I have been referring to the energy in the moving mass as potential energy in relation to the still frame of reference of the windmill.
A better term is kinetic energy. Momentum is the mass times the velocity. Work is the mass times acceleration (deceleration of the wind mass). Kinetic energy is 1/2 the mass times the velocity squared.
When the wind strikes the turbine blades, the momentum of the collision must remain the same. In reality it is less of a collision and more of a sliding pull. But the result is a slight slowing of the air flow. Momentum is transferred from the air to the rotating blades. The slowing of the wind is a reduction of the air’s kinetic energy. This kinetic energy is the source electrical power transferred through the blades, rotor and generator. Each of those devices have some degree of inefficiency and each one generates heat due to the inefficiencies like friction.
There is no cooling, only heating and transfer of kinetic energy to electrical energy.
Forget momentum. Collisions don’t have momentum. The sum of the momentums of two particles that elastically collide remains the same after the collision. Nothing about energy here.
The energy of each molecule is proportional to the square of its velocity. Slow it down and it gives up energy, and the law of conservation of energy says the energy goes somewhere.
In a gas, molecular motion is its energy. The energy is felt as wind and heat. Wind is a net motion in some direction. Heat is random motion.
In our discussions we’re talking about a gas that’s at a constant pressure.