- Email: [email protected]
A trigger mechanism for the Lake Nyos disaster
Journal of Volcanology and Geothermal Research 88 Ž1999. 343–347
A trigger mechanism for the Lake Nyos disaster Aline J. Cotel
)
Department of Mechanical and Industrial Engineering, UniÕersity of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 Received 21 July 1998; accepted 11 January 1999
Abstract The catastrophic release of carbon dioxide gas from Lake Nyos on 21 August 1986 is discussed in the context of the buoyancy reversal instability. Originally proposed by Randall Ž1980. wRandall, D.S., 1980. Conditional instability of the first kind upside-down. Journal of Atmospheric Sciences 37: 125–130.x and Deardoff Ž1980. wDeardoff, J.W., 1980. Cloud-top entrainment instability. Journal of Atmospheric Sciences 37: 131–147.x for the ‘cloud-top entrainment instability’ of stratocumulus clouds, the buoyancy reversal instability has been studied experimentally in water tank experiments by Shy and Breidenthal Ž1990. wShy, S.S., Breidenthal, R.E., 1990. Laboratory experiments on the cloud-top entrainment instability. Journal of Fluid Mechanics 214: 1–15.x, who identified three criteria for instability. The initial disturbance must be sufficiently large, so that its Reynolds number is above the mixing transition, its Richardson number must be less than one to achieve overturning and mixing, and the buoyancy reversal parameter must be greater than a critical value, of order one. The implications and applicability of these criteria to Lake Nyos are discussed. The criterion for the Reynolds number is easily satisfied for typical wind velocities in the Lake Nyos region. Similarly, the Richardson number based on incident turbulence is easily less than unity, and therefore satisfy the second criterion for instability. In the case of Lake Nyos, the continuous release of carbon dioxide at the bottom of the lake increases the value of the buoyancy reversal parameter until it reaches its critical value, at which point an explosion occurs. This instability provides a plausible trigger for the 1986 explosion. After each explosion, the buoyancy reversal parameter returns to below its critical value, only to slowly rise again over time, as CO 2 continues to enter the lake, setting the stage for the next explosion. Future explosions may be avoided if the value of the buoyancy reversal parameter is prevented from approaching its critical value by artificial mixing at the thermocline, such as with a bubble plume. q 1999 Elsevier Science B.V. All rights reserved. Keywords: CO 2 ; Lake Nyos; Buoyancy reversal parameter
1. Introduction Lake Nyos, a remote lake in northwestern Cameroon in West–Central Africa, is located in a region of volcanic activity called the ‘Cameroon Volcanic Line’. It has been active for several million )
Fax: q1-204-275-7507; E-mail: [email protected]
years, and has generated lava flows and pyroclastic material ŽSigvaldason, 1989.. The water of Lake Nyos contains volatile compounds, with carbon dioxide as the most abundant and also the component most likely to reach the surface due to its large quantity and low reactivity. A decade ago, a sudden release of carbon dioxide ŽCO 2 . gas from Lake Nyos killed over 1700 people
0377-0273r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 0 2 7 3 Ž 9 9 . 0 0 0 1 7 - 7
344
A.J. Cotel r Journal of Volcanology and Geothermal Research 88 (1999) 343–347
and many animals. The release of CO 2 was not randomly distributed over the entire lake, but was localized in the form of a fountain, 120 m high. The gas spread up to 25 km from the lake. Associated with the release were surface waves on the lake which rose to a height of 25 m above the lake level and over 80 m at one point ŽKling et al., 1987.. Before the event, the lake had been strongly stratified, with a heavy, CO 2-saturated, lower layer. Some time after the eruption, the lake returned to a similar stratification, with a dense, CO 2-rich, deeper layer, and a chemocline separating a less dense upper layer ŽSigvaldason, 1989.. Proposed mechanisms generally fall into one of the two classes: an internal release of CO 2 from that dissolved in the lake water itself ŽKling et al., 1989; Giggenbach, 1990. or a volcanic release from beneath the lake ŽTazieff, 1988; Sigvaldason, 1989.. A variety of evidence favors the former mechanism. The lake is chemically stratified, the water was relatively cool and nearly isothermal after the eruption, no sulfur and chlorine compounds were found in the lake waters or sediments, as well as no hydrogen fluoride Žthese gases are all common in a volcanic eruption., and finally no suspended sediments were detected in any water sample below 10 m. However, the precise requirements for such a release are not yet clear. If the limnic explanation is accepted, there is no explanation for what caused the significant amount of mixing to be produced and therefore exsolve the CO 2 from the lower layer. Zhang Ž1996. proposed a model for the eruption dynamics combining an equation of state for the gas–liquid mixture and Bernoulli’s equation. Zhang showed that bubbly flows can produce strong explosions, as in the case of Lake Nyos. However, no trigger mechanism was proposed. Another model was put forward by Chau et al. Ž1996. to explain the triggering mechanism. They modeled the gas injection from the bottom of the lake and the degassing process based on a simple diffusion-driven equation, concluding that the outburst was triggered from the lake bottom. However, this is in apparent conflict with the observation that sediments were not disturbed at the lake bottom ŽKling et al., 1987.. Also, they assumed that the trigger mechanism came from an oversaturation of CO 2 in the lake.
The purpose of this paper is to propose both the criteria for instability and the critical physical parameters for the release of gas from the lake water. As we will see, it is in accord with most of the observations, including the fact that after the release, the lake still contained considerable dissolved CO 2 .
2. Cloud-top entrainment instability Randall Ž1980. and Deardoff Ž1980. proposed a new instability for the mixing of dry and cloudy air. Consider the top of a stratocumulus cloud, with dry air of lower density floating on top of higher density, cloudy air. After mixing of dry and cloudy parcels, the evaporative cooling of the cloudy droplets causes the mixture to increase in density. Under certain conditions, parcels of mixed fluid would have a greater density than even the cloudy air below. As a consequence, the heavy mixed parcels would descend, perhaps causing additional mixing between dry and cloudy air during their descent. Additional mixing implies an instability, in which the process accelerates exponentially. Randall and Deardorff suggested that the criterion for instability is simply that the density of the mixed parcels be greater than either parent parcel. Shy and Breidenthal Ž1990. tested this hypothesis in simple water tank experiments. Using water and alcohol mixtures, a stratified interface was formed between two layers. The interface was disturbed, causing mixing between the two layers. The experiments confirmed the existence of the Randall– Deardorff instability. However, in contrast to the original hypothesis, the instability occurred only when the mixed fluid was much denser than the lower fluid. A buoyancy reversal parameter D was proposed, which compared the density increase from mixing to the initial stratification. Only when D was greater than a critical value Dcrit , in this case, 1.3 for the water–alcohol system, did the descending mixture trigger additional mixing. In the laboratory experiments, two other dimensionless criteria for instability were noted. Because this is a mixing instability, the initial disturbance must exceed the mixing transition ŽKonrad, 1977; Breidenthal, 1981. in water. Consequently, the
A.J. Cotel r Journal of Volcanology and Geothermal Research 88 (1999) 343–347
Reynolds number of the disturbance must exceed about 10 3. The mixing occurs at a stratified interface. Therefore, the Richardson number of the disturbance must be less than one for at least some eddy whose Reynolds number is above the mixing transition. Only in this way can a tongue of buoyant fluid be pulled across the interface and around a vortex, permitting its engulfment and mixing ŽRoshko, 1976..
3. Comparison with Lake Nyos In the buoyancy reversal instability, the violent activity begins at a stratified interface. It is independent of the boundaries. This is consistent with the observation that the bottom sediments in the lake were largely undisturbed by the release. The laboratory experiments not only revealed that D must be greater than Dcrit for instability, but also that the instability ceases as soon as D falls again below Dcrit . This occurs as the mixed fluid dilutes the surrounding fluid while descending through it. The amount of mixed fluid from the instability is constrained by the size of the container. Applied to Lake Nyos, this implies that the CO 2 gas eruption will cease as soon as the concentration of dissolved CO 2 is either sufficiently depleted in the lower layer or sufficiently increased in the upper layer such that D falls below Dcrit . Either way, the residual dissolved CO 2 after the release would still be appreciable, in accord with observation. Because the mixed fluid contains bubbles, it would be positively buoyant. Even a relatively small mass release of CO 2 corresponds to a large mean density change. The bubbly mixture would accelerate upward, with an initial motion in the near field unaffected by turbulent entrainment and deceleration. If the horizontal width of the bubbly mixture is comparable to or greater than the thermocline depth, then there is insufficient distance for turbulent entrainment to slow down the buoyant flow, so that it will reach the lake surface at its full hydraulic speed. The mixture would accelerate upward, potentially reaching its hydraulic speed at the lake surface. Such speeds easily account for the large waves inferred from shoreline damage.
345
4. Estimation of the three dimensionless parameters for Lake Nyos The Richardson number is based on the turbulence quantities impinging on the thermocline: Ri s
g Xd w 12
,
where d is the characteristic length scale associated with the incident turbulence, w 1 its characteristic velocity and g X s Ž D rrr . g is based on the density difference D r across the stratified interface, as shown in Fig. 1. Similarly, the Reynolds number is also based on the turbulence at the thermocline: Re s
w1 d Õ
,
where Õ is the kinematic viscosity of water. The buoyancy reversal parameter is defined as: r 1 y rmix D' , r1 y r2 where r 1 is the density of the upper layer, r 2 the density of the lower layer, and rmix is the density of the mixture containing the CO 2 bubbles. Here is an example of the type of disturbance required to trigger the instability in Lake Nyos. Assume a characteristic velocity of 10 cmrs, a characteristic length scale of 10 cm, a density difference of the order of 1 kgrm3 ŽKling et al., 1994., the Reynolds number is then of the order of 10 4 , and the Richardson number is 10y1 . These easily satisfy two of the three criteria. In the case of D, the buoyancy reversal parameter, as the concentration of CO 2 in the lake increases, D also increases, up to the critical value, where a very small perturbation, such as in the example above, can create a significant amount of mixing and therefore initiate the outgassing of the lake.
Fig. 1. Schematic of the impinging turbulence.
346
A.J. Cotel r Journal of Volcanology and Geothermal Research 88 (1999) 343–347
Fig. 2. Solubility of CO 2 vs. temperature at a pressure of 1 atm.
It is proposed that the mixing occurred at the upper thermocline, not at the deeper chemocline, which is in agreement with both the conclusions of Tietze Ž1992. and the fact that bottom sediments were not disturbed. Mixing across the thermocline of heavy, cool water saturated with CO 2 with warmer surface water yields a mixture of intermediate temperature. The maximum amount of dissolved CO 2 is a nonlinear function of water temperature at any given pressure ŽCRC Handbook of Chemistry and Physics, 1996.. If the surface layer water is saturated in CO 2 , then any mixing across the thermocline will yield supersaturated mixtures, illustrated in Fig. 2, which shows the behavior at one atmosphere as an example. Any mixed fluid element must lie somewhere along a straight line between the two parent parcels. Since the saturated curve is concave upward, it is always below the straight line, no matter what the initial temperatures of the two parent parcels. Thus, any mixture is supersaturated. The excess CO 2 in the mixed fluid comes out of solution, forming bubbles. As long as the buoyancy reversal parameter D exceeds its critical value, the laboratory experiments of Shy and Breidenthal demonstrate that the buoyant vortex core will continue to lift up a tongue of heavy water from below the thermocline, mixing it with surface water and feeding itself, growing exponentially. Eventually, D must fall below the critical value, as the supply of unmixed fluid on the two sides of the thermocline is limited.
Note that this mechanism does not rely on pressure changes or large vertical displacements in the water column to trigger the gas release. Only a very small disturbance is required. As a consequence, the proposed buoyancy-reversal mechanism is more likely to occur first. Also note that the Reynolds and Richardson number criteria are more or less continuously satisfied by the random perturbations of wind stress and the like. Consequently, the ability to predict and to prevent CO 2 releases hinges on the behavior of the buoyancy reversal parameter D. According to the proposed mechanism, the time history of D exhibits a sawtooth waveform, slowly rising as CO 2 is added to the lake water until D reaches its critical value, then suddenly dropping during an explosion. As more CO 2 enters the lake, the cycle repeats itself.
5. Conclusions Initially developed for cloud dynamics, the buoyancy reversal entrainment instability is proposed to explain the Lake Nyos CO 2 release. For instability, there are three critical parameters: the Reynolds and Richardson numbers and the buoyancy reversal parameter, which have been estimated for the specific case of Lake Nyos. In a lake, it is relatively easy to
A.J. Cotel r Journal of Volcanology and Geothermal Research 88 (1999) 343–347
generate disturbances with appropriate Reynolds and Richardson numbers. For example, wind blowing in the vicinity of the lake will create shear at the surface, and generate vorticity, which will be transported to the thermocline. The Reynolds number based on the turbulence at the thermocline can easily reach the critical value of 10 3. The shear will produce turbulence at the interface, allowing the Richardson number for some eddy size to drop below its critical value of unity. Similarly, internal wave breaking can allow the Reynolds and Richardson numbers to meet the instability criteria. Internal waves are present anytime there is a density difference and small perturbations. In a case like Lake Nyos, internal wave breaking occurs at the side boundaries of the lake and produce turbulence which can meet the criteria for the Reynolds and Richardson numbers. In particular, internal waves breaking on a slope produces vortices at the boundary, which are strong and equally spaced along the slope ŽSlinn and Riley, 1995.. Since the Reynolds and Richardson numbers easily meet the instability criteria, the controlling criterion is the buoyancy reversal parameter D. The steady buildup of dissolved CO 2 at depth will slowly increase the value of D, until its critical value is reached, at which point an explosion is eminent. It follows that such catastrophes can only be predicted by monitoring the CO 2 concentration, and they can only be prevented by reducing D artificially. This is readily achieved by eliminating the thermocline with artificial mixing across the interface with an air bubble plume. The instability criteria are not only applicable to Lake Nyos, but to any other stratified volcanic lake.
Acknowledgements The author wishes to thank Guillermo Terrones for introducing me to the problem and Robert Breidenthal for useful discussions.
347
References Breidenthal, R.E., 1981. Structure of turbulent mixing layers and wakes using a chemical reaction. J. Fluid Mech. 109, 1–24. Chau, H.F., Knowk, P.K., Mak, L., 1996. A model of gas buildup and release in crater lakes. J. Geophys. Res. 101 ŽB12., 28253–28263. CRC Handbook of Chemistry and Physics, 77th edn., 1996, D.R. Lide ŽEd... Deardoff, J.W., 1980. Cloud-top entrainment instability. J. Atmos. Sci. 37, 131–147. Giggenbach, W.F., 1990. Water and gas chemistry of Lake Nyos and its bearing on the eruptive process. J. Volcanol. Geotherm. Res. 42, 337–362. Kling, G.W., Evans, W.C., Tuttle, M.L., Tanylieke, G., 1994. Degassing of Lake Nyos. Nature 368, 405–406. Kling, G.W., Tuttle, M.L., Evans, W.C., 1989. The evolution of thermal structure and water chemistry in Lake Nyos. J. Volcanol. Geotherm. Res. 39 Ž2r3., 151–165. Kling, G.W., Clark, M.A., Compton, H.R., Devine, J.D., Evans, W.C., Humphrey, A.M., Koenigsberg, E.J., Lockwodd, J.P., Tuttle, M.L., Wagner, G.N., 1987. The Lake Nyos gas disaster in Cameroon, West Africa. Science 236, 169–175. Konrad, J.H., 1977. An experimental investigation of mixing in two-dimensional turbulent shear flows with application of diffusion-limited chemical reactions. PhD thesis, California Institute of Technology. Reprinted as project SQUID Tech. Rep. CIT-8-PU, Dec. 1978. Randall, D.S., 1980. Conditional instability of the first kind upside-down. J. Atmos. Sci. 37, 125–130. Roshko, A., 1976. Structure of turbulent flows: a new look. AIAA J. 14, 1349–1357. Shy, S.S., Breidenthal, R.E., 1990. Laboratory experiments on the cloud-top entrainment instability. J. Fluid Mech. 214, 1–15. Sigvaldason, G.E., 1989. International conference on Lake Nyos disaster, Yaounde, Cameroon, 16–20 March, 1987: conclusions and recommendations. J. Volcanol. Geotherm. Res. 39 Ž2r3., 97–107. Slinn, D.N., Riley, J.J., 1995. Turbulent mixing in the oceanic boundary layer caused by internal wave reflection from sloping terrain. Dyn. Atmos. Oceans 24, 51–62. Tazieff, H., 1988. Mechanisms of the Lake Nyos dioxide and of so-called phreatic steam eruptions. J. Volcanol. Geotherm. Res. 39 Ž2r3., 109–115. Tietze, K., 1992. In: Freeth, Ofoegbu, Onuoha ŽEds.., Natural hazards in West Africa and Central Africa. Vieweg, Braunschweig, pp. 97–107. Zhang, Y., 1996. Dynamics of CO 2-driven lake eruptions. Nature 379, 57–59.
A trigger mechanism for the Lake Nyos disaster Aline J. Cotel
)
Department of Mechanical and Industrial Engineering, UniÕersity of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 Received 21 July 1998; accepted 11 January 1999
Abstract The catastrophic release of carbon dioxide gas from Lake Nyos on 21 August 1986 is discussed in the context of the buoyancy reversal instability. Originally proposed by Randall Ž1980. wRandall, D.S., 1980. Conditional instability of the first kind upside-down. Journal of Atmospheric Sciences 37: 125–130.x and Deardoff Ž1980. wDeardoff, J.W., 1980. Cloud-top entrainment instability. Journal of Atmospheric Sciences 37: 131–147.x for the ‘cloud-top entrainment instability’ of stratocumulus clouds, the buoyancy reversal instability has been studied experimentally in water tank experiments by Shy and Breidenthal Ž1990. wShy, S.S., Breidenthal, R.E., 1990. Laboratory experiments on the cloud-top entrainment instability. Journal of Fluid Mechanics 214: 1–15.x, who identified three criteria for instability. The initial disturbance must be sufficiently large, so that its Reynolds number is above the mixing transition, its Richardson number must be less than one to achieve overturning and mixing, and the buoyancy reversal parameter must be greater than a critical value, of order one. The implications and applicability of these criteria to Lake Nyos are discussed. The criterion for the Reynolds number is easily satisfied for typical wind velocities in the Lake Nyos region. Similarly, the Richardson number based on incident turbulence is easily less than unity, and therefore satisfy the second criterion for instability. In the case of Lake Nyos, the continuous release of carbon dioxide at the bottom of the lake increases the value of the buoyancy reversal parameter until it reaches its critical value, at which point an explosion occurs. This instability provides a plausible trigger for the 1986 explosion. After each explosion, the buoyancy reversal parameter returns to below its critical value, only to slowly rise again over time, as CO 2 continues to enter the lake, setting the stage for the next explosion. Future explosions may be avoided if the value of the buoyancy reversal parameter is prevented from approaching its critical value by artificial mixing at the thermocline, such as with a bubble plume. q 1999 Elsevier Science B.V. All rights reserved. Keywords: CO 2 ; Lake Nyos; Buoyancy reversal parameter
1. Introduction Lake Nyos, a remote lake in northwestern Cameroon in West–Central Africa, is located in a region of volcanic activity called the ‘Cameroon Volcanic Line’. It has been active for several million )
Fax: q1-204-275-7507; E-mail: [email protected]
years, and has generated lava flows and pyroclastic material ŽSigvaldason, 1989.. The water of Lake Nyos contains volatile compounds, with carbon dioxide as the most abundant and also the component most likely to reach the surface due to its large quantity and low reactivity. A decade ago, a sudden release of carbon dioxide ŽCO 2 . gas from Lake Nyos killed over 1700 people
0377-0273r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 0 2 7 3 Ž 9 9 . 0 0 0 1 7 - 7
344
A.J. Cotel r Journal of Volcanology and Geothermal Research 88 (1999) 343–347
and many animals. The release of CO 2 was not randomly distributed over the entire lake, but was localized in the form of a fountain, 120 m high. The gas spread up to 25 km from the lake. Associated with the release were surface waves on the lake which rose to a height of 25 m above the lake level and over 80 m at one point ŽKling et al., 1987.. Before the event, the lake had been strongly stratified, with a heavy, CO 2-saturated, lower layer. Some time after the eruption, the lake returned to a similar stratification, with a dense, CO 2-rich, deeper layer, and a chemocline separating a less dense upper layer ŽSigvaldason, 1989.. Proposed mechanisms generally fall into one of the two classes: an internal release of CO 2 from that dissolved in the lake water itself ŽKling et al., 1989; Giggenbach, 1990. or a volcanic release from beneath the lake ŽTazieff, 1988; Sigvaldason, 1989.. A variety of evidence favors the former mechanism. The lake is chemically stratified, the water was relatively cool and nearly isothermal after the eruption, no sulfur and chlorine compounds were found in the lake waters or sediments, as well as no hydrogen fluoride Žthese gases are all common in a volcanic eruption., and finally no suspended sediments were detected in any water sample below 10 m. However, the precise requirements for such a release are not yet clear. If the limnic explanation is accepted, there is no explanation for what caused the significant amount of mixing to be produced and therefore exsolve the CO 2 from the lower layer. Zhang Ž1996. proposed a model for the eruption dynamics combining an equation of state for the gas–liquid mixture and Bernoulli’s equation. Zhang showed that bubbly flows can produce strong explosions, as in the case of Lake Nyos. However, no trigger mechanism was proposed. Another model was put forward by Chau et al. Ž1996. to explain the triggering mechanism. They modeled the gas injection from the bottom of the lake and the degassing process based on a simple diffusion-driven equation, concluding that the outburst was triggered from the lake bottom. However, this is in apparent conflict with the observation that sediments were not disturbed at the lake bottom ŽKling et al., 1987.. Also, they assumed that the trigger mechanism came from an oversaturation of CO 2 in the lake.
The purpose of this paper is to propose both the criteria for instability and the critical physical parameters for the release of gas from the lake water. As we will see, it is in accord with most of the observations, including the fact that after the release, the lake still contained considerable dissolved CO 2 .
2. Cloud-top entrainment instability Randall Ž1980. and Deardoff Ž1980. proposed a new instability for the mixing of dry and cloudy air. Consider the top of a stratocumulus cloud, with dry air of lower density floating on top of higher density, cloudy air. After mixing of dry and cloudy parcels, the evaporative cooling of the cloudy droplets causes the mixture to increase in density. Under certain conditions, parcels of mixed fluid would have a greater density than even the cloudy air below. As a consequence, the heavy mixed parcels would descend, perhaps causing additional mixing between dry and cloudy air during their descent. Additional mixing implies an instability, in which the process accelerates exponentially. Randall and Deardorff suggested that the criterion for instability is simply that the density of the mixed parcels be greater than either parent parcel. Shy and Breidenthal Ž1990. tested this hypothesis in simple water tank experiments. Using water and alcohol mixtures, a stratified interface was formed between two layers. The interface was disturbed, causing mixing between the two layers. The experiments confirmed the existence of the Randall– Deardorff instability. However, in contrast to the original hypothesis, the instability occurred only when the mixed fluid was much denser than the lower fluid. A buoyancy reversal parameter D was proposed, which compared the density increase from mixing to the initial stratification. Only when D was greater than a critical value Dcrit , in this case, 1.3 for the water–alcohol system, did the descending mixture trigger additional mixing. In the laboratory experiments, two other dimensionless criteria for instability were noted. Because this is a mixing instability, the initial disturbance must exceed the mixing transition ŽKonrad, 1977; Breidenthal, 1981. in water. Consequently, the
A.J. Cotel r Journal of Volcanology and Geothermal Research 88 (1999) 343–347
Reynolds number of the disturbance must exceed about 10 3. The mixing occurs at a stratified interface. Therefore, the Richardson number of the disturbance must be less than one for at least some eddy whose Reynolds number is above the mixing transition. Only in this way can a tongue of buoyant fluid be pulled across the interface and around a vortex, permitting its engulfment and mixing ŽRoshko, 1976..
3. Comparison with Lake Nyos In the buoyancy reversal instability, the violent activity begins at a stratified interface. It is independent of the boundaries. This is consistent with the observation that the bottom sediments in the lake were largely undisturbed by the release. The laboratory experiments not only revealed that D must be greater than Dcrit for instability, but also that the instability ceases as soon as D falls again below Dcrit . This occurs as the mixed fluid dilutes the surrounding fluid while descending through it. The amount of mixed fluid from the instability is constrained by the size of the container. Applied to Lake Nyos, this implies that the CO 2 gas eruption will cease as soon as the concentration of dissolved CO 2 is either sufficiently depleted in the lower layer or sufficiently increased in the upper layer such that D falls below Dcrit . Either way, the residual dissolved CO 2 after the release would still be appreciable, in accord with observation. Because the mixed fluid contains bubbles, it would be positively buoyant. Even a relatively small mass release of CO 2 corresponds to a large mean density change. The bubbly mixture would accelerate upward, with an initial motion in the near field unaffected by turbulent entrainment and deceleration. If the horizontal width of the bubbly mixture is comparable to or greater than the thermocline depth, then there is insufficient distance for turbulent entrainment to slow down the buoyant flow, so that it will reach the lake surface at its full hydraulic speed. The mixture would accelerate upward, potentially reaching its hydraulic speed at the lake surface. Such speeds easily account for the large waves inferred from shoreline damage.
345
4. Estimation of the three dimensionless parameters for Lake Nyos The Richardson number is based on the turbulence quantities impinging on the thermocline: Ri s
g Xd w 12
,
where d is the characteristic length scale associated with the incident turbulence, w 1 its characteristic velocity and g X s Ž D rrr . g is based on the density difference D r across the stratified interface, as shown in Fig. 1. Similarly, the Reynolds number is also based on the turbulence at the thermocline: Re s
w1 d Õ
,
where Õ is the kinematic viscosity of water. The buoyancy reversal parameter is defined as: r 1 y rmix D' , r1 y r2 where r 1 is the density of the upper layer, r 2 the density of the lower layer, and rmix is the density of the mixture containing the CO 2 bubbles. Here is an example of the type of disturbance required to trigger the instability in Lake Nyos. Assume a characteristic velocity of 10 cmrs, a characteristic length scale of 10 cm, a density difference of the order of 1 kgrm3 ŽKling et al., 1994., the Reynolds number is then of the order of 10 4 , and the Richardson number is 10y1 . These easily satisfy two of the three criteria. In the case of D, the buoyancy reversal parameter, as the concentration of CO 2 in the lake increases, D also increases, up to the critical value, where a very small perturbation, such as in the example above, can create a significant amount of mixing and therefore initiate the outgassing of the lake.
Fig. 1. Schematic of the impinging turbulence.
346
A.J. Cotel r Journal of Volcanology and Geothermal Research 88 (1999) 343–347
Fig. 2. Solubility of CO 2 vs. temperature at a pressure of 1 atm.
It is proposed that the mixing occurred at the upper thermocline, not at the deeper chemocline, which is in agreement with both the conclusions of Tietze Ž1992. and the fact that bottom sediments were not disturbed. Mixing across the thermocline of heavy, cool water saturated with CO 2 with warmer surface water yields a mixture of intermediate temperature. The maximum amount of dissolved CO 2 is a nonlinear function of water temperature at any given pressure ŽCRC Handbook of Chemistry and Physics, 1996.. If the surface layer water is saturated in CO 2 , then any mixing across the thermocline will yield supersaturated mixtures, illustrated in Fig. 2, which shows the behavior at one atmosphere as an example. Any mixed fluid element must lie somewhere along a straight line between the two parent parcels. Since the saturated curve is concave upward, it is always below the straight line, no matter what the initial temperatures of the two parent parcels. Thus, any mixture is supersaturated. The excess CO 2 in the mixed fluid comes out of solution, forming bubbles. As long as the buoyancy reversal parameter D exceeds its critical value, the laboratory experiments of Shy and Breidenthal demonstrate that the buoyant vortex core will continue to lift up a tongue of heavy water from below the thermocline, mixing it with surface water and feeding itself, growing exponentially. Eventually, D must fall below the critical value, as the supply of unmixed fluid on the two sides of the thermocline is limited.
Note that this mechanism does not rely on pressure changes or large vertical displacements in the water column to trigger the gas release. Only a very small disturbance is required. As a consequence, the proposed buoyancy-reversal mechanism is more likely to occur first. Also note that the Reynolds and Richardson number criteria are more or less continuously satisfied by the random perturbations of wind stress and the like. Consequently, the ability to predict and to prevent CO 2 releases hinges on the behavior of the buoyancy reversal parameter D. According to the proposed mechanism, the time history of D exhibits a sawtooth waveform, slowly rising as CO 2 is added to the lake water until D reaches its critical value, then suddenly dropping during an explosion. As more CO 2 enters the lake, the cycle repeats itself.
5. Conclusions Initially developed for cloud dynamics, the buoyancy reversal entrainment instability is proposed to explain the Lake Nyos CO 2 release. For instability, there are three critical parameters: the Reynolds and Richardson numbers and the buoyancy reversal parameter, which have been estimated for the specific case of Lake Nyos. In a lake, it is relatively easy to
A.J. Cotel r Journal of Volcanology and Geothermal Research 88 (1999) 343–347
generate disturbances with appropriate Reynolds and Richardson numbers. For example, wind blowing in the vicinity of the lake will create shear at the surface, and generate vorticity, which will be transported to the thermocline. The Reynolds number based on the turbulence at the thermocline can easily reach the critical value of 10 3. The shear will produce turbulence at the interface, allowing the Richardson number for some eddy size to drop below its critical value of unity. Similarly, internal wave breaking can allow the Reynolds and Richardson numbers to meet the instability criteria. Internal waves are present anytime there is a density difference and small perturbations. In a case like Lake Nyos, internal wave breaking occurs at the side boundaries of the lake and produce turbulence which can meet the criteria for the Reynolds and Richardson numbers. In particular, internal waves breaking on a slope produces vortices at the boundary, which are strong and equally spaced along the slope ŽSlinn and Riley, 1995.. Since the Reynolds and Richardson numbers easily meet the instability criteria, the controlling criterion is the buoyancy reversal parameter D. The steady buildup of dissolved CO 2 at depth will slowly increase the value of D, until its critical value is reached, at which point an explosion is eminent. It follows that such catastrophes can only be predicted by monitoring the CO 2 concentration, and they can only be prevented by reducing D artificially. This is readily achieved by eliminating the thermocline with artificial mixing across the interface with an air bubble plume. The instability criteria are not only applicable to Lake Nyos, but to any other stratified volcanic lake.
Acknowledgements The author wishes to thank Guillermo Terrones for introducing me to the problem and Robert Breidenthal for useful discussions.
347
References Breidenthal, R.E., 1981. Structure of turbulent mixing layers and wakes using a chemical reaction. J. Fluid Mech. 109, 1–24. Chau, H.F., Knowk, P.K., Mak, L., 1996. A model of gas buildup and release in crater lakes. J. Geophys. Res. 101 ŽB12., 28253–28263. CRC Handbook of Chemistry and Physics, 77th edn., 1996, D.R. Lide ŽEd... Deardoff, J.W., 1980. Cloud-top entrainment instability. J. Atmos. Sci. 37, 131–147. Giggenbach, W.F., 1990. Water and gas chemistry of Lake Nyos and its bearing on the eruptive process. J. Volcanol. Geotherm. Res. 42, 337–362. Kling, G.W., Evans, W.C., Tuttle, M.L., Tanylieke, G., 1994. Degassing of Lake Nyos. Nature 368, 405–406. Kling, G.W., Tuttle, M.L., Evans, W.C., 1989. The evolution of thermal structure and water chemistry in Lake Nyos. J. Volcanol. Geotherm. Res. 39 Ž2r3., 151–165. Kling, G.W., Clark, M.A., Compton, H.R., Devine, J.D., Evans, W.C., Humphrey, A.M., Koenigsberg, E.J., Lockwodd, J.P., Tuttle, M.L., Wagner, G.N., 1987. The Lake Nyos gas disaster in Cameroon, West Africa. Science 236, 169–175. Konrad, J.H., 1977. An experimental investigation of mixing in two-dimensional turbulent shear flows with application of diffusion-limited chemical reactions. PhD thesis, California Institute of Technology. Reprinted as project SQUID Tech. Rep. CIT-8-PU, Dec. 1978. Randall, D.S., 1980. Conditional instability of the first kind upside-down. J. Atmos. Sci. 37, 125–130. Roshko, A., 1976. Structure of turbulent flows: a new look. AIAA J. 14, 1349–1357. Shy, S.S., Breidenthal, R.E., 1990. Laboratory experiments on the cloud-top entrainment instability. J. Fluid Mech. 214, 1–15. Sigvaldason, G.E., 1989. International conference on Lake Nyos disaster, Yaounde, Cameroon, 16–20 March, 1987: conclusions and recommendations. J. Volcanol. Geotherm. Res. 39 Ž2r3., 97–107. Slinn, D.N., Riley, J.J., 1995. Turbulent mixing in the oceanic boundary layer caused by internal wave reflection from sloping terrain. Dyn. Atmos. Oceans 24, 51–62. Tazieff, H., 1988. Mechanisms of the Lake Nyos dioxide and of so-called phreatic steam eruptions. J. Volcanol. Geotherm. Res. 39 Ž2r3., 109–115. Tietze, K., 1992. In: Freeth, Ofoegbu, Onuoha ŽEds.., Natural hazards in West Africa and Central Africa. Vieweg, Braunschweig, pp. 97–107. Zhang, Y., 1996. Dynamics of CO 2-driven lake eruptions. Nature 379, 57–59.