  Using tools from large deviation theory, we study fluctuations of the heat
current in a model of $d$-dimensional incompressible fluid driven out of
equilibrium by a temperature gradient. We find that the most probable
temperature fields sustaining atypical values of the global current can be
naturally classified in an infinite set of curves, allowing us to exhaustively
analyze their topological properties and to define universal profiles onto
which all optimal fields collapse. We also compute the statistics of empirical
heat current, where we find remarkable logarithmic tails for large current
fluctuations orthogonal to the thermal gradient. Finally, we determine
explicitly a number of cumulants of the current distribution, finding
remarkable relations between them.
