  We examine the probability distribution function (pdf) of energy injection
rate (power) in numerical simulations of stationary two--dimensional (2D)
turbulence in the Lagrangian frame. The simulation is designed to mimic an
electromagnetically driven fluid layer, a well-documented system for generating
two--dimensional turbulence in the laboratory. In our simulations, the forcing
and velocity fields are close to Gaussian. On the other hand, the measured PDF
of injected power is very sharply peaked at zero, suggestive of a singularity
there, with tails which are exponential but asymmetric. Large positive
fluctuations are more probable than large negative fluctuations. It is this
asymmetry of the tails, which leads to a net positive mean value for the energy
input despite the most probable value being zero. The main features of the
power distribution are well described by Craig's XY distribution for the PDF of
the product of two correlated normal variables. We show that the power
distribution should exhibit a logarithmic singularity at zero and decay
exponentially for large absolute values of the power. We calculate the
asymptotic behavior and express the asymmetry of the tails in terms of the
correlation coefficient of the force and velocity. We compare the measured pdfs
with the theoretical calculations and briefly discuss how the power pdf might
change with other forcing mechanisms.
