Data-based Master Equations for the Stratosphere

Three-dimensional data-based master equations are developed and
subsequently used to study climate variability in the stratosphere.
Master equations are used to develop understanding of observed
systems where no dynamic equations are available. Master equations
are used in this thesis as prognostic equations for the probability
density in a discretized phase space spanned by climate variables.
The evolution of the probability density may then reveal
information about the relationship between these variables. The
phase space is partitioned into several hundred boxes of equal grid
size representing at any one time states that the system can
assume. In this discretized version of the phase space, the
coefficients of a master equation may be estimated from the
relative frequencies of transitions observed in a time series of
the variables obtained from observations or numerical model runs.
Data-based master equations are numerical structures whose success
depends among other things on the resolution and volume of the
available time series. These dependencies are studied on the basis
of data from the famous three-component Lorenz convection model
extended with a stochastic forcing. Time series of the desired
length and time resolution can thus be generated easily.
Furthermore, the results can be compared directly. Best results are
obtained through the combination of a long data record and a coarse
time resolution. The choice of the variables and their number also
play a crucial role in the success of a master equation. Time
series of stratospheric climate indices obtained from the
reanalyses ERA-40 lead also to these last results. The stratosphere
serves now as an implementation area. The master equation shows
that during the eastern phase of the quasi-biennial oscillation
(QBO) of equatorial zonal wind the arctic stratosphere is about 2 K
warmer than during the western phase. Thus the relationship between
QBO and arctic stratosphere can be quantified. The influence of the
11-year solar cycle is described by the master equation. It emerges
that the relationship between QBO and temperature anomaly of the
arctic stratosphere shows a dependence on solar variability. The
implications of stratospheric processes on the climate in the
troposphere are analysed with a master equation for a time series
of an index of the Arctic Oscillation (AO) at stratospheric and
tropospheric pressure levels. The master equation captures the main
features of this interaction between stratosphere and troposphere.
It is shown that anomalies of the AO in the middle stratospere
propagate deeply into the troposphere with a time scale of 4 weeks.
Furthermore the master equation shows that the influence of strong
tropospheric AO-anomalies remains confined to the lower
stratosphere.