Nonisothermal equilibrium physicochemical dynamics has been numerically modeled to estimate the effect of reduced asthenosphere fluids on continental lithosphere profiles beneath the Siberian Platform (SP). When the over-asthenosphere continental mantle is metasomatically changed by reduced magmatic fluids, the following sequence of zones forms: (1) zone where initial rocks are intensively sublimated and depleted by most petrogenic components; the restite in this case becomes carbonated, salinated and graphitized; (2) zone of Si and Fe enrichment and carbon deposition in initial rocks depleted in Na, K, P, Mn; (3) zone of diamond-bearing lherzolites enriched with Na; (4) zone of hydrated rocks enriched with K; (5) zone of hydrated rocks not enriched with petrogenic components. Zone 1 can be responsible for the formation of kimberlite melts, zones 3 and 4 can be substrates of alkaline magma melting, and zone 5 can be the source of mafic tholeiitic magma.

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A generalized Hamilton variational principle of the mechanics of two-velocity media is proposed, and equations of motion for homogeneous and heterogeneous two-velocity continua are formulated. It is proved that the convexity of internal energy ensures the hyperbolicity of the one-dimensional equations of motion of such media linearized for the state of rest. In this case, the internal energy is a function of both the phase densities and the modulus of the difference in velocity between the phases. For heterogeneous media with incompressible components, it is shown that, in the case of low volumetric concentrations, the dependence of the internal energy on the modulus of relative velocity ensures the hyperbolicity of the equations of motion for any relative velocity of motion of the phases.Introduction. At least three approaches to constructing mathematical models of two-velocity media are known at present. The averaging method is used most widely, especially to constructing models of motion for heterogeneous two-velocity media. A distinguishing feature of heterogeneous media is that each phase occupies only part of the volume of the mixture, unlike in homogeneous mixtures, in which each phase is uniformly distributed over the entire volume of the mixture. Applying an appropriate averaging operator to the equations of conservation of mass, momentum, etc. that are valid within each phase, one obtains averaged equations of motion. The main problem that arises in this approach consists in closing the resulting system: the system contains more unknowns than equations. Different experimental and theoretical assumptions on the flow structure, the mechanism of interaction between the phases, etc. [1-3] (see also a review [4]), are used for closing. As was noted by many authors [5, 6], if the pressure in the phases coincide, the corresponding equations of motion in a nondissipative approximation turn out to be nonhyperbolic even when the relative difference between the phase velocities is slight. This means that the Cauchy problem for the corresponding nonlinear equations of motion is incorrect.In [7-9], hyperbolic (nonequilibrium in pressure) models of two-layer liquid flows were obtained by the averaging method. For closure of the equations of motion, a series of hypotheses relating the pressure and velocity at the interface between the liquids to their average values in the layer were invoked [7], or the process of mixing of the liquid at the interface was taken into account by introducing a third liquid layer [9]. An interesting two-velocity model of a bubble liquid which takes into account oscillations of bubbles is proposed in [10]. In the approximation of an incompressible liquid and a small bubble concentration, the model is hyperbolic for a low relative velocity of the bubbles and gives steady wave modes. Interesting hyperbolic models are also proposed in [11][12][13][14]. Their hyperbolicity was reached using closing relations, which are usually Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian A...