- Email: [email protected]
Lead-induced convection: a laboratory perspective
Journal of Marine Systems, 4 (1993) 217-230
217
Elsevier Science Publishers B.V., Amsterdam
Lead-induced convection: a laboratory perspective H.J.S. F e r n a n d o and C.Y. Ching Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ 85287-6106, USA (Received September 22, 1992; revised and accepted January 15, 1993)
ABSTRACT
Fernando, H.J.S. and Ching, C.Y., 1993. Lead-induced convection: a laboratory perspective. J. Mar. Syst., 4: 217-230. The polar ice cap often cracks and forms long, narrow channels of open water which are known as leads. These openings play an important role in the heat budget and circulation of polar oceans. In the winter the opening of a lead is associated with the onset of turbulent convection above and below the lead, because of the refreezing of surface water and rejection of heat into the atmosphere, respectively. Lead-induced convection is associated with a rich variety of fluid-dynamical phenomena, yet studying them in field situations is not easy due to the unpredictability of their occurrence and short resident times. As such, laboratory experiments and numerical models can play a major role in lead-related studies. The purpose of this paper is to review the results of some previous laboratory experiments, which appear to be of importance in guiding and interpreting field experiments, and to present the results of some new laboratory experiments dealing with lead-induced motions.
Introduction Leads are sporadic openings of the polar ice cap that form mainly due to the divergence of the ice drift. In general, Arctic pack ice is an aggregate of many separate floating ice sheets and, due to significant deformation, these pieces override one another, breaking off blocks and forming long piles of rubble called pressure ridges. During such deformations, open water is exposed in the form of long, narrow channels of widths of the order of a few to hundreds of meters and lengths from several to hundreds of kilometers. The average distance between the leads vary from 5 km in the marginal ice zone to about 275 km in the central Arctic pack (Wadhams, 1981). Typically leads occupy about 1% of the total area of the ice cap, but they account for about 20% of the new ice production (Koerner, 1973) and a major fraction of the total heat transfer. According to Maykut (1978), thin ice with a thickness less than 1 m contributes as much to heat transfer
as thick ice, although the former accounts for only about 10% of the total area. Most of the species transfer, for example, CO2, moisture and other aerosols, in polar regions takes place through open leads. In the winter, as soon as a lead is opened, refreezing of surface water is initiated, owing to the high temperature differences between the open water and ambient air. Initially, a thin ice layer is formed, which usually breaks in the presence of the wind and releases sea-salt droplets into the air. Frazil ice and skim ice are common features in leads, and sometimes the latter is piled up to produce ridges and hummocks of ice. In the absence of background wind forcing, refreezing proceeds with a decreasing rate. and eventually produces a uniformly thick coating of ice. The resident time of leads can vary from a day to several weeks, depending on the background environmental conditions. Leads should be contrasted from polynyas, which are openings of the ice pack that are al-
0924-7963/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
218
ways partially or totally ice free, or produce icefree conditions recurrently at the same location. Typically they are of rectangular or oval shape and range in size from a few hundred meters to hundreds of kilometers. According to Smith et al. (1990), polynyas can be formed by two mechanisms. In the first, ice can be formed within the region and can be continually removed by winds a n d / o r currents, and the associated latent heat maintains open water (latent-heat polynyas). In the second, oceanic heat sources produce enough heat to maintain certain regions ice free (sensible-heat polynyas). The open waters of polynyas and leads provide sites for brine formation in the polar seas, waterways for navigation without requiring heavy ice breakers, a feeding ground and corridors for migration of polar mammals and a natural laboratory containing a rich variety of biological, chemical and physical processes of interest to oceanographers. Lead-induced convection The key hydrodynamical processes which are operative at leads are driven by the energy derived from thermal convection. Open surface water at a lead is warmer ( ~ - 2 ° C ) compared to the contiguous atmosphere ( ~ - 3 0 ° C ) and, as soon as a lead is opened, the air (water) column above (below) the surface becomes unstable and forms turbulent convective motions. The drop of saline water temperature below the freezing point causes the formation of ice. During this process heavy brine is released as a buoyant mass of water; streams of brine can descend into the interior of the ocean. As is evident from Foster (1972) and Atkinson and Wake (1987), and is discussed by Rudels (1990), initially, the brine forms steady descending (perhaps laminar) streamers, which gradually merge to form largescale turbulent plumes. The wave length of the streamers is the same as that of the most unstable wave mode predicted by stability theory. Lead-induced plumes in oceanic and atmospheric sides are subjected to a variety of background forcing and modifications. Figure 1 shows a schematic of the movement of plumes and their modification by factors such as ice drift (with a
H,J.S.
FERNANDO
AND
C.Y.
CHINCl
,o.i ,p
In;cF,ll,~
-
--J
. clouo~ ~ , J r7 _ _ _ ~ e n g ~ o l J
level
UI(Z ~
m~xea lay(
Fig. 1. A schematic diagram of the convective processes occurring at or near a winter polar lead.
velocity Vi), background (e.g., tidally-induced) turbulence, ice-water and ice-air boundary-layer turbulence, presence of atmospheric and oceanic inversion zones (pycnoclines) and condensation of water vapor. As depicted in Fig. 1, the atmospheric-side convection is driven by the temperature difference between open water and air. The heated air carrying water vapor and other aerosols rises as a turbulent plume, while being subjected to ambient mean wind [Ul(z)] and background turbulence generated by the ice-air boundary layer. If these adverse effects can be overcome, the plume finally arrives at and penetrates into the inversion zone, rebounds and then spreads at its equilibrium density level as a gravity current of velocity Uf. During this interaction process, internal waves can be generated and propagated away. The rising moist, warm air may pass through the condensation level, thus forming clouds (Holmgren and Spears, 1974); the resulting cloud-top convection may carry moisture and other aerosols further into the upper stable layers. In this case the geostrophic winds can transport such matter over long distances and the lead convection can have a significant effect on the climatology of polar regions. If the moisture mixes with dry air advecting over the lead, the condensation (cloud formation)
LEAD-INDUCED CONVECTION: A LABORATORY PERSPECTIVE
may not be possible. Aircraft observations have reported the penetration of Arctic lead(polynya)induced plumes up to depths of 4 km vertically (Schnell et al. 1989). The impingement of plumes on the inversion also causes turbulent mixing, which results in the growth of the depth of the surface mixed-layer. During this phase, warm, humid stable air is entrained into the cold mixing layer and causes supersaturation of the lower atmosphere with respect to water vapor. Ice crystals may form as a result, thus increasing the haziness of the surrounding atmosphere. Excellent reviews on the oceanography and meteorology of leads can be found in Untersteiner (1986), Smith et al. (1990) and Morison et al. (1992). Processes similar to that shown in Fig. 1 occur in the ocean-side of a lead except those corresponding to the condensation of water vapor. Further, in this case, the conditions near the plume source are more complex due to the changes of phase and the rejection of salt. Studies on l e a d - i n d u c e d convection
Several field observational, numerical and laboratory studies have been carried out to investigate convective processes occurring under leads. In the field observational context, such studies have been often plagued by the short resident times of leads, the unpredictability of their occurrence and other logistical constraints. Thus, many available data have been taken in an opportunistic basis, for example when a lead is opened unexpectedly amidst of an ongoing oceanographic study. Such observations have provided some fragmentary evidence for the processes described in Fig. 1. For example, during the Arctic Mixed Layer Experiment (AMLE), that was designed to study the ice-ocean boundary-layer, a lead opened up thus providing an opportunity to make detailed measurements in its surrounding areas (Morison, 1978; Morison et al., 1992). The resulting measurements indicated the deepening of the upper mixed layer, and, the presence of an inward directed jet close to the water surface and an outward flowing jet near the base of the mixed layer. Similar observations have been made during the Arctic Ice Dynamics Joint Experiment
219
(AIDJEX), whose oceanographic component was dedicated to study lead-related processes (designated as ALEX deployments). ALEX 1 was primarily used for instrument calibration and checking. In ALEX 2-4, various other processes, such as the presence of inward jets in the mixed layer, outward flowing gravity currents above the thermocline, the penetration of plumes into the thermocline and the possibility of the modification of the jet structure by the earth's Coriolis effects, were noted (Smith, 1974). In an earlier AIDJEX pilot study, Smith (1973) has found evidence for the existence of a sub-surface jet structure associated with leads. A more concerted effort in understanding lead-related processes has been made during the LEADEX experiment which was completed in April, 1992. The findings of this study are yet to be reported. Several numerical studies on lead-induced convection have been reported. The two-dimensional numerical models of Lo (1986) and Shreffler (1975) have mainly focused on the modification of the atmospheric boundary layer by lead convection. In these studies, the governing equations were solved using eddy-diffusivity closure; quantities pertinent to convection, internal boundary layers, and their dependence on such quantities as wind fetch and roughness height over the leads, were studied. In the oceanic context, the thermodynamic model of Schaus and Gait (1973) treats lead convection as a purely diffusive process superimposed on a constant current; the dynamical interaction between the convection and mean flow was neglected. The model of Kozo (1983) is more elaborate, in the sense that ocean circulation induced by the brine rejection at a lead is modeled using a semi-empirical set of equations. It also included the effects of background mean flow and rotation. The model was run for specific cases of interest, which showed the existence of symmetric convective circulation cells below the lead axis during the refreezing and deepening of the mixed layer. A two-dimensional hydrostatic level model that accounts for atmospheric cooling, ice formation and brine rejection has been recently developed by Smith and Morison (1993). Glendening and Burk (1992) have reported a three-dimensional large
220
H..I.S. F E R N A N D O A N D C.Y. CHIN(~
eddy simulation model to predict the turbulence and upward heat flux which occur downstream of a lead; they found that the turbulent quantities are more pronounced at some distance away from the lead. In large-scale models, the effects of leads must be parameterized as sub-grid scale quantities because they are small-scale features. Using a coupled climate-ocean model, Ledley (1988) studied the influence of leads explicitly, and showed that they can have an appreciable influence on the zonally-averaged surface air temperature. To our knowledge, no laboratory experiments have been reported to model lead-induced convection per se, but there are related studies that can throw light on this problem. In this paper, some such experiments are reviewed and new experimental results are presented. In general, the governing parameters for lead convection are determined by the scales of motion. If the length of the lead L is much greater than its width W, and if the depth D of the thermocline is large compared to W, then leads can be modeled as line buoyancy sources, say of buoyancy flux per unit length qo" On the other hand, when W/D > 1, the convection can be represented as due to a uniformly-distributed buoyancy flux qo over the lead surface. When L is finite, the end effects of the leads are also important. In the remainder of this paper, we will mainly concentrate on laboratory modeling of
plume generator
leads as line buoyancy sources, paying attention to a single physical effect at a time. In addition, previous laboratory results pertinent to convection induced by line thermals and horizontally homogeneous sources are presented. In applying such results to field situations, the conditions under which they are valid should be borne in mind and the appropriate modeling approach should be identified.
Laboratory experiments Descent of the initial front In a series of laboratory experiments, the dense brine plumes that are formed in the ocean-side during lead refreezing were simulated by using a line source of salt water introduced onto a layer of fresh water. The experiments were performed in a Plexiglas tank of dimensions 243 x 31 x 61 (cm) with a total water depth of 60 cm. A twolayer stratification was used with three upperlayer depths D = 20, 30 and 40 cm; see Fig. 2. The line plume was generated by slowly releasing a stream of salt water onto the freshwater layer; a carefully designed plume generator, which spans the entire width of the tank and is located along the short dimension of the tank, at the plane of symmetry, was used. The development of the plume was monitored using the conventional laser-induced fluorescence technique with a sheet
I flow meter
\~ I flow ~
X l
/
control
valve
conductivity probe
Pl £3 laser sheet ~
, ' J " ~ - ~ Uf
'" P2
Ub
~ Z
Fig. 2. A schematic of the experimental apparatus.
LEAD-INDUCED CONVECTION: A LABORATORY PERSPECTIVE
of laser light passing through the center of the tank along its long dimension. The plume fluid was added with fluorescein dye. Figure 3a shows the development of a starting plume. Quantities such as the rate of descend of the plume, the half size of the plume cap lp, the half plume width at the impingement on the density interface 1D, and the depth of penetration of the plume into the pycnocline could be obtained using time-lapse video records. On dimensional grounds, the vertical position zf of the front at a time t after the release can be represented by Zf = C l q l / 3 t ,
(1)
where c~ is a constant; it was assumed that the turbulent line plumes can be described using the single parameter q p (buoyancy flux per unit length). Similarly it is possible to a r g u e lp ~ Zf and l o ~ D. The experiments show c 1 = 1 (Fig. 3b) and l v -- 0.33zf (Fig. 3c), which are consistent with the previous measurements of Tsang (1970). Further, it was found that l D - 0.16D. The above observations are of use in determining whether a lead should be treated as a plume (a continuous source of buoyancy) or a thermal (a buoyant blob of fluid). If the closing time of a lead is Tc, then the plume-like behavior can be expected when D/clqlp/3 << Tc or vice versa. For typical leads q pl / 3 = 8 × 10 2 ms -1, D = 3 0 m and T ~ = 2 0 h, which indicate that the line plume approximation is satisfactory. A detailed treatment of line plumes in confined homogeneous environments is given in Baines and T u r n e r (1969). Similarly, if a line thermal is considered then it should be represented by its initial total buoyancy qT (Turner, 1973), and analogous to Eq. (1) it is possible to write zf = c2qlT/3t 2/3. Previous experiments of Noh et al. (1992) indicate that the constant c 2 = 2. Fernando et al. (1991) have studied the development of turbulent thermal convection over a horizontal flat surface; in this case the governing variable is the buoyancy flux per unit area qo, and the frontal propagation speed is described by zf = c3(qot3) 1/2, where c 3 - 0.3. It is also possible to expect that, at small z, the plume problem can be treated as a horizontally homogenous one with Z f "~ t 3/2, whereas at large z, zf ~ t can be expected in view of z >> W, when D >> W.
221
Based on the previous experiments on turbulen! jets, this transition is expected to occur at zf ~ (3-5)W. Interaction o f plumes with a density interface
Upon impingement, the descending plume cap (and the fluid that follows) penetrates into the interface, rebounds and then propagates outwards as a gravity current (Fig. 1). For a two-layer fluid with a buoyancy jump of Ab across the layers, the depth of penetration d can be estimated approximately using a balance of potential energy stored during the deformation of the pycnocline and the vertical kinetic energy of the plume cap before the impingement. Here, Ab = g A p / p 0 , where, g is the gravitational acceleration, Ap is the density jump across the layers and P0 is a reference density. Using the scaling w ~ ql/3 for the characteristic vertical velocity of the P cap, it is possible to write this balance as a.102 / 3 ~ 8Ab, or ~ / l 0 = c4R/-1, where Ri = A b l D / q 2/3 is the bulk Richardson number and c 4 is a constant. If the thermocline is of finite thickness h, then the proposed balance becomes --~p a 2/3 ~ ( A b / h ) ~ 2, or ~ / l o = c s R i - 1 / 2 ( h / l o ) 1/2. Figure 4a shows a plot of ~ / l o versus Ri for the experiments carried out using very thin interfaces (i.e., (5 >> h); good agreement with the proposed R i - ~ scaling can be seen for Ri < 5, with c 4 - 4. For the case of a thick interfaces, ~ / h < 1, the results are presented in Fig. 4b, and good agreement with the latter scaling can be found, with c 5 -- 4. For the case of thermals penetrating into twolayer fluids, the appropriate Richardson number is defined as Ri = A b l o / w 2, where w o is the velocity of the thermal before the impingement. Dimensional arguments suggest w o = C6qlT/2 D-1/2, and the laboratory experiments of Noh et al. (1992) and Richards (1963) indicate that c 6 1.93 and c 4 = 2. As is evident from Fig. 5, the evolution of the gravity current after the impingement depends on Ri (note that the external parameters for the problem are qp, D and Ab, and D c c l D so that Ri is the governing non-dimensional parameter). Observations show that when Ri > 10, the penetration 8 is small and the plume cap splits into
222
HA.S. FERNANDO
(b) ~/
~>"
,' "
/
4
Zf (era) / ±
/
....
o
I
I
I
lO
20
30
qO
qpl/3t (cm) 25
20
15
] ,
p
tcm
)
AND
C.Y. CHIN(I
two separating vortices, which, upon loosing their initial momentum, collapse to form gravity currents propagating over the interface. These gravity currents have such typical features as a raised-head structure and Kelvin-Helmholtz billows behind the head (c.f., Simpson 1982), but their nose angle is much smaller than the prediction (60°) of the inviscid theory. In the range 1 < Ri < 5, the penetration 8 is significant and the nose angle is close to 60°. When 0.5 < Ri < 1 the nose is symmetric with respect to the interface, possibly because of the entrainment of fluid from the lower layer. When Ri < 0.5, the gravity current never bounces back, but spreads below the pycnocline. Further discussions on the interaction of buoyant plumes with density stratification are given in Turner (1973) and Weil (1982). The experiments of Noh et al. (1992) show that, for the case of thermals, the penetration becomes significant when Ri < 5. When Ri > 10, the propagation of the gravity current shows similar behavior to that over a solid surface, except that the nose angle is smaller. According to Richards (1963), thermals descend into the lower layer indefinitely when Ri < 1.9. Deardorff et al. (1980) have studied the penetration of buoyant parcels into a linearly stratified layer that lies above a turbulently convecting layer, when the driving unstable buoyancy flux is horizontally homogeneous with a buoyancy flux per unit area qo- They found that the depth of penetration is given by 8 = D (0.21 + 1.31 R/- ~), where R i . = ( N 2 ~ ) D / w , is the bulk Richardson number, w . = (qoD) 1/3 is the convection velocity and N is the buoyancy frequency of stratification.
t0
Background velocity field 5
0
5
10
15
20
25
30
35
40
45
50
55
60
zf [cm) Fig. 3a. A photograph of a starting line plume. One cm on the figure represents 12 cm of actual distance, b. A plot of the vertical position of the plume cap zf versus ql/3t, c. The relationship between the half width of the plume cap lp and its vertical position zf.
As was pointed out earlier, the entrainment flow into the descending plume generates an inward jet directed toward the plume (of velocity Ui) close to the surface and an out-flowing gravity current above the thermocline of velocity Uf. Ching et al. (1993) have made measurements of Uf and Ui using the facility shown in Fig. 2. Their results show that Uf/q~/3= 0.95 and Uf/q~/3= 0.7 R i °17, for Ri > 5 and Ri < 5, respectively. Further, Ui/qlv/3= 0.25 for the entire Ri range.
223
LEAD-INDUCED CONVECTION: A LABORATORY PERSPECTIVE
Similar laboratory experiments for the thermal case have been reported by Noh et al. (1992); the results show Uf/(qT/D) 1/2= 0.4-0.5 and Ui/(qT/D) 1/2 0.2-0.3. =
Effects o f background rotation
Since leads are opened for periods of the order of days, it is important to consider the effects of Earth's Coriolis forces on lead-induced motions. The following discussion is based on a line plume of length L and width W. For the case of infinite L, the plume can be affected by rotation in two ways. Firstly, the turbulence within the plume can be affected by the rotation when the plume descends to a critical depth z c. Based on previous laboratory studies (for example, Fernando et al., 1991; Fernando and Ching 1993) it can be inferred that the arrest of turbulent diffusion by the background rotation occurs when the Rossby number, based on the lengthscale l and velocity scale --~p al/3 of turbulence within the plume, becomes of the order unity, viz., R o = q l / 3 / f l ~ 0(1), where f = 2fl is the Coriolis parameter and f~ is the rate of background rotation. Because l ~ z, it is possible to evaluate z c ~ q~p/3/f. Secondly, the entrainment flow of velocity Ui can be
deflected sideways due to Coriolis forces, thus generating a background mean flow in the direction of the long axis of the lead. Obtaining a perfectly two-dimensional plume in the laboratory is not possible because of end effects, which are particularly important in the presence of background rotation. Fernando and Ching (1993) have conducted a series of laboratory experiments on line plumes of finite length descending into homogeneous rotating environments. They reported that the plumes initially descend as if there is no rotation, until their growth is arrested by the Coriolis forces at a time approximately 5.8/.f, during which they descend a vertical distance 6 . 4 ( q p / f 3 ) 1/3. T h e width of the plume at this instance was found to be 2.2(qp/ f3)1/3. No dependence of these quantities on the length of the plume was noted. After this time, the entrainment flow becomes unstable and defleets sideways forming two cyclonic vortices of highly three-dimensional nature. For a typical lead yieldinga buoyancy flux of qp --~ 5 X 10 - 4 m 3 s -3, this means that the Earth's rotation ( f = 1.4 × 1 0 - 4 s - 1 ) becomes important after the plume has descended to a depth of about 3500 m. However, the thermocline in polar oceans is usually shallow (e.g., Huskins, 1974);
(b)&~
(a)&
tD
& 1D
m
~_
%
ID
~ r~.
0
%)
tO-i
I
I
I
I I III
I
I
i
2
3
q 5 6789
2
3
q 5 6789
I
100
Ri
i
i
i o
i i
O
I
2
I01
@
10-2
3
q 5 6789
2 10-i
i
i
3
q 5 6789
I
I I i I
100
Ri ~(~n)'~
Fig. 4a. A plot of the normalized depth of penetration of the plume cap into the interface gilD versus the bulk Richardson n u m b e r this case the interface is thinner than the depth of penetration, b. Same as Fig. 4a, but the abscissa represents the normalized variable Ri-t/2(h/lD) 1f2. In this case the interface is thicker than the depth of penetration.
Ri. In
224
ft.J.S, FERNANDO AND C.Y. (~HIN(I
where r d = Ur/f is the Rossby radius of deformation. In addition, as stated before, the entrainment flow is expected to become unstable at a time 5.8/f. The instability of the entrainment flow is also expected to produce isolated vortices of size ~ r d. The existence of sub-surface eddies under the polar ice cap has been clearly demonstrated in many studies, particularly during the A I D J E X experiment by Huskins (1974). The nature of the flow under a lead at large times is expected to depend on the type of instability that develops first. Because the time necessary for a plume to reach the thermocline is given by T,I = D/clqlo/3 and the time scale for the baroclinic instability of the gravity current is tl, = (5 - 7XUf/f)/Uf = (5 7 ) f - J , the total time for the appearance of the baroclinic eddies can be estimated as (D/clqJo/3 + (5 - 7 ) f - ~). This is usually larger than the time 5.8f-1 required for the deflection of the entrainment flow. In order to check which instability occurs first, we have extended the experiments of Fernando and Ching (1993) to realize the condition in which a plume impinges on the tank bottom before its growth is arrested by rotation. These experiments were performed in a large tank of dimensions 400 × 114.5 × 29 (cm) that was located on a rotating table. Plume generators of lengths 14 cm and 18 cm, located at the middle of the tank, with their axes aligned with the short dimension of the tank, were used; the plume was released on to the surface of a homogeneous water layer of depth 26 (cm). Flow visualization was performed using the laser-induced fluorescence technique. Figure 6 shows a sequence of photographs that was taken during the experiments. The initial spreading of the front was found to be very similar to the case of a nonrotating fluid, but the plume becomes unstable and starts deflecting cyclonically (see arrows) before the gravity current head breaks into eddies. The gravity current near the bottom deflects anti-cyclonically. In fact, the entrainment flow becomes unstable and deflects sideways soon after the plume impinges on the density interface. Similar observations were made in all the experiments that were carried out covering a range of experimental parameters. -
Fig. 5, The a p p e a r a n c e of the gravity c u r r e n t u n d e r different bulk R i c h a r d s o n n u m b e r s , T h e position of the interface is m a r k e d . (a) Ri = 53.2: (b) Ri = 1.2; (c) Ri = 0.95; (d) Ri = 0.4.
hence a plume should reach the thermocline and start spreading horizontally as a gravity current way before the rotational effects become important. This gravity current flow can become baroclinically unstable and break down into unstable vortices. Laboratory studies (e.g., Ivey, 1987) show that such instabilities develop after the gravity current propagates a distance of about (5 - 7)r~,
LEAD-INDUCED CONVECTION: A LABORATORY PERSPECTIVE
Effect of ice movement and background turbulence Drift of the ice pack owing to wind forcing causes the development of a turbulent boundary layer underneath the ice. As is shown in Fig. 1, the background mean flow causes the plumes to bend over, and the interaction with the mean flow is expected to modify their characteristics. Some information on whether the mean flow is passive (i.e., has a purely advective effect) can be obtained by considering the governing variables. If a two-dimensional line plume can be specified by two parameters, qp and W, and, if the turbulent boundary layer under the ice is governed by the ice velocity Vi, the local turbulent boundary layer thickness 6 T and the friction velocity due to under-ice stress u . , then any dependent quan-
225
tity, for example, the depth z e ( < t~T) at which the plume is destroyed, can be expressed as
Ze = f
c d,
,
(2)
where u 2 / V i 2 = c d is the drag coefficient and f , f l . . . . are functions. At high Reynolds numbers c d can assumed to be a constant and, for simplicity, when W<< 6T, the effect of W / 6 T can be neglected. Thus,
Zo
[ v?
)
On this basis the major governing parameter determining the modification of the plume is Vi3/q,. Above a critical value of this parameter,
Fig. 6. A plan view of the flow development when a line plume of finite length impinges on a solid surface and spreads horizontally in the presence of background rotation. T h e plume rea6hes the solid bottom before the background rotation arrests its growth. For this run, qp = 15.2 cm 3 s - 3 and f = 0.44 s -1. T h e pictures were taken at (a) 1.5/f; (b) 2 . 3 / f ; (c) 3 . 1 / f ; (d) 3 . 9 / f ; (e) 4.7/]'; (f) 5.5/f. O n e cm on the figure represents 10 cm of actual distance.
226
which is to be determined by experiments, the plume is expected to be modified by the turbulent boundary layer (forced convection). Cederwall (1971), Roberts (1979) and Roberts et al. (1989a, b, c) have carried out laboratory experiments on line plumes in cross flows with the goal of designing waste-water outfalls. In these experiments a line plume was towed in a quiescent environment with neutral or stable stratification. The experiments indicated that the dilution characteristics of plumes are drastically modified at higher cross flows; extrapolation of their plume dilution results to the present case indicates that the c r i t i c a l Vi3/qp should lie in the range (0.1-0.2). This implies that, for example, for a lead with qp = 5 X 10 -4 m 3 s-3 and Vi = 5 x 10 -2 m / s , Vi3/qo=0.25; according to the above criterion, the forced-convection effects are important in this case. The above analysis should be contrasted with that of Morison et al. (1993). By non-dimensionalizing the governing equations using appropriate scales, they obtained a criterion based on the lead number L o =qoD/CdVi 3 to demarcate between the forced and free convection r e g ~ e s ; note that here qo = qp/W, which is the buoyancy flux per unit area. Forced or free convection was found to prevail depending on L o < 1 or L o > 1. The present arguments assume that the plume is modified mainly within the turbulent boundary layer, and the depth of the mixed layer and the width of the plume W are of secondary importance, when D > t$T > W. If the plume can survive the adverse influence of the turbulent boundary layer, then it descends into the region outside the boundary layer which can also be turbulent (for example, owing to tidal stirring and other environmental sources). To our knowledge, so far no explicit laboratory work has been reported on the interaction of line plumes with environmental turbulence. However, some idea on the level of background turbulent intensity u required for the destruction of the plume can be obtained using the instructive laboratory experiments of Thomas and Simpson (1985). They considered the effect of background turbulence on a gravity current propagating on a solid surface, and delineated a criterion for the modiflca-
H..I.S. F E R N A N D O
A N D C.Y. C H I N G
tion of the current by the turbulence. Extension of this result to the present case suggests that the plume will be unaffected by the turbulence as long as u/qlp/3 < 0.05.
Merger of plumes The influence of one lead on another is an important aspect in modeling the circulation under the ice pack. Intuitively, it is possible to expect that the interaction between leads occurs mainly via the interaction between the entrainment flows; the depletion of fluid between leads due to entrainment causes high fluid velocities and low pressure between descending plume caps, thus resulting in the merger of plumes. Early experiments of Rouse et al. (1952) with parallel line plumes suggest that the combined plume after the merger can be treated as one having a total strength equal to the sum of the strengths of the individual plumes. Only a few studies have been reported on the interaction of turbulent line plumes in homogeneous fluids. Yoshida (i983) and Yoshida and Nagata (1986) carried out experiments on interacting parallel line plumes of equal strengths qp and identified three stages of plume evolution. In stage 1, the plumes descend vertically until the caps of the starting plumes touch each other, whence the two plumes initiate their rapid merger (stage 2). During the merger, the shallowest contact point between the plumes ascends vertically, and finally comes to a quasi-steady state (stage 3). The transition times between the stages were identified as tHi=2.12Xq~ 1/3 and t l H H ~ 4.46Xqff 1/3, where X is the half distance between the sources. In the present work, a series of laboratory. experiments were carried out in a large Plexiglas tank (180 x 15 × 120 era), using two line plumes of equal buoyancy fluxes qp; all other features of the experiment were similar to those described in the context of Fig. 1. In some experiments a homogeneous background fluid was used and in the others a two-layer fluid was employed so that the merger between the plumes occurred after their caps arrive at the density interface. Figure 7a shows the descent and merger of plumes in a
LEAD-INDUCED CONVECTION: A LABORATORY PERSPECTIVE
homogeneous deep fluid layer. One distinct observation made here, that differs from Yoshida (1983), is the initiation of merger before the plume caps come into contact. Figure 7b shows the case where the plumes impinge on the interface before merging. In this case, the space between the two plumes is filled by the plume fluid; this fluid belongs to the two counter-flowing gravity currents that form above the density interface between the plumes. After this fluid is raised to a certain height, the rapid merger is initiated. The time scale for merger Tm can be written in the form
Tm = g ( X , D, v, qp),
(4)
where g, g t . . . are functions, v is the kinematic viscosity, and the effects of the tank side walls have been neglected. Thus, we get X
where
gl
v
'
'
(5)
ql/aX/l~ is a Reynolds number, influence
227
of which can be neglected at high values of T h u s , it is possible to write
ql/3x//v. Traq1/3
-- c o n s t a n t ,
(6)
for merger in deep homogeneous fluid layers, and
x
=g2
B
,
(7)
for two-layer fluids; it is expected that, for high density stratifications, Ri > 5, the buoyancy jump across the interface Ab is unimportant and the interface acts as a rigid wall. Figure 8 shows the horizontal deviation of the center of the plume cap ¢ from the vertical centerline of the plume for experiments with homogeneous fluids. The length and time scales have been selected as X and Xqff 1/3, in view of Eq. (6). Note that this deviation has grown to )2/2 at a time of 2.0Xqp i/3, which can be considered as a time scale for the merger. The line drawn through the data points represents the best-fit line to the data as ¢ / X = 0.11(tql/3/X)2.
Fig. 7a. A photograph that shows the merger of descending parallel line plumes in a deep homogeneous fluid. For this run, qp = 15.1 cm 3 s -3. For (a) and (b), one cm represents 9 cm of actual distance, b. The merger of two line plumes, after they have impinged on a density interface. For this run, qp = 21.0 cm 3 s -3.
228
H..t.S. F E R N A N D O
oo f co
AND C.Y. CHIN(;
. . . . . . D
"I
/
Tmq~/3 X
X
o
, "j
q: --
%
©
l
o 0.0
0.8
1.6
co
2 10-1
3.2
2.q
X D
x
Fig. 9. The dependence of the normalized time for the merger of plumes Tmql/3/X on the normalized distance between the plumes X/D in the presence of a density interface,
Fig. 8. The growth of the horizontal distance between the vertical centerline of a plume and its cap ~/X with the non-dimensional time tqlo/3/X.
Figure 9 shows the variation of the normalized merger time scale with X/D for the experiments carried out with density interfaces. Here the merger time is defined as the time at which the two descending plmnes come into contact at the centefline, without having a rising fluid column between them. Note that the data is somewhat scattered, mainly because of the subjective judgment that had to be used in deciding the merger time. The best-fit line to the data indicates that
Tmqlo/3//S =
2.07 X 102(X//D)
Discussion It is instructive to discuss the possible applications of above results to oceanic lead-related situations. As mentioned earlier, the available observations are f r ~ , and a strait-laced comparison of them with laboratory observations is untenable. Some of the avait~le observations and estimated qumatitics are listed in Table 1. Note that the l e ~ ( D - . - 2 0 m) and time scales of leads are much larBer than those of the labora-
2"6.
TABLE 1
Field observations Location
Experiment
Beaufort A L E X 2 Sea, near Barrow, Alaska
ALEX 3 ALEX 4
Beaufort Sea
AMLE
References
qp(m3s -3)
D (m)
h (m)
Smith (1974) Smith et al.
~ 9 × I0 - 4
15
10
(1990) AIDJEX PI's r e p o r t (1974)
~ 5 x 10 - 4 ~ 5 x 10 - 4
16 16
Morison
~5x10
35
(1980)
Morison et al. (1992)
-4
W (m)
1/i (ms - 1 )
8 (m)
50
1
20
9 9
50 300
1 1
-
20
1400
-
-
Ui(nas ~1)
Ab(ms -l)
Ri
-
3
x 10 - 3
0.8
( 5 - 8 ) × 10 - 2 -
3 x 10 - 2
1 × 10 - 2 1.2 × 10 - 2
4.0 4.8
2x10 -2
2x10 -2
5
4.4
Uf (ms - i ) -
Xl0 -3
LEAD-INDUCED CONVECTION: A LABORATORY PERSPECTIVE
229
TABLE 2 Predictions Experiment
8 (m) based on qp
ALEX3 ALEX4 AMLE
~9 ~9 ~ 20
8 (m) based on
Uf(ms-')
Ui(ms-')
7 x l O -2 7 x l O -2 7 x l O -2
2 × 1 0 -2 2 × 1 0 -2 2 x l O -2
qo(= qp/W) 4.1 3.6 7.5
tory experiments ( D ~ 0.3 m), which naturally raises the question of the applicability of the laboratory results to the ocean. For laboratory experiments q p ~ 5 × 10 -5 m3s -3 and for the ocean qp ~ 5 x 10 -4 m 3 s -3 (Morison et al., 1992); hence, the Reynolds numbers for these cases are of the order 104 and 106, respectively. Since the plumes are turbulent in both cases, based on Reynolds number similarity, it is possible to argue that the laboratory results should have applicability to oceanic situations. Another question of importance is how to model a lead. Earlier, it was argued that a lead ought to be treated as a line plume if W << D, but in most oceanic cases W ~ D. In such cases it is advisable to treat the descending plume as if it is arising from a horizontally homogeneous source with a buoyancy flux per unit area qo. Of course, after the plumes impinge on the interface, the horizontal inhomogeneities need to be considered. Future experiments should investigate the effect of finite W on the evolution of line plumes. Table 2 shows the laboratory-based predictions for the field experiments depicted in Table 1; the agreement appears to be fair (because the stratification observed in A L E X 2 is not compatible with the laboratory experiments described herein, it was omitted from Table 2). However, it should be borne in mind that these comparisons have been made by assuming that the intensity of background turbulence is weak, free convection dominates beneath the lead and the observations were made before the rotational effects become important. No substantial evidence is available to corroborate such assumptions, and hence the comparison between the two cases needs to be done with caution.
q~/3D/v
Acknowledgments We wish to thank the Office of Naval Research (Arctic Sciences and Small-Scale Oceanography) and the National Science Foundation for the financial support. Also, we were immensely benefited by the assistance of Dr. Linus Mofor, Prof. D.F. Jankowski, Dr. P.A. Davies, Mr. Leonard Montenegro and G. Oth in numerous ways.
References AIDJEX Principal Investigators Report, 1974. AIDJEX Bull., 27: 117-120. Atkinson, J.F. and Wake, A., 1987. Double-diffusive convection under sea ice. J. Cold Regions Eng., 2: 89-95. Baines, W.D. and Turner, J.S., 1969. Turbulent buoyant convection from a source in a confined region. J. Huid Mech., 37(1): 51-80. Cederwall, K., 1971. Buoyant Slot Jets into Stagnant or Flowing Environments. W.M. Keck Lab. Hydraul. Water Resour., California Inst. Technol. Rep. No. KH-R-25. Ching, C-Y., Fernando, H.J.S. and Noh, Y., 1993. Interaction of the negatively buoyant line plume with a density interface. Dyn. Atmos. Oceans (accepted). Deardorff, J.W., Willis, G.E. and Stockton, B.H., 1980. Laboratory studies of the entrainment zone of a convective mixed-layer. J. Fluid Mech., 100: 41-64. Fernando, H.J.S. and Ching, C.Y., 1993. Effects of background rotation on turbulent line plumes. J. Phys. Oceanogr. (Accepted) Fernando, H.J.S., Chen, R.-R. and Boyer, D.L., 1991. Effects of rotation on convective turbulence. J. Fluid Mech., 228: 513-547. Foster, T.D., 1972. Haline convection in polynyas leads. J. Phys. Oceanogr., 2: 462-469. Glendening, J.W. and Burk, S.D. 1992. Turbulent transport from an Arctic lead: A large eddy simulation. BoundaryLayer Meteoroi., 59: 315-339.
230 Holmgren, B. and Spears, L. 1974. Sodar investigation of the effect of open leads on the boundary-Layer structure over the Arctic Basin. AIDJEX Bull., 27: 168-179. Huskins, K.L., 1974. Subsurface eddies in the Arctic Ocean. Deep-Sea Res., 21: 1017-1033. Ivey, G.N., 1987. Boundary mixing in a rotating stratified fluid. J. Fluid Mech., 183: 25-44. Koerner, R.M., 1973. The mass balance of the sea ice of the Arctic Ocean. J. Glaciol., 12(65): 173-185. Kozo, T.L, 1983. Initial model results for Arctic mixed layer circulation under a refreezing lead. J. Geophys. Res., 88: 2926-2934. Ledley, T.S., 1988. A coupled energy balance climate-sea ice model; Impact of sea ice and leads on climate. J. Geophys. Res., 93: 15,919-15,932. Ix), A.K., 1986. On the boundary-layer flow over a Canadian archipelago polynya. Boundary-Layer Meteorol., 35: 53-71. Maykut, G.A., 1978. Energy exchange over young sea ice in the central Arctic. J. Geophys. Res., 83: 3646-3658. Morison, J.H., 1978. The Arctic profiling system. Proc. Working Conf. Current Measurements, Univ. Delaware, pp. 313-318. Morison, J.H., 1980. Forced Internal Waves in the Arctic Ocean, Ph.D. Thesis, Dep. Geophys. Univ. Washington, Seattle. Morison, J.H., McPhee, M., Curtin, T. and Paulson, C., 1992. The oceanography of winter leads. J. Geophys. Res., 97 (C7): 11,199-11,218. Nob, Y., Fernando, H.J.S. and Ching, C.Y., 1992. Flows induced by the impingement of a two-dimensional thermal on a density interface. J. Phys. Oceanogr., 22(10): 12071220. Richards, J.M., 1963. Experiments on the motion of isolated cylindrical thermals through unstratified surroundings. Int. J. Air Water Pollut., 7: 17-34. Roberts, P.J.W., 1979. A mathematical model of initial dilution for deepwater ocean outfaUs. Proc. Specialty Conf. Conserv. Util. Water Energy Resour. San Francisco, Aug. 8-11, pp. 218-225. Roberts, P.J.W., Snyder, W.H. and Baumgartner, D.J., 1989a. Ocean outfalls. I: Submerged wastefield formation. J. Hydraul. Eng., 115(1): 1-25. Roberts, P.J.W., Snyder, W.H. and Baumgartner, D.J., 1989b. Ocean outfails. II: Spatial evolution of submerged wastefield. J. Hydraul. Eng., 115(1): 26-48. Roberts, P.J.W., Snyder, W.H. and Baumgartner, D.J., 1989c. Ocean outfalls. II1: Effect of diffuser design on submerged wastefield. J. Hydraul. Eng., 115(1): 49-70.
H.J.S. F E R N A N D O
A N D (~.Y. C H I N ( ;
Rouse, H., Baines, W.D. and Humphreys, H.W., 1952. Free convection over parallel sources of heat. Proc. Phys. Soc., LXVI, 5-B: 393-399. Rudels, B., 1990. Haline convection in the Greenland Sea. Deep-Sea Res., 37(9): 1491-15ll. Schaus, R.H. and Gait, J.A., 1973. A thermodynamic model of an Arctic lead. Arctic, 26(3): 208-221. Schnell, R.C., Barry, R.G., Miles, M.W., Andreas, E.L, Radke, L.F., Brock, C.A., McCormic, M.P. and Moore, J.L, 1989. Lidar detection of leads in Arctic Sea ice. Nature, 339: 530-532. Shreffier, J.H., 1975. A Numerical Model of Heat Transfer to the Atmosphere from an Arctic Lead. Ph.D. Thesis, Dep. Oceanogr., Oregon State Univ., Corvallis, 135 pp. Simpson, J.E, 1982. Gravity currents in the laboratory, atmosphere and ocean. Annu. Rev. Fluid Mech., 14: 213-234. Smith IV, D.C. and Morison, J.H., 1993. A numerical study of haline convection beneath sea ice. (In prep.) Smith, J.D. 1973. Lead-driven convection in the Arctic Ocean. EOS, 54(11): 1108-1109 (abstract). Smith, J.D., 1974. Oceanographic investigations during the AIDJEX lead experiment. AIDJEX Bull., 27: 125-133. Smith, S.D., Muench, R.D. and Pease, C.H., 1990. Polynyas and leads: An overview of physical processes and environment. J. Geophys. Res. 95(C6): 9461-9479. Thomas, N.H. and Simpson, J.E., 1985. Mixing of gravity currents in turbulent surroundings: Laboratory studies and modeling implications. In: J.C.R. Hunt (Editor), Turbulence and Diffusion in Stable Environments. Clarendon Press, Oxford, pp. 61-96. Tsang, G., 1970. Laboratory study of two-Dimensional starting plumes. Atmos. Environ., 4: 519-544. Turner, J.S., 1973. Buoyancy Effects in Fluids. Cambridge Press. Untersteiner, N. (Editor), 1986. The Geophysics of Sea Ice. Plenum Press, N.Y. Wadhams, P., 1981. The ice cover of the Greenland and Norwegian Seas. Rev. Geophys. Space Phys., 19: 345-393. Weil, J.C., 1981. Source Buoyancy Effects in Boundary Layer. Workshop on Parameterization of Mixed Layer Diffusion, Las Cruces, NM. Yoshida, J. and Nagata, Y., 1986. Velocity fields and their variation in the two-dimensional forced plume without and with a vertical wall nearby. J. Hydrosci. Hydraul. Eng., 4(1): 51-64. Yoshida, J., 1983. The behavior of a two-Dimensional dual forced plume. J. Oceanogr. Soc. Jpn., 39: 317-323.
217
Elsevier Science Publishers B.V., Amsterdam
Lead-induced convection: a laboratory perspective H.J.S. F e r n a n d o and C.Y. Ching Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ 85287-6106, USA (Received September 22, 1992; revised and accepted January 15, 1993)
ABSTRACT
Fernando, H.J.S. and Ching, C.Y., 1993. Lead-induced convection: a laboratory perspective. J. Mar. Syst., 4: 217-230. The polar ice cap often cracks and forms long, narrow channels of open water which are known as leads. These openings play an important role in the heat budget and circulation of polar oceans. In the winter the opening of a lead is associated with the onset of turbulent convection above and below the lead, because of the refreezing of surface water and rejection of heat into the atmosphere, respectively. Lead-induced convection is associated with a rich variety of fluid-dynamical phenomena, yet studying them in field situations is not easy due to the unpredictability of their occurrence and short resident times. As such, laboratory experiments and numerical models can play a major role in lead-related studies. The purpose of this paper is to review the results of some previous laboratory experiments, which appear to be of importance in guiding and interpreting field experiments, and to present the results of some new laboratory experiments dealing with lead-induced motions.
Introduction Leads are sporadic openings of the polar ice cap that form mainly due to the divergence of the ice drift. In general, Arctic pack ice is an aggregate of many separate floating ice sheets and, due to significant deformation, these pieces override one another, breaking off blocks and forming long piles of rubble called pressure ridges. During such deformations, open water is exposed in the form of long, narrow channels of widths of the order of a few to hundreds of meters and lengths from several to hundreds of kilometers. The average distance between the leads vary from 5 km in the marginal ice zone to about 275 km in the central Arctic pack (Wadhams, 1981). Typically leads occupy about 1% of the total area of the ice cap, but they account for about 20% of the new ice production (Koerner, 1973) and a major fraction of the total heat transfer. According to Maykut (1978), thin ice with a thickness less than 1 m contributes as much to heat transfer
as thick ice, although the former accounts for only about 10% of the total area. Most of the species transfer, for example, CO2, moisture and other aerosols, in polar regions takes place through open leads. In the winter, as soon as a lead is opened, refreezing of surface water is initiated, owing to the high temperature differences between the open water and ambient air. Initially, a thin ice layer is formed, which usually breaks in the presence of the wind and releases sea-salt droplets into the air. Frazil ice and skim ice are common features in leads, and sometimes the latter is piled up to produce ridges and hummocks of ice. In the absence of background wind forcing, refreezing proceeds with a decreasing rate. and eventually produces a uniformly thick coating of ice. The resident time of leads can vary from a day to several weeks, depending on the background environmental conditions. Leads should be contrasted from polynyas, which are openings of the ice pack that are al-
0924-7963/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
218
ways partially or totally ice free, or produce icefree conditions recurrently at the same location. Typically they are of rectangular or oval shape and range in size from a few hundred meters to hundreds of kilometers. According to Smith et al. (1990), polynyas can be formed by two mechanisms. In the first, ice can be formed within the region and can be continually removed by winds a n d / o r currents, and the associated latent heat maintains open water (latent-heat polynyas). In the second, oceanic heat sources produce enough heat to maintain certain regions ice free (sensible-heat polynyas). The open waters of polynyas and leads provide sites for brine formation in the polar seas, waterways for navigation without requiring heavy ice breakers, a feeding ground and corridors for migration of polar mammals and a natural laboratory containing a rich variety of biological, chemical and physical processes of interest to oceanographers. Lead-induced convection The key hydrodynamical processes which are operative at leads are driven by the energy derived from thermal convection. Open surface water at a lead is warmer ( ~ - 2 ° C ) compared to the contiguous atmosphere ( ~ - 3 0 ° C ) and, as soon as a lead is opened, the air (water) column above (below) the surface becomes unstable and forms turbulent convective motions. The drop of saline water temperature below the freezing point causes the formation of ice. During this process heavy brine is released as a buoyant mass of water; streams of brine can descend into the interior of the ocean. As is evident from Foster (1972) and Atkinson and Wake (1987), and is discussed by Rudels (1990), initially, the brine forms steady descending (perhaps laminar) streamers, which gradually merge to form largescale turbulent plumes. The wave length of the streamers is the same as that of the most unstable wave mode predicted by stability theory. Lead-induced plumes in oceanic and atmospheric sides are subjected to a variety of background forcing and modifications. Figure 1 shows a schematic of the movement of plumes and their modification by factors such as ice drift (with a
H,J.S.
FERNANDO
AND
C.Y.
CHINCl
,o.i ,p
In;cF,ll,~
-
--J
. clouo~ ~ , J r7 _ _ _ ~ e n g ~ o l J
level
UI(Z ~
m~xea lay(
Fig. 1. A schematic diagram of the convective processes occurring at or near a winter polar lead.
velocity Vi), background (e.g., tidally-induced) turbulence, ice-water and ice-air boundary-layer turbulence, presence of atmospheric and oceanic inversion zones (pycnoclines) and condensation of water vapor. As depicted in Fig. 1, the atmospheric-side convection is driven by the temperature difference between open water and air. The heated air carrying water vapor and other aerosols rises as a turbulent plume, while being subjected to ambient mean wind [Ul(z)] and background turbulence generated by the ice-air boundary layer. If these adverse effects can be overcome, the plume finally arrives at and penetrates into the inversion zone, rebounds and then spreads at its equilibrium density level as a gravity current of velocity Uf. During this interaction process, internal waves can be generated and propagated away. The rising moist, warm air may pass through the condensation level, thus forming clouds (Holmgren and Spears, 1974); the resulting cloud-top convection may carry moisture and other aerosols further into the upper stable layers. In this case the geostrophic winds can transport such matter over long distances and the lead convection can have a significant effect on the climatology of polar regions. If the moisture mixes with dry air advecting over the lead, the condensation (cloud formation)
LEAD-INDUCED CONVECTION: A LABORATORY PERSPECTIVE
may not be possible. Aircraft observations have reported the penetration of Arctic lead(polynya)induced plumes up to depths of 4 km vertically (Schnell et al. 1989). The impingement of plumes on the inversion also causes turbulent mixing, which results in the growth of the depth of the surface mixed-layer. During this phase, warm, humid stable air is entrained into the cold mixing layer and causes supersaturation of the lower atmosphere with respect to water vapor. Ice crystals may form as a result, thus increasing the haziness of the surrounding atmosphere. Excellent reviews on the oceanography and meteorology of leads can be found in Untersteiner (1986), Smith et al. (1990) and Morison et al. (1992). Processes similar to that shown in Fig. 1 occur in the ocean-side of a lead except those corresponding to the condensation of water vapor. Further, in this case, the conditions near the plume source are more complex due to the changes of phase and the rejection of salt. Studies on l e a d - i n d u c e d convection
Several field observational, numerical and laboratory studies have been carried out to investigate convective processes occurring under leads. In the field observational context, such studies have been often plagued by the short resident times of leads, the unpredictability of their occurrence and other logistical constraints. Thus, many available data have been taken in an opportunistic basis, for example when a lead is opened unexpectedly amidst of an ongoing oceanographic study. Such observations have provided some fragmentary evidence for the processes described in Fig. 1. For example, during the Arctic Mixed Layer Experiment (AMLE), that was designed to study the ice-ocean boundary-layer, a lead opened up thus providing an opportunity to make detailed measurements in its surrounding areas (Morison, 1978; Morison et al., 1992). The resulting measurements indicated the deepening of the upper mixed layer, and, the presence of an inward directed jet close to the water surface and an outward flowing jet near the base of the mixed layer. Similar observations have been made during the Arctic Ice Dynamics Joint Experiment
219
(AIDJEX), whose oceanographic component was dedicated to study lead-related processes (designated as ALEX deployments). ALEX 1 was primarily used for instrument calibration and checking. In ALEX 2-4, various other processes, such as the presence of inward jets in the mixed layer, outward flowing gravity currents above the thermocline, the penetration of plumes into the thermocline and the possibility of the modification of the jet structure by the earth's Coriolis effects, were noted (Smith, 1974). In an earlier AIDJEX pilot study, Smith (1973) has found evidence for the existence of a sub-surface jet structure associated with leads. A more concerted effort in understanding lead-related processes has been made during the LEADEX experiment which was completed in April, 1992. The findings of this study are yet to be reported. Several numerical studies on lead-induced convection have been reported. The two-dimensional numerical models of Lo (1986) and Shreffler (1975) have mainly focused on the modification of the atmospheric boundary layer by lead convection. In these studies, the governing equations were solved using eddy-diffusivity closure; quantities pertinent to convection, internal boundary layers, and their dependence on such quantities as wind fetch and roughness height over the leads, were studied. In the oceanic context, the thermodynamic model of Schaus and Gait (1973) treats lead convection as a purely diffusive process superimposed on a constant current; the dynamical interaction between the convection and mean flow was neglected. The model of Kozo (1983) is more elaborate, in the sense that ocean circulation induced by the brine rejection at a lead is modeled using a semi-empirical set of equations. It also included the effects of background mean flow and rotation. The model was run for specific cases of interest, which showed the existence of symmetric convective circulation cells below the lead axis during the refreezing and deepening of the mixed layer. A two-dimensional hydrostatic level model that accounts for atmospheric cooling, ice formation and brine rejection has been recently developed by Smith and Morison (1993). Glendening and Burk (1992) have reported a three-dimensional large
220
H..I.S. F E R N A N D O A N D C.Y. CHIN(~
eddy simulation model to predict the turbulence and upward heat flux which occur downstream of a lead; they found that the turbulent quantities are more pronounced at some distance away from the lead. In large-scale models, the effects of leads must be parameterized as sub-grid scale quantities because they are small-scale features. Using a coupled climate-ocean model, Ledley (1988) studied the influence of leads explicitly, and showed that they can have an appreciable influence on the zonally-averaged surface air temperature. To our knowledge, no laboratory experiments have been reported to model lead-induced convection per se, but there are related studies that can throw light on this problem. In this paper, some such experiments are reviewed and new experimental results are presented. In general, the governing parameters for lead convection are determined by the scales of motion. If the length of the lead L is much greater than its width W, and if the depth D of the thermocline is large compared to W, then leads can be modeled as line buoyancy sources, say of buoyancy flux per unit length qo" On the other hand, when W/D > 1, the convection can be represented as due to a uniformly-distributed buoyancy flux qo over the lead surface. When L is finite, the end effects of the leads are also important. In the remainder of this paper, we will mainly concentrate on laboratory modeling of
plume generator
leads as line buoyancy sources, paying attention to a single physical effect at a time. In addition, previous laboratory results pertinent to convection induced by line thermals and horizontally homogeneous sources are presented. In applying such results to field situations, the conditions under which they are valid should be borne in mind and the appropriate modeling approach should be identified.
Laboratory experiments Descent of the initial front In a series of laboratory experiments, the dense brine plumes that are formed in the ocean-side during lead refreezing were simulated by using a line source of salt water introduced onto a layer of fresh water. The experiments were performed in a Plexiglas tank of dimensions 243 x 31 x 61 (cm) with a total water depth of 60 cm. A twolayer stratification was used with three upperlayer depths D = 20, 30 and 40 cm; see Fig. 2. The line plume was generated by slowly releasing a stream of salt water onto the freshwater layer; a carefully designed plume generator, which spans the entire width of the tank and is located along the short dimension of the tank, at the plane of symmetry, was used. The development of the plume was monitored using the conventional laser-induced fluorescence technique with a sheet
I flow meter
\~ I flow ~
X l
/
control
valve
conductivity probe
Pl £3 laser sheet ~
, ' J " ~ - ~ Uf
'" P2
Ub
~ Z
Fig. 2. A schematic of the experimental apparatus.
LEAD-INDUCED CONVECTION: A LABORATORY PERSPECTIVE
of laser light passing through the center of the tank along its long dimension. The plume fluid was added with fluorescein dye. Figure 3a shows the development of a starting plume. Quantities such as the rate of descend of the plume, the half size of the plume cap lp, the half plume width at the impingement on the density interface 1D, and the depth of penetration of the plume into the pycnocline could be obtained using time-lapse video records. On dimensional grounds, the vertical position zf of the front at a time t after the release can be represented by Zf = C l q l / 3 t ,
(1)
where c~ is a constant; it was assumed that the turbulent line plumes can be described using the single parameter q p (buoyancy flux per unit length). Similarly it is possible to a r g u e lp ~ Zf and l o ~ D. The experiments show c 1 = 1 (Fig. 3b) and l v -- 0.33zf (Fig. 3c), which are consistent with the previous measurements of Tsang (1970). Further, it was found that l D - 0.16D. The above observations are of use in determining whether a lead should be treated as a plume (a continuous source of buoyancy) or a thermal (a buoyant blob of fluid). If the closing time of a lead is Tc, then the plume-like behavior can be expected when D/clqlp/3 << Tc or vice versa. For typical leads q pl / 3 = 8 × 10 2 ms -1, D = 3 0 m and T ~ = 2 0 h, which indicate that the line plume approximation is satisfactory. A detailed treatment of line plumes in confined homogeneous environments is given in Baines and T u r n e r (1969). Similarly, if a line thermal is considered then it should be represented by its initial total buoyancy qT (Turner, 1973), and analogous to Eq. (1) it is possible to write zf = c2qlT/3t 2/3. Previous experiments of Noh et al. (1992) indicate that the constant c 2 = 2. Fernando et al. (1991) have studied the development of turbulent thermal convection over a horizontal flat surface; in this case the governing variable is the buoyancy flux per unit area qo, and the frontal propagation speed is described by zf = c3(qot3) 1/2, where c 3 - 0.3. It is also possible to expect that, at small z, the plume problem can be treated as a horizontally homogenous one with Z f "~ t 3/2, whereas at large z, zf ~ t can be expected in view of z >> W, when D >> W.
221
Based on the previous experiments on turbulen! jets, this transition is expected to occur at zf ~ (3-5)W. Interaction o f plumes with a density interface
Upon impingement, the descending plume cap (and the fluid that follows) penetrates into the interface, rebounds and then propagates outwards as a gravity current (Fig. 1). For a two-layer fluid with a buoyancy jump of Ab across the layers, the depth of penetration d can be estimated approximately using a balance of potential energy stored during the deformation of the pycnocline and the vertical kinetic energy of the plume cap before the impingement. Here, Ab = g A p / p 0 , where, g is the gravitational acceleration, Ap is the density jump across the layers and P0 is a reference density. Using the scaling w ~ ql/3 for the characteristic vertical velocity of the P cap, it is possible to write this balance as a.102 / 3 ~ 8Ab, or ~ / l 0 = c4R/-1, where Ri = A b l D / q 2/3 is the bulk Richardson number and c 4 is a constant. If the thermocline is of finite thickness h, then the proposed balance becomes --~p a 2/3 ~ ( A b / h ) ~ 2, or ~ / l o = c s R i - 1 / 2 ( h / l o ) 1/2. Figure 4a shows a plot of ~ / l o versus Ri for the experiments carried out using very thin interfaces (i.e., (5 >> h); good agreement with the proposed R i - ~ scaling can be seen for Ri < 5, with c 4 - 4. For the case of a thick interfaces, ~ / h < 1, the results are presented in Fig. 4b, and good agreement with the latter scaling can be found, with c 5 -- 4. For the case of thermals penetrating into twolayer fluids, the appropriate Richardson number is defined as Ri = A b l o / w 2, where w o is the velocity of the thermal before the impingement. Dimensional arguments suggest w o = C6qlT/2 D-1/2, and the laboratory experiments of Noh et al. (1992) and Richards (1963) indicate that c 6 1.93 and c 4 = 2. As is evident from Fig. 5, the evolution of the gravity current after the impingement depends on Ri (note that the external parameters for the problem are qp, D and Ab, and D c c l D so that Ri is the governing non-dimensional parameter). Observations show that when Ri > 10, the penetration 8 is small and the plume cap splits into
222
HA.S. FERNANDO
(b) ~/
~>"
,' "
/
4
Zf (era) / ±
/
....
o
I
I
I
lO
20
30
qO
qpl/3t (cm) 25
20
15
] ,
p
tcm
)
AND
C.Y. CHIN(I
two separating vortices, which, upon loosing their initial momentum, collapse to form gravity currents propagating over the interface. These gravity currents have such typical features as a raised-head structure and Kelvin-Helmholtz billows behind the head (c.f., Simpson 1982), but their nose angle is much smaller than the prediction (60°) of the inviscid theory. In the range 1 < Ri < 5, the penetration 8 is significant and the nose angle is close to 60°. When 0.5 < Ri < 1 the nose is symmetric with respect to the interface, possibly because of the entrainment of fluid from the lower layer. When Ri < 0.5, the gravity current never bounces back, but spreads below the pycnocline. Further discussions on the interaction of buoyant plumes with density stratification are given in Turner (1973) and Weil (1982). The experiments of Noh et al. (1992) show that, for the case of thermals, the penetration becomes significant when Ri < 5. When Ri > 10, the propagation of the gravity current shows similar behavior to that over a solid surface, except that the nose angle is smaller. According to Richards (1963), thermals descend into the lower layer indefinitely when Ri < 1.9. Deardorff et al. (1980) have studied the penetration of buoyant parcels into a linearly stratified layer that lies above a turbulently convecting layer, when the driving unstable buoyancy flux is horizontally homogeneous with a buoyancy flux per unit area qo- They found that the depth of penetration is given by 8 = D (0.21 + 1.31 R/- ~), where R i . = ( N 2 ~ ) D / w , is the bulk Richardson number, w . = (qoD) 1/3 is the convection velocity and N is the buoyancy frequency of stratification.
t0
Background velocity field 5
0
5
10
15
20
25
30
35
40
45
50
55
60
zf [cm) Fig. 3a. A photograph of a starting line plume. One cm on the figure represents 12 cm of actual distance, b. A plot of the vertical position of the plume cap zf versus ql/3t, c. The relationship between the half width of the plume cap lp and its vertical position zf.
As was pointed out earlier, the entrainment flow into the descending plume generates an inward jet directed toward the plume (of velocity Ui) close to the surface and an out-flowing gravity current above the thermocline of velocity Uf. Ching et al. (1993) have made measurements of Uf and Ui using the facility shown in Fig. 2. Their results show that Uf/q~/3= 0.95 and Uf/q~/3= 0.7 R i °17, for Ri > 5 and Ri < 5, respectively. Further, Ui/qlv/3= 0.25 for the entire Ri range.
223
LEAD-INDUCED CONVECTION: A LABORATORY PERSPECTIVE
Similar laboratory experiments for the thermal case have been reported by Noh et al. (1992); the results show Uf/(qT/D) 1/2= 0.4-0.5 and Ui/(qT/D) 1/2 0.2-0.3. =
Effects o f background rotation
Since leads are opened for periods of the order of days, it is important to consider the effects of Earth's Coriolis forces on lead-induced motions. The following discussion is based on a line plume of length L and width W. For the case of infinite L, the plume can be affected by rotation in two ways. Firstly, the turbulence within the plume can be affected by the rotation when the plume descends to a critical depth z c. Based on previous laboratory studies (for example, Fernando et al., 1991; Fernando and Ching 1993) it can be inferred that the arrest of turbulent diffusion by the background rotation occurs when the Rossby number, based on the lengthscale l and velocity scale --~p al/3 of turbulence within the plume, becomes of the order unity, viz., R o = q l / 3 / f l ~ 0(1), where f = 2fl is the Coriolis parameter and f~ is the rate of background rotation. Because l ~ z, it is possible to evaluate z c ~ q~p/3/f. Secondly, the entrainment flow of velocity Ui can be
deflected sideways due to Coriolis forces, thus generating a background mean flow in the direction of the long axis of the lead. Obtaining a perfectly two-dimensional plume in the laboratory is not possible because of end effects, which are particularly important in the presence of background rotation. Fernando and Ching (1993) have conducted a series of laboratory experiments on line plumes of finite length descending into homogeneous rotating environments. They reported that the plumes initially descend as if there is no rotation, until their growth is arrested by the Coriolis forces at a time approximately 5.8/.f, during which they descend a vertical distance 6 . 4 ( q p / f 3 ) 1/3. T h e width of the plume at this instance was found to be 2.2(qp/ f3)1/3. No dependence of these quantities on the length of the plume was noted. After this time, the entrainment flow becomes unstable and defleets sideways forming two cyclonic vortices of highly three-dimensional nature. For a typical lead yieldinga buoyancy flux of qp --~ 5 X 10 - 4 m 3 s -3, this means that the Earth's rotation ( f = 1.4 × 1 0 - 4 s - 1 ) becomes important after the plume has descended to a depth of about 3500 m. However, the thermocline in polar oceans is usually shallow (e.g., Huskins, 1974);
(b)&~
(a)&
tD
& 1D
m
~_
%
ID
~ r~.
0
%)
tO-i
I
I
I
I I III
I
I
i
2
3
q 5 6789
2
3
q 5 6789
I
100
Ri
i
i
i o
i i
O
I
2
I01
@
10-2
3
q 5 6789
2 10-i
i
i
3
q 5 6789
I
I I i I
100
Ri ~(~n)'~
Fig. 4a. A plot of the normalized depth of penetration of the plume cap into the interface gilD versus the bulk Richardson n u m b e r this case the interface is thinner than the depth of penetration, b. Same as Fig. 4a, but the abscissa represents the normalized variable Ri-t/2(h/lD) 1f2. In this case the interface is thicker than the depth of penetration.
Ri. In
224
ft.J.S, FERNANDO AND C.Y. (~HIN(I
where r d = Ur/f is the Rossby radius of deformation. In addition, as stated before, the entrainment flow is expected to become unstable at a time 5.8/f. The instability of the entrainment flow is also expected to produce isolated vortices of size ~ r d. The existence of sub-surface eddies under the polar ice cap has been clearly demonstrated in many studies, particularly during the A I D J E X experiment by Huskins (1974). The nature of the flow under a lead at large times is expected to depend on the type of instability that develops first. Because the time necessary for a plume to reach the thermocline is given by T,I = D/clqlo/3 and the time scale for the baroclinic instability of the gravity current is tl, = (5 - 7XUf/f)/Uf = (5 7 ) f - J , the total time for the appearance of the baroclinic eddies can be estimated as (D/clqJo/3 + (5 - 7 ) f - ~). This is usually larger than the time 5.8f-1 required for the deflection of the entrainment flow. In order to check which instability occurs first, we have extended the experiments of Fernando and Ching (1993) to realize the condition in which a plume impinges on the tank bottom before its growth is arrested by rotation. These experiments were performed in a large tank of dimensions 400 × 114.5 × 29 (cm) that was located on a rotating table. Plume generators of lengths 14 cm and 18 cm, located at the middle of the tank, with their axes aligned with the short dimension of the tank, were used; the plume was released on to the surface of a homogeneous water layer of depth 26 (cm). Flow visualization was performed using the laser-induced fluorescence technique. Figure 6 shows a sequence of photographs that was taken during the experiments. The initial spreading of the front was found to be very similar to the case of a nonrotating fluid, but the plume becomes unstable and starts deflecting cyclonically (see arrows) before the gravity current head breaks into eddies. The gravity current near the bottom deflects anti-cyclonically. In fact, the entrainment flow becomes unstable and deflects sideways soon after the plume impinges on the density interface. Similar observations were made in all the experiments that were carried out covering a range of experimental parameters. -
Fig. 5, The a p p e a r a n c e of the gravity c u r r e n t u n d e r different bulk R i c h a r d s o n n u m b e r s , T h e position of the interface is m a r k e d . (a) Ri = 53.2: (b) Ri = 1.2; (c) Ri = 0.95; (d) Ri = 0.4.
hence a plume should reach the thermocline and start spreading horizontally as a gravity current way before the rotational effects become important. This gravity current flow can become baroclinically unstable and break down into unstable vortices. Laboratory studies (e.g., Ivey, 1987) show that such instabilities develop after the gravity current propagates a distance of about (5 - 7)r~,
LEAD-INDUCED CONVECTION: A LABORATORY PERSPECTIVE
Effect of ice movement and background turbulence Drift of the ice pack owing to wind forcing causes the development of a turbulent boundary layer underneath the ice. As is shown in Fig. 1, the background mean flow causes the plumes to bend over, and the interaction with the mean flow is expected to modify their characteristics. Some information on whether the mean flow is passive (i.e., has a purely advective effect) can be obtained by considering the governing variables. If a two-dimensional line plume can be specified by two parameters, qp and W, and, if the turbulent boundary layer under the ice is governed by the ice velocity Vi, the local turbulent boundary layer thickness 6 T and the friction velocity due to under-ice stress u . , then any dependent quan-
225
tity, for example, the depth z e ( < t~T) at which the plume is destroyed, can be expressed as
Ze = f
c d,
,
(2)
where u 2 / V i 2 = c d is the drag coefficient and f , f l . . . . are functions. At high Reynolds numbers c d can assumed to be a constant and, for simplicity, when W<< 6T, the effect of W / 6 T can be neglected. Thus,
Zo
[ v?
)
On this basis the major governing parameter determining the modification of the plume is Vi3/q,. Above a critical value of this parameter,
Fig. 6. A plan view of the flow development when a line plume of finite length impinges on a solid surface and spreads horizontally in the presence of background rotation. T h e plume rea6hes the solid bottom before the background rotation arrests its growth. For this run, qp = 15.2 cm 3 s - 3 and f = 0.44 s -1. T h e pictures were taken at (a) 1.5/f; (b) 2 . 3 / f ; (c) 3 . 1 / f ; (d) 3 . 9 / f ; (e) 4.7/]'; (f) 5.5/f. O n e cm on the figure represents 10 cm of actual distance.
226
which is to be determined by experiments, the plume is expected to be modified by the turbulent boundary layer (forced convection). Cederwall (1971), Roberts (1979) and Roberts et al. (1989a, b, c) have carried out laboratory experiments on line plumes in cross flows with the goal of designing waste-water outfalls. In these experiments a line plume was towed in a quiescent environment with neutral or stable stratification. The experiments indicated that the dilution characteristics of plumes are drastically modified at higher cross flows; extrapolation of their plume dilution results to the present case indicates that the c r i t i c a l Vi3/qp should lie in the range (0.1-0.2). This implies that, for example, for a lead with qp = 5 X 10 -4 m 3 s-3 and Vi = 5 x 10 -2 m / s , Vi3/qo=0.25; according to the above criterion, the forced-convection effects are important in this case. The above analysis should be contrasted with that of Morison et al. (1993). By non-dimensionalizing the governing equations using appropriate scales, they obtained a criterion based on the lead number L o =qoD/CdVi 3 to demarcate between the forced and free convection r e g ~ e s ; note that here qo = qp/W, which is the buoyancy flux per unit area. Forced or free convection was found to prevail depending on L o < 1 or L o > 1. The present arguments assume that the plume is modified mainly within the turbulent boundary layer, and the depth of the mixed layer and the width of the plume W are of secondary importance, when D > t$T > W. If the plume can survive the adverse influence of the turbulent boundary layer, then it descends into the region outside the boundary layer which can also be turbulent (for example, owing to tidal stirring and other environmental sources). To our knowledge, so far no explicit laboratory work has been reported on the interaction of line plumes with environmental turbulence. However, some idea on the level of background turbulent intensity u required for the destruction of the plume can be obtained using the instructive laboratory experiments of Thomas and Simpson (1985). They considered the effect of background turbulence on a gravity current propagating on a solid surface, and delineated a criterion for the modiflca-
H..I.S. F E R N A N D O
A N D C.Y. C H I N G
tion of the current by the turbulence. Extension of this result to the present case suggests that the plume will be unaffected by the turbulence as long as u/qlp/3 < 0.05.
Merger of plumes The influence of one lead on another is an important aspect in modeling the circulation under the ice pack. Intuitively, it is possible to expect that the interaction between leads occurs mainly via the interaction between the entrainment flows; the depletion of fluid between leads due to entrainment causes high fluid velocities and low pressure between descending plume caps, thus resulting in the merger of plumes. Early experiments of Rouse et al. (1952) with parallel line plumes suggest that the combined plume after the merger can be treated as one having a total strength equal to the sum of the strengths of the individual plumes. Only a few studies have been reported on the interaction of turbulent line plumes in homogeneous fluids. Yoshida (i983) and Yoshida and Nagata (1986) carried out experiments on interacting parallel line plumes of equal strengths qp and identified three stages of plume evolution. In stage 1, the plumes descend vertically until the caps of the starting plumes touch each other, whence the two plumes initiate their rapid merger (stage 2). During the merger, the shallowest contact point between the plumes ascends vertically, and finally comes to a quasi-steady state (stage 3). The transition times between the stages were identified as tHi=2.12Xq~ 1/3 and t l H H ~ 4.46Xqff 1/3, where X is the half distance between the sources. In the present work, a series of laboratory. experiments were carried out in a large Plexiglas tank (180 x 15 × 120 era), using two line plumes of equal buoyancy fluxes qp; all other features of the experiment were similar to those described in the context of Fig. 1. In some experiments a homogeneous background fluid was used and in the others a two-layer fluid was employed so that the merger between the plumes occurred after their caps arrive at the density interface. Figure 7a shows the descent and merger of plumes in a
LEAD-INDUCED CONVECTION: A LABORATORY PERSPECTIVE
homogeneous deep fluid layer. One distinct observation made here, that differs from Yoshida (1983), is the initiation of merger before the plume caps come into contact. Figure 7b shows the case where the plumes impinge on the interface before merging. In this case, the space between the two plumes is filled by the plume fluid; this fluid belongs to the two counter-flowing gravity currents that form above the density interface between the plumes. After this fluid is raised to a certain height, the rapid merger is initiated. The time scale for merger Tm can be written in the form
Tm = g ( X , D, v, qp),
(4)
where g, g t . . . are functions, v is the kinematic viscosity, and the effects of the tank side walls have been neglected. Thus, we get X
where
gl
v
'
'
(5)
ql/aX/l~ is a Reynolds number, influence
227
of which can be neglected at high values of T h u s , it is possible to write
ql/3x//v. Traq1/3
-- c o n s t a n t ,
(6)
for merger in deep homogeneous fluid layers, and
x
=g2
B
,
(7)
for two-layer fluids; it is expected that, for high density stratifications, Ri > 5, the buoyancy jump across the interface Ab is unimportant and the interface acts as a rigid wall. Figure 8 shows the horizontal deviation of the center of the plume cap ¢ from the vertical centerline of the plume for experiments with homogeneous fluids. The length and time scales have been selected as X and Xqff 1/3, in view of Eq. (6). Note that this deviation has grown to )2/2 at a time of 2.0Xqp i/3, which can be considered as a time scale for the merger. The line drawn through the data points represents the best-fit line to the data as ¢ / X = 0.11(tql/3/X)2.
Fig. 7a. A photograph that shows the merger of descending parallel line plumes in a deep homogeneous fluid. For this run, qp = 15.1 cm 3 s -3. For (a) and (b), one cm represents 9 cm of actual distance, b. The merger of two line plumes, after they have impinged on a density interface. For this run, qp = 21.0 cm 3 s -3.
228
H..t.S. F E R N A N D O
oo f co
AND C.Y. CHIN(;
. . . . . . D
"I
/
Tmq~/3 X
X
o
, "j
q: --
%
©
l
o 0.0
0.8
1.6
co
2 10-1
3.2
2.q
X D
x
Fig. 9. The dependence of the normalized time for the merger of plumes Tmql/3/X on the normalized distance between the plumes X/D in the presence of a density interface,
Fig. 8. The growth of the horizontal distance between the vertical centerline of a plume and its cap ~/X with the non-dimensional time tqlo/3/X.
Figure 9 shows the variation of the normalized merger time scale with X/D for the experiments carried out with density interfaces. Here the merger time is defined as the time at which the two descending plmnes come into contact at the centefline, without having a rising fluid column between them. Note that the data is somewhat scattered, mainly because of the subjective judgment that had to be used in deciding the merger time. The best-fit line to the data indicates that
Tmqlo/3//S =
2.07 X 102(X//D)
Discussion It is instructive to discuss the possible applications of above results to oceanic lead-related situations. As mentioned earlier, the available observations are f r ~ , and a strait-laced comparison of them with laboratory observations is untenable. Some of the avait~le observations and estimated qumatitics are listed in Table 1. Note that the l e ~ ( D - . - 2 0 m) and time scales of leads are much larBer than those of the labora-
2"6.
TABLE 1
Field observations Location
Experiment
Beaufort A L E X 2 Sea, near Barrow, Alaska
ALEX 3 ALEX 4
Beaufort Sea
AMLE
References
qp(m3s -3)
D (m)
h (m)
Smith (1974) Smith et al.
~ 9 × I0 - 4
15
10
(1990) AIDJEX PI's r e p o r t (1974)
~ 5 x 10 - 4 ~ 5 x 10 - 4
16 16
Morison
~5x10
35
(1980)
Morison et al. (1992)
-4
W (m)
1/i (ms - 1 )
8 (m)
50
1
20
9 9
50 300
1 1
-
20
1400
-
-
Ui(nas ~1)
Ab(ms -l)
Ri
-
3
x 10 - 3
0.8
( 5 - 8 ) × 10 - 2 -
3 x 10 - 2
1 × 10 - 2 1.2 × 10 - 2
4.0 4.8
2x10 -2
2x10 -2
5
4.4
Uf (ms - i ) -
Xl0 -3
LEAD-INDUCED CONVECTION: A LABORATORY PERSPECTIVE
229
TABLE 2 Predictions Experiment
8 (m) based on qp
ALEX3 ALEX4 AMLE
~9 ~9 ~ 20
8 (m) based on
Uf(ms-')
Ui(ms-')
7 x l O -2 7 x l O -2 7 x l O -2
2 × 1 0 -2 2 × 1 0 -2 2 x l O -2
qo(= qp/W) 4.1 3.6 7.5
tory experiments ( D ~ 0.3 m), which naturally raises the question of the applicability of the laboratory results to the ocean. For laboratory experiments q p ~ 5 × 10 -5 m3s -3 and for the ocean qp ~ 5 x 10 -4 m 3 s -3 (Morison et al., 1992); hence, the Reynolds numbers for these cases are of the order 104 and 106, respectively. Since the plumes are turbulent in both cases, based on Reynolds number similarity, it is possible to argue that the laboratory results should have applicability to oceanic situations. Another question of importance is how to model a lead. Earlier, it was argued that a lead ought to be treated as a line plume if W << D, but in most oceanic cases W ~ D. In such cases it is advisable to treat the descending plume as if it is arising from a horizontally homogeneous source with a buoyancy flux per unit area qo. Of course, after the plumes impinge on the interface, the horizontal inhomogeneities need to be considered. Future experiments should investigate the effect of finite W on the evolution of line plumes. Table 2 shows the laboratory-based predictions for the field experiments depicted in Table 1; the agreement appears to be fair (because the stratification observed in A L E X 2 is not compatible with the laboratory experiments described herein, it was omitted from Table 2). However, it should be borne in mind that these comparisons have been made by assuming that the intensity of background turbulence is weak, free convection dominates beneath the lead and the observations were made before the rotational effects become important. No substantial evidence is available to corroborate such assumptions, and hence the comparison between the two cases needs to be done with caution.
q~/3D/v
Acknowledgments We wish to thank the Office of Naval Research (Arctic Sciences and Small-Scale Oceanography) and the National Science Foundation for the financial support. Also, we were immensely benefited by the assistance of Dr. Linus Mofor, Prof. D.F. Jankowski, Dr. P.A. Davies, Mr. Leonard Montenegro and G. Oth in numerous ways.
References AIDJEX Principal Investigators Report, 1974. AIDJEX Bull., 27: 117-120. Atkinson, J.F. and Wake, A., 1987. Double-diffusive convection under sea ice. J. Cold Regions Eng., 2: 89-95. Baines, W.D. and Turner, J.S., 1969. Turbulent buoyant convection from a source in a confined region. J. Huid Mech., 37(1): 51-80. Cederwall, K., 1971. Buoyant Slot Jets into Stagnant or Flowing Environments. W.M. Keck Lab. Hydraul. Water Resour., California Inst. Technol. Rep. No. KH-R-25. Ching, C-Y., Fernando, H.J.S. and Noh, Y., 1993. Interaction of the negatively buoyant line plume with a density interface. Dyn. Atmos. Oceans (accepted). Deardorff, J.W., Willis, G.E. and Stockton, B.H., 1980. Laboratory studies of the entrainment zone of a convective mixed-layer. J. Fluid Mech., 100: 41-64. Fernando, H.J.S. and Ching, C.Y., 1993. Effects of background rotation on turbulent line plumes. J. Phys. Oceanogr. (Accepted) Fernando, H.J.S., Chen, R.-R. and Boyer, D.L., 1991. Effects of rotation on convective turbulence. J. Fluid Mech., 228: 513-547. Foster, T.D., 1972. Haline convection in polynyas leads. J. Phys. Oceanogr., 2: 462-469. Glendening, J.W. and Burk, S.D. 1992. Turbulent transport from an Arctic lead: A large eddy simulation. BoundaryLayer Meteoroi., 59: 315-339.
230 Holmgren, B. and Spears, L. 1974. Sodar investigation of the effect of open leads on the boundary-Layer structure over the Arctic Basin. AIDJEX Bull., 27: 168-179. Huskins, K.L., 1974. Subsurface eddies in the Arctic Ocean. Deep-Sea Res., 21: 1017-1033. Ivey, G.N., 1987. Boundary mixing in a rotating stratified fluid. J. Fluid Mech., 183: 25-44. Koerner, R.M., 1973. The mass balance of the sea ice of the Arctic Ocean. J. Glaciol., 12(65): 173-185. Kozo, T.L, 1983. Initial model results for Arctic mixed layer circulation under a refreezing lead. J. Geophys. Res., 88: 2926-2934. Ledley, T.S., 1988. A coupled energy balance climate-sea ice model; Impact of sea ice and leads on climate. J. Geophys. Res., 93: 15,919-15,932. Ix), A.K., 1986. On the boundary-layer flow over a Canadian archipelago polynya. Boundary-Layer Meteorol., 35: 53-71. Maykut, G.A., 1978. Energy exchange over young sea ice in the central Arctic. J. Geophys. Res., 83: 3646-3658. Morison, J.H., 1978. The Arctic profiling system. Proc. Working Conf. Current Measurements, Univ. Delaware, pp. 313-318. Morison, J.H., 1980. Forced Internal Waves in the Arctic Ocean, Ph.D. Thesis, Dep. Geophys. Univ. Washington, Seattle. Morison, J.H., McPhee, M., Curtin, T. and Paulson, C., 1992. The oceanography of winter leads. J. Geophys. Res., 97 (C7): 11,199-11,218. Nob, Y., Fernando, H.J.S. and Ching, C.Y., 1992. Flows induced by the impingement of a two-dimensional thermal on a density interface. J. Phys. Oceanogr., 22(10): 12071220. Richards, J.M., 1963. Experiments on the motion of isolated cylindrical thermals through unstratified surroundings. Int. J. Air Water Pollut., 7: 17-34. Roberts, P.J.W., 1979. A mathematical model of initial dilution for deepwater ocean outfaUs. Proc. Specialty Conf. Conserv. Util. Water Energy Resour. San Francisco, Aug. 8-11, pp. 218-225. Roberts, P.J.W., Snyder, W.H. and Baumgartner, D.J., 1989a. Ocean outfalls. I: Submerged wastefield formation. J. Hydraul. Eng., 115(1): 1-25. Roberts, P.J.W., Snyder, W.H. and Baumgartner, D.J., 1989b. Ocean outfails. II: Spatial evolution of submerged wastefield. J. Hydraul. Eng., 115(1): 26-48. Roberts, P.J.W., Snyder, W.H. and Baumgartner, D.J., 1989c. Ocean outfalls. II1: Effect of diffuser design on submerged wastefield. J. Hydraul. Eng., 115(1): 49-70.
H.J.S. F E R N A N D O
A N D (~.Y. C H I N ( ;
Rouse, H., Baines, W.D. and Humphreys, H.W., 1952. Free convection over parallel sources of heat. Proc. Phys. Soc., LXVI, 5-B: 393-399. Rudels, B., 1990. Haline convection in the Greenland Sea. Deep-Sea Res., 37(9): 1491-15ll. Schaus, R.H. and Gait, J.A., 1973. A thermodynamic model of an Arctic lead. Arctic, 26(3): 208-221. Schnell, R.C., Barry, R.G., Miles, M.W., Andreas, E.L, Radke, L.F., Brock, C.A., McCormic, M.P. and Moore, J.L, 1989. Lidar detection of leads in Arctic Sea ice. Nature, 339: 530-532. Shreffier, J.H., 1975. A Numerical Model of Heat Transfer to the Atmosphere from an Arctic Lead. Ph.D. Thesis, Dep. Oceanogr., Oregon State Univ., Corvallis, 135 pp. Simpson, J.E, 1982. Gravity currents in the laboratory, atmosphere and ocean. Annu. Rev. Fluid Mech., 14: 213-234. Smith IV, D.C. and Morison, J.H., 1993. A numerical study of haline convection beneath sea ice. (In prep.) Smith, J.D. 1973. Lead-driven convection in the Arctic Ocean. EOS, 54(11): 1108-1109 (abstract). Smith, J.D., 1974. Oceanographic investigations during the AIDJEX lead experiment. AIDJEX Bull., 27: 125-133. Smith, S.D., Muench, R.D. and Pease, C.H., 1990. Polynyas and leads: An overview of physical processes and environment. J. Geophys. Res. 95(C6): 9461-9479. Thomas, N.H. and Simpson, J.E., 1985. Mixing of gravity currents in turbulent surroundings: Laboratory studies and modeling implications. In: J.C.R. Hunt (Editor), Turbulence and Diffusion in Stable Environments. Clarendon Press, Oxford, pp. 61-96. Tsang, G., 1970. Laboratory study of two-Dimensional starting plumes. Atmos. Environ., 4: 519-544. Turner, J.S., 1973. Buoyancy Effects in Fluids. Cambridge Press. Untersteiner, N. (Editor), 1986. The Geophysics of Sea Ice. Plenum Press, N.Y. Wadhams, P., 1981. The ice cover of the Greenland and Norwegian Seas. Rev. Geophys. Space Phys., 19: 345-393. Weil, J.C., 1981. Source Buoyancy Effects in Boundary Layer. Workshop on Parameterization of Mixed Layer Diffusion, Las Cruces, NM. Yoshida, J. and Nagata, Y., 1986. Velocity fields and their variation in the two-dimensional forced plume without and with a vertical wall nearby. J. Hydrosci. Hydraul. Eng., 4(1): 51-64. Yoshida, J., 1983. The behavior of a two-Dimensional dual forced plume. J. Oceanogr. Soc. Jpn., 39: 317-323.