In article [1] presents an overview of the state of immersion and convective heat transfer problems lithospheric plate in the subduction zone. In spite of a great number of publications dealing with a study of the natural mantle convection, the questions concerning the character, structure and the primary reason for occurrence of the convection remain still open. In fact, the influence of convection on the geometry of plates and, also, differences in the character of convection caused by using the Newtonian and non-Newtonian models of Earth’s mantle, has not been studied yet.
Undoubtedly, solution of these questions will entail great difficulties as at the stage of setting the problems so at the stage of their numerical solution. The above difficulties are caused by non-linearity of the problem, that is, the temperature and pressure dependence, practically, of all parameters of the mantle, lack of information on a lot of rheological parameters, restriction to the stability of the numerical solution due to prevalence of viscosity members over the inertial ones in the Navier-Stokes equations.
The main shortcoming of the above mathematical models is the fact that phases transitions at 670 km depth causing jumps in viscosity and density of the mantle substance which, in the opinion of some researchers, may lead to localization of the convection in the upper mantle, are not taken into account.

2. Formulation of the Problem

The paper considers a 2-D model of the continuous convection of the lithospheric plate close to oceanic trench with regard for the heat of the phase transition. The subduction zone is considered, where lithospheric plate collides with continental, and then on a trough, which axis is located under an angle to the land surface, is immersed in mantle. The depth of immersing of lithospheric plate in mantle is determined as a result of the task decision. Constructing the mathematical model, the following assumptions have been adopted [1]: the lithospheric plate and the underlying mantle are considered as the non-compressible Newtonian liquid with a very high viscosity. The temperature at a boundary between the mantle and the plate is constant and equals to the temperature of solid state T_s. The thermal conductivity , the viscosity of the substance and the heat flux q_v are determined with account of their temperature dependence: 
1. 
Index 1 denotes the lithosphere parameters, 2 – the mantle parameters. The dependence of the density p on the temperature of the medium is assumed to be

2. Border between lithosphere and mantle is isotherm of solid phase with value of temperature Ts.
3. The relative quantity of the solid phase (characterizing melting and crystallization of the substance) contained in the mantle or lithosphere is determined depending on the state of the substance (a solid state with a temperature T_S or a liquid state with a temperature T_L). The relative quantity of the solid phase , depending of temperature, approximates by cubic spline [1]:

At < 0.95 it is assumed that the substance is in the melted (liquid) state, while at > 0.95 – in the solid state. A set of parameters for the lithosphere and the mantle used for the calculations are listed in the Table.
Following the hypothesis propounded in 1960 by H. Hess, the lithospheric plate motions are going on in a rectilinear manner starting from the areas of formation of the plates up to the zones of under thrusting, which allows us to solve the problem in a plane vertical layer. It results from systematic investigations of the island arc and continental margin zones that the depth of subsidence of the under thrusting plate does not exceed 700 km. Therefore, the extent of the area calculated along the Y-axis is assumed to be 1000 km. According to the results obtained, the extent of the calculated area along the X-axis is assumed to be 3000 km.

3. Mathematical Model

Fig. 1 shows physical setting of a problem [1]. The horizontal oceanic plate moves towards the continental plate with a constant velocity U₀, and subsides in the asthenosphere in the trench zone at an angle to the land surface with the same velocity. In turn, the continental plate moves towards the oceanic plate with a velocity U_k. Most often the lithospheric plates are subsiding in the subduction zones at an angle of 45°, though in some sectors of the island arcs the angles of subsidence from 30° to 90° have been marked. Here U_k - velocity of a continental plate, U₀ - velocity of oceanic (lithospheric) plate; index 1 – lithosphere parameters, 2 – mantle parameters; – angle of immersing of oceanic plate; Г1, Г2, Г3 - right, lower and left border of a trench, on which the plate is immersed, accordingly. 

Fig. 1. Physical formulation of the problem

Location of trough, on which the plate is immersed, is considered equal Lx /2. The lower border Г2, up to which the velocity of immersing of a plate is known, is determined on the interval of temperature melting. At achievement of melting temperature (liquid) T_L the lithospheric plate melts and depth of its immersing in mantle is determined.
The mathematical statement of a task describing convection in subduction zone includes the equations of movement of an incompressible liquid in the Boussinesq approximation, continuity and energy with account internal heat sources. The governing dimensionless equations can be written as [1]:

,
,
.

Where
, , , , , , , , , , , , , , . , , ,  Stefan, Reynolds, Grashof and Peclet numbers.
 .

.
, H = [0;1] the dimensionless mantle depth. 

Boundary conditions: 
 = 0 at Y = 1;  = 1 at Y = 0;  at X = 0 and X = Lx/Ly.
 , (k = 1,2,3) at border Г1, Г2, Г3 (fig. 1).
 at border Г1, Г3;  at border Г2.
 at X = [0; (Lx/2Ly - Lp/2Ly)]; Y = 1.
 at X = [(Lx/2Ly - Lp/2Ly); Lx/Ly]; Y = 1.
,  at X = [0; Lx/Ly];  at;  at Y = 0.
 at  (fig. 1). 
 on Г1, Г3 (fig. 1) and inside a trench.
For intensity of a vortex W the boundary conditions were calculated with the use of boundary conditions for stream function. In accounts the non-uniform grid with a condensation in trench region and at the upper boundary of the area (Y = 1) was applied.
Dimensionless local Nusselt numbers on top (Y=1) and bottom (Y=0) to border of calculated area were determined by the expression  (the derivative was calculated on three points with the second order of approximation).
The calculations were performed on a nonuniform grid with concentration nodes in the chute region, day surface and vertical boundaries of the computational domain (fig. 2).

Fig. 2. The computational grid

The values of temperature at upper and lower border are accepted equal T_c = 1400 K, T_h = 2400 K. The extent of area on an axis х is accepted equal 3000 km, and on an axis y – 1000 km. Lithosphere thickness Ln - 100 km. The Temperatures of solid state Ts and liquid state T_L are 1800К and 1900К accordingly. Velocity of movement of a continental plate Uк = 2 сm/year, and velocity of oceanic plate U₀ 1 сm/year. Re=1,0510^-19; Gr= 3,210^-14; Pe = 251; Ste = 3. Gravitational acceleration value g = 9,8 is constant. 
In fig. 1 the results of calculation for velocity of movement oceanic plate 1 сm/year and angle of a feat under continental plate 30^o are submitted. 

In trench region isothermals “bend” to lower boundary of the area. Under oceanic and continental plates two large-scale convective cells are formed, the liquid in which is gone in opposite directions. And besides convective cell and vortex under continental plate flow round immersed lithosphere and penetrate into area of oceanic convective cell and vortex, located to the right of subduction zone, moving them to the right. Immersion depth of the lithosphere reaches about 300 km. The distributions of Nusselt numbers on the upper and lower boundaries of the area are given. At value x ~ 1700 km in the collision region of plates takes place minimum heat flow on the upper border of calculated area that will be agreed with known experimental data [1, 2].
In fig. 2 the results of calculation for velocity of movement oceanic plate 1 сm/year and angle of a feat under continental plate 45^o are submitted.

 

In fig. 3 the results of calculation for velocity of movement oceanic plate 1 сm/year and angle of a feat under continental plate 60^o are submitted.

Comparing the results shown in fig. 1, 2, 3, we can see that the immersion depth of lithospheric plate for Uк = 2 сm/year and U₀ = 1 сm/year is weakly dependent from the angle of immersing of oceanic plate. With increasing angle of immersing the isotherms deflection in the chute region increases (figs. 1, 2, 3).
The intensity of the stream functions as for cell continental and oceanic increases with increasing angle of immersing. For want of it has a place a small decrease in the continental scale cells and a small increase in the scale of the oceanic cell.
The intensity of the vortex as for a continental cell and oceanic cell will increase, with increasing angle of immersing from 30⁰ to 45⁰ (figs. 1, 2). For the angle 60⁰ (fig. 2) the intensity of the vortex as for a continental cell and oceanic cell decreases on a comparison with the results shown in fig. 3. For all modes scale of continental vortex is decreases and scale of oceanic vortex is increases.
The interval of a modification of significance of the Nusselt number on the upper and lower boundary of the computational domain increases with increasing angle. Minimum heat exchange at the upper boundary of the computational domain and a maximum heat exchange at the lower boundary of the computational domain are at the value of x 1700 km.
In fig. 4 the structure of current of a liquid in a subduction zone for various angle values of subsidence of lithosphere is given. Velocity of movement of a continental plate U_k = 2 сm/year.

Analyzing the results of modeling we can make the following conclusion:
- with the increasing angle of immersing deflection of isotherms in the chute region is increased;
- with the increasing angle of immersing the intensity of a stream function is increased. Scale of a continental cell decreases, and oceanic – increases; 
- the intensity of vortex increases with increasing angle of immersing from 30⁰ to 45⁰, and for angle of immersing 60⁰ decreases on a comparison with results for angle 45⁰. For all modes scale of a continental vortex decreases, and oceanic – is increased;
- with the increasing of immersing angle the values ​​of the Nusselt numbers will increase;
- the received decisions as it is qualitative, and will quantitatively be agreed on depth of immersing of a plate in a mantle and under the law of change of a heat flow on a surface of the Earth with known experimental and theoretical results of other authors.Thus, suggested the mathematical model and the results obtained contribute to the available information on investigation of convection in the Earth’s interior and may be of use for better understanding and explanation of such phenomena as subduction, lithospheric plate motion and heat flow change at Earth’s surface.