On the distribution and separation of crystals in convecting magma

Abstract Fractionation of crystals from magma stands squarely as the central principle of igneous petrology, and the critical balance is the competition between crystal settling (or rising) velocity and magmatic convective velocity; a ratio we call S . The convective velocity is intimately related to the rate of heat transfer from the system. Although the theory of sediment distribution in rivers establishes the critical parametric balance and it has been often applied to igneous systems, this method turns out to be valid only for small concentrations of crystals and it is not clear if it is accurate for other than turbulent flows where the Prandtl number is near unity. A generalized theory of particle (crystal or bubble) behavior is presented. Scaling of the generalized result shows for the magmatic conditions of small crystallinities, small particle Reynolds number, and Newtonian viscosity that the exact distribution of crystals in a convecting magma can be found analytically. Regardless of the flow field, critical retention occurs only for 0 ⩽ S S = 0. For sinking crystals, retention is centered in the region of maximum updraft. For floating particles retention occurs in regions of downflow, which for a simple cellular convective pattern produces a torus-shaped retention region against the walls about the midsection of the body. For a stack of flattened convective cells, which in the crudest sense is analogous to double-diffusive convection, the zones of retention form, depending on the exact flow field, either a stack or staggered stack of pillows through the central region. Crystals settling from top to bottom around this stack take a tortuous path and may exhibit a complex but regular compositional zoning. Additional effects of fluids possessing a yield strength, turbulence, particle shape and wall and particle dispersive effects are more qualitatively evaluated. For each flow field the amount of retention and the effective settling velocity of all crystals is found as a function of S . The critical parameter S controlling retention can be estimated from a parametric representation involving Stokes's Law and the Rayleigh number. This shows that some retention probably occurs in all magmas, and it increases with approach of the magma to the earth's surface. Although both crystal settling rate and convection decrease with increasing magmatic viscosity, the overall effect is to increase retention (decrease S ). Magmas having a viscosity near and beyond about 10 4 poise and chambered in the nearsurface environment should show large degrees of crystal retention. Retention is, however, never complete for magmas, and the flux of settling crystals leading to differentiation varies as a function of crystallinity. This flux or flux probability ( P F ) is zero both at the liquidus and the point of maximum packing of crystals, which for basalts is about 60%. The importance of differentiation is regulated by the amount of time spent in the high crystal-flux state, and a relative measure of this time is given by the thermal probability ( P T ) introduced previously. Thus the probability of differentiation ( P D ) is given by the product P F P T , which shows that differentiation must occur over a restricted range of crystallinities from about 20 to 45% for basalts and from about 10 to 30% for rhyolitic magmas. The most basic lessons learned here are that the full range of crystallinities is not equally accessible to crystal fractionation, and the most fundamental information describing the dynamic state and behavior of magma may well be the distribution of crystal sizes found in lavas.