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Aside from the direct surface cooling that sulfate geoengineering (SG) would produce, investigations of the possible side effects of this method are still ongoing, such as the exploration of the effect that SG may have on upper tropospheric cirrus cloudiness. The goal of the present study is to better understand the SG thermodynamical effects on the freezing mechanisms leading to ice particle formation. This is undertaken by comparing SG model simulations against a Representative Concentration Pathway 4.5 (RCP4.5) reference case. In the first case, the aerosol-driven surface cooling is included and coupled to the stratospheric warming resulting from the aerosol absorption of terrestrial and solar near-infrared radiation. In a second SG perturbed case, the surface temperatures are kept unchanged with respect to the reference RCP4.5 case. When combined, surface cooling and lower stratospheric warming tend to stabilize the atmosphere, which decreases the turbulence and updraft velocities (-10% in our modeling study). The net effect is an induced cirrus thinning, which may then produce a significant indirect negative radiative forcing (RF). This RF would go in the same direction as the direct effect of solar radiation scattering by aerosols, and would consequently influence the amount of sulfur needed to counteract the positive RF due to greenhouse gases. In our study, given an 8 Tg-SO2 yr(-1) equatorial injection into the lower stratosphere, an all-sky net tropopause RF of -1.46Wm(-2) is calculated, of which -0.3Wm(-2) (20 %) is from the indirect effect on cirrus thinning (6% reduction in ice optical depth). When surface cooling is ignored, the ice optical depth reduction is lowered to 3 %, with an all-sky net tropopause RF of -1.4Wm(-2), of which -0.14Wm(-2) (10 %) is from cirrus thinning. Relative to the clear-sky net tropopause RF due to SG aerosols (-2.1Wm(-2)), the cumulative effect of the background clouds and cirrus thinning accounts for +0.6Wm(-2), due to the partial compensation of large positive shortwave (+1.6Wm(-2)) and negative longwave adjustments (-1.0Wm(-2)). When surface cooling is ignored, the net cloud adjustment becomes +0.8Wm(-2), with the shortwave contribution (+1.5Wm(-2)) almost twice as much as that of the longwave (-0.7Wm(-2)). This highlights the importance of including all of the dynamical feedbacks of SG aerosols.