)
Abstract -A laboratory model has been developed to simulate the mixing at an oceanic front across which there are large T S anomalies but a small net horizontal density difference. Identical stable density distributions are established on the two sides of a central vertical barrier, using the analogue system of salt solution at one end of a tank and sugar solution at the other. When the barrier is removed, local density anomalies are produced by double-diffusive transports across the frontal region. These lead to the spontaneous formation of interleaving layers over the whole depth of the tank, with a scale directly proportional to the horizontal concentration differences across the front and inversely proportional to the vertical density gradient. This result is explained using a theoretical argument based on a comparison of the potential energy in the initial and final distributions and the mechanism of energy release by double-diffusive transports across the interfaces between alternate layers. The available data on the scale of intrusive layers near oceanic fronts are also broadly consistent with this energy argument.
1. I N T R O D U C T I O N
THERE ARE n o w m a n y o b s e r v a t i o n s in the ocean that d e m o n s t r a t e the i m p o r t a n c e of interleaving processes in d e t e r m i n i n g the rate of h o r i z o n t a l mixing between water masses. P a r t i c u l a r l y when there is a large c o n t r a s t in T - S p r o p e r t i e s in the h o r i z o n t a l b u t a small net density difference between the w a t e r masses, interleaving l a m i n a e a n d a s s o c i a t e d local t e m p e r a t u r e inversions are very c o m m o n . FEDOROV (1978) surveyed this field in some detail a n d p r e s e n t e d m u c h evidence to show that large intrusions (such as the M e d i t e r r a n e a n outflow into the Atlantic) b r e a k up into a n u m b e r of thinner layers t h a t then interleave with the s u r r o u n d i n g water. These effects are m o s t p r o m i n e n t near s h a r p fronts, which suggests that the interleaving is in some w a y driven by the h o r i z o n t a l g r a d i e n t s of p r o p e r t i e s at a rate related to the m a g n i t u d e of the gradients. The fact that interleaving layers are distinguished by b o t h t e m p e r a t u r e a n d salinity a n o m a l i e s (which m u s t be nearly c o m p e n s a t i n g to preserve static stability) i m m e d i a t e l y suggests that double-diffusive processes, i.e. those that d e p e n d essentially on the different m o l e c u l a r diffusivities of heat a n d salt, c o u l d be i m p o r t a n t for their f o r m a t i o n a n d p r o p a g a t i o n . M o s t of the detailed l a b o r a t o r y e x p e r i m e n t s on d o u b l e diffusion have c o n c e n t r a t e d on vertical t r a n s p o r t processes across interfaces between layers with different T S properties. TURNER a n d CHEN (1974) p o i n t e d out, however, t h a t these m a y n o t always be directly a p p l i c a b l e to the ocean, a n d they carried o u t a n u m b e r of e x p l o r a t o r y experiments that i n c l u d e d t w o - d i m e n s i o n a l effects. M o r e recently, TURNER (1978) explicitly studied the i n t r u s i o n of one fluid into a g r a d i e n t o f a n o t h e r a n d s u m m a r i z e d the features of ocean o b s e r v a t i o n s that can be e x p l a i n e d in terms of this a n d related processes. JOYCE, ZENK a n d TOOLE (1978) have now presented direct evidence (from m e a s u r e m e n t s
* Research School of Earth Sciences, Australian National University, P.O. Box 4, Canberra, Australia. 903
904
13. R. RUDDICK and J. S. TURNER
across the Antarctic Polar Front) of the importance of double-diffusion in driving the horizontal mixing of heat and salt across the front. The dynamics of this intrusive process are still far from understood, however, and there has as yet been no satisfactory quantitative explanation of even the most obvious property, namely, the vertical length scale of the individual layers. Various suggestions have been made about the mechanism that might determine this scale (see TURNER, 1978). It could be due to distortions by internal waves (the vertical wavelength of which sets the scale); to a process analogous to the 'side wall heating' mechanism described by THORPE, HUTT and SouLsav (1969) and CHEN, BRIGGSand WIRTZ (1971); or to some inherent instability that depends only on the presence of nearly compensating vertical property gradients, not on the specific mechanism that produced them (LINDEN, 1976). None of these explanations has yet been properly tested because of various inadequacies in all the two-dimensional laboratory work to date. The new experiments described below, however, have achieved a much greater degree of control over the important parameters and have produced clear-cut results that can be compared with a simple theory related specifically to the experimental geometry. Briefly, a vertical front has been set up by withdrawing a barrier separating regions containing sugar and salt solutions having the same vertical density distribution. After the initial wave disturbances and small scale turbulence have died away, a series of interleaving layers develops and gradually extends horizontally. In the present paper we will describe these observations qualitatively and then concentrate on the vertical length scale and its theoretical explanation. 2. LABORATORYEXPERIMENTS ON FRONTS It will be helpful first to sketch the steps that have led to the present experiment in order to emphasize the advantages of the technique adopted. Consider what happens when a source of fluid (say sugar solution) is injected into a tank containing a homogeneous solution of a second property (say salt), which has a different molecular diffusivity but the same density. TURNER and CHEN (1974) have shown that vigorous vertical convection ensues, with rapid mixing between the injected fluid and its surroundings. When a source of one fluid is released at its neutral buoyancy level into a gradient of the second, similar vertical convection is observed before the fluid spreads out at several levels (TURNER, 1978). This behaviour implies, however, that the properties of the fluid as it begins to intrude are not well known, because a rather uncontrolled mixing process has preceded the interleaving, which is our major concern. There have also been experiments that (to some extent at least) model more directly the effect of a discontinuity of properties across a vertical or inclined frontal surface. TURNER and CHEN (1974) and LINDEN and WEBER (1977) have shown how the introduction of an inclined boundary can produce a series of extending 'noses' in a tank containing opposing vertical gradients of two properties, which was previously quite stable. The disadvantage of this method is clearly the solid boundary, which inhibits horizontal motions. Horizontal differences have also been produced just by adding an excess of one component to the end of a previously stratified tank, using a nozzle moved up and down to produce rapid mixing over the whole depth. This can also be done with a central barrier in place, which is later removed to allow the different mixtures to communicate. It is difficult to avoid a substantial density mismatch and this inevitably produces intruding motions related (at least initially) to the overall density structure and not to the slower double-diffusive processes.
(a)
(b)
(c) Fig. 2(a)-(c).
[facing p. 904]
(d)
(e)
if) Fig. 2(d)-(f).
Fig. 3. (a) Another experiment in which the initial distortion of the front was small, showing the smooth increase in layer scale with increasing depth. (b) An experiment in which fibs across the front was constant in depth (see Fig. 4). The layer scale shows no systematic variation with depth.
Fig, 2, (Previo~ two pa~e:i.) Shadowgraph record O~alypical experiment, with a bottom density of 1.0" g c m -3. The scale is in centimetres, measured f r o ~ the bottom. (a) (t = 0 m 0 s) the barrier is wi .~drawn. (b) (t ~ m 55 s) the distorted fron t after the density anomalies have been adjusted by i r ~ r n a l wave motions. (c) (t = 8 m 28 s) interleaving mO~ta~ have formed, with alternating diffusive interfaces and fingering regions. (d) (t = 12 m 36 s) thelayers sprea~ horizontally and become more organized in appearance. (e), (f) (t = 25 m 25 s and 25 m 45 ~; th~-ell-developed interleaving layers. The motion of the dye streaks shows flow to the right just below the diffusive interfaces, and flow to the left just above them. The velocity of this flow is several times the advance speed of the layers. Note the increase in layer thickness with increasing del: and BAS.
The vertical length scale of double-diffusiveintrusions
905
In the present experiments, great care was taken to minimize these disturbances by producing density distributions on the two sides of the barrier as nearly identical as possible. With the 'dam' in place, the two ends of the tank are filled simultaneously using duplicate 'double bucket' devices to produce the same linear gradient of salt solution on one side and sugar on the other (with, of course, equal densities at the top and bottom). A schematic diagram of the apparatus is shown in Fig. 1. A small hole in the top of the dam just below the final depth allows an adjustment of water level if one side has been slightly overfilled. Before starting the experiment, the dam is lifted about 3 mm to allow the pressure at the bottom to be equalized. When the experiment proper is started, by raising the dam smoothly (at a typical rate of 2 mm s- 1), the behaviour is as shown in the sequence of photographs of Fig. 2. As the barrier is raised, small-scale mixing is generated locally and there are also larger-scale wave-like motions, generated by the remaining internal density anomalies [Fig. 2(a)]. These motions rapidly die away and we are left with a slightly distorted but essentially vertical front between fluids of different diffusivity [Fig. 2(b)]. It is important to emphasize from the start that the subsequent motion is insensitive to either the initial distortion or the small-scale mixing; different devices for mixing the interfacial region were added to the trailing edge of the dam, but the final state depended only on the property differences across this region and not on its width or detailed structure. As time goes on, the interleaving layers shown in Fig. 2(c)-(f) spontaneously form and extend. The interfaces above and below them are alternately 'finger' and 'diffusive' in character, and the motion is driven by local density anomalies produced by the release of potential energy due to the double-diffusive transports between layers. The extending layers have a small tilt to the horizontal, which can be attributed to the fact that the buoyancy flux through the finger interfaces is greater than that through the diffusive interfaces. Qualitatively, the motion in these layers can be explained in the same terms as the multiple
~4
MIXER t
. . . . .
REMOV~E
K -,ER
IL
._
_
//
T
S
t~
1,8m
0.3m
~'~
Fig. 1. Sketchof the experimentalapparatus, showingthe tank, the removablebarrier which divides the tank into two equal halves, and the two identical 'double bucket' filling systems, one for each half of the tank.
906
B . R . RUDDICK and J. S. TURNER
intrusions described by TURNER(1978), but there is the important difference that the initial conditions are now carefully controlled and measured. Both the size of the layers and their rate of horizontal extension increases systematically with depth, i.e. with the concentration anomaly across the front. Figure 3(a) shows another experiment, similar to the one just described but in which the density gradients on either side of the barrier were particularly well-matched. The initial distortions of the front were small, allowing the 'natural' scale to develop more readily. Although the systematic increase in the layer scale with depth is more regular and clearly visible, the scale of the developed intrusions is nearly the same in the two experiments just described. The initial distortion of the front in Fig. 2(b) has to a large extent been forgotten by the later stages [Fig. 2(e), (f)]. Another type of experiment, in which flAS across the front and dp/dz were independent of depth, is illustrated in Fig. 3(b). (The initial distributions of S and T corresponding to this case are sketched in Fig. 4.) The layer scale and rate of extension do not vary systematically with depth but now tend to be constant. The only results that will be considered quantitatively here are those for the vertical length scale of the layer. A series of experiments with different linear density gradients and top and bottom concentrations is summarized in Fig. 5. The observed layer scale H within each experimental run is, with some experimental scatter, directly proportional to the horizontal concentration difference across the front, and the various runs are brought together if H is inversely proportional to the vertical density gradient. (Note that dp/dz varies by a factor of 20 in these experiments.) This implies that, for experiments where the surface water is fresh, the layer scale is directly proportional to the depth below the free surface and is independent of the actual density at the bottom. More generally, if there is only one component on each side, with a higher concentration at the surface, the layer scale is proportional to the distance below the level at which the extrapolated linear concentration profile would fall to zero. The constant of proportionality and its relation to the mechanism of layer formation and extension will be considered in the following section.
~'S
,,T
~s
aT
f ~A S
BARRIER
,BAS
Fig. 4. Sketch of the initial profiles on either side of the barrier for the experiment in Fig. 3(b), in which [3AS was constant with depth. A basic density stratification (on both sides) is altered by the addition of sugar to both buckets of the left-hand filling system and a corresponding amount of salt to both buckets of the right-hand filling system. The extra sugar and salt increase the mean density on either side without altering the density gradient.
907
The vertical length scale of double-diffusive intrusions
•
~\'~ ,
B
/
N4
B//
A
I
~'r MIXED LAYERS
kx'~
I
I
20 d= /~AS /de P dz
GRADIENT
.o"
/
I
I0
Fig. 5.
,~j~
°/
d
LINEAR
,
30
(cm.)
Plot of the observed layer thickness, H versus
= flAS
/;o
dz
for the experiments to date. The data points A J are from the experiments described in Section 3, with dp/dz constant but flAS increasing in depth. Each data point represents one layer. The solid points (solid circle, square, triangle) are from experiments with flAS and dp/dz constant in depth. The layer thickness was also constant in depth, and the points represent an average over several layers. The experimental parameters are summarized below. The solid lines are described in the text. Data point 1 dp
A
-- - - x 103 (cm 1) P0 dz
flAS
B
0.33 1.1
C
D
E
F
G
H
I
J
1.5
1.9
2.0
3.0
3.5
3.7
5.6
7.5
Increases with depth
•
•
•
1.3
1.2
0.67
0.025 0.003 0.011
3. AN ENERGY A R G U M E N T FOR THE LAYER D E P T H
The argument developed below is based on a comparison of the potential energy in the initial distribution of properties with that corresponding to the final distribution. The latter is produced by the coupled transports through the quasi-horizontal interfaces between alternate layers, which are set up by incipient intrusive motions of an assumed form. By equating these energies, one arrives at a kind of finite amplitude stability criterion, which in fact includes a prediction of layer depth in terms of the given parameters of the system• In spirit, the theory is similar to the energy arguments used by TURNER (1968) to predict the mixed-layer depth produced when a stable salinity gradient is heated from below• Explicitly, suppose (as in our experiment) that there are identical linear gradients of two properties on each side of a barrier [Fig. 6(a)]. Denote by T the property with higher molecular diffusivity and by S that with the lower diffusivity, and let p = po(1 +flAS) = po(1 +~AT) be the density at the level z o on which we fix our attention,
908
B.R. RUDDICK and J. S. TURNER
a) Profiles before intrusion (Left)
(Right)
b) Profiles in centre immediately after intrusion I
.
,
'
.'I
|':.".:..
/9S(z)
cIT(z)
-~ O A S ~
-NaZ~T le
p(z)
oriltnal
gradient
c) Profiles after fingers have "run down" the intrusion BS(z)
GtT(z)
p(z)
oAT
-I
7, r~AT(l-n)
Fig. 6. Schematic diagram of the intrusion process. (al Profiles offlS(z) (left of barrier) and aT(z) (to the right of barrier) before intrusions occur. (b) Hypothetical profiles offlS(z), aT(z), and p(z) after an intrusion has caused a layer'of S to flow over a layer of T. (c) Hypothetical profiles of S, T, and p after the finger fluxes have 'run down' the S contrast between the upper and lower halves of the layer.
at a distance d below that at which the concentration difference is zero. The density gradient is thus 1/po(dp/dz ) = 7 = ~AS/d. When the barrier is removed, suppose that the motions that develop are such that locally a layer o f S, originally of depth H, intrudes on top of T to produce a layer of half that depth, as shown in Fig. 6(b). (The a r g u m e n t can be carried through using alternative assumptions a b o u t the intrusion process, which will be referred to below.) The situation shown is clearly unstable, but double-diffusive transports driven by the potential energy in the S distribution will rapidly cause it to run down towards a state where the S is uniformly distributed and the T (and therefore the net density) are hydrostatically stable. The 'final state' is shown in Fig. 6(c); it is implied here that the transport across the finger interface formed by the initial interleaving is so m u c h larger than that across the diffusive interfaces above and below that the latter can be ignored. It has also been assumed that over the whole period of rundown, the flux ratio of heat to salt is (in density terms)
BFs
= n = constant < 1.
(1)
The total changes in S and T and in net density shown on Fig. 6(c) have been plotted using (1). Using this notation, the initial potential energy, relative to the level z o, of the region
The vertical length scale of double-diffusiveintrusions
909
z = +_H/2 in which the density is p = po(1 + T z ) is Eo = - g P o
f
ill2
(1 +yz)z dz
, ) - HI2
(2)
= -gpoTH3/12.
The final potential energy of the stepped density distribution shown in Fig. 6(c) is E = t~gPo flAS(n + 1 )H 2 - l~gpoflAS(3 - n)H z = ~gpoflAS(n - 1 )H 2.
(3)
The potential energy remains unchanged or has decreased, so that it is possible to achieve the final state using only energy available internally if E o >~ E~, or - ~27H 3 >~ - ~(1 - n ) f l A S H 2 n ~< -~(1- n ) f l A S / 7 .
(4)
This has the same form as the laboratory experimental result, with a numerical constant that depends on the flux ratio n. A lower limit to the layer depth can be obtained as follows : The density in the upper half layer of the 'final' stepped density distribution shown in Fig. 6(c) is Pl
~--p0[1 + flAS(1 + n)/2].
(5)
The density of the lower half of the (assumed similar) pair above this is P2 = Po[ 1 + f l A S ( 3 - n ) / 2 - T H ] .
(6)
The condition that the interface between the two be statically stable leads to H > (flAS/7)(1 - n ) .
(7)
Thus H is bounded above and below: ( f l A S / 7 ) ( 1 - n ) < H < {(flAS/?)(1 - n ) .
(8)
Many variations on this argument are possible, and they lead to forms similar to (8) but with different (generally larger) numerical constants. Two examples will be described here. Suppose that the final density distribution is not stepped, as shown in Fig. 6(c), but remains linear (while redistributing the S component within the layer). This might be the case in the laboratory experiments, where fingers fill the entire region between diffusive interfaces. Then it is found that the two limits in (8) coalesce, giving H = 2(1 - n)flAS/7.
(9)
Thus allowing the fingers to fill the entire region only changes the numerical factor from 3/2 to 2. It can be argued that the intermediate state drawn in Fig. 6(b) is unrealistic in that it implies a temporary increase in potential energy. If instead we suppose that the gradient of T is just replaced over a depth H/2 by the S gradient (with no change in density gradient), then the run-down state is slightly different and (1 - n)flAS
(¼-- ¼n)7
< H <
(1 - n)flAS
( ~ - - ¼n)), '
(10)
910
B . R . RUDDICK and J. S. TURNER
If the basic stratification is double-diffusive, with some available potential energy in the vertical stratification, the predicted layer thickness is only weakly increased. This is because the S and T contrasts due to horizontal advection tend to be stronger than the contrasts already present in the stratification.
4. C O M P A R I S O N WITH O B S E R V A T I O N S
(a) Laboratory experiments Although the estimates (8-10) all give the same form for the predicted layer thickness, there is a considerable variation in the numerical factors according to the particular assumptions one makes in the arguments of Section 3. In what follows we use the experimental results to estimate the 'best' value of the numerical factor. GRIFFITHSand RUDDICK (1979) recently measured, more accurately than was previously possible, the ratio of buoyancy fluxes for sugar/salt fingers. They found that as the density anomaly ratio Rp = ctAT/flAS increases during a rundown experiment, n decreases rapidly and approaches a nearly constant value ofn = 0.88+0.01 for Ra > 1.3. We take this value for n rather than the earlier value of 0.92 measured by LAMBERTand DEMENKOW(1972). Note that the quantity 1 - n, which appears in equations (8)-(11 ), differs by 50% according to which estimate of n is used. In Fig. 5 are plotted the observed layer thicknesses, H, vs
d = flAS/ ,'
1 dp Po dz
for the experiments to date. The cross-frontal salinity difference for each layer, AS, was proportional to the depth below the free surface in experiments A to J. Each data point represents a single layer. The solid points (solid square, circle, triangle) are from experiments in which AS and dp/dz were constant with depth, as in Fig. 4. Each such point represents an average over several layers. Also plotted in Fig. 5 are the straight lines corresponding to equations (8) and (9) using n = 0.88. With the exception of experiment B, the data are contained by the three lines. (In experiment B, exceptionally strong distortions of the initial front pre-set the scale of interleaving. Although the scale decreased slowly with time, the initial structure persisted for the duration of the experiment. In the other experiments, the distortions were weak enough for their effects to diminish with time.) We take the central line, H = 3(l-n)d,
(11)
as the 'best fit' to the experimental data. TURNER (1978) described experiments in which an intrusion produced by a source of S in a gradient of T (or vice versa) broke up into a number of sloping layers, with a structure similar to the layers formed at a front. The observed layer scale in those experiments was considerably smaller than that predicted by equation (9). We can now attribute this to the intense mixing at the source, which diluted the incoming fluid with ambient fluid, causing the effective contrast flAS to be much smaller. However, the sense of the changes in layer scale with density gradient and source input rate described by TURNER is consistent with equation (9). THORPE et al. (1969) and CUEN et al. (1971) described experiments in which a salinity
The vertical length scale of double-diffusive intrusions
911
gradient was heated from the side, producing a series of layers with alternating diffusive and finger interfaces. Their analysis predicted a layer scale of ~ A T ' / 1 a_p, while experimentally /P
O2
the constant of proportionality was found to be 0.81 + 0.10. For heat/salt fingers,* equation (11) predicts a scale of 0.66eAT
Although the mechanism of layer formation is
apparently very different in the 'heated side wall' experiments and ours, it is worth remarking on the identical form and the close numerical agreement of the two results.
(b) Oceanic observations In his study of observations of oceanic finestructure, FEDOROV (1978, Fig. 6) described a front in the surface layer of the tropical Atlantic Ocean that exhibits several of the features found in the laboratory fronts. These properties, summarized by TURNER (1978), are directly attributable to the double-diffusive nature of the cross-frontal interleaving. They are (1) the presence of two or more temperature inversions, indicative of diffusive interfaces on either side of a salt finger interface ; (2) the slope of these interfaces upward away from the warm salty water mass ; and (3) changes of temperature and salinity along the layer in a manner consistent with the changes produced by the salt finger fluxes. Other fronts in which double diffusion may be the driving mechanism have been reported by JOVCE et al. (1978), POSMENTIERand HOUGHTON(1978), HORNE (1978), VOORHIS, WEaa and MILLARD (1976), and NACATA (1970). In Table 1 we compare the vertical scale predicted by equation (11 ) with the scale of interleaving actually observed. The computed and measured scales agree within a factor of two for all cases except that of Horne. [Horne's calculations for the front off N o v a Scotia indicate that the diffusive fluxes, neglected in the calculation leading to equation (11), dominate over the finger fluxes. Thus our simple estimate may not apply there.]
Table 1. A comparison between the vertical length scale of interleaving layers computed JJ'om equation (11) and the observed scale
Layer depth (m) Reference
FEDOROV(19781 JOVCEet al. (1978) POSMENTIER and
HOUGHTON(1978) HORNE(1978) VOORHISet al. (1976) NAGATA(1970)
* n = 0.56 (TURNER,1967).
AT
~
(C)
(°C)- 1 x 104
dp -p dz (m- 1t
2.9 0.95 1.7 1.7 1.55 2.0 1.43
1.6× 10-5 4.9 × 10-7 1.6×10 5 2.8×10 5 9.1 × 1 0 - 6 1.6×10 4 6.5 × 10 6
0.4 1.25 8 8 4 7 5
1
Computed
Observed
0.66d
H
4.8 160 56 32 45 5.8 73
10 100 28 23 10 11 120
912
B.R. RUDDICKand J. S. TURNER
Overall the agreement is good, considering that the laboratory data exhibit a scatter of a factor of two and were carried out using the sugar/salt system. Dimensional arguments suggest that
d = flAS/ is a natural vertical scale for intrusions (regardless of the driving mechanism), but our comparison with the observations seems to imply more than that. The multiplying term (1 - n ) in equation (11) changes by a factor of 4 from the laboratory value of 0.12 for sugar/salt to the 'oceanic' value of 0.44 for heat/salt. The good numerical agreement thus supports the mechanism we have proposed, whereby double-diffusive fluxes release the energy used for frontal interleaving.
5. CONCLUSIONS From the laboratory experiments modelling horizontal interleaving at a front, combined with a simple energy argument based on the double-diffusive driving mechanism, one can make the following estimate of the vertical length scale of cross-frontal intrusions :
H = ~(1-n)/~AS
(11)
1 dp p dz where n = density flux ratio, 0.56 for heat/salt fingers, 0.88 for sugar/salt fingers AS = the salinity contrast, measured across the front along an isopycnal (note that flAS = sAT)
1 dp - the vertical density gradient, and p dz 1 @ p ~S This estimate is probably accurate to about a factor of two. When extrapolated from laboratory scales (a few centimetres) to several examples of oceanic fronts (where the scale is tens or hundreds of metres), the observed interleaving scale matches that predicted by equation (i 1) to the expected accuracy. This agreement suggests that double-diffusive processes play a major role in mixing at oceanic fronts and that the layer scales are determined by the mechanism we have described. It has become apparent that observations near oceanic fronts should be made in enough detail to resolve the interleaving and that the interpretation of such data must take into account the double-diffusive nature of the process before one can properly estimate the various fluxes. In particular, the unrealistic concept of eddy diffusivities should be avoided. More laboratory experiments are in progress, with the aim of obtaining a better understanding of the horizontal fluxes of heat and salt associated with such interleaving systems of layers.
Acknowledgements--We are grateful to DEREKCORRIGANand Ross WYLDE-BROWNE for their assistance in building the apparatus and carrying out the experiments.
The vertical length scale of double-diffusive intrusions
913
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