Buoyant instability of a viscous film over a passive fluid

Abstract

In certain geophysical contexts such as lava lakes and mantle convection, a cold, viscous boundary layer forms over a deep pool. The following model problem investigates the buoyant instability of the layer. Beneath a shear-free horizontal boundary, a thin layer (thickness d 1 ) of very viscous fluid overlies a deep layer of less dense, much less viscous fluid; inertia and surface tension are negligible. After the initial unstable equilibrium is perturbed, a long-wave analysis describes the growth of the disturbance, including the nonlinear effects of large amplitude. The results show that nonlinear effects greatly enhance growth, so that initial local maxima in the thickness of the viscous film grow to infinite thickness in finite time, with a timescale 8µ/∆ρ gd 1 . In the final catastrophic growth the peak thickness is inversely proportional to the remaining time. (A parallel analysis for fluids with power-law rheology shows similar catastrophic growth.) While the small-slope approximation must fail before this singular time, the failure is only local, and a similarity solution describes how the peaks become downwelling plumes as the viscous film drains away. BUOYANT INSTABILITY OF A VISCOUS FILM OVER A PASSIVE FLUID INTRODUCTIONThis work examines the strongly nonlinear effects of finite amplitude in the RayleighTaylor instability of a horizontal viscous film under a shear-free boundary and over a much less viscous fluid. Inertia and surface tension are neglected, and, in the parameter range considered, the motion is limited by normal stresses in the more viscous fluid. The analysis exploits the fact that the most unstable wavelengths are long compared to the thickness of the film. The results show how the growth of disturbances to the interface becomes greatly enhanced when the disturbance amplitude becomes large, leading to the formation of downwelling sheets or plumes in a finite time.The motivation for this problem comes from certain geophysical situations, particularly the stability of the Earth's lithospheric plates. In simplified terms, the oceanic lithosphere (tectonic plates) can be considered a cold, stiff thermal boundary layer above the convecting mantle. Where two plates come together, one subducts under the other and flows downward due to its negative buoyancy. The question of how a new subduction zone is formed, how one large plate may break into two and thus allow some of the dense material to flow back down into the mantle, is not yet resolved. Other closely related geophysical situations include the surfaces of lava lakes, thermal convection in the mantles of other planets, and possibly convection in the Earth's solid core.This work examines a simple model of one possible mechanism for the initiation of subduction: the Rayleigh-Taylor instability. In this model, the lithosphere and the mantle are treated as distinct, highly viscous fluids, the lithosphere being denser (and much more viscous) than the mantle. In this unstable configuration, any variations in the lithosphere thickness t...

A rapid mechanism to remobilize and homogenize highly crystalline magma bodies

“…The growth of a perturbation of amplitude w(t) becomes 30 : Estimating the strain rate and the shear stress applied to the mush during unzipping is useful to determine which rheology applies best to the unzipping process. To first order, the strain rate can be obtained by evaluating stress caused by the growth of the instability near its tip.…”

Laboratory Studies of Mantle Convection

“…This is the classic form of the so-called Rayleigh-Taylor instability (hereafter RTI), first studied theoretically by Rayleigh (1883) and later by Taylor (1950) (see Chapter 7.04). RTIs have been used to model a number of geophysical processes, including the formation and distribution of salt domes (e.g., Nettleton, 1934;Selig, 1965;Biot and Ode, 1965;Ribe, 1998), the emplacement of gneissic domes and granitic batholiths (Fletcher, 1972), instability of continental lithosphere beneath mountain belts (Houseman and Molnar, 1997), subduction of oceanic lithosphere (Canright and Morris, 1993), the temporal and spatial periodicity of volcanic activity in a variety of geological settings, namely island arcs (Marsh and Carmichael, 1974;Fedotov, 1975;Marsh, 1979;Kerr and Lister, 1988), continental rifts (Mohr and Wood, 1976;Bonatti, 1985;Ramberg and Sjostrom, 1973), Iceland (Sigurdsson and Sparks, 1978), and mid-ocean ridges (Whitehead et al, 1984;Shouten et al, 1985;Crane, 1985;Whitehead, 1986;Kerr and Lister, 1988), segregation and mixing in the early history of Earth's core and mantle (Jellinek et al, 1999), and the initiation of instabilities deep in the mantle (Ramberg, 1972;Whitehead and Luther, 1975;Stacey and Loper, 1983;Loper and Eltayeb, 1986;Ribe and de Valpine, 1994;Kelly and Bercovici, 1997;Bercovici and Kelly, 1997). Laboratory experiments have proved to be powerful tools for studying the development and morphology of RTI.…”

“…A large number of experimental studies using a variety of materials have been performed. The characteristic spacing and growth rate of RTI strongly depend on the viscosity of the materials (Selig, 1965;Whitehead and Luther, 1975;Canright and Morris, 1993;Bercovici Olson and Nam (1986). Early experiments with putty and other non-Newtonian fluids have been extensively photographed and compared with geological formations (esp.…”

Analytical Approaches to Mantle Dynamics

“…The R-T instability has been used to model a variety of mantle processes. Canright and Morris (1993) performed a detailed scaling analysis of the instability for two layers of finite depth (see Section 7.04.4.3), and studied the nonlinear evolution of a Newtonian or power-law layer above an effectively inviscid half-space as a model for the initiation of subduction (see Section 7.04.8.3.) Lister and Kerr (1989a) studied analytically the instability of a rising horizontal cylinder of buoyant fluid, motivated in part by suggestions that the R-T instability in this geometry might explain the characteristic spacing of island-arc volcanoes (Marsh and Carmichael, 1974) and of volcanic centers along mid-ocean ridges (Whitehead et al, 1984).…”