Thermal boundary layer convection in silicic magma chambers: Effects of temperature‐dependent rheology and implications for thermogravitational chemical fractionation

Solution of the nonlinear boundary value problem describing the thermomechanical structure of a boundary layer adjacent to an isothermal cooled vertical wall in a strongly temperature dependent rheological medium is presented. The analysis and boundary conditions chosen are applicable to large Rayleigh number convection in a magma chamber. Numerical solution of the boundary layer equations by parallel shooting have been obtained for viscosity contrast up to 1011. Parameterization of these results allows extrapolation to higher viscosity contrasts. The non-Arrhenius viscosity function μ = μ∞ exp (−a(T∞ - T), where a is the rheological parameter and T is the absolute temperature, is based on experimental data and accounts for the effects of temperature and crystallinity on magma viscosity. The calculations clearly show the importance of explicit consideration of the viscosity-temperature relationship in determining the quantitative features of boundary layer convection. For example, for anhydrous rhyolite (μ∞ = 109 P), the thermal boundary layer thickness (δT), wall heat flux (q0), wall viscous shear stress (τ0), and maximum vertical convective velocity (umax) are 10 m, 700 HFU, 5×10−3 bar, and 2.5 km yr−1 when μ is taken as constant (i.e., a = 0). However, for a viscosity contrast across the thermal layer of 1012 (i.e., a = 0.07 K−1), δT = 120 m, q0 = 350 HFU, τ0 = 0.36 bar, and umax = 400 m yr−1. Extreme caution must be exercised in using results from isoviscous boundary layer theories for the prediction of the thermomechanical and heat transfer parameters of magma chamber convection. For a wall temperature of 600°C (country rock temperature far from intrusion is ∼200°C) and a core temperature of 1000°C and adopting rheological parameters characteristic of rhyolitic magma containing 3 wt % dissolved H2O, we find at distances of 1 and 10 km from the top of the convecting zone viscous shear stresses, τ0 around 6×10−2 and 3×10−2 bar, maximum vertical velocities wmax of 10 and 25 km/yr, boundary layer thicknesses δT of 20 and 50 m, and marginal heat flows q0 of 1200 and 700 HFU, respectively. Transport parameters depend markedly on H2O content; for anhydrous rhyolite with identical boundary conditions q0 ∼ 600 HFU, umax ∼0.5 km yr−1 and τ0 ∼ 10−1 bar at a distance of 1 km along the vertical wall. Maximum vertical convective velocities usually exceed crystal settling rates by several orders of magnitude; crystal settling cannot be important in vigorously convecting chambers except in local regions. The petrological and geothermal implications of the calculations are discussed in terms of an extreme (but plausible) type of magmatic system, the constant enthalpy open magma chamber. In this case, heat losses due to dissipation of magmatic heat to the country rock are precisely balanced by heat input by injection of hot, mafic magma into the roots of the chamber. The requirements of the thermal steady-state chamber permit an estimation of the mass flow rate into the roots of the chamber. Results agree well with known rates of basaltic magmatism along mid-ocean ridges and at intraplate ‘hot spot’ sites. Semi-quantitative evaluation of the magnitude of chemical fractionation in a constant enthalpy magma chamber due to coupling of rapid (km yr−1) vertical convective flow with slow horizontal Soret diffusion across thin marginal thermal layers suggests that on a 106 year time scale, significant chemical gradients can be generated. An analytical approach is suggested for answering the question of whether or not such fractionated melt can maintain its integrity (i.e., not become remixed).