Measurements relevant for the climate need to be done for long time series. While the measurements for the daily weather a often of a relatively high resolution, climate relevant measurements of more than 100 years are often available for just a few measurement sites. Nontheless such isolated measurements are used to analyse the climate. An example is the

Real 3-d climate is often analysed under the assumption that the
climate state is constant and the weather is a deviation from this
climate state. This means a point climate or
climate dimension 0 in phase space. A more advanced analysis would
assume that the climate state is not a point in phase space but
rather a state depending on parameters
`c1, ..., cn`,
where n is a small number. From the argument given above it
follows that n must be small in order to analyse it
using just a few available measurements. Very little work has
been done beyond the point climate model for the real atmosphere.

However for simplified climate systems this is more easy to
analyse and the Lorenz model is an example often used to
imagine howw the real climate may look like. In particular
the Lorenz model shows __climate bifurcation__, which should be a
point of concern, if in the real climate it would occur also.
In this place we try to give a system of looking at
the **Lorenz climate bifurcation** without a detailed
mathematical understanding and no programming skills in Unix
and Fortran and still alto do climate runs of his / her own.

Fig. 1: The Lorenz bifurcation diagram shows the time development of the three amplitudes used in the Lorenz model.

While sometimes the Lorenz model is presented as a purely
algebraic construction, we here want to emphasize that it
describes real atmospheric flow for a
simplified geometry. The Lorenz model describes the
**circulation in a rectangular cave**, which is heated from
below and cooled above. The circulation depends on
parameters which describe the flow, such as length, height and
viscosity of the air. Commonly these are given by non-dimensional
parameters. For our rectangular cave these are: Reynolds number Re,
describing the viscosity of the air, the Rayley number Ra being
connected to the temperature difference between top and bottom and the
Prantl number Pr conncted to the surface friction. Note that the
physical situation and the flow inside the area are the same
whenever these three numbers are the same. This means that
caves of the different size may have the same flow
solution when the non-dimensional parameters are equal.

The theory is described in [1] and [2]. The systems have been
experimentally realized and simulated by high dimensional
numerical models and for some values of Re,Ra, Pr the low
dimensional Lorenz model simulates the flow well.
When using the same terminology as with the real climate
system on the sphere, we call the high dimensional model a
**general circulation model** and the Lorenz model
the **climate dynamics model**.

The reader with an interest in this theory and some (small) programming skills can obtain a account on our server and do some Lorenz experiments on his/her own. In particular the dependence of the the climate bifurcation on Re,Ra,Pr can be explored using the software provided.

In this WEB application
we want to give the
opportunity to experiment with initial values of the
Lorenz models with only **rudimentary mathematical
skills** and **no knowledge of Unix/Linux and Fortran**.
The phase space the Lorenz model is three dimensional and the
amplitudes are called
A, T1 and T2.
These amplitudes are time
dependent describing the climate development in time.
For A,T1 and T2 initial values must be provided and
the time development is computed in time steps of dt,
where here a default value of dt=.1 is used.

The initial values can be chosen from the whole 3 dimensional space. However, the solution curves do not cover the whole 3-d area, but rather fill two two dimensional surfaces, which are connected and the solution curve changes between these surface. While at first sight the cuves in each sheet seem to obe periodic, they are not and never come back to the same value. They are also chaotic, meaning unpredictable. After some prediction time the two solutions differ by a large amount, when at the start time the solutions have only a tiny difference. There is a predictive time of total chaos. This means that when the difference of two initial values goes to 0, the curves obtain a maximum difference.

In the fig. 1 above the initial value was chosen inside the attractor. Any value can be chosen and the solution the goes rather fast towards the attractor. The simple Lorenz model can be used to experiment with different initial values, estimate the predictice time and the chaotic regime and more.

References:

[1]

Contact: send mail to Juergen Steppeler

Design and programming by EH