  We present a method for studying the evolution of plasma turbulence by
tracking dispersion relations in the energy spectrum in the
wavenumber-frequency domain. We apply hybrid plasma simulations in a simplified
two-dimensional geometry to demonstrate our method and its applicability to
plasma turbulence in the ion kinetic regime. We identify four dispersion
relations: ion-Bernstein waves, oblique whistler waves, oblique
Alfv\'en/ion-cyclotron waves, and a zero-frequency mode. The energy partition
and frequency broadening are evaluated for these modes. The method allows us to
determine the evolution of decaying plasma turbulence in our restricted
geometry and shows that it cascades along the dispersion relations during the
early phase with an increasing broadening around the dispersion relations.
