Remember that P is a function of time (P = P„(l + sin(M))), while L is constant. In this case, [X]ss is set by the instantaneous production rale and loss frequency, and will vary throughout the day as the production rate varies. In other words, the chemistry is so fast that the abundance of X reacts essentially instantly to the variation of P with time, and [X]ss is determined by the instantaneous values of P and L. This is often referred to as "photochemical steady state".
If L « 0), then the lifetime of X is much longer than the time-scale for production to vary, and
where we have used the fact that P(,co » LPltO) cosí a)/) in the simplification. In this case, the chemistry is slow compared to the rate of variation of production. As a result, the abundance of X is determined by the average value of P and L. In other words, the production rate equals the loss rate when averaged over I/o». The instantaneous production rate will not, in general, equal instantaneous loss rate. We refer to this condition as "diurnal steady state".
As stated earlier, production and loss rates both tend to vary strongly during the day as photolysis rates change. This means that if the loss lifetime ML is much shorter than this, then the system will be in photochemical steady state, and |X] can vary during the day—i.e. [X] will display a diurnal cycle. If the loss lifetime 1/L is comparable to or longer than 24 h, then the system will be in diurnal steady state, and [X| will not vary significantly during the day—i.e. fXJ will display little diurnal variations. Note that in no case can ¡X] change significantly over periods of time much shorter than \/L.
Finally, this example was based on the assumption that the transport term of the continuity equation was small compared to the production and loss terms, a situation that arises frequently in the stratosphere. However, it is worth emphasizing that this assumption does not mean that transport is, in general, unimportant. Even for species for which transport of that species is unimportant, transport can be important for other constituents that determine the production rate and loss frequency for X.
2.3 Lifetimes, Time-scales, and Time Constants
In the last section we introduced the term "lifetime". In this section we discuss this further and introduce the related quantities "time-scale" and "time constant". These three quantities, which have units of time and are often used interchangeably, are important and used frequently in the literature and in colloquial contexts.
2.3 Lifetimes, time-scales, and time constants
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