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Convection in the mantle: A laboratory model with temperature-dependent viscosity
EARTH AND PLANETARY LETTERS 17 (1973) 369- 374. NORTH-HOLLAND PUBLISHING COMPANY
CONVECTION
IN T H E M A N T L E : A L A B O R A T O R Y
TEMPERATURE-DEPENDENT
MODEL WITH
VISCOSITY
J.S. TURNER Department of Applied Mathematics and TheoreticalPhysics, University of Cambridge, Cambridge, UK
Received 6 October 1972 Revised version received 23 October 1972
Some exploratory laboratory experiments are described, using a working fluid (glycerine) whose viscosity depends strongly on temperature. Surface cooling produces a thin, viscous sheet which is driven away from regions of upward convection, and then plunges down steeply into the less viscous interior, maintaining its identity as it does so. When there are gradients of composition as well as temperature, convective motions can also be produced above the plunging sheets by a "double-diffusive" mechanism. The behaviour is quite unlike that observed previously in a fluid of constant properties, but it closely resembles some features of recent plate model of the Earth's mantle.
It is now widely accepted that convective motions are occurring in the Earth's mantle. Such motions are consistent with the measured distributions of the surface heat flux [1] and can provide the forces necessary to drive sea floor spreading and continental drift [2]. The earliest theories [ 3 - 5 ] which applied convection to this problem were, however, based largely on experience with fluids whose behaviour bears little relation to the geophysical situation. In particular, constant fluid properties have usually been assumed, whereas there can certainly be enormous variations in viscosity of the mantle material over the relevant range in temperature. (The viscosity at the surface may be 104 times that at 200 km). The present work is a first step towards making such laboratory experiments more realistic, using a fluid (glycerine) whose viscosity, like that of the mantle material, depends strongly on temperature. Rapid progress has been made recently in the field of plate tectonics [6, 7], which concentrates on the kinematics of the motion of rigid plates. Less attention has been given to the associated dynamical problems, though it is implied that the plates are being moved about by convection occurring in a separate region below them. The discussions of the dynamics of the motion have sometimes invoked various geo-
chemical processes, especially in the regions of downthrust [8]. The laboratory results presented below suggest, however, that a similar qualitative behaviour can be achieved entirely as a result of physical effects in a fluid with continuously variable properties. The need for models incorporating variable properties has certainly been recognised before, and some progress has been made theoretically using a numerical model [9] which shows that the dependence of viscosity on temperature is likely to be the most important effect. A thin region of high viscosity at the surface, overlying a deeper, hotter and less viscous region, is predicted in this way, and the more recent theories bring out the importance of concentrated convection plumes and thermal boundary layers. Elder [10] has investigated, both numerically and experimentally, the role of thermal turbulence in mantle convection, and has discussed the effect of variations in viscosity. He has also studied the behaviour of plastic sheets of non-uniform thickness floating on the surface of a liquid heated from below [11] The latter experiment showed how the resulting distribution o f heat flux produces a convection pattern which can propel the surface layers about; but of Course it used two very different substances to produce this effect. Whitehead [12] has reported related
370
J.S. Turner, Convection in the mantle: a laboratory model
~.-•
.
'_ ' Z i.~
.
.
•
~ ,
Q
m
Fig. 1. Shadowgraph photograph of a plunging sheet of cold, viscous glycerine, resulting from the combination of a strong circulation and surface coolinz produced by bubbles rising from a piece of dry ice on the far right of the experimental tank.
l:ig. 2. Convection driven by a central source. The sheet on the left is forced under at the edge of a barrier on the surface; note the convection occurring above the sinking sheet. laboratory experiments using heat sources which can be carried about by the convective motions they produce. Recently it has been suggested [13] that the properties of molten glass model those in the mantle, but that system is much more difficult to work with than the one described below. The concept behind the exploratory experiments reported here is very simple. In a fluid whose viscosity
varies widely over a temperature range available in the laboratory, we sought to produce strong convection in a region of higher temperature (low viscosity), combined with strong cooling at the surface to form a layer of high viscosity• An obvious choice of working fluid is glycerine, whose viscosity increases by a factor of 100 as the temperature is lowered from +30°C to - 15°C, and by another factor of 100 at
J.S. Turner, Convection in the mantle: a laboratory model
371
Fig. 3. Convection driven from both ends of the tank. Symmetrical plunging sheets of viscous fluid, with convection above them.
Fig. 4. The pattern of motion (as shown by a shadowgraph) produced by heating glycerine at a line source located at the bottom of the tank. A slow surface spreading, and a more rapid closed circulation, are clearly seen. - 4 0 ° C . Even in the easily accessible laboratory range 3 0 - 0 ° C , the viscosity increases by a factor o f 20. In early trials with purely thermal convection, however, it was found difficult to produce high enough velocities with the available temperature differences, and eventually a different technique was adopted. An appropriate form o f convection was produced by releasing a stream o f bubbles in the fluid, a method
which had previously been used in modelling atmospheric phenomena [14, 15]. A lump of dry ice (about 2 cm 3) was dropped through a 12-cm deep layer of glycerine, contained in a perspex tank 25 cm × 7 cm in cross section. The glycerine was either at room temperature or gently heated, so that the viscosity was relatively low (of order 10 poise). The dry ice adhered to the b o t t o m ,
372
J.S Turner, Convection in the mantle: a laboratory model
t.ig. 5. A single plunging sheet, with a light wedge of fluid formed by convection above it. ]'his is held in equilibrium by the outflow associated with the convection driven by the bubbles. ~~.
•
".
o-~. ~"
Fig. 6. The cxperiment of fig. 5, shortly after the bubbling has stopped. The wedge of light fluid floats over the top.
held by a cold viscous layer of glycerine, and released a stream of bubbles wtfich rose through the fluid and produced a strong, almost two-dimensional circulation in it. It is of course this circulation which isi an essential feature of the model, not the particular method (in this case the bubbles) used to dave it. ~,t the same time, the bubbles o f cold carbon dioxide breaking at the surface produced the desired increase
of viscosity there. In some experiments the surface of the glycerine was also cooled in a more controlled manner, through a thin layer of light, less viscous oil floating on its surface. With a typical temperature difference of 30°C, the most important new observation was that the viscous material near the surface was pushed outwards as a nearly rigid sheet and then plunged down steeply into the interior, maintaining
J.S. Turner, Convection in the mantle: a laboratory model
its identity as it did so. Buoyant and viscous forces must both contribute to this process, though the mechanism is not yet fully understood. The behaviour is illustrated in the shadowgraph photographs (figs. 1-3) for various geometries and combinations of convective sources, which show the striking contrast with convection in fluids of nearly constant properties. In fig. 1 there is a single bubble source at the far right of the tank, and a plunging viscous sheet leaving the surface at a sharp angle. Fig. 2 shows a central source, with unsymmetrical downflow, the sheet on the left being pushed under prematurely at the edge of a solid obstacle (a piece of plywood) floating on the surface. The result of driving with two streams of bubbles, one at either end of the tank, is shown in fig. 3. A symmetrical flow is produced, with again the plunging sheets leaving the surface at a sharp angle. There is a second effect visible in figs. 2 and 3 which may also be relevant in the context of mantle convection, and which arises in the following way. When the top is not protected by a layer of oil, water is rapidly absorbed from the air into the surface of the glycerine (and in later experiments it was added to enhance the effect). The downflow therefore consists of cold, viscous and slightly diluted glycerine. Thermal diffusion raises the temperature of this downflow more nearly to that of the surroundings; in the experiments described here, this happened in a time of order 10 sec, which is consistent with the observed sheet thickness of about 1 ram. The sheet will then be lighter than the pure glycerine above it and convection will begin, as shown in the photographs. Such "double-diffusive" processes, which depend on the different rates of diffusion of two substances (in this case heat and water in glycerine) have recently become familiar in other contexts, especially in oceanography [16, 17]. In the experiment pictured in fig. 5 for example, a quasi-steady state has been achieved in this way, with a wedge of lighter fluid floating on top of the more rigid plunging sheet. The flow across the surface, driven by the upcurrent, supplies new viscous material to the downflow, and at the same time prorides a viscous force which counteracts the tendency for the lighter wedge to float over the top. When the driving is removed (i.e., when the dry ice exhausted, as in fig. 6, taken at a later stage of the same experiment) this is indeed what happens.
373
Though steady circulations of the kind described above have not yet been produced in experiments which use convection driven by heating and cooling alone, one observation seems worth reporting now. Fig. 4 is a shadowgraph picture of the convection pattern produced by heating glycerine at room temperature (at a rate of 60 W supplied through a coil extending across the width of the tank). The flow is made visible by inhomogeneities of temperature and composition, and consists of two parts. There is a slow surface outflow (which did not plunge down again over the range of conditions tried, but might do so in other circumstances). In addition there is a much more rapid closed circulation near the heated plume; clearly this is a warmer region of lower viscosity in which downward motion can occur more easily than in the colder surrounding fluid. This feature is worth bearing in mind as numerical models with higher resolution are developed to describe motion in fluids with variable viscosity. Until more quantitative experiments have been carried out, it is hardly appropriate to make detailed comparisons between these experiments and mantle convection. Qualitatively, however, they do bear a remarkable resemblance to the interpretation of the observations summarized by Ringwood [8] (see especially his fig. 5). The important point to emphasize now is that, whereas more complicated geochemical processes are often involved in explanations of the behaviour of spreading and descending plates, comparable flows might occur as a result of purely physical effects without phase changes or pressure variations. These laboratory experiments suggest a natural way in which it may be possible to bridge the gap between the kinematic plate models on the one hand rather oversimplified convection theories on the other.
Acknowledgements This work was begun while the author was on study leave visiting the Cloud Physics Section of CSIRO, Australia, and he is grateful for the support and facilities provided during that time.
374
J.S. Turner, Convection in the mantle." a laboratory model
References [ 1 ] E.C. Bullard, A.E. Maxwell and R. Revelle, Heat flow through the deep sea floor, Adv. Geophys. 3 (1956) 153. 12J F.J'. Vine and D.H. Mathews, Magnetic anomalies over oceanic ridges, Nature 199 (1963) 947. [3] A.L. Hales, Convection currents in the earth, Mon. Not. R. Astron. Soc., Geophys. Suppl. 3 (1936) 372. 14] S.K. Runcorn, Satellite gravity measurements and a laminar viscous flow model of the earth's mantle, J. Geophys. Res. 69 (1964) 4389. [5] L. Knopoff, The convection current hypothesis, Rev. Geophys. 2 (1964) 89. 161 I.R. Heirtzler, G.O. Dickson, E.M. Herron, W.C. Pitman and X. Le Pichon, Marine magnetic anomalies, geomagnetic field reversals, and motions of the ocean floor and continents, J. Geophys. Res. 73 (1968) 2119. [7] D.P. McKenzie, Speculations on the consequences and causes of plate motions, Geophys. J. 18 (1969) 1. [8] A.E. Ringwood, Composition of the crust and upper mantle, in: ed. P.J. Hart, The Earth's Crust and Upper Mantle, Geophys. Monogr. 13, Am. Geophys. Union (I 969) I.
[9] K.E. Torrance and D.L. Turcotte, Thermal convection with large viscosity variations, J. Fluid Mech. 47 (1971) 113. [I0] J.W. Eider, Thermal turbulence and its role in the Earth mantle, in: ed. S.K. Runcorn, Mantles of the Earth and TerrestriM Planets (Wiley, London, 1967) 525. [11 ] J.W. Elder, Convective self-propulsion of continents, Nature 214 (1967) 657. [12] J.A. Whitehead, Moving heaters as a model continental drift, Phys. Earth Planet. Inter. 5 (1972) 199. [ 13] I. Peych~s and M. Zortea, Glass tanks as model for convection in the upper mantle, J. Geophys. Res. 76 (1971 ) 1416. [14] J.S. Turner and D.K. Lilly, The carbonated water tornado vortex, J. Atmos. Sci. 20 (1963)468. [15] J.S. Turner, The circulation driven by ring convection in a rotating system, Q. J.R. Met. Soc. 94 (1968) 589. [ 16J J.S. Turner and Henry Stommel, A new case of convection in the presence of combined vertical salinity and temperature gradients, Proc. Nat. Acad. Sci. 52 (1964) 49. [17~ M.E. Stern and J.S. Turner, Salt fingers and convecting layers, Deep-Sea Res. 16 (1969) 497.
CONVECTION
IN T H E M A N T L E : A L A B O R A T O R Y
TEMPERATURE-DEPENDENT
MODEL WITH
VISCOSITY
J.S. TURNER Department of Applied Mathematics and TheoreticalPhysics, University of Cambridge, Cambridge, UK
Received 6 October 1972 Revised version received 23 October 1972
Some exploratory laboratory experiments are described, using a working fluid (glycerine) whose viscosity depends strongly on temperature. Surface cooling produces a thin, viscous sheet which is driven away from regions of upward convection, and then plunges down steeply into the less viscous interior, maintaining its identity as it does so. When there are gradients of composition as well as temperature, convective motions can also be produced above the plunging sheets by a "double-diffusive" mechanism. The behaviour is quite unlike that observed previously in a fluid of constant properties, but it closely resembles some features of recent plate model of the Earth's mantle.
It is now widely accepted that convective motions are occurring in the Earth's mantle. Such motions are consistent with the measured distributions of the surface heat flux [1] and can provide the forces necessary to drive sea floor spreading and continental drift [2]. The earliest theories [ 3 - 5 ] which applied convection to this problem were, however, based largely on experience with fluids whose behaviour bears little relation to the geophysical situation. In particular, constant fluid properties have usually been assumed, whereas there can certainly be enormous variations in viscosity of the mantle material over the relevant range in temperature. (The viscosity at the surface may be 104 times that at 200 km). The present work is a first step towards making such laboratory experiments more realistic, using a fluid (glycerine) whose viscosity, like that of the mantle material, depends strongly on temperature. Rapid progress has been made recently in the field of plate tectonics [6, 7], which concentrates on the kinematics of the motion of rigid plates. Less attention has been given to the associated dynamical problems, though it is implied that the plates are being moved about by convection occurring in a separate region below them. The discussions of the dynamics of the motion have sometimes invoked various geo-
chemical processes, especially in the regions of downthrust [8]. The laboratory results presented below suggest, however, that a similar qualitative behaviour can be achieved entirely as a result of physical effects in a fluid with continuously variable properties. The need for models incorporating variable properties has certainly been recognised before, and some progress has been made theoretically using a numerical model [9] which shows that the dependence of viscosity on temperature is likely to be the most important effect. A thin region of high viscosity at the surface, overlying a deeper, hotter and less viscous region, is predicted in this way, and the more recent theories bring out the importance of concentrated convection plumes and thermal boundary layers. Elder [10] has investigated, both numerically and experimentally, the role of thermal turbulence in mantle convection, and has discussed the effect of variations in viscosity. He has also studied the behaviour of plastic sheets of non-uniform thickness floating on the surface of a liquid heated from below [11] The latter experiment showed how the resulting distribution o f heat flux produces a convection pattern which can propel the surface layers about; but of Course it used two very different substances to produce this effect. Whitehead [12] has reported related
370
J.S. Turner, Convection in the mantle: a laboratory model
~.-•
.
'_ ' Z i.~
.
.
•
~ ,
Q
m
Fig. 1. Shadowgraph photograph of a plunging sheet of cold, viscous glycerine, resulting from the combination of a strong circulation and surface coolinz produced by bubbles rising from a piece of dry ice on the far right of the experimental tank.
l:ig. 2. Convection driven by a central source. The sheet on the left is forced under at the edge of a barrier on the surface; note the convection occurring above the sinking sheet. laboratory experiments using heat sources which can be carried about by the convective motions they produce. Recently it has been suggested [13] that the properties of molten glass model those in the mantle, but that system is much more difficult to work with than the one described below. The concept behind the exploratory experiments reported here is very simple. In a fluid whose viscosity
varies widely over a temperature range available in the laboratory, we sought to produce strong convection in a region of higher temperature (low viscosity), combined with strong cooling at the surface to form a layer of high viscosity• An obvious choice of working fluid is glycerine, whose viscosity increases by a factor of 100 as the temperature is lowered from +30°C to - 15°C, and by another factor of 100 at
J.S. Turner, Convection in the mantle: a laboratory model
371
Fig. 3. Convection driven from both ends of the tank. Symmetrical plunging sheets of viscous fluid, with convection above them.
Fig. 4. The pattern of motion (as shown by a shadowgraph) produced by heating glycerine at a line source located at the bottom of the tank. A slow surface spreading, and a more rapid closed circulation, are clearly seen. - 4 0 ° C . Even in the easily accessible laboratory range 3 0 - 0 ° C , the viscosity increases by a factor o f 20. In early trials with purely thermal convection, however, it was found difficult to produce high enough velocities with the available temperature differences, and eventually a different technique was adopted. An appropriate form o f convection was produced by releasing a stream o f bubbles in the fluid, a method
which had previously been used in modelling atmospheric phenomena [14, 15]. A lump of dry ice (about 2 cm 3) was dropped through a 12-cm deep layer of glycerine, contained in a perspex tank 25 cm × 7 cm in cross section. The glycerine was either at room temperature or gently heated, so that the viscosity was relatively low (of order 10 poise). The dry ice adhered to the b o t t o m ,
372
J.S Turner, Convection in the mantle: a laboratory model
t.ig. 5. A single plunging sheet, with a light wedge of fluid formed by convection above it. ]'his is held in equilibrium by the outflow associated with the convection driven by the bubbles. ~~.
•
".
o-~. ~"
Fig. 6. The cxperiment of fig. 5, shortly after the bubbling has stopped. The wedge of light fluid floats over the top.
held by a cold viscous layer of glycerine, and released a stream of bubbles wtfich rose through the fluid and produced a strong, almost two-dimensional circulation in it. It is of course this circulation which isi an essential feature of the model, not the particular method (in this case the bubbles) used to dave it. ~,t the same time, the bubbles o f cold carbon dioxide breaking at the surface produced the desired increase
of viscosity there. In some experiments the surface of the glycerine was also cooled in a more controlled manner, through a thin layer of light, less viscous oil floating on its surface. With a typical temperature difference of 30°C, the most important new observation was that the viscous material near the surface was pushed outwards as a nearly rigid sheet and then plunged down steeply into the interior, maintaining
J.S. Turner, Convection in the mantle: a laboratory model
its identity as it did so. Buoyant and viscous forces must both contribute to this process, though the mechanism is not yet fully understood. The behaviour is illustrated in the shadowgraph photographs (figs. 1-3) for various geometries and combinations of convective sources, which show the striking contrast with convection in fluids of nearly constant properties. In fig. 1 there is a single bubble source at the far right of the tank, and a plunging viscous sheet leaving the surface at a sharp angle. Fig. 2 shows a central source, with unsymmetrical downflow, the sheet on the left being pushed under prematurely at the edge of a solid obstacle (a piece of plywood) floating on the surface. The result of driving with two streams of bubbles, one at either end of the tank, is shown in fig. 3. A symmetrical flow is produced, with again the plunging sheets leaving the surface at a sharp angle. There is a second effect visible in figs. 2 and 3 which may also be relevant in the context of mantle convection, and which arises in the following way. When the top is not protected by a layer of oil, water is rapidly absorbed from the air into the surface of the glycerine (and in later experiments it was added to enhance the effect). The downflow therefore consists of cold, viscous and slightly diluted glycerine. Thermal diffusion raises the temperature of this downflow more nearly to that of the surroundings; in the experiments described here, this happened in a time of order 10 sec, which is consistent with the observed sheet thickness of about 1 ram. The sheet will then be lighter than the pure glycerine above it and convection will begin, as shown in the photographs. Such "double-diffusive" processes, which depend on the different rates of diffusion of two substances (in this case heat and water in glycerine) have recently become familiar in other contexts, especially in oceanography [16, 17]. In the experiment pictured in fig. 5 for example, a quasi-steady state has been achieved in this way, with a wedge of lighter fluid floating on top of the more rigid plunging sheet. The flow across the surface, driven by the upcurrent, supplies new viscous material to the downflow, and at the same time prorides a viscous force which counteracts the tendency for the lighter wedge to float over the top. When the driving is removed (i.e., when the dry ice exhausted, as in fig. 6, taken at a later stage of the same experiment) this is indeed what happens.
373
Though steady circulations of the kind described above have not yet been produced in experiments which use convection driven by heating and cooling alone, one observation seems worth reporting now. Fig. 4 is a shadowgraph picture of the convection pattern produced by heating glycerine at room temperature (at a rate of 60 W supplied through a coil extending across the width of the tank). The flow is made visible by inhomogeneities of temperature and composition, and consists of two parts. There is a slow surface outflow (which did not plunge down again over the range of conditions tried, but might do so in other circumstances). In addition there is a much more rapid closed circulation near the heated plume; clearly this is a warmer region of lower viscosity in which downward motion can occur more easily than in the colder surrounding fluid. This feature is worth bearing in mind as numerical models with higher resolution are developed to describe motion in fluids with variable viscosity. Until more quantitative experiments have been carried out, it is hardly appropriate to make detailed comparisons between these experiments and mantle convection. Qualitatively, however, they do bear a remarkable resemblance to the interpretation of the observations summarized by Ringwood [8] (see especially his fig. 5). The important point to emphasize now is that, whereas more complicated geochemical processes are often involved in explanations of the behaviour of spreading and descending plates, comparable flows might occur as a result of purely physical effects without phase changes or pressure variations. These laboratory experiments suggest a natural way in which it may be possible to bridge the gap between the kinematic plate models on the one hand rather oversimplified convection theories on the other.
Acknowledgements This work was begun while the author was on study leave visiting the Cloud Physics Section of CSIRO, Australia, and he is grateful for the support and facilities provided during that time.
374
J.S. Turner, Convection in the mantle." a laboratory model
References [ 1 ] E.C. Bullard, A.E. Maxwell and R. Revelle, Heat flow through the deep sea floor, Adv. Geophys. 3 (1956) 153. 12J F.J'. Vine and D.H. Mathews, Magnetic anomalies over oceanic ridges, Nature 199 (1963) 947. [3] A.L. Hales, Convection currents in the earth, Mon. Not. R. Astron. Soc., Geophys. Suppl. 3 (1936) 372. 14] S.K. Runcorn, Satellite gravity measurements and a laminar viscous flow model of the earth's mantle, J. Geophys. Res. 69 (1964) 4389. [5] L. Knopoff, The convection current hypothesis, Rev. Geophys. 2 (1964) 89. 161 I.R. Heirtzler, G.O. Dickson, E.M. Herron, W.C. Pitman and X. Le Pichon, Marine magnetic anomalies, geomagnetic field reversals, and motions of the ocean floor and continents, J. Geophys. Res. 73 (1968) 2119. [7] D.P. McKenzie, Speculations on the consequences and causes of plate motions, Geophys. J. 18 (1969) 1. [8] A.E. Ringwood, Composition of the crust and upper mantle, in: ed. P.J. Hart, The Earth's Crust and Upper Mantle, Geophys. Monogr. 13, Am. Geophys. Union (I 969) I.
[9] K.E. Torrance and D.L. Turcotte, Thermal convection with large viscosity variations, J. Fluid Mech. 47 (1971) 113. [I0] J.W. Eider, Thermal turbulence and its role in the Earth mantle, in: ed. S.K. Runcorn, Mantles of the Earth and TerrestriM Planets (Wiley, London, 1967) 525. [11 ] J.W. Elder, Convective self-propulsion of continents, Nature 214 (1967) 657. [12] J.A. Whitehead, Moving heaters as a model continental drift, Phys. Earth Planet. Inter. 5 (1972) 199. [ 13] I. Peych~s and M. Zortea, Glass tanks as model for convection in the upper mantle, J. Geophys. Res. 76 (1971 ) 1416. [14] J.S. Turner and D.K. Lilly, The carbonated water tornado vortex, J. Atmos. Sci. 20 (1963)468. [15] J.S. Turner, The circulation driven by ring convection in a rotating system, Q. J.R. Met. Soc. 94 (1968) 589. [ 16J J.S. Turner and Henry Stommel, A new case of convection in the presence of combined vertical salinity and temperature gradients, Proc. Nat. Acad. Sci. 52 (1964) 49. [17~ M.E. Stern and J.S. Turner, Salt fingers and convecting layers, Deep-Sea Res. 16 (1969) 497.