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In this paper, we investigate vortex dynamics in a two-dimensionalBose-Hubbard model coupled with a weak artificial magnetic field, a randomwhite noise and a dissipation. Origin of the noise and dissipation isconsidered as thermal fluctuations of atoms that do not participate theBose-Einstein condensation (BEC). Solving a stochastic Gross-Pitaevskiiequation to this system, we show that the interplay of the magnetic field andthe white noise generates vortices in the bulk of the BEC and stable steadystates of vortices form after a transition period. We calculate theincompressible part of the kinetic-energy spectrum of the BEC. In thetransition period, a Kolmogorov $k^{-5/3}$ spectrum appears in the infraredregime with the wave number $k$, $k<\zeta^{-1}$, where $\zeta$ is the healinglength, whereas in the ultraviolet region, $k>\zeta^{-1}$, the spectrum behavesas $k^{-3}$. On the other hand in the steady states, another scaling lowappears. We find a relationship between the above mentioned kinetic-energyspectra and the velocity of vortices. By an inverse cascade, the large velocityof a few created vortices develops the Kolmogorov $k^{-5/3}$ spectrum.

Vortex dynamics in lattice Bose gases in a synthesized magnetic field  with a random noise and a dissipation: Study by the stochastic  Gross-Pitaevskii equation