Fluid dynamic implications for massive sulphide deposits of hot saline fluid flowing into a submarine depression from below

Deep-SeaResearch.Vol. 3 I, No. 2. pp. 145 to 170. 1984. Printed in Great Britain.

0198-0149/84 $3.00 + 0.00 ('~ 1984 Pergamon Press Ltd.

Fluid dynamic implications for massive sulphide deposits of hot saline fluid flowing into a submarine depression from below

TREVOR J. M c D O U G A L L *

(Received 25 March 1983; accepted 5 August 1983; final revision receil,ed 2 September 1983) A b s t r a c t - - S o m e massive sulphide deposits are thought to have formed from dense hot saline brine solutions that have ponded in a local depression o f the sea floor. After the source has been flowing for some time the level o f the warm fluid in the depression reaches the height of a weir (or sill) and thereafter the ponded fluid spills over the weir at a flow rate equal to that of the inflowing source fluid. A double-diffusive fluid interface forms at the height o f the weir and a steady-state balance is achieved where the fluxes o f heat and salt across the 'diffusive' interface are equal to the corresponding advective fluxes due to the source fluid being warmer and more saline than the fluid that spills over the weir. In this paper we consider the case where the source is below the height of the sill. The salient feature o f our fluid dynamical model is that it provides a mechanism whereby the temperature of the fluid in the depression may be substantially reduced (by double diffusion across the sharp fluid interface at the sill height), but the salinity o f the fluid is only slightly less than that of the source fluid. Both the low temperature and high salt concentrations are conducive t o the precipitation o f sulphide sediments. Another important result is that the source fluid need not be denser than the surrounding seawater for the occurrence of ponding in the depression. In fact, taking a temperature difference of 300°C between the source fluid and seawater, we show that a salinity excess in the source o f just 5 x 10 -3 (or just 0.1 M NaCI) will enable a ponded layer of fluid to exist in the vicinity o f the vent and so help to explain the confined lens-like property o f a massive sulphide deposit. We also discuss application to the Red Sea deep brine pools and develop a model o f a diffusive therrnohaline interface that explains how the interfaces in the Red Sea persist in a quasi-steady state with quite large values of the stability ratio Rp.

1. I N T R O D U C T I O N

TURNER and GUSTAFSON(1978) reviewed much of the existing literature of geophysical fluid dynamics applicable to the flow of hot saline solutions from vents in the sea floor. The physical processes they discussed include several geometries in which double-diffusive convection is important, the effects of non-linear mixing on the behaviour of turbulent plumes, and the Idling-box process for producing density gradients due to a source of buoyancy in a confined environment. In this paper we examine the fluid dynamic processes that occur when a hot saline source flows into a submarine depression from below the level of the sill (Fig. 1). The analogous case in which the source is above the sill height exhibits different behaviour and will be the subject of a subsequent paper.

* Research School o f Earth Sciences, Australian National University, P.O. Box 4, Canberra, A.C.T. 2601 Australia. Present address: CSIRO Division of Oceanography, P.O. Box 1538, Hobart, Tasmania 7001, Australia. 145

146

T.J. McDoUGALL

X TeS ' eA Q,Ti ,Si

Te,S e

eTt tFs

"'~r'rT-rr~'

"

,~ (bl

Q,Ti ,Si

,!,::)' y Q,1 ,SI Fig. 1. Sketches showing the progression of events after flow from the source begins.

A massive sulphide sediment several metres thick implies that a large volume of source fluid has been cycled through the sea-floor depression (or reservoir) because the source fluid contains only trace amounts of the heavy metal ions. In this paper we concentrate on the physical processes that occur when the hot saline fluid continues to flow for a long time so that the total volume emitted is many times the volume of the reservoir. The ponded fluid loses heat (and a little salt) to the seawater across a sharp double-diffusive interface, which forms at the height of the weir. The continual flux of buoyancy from the reservoir makes it denser than the inflowing source fluid. Although the source fluid may be denser than the seawater (Fig. I a), in the steady state it rises as a turbulent plume through the even denser reservoir fluid (Fig. lc). A steady state balance is achieved between the double-diffusive fluxes of heat and salt out of the reservoir and the advective fluxes of heat and salt due to the source fluid being warmer and more saline than the reservoir fluid which spills over the weir. We examine the stability of such a steady state and Fred it to be stable. In Section 5 we describe laboratory experiments that confirm the theoretical results. Once fluid has ponded in the depression, it is no longer necessary for the source fluid to be denser than the seawater; this feature of the model may prove attractive as studies on fluid inclusion studies of rock samples taken near a vent do not always give a sufficiently high

Sulphide deposits from hot saline fluid

147

salinity for the source fluid to have been more dense than the surrounding seawater. Even when the source fluid is less dense than the seawater, the model provides a mechanism for the localization of the sediments in the depression, rather than the more widespread deposition that would result were a buoyant plume discharged directly into seawater and then rose to a considerable height before the suspended sediment fell out of the plume. The steady-state model of this paper is extended in Section 4 to include crystallization, in which case the temperature and concentration of the crystallizing component is necessarily on a saturation curve. With application to the Red Sea hot brines in mind we develop a model for diffusive doublediffusive convection across a sharp interface, which demonstrates how such a diffusive interface can exist in a quasi-steady state even at the large values of the density ratio observed in the Red Sea.

2. T H E S T E A D Y - S T A T E

MODEL

We assume the source of hot saline fluid to be near the bottom of a depression on the sea floor (Fig. 1). The source flow rate is Q and the temperature is Ti (i for input) and salinity Si. The seawater environment has temperature Te and salinity S,.. The source fluid is assumed (for the present) to be denser than the ambient seawater and initially to flow out along the bottom of the depression (Fig. la). The ponded fluid loses heat and salt by double-diffusive convection and the density of the lower fluid increases. The incoming source fluid is then relatively less dense than the ponded fluid and so it rises as a plume through the lower layer (Fig. lb). After some time the total volume of incoming fluid is equal to the volume of the depression and then fluid spills over the weir (Fig. lc). A steady state is then reached in which the reservoir properties do not change with time and there is a balance between the doublediffusive fluxes of heat and salt and the fluxes due to the inflowing fluid having a higher temperature and salinity than that leaving the depression over the weir. The reservoir properties are homogeneous in this steady state and this can be understood by considering the equation for the conservation of density in the environment of the reservoir. Following BAI~m and TURNER(1969) this is ~+W~=O, 3t 3z where W is the vertical velocity of fluid in the reservoir, p is the fluid density, and z is the vertical spatial coordinate. Because it is steady (i.e., bp/3t = 0) and W =~ 0 (the downwards volume flux in the reservoir environment is equal to the upward volume flux in the plume at each heigh0, the vertical density gradient, 3p/~z, must be zero and so the reservoir is homogeneous. 2.1 The reservoir properties in the steady state Let the volume of the depression (or reservoir) below the lowest weir depth be Vw. The conservation equations for T and S are then dT Vw ~ = Q(T~ - T) - A wFr

(1)

148

T.J. McDOUGALL

and dS vw __

d/

= Q(Si - s) - A w~'s,

(2)

where A w is the horizontal area of the reservoir at the depth of the weir. In the 'diffusive' type of double-diffusive convection, the buoyancy flux ratio R I is defined by RI = l/Fs/aFr, where a and l/are the coefficients in the equation of state p = P0(l - a T + l/S) and Po is a reference density. In the steady state, (1) and (2) lead to

~ S i - S) a(T, - 7")

= R~.

(3)

This relation has a simple geometrical representation on the temperature-salinity (T-S) diagram. Figure 2 shows such a diagram with the temperature and salinity of the environment represented by the point e and the source fluid by the point i. Equation (3) then implies that the steady-state solution point must lie on a line from point i with slope (fl/a)R~ I (Fig. 2). To determine where the solution point lies on this line, we need to use an expression for the flux of heat, Fr, across a diffusive interface. HUPPERT (1971) proposed the following expression for Fr (which is the 'flux of temperature', being the flux of heat divided by the density and the specific hea0

aFr = b[a(T-

Te)]413Rf 2,

(4)

where Rp is the density anomaly ratio of the fluid interface, defined by # ( S - S,) Rp = a ( T - T,)

¢

(5)

}

O

> S{x I0-3} Fig. 2. Temperature-salinity diagram showing the environmental point e, the source point i, and the solution point with several trajectories from the linear stability analysis (Section 2.2) converging on it. The slope of the line from point i marked R / i s d T / d S = (fl/a)R? .

149

Sulphide deposits from hot saline fluid

and b is a dimensional constant equal to 0.323 (gtc~-[v)~, where the symbols have their usual meaning. For water at room temperature b can be taken as 1.9 x 10-3 m s -~. We define dimensionless temperature and salinity variables 0 and t! by Ti-T 0- - - ,

Si-S '1 ~ - - Si - S<.

T i -- 7",.

(6)

The steady-state balance between the fluxes of heat and salt due to double-diffusive convection (A , F r and A wFs) and due to advection IQ(T; - 7) and Q(S~ - S)l can then be written in terms of 0 and t/ as

Rf 0s = C(I - 0,)1°/3

) -2

1 - -Aft 0, RI,

Rj. ;

r/< = 0~ 0" ' Rp

(7)

where R ° is the density anomaly ratio based on the environment and source fluids, Ro

_

- se)

a(T, -

T,)

"

and C is the dimensionless constant that embodies the various experimental parameters and is defined by C = bla(Ti - T,)IiA wQ-m(R°p) -e.

(8)

The subscripts s on 0, and r/, emphasize that these are the steady-state solutions. Figure 3 shows the relationship (7) between the nondimensional temperature 0 and C for R f i R ° = 0.0 and 0.1. From (7) it is evident that for small values of C, 0, _-__C, while for large values of C, 0s ~ 1 - C-°3(1 - RI/R~) °'~. The primary parameter in determining the steady-state value 0, is C (rather than RI/R~) and this must be >1 if double-diffusive convection is to have a large influence on the steady-state reservoir temperature. We now briefly consider the question of how much entrainment of environment fluid occurs across the sharp double-diffusive interface due to the impingement of the plume from below (Fig. le). BAriums (1975) studied entrainment in such geometry and found that the flux of buoyancy due to entrainment across the interface F*, divided by the buoyancy flux from the source, Fo, is given by F.

+

= 0.4Fr

L

J

"

where Fr is the Froude number of a normal plume, p, is the density of the reservoir fluid, Pe is the density of the environment (above the interface), and p ~ is the density of the plume at the interface. The Froude number of an axisymmetric plume is simply {(8/5)a] -t, where a is the entrainment constant (TitulaR, 1973). We take a to be 0.1, which makes the factor 0.4Fr in the above equation equal to 1.0 (fortuitously). The density difference ( P l - P , ) between a plume and its surroundings decreases as z -an with distance z from the source of buoyancy and typically the ratio of density differences (Pro-P,)/(Pr-Pc) will be very small at the interface. Thus we do not expect the entrainment of fluid across the interface to be an important mechanism in the reservoir and it will be ignored in the remainder of this paper. It is of interest to find the density anomaly ratio R2 nt of the double-diffusive interface in the

150

T.J. McDouGALL 1.0

Os 0.8

0.6

0.4

0.2

0

i

i

i

5

10

15

20

C Fig. 3. Graph of the steady-state nondimensional temperature Os plotted against C for Rf/R0 = 0.0 (solid line) and 0. I (dashed line). The circled points are from laboratory experiments described in Section5.

steady state. This can be shown to be R / ' - R o _ R~Os 1 -

O,

where R ° is the stability ratio defined using the temperature and salinity differences between the source and the environment fluids and 0, is the steady-state value o f the nondimensional temperature. To estimate 0s for a massive sulphide deposit we need to find a value for the parameter C given by (8). Taking the volume flow rate Q of source fluid as 1 1 s -~ at a temperature of 200°C with R ° ~ 1 (i.e., the source fluidis only slightly denser than the surrounding seawater) and a reservoir diameter of 200 m at the height of the weir, we have C ~ 2 x 104. By (7), the steady-state nondimensional temperature 0s is approximately 0.95, which means that the reservoir fluid is warmer than the seawater environment by just 10°C. The above expression for Rpm' shows this to be about 17 for this example--quite a large value. We examine the relationship between R ~ t and R ° in more detail in Section 6. L~VEN and SHmTCUFFF (1978) developed a theoretical model o f a diffusive thermohaline interface that has steps of properties at the upper and lower edges of the interface with smooth linear gradients in the centre. The flux of heat given by the model agrees well with that found in laboratory experiments in the range 2 < Rp < 7. For lower values of RI,, direct turbulent entrainment of fluid across the interface causes a larger flux of heat than in Linden and Shirtcliffe's model, while as Rp approaches (Xs/ICr)-~ ( ~ 9 for heat and salt) the heat flux of the model goes to zero. Laboratory experiments on a heat-salt diffusive interface have been conducted by MARMORINO and CALDWELL (1976) with values of Rp up to 12, and many

Sulphidedeposits from hot salinefluid

15 1

researchers have observed sharp interfaces in the Red Sea deep brines with Rp between 15 and 18 (MoNIN and PLAKmN, 1982). In the Appendix we develop a modification of Linden and Shirtdiffe's model that extends its range of applicability from Rp = 9 to beyond R I, = 20. This is done by including the effects of a small amount of entrainment that occurs at both the upper and lower edges of the interracial region. The entrainment is due to the convective turbulence, which exists in the well-mixed layers above and below the interface. The main point of the model is that it shows that a diffusive interface can exist in a quasi-steady state with Rp much greater than Linden and Shirtcliffe's "critical' value of (Xs/Kr)-¢ and that if there is a value of R~ beyond which a steady diffusive interface cannot exist then such a value is much larger than (Ks/Kr) -½. 2.2 The stability of the steady state In terms of nondimensionai variables, (1) and (2) are d0

= - 0 + C(1 - 0)1°/3(1 - r/) -2

(9)

dr and

dr~ =

-r/+ ~R f C(I - 0) io/3(I- r/)-2,

(I0)

where r is t/t,,the nondimensional time and tr is the renewal timescale, t, = Vw/Q. The nondimensional time z is I when fluid fn'stspillsover the weir. W e linearize the equations about the steady-state solutions 0s, r/sand obtain the following matrix equation for the deviations 0', r/' from the steady-state solutions.

[

10

20s ]le

0s

-l

idol: dr/'

I0 R/

3 .-o,) 0,

/dr_l

3 R ° (I-0,)

(z -

(II) R: os -1 + 2 Ro ~ - r / , )

r/'

The two eigenvalues of the matrix are

],|=--I 6, R/ O, - + 23 (l - 0,) R ° (I - t/,)

10 2 2 =-I

=-1 (1 - OsX1 - r/s)

-3"

Ra

7J"

Both eigenvalues are negative and so the steady-state solution is stable. The second arrangement of the expression for '1.2 shows that 1/121> I~.~[ because R//R ° < L 0, < 1, and r/s < 1.

152

T.J. MCDOUGALL

The eigenvector corresponding to ~,, is shown in Fig. 2 by the line with double arrows and has a slope on the T - S diagram of 0.6 of the slope of the line joining the steady-state solution point to the environment point e, i.e., adT palS

0.6 ( 1 - 0 , ) -

R°

(I - .,)

l dO -

R ° d.

along this eigenvector. It can be shown that to first order in 0' the double-diffusive fluxes FT and Fs are constant for reservoir properties (T, S) on this eigenvector and so the linearized stability analysis simply gives the exponentially decreasing renewal process, exp (-r) for disturbances from (0s, th) along this direction. The second eigenvector, corresponding to 22, is along the line joining the steady-state solution point to i with slope given by a dT/fldS = R] ~ (Fig. 2, the line with triple arrows). As 122 I ~ 12! I (e.g., with C ~ 6 so that 0~ ~0.5, 22 ~ --4.2), disturbances from (0,, ~h) in the direction of the second eigenvector decay faster than in the direction of the first eigenvector (as indicated by the trajectories in Fig. 2). The stability of this system makes an interesting contrast with that considered by WELANDER (1982). He studied the stability of the temperature and salinity of the ocean surface layer as it exchanges properties with the air above and also, by entrainment, with the seawater below. The terms in (1) and (2) proportional to the volume flow rate Q also appear in Welander's model, but where we have the double-diffusive fluxes Fr and Fs, Welander had an entrainment function that is a single-valued function of the mixed-layer density p = p0(l - a T + ~S). He found that sustained oscillations could occur if the entrainment function varied strongly with p. The double-diffusive fluxes Fr and Fs are not functions of density alone but depend on both ( T - T~) and (S - S~); in contrast to Welander we have found that the steady-state solution is always linearly stable. 3. THE E V O L U T I O N OF THE R E S E R V O I R TO T H E S T E A D Y S T A T E

We now seek the reservoir properties T and S as functions of time from when the dense, hot, saline source begins to flow into the reservoir. We treat the properties of the deepening reservoir fluid as uniform and now we justify this assumption. Without double-diffusive convection the ponded fluid would have the temperature Ti and salinity Si of the input fluid. The double diffusion provides a flux of buoyancy across the sharp fluid interface, which tends to keep the ponded fluid well mixed, but the change in density of the reservoir fluid caused by the double-diffusive convection means that the inflowing source fluid rises through the ponded fluid as a buoyant plume. Such a plume flowing into a confined environment normally stratifies the environment by the Idling-box mechanism (BAIm~s and TURNER, 1969). In the present case density differences in the reservoir are caused in the first instance by double diffusion across the sharp fluid interface, and then the idling-box mechanism can work to redistribute density differences in the vertical. The time scale for changes in the mean density of the reservoir fluid is the renewal time scale tr (see later in this section) and the filling-box time scale is much smaller than L, typically ~0.01tr. Considering the equation for conservation of density Op/~t + W ~p/~)z = 0, we have a value of ~p/~t much smaller than that of a normal filling-box process (because of the different time scale tr involved in ~)p/t~t), and so the vertical density gradient Op/Oz in the reservoir must be very small. In the experiments, the reservoir fluid remained well mixed and we suspect that the strength of double-diffusive convection will always be sufficient to overcome the small density gradient that would arise from

Sulphidedeposits from hot saline fluid

153

the filling-box process alone. In the remainder of this section we treat the fluid in the depression as homogeneous but with properties that are functions of time. The solutions, T(t) and S(t), will depend on the geometry of the reservoir and we consider just two cases, firstly when the area of the reservoir is independent of height and secondly when the area is proportional to height.

3. l Reservoir area independent of height Equations (l) and (2) still apply to this case even though there is no flow over the weir at this stage, but now the volume of fluid Vw in the reservoir needs to be replaced by Qt. The same steady-state temperature and salinity given by O, and tl~ from (7) now satisfy (1) and (2) for all time and so the temperature and salinity of the ponded fluid are constant as the fluid level rises towards the weir. The linear stability analysis proceeds exactly as before except that now the time derivatives on the left-hand side of (11) are r d 0 ' / d r and rd~/'/dr. The eigenvalues 2~, 22 and the eigenvectors are exactly the same as before but now any disturbances from the steady-state solutions decay as power laws instead of exponentials. This can most easily be seen by changing the time variable from r to In r. A general disturbance from 0, for r < 1 decays as 0]r a, +0~r a'- while the depression is filling, instead of as 0~ exp (2tr) + O~ exp (220, which is appropriate once fluid spills over the weir, that is, for r>l. 3.2 Reservoir area proportional to height This variation of reservoir area with height IA(z) = az] is that appropriate to the body of revolution formed by a parabolic sidewall. To a first approximation this is the geometry at the bottom of a sphere, for example, and so corresponds to reservoir shapes most likely to be found in practice. The geometrical relationship 2

V= Qt = f 0 A(z) dz = ~ z 2

(12)

implies that A(z)/A w = r ½, where r ~ tQ/Vw and the subscript W means the value at the weir height. The changes in the reservoir temperature and salinity from Ti, S,. occur only by double-diffusive convection and so during the evolution the nondimensional salinity and temperature are related by r / = (Rf/R°)O. The equation for the conservation of heat then is

dO

~ d - T = - 0 + r½C(l -

0),0/3(l - - ~R.r ) -2, 0 O

(13/

where C is defmed as in (8) using the area A ~, at the weir. For small times the approximate solution to (13) is 2

O,~,jCr½7"

10

- - -9C 2 r

2 R/ 2 + -3- -Rp ~ C r.

(14)

Equation (13) was integrated by the Rtmge---Kutta method using (14) as a starting value. The integration was begun at a time r corresponding to 0 = 0.01. To reduce the size of the temporal derivative of 0 at small times and so facilitate the numerical integration, r was actually transformed to r½ in (13) and the computer integrated (13) with r ~ as the independent variable. The details of the transformation are omitted in the interests of brevity. For r > 1 the

154

T . J . McDOuGALL

1.0

0 0.8 C-25.0 0.6

-" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....

0.4

~

O.2

~

j

~

~

~

'

f

f

O

0

I

0.5

"

-

"

c

~.s..o .........

C - 1.0

~ ~ - " " ~ " = ' ~ ° '-' "- - - c-o.2 I

1.0

I

I

1.5

2.0

T

Fig. 4. Graph o f the nondimensionai temperature O as a function o f the dimensionless time r for C = 0.2, 1.0, 5.0, and 25.0 in a reservoir that has its horizontal area proportional to heighL The full

lines of each pair have Ry/X° = 0 and the dashed lines have R//R °

fluid discharges over the weir and the differential equation in the numerical integration is changed to (9) with r / = (Ry/R°)0. Figure 4 shows the results o f the numerical integration for four different values of C and for R fiR ° = 0.0 and 0.1. The solution 0 is not very sensitive to the value of Ry/R °. It is natural to enquire into the stability of the evolving solution. Mathematically, this is a difficult problem as the differential equations for 0 and v/are now nonautonomous. We can however answer the question of stability with the help of the geometry of the T - S diagram. Figure 5 shows the T - S diagram with the solution locus at various nondimensional times r. At r >/1, fluid flows over the weir and the solution in Section 2.2 applies. The reservoir properties will then quickly progress from the r = 1 point to the r = oo point as an exponential decay, exp (~12r), where A2 will normally be in the range - 7 to -2. We are concerned now with the evolution of the reservoir temperature and salinity after a disturbance has perturbed the evolving properties at r < 1. Such a (7', S) point is shown by point (a) in Fig. 5, which is to the left of the line with slope given by p d $ / a d T = R/. There are two separate physical processes that act to change the temperature and salinity of the reservoir fluid, firstly the process of mixing of source fluid into the reservoir and secondly, double-diffusive convection. The two processes cause the (T, S) locus to move in the directions shown by the arrows in Fig. 5 labelled 'mix' for the mixing process and 'd.d' for double-diffusive convection. The slope of the d.d. vector is set by the flux ratio Rf, whereas the slope of the mixing vector depends on the location of the point a in relation to point L By taking the vector sum of the two vectors, 'mix' and 'd.d.', it is evident that deviations from the R f line decay stably back towards the line. Given this geometrical insight into the problem, we need only consider disturbances along the line. We consider a small positive disturbance imposed on 0 at time r and we use 0 3 ) to determine whether d0/dr will increase or decrease due to the presence of the disturbance. If the disturbance to 0 is small enough to use the binomial expansion about the undisturbed values

Sulphide deposits from hot saline fluid

15 5

T (°C) t

T:O-5

e~¢-"

t I Rf

)

SIx 10-31 Fig. 5. Tempcrature-safinity diagram showing the direction of the vectors due to mixing of source fluid into the reservoir ('mix') and duc to double-diffusive convection ('d.d.'). This diagram shows that disturbances tend to decay towards the R£ line. The points shown for r = 0, 0.5, 1.0, and oo indicate the solution points (T, S) for the case ~'ith the horizontal area of the reservoir proportional to height.

of (1 - 0 ) and 11 -(RI/R°)OI for the two powers in (13), it is straightforward to show that d0/dr is decreased due to a positive 0 disturbance and is increased by a negative 0 disturbance, hence showing that the solutions 0(0 we have found for the variable area reservoir are indeed stable in the sense that deviations from the solution decay towards it. 4. THE S T E A D Y - S T A T E S O L U T I O N WITH C R Y S T A L L I Z A T I O N

4.1 Differential equations

We now extend the analysis of Sections 2.1 and 2.2 to the situation where the reservoir is saturated with a salt and crystallization is occurring. Let L be the latent heat of crystallization of the salt, Cp the speazifie heat of the solution, and ~ the rate of crystallization in units of weight per unit time. The conservation equations for T and S are dT

L

Vw-"S-= Q(T,- T)-AwFr llt

+

E

(15)

poC,

dS

1

V w - - ~ = Q ( S , - S) - A w F s - - - E. Po

(16)

s = P(T~

(17)

A third equation

represents the saturation curve of the salt as a function of temperature. The rate of crystallization, 6, is not set a priori but is a property of the solution. Figure 6a shows the T - S diagram

156

T.J. MCDoUGALL

(a)

T

i

(°C) I

S ( x 10-3)

(b) Cry

/

--I 0

I

2

9

PosJtlo~ on saturation line

J Fig. 6. (a) Temperature-salinity diagram for the case where the salt S is saturated. The saturation curve S ----P(T) is assumed to be the straight line as shown. Point m would be the steady-state solution (from Section 2.1) if the saR were not subject to the saturation constraint. The arrowed trajectories also apply to the imaginary case where the salt is not constrained by crystallization. (b) Sketch of the value of the latent heat parameter I as a function of the position of the solution point on the saturation line of (a). The numbered points 1, 2 . . . 9 are shown on both (a) and (b). Doublediffusive convection is not possible to the left of point 2. The physically relevant solutions lie in the ranges _p-I <~ I'~ +0o, d ~> 0 between points 4 and 8.

Sulphidedeposits from hot salinefluid

157

with crystallization. The saturation curve S =/'(73 is drawn as a straight line in the figure. Point m is the solution point derived in Section 2.1 where the salt was not subject to the present saturation condition, and because this point is now in the supersaturated region of the T - S plane, the actual solution point must lie somewhere on the saturation line marked by points 1, 2, 3 . . . 9. We address the problem of finding the location of the solution along the saturation line and the rate of crystallization for various values of the latent heat L. We eliminate d between (15) and (16) to obtain the equation

(I - 0 ) 1°/3 c

(1 -,7)

(18)

(i +/Re) = (0 +/R,°,q),

which must hold in the steady state. Here C is defined by (8), 0 and q are the in situ nondimensional temperature and salinity, respectively (see 6), and l = aL/pCp. The linear stability analysis is now a one-dimensional problem (along the saturation curve) and perturbations 0' from the steady-state value 0 (we do not use the subscript s for steady state here for convenience) obey 1 d0' - 0' dr

--

1 +

(O+lRffq) f l 3 1 -(1 +lp) (1 0)

2p 1 ] R° ( l - - q )

,

(19)

where p -- ~/a)dP/dT. By comparing the slope of the saturation curve with the slope of the line e--m it can be shown that the square bracket in (19) is positive and this assists in the stability analysis. 4.2 The location of the solution points on the saturation curve We begin by considering the case where L = 1 = 0. Equation (15) then shows that in the steady state there is a balance between the mixing term Q(Ti - T) and the double-diffusive term A wFr. The same balance occurs at point m without crystallization, but now a non-zero b is required to balance the salt conservation equation. We lineafize the nondimensional equation representing the heat balance about point m to obtain the slope of the line from m on which Q(Ti - 7) = A wFr. This gives dO dr/

20,.

(l - 0,.)

(1 + 7/3 0,.) (1 -r/,,)

which is less than 0.6 (1 -0.,)/(1 - r/.,), this being the slope of the separatice trajectory from m to point 5, which has F r = constant (Section 2.2). The solution point for ! = 0 (labelled point 6 in Fig. 6a) lies between points 5 and 7, which has T~ = T,.. The curved trajectories in Fig. 6a are the loci of evolution of the solution (T, S) from Section 2.2 without crystallization, and at point 6 the trajectory is horizontal The trajectory of the crystallization process by itself is a dT/p d S - - - 4 , and in the steady state the slope must be parallel to the trajectory a dT/p d S sketched in Fig. 6a, which is the resuR of double-diffusive convection and advection without crystallization. When double-diffusive convection, advection, and crystallization occur together, a balance is possible for a suitable value of d if the above two trajectories are parallel. For 1= 0, both slopes are zero at point 6 in Fig. 6a and one can show that the rate of crystallization, d, is positive there. For 1 > 0 the solution point moves to the fight from point 6 until as 1 --, +oo, point 8 is approached where d -, 0 +, and a balance occurs between Q(S; - S) and A wFs. An expansion

158

T.J. McDouO^LL

about point m (where the balance also holds) to find the line from m, where Q(Si - S) = A wFs gives dO

0.3(1

d~

-

O~Xl- 3~.)

~.(l - 0.)

which is a large negative slope and is the slope of the line from point m to point 8. The trajectory at point 8 in the absence of crystallization is vertical. For I < 0 the solution point moves to the leftof point 6, but the rate of crystallization 6 remains positive. At point 4 the trajectory is tangent to the saturation curve and so the value of I appropriate to a steady-state solution is -p-t. Figure 6b shows the steady-state value of I as a function of distance along the saturation curve. To the leftof point 4 and to the right of point 8, I<-p-J. Between points 3 and 4 the steady-state solution is unstable while to the right of point 8, the rate of crystallization is negative. For thermodynamic regions it is believed that I should be >-p-J ; otherwise an isolated mixture of crystals and solution can spontaneously crystallize because the sign and magnitude of the heat of crystallization are such as to drive the system towards supersaturation. For this reason we restrictattention to the range -p-~ < I < +oo. The solution point on the T - S diagram (Fig. 6a) for this range of I lies between points 4 and 8, with b > 0 in the whole of the range. The solutions are stable steady-state solutions. 4.3 The rate of crystallization, We now address the question of finding the rate of crystallization 6 as a function of the latent heat parameter/, or equivalently, as a function ofthe location of the steady-state solution point on the saturation curve. For large positive/, the solution point is just to the left of point 8 in Fig. 6a and b ~ 0. At point 5, where 1 < 0, the double-diffusive fluxes, Fr and Fs, are the same as those at m, i.e., A wFs = Q(Si - Sin), and from (16), ~ =poQ(Sm - S). It can be shown that at point 6, where 1 = 0 tic = P o Q [ ~ S

, - S) - Rfa( T i -

T)],

and at point 4, where l = - p -~, Be = PoQ[~(S, - S) - Rfa(Ti -

(

T)] \ I -

Rf )-l. 7-

It is evident from consideration of the selected points on the saturation curve that as I decreases from +oo through zero to -p-J, the rate of crystallization, ~, increases monotonically from zero and remains finite at point 4. 4.4 The effect of variations of C Consider what happens for a given latent heat parameter I as the 'double-diffusive strength' parameter C defined by (8) is slowly varied. The variation is assumed to be slow enough for the system to evolve through a series of quasi-steady states. Assume that the temperatures and salinities of both the source fluid and the environment remain constant, but the flow rate of source fluid Q slowly decreases. Initially, for a large value o f Q, the non-crystallizing solution point m will lie above the saturation curve on the Rf line from i (Fig. 6a). As Q decreases, the solution point moves down the line until point 9 is reached. As Q is further reduced the solution moves to the left down the saturation curve S = P(T) such that the slope of the line

15 9

Sulphide deposits from hot saline fluid

(b)

(a) T(°C}

T(oC) '

I

/

/

• =P(T)

T

e )

(ppm)

S ( x I0 "5)

Fig. 7.

Temperature-salinity diagram for the major salt (a) and (b) for a minor salt ~: which is present in only trace amounts,

from the solution point to the appropriate point m is a monotonic function of the latent heat parameter l. 4.5 Crystallization o f a very dilute component If the crystallizing substance is present in the source fluid in trace amounts, the temperature and total salinity of the reservoir will be set by the balance between double-diffusive convection and the steady flow of the more abundant salt just as in Section 2.1 and Fig. 7a. Having determined the temperature T,, of the reservoir in this way, we need to find the concentration of the minor component in the reservoir and its rate of crystallization. If the minor component is crystallizing, its concentration e must be on the saturation curve e = P(T) at the known temperature T,, (Fig. 7b). From the conservation equation of the minor component, the rate of crystallization e is given by (20)

e/Po = Q(e, - c) - A w F ,

where F, is the flux per unit area of component e across the double-diffusive interface; the flux can be estimated by the theory of GRwFrrHs (1979). His equation (2.7) shows that

(s-se)

Fs < \ X s /

( S - S~)'

(21)

depending on the stability ratio Rp of the interface. If, on finding e = P(T,.) from Fig. 7b and estimating F, from GgwFrrns (1979), the rate of crystallization ~ (from 20) was negative, crystallization would not occur and the value of e would be less than that shown in Fig. 7b. 5. L A B O R A T O R Y E X P E R I M E N T S

A series of laboratory experiments was performed to verify the steady-state model of the reservoir properties developed in Section 2. Figure 8 is a sketch of the experimental tank. Source fluid at known temperature and salinity enters horizontally at the bottom of the tank. A vertical barrier 0.2 m high forms the weir over which fluid from the reservoir flows and a

160

T.J. McDOUGALL 1_

]C source

= , * - - fresh

~

taDwatet

we,,

fluid

"""

.

" " ' '

~to

waste

....

" . . . . . . .

" . . . . . .

''''"

'

Fig. 8. Sketch of the laboratory tank. The reservoir fluid that spills over the weir is continually siphoned to waste and cool, fresh tap water is added to the tank as shown. The tank is 2.0 rn long, 0.4 m high, and 0.2 m deep. The length of the reservoir region was varied between 0.6 and 1.0 m and the height of the weir was 0.2 m.

sharp interface forms at the height of the weir. The overflowing warm, saline fluid is trapped by a second smaller vertical barrier and is siphoned to waste from this region at a large flow rate (typically 20 times the source flow rate). The total depth of water in the tank is maintained at about 0.4 m by the inflow of fresh, cool tap water (Fig. 8). The purpose of this source and sink arrangement is to maintain the temperature and salinity of the fluid above the reservoir relatively constant by not allowing the fluid that spills over the weir to contaminate the environment. The horizontal direction of inflowing hot, saline fluid ensures that any vertical momentum of the plume is due to its buoyancy flux rather than to its initial vertical momentum flux. In the application to a vent on the sea floor, we expect the buoyancy flux of such a plume to be more important than vertical momentum and so the horizontal nature of our initial plume motion was a desirable (although by no means necessary) feature of the experiment. The experimental conditions are given in Table 1; A T is the temperature difference between the source fluid and the environment, Q is the flow rate from the source, R ° is the density anomaly ratio defined using the source and environment fluids, t, is the renewal time scale for the experiment, and rm,~ is the value of the nondimensional time r (equal to time divided by Ir) at the end of the experiment; C is the value of this parameter calculated using the known experimental parameters, 0 (theory) is found from (7) and 0 (meas) is the value of 0 measured at the end of the experiment. The results are shown by the seven data points in Fig. 3. The general agreement between the theoretical and experimental results is obvious; if anything the nondimensional temperature 0 was larger in the experiments than the theoretical predictions, which implies Table 1. Table of experimental conditions

Expt No. I 2 3 4 5 6 7

AT

Q

(°C)

(cm ~ s-')

9.78 7.90 5.35 19.30 6.54 5.61 13.2

9.2 9. I 4.0 3.7 8.4 3.7 3.9

ROp

tr

rmax

C

2.0 3.3 2.1 1.33 3.79 3.15 1.8

2.19 1.40 1.19 14.7 0.60 1.62 7.82

0(theory)

O(meas)

(h) 1.48 1.77 2.72 1.03 2.23 1.84 1.18

1.08 1.06 2.40 2.64 0.73 1.53 2.03

0.405 0.345 0.32 0.625 0.24 0.375 0.565

0.42 0.39 0.36 0.66 0.22 0.44 0.58

ill!!• m .~ ~ :•_i i,: H ~ :

Fig. 9.

~

~

Shadowgraphs taken at (a) r = 0.29, (b) 0.53, and (c) 1.08 during experiment number six.

Sulphide deposits from hot saline fluid

163

that the flux of heat due to double-diffusive convection was larger (by say 20%) than that given by (4). During the experiments there was considerable shear across the double-diffusive interface. The interface provides a flux of buoyancy (mainly due to heat) to the upper layer fluid and a vigorous circulation is driven in the upper layer in the direction shown in Fig. 8. The flow is similar to that addressed by the theory of PHILLIPS (1966). The mean shear across the diffusive interface may well be the reason for the increased double-diffusive flux of heat across the interface. Figure 9 shows three photographs at r = 0.29, 0.53, and 1.08 during experiment number 6. The first two photographs were during the filling, when the reservoir fluid lay below the height of the weir; the third photograph is at r > l and fluid can be seen spilling over the weir. Each photograph shows that dye had been added to the plume shortly before each photograph. In each case the dye rose to the interface (through the turbulent reservoir fluid), indicating that the reservoir fluid was denser than the incoming source fluid. Figure l0 shows the T - S diagram for the experiment. The source (i) and environment (e) points are shown, as is an isopycnal through i and the line marked Rf from point i with slope given by a dT/fl d S = R-fI. The temperature of the environment varied during the experiment in the range indicated (Fig. 10). The temperature and salinity of the reservoir fluid at several times r are shown by the series of circles near the Rf line and the square marks the theoretical solution point for C = 1.62, which applies for all times r. The reservoir properties lie quite close to the R f---- 0.15

25

24

T(°C) 23

22

I

!

21

I

!

/

20

Rt - 0 . 1 5 19

0

1

2

3

4

S (x I0 "3) Fig. 10. Temperature-salinity diagram for experiment number six. The circled points beginning with the one labelled r = 0.1 and ending with r = 3.15 show some measurementsof the temperature and salinity of the reservoir fluid during the course of the experiment. The square point is the solution point given by the theory.

164

T . J . MCDOUGALL

line, indicating that the entrainment of environment fluid into the reservoir is small. Between r = 0.1 and 0.93 the temperature o f the reservoir rose from 21.8 to 22.7°C, but it then settled down at longer times to 22.3°C. The initially cool solution at r = 0.1 may have been due to the transfer of heat to the cooler glass bottom and walls (which were initially at Te = 19°C). Once the reservoir fluid rose to the height of the weir, the interface was subjected to the significant shear flow of the upper layer fluid. The increased flux of heat due to the shear can explain the drop in temperature (implying more active double diffusion) from r = 0.93 to 3.15. The theoretical prediction for a reservoir such as ours with constant cross-sectional area (see Section 3.1) is that the reservoir properties remain fixed for all values of r > 0. The cluster of points from the experiment for r > 0.I in Fig. 10 support this prediction. 6. DISCUSSION

The general property of the fluid dynamic models we have considered is the balance between double-diffusive convection and a mean through-flow (or advection) that enters and leaves the control volume at different temperatures and salinities. In the steady state, the balance allows a large heat loss to the environment across the double-diffusive interface while only a small amount of the solute is transferred across the interface. We have restricted our discussion to the geometry shown in Fig. 1, where the source of hot saline fluid lies below the height of the weir (or sill) of the depression on the sea floor. At least initially the source fluid is assumed to be denser than the surrounding seawater and it ponds in the bottom of the depression. Double-diffusive convection then increases the density of the ponded fluid and the source fluid rises as a turbulent plume through the fluid in the depression. This ensures that the incoming source fluid is significantly diluted by turbulent mixing in the plume and that the fluid in the depression is homogeneous. The sudden cooling of the source fluid in the plume is assumed to cause a fine suspension of particles to form close to the vent, much as in the present-day hot (black) smokers recently observed at oceanic spreading centres (EDMOND, 1981). The f'me suspension of particles will mix uniformly throughout the reservoir and some will flow over the sill. Near the bottom of the depression where the flow velocities are small, the free particles can fall out of suspension and form massive sulphide sediments. Further research is required to investigate the amount of suspended sediment that exits over the weir as a fraction of that deposited on the floor of the depression. The ratio will depend on the equilibrium concentration of particulate matter in the reservoir fluid, which in turn will be a function o f the terminal falling speed of the particles. The present model predicts the trapping of the particles in the depression for a considerable period of time, allowing slow sedimentation to occur. If there is insufficient reduced sulphur in the source fluid, because of the low temperature of the reservoir fluid, it will be a suitable environment for bacteria that can reduce the sulphur in the sulphate ion. We now consider the density of the source fluid. The model requires that it is initially denser than the surrounding seawater, at least until the fluid interface in the depression rises past the level of the source. Having formed the ponded layer, the density of the plume may decrease and become less than that of seawater. The double-diffusive buoyancy flux continually acts to increase the density of the ponded fluid and so long as the density of the plume at the interface height is greater than that of seawater, the ponded layer will remain intact. In Section 2.1 the expression for R~ "t was derived, i.e., the density anomaly ratio for the doublediffusive interface at the top of the reservoir. The same expression can be used to estimate the limiting value of R ° (the stability ratio defined using the source fluid and seawater properties)

Sulphide deposits from hot saline fluid

165

!

T I

%'

(°C]

I

il

I

SIx 1(3~1 Fig. 11. Temperature-salinity diagram showing the environmental point e, the reservoir point m, and two different source points i I and i 2. Source point i i is denser than the seawater environment while point i 2 is less dense than e.

at which Rpi"t = 1. This will yield the value of R ° at which the ponded fluid and seawater densities are equal. The limiting value of R ° is R ° = R~"(I

-

0,) + Ry0~

(22)

and for 0, = 0.95 and R / = 0.15, R ° is 0.2 for R~" = 1, while for Rpi"' = 2, R ° = 0.25. Such small values of R ° mean that the source fluid need not be much more saline than seawater. For example, taking a source temperature difference AT of 300°C, a = 0.5 x 10-4(°C) -I appropriate to seawater at 0°C and p = 0 . 7 6 x 10-3(%o)-', R ° - - p A S ] a A T = 0 . 2 5 gives AS -----5 x 10 -3. The example indicates that for A T = 300°C, a source that is only 5 x 10-3 more saline than seawater (which has a salinity of approximately 35 x 10-3) is sufficient to give R ° = 0.25 and Rp~"t = 2 (using 0, = 0.95). Figure 11 shows the T - S diagram for a case with the source fluid (i2) less dense than the environment (e). The other source point (il) is the case discussed in the rest of the paper where the source density is greater than that of the seawater. As the steady-state balance between double diffusion and advection (1 and 2) depends only on the fluxes Q ( T I - T) and Q ( S I - S) and not on the properties Ti and Si separately, it is possible to arrive at the same solution point (m) for the reservoir properties so long as Ql(Tll -- T,,)= Q2(TI2 - T,,) and if both ii and i2 lie on the Rf line through point m. In a review o f fluid inclusions in exhalative ore deposits, SOLOMON (1983) showed that at Silvermines in Ireland and Mt. Chalmers in Queensland, the source fluid temperatures and salinities were such that the incoming fluid would have been denser than seawater. LARGE and BOTH (1980) have explained the localized nature of the massive sulphide deposit at Mt. Chalmers on the basis of the source fluid being denser than the surrounding seawater. One of the important conclusions of this paper is that the source fluid need not have a density greater than or even close to that o f the seawater. The above example using a source temperature of 300°C shows that the salinity of the source must be at least 40 x 10-3 (corresponding to

166

I". J. McDouGALL

0.6 M NaCI, that is, only 0.1 M NaCI more than seawater) for the ponded fluid to persist. This is far from the concentration of 3.5 M NaC! required for the source fluid to be as dense as seawater. However, it is still necessary in our model for the source fluid to have been more saline and denser than seawater for some short time in its history. The advantage of a model in which the source of heavy metal ions discharges into a confined reservoir is that it naturally leads to a confined ore body. The large buoyancy fluxes from the present-day hot smokers (EDMOND, 1981) will only lead to a confined, lens-shaped ore body if the water with which the plume mixes is an isolated body of reservoir fluid and if the salinity of the source fluid is sufficient. If, on the other hand, the hot smokers discharge directly into seawater, the buoyancy flux is such that the plume will rise to a height of order 1 km (there is some evidence for this in the helium concentrations from the GEOSECS programme) and any subsequent sedimentation from such a height would be spread over a large horizontal area. If the fluid exhaled from a hydrothermal vent is saltier than seawater but lighter because of its high temperature, the first fluid exhaled may well be denser than seawater. This is because the fwst fluid is cooled to close to 0°C by conduction (not mixing) as it passes through cold rocks close to the sea floor. It is not until an appreciable volume of fluid has passed through the system and the rocks adjacent to the fractures that feed the vent have been heated to high temperatures that >300°C fluid can be exhaled. We now apply this work to the deep brine pools at the bottom of the Red Sea. There is little doubt that there are hot saline sources in the pools, but the sources have not been located and so their properties are unknown. SCHOELLand HARTMANN(1973) observed a small increase in temperature immediately below the lowermost brine interface in several locations in the Red Sea, and Voomms and DORSON (1975) identified slow vertical velocities associated with vertical convection in the layers. Both of these observations indicate the presence of convective plumes in the deep brines. The direct application of our model places a constraint on the T - S properties of the source fluid (Fig. 11). If we interpret point m on the figure as the point corresponding to the lowest brine pool in the Atlantis II Deep (62°C, 188 g C! l-Z), the source fluid must lie on the R f fine at a source point like i~ or/2. Because the double-diffusive flux ratio R/- is only 0.15, the source fluid may well have a temperature of several hundred degrees Celsius, while its salinity may not be much greater than that of the lowest brine. It is likely that the Red Sea brine sources have been flowing for 18,000 years and so we would expect the system to be in a steady state with fluid continually flowing over a weir at the interface height. HARTr~ANN (1980) however found that the interface height has increased since the year 1965 by 2 m; this would seem to cast doubt on the applicability of a steady-state model to the Red Sea brine pools. The observed unsteadiness makes detailed application of our model unwarranted, but we nonetheless expect the above constraint on source properties to apply, namely the source fluid is not much more saline than the lowest brine while it can be considerably hotter. MONIN and PLAKHm (1982) have documented many diffusive interfaces in the Red Sea with Rp near 18 and this suggested our model (in the Appendix), which shows that even a small entrainment ofinterfacial fluid into each of the well-mixed layers can explain the diffusive, double-diffusive convection process at such large values ofRp. REFERENCES

BAritES W.D. (1975) Entrainment by a plume or jet at a density interface. Journal of Fluid Mechanics, 68, 309-320.

Sulphide deposits from hot saline fluid

167

BA|NES W. D. and J. S. TuaNl~a (i 969) Turbulent buoyant convection from a source in a confined region. Journal oJ Fhdd Mechanics, 37, 51--80. EOMOND J. M. (1981) Hydrothermal activity at mid-ocean ridge axes. Nature, London, 290, 87-88. GRtFFtTHS R. W. (1979) The transport of multiple components through thermohaline diffusive interfaces. DeepSea Research, 26, 383-397. I-L~RTMANN M. (1980) Atlantis !1 deep geothermal brine system. Hydrographic situation in 1977 and changes since 1965. Deep-Sea Research, 27, 161-171. HUPPERT H. E. (1971) On the stability of a series of double-diffusive layers. Deep-Sea Research, 18, 1005-1021. KANTHA L. H. (1980) Turbulent entrainment at a buoyancy interface due to convective turbulence, in: Fjord oceanograph.r.H.J. FREELAND, D.M. FARMER and C. D. LEVINGS, editors, Plenum Press, New York, pp. 205-213. LARGE R. R. and R.A. BOtH (1980) The volcanogenic sulphide ores at Mount Chalmers, Eastern Queensland. Economic Geology 75, 992-1009. LINon~ P. F. and T. G. L. SHmrCtaFFE (1978) The diffusive interface in double-diffusive convection. Journal of Fluid Mechanics, 87, 417-432. MARMOatNO G.O. and D.R. C^LDWELt. (1976) Equilibrium heat and salt transport through a diffusive thermohaline interface. Deep-Sea Research, 23, 59-68. MONtN A. S. and E. A. Pt~KmN (1982) Stratification and space-time variability of Red Sea hot brines. Deep-Sea Research, 29, 1271-1291. PHtLLtPS O. M. (1966) On turbulent convection currents and the circulation of the Red Sea. Deep-Sea Research, 13, 1149-1160. SCHOELL M. and M. HARTMANN(1973) Detailed temperature structure of the hot brines in the Atlantis 1I deep area (Red Sea). Marine Geology, 14, 1-14. SOLOMON M. (1983) Fluid inclusions in exhalative ore deposits and the origin of stockworks. In: Fluid inclusions--their nature and investigation, A.M. EVANS, editor, in press. STULL R. B. (1976) The energetics of entrainment across a density interface. Journal ofA tmospheric Sciences, 33, 1260-1267. TURNER J. S. (1973) Buoyancy effects influids, Cambridge University Press, 367 pp. TURNER J. S. and L. B. GOSTAFSON(1978) The flow of hot saline solutions from vents in the sea floor--some implications for exhaladve massive sulphide and other ore deposits. Economic Geolog); 73, 1082-1100. VOORHIS A. D. and D.L. DORSON (1975) Thermal convection in the Atlantis II hot brine pool. Deep-Sea Research, 22, 167-175. WELANDER P. (1982) A simple heat-salt oscillator. Dynamics of Atmospheres and Oceans, 6, 233-242. APPENDIX

A model of a diffusive thermohaline interface We modify LINDEN 8Ild SHlgTCLIPFE'S (1978) mechanistic model of a diffusive interface by including a small amount of entrainment at the upper and lower edges of the interfacial region. This significantly extends the range of stability ratio Rp, where steady convection is possible from r -½ (where r is the ratio of the diffusivity of salt to that of heat, Xs/Kr), which is about 9 for the heat-salt system, to beyond 2r -~ . The main result is that it is no longer necessary to regard an observed diffusive interface with Rp > 9 (e.g., in the Red Sea hot brine pools) as necessarily being unsteady and extraordinary. Figure AI shows the assumed prof'des of temperature, salinity, and density through the diffusive interface. The sharp steps of properties at the upper and lower edges of the interfscial region are the sites of intermittent growth of boundary layers by diffusion of each property. The diffusion part of the process is followed by the removal of the boundary region, whereupon the prof'des of Fig. AI are re-established. Linden and Shirtcliffe assumed that the density step J~p at the edges of the interface (Fig. A l ) was zero, giving { a ~ T = J ~ S , whereas we assume that ~s a~T

= P,

(A1)

where P ~ 1. Figure A2a shows the error function profiles of aT and pS that form by diffusion of both temperature and salinity above the top step region of the interface, and Fig. A2b shows a triangular spproxinmtion to the error function profiles that preserves the area under each profde. In this way, the diffusion distance on the vertical side of the triangle is 4

4

168

T . J . McDouGALL

!

1 '

Fig. AI.

Sketch of the temperature, salinity, and density profiles in the model of a diffusive interface. Note the steps in properties at the edge of the linear gradient region.

after a diffusion time t,. The buoyancy flux ratio, R / i s given by the ratio of the areas of the triangles in Fig. A2b, namely

fl~s [ xs \½ The ratio of the fluxes of density due to salt and heat must be the same as that due to molecular diffusion down the smooth gradients in the centre of the interface, i.e., Pr ½ = x s fl(AS - 3S) x r a(AT-

~T)

(A3)

where AS and AT are the total differences of salinity and temperature between the well-mixed layers. On rearranging the equation we find an expression for the ratio of the temperature step ( ~ T ) at the edge of the interface to the total AT 6T

[ 1 - (r½Rp/P)]

AT

(1 - r ½)

-"

(A4)

where R,, = fl A S / a A T is the density anomaly ratio between the well-mixfd layers. The model will then predict that ~ T ~ 0 and hence the flux of heat tends to zero as Rp -, R~ = Pr --t. We now estimate P from the following energy argument. After the properties have diffused for time t, ~ is assumed that the staticly unstable region of Fig. A2b exceeds a stability criterion based on a Rayleigh number and undergoes a 'thermal burst' in a similar fashion to Linden and Shirtcliffe's model. The amount of density deficiency causing the burst is given by the area A 1 in Fig. A2b and is restricted to distances z > z c above the initial position of the sharp steps at z -- 0. The flux of buoyancy into the upper well-mixed layer will drive a certain level of turbulent activity in the layer and some of the kinetic energy will he used to entrain fluid from 0 < z < z c into the upper well-mixed layer. The energy used to entrain the relatively dense fluid into the well-mixed layer is assumed to he a constant fraction k of the energy put into the well-mixed layer by the prior release of the buoyant fluid from the region z > z c. The energy argument is similar to that used to estimate the entrainment at the atmospheric inversion layer caused by a flux of buoyancy into the atmospheric mixed layer from the ground.

169

Sulphide deposits from hot saline fluid

(a)

Z ¸

z

Zc--

(b)

-

t=O

t=O

aT

aT

,8S

BS

Fig. A2. (a) Profiles caused by the diffusion of heat and salt from the initially sharp steps ½~iTand ½~S at t = 0. The error function profiles are approximated by triangular shapes in (b).

KArCrHA (1980) estimated k to be 0.2 in a laboratory experiment, and STULL (1976) found that k = 0.1 agrees with various data sets from the atmosphere. In terms of Fig. A2b, the above energy argument implies that the excess density given by area A 2 is a fraction k of the density deficiency given by area A l, which drives the whole process. The geometry of the figure gives ( P - 1)

4

Zc = -'~ (Xrt*)} I(P/r ~') - l l

(AS)

and using this, the ratio o f the areas can be found to be A2 - -

r ½ ( p - 1) 2 =

k --

A1

(A6)

P (I-r~') 2

This is a quadratic equation for P and the solution with P > 1 is given by

P= 1 +--+

+ l,

where l = kr-4(l - r ½ ) 2.

(A7)

2 For small valu~ o f k and hence ~ ( P - 1) ~ / ~ , while for large values of k, (P - 1) ~ L For example, with the heat--~alt system P = 1 for k = 0, P = 1.8 at k = 0.05, P = 2.27 at k = 0.1, and P = 3.10 at k = 0.2. It is difficult to estimate a value o f k that is appropriate to this situation because the source of the buoyancy flux is the same region as that where the entrainment is occurring, but a value close to k = 0.1 is probably realistic. This leads to the 'critical' value o f the dendty anomaly ratio, R c , equal to 2.27 x 9 g 20 and a buoyancy flux ratio R f o f 0 . 2 5 (from A2). Choosing a lower value o f k o f say 0.05 still leads to quite a large Rpc of 16.2 and a buoyancy flux ratio of 0.20. The flux of heat through the interface can be obtained by applying a critical Rayleigh number criterion to the intermittently unstable boundary layer. Following Linden and Shirteliffe, we define the relevant length scale that appears in the Rayleigh number as (1 - r~)(rtrrt,) t. The depth-integrated density deficiency at time t, is given by the area A I in Fig. A2b. The intermittent sweeping away of fluid is assumed to occur when a critical Rayleigh

170

number

1". J. McDOUGALL

Rc is reached. This leads to the following expression for the flux of heat 1 ( g )~,I-(r½Rp/P)I 4/3 a F r = ~ Kr ~xr ii_(r~/P)l ~ (aAT) 4/3,

(AS)

which reduces to Linden and Shirtcliffe's expression when P = i. The main conclusion from this model is that diffusive interfaces can exist in a quasi-steady state for We have shown that a small amount of entrainment at the upper and lower edges of the interracial region can extend the range for a steady diffusive interface from 9 to >20 (for the heat-salt system).

Rp > r-~.

of Rp