  In this study we investigate the statistics of two-dimensional stationary
turbulence using a Markovian forcing scheme, which correlates the forcing
process in the current time step to the previous time step according to a
defined memory coefficient. In addition to the Markovian forcing mechanism, the
hyperviscous dissipation mechanism for small scales and the Ekman friction type
of linear damping mechanism for the large scales are included in the model. We
examine the effects of various dissipation and forcing parameters on the
turbulence statistics in both wave space and physical space. Our analysis
includes the effects of the effective forcing scale, the bandwidth of the
forcing, the memory correlation coefficient, and the forcing amplitude, along
with the large scale friction and small scale dissipation coefficients. Scaling
exponents of structure functions and energy spectra are calculated, and the
role of the parameters associated with the Markovian forcing is discussed. We
found that the scaling exponents are approximately invariant and show a
universal behavior for the various forms of forcing schemes used. We found,
however, that the final states strongly depend on the large scale friction
mechanism considered. When the large scale friction mechanism is included in
the model with a high friction coefficient, we demonstrate that the behavior is
no longer universal. Our analysis also shows that the second-order vorticity
structure function has an asymptotic scaling exponent for larger dissipation.
Additionally, we confirmed that vorticity behaves as a passive scalar when the
dissipation mechanism becomes less effective. Finally, although turbulence is
not believed to have a separation of time scales in the dynamics of the
velocity field, we conjectured that a separation of time scales exists in the
dynamics of the energy spectrum.
