Norman H. Sleep

Shear from flow in the mantle tilts the conduits of mantle plumes. The density stratification at the top of the plume conduit is then unstable in cross section with cold material over hot. This stratification drives cross‐stream convection. Such convection including the Stokes' law rise of the plume conduit through the ambient mantle readily mixes isoviscous thermal plumes with their surroundings. Laboratory experiments by Kerr and Mériaux (2004) indicate that cross‐stream convection is weak within plumes with strongly temperature‐dependent viscosity and significant ∼56 viscosity contrasts. Analogy with stagnant‐lid convection provides scaling relationships with a Rayleigh number for cross‐stream convection within tilted plume conduits. Only material within a factor of a few e of the central viscosity of the plume convects significantly. The scaling analysis shows that the laboratory experiments by Kerr and Mériaux (2004) had the proper parameter range for the Hawaiian plume. The petrological conclusion that the plume conduit has been only mildly stirred by cross‐stream convection is reasonable. Cross‐stream convection is less likely for low‐flux plumes, like Galapagos, than high‐flux plumes and plumes in the deep mantle compared with those in the upper mantle.

Frictional sliding dilates gouge or increases the separation of sliding surfaces. A common hypothesis is that the rate of dilatational strain scales linearly with the rate of shear strain. The proportionality constant is called the dilatancy coefficient. Real contact theory of friction may explain this feature. Moving contact asperities produce damaged regions with elastic strains scaling to the ratio of the real strength of asperities to the elastic modulus. Balance between this local strain‐energy production rate and macroscopic work against normal traction indicates that the dilatancy coefficient scales with this ratio. So do the average slopes on a mature rough sliding surface if opening‐mode cracks are unimportant. The result is compatible with the observed dilatancy coefficient of quartz gouge, ∼4%.

The well‐known formalism of rate‐ and state‐dependent friction represents the transition between static and dynamic friction. The instantaneous coefficient of friction is proportional to the logarithm of the sliding velocity V and the logarithm of a state variable ψ whose inverse is a measure of the current amount of damage within a shearing gouge layer or on a sliding surface. Evolution equations that represent the change of the state variable with time include damage from sliding and healing. The Dieterich (1979) evolution law contains separate terms for the two effects; the Ruina (1983) law groups the two effects into one term proportional to V ln(V). In addition, the state variable depends exponentially on porosity as exp[(ϕ − f)/C_ɛ], where ϕ is the porosity at reference conditions and C_ɛ is a dimensionless constant. I obtain these relationships as well as relationships between friction‐law parameters and physical parameters by representing asperity contacts as rectangular deformable regions between two rigid sheets. The creep rate within an asperity depends exponentially on the second invariant of the deviatoric stress . In terms of asperity‐scale quantities the invariant includes normal traction and shear traction. The derivations assume that contact stresses scale with macroscopic stresses. This yields simple expressions analogous to those in rate and state friction. The Ruina (1983) evolution law in general and the decrease in compaction rate during a hold in a slider experiment in particular arise because a sudden decrease in sliding velocity implies that the shear traction and hence the invariant decrease. The porosity‐state relationship arises when real stresses at contacts (or the inverse real contact area) depend linearly on the separation of the sheets and by analogy the porosity. Conversely, the Ruina (1983) evolution law is inapplicable when asperity‐scale stresses within a deforming material do not scale with macroscopic stresses.

The well‐known formalism of rate and state dependent friction represents the transition between starting friction and sliding friction. It expresses the instantaneous coefficient of friction in macroscopic quantities as μ = μ₀ + a ln (V/V₀) + b ln (ψ/ψ_norm), where μ₀ is the first‐order coefficient of friction, a and b are small (∼0.01) dimensionless constants, V is sliding velocity, V₀ is a reference sliding velocity, and the inverse of the state variable 1/ψ represents damage. The normalizing state variable ψ_norm ≡ (P/P₀)^α/b represents the effect of the normal traction P on stress concentrations, where P₀ is a constant with dimensions of pressure and α ≫ b is another dimensionless constant. Evolution equations represent the combined effect of damage from sliding and healing on the state variable. The Ruina (1983) evolution equation implies that the state variable does not change (no healing) during holds when sliding is stopped. It arises from exponential creep within gouge when the concentrated stress at asperities scales with the macroscopic quantities. The parameter b − a being positive is a necessary condition for a spring‐slider system becoming unstable. Microscopically, this parameter represents the tendency of asperities accommodating shear creep to persist longer than asperities of compaction creep at high sliding velocities. In the Dieterich (1979) evolution law, healing occurs when the sample is at rest. It is a special case where creep that produces shear and creep that produces compaction occur at different microscopic locations at a subgrain scale. It also applies qualitatively for compaction at a shear traction well below that needed for frictional sliding. Chemically, it may apply when shear sliding occurs within weak microscopic regions of hydrated silica while compaction creep occurs within comparatively anhydrous grains.

Studies of xenolith suites yield depth‐temperature arrays for the lithosphere at the time of eruption. These studies constrain the conductive heat flow through the lithosphere and limit the temperature range across the rheological boundary layer to <300 K. Numerical models of convection illustrate features that may be found within suites of samples. An individual dike may sample the geotherm along a finite length of its strike having variable geotherms, causing scatter in the depth‐temperature array. In a suite from the Slave Province of Canada, local heating within a deep stockwork is a more likely cause of scatter. Plume material ponded beneath a craton may yield a linear array extrapolating to the shallow geotherm without any noticeable kink or curvature. This situation existed beneath Lesotho in South Africa at ∼90 Ma. Convection at the base of the lithosphere may produce stress concentrations that trigger dike formation. Dikes may stagnate and form stockworks in low‐stress regions above the thermal boundary layer.