In theory, changes in mirror shape will have a much greater effect as a shift d in the surface makes a 2 * d wavefront error rather than a (n-1) * d error - so with a plate glass mirror, temperature effects in the window will be unimportant. A low expansion mirror will have an expansion coefficient half that of the window (pyrex) or almost zero (zerodur), and in this case one may have to worry about the window temperature. The problem (if there is one, I am only theorising!) is essentially that one cannot get low expansion optical glass. Mirrors also suffer more because any thermal stressing will distort the whole blank.
What one needs to is make a dummy telescope with an aluminised mirror blank and plate glass window, stick thermocouples over them, leave outside at night and see how large the temperature differences can be!
Obviously if one can maintain the scope at near night time temperatures throughout the day there will be no problem. .............................................................
Regarding the general problem of tube currents, the problem is not that the air is moving (air is invisible, after all) but that the convection currents move because of temperature differences, and the refractive index of air changes with temperature. What you have is what would be known by fluid mechanicists as a "thermal boundary layer". Now boundary layers are commonly removed by having a porous surface (or one peppered with tiny holes) and sucking air through it. I suspect that one would get seeing as good as with an optical window by making a double skinned tube with holes in the inner wall and sucking this warm away; for good measure one would have a central hole in the mirror and suck here also, in the shadow of the secondary. It would be much easier to make some holes in a baffle plate than to make an optical window.
I believe some observatories do this on a larger scale, sucking air downwards through the dome slit and then passing it through a large pre-cooled "cold room" so that when it emerges again, some distance away, it is at ambient temperature and does not make a plume of hot air.
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