A power-law non-Newtonian fluid is usually assumed to model slow flows in the mantle and, in particular, convective flows. However, the power-law fluid has no memory, in contrast to a real material. A new non-linear integral ( having a memory) model is proposed to describe the rheology of rocks. The model is consistent with the theory of simple fluids with fading memory and with laboratory studies of rock creep. The proposed model reduces to the power-law fluid model for stationary flows and to the Andrade model for flows associated with small strains. Stationary convection beneath continents has been studied by Fleitout & Yuen (1984), who used the power-law fluid model and obtained the cold immobile boundary layer (continental lithosphere). In a stability analysis of this layer, the Andrade model must be used. The analysis shows that the lithosphere is overstable (the period of oscillation is about 200 Ma). In the present study, it is suggested that these thermoconvective oscillations of the lithosphere are a mechanism for sedimentary basin formation. The vertical crustal movement in sedimentary basins can be considered as a slow subsidence on which small-amplitude oscillations are superimposed. The longest period of oscillatory crustal movement is of the same order of magnitude as the period of convective oscillation of the lithosphere found in the stability analysis. Taking into account the difference between depositional and erosional transport rates we can explain the permanent subsidence as well as the oscillations.

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For flows associated with small strains, the rheology of rocks is described by the linear integral (having a memory) law, which reduces to the Andrade law in the case of constant stress. A continental lithosphere with such a rheology is overstable. Thermoconvective waves that propagate through the lithosphere with minimal attenuation have a period of about 200 Myr and a wavelength of the order of 400 km. An initial temperature point‐concentrated perturbation in the lithosphere excites amplitude‐modulated thermoconvective waves (wave packets). When the initial perturbation occurs in a finite area, thermoconvective waves propagate outwards from this area, and thermoconvective oscillations (standing waves) are established inside the area. Thermoconvective waves induce oscillations of the Earth’ surface, accompanied by sedimentation and erosion, and can be considered as a mechanism for the distribution of sediments on continental cratons.

SUMMARY Laboratory experiments with rock samples show that transient creep, at which strain grows with time and strain rate decrease at constant stress, occurs while creep strains are sufficiently small. The transient creep at high temperatures is described by the Andrade rheological model. Since plate tectonics allows only small deformations in lithospheric plates, creep of the lithosphere plates is transient whereas steady‐state creep, described by non‐Newtonian power‐law rheological model, takes place in the underlying mantle. At the transient creep, the effective viscosity, found in the study of postglacial flows, differs significantly from the effective viscosity, which characterizes convective flow, since timescales of these flows are very different. Besides, the transient creep changes the elastic crust thickness estimated within the power‐law rheology of the lithosphere. Two problems of convective stability for the lithosphere with the Andrade rheology are solved. The solution of the first problem shows that the state, in which large‐scale convective flow in the mantle occurs under lithospheric plates, is unstable and must bifurcate into another more stable state at which the lithospheric plates become mobile and plunge into the mantle at subduction zones. If the lithosphere had the power‐law fluid rheology, the effective viscosity of the stagnant lithospheric plates would be extremely high and the state, in which large‐scale convection occurs under the stagnant plates, would be stable that contradicts plate tectonics. The mantle convection forms mobile lithospheric plates if the effective viscosity of the plate is not too much higher than the effective viscosity of the underlying mantle. The Andrade rheology lowers the plate effective viscosity corresponding to the power‐law fluid rheology and, thus, leads to instability of the state in which the plates are stagnant. The solution of the second stability problem shows that the state, in which the lithospheric plate moves as a whole with constant velocity, is stable but small‐amplitude oscillations are imposed on this motion in regions of thickened lithosphere beneath continental cratons (subcratonic roots) where the thickness of the lithosphere is about 200 km. These oscillations create small‐scale convective cells (the horizontal dimensions of the cells are of the order of the subcratonic lithosphere thickness). Direction of motion within the cells periodically changes (the period of oscillations is of the order of 108 yr). The small‐amplitude convective oscillations cause small strains and do not destroy the thickening of the lithosphere beneath cratons. Thus, the transient creep of the lithosphere explains not only mobility of the lithospheric plates but longevity of subcratonic roots as well.

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