On the formation of the attractors dimension of two-dimensional viscous hydrodynamics equations of the rotating sphere (Q2840187)

The author investigates an equation which describes the dynamics of a two-dimensional viscous incompressible flow onto a rotating sphere. An earlier found evaluation of the Hausdorff dimension of the attractor of this equation reads NEWLINE\[NEWLINE\dim_H \leq aG^{2/3}(b + \ln G)^{1/3},NEWLINE\]NEWLINE where \(G\) is the Grasshoff number and \(a, b\) are constants. The basic question is to find the physical model which enables to explain the above formula. To this end, the following estimations of the lower and upper dependences of the attractors on the Grasshoff number are obtained: NEWLINE\[NEWLINEN_{\eta} \leq G^{2/3}, \quad N_{\epsilon} \leq G,NEWLINE\]NEWLINE where \( N_{\eta}\) refers to the Kraichnan scale and \( N_{\eta}\) refers to the Kolmogorov scale.