The fundamental equations of two-dimensional layer flows

Abstract

In many studies on two-dimensional flows in field of atmosphere and ocean the equations which are extension of river-hydraulic equations

$$\frac{\partial }{{\partial t}}U_\alpha + U_\beta \frac{{\partial U_\alpha }}{{\partial x_\beta }} = - g\frac{{\partial h}}{{\partial x_\alpha }} + g\left( {i_\alpha \frac{{| \cup |U_\alpha }}{{c^2 R}}} \right) + F_\alpha $$

or Navier-Stokes equations

$$\frac{\partial }{{\partial t}}U_\alpha + U_\beta \frac{{\partial U_\alpha }}{{\partial x_\beta }} = - g\frac{{\partial h}}{{\partial x_\alpha }} + gi_\alpha + F_\alpha + \frac{\partial }{{\partial x_\beta }}\left( {v\frac{{\partial U_\alpha }}{{\partial x_\beta }}} \right)$$

are usually used. In these equations$- g\frac{{| \cup |U}}{{c^2 R}}$ or$\frac{\partial }{{\partial x_\beta }}\left( {v\frac{{\partial U_\alpha }}{{\partial x_\beta }}} \right)$ 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.