This paper is devoted to the study of a turbulent circulation model. Equations are derived from the “Navier-Stokes turbulent kinetic energy” system. Some simplifications are performed but attention is focused on non linearities linked to turbulent eddy viscosity $\nu _{t}$. The mixing length $\ell $ acts as a parameter which controls the turbulent part in $\nu _{t}$. The main theoretical results that we have obtained concern the uniqueness of the solution for bounded eddy viscosities and small values of $\ell $ and its asymptotic decreasing as $\ell \rightarrow \infty $ in more general cases. Numerical experiments illustrate but also allow to extend these theoretical results: uniqueness is proved only for $\ell $ small enough while regular solutions are numerically obtained for any values of $\ell $. A convergence theorem is proved for turbulent kinetic energy: $k_{\ell }\rightarrow 0$ as $\ell \rightarrow \infty ,$ but for velocity $u_{\ell }$ we obtain only weaker results. Numerical results allow to conjecture that $k_{\ell }\rightarrow 0,$$\nu _{t}\rightarrow \infty $ and $u_{\ell }\rightarrow 0$ as $\ell \rightarrow \infty .$ So we can conjecture that this classical turbulent model obtained with one degree of closure regularizes the solution.