Entrainment due to a thermal impinging on a stratified interface with and without buoyancy reversal

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 105, NO. D12, PAGES 15,457-15,467, JUNE 27, 2000 Entrainment due to a thermal impinging on a stratified interface with and without buoyancy reversal Qing Zhang Department of Mechanical and Industrial Engineering, University of Manitoba, Winnipeg, Manitoba, Canada Aline Corel Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor Abstract. The aim of the present paper is to understand the interaction between a rising thermal and an inversion in the atmosphere and to quantify the turbulent entrainment rate due to a thermal impingement on a stratified interface. This problem was simulated in the laboratory using a water tank. The thermal is created by releasing a small volume of buoyant fluid into a stratified environment composed of two layers of different densities. A thin interface separates the lower layer from the lighter upper layer. The entrainment of upper layer fluid into the thermal is investigated using a passive dye flow visualization technique. The entrainment rate is found to obey a Ri -3/ power law, as predicted by Cotel and Breidenthal [1997]. The effect of simulated evaporative cooling on the entrainment of a thermal impinging on a stratified interface is also investigated experimentally. Evaporative cooling in atmospheri clouds is simulated in the laboratory using alcohol-water mixtures, so that the mixed fluid is denser than either parent parcel. This is realized in the laboratory by releasing a mixture of ethyl alcohol and ethylene glycol in an aqueous solution. It rises first through a relatively dense lower layer fluid and then impinges on a thin stratified interface, above which is a layer of relatively light fluid. The entrainment rate for values of the buoyancy reversal parameter D* between 0 and 0.5 was found to obey a Ri -3/2 power law. The entrainment rate is independent of D* between 0 and 0.5 for a range of Richardson numbers Ri from 3 to 25. This is consistent with the behavior of the buoyancy-reversing thermal in an unstratified environment observed by Johari [1992]. 1. Introduction Most fluid motions in the atmosphere and oceans are strongly influenced by the dynamics of stratified flows. It is therefore essential to understand their dynamics in order to comprehend the environment. This paper focuses on understanding the turbulent entrainment of a thermal impinging on a stratified interface. The goal of the present study is to quantify the entrainment rate due to the impinging thermal with and without buoyancy reversal. A direct application of this work is to study the interaction between clouds and the tropopause. Accurate modeling of mass transport across the tropopause is necessary to understand the concentration of ozone and other chemicals in the stratosphere. Some cumulus clouds contain pollutant air, and their penetration through atmospheric inversions may affect the ozone budget. The stratosphere is just like a "reservoir" for certain types of atmospheric pollution. The quantity of pollutants penetrating into the stratosphere might reduce the equilibrium concentration of ozone, thereby allowing more ultraviolet radiation to reach the Earth's surface [Wallace and Hobbs, 1977]. This is considered to be an important factor in the issue of global warming. It is necessary to gain further insight into stratified entrainment to yield better models of atmospheric dynamics in global climate models as well as Copyright 2000 by the American Geophysical Union. Paper number 2000JD /00/2000JD weather forecasts. The study of a rising thermal will also enhance our understanding of atmospheric convection, an essential part of atmospheric dynamics [Emanuel, 1994]. Among turbulent flows the motion of buoyant fluid known as the "thermal" has been studied theoretically, experimentally, and computationally for several decades. However, only preliminary work has been done to study the interaction between a stratified interface and a thermal, and no entrainment measurements have been performed for this specific case Background on Thermals The term thermal is used to denote a parcel of isolated buoyant fluid suddenly released from rest, which subsequently moves under the action of buoyancy forces alone. An isolated thermal is the ultimate form of convective elements in the atmosphere [Woodward, 1959]. From the motion of cumulus clouds convection, which is produced by buoyancy forces, appears to consist largely of more or less isolated masses of buoyant air rising into and mixing with their surroundings [Scorer, 1957]. A thermal is characterized by its total buoyancy force and does not possess any circulation, kinetic energy, or momentum at the moment of release. The circulation is generated entirely by buoyancy after the thermal is released. Entrainment in thermals is accomplished through engulfment of fluid in the rear of thermals because the induced velocity from the vorticity field causes ambient fluid to enter the backside of the thermal. The motion within a thermal is believed to be composed of a rising motion in the middle and a descending 15,457

2 15,458 ZHANG AND COTEL: ENTRAINMENT DUE TO AN IMPINGING THERMAL where Ri = (2) w ' Figure 1. Cloud entrainment. one on the sides, similar to a vortex ring [Woodward, 1959; Turner, 1963]. The advancing front of a thermal has traditionally been used for measurementsince it is a relatively distinctive point [Johari, 1989]. A self-similar thermal has a width of about half the distance the front traveled [Johari, 1992] Background on Stratified Entrainment A. I. Taylor first described the entrainment hypothesis at the Pacific Science Association meeting in 1949; however, Batchelor [1954] gave it wider exposure through a review lecture. The basic form of this hypothesis is that the mean inflow velocity across the edge of a turbulent flow is assumed to be proportional to the local characteristic velocity, for a thermal, the rising speed (Figure 1). The total volume inflow depends on the surface area, the geometry, and the dynamics of the flow [Turner, 1986]. Scorer [1957], Woodward [1959], and Turner [1963] performed tank experiments involving heavy brine parcel sinking through freshwater. They observed that the diameter increases linearly with vertical distance from the source. Their results seemed to confirm the applicability of similarity solutions. Grabowski and Clark [1991, 1993] used two- and threedimensional numerical experiments to simulate thermals in a stratified environment. The subject of their studies, namely, the cloud-environment interface instability, is considered by them a primary entrainment mechanism. In stratified turbulence the interaction between stratified interfaces and turbulent flows is a critical problem. Experiments have shown that in general the entrainment rate declines with increasing stratification. The relationship between the entrainment rate and the Richardson number is observed to obey a power law [Turner, 1973] when the Richardso number Ri is between one and a few hundreds. The relationship is often defined as: g': ---. (3) p Here g' is the buoyancy acceleration due to the density difference A9; the acceleration of gravity is g; A 9 is the relevant density difference across the interface or between the inside and outside of a thermal; 9 is the density of the surrounding fluid; (5 is the length scale of the incident turbulence at the interface; and w i is its characteristic velocity (Figure 2). For Ri < 1 the kinetic energy of the largest eddies is greater than the potential energy investment in engulfing a tongue of fluid, so there is little or no effect of stratification on entrainment. When Ri is very large, the interface becomes flat, and so, potential energy can no longer affect the entrainment. Therefore the entrainment rate becomes independent of Ri [Cotel and Breidenthal, 1997]. Diffusion becomes the only available entrainment mechanism. The entrainment velocity we represents the rate at which the interface is rising because of entrainment of upper layer fluid into the turbulent flow emerging from the lower layer (see Figure 2). The dimensionless entrainment rate is defined as the ratio of the entrainment velocity w e to the characteristic velocity w 1- In the literature the value of the entrainment expo- nent a has been found to be -1/2, -1, -3/2 or, in one case, -2. Linden [1973] concluded from theoretical arguments that a should be -3/2 for a vortex ring. Kumagai [1984] measured a for a plume, finding it close to -3/2. For a jet impinging on a stratified interface [Cotel et al., 1997], a was measured to be -1/2. Turner [1973] also found that the value of a seemed to depend on the Schmidt or the Prandtl number from his stirring grid experiments with heat and salt. The Schmidt number Sc is defined to be SC ---Din, (4) where Dm is the molecular diffusivity and v is the kinematic viscosity. The Prandtl number Pr is Interface Pr = Dr' (5) We w1 where w e is the entrainment velocity across the interface, w i is the characteristic velocity of the impinging turbulent flow, and c and a are dimensionless constants. Ri (the ratio of potential energy to kinetic energy) is defined in terms of the impinging turbulence quantities: Figure 2. Schematic of a thermal impinging on a stratified interface.

3 ZHANG AND COTEL: ENTRAINMENT DUE TO AN IMPINGING THERMAL 15,459 where D r is the thermal diffusivity. When Ri < 7, the value of a equals -3/2. If Ri > 7, the value of a remains -3/2 for salt stratification but becomes -1 for heat stratification in the case of stirring grid turbulence [Turner, 1973]. Cotel and Breidenthal [1997] suggested that the persistence of the entraining eddies is a critical parameter in determining the entrainment regime, in addition to the conventional parameters, such as Ri, Reynolds number Re, and Sc. Ri and Sc have been defined previously. The Reynolds number Re is defined by w 8 Re = --, (6) PA PB I Ptl Lid Plate PB>PA>Pt where 15 and w represent the characteristic lengthscale and Cylinder containing the thermal fluid velocity scale of the flow, respectively, and v is the kinematic viscosity. The dimensionless vortex persistence parameter T is defined to be the number of rotations a vortex makes during Figure 3. Sketch of the water tank. the time it moves its own diameter with respect to the inter- [1995]. According to Johari [1992] and Grabowski and Clark face. T is proportional to the ratio of the rotational to the [1993], the entrainment should not be influenced dramatically translational speed of the entraining eddy. The translational by buoyancy reversal as long as it is weak. Moreover, the speed is the speed at which the eddy is moving with respect to entrainment rate should decrease in general with increasing the interface. The rotational speed is unambiguous to any stratification as is the case for ordinary thermals. However, the inertial observer. T is independent of Ri, Re, or Sc since they effect of evaporative cooling has not yet been tested in the case all can be varied separately while T is held constant. According of impinging turbulence on a stratified interface. to Corel and Breidenthal's [1997] investigation a thermal has a A water tank with a thermal release mechanism was used for persistence parameter of unity or less because of its nonstathe experiments. The laboratory experiments and results for tionary character as it impinges on the interface, similar to that the classical thermals impinging on a stratified interface are of a vortex ring. The entrainment rate of the thermal is then presented in section 2. The buoyancy reversal case is discussed predicted to be proportional to Ri -3/2 for a thermal for a in section 3. Finally, the summary and conclusions are prerange ofri > 1 but <Ri = Re /4, at which point the interface becomes flat. Little laboratory work has been done on thermals interacting with a stratified interface. In the early 1960s, Richards [1962] and Saunders [1962] observed the penetration of thermals through an interface between different density fluids. However, they did not quantify the entrainment rate of thermals Background on Evaporative Cooling Evaporative cooling is another factor that affects the dynamics of clouds in the atmosphere. It occurs when dry environmental air is entrained into positively buoyant cumulus clouds, the resulting evaporation reducing or even completely reversing the positive buoyancy. Buoyancy reversal from evaporative cooling has been considered to have a dramatic effect on the global motions of clouds but is believed to have a lesser effect on the entrainment rate and mixing of buoyant fluid [Johari, 1992; Grabowski and Clark, 1993]. Grabowski [1995] suggested that the buoyancy reversal affects the entrainment in two different ways. First, buoyancy reversal directly enhances the kinetic energy of small-scal entraining eddies so the enhanced entrainment is thought to occur when the magnitude of the reversal is comparable to the initial stratification. Second, buoyancy reversal indirectly leads to more entrainment and mixing through large-scal eddies entraining air into cloud. In other words, enhancement of entrainment occurs through the enhancement of global flow structures. Furthermore, Turner and Yang's [1963] early laboratory experiments found that the rate of entrainment at the top of stratocumulus clouds was not enhanced significantly by modest buoyancy reversal. Buoyancy reversal has a stronger effect on the large-scale structure of the flow than on the small-scale dynamics, as observed in the laboratory by Johari [1992] and numerically by Grabowski sented in section Thermals Impinging on a Stratified Interface Without Buoyancy Reversal As discussed in section 1, there is an obvious need to measure the entrainment rate of a thermal impinging on a stratified interface Experimental Apparatus An ideal mechanism for releasing thermals would simulate the atmospheric release process as closely as possible. The dimensions of the tank are 30 x 30 x 65 cm in height. A vertical cylinder, 2.5 cm inside diameter and 2.9 cm high, is located at the bottom of the tank and covered by a thin, horizontal sliding stainles steel lid. For the purpose of generating a thin stratified interface the tank is separated into two chambers by a thin, horizontal sliding stainles steel plate. The material of the tank walls consists of chemically resistant lucite to give an unobstructed view of the flow. Thermals are released from the bottom of the tank as the rising clouds in the atmosphere. The apparatus is sketched in Figure Experimental Techniques Set up. The top and bottom layers are composed of two fluids of different salt concentrations. The density of the lower layer fluid Pa is greater than that of the upper layer fluid 9.4- The cylinder is filled with a mixture of pure water and water soluble dye, and the lid is slid back over the cylinder. Then the lower section of the tank is filled with a fluid of density Pa. The sliding plate is pushed in, and finally, a fluid with density 9.4 is poured into the upper section. The sliding plate and the lid are used to minimize mixing between the

4 15,460 ZHANG AND COTEL: ENTRAINMENT DUE TO AN IMPINGING THERMAL Figure 4. Photograph of (a) a classical thermal just after release, (b) a classical thermal before impinging on the stratified interface, (c) a classical thermal impinging on the stratified interface, and (d) a classical thermal after impingement on the stratified interface. different fluids before a run. The density of the thermal p, is always less than both pu and P Experimental procedure. First, the sliding plate, located in the middle of the tank, is slowly retracted. In order to obtain sufficient entrainment and to minimize wall effects, the tank is then slowly drained so that the interface is positioned at a desired level. The initial level of the interface is set at 30 cm; the final position is between 20 and 30 cm from the tank bottom, depending on the conditions. The interface remains thin throughout the draining process. Then the lid above the cylinder is withdrawn to begin a run. Considerable attention is necessary at this stage so that minimal initial circulation is imposed on the thermal Experimental Results The evolution of a thermal rising toward the interface is shown in a sequence of photographs from a single run (Figures 4a-4d). The initial interface is marked with dots in the photographs. The interface is thin compared with the thermal eddy size. The buoyant thermal fluid is released from the cylinder at the bottom of the tank (Figure 4a). In Figure 4b the thermal becomes approximately self-similar as it has advanced 3.5 times the effective diameter of the released fluid [Johari, 1992]. The thermal penetrates through the interface, rebounds from it, and spreads laterally (Figure 4c). At this point in the exper- iment the entraining eddy may not be the largest one because of stratification. In an unstratified incompressiblenvironment with no external forcing applied the largest eddies are the most efficient eddies at entraining and mixing fluid [Roshko, 1976]. When stratification is present, the largest eddies' Ri is now greater than unity, and entrainment by these vortices is impeded. The most efficient entraining eddy is now the largest eddy with a Ri still below unity. This entrainment process causes the interface to rise by a certain amount. Finally, the motion decays, and the interface becomes flat (Figure 4d). Although a single thermal impinging on the interface described above is a single discrete event, by considering the process to be repeated many times we can define a corresponding average entrainment velocity: Mir We : --, (7) tit where H is the change in interface height due to thermal entrainment measured at t - t (where t represents the duration of the entrainment process) from the time the thermal is released until the thermal stops entraining fluid from the upper layer. The entrainment process starts shortly after the release. There is a small error in the time measurement, accounted for in the error bars. A stopwatch is used to record the different times. The thermal velocity is

5 ZHANG AND COTEL: ENTRAINMENT DUE TO AN IMPINGING THERMAL 15,461 d tr W1 = tt r, where d tr is a distance of 2 cm just below the interface and t tr is the time the thermal traveled the distance dtr recorded with a stopwatch. Thus w i represents the thermal velocity just before impingement. The height of the interface and the distance traveled by the thermal are measured visually from a ruler on the side of the tank. The width of the thermal 8 is measured just before the thermal impinging on the interface (Figure 2). For all experiments the interface is reached by a thermal in a few seconds after release, but the entrainment process at the interface takes at least 40 s. The dimensionless entrainment rate we/w i versus the Ri is shown in log-log plots (Figure 5). The entrainment rate we/w 1 is the velocity at which the interface is rising normalized by the measured thermal velocity as it approaches the interface. Ri is based on the density difference between the upper and the lower layer fluid, and the density p is the density of the upper layer fluid (equation(2)). We found that the entrainment rate declines with increasing Ri as in (1). The value of the exponent determined by curve fitting is found to be -3/2. There is -_ % error due to lack of accuracy when measuring the time. Re defined in (5), is based on the thermal quantities: its diameter 8 and velocity w 1. The value of Re for all experiments is Discussion The results are similar to those obtained for stirring grid turbulence [Turner, 1973] and for a plume impinging on a stratified interface [Kumagai, 1984]. There is almost no entrainment at high Ri; because of the lack of measurement accuracy for such small values of the entrainment rate, no data are plotted above Ri - 30 in Figure 5. On the other hand, no matter which density difference is used in the definition of Ri, the same results are obtained ø 101 Ri 10 2 Figure 5. Entrainment rate w /w i versus Richardson number Ri for the classical thermal. 3. Evaporative Cooling Effects on Thermal Entrainment Evaporative cooling happens when a positively buoyant cloud parcel becomes negatively buoyant because of a change in density due to the evaporation [Johari, 1992]. For instance, atmospheric moist convection is an example of a buoyancy reversing system because the mixing of positively buoyant cloudy air and dry environmental air usually leads to the formation of negative buoyancy [Grabowski, 1995] Buoyancy Reversal Parameter In order to simulate the density changes accompanying evaporative cooling in a water tank, Johari [1992] used mixtures of ethyl alcohol and ethylene glycol as thermals, which were released in a uniform environment from the bottom of a The results confirm the prediction that the entrainment rate tank. Johari [1992] found that the behavior of thermals deof a thermal impinging on a stratified interface is proportional pends strongly on a dimensionless parameter, called the buoyto Ri -3/2 [Cotel and Breidenthal, 1997]. The present experi- ancy reversal parameter: ments reinforce the importance of the persistence parameter. As we mentioned in section 1, a thermal has a nonpersistent character, moving relatively rapidly away from the interface. p* - p(0) D* p(0) - p(1)' (8) Let us compare a thermal with a vortex ring, the former The density is a function of the volume fraction of released driven by buoyancy and the latter driven by momentum. Asthermal fluid. Thus p(0) is the density of the environment, and suming the Ri, Re, and Sc are the same for both cases, the p(1) is that of the initial thermal fluid. The peak density of a entrainment regime is completely independent of momentum given alcohol-glycol mixture is p*. or buoyancy. For both the vortex ring and the thermal the The same technique was used in our experiments. Several persistence parameter is the same, less than its presumed crit- curves (Figure 6) describe the density of thermals against the ical value of order unity, and the same entrainment law is volume fraction of thermal fluid P. The horizontal line repremeasured in spite of the different origins of the flows. As sents the density of the lower layer fluid initially surrounding further examples, a steady jet is persistent while a plume is not the thermal. The curves above and below this line show the since the plume fluctuates much more than the jet [Dai et al., 1995]. There is a direct correlation between the value of the persistence parameter and the entrainment regime. It does not matter whether the flow originates from buoyancy or momentum. A fully accurate laboratory simulation of atmospheric thermals would include the effect of evaporative cooling and latent heat. We choose to add the effect of buoyancy reversal to a thermal in order to simulate evaporative cooling in the laboratory. Evaporative cooling is more practical to simulate in the laboratory than latent heat release at this stage of our study. thermal with negative and positive buoyancy, respectively. The pure thermal fluid is represented by P = 1, and the ambient fluid is represented by P = 0. When P equals P*, the alcoholglycol mixture reaches its peak density p*. A thermal would then have maximum negative buoyancy. The dimensionless parameter D* (equation(8)) determines the intensity of buoyancy reversal. In the absence of any buoyancy reversal, D* = 0. Thermals rise continuously in a uniform environment where D* < 0.5. Within this regime, thermals do not stop or turn back even though they decelerate. They also leave some fluid

6 15,462 ZHANG AND COTEL: ENTRAINMENT DUE TO AN IMPINGING THERMAL ø ø ' I Density of Water (gm/cc)... D*= D*= D*=0.471 _ m _ D*=0 I I i I I I I I I I II I,, I!, I I ø 'l ø Volume Fraction P Figure 6. Density p as a function of volume fraction P. behind in their wake. For the values of D* close to 0.5, thermals detrain a large volume of fluid [Johari, 1992]. In general, the trajectory of thermals with buoyancy reversal depends strongly on D*, and the "evaporating" thermals do not penetrate as far as the thermals without buoyancy reversal. The range of D* for the present experiments is from 0 to 0.5. Within this range, thermals rise toward the top of the tank and do not break into two parts or turn down shortly after having been released, as discussed above. There is a small amount of entrainment across the interface when the range of D* is >0.5, so that no accurate measurements could be performed. We chose to restrict the range of D* so that the thermals produced are representative of the real atmospheri case [Siems et al., 1990]. alcohol varies from 0.04 to 0.17% in this layer for different runs. The lower layer fluid is pure water with density greater than p.. The release mechanism is a cylinder covered with a small sliding lid to prevent the alcohol-glycol mixture from mixing with the surrounding fluid. The mixture is visualized by adding water soluble dye. The volume ratio of alcohol to glycol in the initial thermals is 94 to 71% for different runs. The densities of the two fluids, p. and pb, are always greater than the thermal density lot. The experimental procedure used in the nonevaporative case is repeated for the evaporative case. To ensure that thermals are reaching the interface without wall effects, the interface is lowered by draining the water from the lower chamber. The appropriate position was found to be at a height between 14 and 20 cm from the bottom of the tank for different runs. The draining process must be slow to avoid disturbing the interface. Then the lid is pulled out to begin a run Results The buoyant thermal fluid mixture rising from the cylinder at the bottom of the tank is shown in Figure 7a. When the initial positively buoyant thermal begins to mix with the ambient fluid, the density of the thermal is increased but is still less than that of the lower layer (Figure 7a). A partially negatively buoyanthermal is generated as shown in Figure 7b. It appears that the negative buoyancy decelerates the thermal. The thermal continues to travel upward, because the total density of the thermal is still less than that of the environment, penetrates through the interface, rebounds from it, and spreads laterally along the interface (Figure 7c). The interface rises a certain amount because of turbulent entrainment of the upper layer fluid into the thermal. The thermal collapsed after several seconds. Finally, the thermal entrainment process is finished (Figure 7d). The measurement techniques used in the classical thermal case are repeated here, and the same quantities are measured, i.e., the instantaneous width of the thermal below the interface, the velocity of the thermal, and the entrainment velocity. Johari's [1992] experiment show that the molecular-scale mixing features of these thermals are similar to those of rising thermals without buoyancy reversal as long as these thermals The dimensionless entrainment rate W e/w versus Ri is shown in six log-log plots (Figures 8a-8e), and the value of the exponent is determined from curve fitting to between maintain upward motion, even though the trajectory is strongly and -1.63, following (1). The entrainment rate W e/w is deaffected by D*. This implies that buoyancy reversal does not fined as before, i.e., on the basis of the thermal quantities influence the fundamental mixing and entrainment dynamics below the interface. Ri is based on the same thermal quantiof convective elements; that is, entrain tongues of ambient fluid at each vortex rotation independent of buoyancy reversal. ties. The entrainment rates clearly decline with increasing Ri (Figures 8a-8e) for different values of D*. The relationship The same result was also found in numerical experiments. between the entrainment rate W e/w and the dimensionless Grabowski and Clark [1993] and Grabowski [1995] believe that entrainment is driven by the baroclinic instability and not by parameter D* between 0 and 0.5 is plotted in log-log plots for several values of Ri (Figures 9a-9c). The slope between We/W the buoyancy reversal process. They suggested that buoyancy and D* is from 0.03 to 0.08, obtained by curve fitting. The reversal associated with evaporative cooling affects the overall error is of the order of _+10-20% because of the lack of flow evolution but has small effects on the dynamics of the entraining eddies. In other words, buoyancy reversal has little effect at the initial stages of cloud development, but the entrainment rate is likely to change significantly when the buoyancy reversal becomes very strong. accuracy when measuring the time intervals. Re based on the thermal quantities at the interface is 104. The possibility of buoyancy reversal between the lower and upper layer fluids has been tested. No buoyancy reversal between those two layers was observed Experimental Procedure The apparatus is identical to that used for the experiments of the classical thermals (Figure 3). Two fluids with different densities represent two layers separated by a stainles steel sliding plate in the middle of the tank. The upper layer fluid is a mixture of water and alcohol with density p.. The content of 3.4. Discussion The exponent a of the power law describing the entrainment rate as a function of Ri is similar to the one found for classical thermals. A difference in the magnitude of the dimensionless coefficient c for the entrainment rate with and without buoyancy reversal of a factor of 10 (Figure 10) exists. This can be

7 ZHANG AND COTEL: ENTRAINMENT DUE TO AN IMPINGING THERMAL 15,463 Figure 7. (a) a thermal with buoyancy reversal (D* = 0.16) just after release, (b) a thermal with buoyancy reversal (D* = 0.16) before impinging on the stratified interface, (c) a thermal with buoyancy reversal (D* = 0.16) impinging on the stratified interface, and (d) a thermal with buoyancy reversal (D* = 0.16) after impingement on the stratified interface.

8 15,464 ZHANG AND COTEL: ENTRAINMENT DUE TO AN IMPINGING THERMAL Figure 7. (continued) explained by the decrease in size and acceleration of the thermal due to buoyancy reversal. As seen in Figure 7c, some of the thermal fluid is left behind as the thermal rises. A net decrease in velocity is also easily observed. The entrainment velocity at large Ri could not be measured because of low measurement accuracy. These results indicate that the relationship between the entrainment rate and Ri follows the same power law for thermals with or without buoyancy reversal, impinging on a stratified interface. The results in Figures 9a-9c demonstrate that the entrain-

9 ... ZHANG AND COTEL: ENTRAINMENT DUE TO AN IMPINGING THERMAL 15,465 a loø2 C 10-2 D*=0.053 D*=0.16 lo a.46 lo l0 ø l01 ] ø 10' 102 gi d 10-2 Ri b 10'2 D*=0.078 D*=0.214 l I i i i,,,,, i,,,,,!,, 10 ø RI ø l0 t 10 2 D*=0.471 Ri 10-4 I ø 101 Ri lo 2 Figure 8. Entrainment rate We/Wl versus Ri for different values ofd*' (a) D* = 0.053, (b) D* = 0.078, (c) D* = 0.16, (d)d* = 0.214, and (e)d* = ment rates can be assumed to be independent of the buoyancy reversal parameter for 0 < D* < 0.5 for the case of thermals impinging on a stratified interface. This is in agreement with Grabowski and Clark [1993], who stipulate that the effects of buoyancy reversal are negligible on the dynamics of entraining thermals when the buoyancy reversal parameter is small. These results are also supported by Johari's [1992] laboratory experiments. The mixing and entrainment dynamics of convective elements do not depend on the presence of the buoyancy reversal phenomenon as long as upward motion is maintained. Therefore that buoyancy reversal does not significantly affect the rate of mixing between convecting fluid and its environ- ment is believed. 4. Summary and Conclusions Laboratory experiments provide a powerful tool for investigating the fundamentals of stratified turbulence, leading to new insight in natural atmospheric phenomena. The present experiments aim to improve the current understanding of transport across the tropopause by making the first measurements of entrainment across an interface due to a rising ther- mal with and without evaporative cooling. For thermals without buoyancy reversal the entrainment rate was found to be proportional to Ri - /2, as predicted by Corel and Breidenthal [1997]. The effect of evaporative cooling on thermal entrainment was investigated in the laboratory. The

10 15,466 ZHANG AND COTEL: ENTRAINMENT DUE TO AN IMPINGING THERMAL l0-2 Ri=4 10 ø D ß Ri=6 I i I I I I I II I I I I I,, I ø D*,, I, i, i Ri=10 of 10) between the dimensionless entrainment rate coefficients with and without buoyancy reversal because of the deceleration and the reduced size of the thermal in the buoyancy reversal case. Ri ranged between 2.5 and 25, Re was of the order of 104, and S c was around 600 for the experiments performed. The large differences in Re and Sc between the laboratory and the atmosphere are important issues for geophysical interpretation of laboratory results. Re in the atmosphere ( ) is much larger than Re in the laboratory (of the order of 104). The results from laboratory experiments are expected to be applicable to the atmosphere as long as Re is above a critical value of order 103 called the mixing transition [Breidenthal and Baker, 1985], which is necessary for the development of small-scale turbulence. Furthermore, the entrainment rate of a jet impinging on a stratified interface was found to be independent of Re for Ri between 1 and 100 [Cotel, 1995]. Moreover, Siems et al. [1990] analyzed soundings from atmospheric data on stratocumulus and trade cumulus clouds, taken from the First International Satellite Cloud Climatology Project (ISCCP) Regional Experiment (FIRE) in 1987 and Joint Hawaiian Warm Rain Project experiment in 1985, respectively. They suggesthat D* observed in stratocumulus is often <1.3. The buoyancy reversal parameter in the laboratory is of the same order as that in the atmosphere [Siems et al., 1990]. The laboratory experiments represent a case of weak buoyancy reversal. The experiments reported here represent the first step in quantifying turbulent entrainment of thermals impinging on a stratified interface. The logical next step would be to include latent heat to simulate completely the atmosphericase in the laboratory. Also, the amount of mixing produced by a thermal penetrating in a stratified environment needs to be quantified using laser-induced fluorescence. Particle image velocimetry measurements of the velocity and vorticity fields would be useful in determining the interface dynamics. These future steps would further our understanding of thermals in a strati- fied environment I I i I I i i i I i i i i i i i i D* 10-3 Figure 9. Entrainment rate We/W1 versus buoyancy reversal parameter D* for different values of Ri: (a) Ri = 4, (b) Ri = 6, and (c) Ri = 10. entrainment rate was also found to be proportional to Ri -3/2. In our experiments the value of the entrainment rate exponent does not depend on the buoyancy reversal parameter D* for the range of 0-0.5, which is representative of the atmosphere [Siems et al., 1990]. There is a difference in magnitude (factor 10-5 Thermal with Buoyancy Reversal D*=0.053 i, i i i i I! i, I, i i, 10 ø Ri Figure 10. Comparison of the entrainment rate We/W versus Ri with (D* = 0.053) and without buoyancy reversal.

11 ZHANG AND COTEL: ENTRAINMENT DUE TO AN IMPINGING THERMAL 15,467 Acknowledgments. This work was supported by the Natural Sciences and Engineering Research Council of Canada and a University Linden, P. F., The interaction of a vortex ring with a sharp density interface, J. Fluid Mech., 60, , of Manitoba Research Grant. Richards, J. M., Experiments on the penetration of an interface by References buoyant thermals, J. Fluid Mech., 11, , Roshko, A., Turbulent shear flow: A new look, A/AA J., 14, , Batchelor, G. K., Heat convection and buoyancy effects in fluids, Saunders, P.M., Penetrative convection in stably stratified fluids, Tellus, 14, , Q. J. R. Meteorol. Soc., 80, , Breidenthal, R. E., and M. B. Baker, Convection and entrainment Scorer, R. S., Experiments on convection of isolated masses of buoyant fluid, J. Fluid Mech., 2, , across stratified interfaces, J. Geophys. Res., 90, 13,055-13,062, Cotel, A. J., Entrainment and detrainment of a jet impinging on a Siems, S. T., S. B. Christopher, M. B. Baker, S.S. Shy, and R. E. stratified interface, Ph.D. thesis, Univ. of Wash., Seattle, Breidenthal, Buoyancy reversal and cloud-top entrainment instabil- Cotel, A. J., and R. E. Breidenthal, A model of stratified entrainment ity, Q. J. R. Meteorol. Soc., 116, , Turner, J. S., Model experiments relating to thermals with increasing using vortex persistence, Appl. Sci. Res., 57, , Cotel, A. J., J. A. Gjestvang, N. N. Ramkhelawan, and R. E. Brebuoyancy, Q. J. R. Meteorol. Soc., 89, 62-74, Turner, J. S., Buoyancy Effects in Fluids, 368 pp., Cambridge Univ. identhal, Laboratory experiments of a jet impinging on a stratified Press, New York, interface, Exp. Fluids, 23, , Turner, J. S., Turbulence entrainment: The development of the en- Dai, Z., L. K. Tseng, and G. M. Faeth, Velocity statistics of round, trainment assumption, and its application to geophysical flows, J. fully-developed, buoyant turbulent plumes, J. Heat Transfer, 117, Fluid Mech., 173, , , Turner, J. S., and I. K. Yang, Turbulent mixing at the top of stratocu- Emanuel, K. A., Atmospheric Convection, 580 pp., Oxford Univ. Press, mulus clouds, J. Fluid Mech., 17, , New York, Wallace, J. M., and P. V. Hobbs, Atmospheric science: An introduc- Grabowski, W. W., Entrainment and mixing in buoyancy-reversing tory survey, Academic, San Diego, Calif., convection with applications to cloud-top entrainment instability, Woodward, B., The motion in and around isolated thermals, Q. J. R. Q. J. R. Meteorol. Soc., 121, , Meteorol. Soc., 85, , Grabowski, W. W., and T. L. Clark, Cloud-environment interface instability: Rising thermal calculations in two spatial dimensions, J. Atmos. Sci., 48, , A. Cotel, Department of Civil and Environmental Engineering, Uni- Grabowski, W. W., and T. L. Clark, Cloud-environment interface versity of Michigan, Ann Arbor, MI (acotel@engin. instability, part II, Extension to three spatial dimensions, J. Atmos. umich.edu) Sci., 50, , Q. Zhang, Department of Mechanical and Industrial Engineering, Johari, H., An experimental investigation of mixing in buoyant flows, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2. Ph.D. thesis, Univ. of Wash., Seattle, Johari, H., Mixing in thermals with and without buoyancy reversal, J. Atmos. Sci., 49, , Kumagai, M., Turbulent buoyant convection from a source in a confined two-layer region, J. Fluid Mech., 147, , (Received June 23, 1999; revised November 15, 1999; accepted December 18, 1999.)