The relationship between age of air and the diabatic circulation of the stratosphere

The strength of the Brewer-Dobson circulation is difficult to estimate using observations. Trends in the age of stratospheric air, deduced from observations of transient tracers, have been used to identify trends in the circulation, but there are ambiguities in the relationship between age and the strength of the circulation. This paper presents a steady-state theory and a time-dependent extension to relate age of air directly to the diabatic circulation of the stratosphere. In steady state, it is the difference between the age of upwelling and downwelling air through an isentrope and not the absolute value of age that is a measure of the strength of the diabatic circulation through that isentrope. For the time-varying case, expressions for other terms that contribute to the age budget are derived. An idealized atmospheric general circulation model with and without a seasonal cycle is used to test the time-dependent theory and to find that these additional terms are small upon annual averaging. The steady-state theory holds as well for annual averages of a seasonally varying model as for a perpetual-solstice model. These results are a step toward using data to quantify the strength of the diabatic circulation.