MANTLE CONVECTION tic techniques paved the way for the discovery of sea-ﬂoor spreading [Hess, 1962; Vine and Synonyms Matthews, 1963] and the birth of the grand uni- fying theory of Plate Tectonics in the 1960s; Mantle dynamics. Mantle circulation. this consequently revived interest in mantle con- vection as the driving mechanism for plate mo- Deﬁnition tions [Runcorn, 1962a,b] as well as non-plate- tectonic volcanism such as the possible Hawai- Mantle convection: Thermal convection in the ian plume [Morgan, 1971]. The success of man- terrestrial planetary mantles, the rocky layer be- tle convection theory in explaining plate veloc- tween crust and core, in which hot material rises, ities, sea-ﬂoor subsidence, volcanism, gravity cold material sinks and the induced ﬂow governs anomalies, etc., lead to its further application to plate tectonic and volcanic activity, as well as other terrestrial planets such as Venus and Mars, chemical segregation and cooling of the entire which also sustained unique forms of mantle planet. convection, evident from volcanic activity.
Mantle convection Basics of thermal or free convection Introduction and History Rayleigh-Benard´ Convection All planetary bodies retain some heat from their The simplest form of thermal convection is early formation but are inexorably cooling to referred to as B´enard convection named after space. Planetary surfaces are therefore cold the French experimentalist Henri B´enard who relative to their hotter interiors, and thus un- in 1900 performed the ﬁrst systematic experi- dergo thermal convection wherein cold material ments on convection in thin layers of oil (sper- is dense and sinks while hot material is light and macetti) and recognized both the onset of con- rises (liquid water near freezing being one of the vection from a static conductive state and the rare exceptions to this process). Planetary at- regular patterns formed in a convecting layer mospheres, oceans, rocky mantles and metallic [Benard´ , 1900, 1901]. Fifteen years later, the liquid cores convect and are subject to unique British theoretical physicist and mathematician patterns of circulation in each domain. Silicate Lord Rayleigh (William John Strutt), attempted mantles however tend to be the most massive to explain B´enard’s results for the onset of con- and sluggish part of terrestrial planets and there- vection [Strutt, John William (Lord Rayleigh), fore govern how planetary interiors evolve and 1916] – the delay in communication between cool to space. (See Fig 1.) them being caused by the First World War. The theory of mantle convection was origi- However, the mismatch between theory and ex- nally developed to understand the thermal his- periment was profound, and not resolved until tory of the Earth and to provide a driving mech- the late 1950s [Pearson, 1958] when it was in- anism for Alfred Wegener’s theory of Continen- ferred that B´enard’s experiments were strongly tal Drift in the 1930s [see Schubert et al., 2001; inﬂuenced by surface tension or Marangoni ef- Bercovici, 2007]. Interest in mantle convection fects not included in Rayleigh’s theory (al- waned for decades as Wegener’s theory was crit- though B´enard himself was aware of these ef- icized and apparently discredited. However, the fects). Because Rayleigh’s work provided the accumulation of sea-ﬂoor sounding data during framework for nearly all thermal convection the- War World II and reﬁnement of paleomagen- ory to follow, the simple B´enard convective sys-
1 Encyclopedia of Solid Earth Geophysics Mantle Convection, David Bercovici
Figure 1: Graphic renditions of cut aways of Earth’s structure showing crust, mantle and core (left) and of the convecting mantle (right). The relevant dimensions are that the Earth’s average radius is 6371km; the depth of the base of the oceanic crust is about 7km and continental crust about 35km; the base of the lithosphere varies from 0 at mid-ocean ridges to about 100km near subduction zones; the base of the upper mantle is at 410km depth, the Transition Zone sits between 410km and 660km depths; the depth of the base of the mantle (the core-mantle boundary) is 2890km; and the inner core-out core boundary is at a depth of 5150km. Left frame adapted from Lamb and Sington . Right frame, provenance unknown. tem is also referred to as Rayleigh-B´enard con- tationaly potential energy and go to a minimum vection. energy state, the layer is induced to turn over. Although B´enard’s experiments were in a metal cavity, Rayleigh-B´enard convection actu- Convective onset and the Rayleigh number ally refers to Rayleigh’s idealized model of a While the ﬂuid in a Rayleigh-B´enard layer thin ﬂuid layer inﬁnite in all horizontal direc- might be gravitationally unstable, it is not nec- tion such that the only intrinsic length scale in essarily convectively unstable. Convective over- the system is the layer thickness. The Rayleigh- turn of the layer is forced by heating but resisted B´enard system is heated uniformly on the bot- or damped in two unique ways. Clearly the ther- tom by a heat reservoir held at a ﬁxed temper- mal buoyancy (proportional to density contrast ature (i.e., the bottom boundary is everywhere times gravity) of a hot ﬂuid parcel rising from isothermal) and the top is likewise held at a the bottom surface through colder surroundings ﬁxed colder temperature by another reservoir acts to drive convective overturn. However, vis- (see Figure 2). If the layer were not ﬂuid, heat cous drag acts to slow down this parcel, and would ﬂow from the hot boundary to the cold thermal condcution, or diffusion, acts to erase its one by thermal conduction. But since the ﬂuid hot anomaly (i.e., it loses heat to its colder sur- near the hotter base is (typically) less dense than roundings). Thus while the ﬂuid layer might be the ﬂuid near the colder surface, the layer is gravitationally unstable, hot parcels rising might gravitationally unstable, i.e., less dense material move too slowly against viscous drag before be- underlies more dense material. To release gravi- ing erased by thermal diffusion. Similar argu-
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1.0 0.8 0.6 Z 0.4 0.2 0.0 0 1 2 3 4 5 6 7 8 X Figure 2: Rayleigh-Benard convection: initially conducting layer (top) and numerical simulation of con- vection (bottom).
ments can be made for cold material sinking of size a and temperature anomaly ∆T . Dimen- from the top surface through warmer surround- sional analysis readily shows that the typical as- ings. The competition between forcing by ther- cent rate of the parcel is ρgα∆T a2/µ (with units mal buoyancy, and damping by viscosity and of m/s). However the rate that heat diffuses out thermal diffusion, is characterized in dimension- of the parcel is κ/a (smaller parcels lose heat less ratio called the Rayleigh number faster). The critical state occurs when these two rates are equal; i.e., if the buoyant ascent rate ρgα∆Td3 Ra = (1) just exceeds the diffusion rate, the parcel should µκ rise without being erased, but if the ascent rate is less than the diffusion rate it will not rise very where ρ is ﬂuid density, g is gravity, α is ther- far before being lost. Therefore the critical state mal expansivity (units of of K−1), ∆T is the dif- occurs if ρgα∆T a3/(µκ) 1. Scaling purely ference in temperature between the bottom and by orders of magnitude, a≈ small parcel of ﬂuid top surfaces, d is the layer thickness, µ is ﬂuid can be assumed to be of order 10 times smaller viscosity (units of Pa s) and κ is ﬂuid thermal 2 −1 than the entire layer; thus assuming a d/10 diffusivity (units of m s ). leads to a critical condition for onset of≈ convec- Even though ∆T > 0 (i.e., heating is from be- tion of ρgα∆Td3/(µκ) 1000. low and causes gravitational instability), Ra still ≈ must exceed a certain value, called the critical For the Earth’s mantle, the typical average Rayleigh number Rac for convection to occur. properties from which the Rayleigh number is 3 2 For Ra < Rac the layer is stable and transports constructed are ρ 4000kg/m , g = 10m/s , −5 −1 ≈ heat by conduction; for Ra > Rac the layer will α =3 10 K , ∆T 3000K, d = 2900km, be convectively unstable and transport heat more µ = 10×22Pa s (dominated≈ by the lower man- rapidly via convection. See Figure 3. tle), and κ = 10−6m2/s [see Schubert et al., Although Rac varies depending on the me- 2001]. Taken together these lead to a Rayleigh chanical nature of the horizontal boundaries number of approximately 107, which is well be- (whether rigid or a free surface) it is typically yond supercritical; although the mantle viscos- of order 1000. This value is easily understood ity is extremely high, the mantle is also very hot by considering the fate of a hot (or cold) parcel and very large and hence convecting vigorously.
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2000 most unstable stable mode 1500
super−critical heating 1000 min heating to induce convection
500 not enough heating
0 critical Ra (dimensionless T drop or heating) 0 2 4 6 8 10 12 size (wavelength) of perturbation (T fluctuation)
Figure 3: The critical Rayleigh number Ra for the onset of convection is a function of wavelength or size of the thermal perturbation to a static conductive state. For a layer with isothermal and free-slip (shear- 2 2 3 2 stress free) top and bottom boundaries, the relationship is Racrit = (k + π ) /k where k = 2π/λ and λ is wavelength [Chandrasekhar, 1961]. Values of Ra above the Racrit curve are associated with the conductive layer being convectively unstable (perturbations grow), while below the curve the layer is stable (perturbations decay). The minimum in the Racrit curve occurs at the wavelength of the ﬁrst perturbation to go unstable as heating and Ra is increased, often called the most unstable mode.
Thermal boundary layers and the Nusselt must drop from the well-mixed warm interior to number the cold temperature at the top, and to the hot- For a Rayleigh number Ra above the criti- ter temperature at the bottom. The narrow re- cal value Rac, convective circulation will mix gions accomodating these jumps in temperature the ﬂuid layer, and the mixing and homogeniza- are called thermal boundary layers (Figure 4). tion of the ﬂuid will become more effective the Thermal boundary layers are of great impor- more vigorous the convection, i.e., as Ra is fur- tance in thermal (and mantle) convection for two ther increased. With very large Ra and vig- reasons. First, most of the buoyancy of the orous mixing most of the layer is largely uni- system is bound up in thermal boundary layers form and isothermal. (In fact, if the layer is since these are where most of the cold material deep enough such the pressures are comparable at the top and hot material at the bottom resides to ﬂuid incompressibility, the ﬂuid layer is not and from where hot and cold thermals or cur- isothermal but adiabatic, wherein even without rents emanate. Moreover, with regard to con- any heating or cooling, the temperature would vection in the mantle itself, the top cold thermal drop with ﬂuid expansion on ascent and increase boundary layer is typically associated with the with ﬂuid compression on descent.) Most of Earth’s lithosphere, the 100km or so thick layer the ﬂuid in the Rayleigh-B´enard system is at of cold and stiffer mantle rock that is nominally the mean temperature between the two bound- cut up into tectonic plates. ary temperatures. However the temperature still Second, since ﬂuid in these boundary layers
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temperature all bottom heated
add internal heating height
Figure 4: Sketch of temperature proﬁles, showing how convective mixing homongenizes the conductive mean temperature into a nearly isothermal state (if the fuid is incompressible) with thermal boundary layers connecting it to the cold surface and hot base (top frame). With no internal heating the interior mean temperature is the average of the top and bottom temperatures; the effect of adding internal heating (bottom frames) is to increase the interior mean temperature and thus change the relative size and temperature drop across the top and bottom thermal boundary layers. is near a horizontal boundary, most of the ﬂuid ary, where the heat is then transported out of motion is horizontal and thus heat is only trans- the system by conduction across the top bound- ported vertically across these thin layers by con- ary layer. The eventual heat ﬂow (power out- duction; but since the layers are very thin such put per unit area) out of the well mixed layer conductive transport is rapid. Indeed, the en- is essentially k∆T/δ where k is thermal con- tire cycle of heat transport occurs by heat con- ductivity (units of W K−1m−1), ∆T/2 is the ducted in rapidly through the bottom boundary temperature drop from the well mixed interior to layer, after which hot ﬂuid in this layer will, in the surface and we deﬁne δ/2 is the thickness of various spots, become unstable and rise to form the boundary layer. By comparison, the thermal a convective upwelling that carries heat out of conduction across a static non-convecting layer the boundary layer rapidly across most of the is k∆T/d. The ratio of heat ﬂow in the con- ﬂuid layer and deposits it at the upper bound- vectively well mixed layer to the purely conduc-
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tive layer is thus d/δ, which is called the Nusselt and thickening lithosphere. number Nu (named after the German engineer, Wilhelm Nusselt 1882-1957). The relation be- Patterns of convection, structure of up- tween Nu and convective vigor parameterized wellings and downwellings: plumes, and by Ra is important for understanding how con- slabs vection transports heat and cools off bodies in- When convection occurs upwellings and cluding planets. Convective heat transport is of- downwellings will be separated horizontally by ten written as Nu(k∆T/d) and in considering some optimal distance. If they are too close to this relation Howard  argued that vigor- each other they can induce too much viscous ous convective heat transport across the bulk of drag on each other and/or lose heat rapidly to the well-mixed ﬂuid layer is so fast it is not the each other; if they are too far apart they must roll rate limiting factor in releasing heat (only con- too much mass between them. The separation duction across the thermal boundary layer is), distance is also determined by heat transport in and thus heatﬂow should be independent of ﬂuid the thermal boundary layer between upwellings depth d; this implies that since Ra d3 then and downwelling. When hot upwelling ﬂuid 1 3 ∼ Nu Ra / , which yields a convective heat- reaches the surface it spreads laterally into the ﬂow ∼Nu(k∆T/d) that is independent of d. In thermal boundary layer. As it travels horizon- general, since the ﬂuid is conductive for Ra tally it cools to the surface and eventually gets 1≤/3 Rac, one often writes that Nu = (Ra/Rac) cold and sinks into the downwelling; thus the (although Nu = 1 for Ra < Rac), which upwelling-downwelling separation is also deter- is a reasonably accurate relationship born out mined by the distance it takes for ﬂuid to cool by simple experiments and computer modeling. off and become heavy enough to sink. This relationship also implies that the ratio of The upwelling-downwelling separation dis- thermal boundary width to ﬂuid layer depth is tance or convection cell size is predicted by con- δ/d Ra−1/3, which shows that the bound- vective stability theory to approximately equal ∼ ary layers become increasingly thin as convec- to the layer depth d (a bit larger at the onset of tive mixing of the layer becomes more vigorous. convection but identically d as Ra becomes very The relation of δ Ra−1/3 applies to the large); i.e., the cell that is either least stable and horizontally averaged∼ boundary layer thickness. thus most likely to convect – or equivalently the However, boundary layers change with time or cell that optimizes gravitational potential energy distance from their ﬁrst formation, e.g., where (and thus heat) release – is usually the cell that an upwelling impinges on the top boundary. As is as wide as it is deep. the ﬂuid in the boundary layer moves from an Viewing a convecting layer from above, the upwelling to a downwelling it cools and the upwelling and downwellings may be separated boundary layer thickens as more material cools in various patterns, such as two-dimensional next to the cold surface. The thickening de- rolls (sheets of upwelling rolling over into sheets pends on the thermal diffusivity κ (with units of downwelling, and each cell counter-rotating of m2/s) and the residence time or age t near with its neighboring cell). In the Rayleigh- the cold boundary (i.e., time since leaving the B´enard layer, which is inﬁnite horizontally, no upwelling). Simple dimensional considerations one location of the layer should be different than show that the boundary layer thickness goes as any other one so ideally the pattern should be √κt; this corresponds to the well known √age a regular repeating tile; as there are only so law for subsidence of ocean sea ﬂoor with age many polygons that can form a repeating tile, since formation at mid-ocean ridges, implying the patterns usually involve convection cells in that sea-ﬂoor gets deeper because of the cooling the shapes of rolls (already mentioned), squares,
6 Encyclopedia of Solid Earth Geophysics Mantle Convection, David Bercovici hexagons and triangles (Figure 5). Of course Energy sources for mantle convection and non-ideality and irregularities can occur due to Earth’s thermo-chemical history tiny imperfections for example in the boundaries leading to irregular patterns. Heatﬂow coming out of the Earth is measured by heat-ﬂow gauges (measuring conductivity of In many instances the 3-D pattern of convec- rocks ﬁrst and then thermal gradients in bore- tion, especially in ﬂuids where hot ﬂuid is less holes) both in continents and oceans [see Tur- viscous than cold ﬂuid (as is true in many ma- cotte and Schubert, 1982]. The total heat ﬂow- terials, including the mantle) the upwelling is in ing out from beneath the Earth’s surface is ap- the form of a cylindrical plume at the center of a proximately 46TW (46 trillion Watts) [Jaupart canopy of sheet-like downwellings, much like a et al., 2007], which is in fact not a large num- fountain. ber given the surface area of the Earth, and is actually tens of thousands of times smaller than the heat absorbed from the Sun. Nevertheless it represents the source of power driving dynamics Plumes and Slabs in the mantle: Simple inside our planet, including mantle convection view (and hence tectonic, volcanological and other The common occurence of sheet-like down- geological activity) as well as core cooling and wellings and columnar upwellings in simple ﬂow. convection is crudely applicable to mantle con- The source of the Earth’s internal heat is a vection. Subducting slabs are where tectonic combination of primordial heat, i.e., left over plates sink into the mantle; these are analagous from accretion (gravitational potential energy to the cold sheet-like downwellings seen in 3- from formation and collisions), and heat gener- D convection, although much more complicated ated by unstable radioactive isotopes, especially by rheology as discussed below. Deep hot nar- the isotopes of uranium ( 238U), thorium ( 232Th) row upwelling plumes are infered to explain and potassium ( 40K), although the 40K half- anomalous intraplate volcanism as in Hawaii, as life is much shorter than the others and so gen- well as the ﬁxity of these volcanic hotspots rela- erated a large heat pulse primarily in the early tive to each other (which suggests deep anchor- Earth. Because continental crust is originally ing in the mantle). These mantle plumes are os- formed by partial melting and chemical segrega- tensibly analogous to the pipe-like upwelling in tion of early mantle material (indeed the chem- simple 3-D convection, but again more compli- ical separation allows another energy source in cated by unique mantle properties. (See “Mantle the release of gravitational potential energy, but Plumes” essay by Farnetani and Hofmann.) other than early core formation this is a rela- While mid-ocean ridges or spreading centers tively minor contribution), these radioactive ele- transport material vertically to surface, they are ments tend to be concentrated in crust (i.e., melt very narrow features and for the most part in- more readily dissolves these elements than does volve shallow upwelling (inferred from their solid rock, so they partition toward the melt). weak gravity anomalies that suggest shallow Thus the crust itself produces a signiﬁcant frac- isostatic compensation, i.e., they are ﬂoating on tion of the net heat output through the surface; a shallow buoyant root). Ridges are likely best removing the crustal component leaves approx- explained as being pulled and rifted apart pas- imately 31TW emanating from the mantle and sively from a distant force (ostensibly slabs) core [Schubert et al., 2001; Jaupart et al., 2007]. rather than involving a deep convective up- The relative contributions of primordial cool- welling that pries them open. ing and radiogenic heating to the mantle (and
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