Energy spectrum in the inertial and dissipation ranges of two-dimensional steady turbulence

The energy spectrum in the inertial and dissipation ranges in two-dimensional steady turbulence is examined theoretically and by high resolution direct numerical simulations (DNS) up to N=4096^2. A theoretical spectrum smoothly joining the two ranges is derived using the K\'arm\'an-Howarth-type equation. In the inertial range we obtain an asymptotic form of the energy spectrum as E (k) =C^2/3k^-3 (k/k₃) ^-ln (k/k₈) ^- (2-) / (6-) with small. It is found from the DNS that decreases slowly with the microscale Reynolds number and the constant C is of the order of unity but increases with the microscale Reynolds number. In the far dissipation range, we derive E (k) k^- (3+) /2e^-{₂ (k/k₃) }, which agrees with the DNS results. The slope ₂ depends explicitly on the microscale Reynolds number and agrees with the DNS values. Universality of the spectrum in the two ranges is also discussed.