- Email: [email protected]
Salt fingers and convecting layers
Deep-Sea Research, 1969, Vol. 16, pp, 497 to 511, Pergamon Press. Printed in Great Britain
Salt fingers and conveeting layers MELVIN E. STERN* a n d J. STEWART TURNERt
(Received 10 March 1969) Abstract--It is shown that convective ' fingers ' can form not only in the salt-hcat system, but also in a fluid containing two solutes with much closer diffusivities (for example sugar above a salt solution). This experimental device is used to explore various phenomena which cannot easily be produced in the laboratory with salt and heat. We have investigated: (a) the behaviour of an interface containing fingers and separating two convecting layers, (b) the production of a series of steps from a smooth gradient by imposing a flux at the top, (c) and the mechanism of formation of layers from two opposing gradients of the solutes. In each case the related theoretical ideas are summarized, and it is suggested how these can be extended to describe the new features of the observations.
1. INTRODUCTION MANY different pieces of evidence now support the view that a deep region of fluid containing opposing vertical gradients of temperature and salinity does not remain smoothly stratified, but instead breaks up into a series of steps. In the ' salt finger' regime, where temperature is stabilizing and salinity destabilizing, several oceanographic observations have been attributed to this effect. TAIT and HOWE (1968) recorded steps underneath the Mediterranean outflow, with well mixed-layers separated by interfaces which were at times too sharp to resolve with the technique they used. COOPER and STOMMEL(1968) found a stepped structure in the main thermocline near Bermuda, but with the well mixed and gradient regions more nearly comparable in depth. Some understanding of the physical processes governing this behaviour has already been obtained theoretically and through laboratory experiments. STERN (1969) (referred to as I below) has shown that a field of finite amplitude salt fingers becomes unstable above a critical value of the salt flux, for a given temperature gradient. Energy is fed into relatively long internal waves, leading eventually to a local overturning, with complete mixing in regions of large shear and quasi-laminar salt fingers retained in thin regions of small vertical shear. He has also made predictions about the dependence of the thickness of the layers, and the fluxes of heat and salt between them, on the molecular properties. TURNER (1967) (called II in the present paper) has studied experimentally the transports across an interface containing salt fingers, and has shown how convection can be maintained in the deep layers on each side of the interface by the net unstable buoyancy flux through it. He has also speculated about the formation of layers from a smooth stable temperature gradient, when salt is added to the top. The two papers just cited contain suggestions which must be developed further * Graduate School of Oceanography, University of Rhode Island. t Department of Applied Mathematics and Theoretical Physics, University of Cambridge.
497
498
MELVIN E. STERN and J. STEWARTTURNER
and tested in more detail before they can be applied to the ocean. Experimental difficulties have up till now prevented us from doing this; the main problems arise from the need to maintain a large stable temperature gradient over a deep enough layer, while avoiding spurious effects due to side wall heating or cooling. In the present paper we will describe a new technique which eliminates many of these difficulties. We show that similar 'finger convection' phenomena can readily be produced in a system containing two solutes, for example sugar and salt, which have diffusivities much closer together than the 100:1 ratio for heat and salt. There are therefore no side wall losses, and there is the added advantage that much larger density gradients can be used. We will first describe in qualitative terms the various phenomena observed, many of which are completely inaccessible to laboratory study using the heat-salt system. Secondly, we will set down a summary of certain theoretical ideas which seem consistent with these observations. We deduce some preliminary numerical values of important parameters, but a detailed quantitative study of the dependence on molecular properties remains to be carried out in the future. 2.
THE TWO LAYER SYSTEM
2.1. Previous work and some immediate deductions The simplest situation is obtained by placing hot water above cold water, sharpening the interface by mechanical stirring above and below and then adding some salt solution to the top layer, as described in II. When stirring is stopped salt fingers quickly develop across the interface and seem to be limited in extent by the large scale convective motion which they drive in the deep layers. These measurements supported the relation suggested by a dimensional argument, namely, that the salt flux (p0 Fs gm/cm2/sec) or more conveniently the associated buoyancy flux flFs (cm/sec) has the following dependence on the parameters: flFs : (flAS)4/a (gv)~f~-AS,
= C(flAS) 4j~.
(1)
In the foregoing p0 is the mean water density, AS is the salinity excess and AT is the temperature excess of the upper layer, ~ A T - flAS is the density deficit, g is the acceleration of gravity, v is the kinematic viscosity, ~T is the thermal diffusivity, ~s is the salt diffusivity, a n d f i s an incompletely measured universal function. The measured value of the constant on the right of (1) was about C = 0.1 era/see when ~AT/flAS was about 2, but C decreases only slowly with more stable values of ctAT/flAS. The most important implication of the observed 4/3 power law is that there is a quasi-equilibrium value of the salt flux which is independent of any length scale, either the thickness of the convecting layers or the initial thickness of the interface separating the two fluids. The latter is a quantity which approaches an equilibrium thickness (h) determined by AS and the molecular coefficients [see Section (2.3) for a direct test of this idea]. An explanation of this dependence has been given in I; it was shown there that stable quasi-laminar salt fingers can occur in a region only if the salt flux is less than a critical value given by
vo~~ l ~ z ~- Ncritical "~ !.
(2)
The critical value of the non-dimensional number in (2) is of order unity if the ratio
Salt fingers and convecting layers
499
of the mean temperature and salinity gradients (i.e. s3f'13z (fl~Sl3z) -1) is of order unity and if v/Kr and/or v/Ks are large compared to unity. We can now extend these results by applying (2) to the interface region (containing salt fingers) between the two convecting layers, and assuming that the fingers at the edges of the interface are always close to the critical state. Setting 3~/3z ~ AT/h in (2) and using (1) gives
h~
vsAT v ~Fs ~'~ C ~ A S ) ~ '
(3)
If H denotes the thickness of the ' deep ' convccting layers (assumed to be the same) then in the course of time (t) the salinity difference between the layers will decrease at the rate d (flAS)/dt = -- 2fl Fs/H = -- 2C (fiAS) 4/3 H -1, or
2Ct (flAS)-~ = 3-H ÷ constant.
(4)
Notice that in the slow decay, the system passes through a succession of quasi equilibrium states with
dh/d(AS) dt/
dt
independent of time. The increase of the thickness of the region containing salt fingers is a striking feature of the middle and final periods of decay in all the experiments reported below. In addition (4) will be examined quantitatively in one experiment using sugar and salt. Previous measurements have also been made (ll) of the buoyancy flux ratio R~ ~
aFT 5Fs
(5)
as a function of the density anomaly ratio sAT Rp--: 5XS
(6)
where FT is the horizontal average of the product of vertical velocity and temperature. On energetic grounds RI < 1 and the experiments (I1) gaveRf - 0.56 over a wide range of R o in the heat-salt experiment. 2.2. The dam break method The usefulness of the grid-stirring technique for producing the initial sharp interface was limited by the small range of density differences which can be achieved by heating, of order sAT ~,~ ½~ with AT ~,~ 20°C. With such small density differences it was difficult to set up an experiment with sAT,/f3AS less than 2, and so another technique was devised to reach this range. A vertical watertight barrier was inserted at the centre of a channel about 10 cm wide, 15 cm deep and 150 em long. One half of the channel was filled with hot s/dt water, and the other with cold fresh water, with densities and levels adjusted to be very nearly equal. When the barrier was withdrawn the slightly lighter fluid flowed above and across the denser half, producing either the salt finger regime or the inverse case (heating a stable salt gradient from below), depending upon the exact density ratio of the two fluids in the partitioned channel. Little quantitative work has been done using this ' dam break ' technique, but a movie
500
MELVIN ]~. STERN and J. STEWARTTURNER
was made which dearly illustrated the various possible types of behaviour. When the hot salty water is slightly light, salt fingers were observed immediately after the partition was removed, even though there was considerable shear due to the counter flows and to the internal waves reflected from the ends of the channel. The latter soon subside and the system reaches a state of vertical convection similar to that obtained with the grid-stirring technique. The system was allowed to run down towards a state of vertically uniform salinity. As this stable state was approached the temperature difference fell to about one-half its initial value, which is what one would expect if Rf -- ½. When the possibility of using two different solutes at room temperature became apparent, our first experiment was in this same long channel, using the dam break technique. In a typical run common salt solution with specific gravity about 1.050 and diffusivity ~ -- 1.5 × 10 -5 cm2/s was put on one side of the barrier, and sugar solution with specific gravity 1.045 and K =: 0.5 × 10 -5 cm2/s was put on the other side. After the barrier was withdrawn the sugar solution flowed over the salt solution, with fine fingers falling out. The internal waves die out rapidly and one clearly sees a thin horizontal interfacial layer with surprisingly strong large scale convection in the layers on either side. This system with low diffusivity sugar on top of higher diffusivity salt is analogous to the ordinary salt finger convection in the heat system.* The thickness of the interfacial transition region was about 1-2 mm, a few minutes after the start. Optical manifestations of the fine fingers (due to the variations of index of refraction) were easily visible. These quasi-steady structures were sharply limited, above and below, by the convective regions. Over a much longer time interval the interface region gradually thickened and the intensity of the convection diminished. Sometimes strong horizontal motions were observed in the convective regions, persisting for an hour or more. Section (2.4) describes some measurements of density flux and colorimetric estimates of sugar flux made in this long channel. 2.3. A more direct technique Among the other advantages of the sugar-salt system is the fact that one can work with much larger density differences, typically 10-30 times that achieved with heat and salt. Since it is relatively easy to set up the two solute system with a ' densityratio ' close to unity the dam break technique is not really necessary. Instead, sugar solution can be poured carefully on a thin piece of wood floating on the salt layer, and this was the method used for setting up most of the later runs in a deeper narrower tank. This is a more convenient geometry for most purposes, including photography. Many of the previously mentioned effects were confirmed using this method and studied further in a more controlled way. To examine the evolution of the interfacial region we placed an intermediate layer between a deep upper layer of sugar and a deep bottom layer of salt. The concentration of sugar and salt in this moderately thick mid-layer was also intermediate between that of the deeper layers. Quasi-laminar salt fingers less than 1 m m thick form in the intermediate layer; it is difficult to photograph these directly but they are always clear to the eye. The convection on either side of the interface sharpens the edge of * With a slightly lighter salt solution riding over the sugar solution we have the case analogous to heating a stable concentration gradient from below. Vertical exchange was then observed to take place in the form of fluid sheets which formed intermittently at convergence zones on a paper-thin interface.
Fig. 1. Photograph of a thick interface containing fingers between two well mixed convecting layers. The sugar (upper) layer is dyed, and the tank is lit from below. Note the sharp edgesof the inter facial layer. [facing p.
500]
0,~
q:~
.._~
~
~
~
0 ~..-,
>
e-,
~
=o.~
E
.~.~ ~ ~
"-g g ?Z~ ~ . ~
,.~"~
. ~,~
~
.
~'~ ~
Fig. 6. A photograph of a convecting layer formed from two opposing gradients, made visible with aluminium powder. The pipe used for filling the tank from the bottom is visible in one corner next to the scale.
Salt fingers and convecting layers
501
that region as shown in the photograph of Fig. 1. The thickness of the interface decreases with time in the early stages due to the erosion by the relatively strong convection in the layers. In the final stages the intensity of convection diminishes and the interface region broadens. Time lapse movies showed that the convection in the thick layers has an intermittent character, with buoyant elements (generated by the salt fingers) accumulating and moving through the entire layer. These elements retain their identity as they penetrate to the extreme boundaries of the apparatus, and (as suggested by BMNES and TURNER (1969) in another context) they could in this way produce reversals in the sugar and salt gradients. We have not yet made any gradient measurements to check this impression. They would be of interest in connection with oceanic profiles [TA1T and HOWE (1968), and some unpublished results of these authors] exhibiting reversals or inversions in so called ' well mixed layers.' Qualitative experiments were also carried out using other pairs of diffusing substances. It is especially interesting to note that a moderately concentrated solution of NaC1 (~ 1-5 × 10-5 cm2/sec) carefully placed on a slightly heavier KC1 solution (K:= 1.9 : 10.'5 cm2/sec) will produce salt fingers and convection even though the diffusivities are so close. 2.4. Preliminary tests of theoretical ideas To avoid confusion with terminology and symbolism in past heat/salt experiments and present salt/sugar experiments we adopt the following convention. We call ' T - s t u f f ' the substance with the larger diffusivity and " S-stuff" the substance with the smaller diffusivity. Some quantitative measurements of the Fs, FT fluxes were made using the dam break technique with sugar (S-stuff) and salt (T-stuff). These provide additional support for equations (1) and (3) and information on C which is necessary for section (3.2).
Table 1. The measured sugar concentrations and density differences during one run (Run 8, 30 July 1968), and the deduced density ratio Rp and flux ratio R~ as functions of time. Change in Changein Time (fl~S)% A,% Ro (ftAS)% 40°//0 Rt 9.33 a.m. 1.0 × 4-56 0'21 1-047 0"2 × 4-56 0'08 0'91 9.37 0.8 × 4'56 0.29 1.080 0"2 × 4"56 0-08 0.91 9.42 0.6 × 4'56 0.37 1"135 0-2 × 4'56 0.07 0.92 9.47 0"4 × 4'56 0-44 1'24 0.2 ~." 4.56 0.08 0"91 9.58 0.2 × 4"56 0.52 1"57 Table 1 gives the density and S-stuff concentration as a function of time in a two layer system with initial specific gravities of 1-0456 and 1.0477, respectively. Each of the layers, set up by the dam break method, was 4.4 cm deep and the top layer also had a dilute dye (of the kind used for colouring food) in solution. A colorimetric technique was devised to compare the dye concentrations at any time against samples diluted to known concentrations, using sample containers with the same thickness as the tank viewed against the same uniformly illuminated background.* The time was *The assumption has been made that the dye is transported in the same manner as sugar but a more careful study of multi-component systems would be needed to check this.
502
MELVINE. SXERNand J. SIEWARITURNER
recorded when dye concentrations of 10 ~ , 20 ~ , etc. were reached in the b o t t o m layer. At these same times samples were withdrawn for later weighing, a n d the interface thickness measured. In this way we could measure the density flux and F,s in the intervals between measurements, and Table 1 shows the c o m p u t e d flux ratio R I = ~ Ftt/fl F s a n d the density ratio R o -- ~AT/fl2xS at each stage. R o increases slowly f r o m 1.05 to 1.57; it is always considerably lower than in any experiment carried out with heat and salt. One might expect that the ' final state ' of the system will be Rp = KT/KS ~, 3, since this is the state of marginal stability for salt fingers (STERN (1960)). It w o u l d have been impossible to distinguish visually between such a distribution and complete uniformity o f the dye (or S-stuff). I
I I
6-
O
\
o .El
4O
~O \
E
u
Y_ En
\ O
2-
I
"1"
0 4.0
z,.5
5"0
Density difference (% heavier than fresh water) Fig. 2. Density distributions measured in the sugar-salt finger experiment described in Table 1. (a) Shows the original two layer system set up using the "dam break" technique; and (b) is the profile made five hours later by withdrawing samples at 1 cm intervals and weighing them. RI is constant at 0,91, to within the experimental error, over the range covered. This value (which is supported by our other qualitative runs) is larger than the value (0.56) found for heat and salt, implying that a smaller fraction of the potential energy released by the sugar convection is used in viscous dissipation and a larger fraction is used to raise the center of gravity o f the T-stuff. This is made clear too by Fig. 2 in which the initial and final (i.e. measured after five hours) density distributions are plotted. The density difference increased in time, but by only a small fraction of the separate contributions to the density differences.* * In connection with the observed increase of ~ Fn/flFs with KS/KT'we point out that on general thermodynamic grounds ~ Fn/flFs -+ 1 as ,~s --> ,¢T with Ro ~ 1. This is because the two dissipations ,or (VT) 2 x2and xs (VS)~/~2must be of equal average value. The production of ~T1and/3S variances must then be equal and it then follows that the vertical fluxes are equal. See SXERN(1968) for elaboration.
Salt fingersand convectinglayers
503
In Fig. 3, (AS) -t and h (which has not been included in the table because it was not measured at the same times) are plotted against time. The points can be reasonably fitted by straight lines, supporting the predicted forms (4) and (3). The slope of the line drawn through the concentration measurements allows us to make the estimate
J
I
2'0
10
/ o J
!
(As)
E 1,5
c-
/f
-
5
to to
1.0
9.40
9.30
Time
9. 50
10.00
Fig. 3. The measured values of (AS)--t (taking the initial sugar concentration difference as unity) and the finger interface thickness h, plotted against time for the experiment set out in Table 1.
C = 10-2 cm/s, a factor of ten smaller than the previously obtained value of C for heat and salt. Note that this change in C with molecular coefficients is not inconsistent with the idea that the ratio (2) is independent of the molecular coefficients. The observed h seems to be predicted by (3) to better than an order of magnitude both in these experiments with sugar and salt and those reported in II using salt and heat. It should be kept in mind, however that theoretically there is still some uncertainty about whether to use the vertical density gradient rather than ~ ~/~z in (2), and whether the whole temperature or density step or only part of it should be used in estimating the mean gradient through the interface region. Questions such as these can only be answered by a more detailed study. We emphasize again the exploratory nature of the present paper, in which we aim to describe a range of phenomena which must be important for thermohaline convection, and do not pretend to have solved all the problems raised by these experiments. 3.
THE FORMATION OF LAYERS FROM A GRADIENT
3.1. Laboratory observations In all previous salt finger experiments the scale of the convective layers has been determined in advance by setting up two or more homogeneous layers. In this section we shall show how layers (or steps) separated by fingers can form naturally on a stable
504
MELVINE: STERNand J. STEWARTTURNER
linear gradient when a downward Fs-flux is applied at the top. This is probably the most important result reported in this paper. Uniform gradients (of dT/dz) were set up using the method described by OSTER (1965) and used by TURNER (1968) in the inverse problem (viz., heating a stable salt gradient from below and studying the resulting layer formation.) This method for producing linear solute stratification uses two identical buckets containing fluid with the concentration that is desired at the top and bottom of the stratification. The buckets are joined by a siphon, the bucket containing the lighter fluid is stirred, and the mixture therein is led into the bottom of the tank in which the stratified fluid is to be formed. After a stratification of T-stuff has been produced in this way a uniform layer of S-stuff, with a density slightly less than the topmost T-stuff, is then carefully poured on top. Complex finger motions are observed in this initial stage of mechanical mixing. When the gradient ~ b~/~z was relatively large (typically 4 × 10-:~ c m 4 with flAS = 10~o) all horizontal motion in the gradient region rapidly disappears and one sees vertical fingers, which persist for a long time, extending all the way to the bottom of the container. However, with a smaller stabilizing gradient and a comparable concentration of S-stuff (c~ b~/~z - 1 "~ 10-acm -1 with pAS = 7~o), we observed that the quasi-laminar salt fingers, just below the imposed S-layer, gave way to a new (first) convective layer. The salt fingers seem to become unstable with respect to larger scale convection over a definke vertical extent. The first convective layer, so formed, deepens somewhat but it is always bounded below by a vertical finger regime. When the stabilizing gradient was further reduced (~. bT/~z -- 5 /. 10`4 cm- 1, flAS ~ 10~) a first convecting layer again formed but somewhat later a second convecting layer also formed, beneath the first one. Again, quasi-laminar salt fingers are found beneath the second layer. F'igures 4a and 4b show different stages of this process of layer formation. We did not try to produce more than two layers but fecl confident that this could be done by suitable variation of experimental conditions. Tracer observations by eye and time lapse movies reveal the behaviour of the fingers as a new layer is formed. Intermittent sideways oscillations of the fingers arc followed by stronger horizontal motions until large scale vertical convection is established in the region. Several runs were followed for a long time (up to a week in one case) and we could then observe the decay of the convecting layers. As the concentration differences across the interfacial layer become smaller with time, the thickness of the interfacial layer and the salt fingers therein increase at the expense of the convecting layers [cf. Section 2.1 and equation (3)]. After three days the first (upper) convecting layer had disappeared completely, while the second one was still in existence, bounded on either side by long fingers extending throughout the 1 meter deep cylindrical container. A photograph of this state, taken a few seconds after dye crystals had been dropped through the tank, is shown in Fig. 4c. Note the distortion of the vertical streaks in the convective layers and their relatively undisturbed character in the finger regions. When no further motion was detectable, measured density profiles still showed the kinks in the original linear profile which had been produced and left behind by the convecting layers. Figure 5 contrasts density measurements made before and after runs in which convecting layers formed (5a), or did not form (5b). In both cases there is a general increase in density through the original gradient region (except right at the bottom), and a decrease of density in the well-mixed layer at the surface.
Salt fingers and cortvecting layers
I
I
I
505
I
@
0
3
_1 --0
I
I
I
~-
._
-
I
~
I
.~
~_~ ~ o
'g ~
o._U
0 m
-
~
I
&,o
~
0
I
I
iO
_
0
I
(uao~oq ~^oqo ua~) lqS!~H
I
~o
0
506
MELVINE. STERNand J. STEWARTTURNER
3.2. Theories o f layer formation We have to explain why salt fingers are realized in the experiments described when ~ l b z is large and why they break down into a convecting layer at some critical value of the gradient. In the former case the order of magnitude of the vertical flux produced by a finger regime may be obtained by multiplying their typical vertical velocity w by the typical value of the horizontal variation of S (or T) in the fingers. Considering the formation of fingers at the top of the initial gradient region it seems reasonable to suppose that the horizontal variation of S is proportional to the vertical step AS. Moreover the viscous and buoyancy forces are of the same order in the fingers and therefore vwL -2 ~ g f l A S
where
L ,---
is the width of the salt finger (see I for an elaboration of these ideas, which are only given in outline here). It follows that the buoyancy flux associated with the S-stuff is /7Fs ~ w f l A S ~ , (flAS)2 o¢ bz ] Equation 7 is not inconsistent with t h e " 4/3-power law." A more general dimensional argument leads to a relationship of the form (7) bm with an extra dependence on Ks i.e. it predicts that the salt finger flux should be proportional to g~ (flAS)2 (~ 5~l~z) - i with an unknown co-efficient of proportionality that depends on V[KT and v/Ks. Now according to eq. (2) these salt fingers will become unstable when Fs is greater than some critical value of order a~ 3Fs ~ vo¢ - - . (8) 3z Combining the two preceding equations leads to (/7AS) ---- B vt (g KT)-i (0¢ a'/~i3Z)t,
(9)
where
is the unknown proportionality constant. Equation (9) gives the critical value of AS as a function of ~ / ~ z for the formation of a new convecting layer within the gradient region. When this convecting layer forms, it will be bounded above by a thin layer or interface (of thickness h) containing salt fingers. The T gradient in the h-layer must be much larger than ?~/~z, and the flux through this interface will be described by (1). The first convecting layer to form was observed to grow, and approach a maximum (equilibrium) thickness//m before a second layer formed below it. We can obtain an expression for Hm if AS in (9) is now interpreted as the S-step between this first convecting layer and the fluid below. The corresponding difference in T-stuff is of order Hm ~ / ~ z , assuming that the convecting layer is well-mixed. The net density step at this time must be small (see TURNER, 1968), SO f l A S ~_ Hm ~ 3z
(10)
Salt fingers and convecting layers
507
where ~ denotes equality within a factor of two (say). Combining (9) and (10)we obtain the estimate
Hm ~ B v~ (g xT) -i (~ 3~/bZ)-~ _ B4/3 ~ (g K~)-~ (/3AS)-~
(1])
for the equilibrium thickness of the first convecting layer. The first form of (11) shows that Hm will depend weakly on the original T-gradient and on the molecular properties, but not at all on the way in which the destabilizing flux was originally applied. It is consistent with the prediction made in I (which was based on quite a different detailed model) that the layer thickness should be proportional to (3q~/3z)-~. Moreover from (1) and (10) we get
Hm ~- C -~ Fs ~ (~ ~ / b z ) -1 which has a form similar to the criterion found by TURNER (1968) for layers formed by heating a stable salt gradient at a constant rate from below. This general property of independence of the formative mechanism would also seem to be important if we are ever to relate our laboratory models to layers in the ocean, such as those observed under the Mediterranean Outflow. If we define a mean transfer coefficient by
Ks = flFs \ nra] then we see from (1) and the second equation in (11) that Ks (at the time of formation of a new layer and interface) is independent of flAS i.e. is a function only of the molecular properties. This is also in accord with the completely different model in I, where the transfer coefficient was found to depend only on V/KT, V/KS and the ratio of the statistical averages of ~ (b~/bz) and fl (bS/bz). In the present model where Fs and AS decrease with time, Ks must also decrease, and it will be of interest in the future to measure the resulting change in the transfer coefficient. We can also relate Ks at a general time to Hm and the thickness h of the salt finger interface between two convecting layers. From (3) and (11) it follows that Hm/h is an alternative measure of Ks~Ks. This too is a maximum (and independent of AS and the salt flux) when the convecting layers are first formed, and decreases like (AS)~ if h increases and Hm stays fixed. 3.3. Experimental estimates of the parameters The assumptions involved in arriving at criterion (9) are in qualitative agreement with our observations, and we can now use the values quoted in section 3.1 to obtain an upper bound on B. If it is assumed for the moment that AS refers to the total step of S-stuff applied at the top, these observations imply that stable salt fingers occur when the factor replacing B in (9) was about 60, and that the fingers had definitely broken down when it reached 120. The implied critical value of B ~ 102 is likely to be an overestimate, since the second convective layer formed when the (visually estimated--see the photographs of Fig. 4) concentration of the first convective layer was only a few tenths of the value necessary for the first instability. Another estimate of B can be obtained from (11). The observed layer thickness was Hm ~ 10cm when ~3~/3z ---- 5 × l0 -a and substitution in (11) gives B _ 15.
508
MELVINE. STERNand J. STEWARTTURNER
Other estimates, made using observed h values and theoretical assumptions, fall within the range given above. Neither this value nor the presumed relationship between Hm/h and Ks[Ks can be taken over to the oceanic case, of course. But with laboratory experimentation using other solutes we may learn how to make a reasonable extrapolation. 4. THE DOUBLE GRADIENT SYSTEM 4.1. Observations Using a third type of experiment with sugar and salt we have observed an elusive effect which would be impossible to realize in the laboratory using heat and salt. By filling one bucket with sugar and one bucket with salt and using the mixing technique described in Section 3.1, one can establish a stratification with S-stuff increasing linearly from the bottom to the top of the container and stabilizing T-stuff decreasing linearly. Typically we used ~ 3~/bz -- 3 × 10-3 cm -1 and a gradient of S-stuff which resulted in a stable density gradient less than one tenth of this figure. The rectangular tank of cross section 15 cm × 22 cm was filled to a depth of 40 cm in about 30 min. Vertical salt fingers were always observed and if the gradients were set up carefully with a uniform rate of inflow then the fingers could be seen from top to bottom of the tank and these remained undisturbed. However if the filling was carried out more rapidly, or if the rate was deliberately increased during the latter half of the filling process the disturbance so caused could grow and lead to a striking change in the state of the convection (Fig. 6). The aluminium powder in this photograph shows a well-developed convection layer which had previously formed in the centre of the gradient region. The horizontal boundaries of this region are well marked and the salt fingers (not visible in the photo) appear outside these boundaries. The most remarkable thing about these convective layers is that, although we had difficulty in finding a controllable procedure for setting them up, the convective region, once it did occur, had a very great stability. In one case of careful filling we observed no convective layer and then tried to induce one by rather strong mechanical stirring in one spot. The result was negative since the system went back to vertical salt fingers. In another case where the convecting region formed after a rather weak and uncontrolled perturbation we tried to reverse the direction of the convection cell by stirring mechanically. A short time after this large perturbation the state of the system was indistinguishable from its prior state. From observations on time lapse movies we are convinced that we are not dealing with an artifact, but that we are on the parametric margin of a qualitative effect which is important in the thermohaline convective regime. 4.2. A theory for the growth of a layer in a double gradient Suppose that a well mixed convecting layer of depth H has been formed somewhere in the gradient, giving the distribution of S shown in Fig. 7. A necessary condition for the increase of this H with time can be found by comparing the relative magnitude of the finger flux above the layer with the convective flux within the layer. If the latter is larger, then the difference must be made up by the layer growing into the gradient region. At the time of instability of the salt fingers initially in the region H, the salt flux is of order
Salt fingersand convectinglayers ~FsO ~ .~ ~--~ ~
~--~
509 (12a)
according to the criterion (2) and the experimental condition that fl 3S[bz is only slightly less than ~ ~q~l~,z. After the onset of instability the flux in the H-region changes to flFs' = C (flAS)4,'3 (12b) where AS ~ ½ H b~ (13) 3z is the order of the salinity step in passing from the top of the H layer, through the interface, and into the gradient region. The salt fingers at the top of the interface or bottom of the gradient region are marginally stable so the flux there is still given by (12a). Consequently if Fs' > Fs ° the upper interface will move upwards. From (12a) (12b) and (13) we see that this is possible only if H exceeds a critical value of order H e ~ v ~ C --~ (fi 3,~/3z) -i.
(14)
7/ / // /
H
///
/
/
..1
..
Fig. 7. A sketch of the S distribution produced when a convectinglayer forms in a gradient. Using the experimental values we estimate that H e is between 1 and 10 cm. It is tempting to use this result to rationalize the experimental difficulties in initiating the convective layer. But it is wiser to admit that both the experimental and theoretical problems are exceptionally difficult to come to grips with. At a large time after the initial instability, AS increases and the convective flux is much larger than the finger flux. The rate of increase of the thickness is then given by flAS d (½H)/dt = fl Fs = C (flAS) 413. Using (13) and solving for H gives H = 2 (~- C)~ ( f l ~bsY' z]
t,.
(15)
510
MELVIN E. STERN and J. STEWART TURNER
If typical values are inserted from our laboratory experiment, we find that the time required to reach a depth of 2 cm will be about 20 min so the growth rate on this argument is slow enough to account for the metastability. 5.
SUMMARY AND D I S C U S S I O N
Now let us summarize the various results which, though they have been obtained using the more convenient sugar-salt system, seem to have a wider validity. They should be applicable at least in a qualitative way to the well mixed layers recorded in the ocean; only the values of certain numerical factors will differ when the molecular properties are changed. Our observations suggest that a 'salt finger' interface, however it has been formed, will have an equilibrium thickness which is determined mainly by the salinity difference across it (with a weaker dependence on the temperature difference). If it is thickened by some external process, such as mixing due to shear, then the interface will be sharpened again by the convective stirring in the deeper layers on each side. As the salinity difference decreases due to the transfer of salt through it, the interface will gradually spread out according to (3). In the equilibrium state the flux is the same as that through the deeper convecting layers on each side, varying with the 4/3 power of the salinity difference [equation (1)], and the temperature gradient through the interface is proportional to the flux (2). When a linear stable temperature gradient has a salinity step imposed at the top, it will become unstable if the step exceeds a certain minimum value (9) in relation to the gradient. A convecting layer is formed, which grows by incorporating fluid from below until the new salinity step reaches the critical magnitude. The maximum depth Hm of such a convecting layer, which is achieved at the time another forms below it, depends only on the temperature gradient and the molecular properties (11). At the time of formation of a second layer the ratio of the layer to the interface thickness Hm/h is a function only of the molecular properties. It also follows that at later times this ratio is a measure of the ' mean eddy transport coefficient' Ks. Our conclusions about the system set up with two opposing gradients are much less definite. All we can yet say for certain is that this is unstable to finite amplitude disturbances, and that a convecting layer once formed can grow at the expense of the finger regions above and below it. We have no theory and can make no quantitative predictions regarding the dynamics of the instability. We must emphasize again that the main results of this paper are the qualitative ones: we have shown by direct experiment that phenomena about which there has been some previous speculation do actually occur, and that they can be investigated conveniently in the laboratory with the sugar-salt system. Much more work is needed, using substances of different diffusivity and where possible also salt and heat, to explore the dependence of all the theoretical relations on the molecular properties, and to make direct evaluations of the various numerical factors. There are nevertheless some things that can be said now about the related phenomena in the ocean. The application of the above qualitative conclusions is obvious, and we will just emphasize the possibility that the comparison of the interface and layer thicknesses, with a knowledge of the salinity and temperature steps, could give a direct measure of the vertical fluxes. We can also tentatively apply the relations in a quantitative way to typical oceanic data, to see if they already produce reasonable
Salt fingers and convecting layers
511
numbers. The parameter C has previously been estimated in the laboratory for the salt-heat system (see Section 2.4); this is clearly a function of the molecular properties though its explicit form remains to be investigated. The other factors appearing in the suggested theoretical relations can also in principle be functions of the molecular properties, but as a tentative assumption let us suppose that they can be held constant from one system to another. Consider each of the testable relations in turn. It has already been shown (TURNER, 1967; TAIT and HOWE, 1968) that the application of (1) to the step structure under the Mediterranean outflow gives a reasonable value of Ks ~ 5 cm2/s. This calculation however was based on the directly measured layer thickness, and on the salt-heat value of C measured in the laboratory, so it gives no indication of the possible variation due to the other multiplying factors which could depend on the molecular coefficients. Using the same data in (3) gives an interface thickness h of a few centimeters, and in (11) (with B = 15, as suggested by the sugar-salt experiments) it predicts that Hm should be 10 cm. It is possible that the interface thickness on formation could be as small as this but the layer depth is definitely wrong! Finally, applying (14) to the heat-salt problem with fl b~/~z := 10-s (a typical oceanic value) one obtains a value of Hc of order 10 cm. With the same assumption, (15) predicts a layer depth of H = 4 m after a time interval of one day. We conclude, in confirmation of the warnings given earlier, that the same numerical values (of B as well as C) should not be carried over from the sugar-salt experiments to the salt-heat phenomena observed in the ocean. Though the relations involving C alone seem to give plausible numbers, those involving B (for which we have no estimate in the heat-salt system) certainly do not, and each relation must be carefully investigated before it is applied quantitatively. The most likely reason for the differences seems to lie in the unknown dependence on Ks which has been neglected entirely here, but which must surely enter when Ks becomes comparable in magnitude to
Kr[ ' .
AcknowledgementsIThe experimental work described here was carried out while both of us were visiting the Woods Hole Oceanographic Institution, and this paper is Contribution 2282 from that Institution. During this time the first author received partial support from Office of Naval Research grant 98-460 (with the University of Rhode Island) and the Geophysical Fluid Dynamics Summer Program (sponsored by National Science Foundation). The second author is supported by a grant from the British Admiralty, and his visit to Woods Hole was sponsored by the Officeof Naval Research through Contract No. CO 241. We are grateful to Mr. ROBERTE. FRAZELfor his assistance in setting up and carrying out these experiments. REFERENCES
BAINES W. D. and J. S. TURNER (1969) Turbulent buoyant convection from a source in a confined region. J. fluid Mech., 37, 51-80. COOPER J. W. and H. STOMMEL(1968) Regularly spaced steps in the main thermocline near Bermuda. J. geophys. Res., 73, 5849-5854. OSTER G. (1965) Density gradients. Scient. Am., 213, 70--76. STERN M. E. (1960) The "salt-fountain" and thermohaline convection. Tellus., 12, 172-175. STERN M. E. (1968) T-S gradients on the micro-scales. Deep-Sea Res., 15, 245-250. STERN M. E. (1969) Collective instability of salt fingers. J. fluid Mech., 35, 209-218. TAIT R. I. and M. R. HOWE (1968) Some observations of thermohaline stratification in the deep ocean. Deep-Sea Res., 15, 275-280. TURNER J. S. 0967) Salt fingers across a density interface. Deep-Sea Res., 14, 599-611. TURNER J. S. (1968) The behaviour of a stable salinity gradient heated from below. J. fluid Mech., 33, 183-200.
Salt fingers and conveeting layers MELVIN E. STERN* a n d J. STEWART TURNERt
(Received 10 March 1969) Abstract--It is shown that convective ' fingers ' can form not only in the salt-hcat system, but also in a fluid containing two solutes with much closer diffusivities (for example sugar above a salt solution). This experimental device is used to explore various phenomena which cannot easily be produced in the laboratory with salt and heat. We have investigated: (a) the behaviour of an interface containing fingers and separating two convecting layers, (b) the production of a series of steps from a smooth gradient by imposing a flux at the top, (c) and the mechanism of formation of layers from two opposing gradients of the solutes. In each case the related theoretical ideas are summarized, and it is suggested how these can be extended to describe the new features of the observations.
1. INTRODUCTION MANY different pieces of evidence now support the view that a deep region of fluid containing opposing vertical gradients of temperature and salinity does not remain smoothly stratified, but instead breaks up into a series of steps. In the ' salt finger' regime, where temperature is stabilizing and salinity destabilizing, several oceanographic observations have been attributed to this effect. TAIT and HOWE (1968) recorded steps underneath the Mediterranean outflow, with well mixed-layers separated by interfaces which were at times too sharp to resolve with the technique they used. COOPER and STOMMEL(1968) found a stepped structure in the main thermocline near Bermuda, but with the well mixed and gradient regions more nearly comparable in depth. Some understanding of the physical processes governing this behaviour has already been obtained theoretically and through laboratory experiments. STERN (1969) (referred to as I below) has shown that a field of finite amplitude salt fingers becomes unstable above a critical value of the salt flux, for a given temperature gradient. Energy is fed into relatively long internal waves, leading eventually to a local overturning, with complete mixing in regions of large shear and quasi-laminar salt fingers retained in thin regions of small vertical shear. He has also made predictions about the dependence of the thickness of the layers, and the fluxes of heat and salt between them, on the molecular properties. TURNER (1967) (called II in the present paper) has studied experimentally the transports across an interface containing salt fingers, and has shown how convection can be maintained in the deep layers on each side of the interface by the net unstable buoyancy flux through it. He has also speculated about the formation of layers from a smooth stable temperature gradient, when salt is added to the top. The two papers just cited contain suggestions which must be developed further * Graduate School of Oceanography, University of Rhode Island. t Department of Applied Mathematics and Theoretical Physics, University of Cambridge.
497
498
MELVIN E. STERN and J. STEWARTTURNER
and tested in more detail before they can be applied to the ocean. Experimental difficulties have up till now prevented us from doing this; the main problems arise from the need to maintain a large stable temperature gradient over a deep enough layer, while avoiding spurious effects due to side wall heating or cooling. In the present paper we will describe a new technique which eliminates many of these difficulties. We show that similar 'finger convection' phenomena can readily be produced in a system containing two solutes, for example sugar and salt, which have diffusivities much closer together than the 100:1 ratio for heat and salt. There are therefore no side wall losses, and there is the added advantage that much larger density gradients can be used. We will first describe in qualitative terms the various phenomena observed, many of which are completely inaccessible to laboratory study using the heat-salt system. Secondly, we will set down a summary of certain theoretical ideas which seem consistent with these observations. We deduce some preliminary numerical values of important parameters, but a detailed quantitative study of the dependence on molecular properties remains to be carried out in the future. 2.
THE TWO LAYER SYSTEM
2.1. Previous work and some immediate deductions The simplest situation is obtained by placing hot water above cold water, sharpening the interface by mechanical stirring above and below and then adding some salt solution to the top layer, as described in II. When stirring is stopped salt fingers quickly develop across the interface and seem to be limited in extent by the large scale convective motion which they drive in the deep layers. These measurements supported the relation suggested by a dimensional argument, namely, that the salt flux (p0 Fs gm/cm2/sec) or more conveniently the associated buoyancy flux flFs (cm/sec) has the following dependence on the parameters: flFs : (flAS)4/a (gv)~f~-AS,
= C(flAS) 4j~.
(1)
In the foregoing p0 is the mean water density, AS is the salinity excess and AT is the temperature excess of the upper layer, ~ A T - flAS is the density deficit, g is the acceleration of gravity, v is the kinematic viscosity, ~T is the thermal diffusivity, ~s is the salt diffusivity, a n d f i s an incompletely measured universal function. The measured value of the constant on the right of (1) was about C = 0.1 era/see when ~AT/flAS was about 2, but C decreases only slowly with more stable values of ctAT/flAS. The most important implication of the observed 4/3 power law is that there is a quasi-equilibrium value of the salt flux which is independent of any length scale, either the thickness of the convecting layers or the initial thickness of the interface separating the two fluids. The latter is a quantity which approaches an equilibrium thickness (h) determined by AS and the molecular coefficients [see Section (2.3) for a direct test of this idea]. An explanation of this dependence has been given in I; it was shown there that stable quasi-laminar salt fingers can occur in a region only if the salt flux is less than a critical value given by
vo~~ l ~ z ~- Ncritical "~ !.
(2)
The critical value of the non-dimensional number in (2) is of order unity if the ratio
Salt fingers and convecting layers
499
of the mean temperature and salinity gradients (i.e. s3f'13z (fl~Sl3z) -1) is of order unity and if v/Kr and/or v/Ks are large compared to unity. We can now extend these results by applying (2) to the interface region (containing salt fingers) between the two convecting layers, and assuming that the fingers at the edges of the interface are always close to the critical state. Setting 3~/3z ~ AT/h in (2) and using (1) gives
h~
vsAT v ~Fs ~'~ C ~ A S ) ~ '
(3)
If H denotes the thickness of the ' deep ' convccting layers (assumed to be the same) then in the course of time (t) the salinity difference between the layers will decrease at the rate d (flAS)/dt = -- 2fl Fs/H = -- 2C (fiAS) 4/3 H -1, or
2Ct (flAS)-~ = 3-H ÷ constant.
(4)
Notice that in the slow decay, the system passes through a succession of quasi equilibrium states with
dh/d(AS) dt/
dt
independent of time. The increase of the thickness of the region containing salt fingers is a striking feature of the middle and final periods of decay in all the experiments reported below. In addition (4) will be examined quantitatively in one experiment using sugar and salt. Previous measurements have also been made (ll) of the buoyancy flux ratio R~ ~
aFT 5Fs
(5)
as a function of the density anomaly ratio sAT Rp--: 5XS
(6)
where FT is the horizontal average of the product of vertical velocity and temperature. On energetic grounds RI < 1 and the experiments (I1) gaveRf - 0.56 over a wide range of R o in the heat-salt experiment. 2.2. The dam break method The usefulness of the grid-stirring technique for producing the initial sharp interface was limited by the small range of density differences which can be achieved by heating, of order sAT ~,~ ½~ with AT ~,~ 20°C. With such small density differences it was difficult to set up an experiment with sAT,/f3AS less than 2, and so another technique was devised to reach this range. A vertical watertight barrier was inserted at the centre of a channel about 10 cm wide, 15 cm deep and 150 em long. One half of the channel was filled with hot s/dt water, and the other with cold fresh water, with densities and levels adjusted to be very nearly equal. When the barrier was withdrawn the slightly lighter fluid flowed above and across the denser half, producing either the salt finger regime or the inverse case (heating a stable salt gradient from below), depending upon the exact density ratio of the two fluids in the partitioned channel. Little quantitative work has been done using this ' dam break ' technique, but a movie
500
MELVIN ]~. STERN and J. STEWARTTURNER
was made which dearly illustrated the various possible types of behaviour. When the hot salty water is slightly light, salt fingers were observed immediately after the partition was removed, even though there was considerable shear due to the counter flows and to the internal waves reflected from the ends of the channel. The latter soon subside and the system reaches a state of vertical convection similar to that obtained with the grid-stirring technique. The system was allowed to run down towards a state of vertically uniform salinity. As this stable state was approached the temperature difference fell to about one-half its initial value, which is what one would expect if Rf -- ½. When the possibility of using two different solutes at room temperature became apparent, our first experiment was in this same long channel, using the dam break technique. In a typical run common salt solution with specific gravity about 1.050 and diffusivity ~ -- 1.5 × 10 -5 cm2/s was put on one side of the barrier, and sugar solution with specific gravity 1.045 and K =: 0.5 × 10 -5 cm2/s was put on the other side. After the barrier was withdrawn the sugar solution flowed over the salt solution, with fine fingers falling out. The internal waves die out rapidly and one clearly sees a thin horizontal interfacial layer with surprisingly strong large scale convection in the layers on either side. This system with low diffusivity sugar on top of higher diffusivity salt is analogous to the ordinary salt finger convection in the heat system.* The thickness of the interfacial transition region was about 1-2 mm, a few minutes after the start. Optical manifestations of the fine fingers (due to the variations of index of refraction) were easily visible. These quasi-steady structures were sharply limited, above and below, by the convective regions. Over a much longer time interval the interface region gradually thickened and the intensity of the convection diminished. Sometimes strong horizontal motions were observed in the convective regions, persisting for an hour or more. Section (2.4) describes some measurements of density flux and colorimetric estimates of sugar flux made in this long channel. 2.3. A more direct technique Among the other advantages of the sugar-salt system is the fact that one can work with much larger density differences, typically 10-30 times that achieved with heat and salt. Since it is relatively easy to set up the two solute system with a ' densityratio ' close to unity the dam break technique is not really necessary. Instead, sugar solution can be poured carefully on a thin piece of wood floating on the salt layer, and this was the method used for setting up most of the later runs in a deeper narrower tank. This is a more convenient geometry for most purposes, including photography. Many of the previously mentioned effects were confirmed using this method and studied further in a more controlled way. To examine the evolution of the interfacial region we placed an intermediate layer between a deep upper layer of sugar and a deep bottom layer of salt. The concentration of sugar and salt in this moderately thick mid-layer was also intermediate between that of the deeper layers. Quasi-laminar salt fingers less than 1 m m thick form in the intermediate layer; it is difficult to photograph these directly but they are always clear to the eye. The convection on either side of the interface sharpens the edge of * With a slightly lighter salt solution riding over the sugar solution we have the case analogous to heating a stable concentration gradient from below. Vertical exchange was then observed to take place in the form of fluid sheets which formed intermittently at convergence zones on a paper-thin interface.
Fig. 1. Photograph of a thick interface containing fingers between two well mixed convecting layers. The sugar (upper) layer is dyed, and the tank is lit from below. Note the sharp edgesof the inter facial layer. [facing p.
500]
0,~
q:~
.._~
~
~
~
0 ~..-,
>
e-,
~
=o.~
E
.~.~ ~ ~
"-g g ?Z~ ~ . ~
,.~"~
. ~,~
~
.
~'~ ~
Fig. 6. A photograph of a convecting layer formed from two opposing gradients, made visible with aluminium powder. The pipe used for filling the tank from the bottom is visible in one corner next to the scale.
Salt fingers and convecting layers
501
that region as shown in the photograph of Fig. 1. The thickness of the interface decreases with time in the early stages due to the erosion by the relatively strong convection in the layers. In the final stages the intensity of convection diminishes and the interface region broadens. Time lapse movies showed that the convection in the thick layers has an intermittent character, with buoyant elements (generated by the salt fingers) accumulating and moving through the entire layer. These elements retain their identity as they penetrate to the extreme boundaries of the apparatus, and (as suggested by BMNES and TURNER (1969) in another context) they could in this way produce reversals in the sugar and salt gradients. We have not yet made any gradient measurements to check this impression. They would be of interest in connection with oceanic profiles [TA1T and HOWE (1968), and some unpublished results of these authors] exhibiting reversals or inversions in so called ' well mixed layers.' Qualitative experiments were also carried out using other pairs of diffusing substances. It is especially interesting to note that a moderately concentrated solution of NaC1 (~ 1-5 × 10-5 cm2/sec) carefully placed on a slightly heavier KC1 solution (K:= 1.9 : 10.'5 cm2/sec) will produce salt fingers and convection even though the diffusivities are so close. 2.4. Preliminary tests of theoretical ideas To avoid confusion with terminology and symbolism in past heat/salt experiments and present salt/sugar experiments we adopt the following convention. We call ' T - s t u f f ' the substance with the larger diffusivity and " S-stuff" the substance with the smaller diffusivity. Some quantitative measurements of the Fs, FT fluxes were made using the dam break technique with sugar (S-stuff) and salt (T-stuff). These provide additional support for equations (1) and (3) and information on C which is necessary for section (3.2).
Table 1. The measured sugar concentrations and density differences during one run (Run 8, 30 July 1968), and the deduced density ratio Rp and flux ratio R~ as functions of time. Change in Changein Time (fl~S)% A,% Ro (ftAS)% 40°//0 Rt 9.33 a.m. 1.0 × 4-56 0'21 1-047 0"2 × 4-56 0'08 0'91 9.37 0.8 × 4'56 0.29 1.080 0"2 × 4"56 0-08 0.91 9.42 0.6 × 4'56 0.37 1"135 0-2 × 4'56 0.07 0.92 9.47 0"4 × 4'56 0-44 1'24 0.2 ~." 4.56 0.08 0"91 9.58 0.2 × 4"56 0.52 1"57 Table 1 gives the density and S-stuff concentration as a function of time in a two layer system with initial specific gravities of 1-0456 and 1.0477, respectively. Each of the layers, set up by the dam break method, was 4.4 cm deep and the top layer also had a dilute dye (of the kind used for colouring food) in solution. A colorimetric technique was devised to compare the dye concentrations at any time against samples diluted to known concentrations, using sample containers with the same thickness as the tank viewed against the same uniformly illuminated background.* The time was *The assumption has been made that the dye is transported in the same manner as sugar but a more careful study of multi-component systems would be needed to check this.
502
MELVINE. SXERNand J. SIEWARITURNER
recorded when dye concentrations of 10 ~ , 20 ~ , etc. were reached in the b o t t o m layer. At these same times samples were withdrawn for later weighing, a n d the interface thickness measured. In this way we could measure the density flux and F,s in the intervals between measurements, and Table 1 shows the c o m p u t e d flux ratio R I = ~ Ftt/fl F s a n d the density ratio R o -- ~AT/fl2xS at each stage. R o increases slowly f r o m 1.05 to 1.57; it is always considerably lower than in any experiment carried out with heat and salt. One might expect that the ' final state ' of the system will be Rp = KT/KS ~, 3, since this is the state of marginal stability for salt fingers (STERN (1960)). It w o u l d have been impossible to distinguish visually between such a distribution and complete uniformity o f the dye (or S-stuff). I
I I
6-
O
\
o .El
4O
~O \
E
u
Y_ En
\ O
2-
I
"1"
0 4.0
z,.5
5"0
Density difference (% heavier than fresh water) Fig. 2. Density distributions measured in the sugar-salt finger experiment described in Table 1. (a) Shows the original two layer system set up using the "dam break" technique; and (b) is the profile made five hours later by withdrawing samples at 1 cm intervals and weighing them. RI is constant at 0,91, to within the experimental error, over the range covered. This value (which is supported by our other qualitative runs) is larger than the value (0.56) found for heat and salt, implying that a smaller fraction of the potential energy released by the sugar convection is used in viscous dissipation and a larger fraction is used to raise the center of gravity o f the T-stuff. This is made clear too by Fig. 2 in which the initial and final (i.e. measured after five hours) density distributions are plotted. The density difference increased in time, but by only a small fraction of the separate contributions to the density differences.* * In connection with the observed increase of ~ Fn/flFs with KS/KT'we point out that on general thermodynamic grounds ~ Fn/flFs -+ 1 as ,~s --> ,¢T with Ro ~ 1. This is because the two dissipations ,or (VT) 2 x2and xs (VS)~/~2must be of equal average value. The production of ~T1and/3S variances must then be equal and it then follows that the vertical fluxes are equal. See SXERN(1968) for elaboration.
Salt fingersand convectinglayers
503
In Fig. 3, (AS) -t and h (which has not been included in the table because it was not measured at the same times) are plotted against time. The points can be reasonably fitted by straight lines, supporting the predicted forms (4) and (3). The slope of the line drawn through the concentration measurements allows us to make the estimate
J
I
2'0
10
/ o J
!
(As)
E 1,5
c-
/f
-
5
to to
1.0
9.40
9.30
Time
9. 50
10.00
Fig. 3. The measured values of (AS)--t (taking the initial sugar concentration difference as unity) and the finger interface thickness h, plotted against time for the experiment set out in Table 1.
C = 10-2 cm/s, a factor of ten smaller than the previously obtained value of C for heat and salt. Note that this change in C with molecular coefficients is not inconsistent with the idea that the ratio (2) is independent of the molecular coefficients. The observed h seems to be predicted by (3) to better than an order of magnitude both in these experiments with sugar and salt and those reported in II using salt and heat. It should be kept in mind, however that theoretically there is still some uncertainty about whether to use the vertical density gradient rather than ~ ~/~z in (2), and whether the whole temperature or density step or only part of it should be used in estimating the mean gradient through the interface region. Questions such as these can only be answered by a more detailed study. We emphasize again the exploratory nature of the present paper, in which we aim to describe a range of phenomena which must be important for thermohaline convection, and do not pretend to have solved all the problems raised by these experiments. 3.
THE FORMATION OF LAYERS FROM A GRADIENT
3.1. Laboratory observations In all previous salt finger experiments the scale of the convective layers has been determined in advance by setting up two or more homogeneous layers. In this section we shall show how layers (or steps) separated by fingers can form naturally on a stable
504
MELVINE: STERNand J. STEWARTTURNER
linear gradient when a downward Fs-flux is applied at the top. This is probably the most important result reported in this paper. Uniform gradients (of dT/dz) were set up using the method described by OSTER (1965) and used by TURNER (1968) in the inverse problem (viz., heating a stable salt gradient from below and studying the resulting layer formation.) This method for producing linear solute stratification uses two identical buckets containing fluid with the concentration that is desired at the top and bottom of the stratification. The buckets are joined by a siphon, the bucket containing the lighter fluid is stirred, and the mixture therein is led into the bottom of the tank in which the stratified fluid is to be formed. After a stratification of T-stuff has been produced in this way a uniform layer of S-stuff, with a density slightly less than the topmost T-stuff, is then carefully poured on top. Complex finger motions are observed in this initial stage of mechanical mixing. When the gradient ~ b~/~z was relatively large (typically 4 × 10-:~ c m 4 with flAS = 10~o) all horizontal motion in the gradient region rapidly disappears and one sees vertical fingers, which persist for a long time, extending all the way to the bottom of the container. However, with a smaller stabilizing gradient and a comparable concentration of S-stuff (c~ b~/~z - 1 "~ 10-acm -1 with pAS = 7~o), we observed that the quasi-laminar salt fingers, just below the imposed S-layer, gave way to a new (first) convective layer. The salt fingers seem to become unstable with respect to larger scale convection over a definke vertical extent. The first convective layer, so formed, deepens somewhat but it is always bounded below by a vertical finger regime. When the stabilizing gradient was further reduced (~. bT/~z -- 5 /. 10`4 cm- 1, flAS ~ 10~) a first convecting layer again formed but somewhat later a second convecting layer also formed, beneath the first one. Again, quasi-laminar salt fingers are found beneath the second layer. F'igures 4a and 4b show different stages of this process of layer formation. We did not try to produce more than two layers but fecl confident that this could be done by suitable variation of experimental conditions. Tracer observations by eye and time lapse movies reveal the behaviour of the fingers as a new layer is formed. Intermittent sideways oscillations of the fingers arc followed by stronger horizontal motions until large scale vertical convection is established in the region. Several runs were followed for a long time (up to a week in one case) and we could then observe the decay of the convecting layers. As the concentration differences across the interfacial layer become smaller with time, the thickness of the interfacial layer and the salt fingers therein increase at the expense of the convecting layers [cf. Section 2.1 and equation (3)]. After three days the first (upper) convecting layer had disappeared completely, while the second one was still in existence, bounded on either side by long fingers extending throughout the 1 meter deep cylindrical container. A photograph of this state, taken a few seconds after dye crystals had been dropped through the tank, is shown in Fig. 4c. Note the distortion of the vertical streaks in the convective layers and their relatively undisturbed character in the finger regions. When no further motion was detectable, measured density profiles still showed the kinks in the original linear profile which had been produced and left behind by the convecting layers. Figure 5 contrasts density measurements made before and after runs in which convecting layers formed (5a), or did not form (5b). In both cases there is a general increase in density through the original gradient region (except right at the bottom), and a decrease of density in the well-mixed layer at the surface.
Salt fingers and cortvecting layers
I
I
I
505
I
@
0
3
_1 --0
I
I
I
~-
._
-
I
~
I
.~
~_~ ~ o
'g ~
o._U
0 m
-
~
I
&,o
~
0
I
I
iO
_
0
I
(uao~oq ~^oqo ua~) lqS!~H
I
~o
0
506
MELVINE. STERNand J. STEWARTTURNER
3.2. Theories o f layer formation We have to explain why salt fingers are realized in the experiments described when ~ l b z is large and why they break down into a convecting layer at some critical value of the gradient. In the former case the order of magnitude of the vertical flux produced by a finger regime may be obtained by multiplying their typical vertical velocity w by the typical value of the horizontal variation of S (or T) in the fingers. Considering the formation of fingers at the top of the initial gradient region it seems reasonable to suppose that the horizontal variation of S is proportional to the vertical step AS. Moreover the viscous and buoyancy forces are of the same order in the fingers and therefore vwL -2 ~ g f l A S
where
L ,---
is the width of the salt finger (see I for an elaboration of these ideas, which are only given in outline here). It follows that the buoyancy flux associated with the S-stuff is /7Fs ~ w f l A S ~ , (flAS)2 o¢ bz ] Equation 7 is not inconsistent with t h e " 4/3-power law." A more general dimensional argument leads to a relationship of the form (7) bm with an extra dependence on Ks i.e. it predicts that the salt finger flux should be proportional to g~ (flAS)2 (~ 5~l~z) - i with an unknown co-efficient of proportionality that depends on V[KT and v/Ks. Now according to eq. (2) these salt fingers will become unstable when Fs is greater than some critical value of order a~ 3Fs ~ vo¢ - - . (8) 3z Combining the two preceding equations leads to (/7AS) ---- B vt (g KT)-i (0¢ a'/~i3Z)t,
(9)
where
is the unknown proportionality constant. Equation (9) gives the critical value of AS as a function of ~ / ~ z for the formation of a new convecting layer within the gradient region. When this convecting layer forms, it will be bounded above by a thin layer or interface (of thickness h) containing salt fingers. The T gradient in the h-layer must be much larger than ?~/~z, and the flux through this interface will be described by (1). The first convecting layer to form was observed to grow, and approach a maximum (equilibrium) thickness//m before a second layer formed below it. We can obtain an expression for Hm if AS in (9) is now interpreted as the S-step between this first convecting layer and the fluid below. The corresponding difference in T-stuff is of order Hm ~ / ~ z , assuming that the convecting layer is well-mixed. The net density step at this time must be small (see TURNER, 1968), SO f l A S ~_ Hm ~ 3z
(10)
Salt fingers and convecting layers
507
where ~ denotes equality within a factor of two (say). Combining (9) and (10)we obtain the estimate
Hm ~ B v~ (g xT) -i (~ 3~/bZ)-~ _ B4/3 ~ (g K~)-~ (/3AS)-~
(1])
for the equilibrium thickness of the first convecting layer. The first form of (11) shows that Hm will depend weakly on the original T-gradient and on the molecular properties, but not at all on the way in which the destabilizing flux was originally applied. It is consistent with the prediction made in I (which was based on quite a different detailed model) that the layer thickness should be proportional to (3q~/3z)-~. Moreover from (1) and (10) we get
Hm ~- C -~ Fs ~ (~ ~ / b z ) -1 which has a form similar to the criterion found by TURNER (1968) for layers formed by heating a stable salt gradient at a constant rate from below. This general property of independence of the formative mechanism would also seem to be important if we are ever to relate our laboratory models to layers in the ocean, such as those observed under the Mediterranean Outflow. If we define a mean transfer coefficient by
Ks = flFs \ nra] then we see from (1) and the second equation in (11) that Ks (at the time of formation of a new layer and interface) is independent of flAS i.e. is a function only of the molecular properties. This is also in accord with the completely different model in I, where the transfer coefficient was found to depend only on V/KT, V/KS and the ratio of the statistical averages of ~ (b~/bz) and fl (bS/bz). In the present model where Fs and AS decrease with time, Ks must also decrease, and it will be of interest in the future to measure the resulting change in the transfer coefficient. We can also relate Ks at a general time to Hm and the thickness h of the salt finger interface between two convecting layers. From (3) and (11) it follows that Hm/h is an alternative measure of Ks~Ks. This too is a maximum (and independent of AS and the salt flux) when the convecting layers are first formed, and decreases like (AS)~ if h increases and Hm stays fixed. 3.3. Experimental estimates of the parameters The assumptions involved in arriving at criterion (9) are in qualitative agreement with our observations, and we can now use the values quoted in section 3.1 to obtain an upper bound on B. If it is assumed for the moment that AS refers to the total step of S-stuff applied at the top, these observations imply that stable salt fingers occur when the factor replacing B in (9) was about 60, and that the fingers had definitely broken down when it reached 120. The implied critical value of B ~ 102 is likely to be an overestimate, since the second convective layer formed when the (visually estimated--see the photographs of Fig. 4) concentration of the first convective layer was only a few tenths of the value necessary for the first instability. Another estimate of B can be obtained from (11). The observed layer thickness was Hm ~ 10cm when ~3~/3z ---- 5 × l0 -a and substitution in (11) gives B _ 15.
508
MELVINE. STERNand J. STEWARTTURNER
Other estimates, made using observed h values and theoretical assumptions, fall within the range given above. Neither this value nor the presumed relationship between Hm/h and Ks[Ks can be taken over to the oceanic case, of course. But with laboratory experimentation using other solutes we may learn how to make a reasonable extrapolation. 4. THE DOUBLE GRADIENT SYSTEM 4.1. Observations Using a third type of experiment with sugar and salt we have observed an elusive effect which would be impossible to realize in the laboratory using heat and salt. By filling one bucket with sugar and one bucket with salt and using the mixing technique described in Section 3.1, one can establish a stratification with S-stuff increasing linearly from the bottom to the top of the container and stabilizing T-stuff decreasing linearly. Typically we used ~ 3~/bz -- 3 × 10-3 cm -1 and a gradient of S-stuff which resulted in a stable density gradient less than one tenth of this figure. The rectangular tank of cross section 15 cm × 22 cm was filled to a depth of 40 cm in about 30 min. Vertical salt fingers were always observed and if the gradients were set up carefully with a uniform rate of inflow then the fingers could be seen from top to bottom of the tank and these remained undisturbed. However if the filling was carried out more rapidly, or if the rate was deliberately increased during the latter half of the filling process the disturbance so caused could grow and lead to a striking change in the state of the convection (Fig. 6). The aluminium powder in this photograph shows a well-developed convection layer which had previously formed in the centre of the gradient region. The horizontal boundaries of this region are well marked and the salt fingers (not visible in the photo) appear outside these boundaries. The most remarkable thing about these convective layers is that, although we had difficulty in finding a controllable procedure for setting them up, the convective region, once it did occur, had a very great stability. In one case of careful filling we observed no convective layer and then tried to induce one by rather strong mechanical stirring in one spot. The result was negative since the system went back to vertical salt fingers. In another case where the convecting region formed after a rather weak and uncontrolled perturbation we tried to reverse the direction of the convection cell by stirring mechanically. A short time after this large perturbation the state of the system was indistinguishable from its prior state. From observations on time lapse movies we are convinced that we are not dealing with an artifact, but that we are on the parametric margin of a qualitative effect which is important in the thermohaline convective regime. 4.2. A theory for the growth of a layer in a double gradient Suppose that a well mixed convecting layer of depth H has been formed somewhere in the gradient, giving the distribution of S shown in Fig. 7. A necessary condition for the increase of this H with time can be found by comparing the relative magnitude of the finger flux above the layer with the convective flux within the layer. If the latter is larger, then the difference must be made up by the layer growing into the gradient region. At the time of instability of the salt fingers initially in the region H, the salt flux is of order
Salt fingersand convectinglayers ~FsO ~ .~ ~--~ ~
~--~
509 (12a)
according to the criterion (2) and the experimental condition that fl 3S[bz is only slightly less than ~ ~q~l~,z. After the onset of instability the flux in the H-region changes to flFs' = C (flAS)4,'3 (12b) where AS ~ ½ H b~ (13) 3z is the order of the salinity step in passing from the top of the H layer, through the interface, and into the gradient region. The salt fingers at the top of the interface or bottom of the gradient region are marginally stable so the flux there is still given by (12a). Consequently if Fs' > Fs ° the upper interface will move upwards. From (12a) (12b) and (13) we see that this is possible only if H exceeds a critical value of order H e ~ v ~ C --~ (fi 3,~/3z) -i.
(14)
7/ / // /
H
///
/
/
..1
..
Fig. 7. A sketch of the S distribution produced when a convectinglayer forms in a gradient. Using the experimental values we estimate that H e is between 1 and 10 cm. It is tempting to use this result to rationalize the experimental difficulties in initiating the convective layer. But it is wiser to admit that both the experimental and theoretical problems are exceptionally difficult to come to grips with. At a large time after the initial instability, AS increases and the convective flux is much larger than the finger flux. The rate of increase of the thickness is then given by flAS d (½H)/dt = fl Fs = C (flAS) 413. Using (13) and solving for H gives H = 2 (~- C)~ ( f l ~bsY' z]
t,.
(15)
510
MELVIN E. STERN and J. STEWART TURNER
If typical values are inserted from our laboratory experiment, we find that the time required to reach a depth of 2 cm will be about 20 min so the growth rate on this argument is slow enough to account for the metastability. 5.
SUMMARY AND D I S C U S S I O N
Now let us summarize the various results which, though they have been obtained using the more convenient sugar-salt system, seem to have a wider validity. They should be applicable at least in a qualitative way to the well mixed layers recorded in the ocean; only the values of certain numerical factors will differ when the molecular properties are changed. Our observations suggest that a 'salt finger' interface, however it has been formed, will have an equilibrium thickness which is determined mainly by the salinity difference across it (with a weaker dependence on the temperature difference). If it is thickened by some external process, such as mixing due to shear, then the interface will be sharpened again by the convective stirring in the deeper layers on each side. As the salinity difference decreases due to the transfer of salt through it, the interface will gradually spread out according to (3). In the equilibrium state the flux is the same as that through the deeper convecting layers on each side, varying with the 4/3 power of the salinity difference [equation (1)], and the temperature gradient through the interface is proportional to the flux (2). When a linear stable temperature gradient has a salinity step imposed at the top, it will become unstable if the step exceeds a certain minimum value (9) in relation to the gradient. A convecting layer is formed, which grows by incorporating fluid from below until the new salinity step reaches the critical magnitude. The maximum depth Hm of such a convecting layer, which is achieved at the time another forms below it, depends only on the temperature gradient and the molecular properties (11). At the time of formation of a second layer the ratio of the layer to the interface thickness Hm/h is a function only of the molecular properties. It also follows that at later times this ratio is a measure of the ' mean eddy transport coefficient' Ks. Our conclusions about the system set up with two opposing gradients are much less definite. All we can yet say for certain is that this is unstable to finite amplitude disturbances, and that a convecting layer once formed can grow at the expense of the finger regions above and below it. We have no theory and can make no quantitative predictions regarding the dynamics of the instability. We must emphasize again that the main results of this paper are the qualitative ones: we have shown by direct experiment that phenomena about which there has been some previous speculation do actually occur, and that they can be investigated conveniently in the laboratory with the sugar-salt system. Much more work is needed, using substances of different diffusivity and where possible also salt and heat, to explore the dependence of all the theoretical relations on the molecular properties, and to make direct evaluations of the various numerical factors. There are nevertheless some things that can be said now about the related phenomena in the ocean. The application of the above qualitative conclusions is obvious, and we will just emphasize the possibility that the comparison of the interface and layer thicknesses, with a knowledge of the salinity and temperature steps, could give a direct measure of the vertical fluxes. We can also tentatively apply the relations in a quantitative way to typical oceanic data, to see if they already produce reasonable
Salt fingers and convecting layers
511
numbers. The parameter C has previously been estimated in the laboratory for the salt-heat system (see Section 2.4); this is clearly a function of the molecular properties though its explicit form remains to be investigated. The other factors appearing in the suggested theoretical relations can also in principle be functions of the molecular properties, but as a tentative assumption let us suppose that they can be held constant from one system to another. Consider each of the testable relations in turn. It has already been shown (TURNER, 1967; TAIT and HOWE, 1968) that the application of (1) to the step structure under the Mediterranean outflow gives a reasonable value of Ks ~ 5 cm2/s. This calculation however was based on the directly measured layer thickness, and on the salt-heat value of C measured in the laboratory, so it gives no indication of the possible variation due to the other multiplying factors which could depend on the molecular coefficients. Using the same data in (3) gives an interface thickness h of a few centimeters, and in (11) (with B = 15, as suggested by the sugar-salt experiments) it predicts that Hm should be 10 cm. It is possible that the interface thickness on formation could be as small as this but the layer depth is definitely wrong! Finally, applying (14) to the heat-salt problem with fl b~/~z := 10-s (a typical oceanic value) one obtains a value of Hc of order 10 cm. With the same assumption, (15) predicts a layer depth of H = 4 m after a time interval of one day. We conclude, in confirmation of the warnings given earlier, that the same numerical values (of B as well as C) should not be carried over from the sugar-salt experiments to the salt-heat phenomena observed in the ocean. Though the relations involving C alone seem to give plausible numbers, those involving B (for which we have no estimate in the heat-salt system) certainly do not, and each relation must be carefully investigated before it is applied quantitatively. The most likely reason for the differences seems to lie in the unknown dependence on Ks which has been neglected entirely here, but which must surely enter when Ks becomes comparable in magnitude to
Kr[ ' .
AcknowledgementsIThe experimental work described here was carried out while both of us were visiting the Woods Hole Oceanographic Institution, and this paper is Contribution 2282 from that Institution. During this time the first author received partial support from Office of Naval Research grant 98-460 (with the University of Rhode Island) and the Geophysical Fluid Dynamics Summer Program (sponsored by National Science Foundation). The second author is supported by a grant from the British Admiralty, and his visit to Woods Hole was sponsored by the Officeof Naval Research through Contract No. CO 241. We are grateful to Mr. ROBERTE. FRAZELfor his assistance in setting up and carrying out these experiments. REFERENCES
BAINES W. D. and J. S. TURNER (1969) Turbulent buoyant convection from a source in a confined region. J. fluid Mech., 37, 51-80. COOPER J. W. and H. STOMMEL(1968) Regularly spaced steps in the main thermocline near Bermuda. J. geophys. Res., 73, 5849-5854. OSTER G. (1965) Density gradients. Scient. Am., 213, 70--76. STERN M. E. (1960) The "salt-fountain" and thermohaline convection. Tellus., 12, 172-175. STERN M. E. (1968) T-S gradients on the micro-scales. Deep-Sea Res., 15, 245-250. STERN M. E. (1969) Collective instability of salt fingers. J. fluid Mech., 35, 209-218. TAIT R. I. and M. R. HOWE (1968) Some observations of thermohaline stratification in the deep ocean. Deep-Sea Res., 15, 275-280. TURNER J. S. 0967) Salt fingers across a density interface. Deep-Sea Res., 14, 599-611. TURNER J. S. (1968) The behaviour of a stable salinity gradient heated from below. J. fluid Mech., 33, 183-200.