We compute solutions of the Lagrangian-averaged Navier-Stokes alpha - (LANS alpha ) model for significantly higher Reynolds numbers (up to Re approximately 8300 ) than have previously been accomplished. This allows sufficient separation of scales to observe a Navier-Stokes inertial range followed by a second inertial range specific to the LANS alpha model. Both fully helical and nonhelical flows are examined, up to Reynolds numbers of approximately 1300. Analysis of the third-order structure function scaling supports the predicted l3 scaling; it corresponds to a k-1 scaling of the energy spectrum for scales smaller than alpha. The energy spectrum itself shows a different scaling, which goes as k1. This latter spectrum is consistent with the absence of stretching in the subfilter scales due to the Taylor frozen-in hypothesis employed as a closure in the derivation of the LANS alpha model. These two scalings are conjectured to coexist in different spatial portions of the flow. The l3 [E(k) approximately k-1] scaling is subdominant to k1 in the energy spectrum, but the l3 scaling is responsible for the direct energy cascade, as no cascade can result from motions with no internal degrees of freedom. We demonstrate verification of the prediction for the size of the LANS alpha attractor resulting from this scaling. From this, we give a methodology either for arriving at grid-independent solutions for the LANS alpha model, or for obtaining a formulation of the large eddy simulation optimal in the context of the alpha models. The fully converged grid-independent LANS alpha model may not be the best approximation to a direct numerical simulation of the Navier-Stokes equations, since the minimum error is a balance between truncation errors and the approximation error due to using the LANS alpha instead of the primitive equations. Furthermore, the small-scale behavior of the LANS alpha model contributes to a reduction of flux at constant energy, leading to a shallower energy spectrum for large alpha. These small-scale features, however, do not preclude the LANS alpha model from reproducing correctly the intermittency properties of the high-Reynolds-number flow.