  Although thermal pressure is unimportant dynamically in most molecular gas,
the temperature is an important diagnostic of dynamical processes and physical
conditions. This is the first of two papers on thermal equilibrium in molecular
clouds. We present calculations of frictional heating by ion-neutral (or
ambipolar) drift in three-dimensional simulations of turbulent, magnetized
molecular clouds.
  We show that ambipolar drift heating is a strong function of position in a
turbulent cloud, and its average value can be significantly larger than the
average cosmic ray heating rate. The volume averaged heating rate per unit
volume due to ambipolar drift, H_AD ~ |JxB|^2 ~ B^4/L_B^2, is found to depend
on the rms Alfvenic Mach number, M_A, and on the average field strength, as
H_AD ~ M_A^2<|B|>^4. This implies that the typical scale of variation of the
magnetic field, L_B, is inversely proportional to M_A, which we also
demonstrate.
