Mantle convection Encyclopedia of Solid Earth Geophysics, Harsh Gupta (ed.), Springer David Bercovici Professor & Chair Department of Geology & Geophsyics Yale University New Haven, Connecticut 06520-8109, USA E-mail: [email protected] Tel: (203) 432-3168 Fax: (203) 432-3134 December 26, 2010 Encyclopedia of Solid Earth Geophysics Mantle Convection, David Bercovici 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 [1998]. 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- 2 Encyclopedia of Solid Earth Geophysics Mantle Convection, David Bercovici 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. 3 Encyclopedia of Solid Earth Geophysics Mantle Convection, David Bercovici 4000 3500 3000 unstable 2500 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.