If one centers the gyro-copter in a thermal and then simulates a hover facing headwind; can the gyro-copter climb up like that ? Is it technically possible or is it advisable....
I need to work out the math for this, but an issue complicating the question is that the L/D of the gyrocopter would change in the updraft. So, like a helicopter in autorotation, a gyrocopter derives lift from flow passing upwards through the rotor as it flies forward on its propeller. However, if the aircraft is in a vertical updraft, the freestream is no longer directed horizontally, but more vertically, increasing the angle of attack on the rotor blades and directing your vehicle's drag force in the upward direction (magnitude and direction, naturally, depends on your flight speed, thermal strength, etc.).
Now, gyrocopters have at cyclic and collective pitch control (generally speaking), so you could vary the collective in order to maintain a lower angle of attack, gain stall margin, and maintain altitude...or you could take advantage of the updraft on your rotor to gain altitude for "free"! This also assumes that you have sufficient collective pitch travel to get your blades at a low enough angle of attack to stop producing upward-directed thrust.
You could try to pitch over too, thus redirecting the rotor thrust vector forward and spoiling a lot of the inflow, but this is starting to get into the downside of thermalling in helicopters: namely, that, if the updraft is sufficiently strong to stall your blades (no matter the pitch setting) but also strong enough that the vertical drag on the airframe can induce a climb...it's not going to be a good day. I'm thinking primarily of updrafts that occur in the presence of thunderstorms--they have been known to be strong enough to suck aircraft up into the clouds. Similarly, if you make your way out of the thermal, they are generally surrounded by regions of strong sink because of the recirculating air (http://www.drjack.info/INFO/DELMONTE/thermal.page.html), which not only is going to load the blades completely the opposite of how they were in the thermal (so your inertial loads spike, provided you don't have mast bumping problems...in which case you lose the aircraft right then), but, if the sink is strong enough, you might find yourself in vortex-ring-state or the like and lose control--if only temporarily. At least from my brief experience in an S-300Cbi and R22, helicopters don't like to fly, so even brief periods of control loss would be bad.
So yeah, I think it's technically feasible, and, in some cases, very dangerous. I wouldn't try to set an endurance record in a rotary winged aircraft doing this.