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 Karman-Howarth-type equation. In the inertial range we obtain an asymptotic form of the energy spectrum as E(k)=C eta(2/3)k(-3)(k/k(d))(-delta)[ln(k/k(l))](-(2-delta)/(6-delta)) with small delta. It is found from the DNS that delta 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)proportional to k(-(3+delta)/2)e(-alpha 2(k/kd)), which agrees with the DNS results. The slope alpha(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.