Characteristics of the turbulent boundary and passive layers of an isothermal dry granular avalanche with incompressible grains are studied by the proposed zero-order turbulence closure model. The first and second laws of thermodynamics are applied to derive the equilibrium closure relations satisfying turbulence realizability conditions, with the dynamic responses postulated within a quasi-static theory. The established closure model is applied to analyses of a gravity-driven stationary avalanche down an incline to illustrate the distributions of the mean solid content, mean velocity, turbulent kinetic energy and dissipation across the flow layer, and to show the influence of turbulent fluctuation on the mean flow features compared with laminar flow solutions. In this paper, detailed thermodynamic analysis and equilibrium closure relations are summarized, with the dynamic responses, the complete closure model and numerical simulations reported in the second part.