On the energy spectrum of forced two-dimensional turbulence

We study the energy spectrum of forced two-dimensional turbulence in a doubly periodic domain damped by a general (linear) dissipation and driven by a forcing, for which the characteristic wavenumber either approaches, or fluctuates in time about, a constant. It is shown that the energy spectrum $E(k)$ satisfies $\sum_k(k^2-s^2)D(k)E(k)=0$, where $s$ is the characteristic forcing wavenumber and $D(k)$ is the spectral dissipation function. This result places strong constraints on the admissible forms of $E(k)$ for a given $D(k)$. We theoretically exhaust all possible power laws of the energy spectrum for $D(k)\propto k^{2\mu}$, where $\mu\geq 0$. The combination $D(k)=\nu_{\mu}k^{2\mu}+\nu_{\mu'}k^{2\mu'}$, which could give rise to a dual cascade, is also considered. Finally, we numerically demonstrate various predictions from the theoretical considerations.