Theoretical derivations are given on the change of upper tropospheric humidity (UTH) in a warming climate. The considered view is that the atmosphere, which is getting moister with increasing temperatures, will retain a constant relative humidity. In the present study, we show that the upper tropospheric humidity, a weighted mean over a relative humidity profile, will change in spite of constant relative humidity. The simple reason for this is that the weighting function that defines UTH changes in a moister atmosphere. Through analytical calculations using observations and through radiative transfer calculations, we demonstrate that two quantities that define the weighting function of UTH can change: the water vapour scale height and the peak emission altitude. Applying these changes to real profiles of relative humidity shows that absolute UTH changes typically do not exceed 1?%. If larger changes would be observed they would be an indication of climatological changes of relative humidity. As such, an increase in UTH between 1980 and 2009 in the northern midlatitudes, as shown by earlier studies using the High-resolution Infrared Radiation Sounder (HIRS) data, may be an indication of an increase in relative humidity as well.
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