The fundamental equations of two-dimensional layer flows

In many studies on two-dimensional flows in field of atmosphere and ocean the equations which are extension of river-hydraulic equations ∂/∂tU_α+U_β∂U_α/∂X_β=-g∂h/∂X_α+g(i_α-|U|U_α/c²R)+F_α or Navier-Stokes equations ∂/∂tU_α+U_β∂U_α/∂X_β=-g∂h/∂X_α+gi_α+F_α+∂/∂X_β(v_t ∂U_α/∂X_β) are usually used. In these equations -g|U|U_α/c²R or ∂/∂X_β(v_t ∂U_α/∂X_β) stand for turbulent resistance. Obviously use of these equations in practice may lead to contradiction. In this paper the average of Reynolds equations over depth is taken. The motion equations, continuity equation and diffusion equation are obtained for the average physical variables.