Turbulence in astrophysical and space plasmas is dominated by the nonlinear interaction of counter propagating Alfven waves. Most Alfven wave turbulence theories have been based on ideal plasma models for Alfven waves at large scales. However, in the inertial Alfven wave regime, relevant to magnetospheric plasmas, how the turbulent nonlinear interactions are modified by the dispersive nature of the waves remains to be explored. Here, we present the first laboratory evidence of the nonlinear interaction in the inertial regime. A comparison is made with the theory for MHD Alfven waves.