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The Lorentz Model
A simple and frequently used analog for a climate system is the dynamical model of thermal convection developed by Edward Lorentz (1964). The modified model is specified by the system of coupled, ordinary differential equations with three degrees of freedom: dX/dt = a (X-Y)+F1(t) with forcing: F1(t) = d7 * integral( x(t1)*exp(-d8*(t-t1)) dt1 ) This system does not admit to analytic solution, but is readily integrable, numerically. Here X is proportional to the intensity of convective motion, Y is proportional to the temperature difference between ascending and descending currents, and Z is proportional to the distortion of the average vertical profile of temperature from linearity. Parameter a is the Prandtl number. Parameter b is a measure of the stability of the system, while parameter c is a function of the wavenumber. The implementation of the model in IDL with its powerful graphics capabilities is useful in illustrating the solution. For example, changing the size of the integration time step affects the convergence of the climate system. In the absence of forcing, the coefficients a,b and c can cause the climate to decay, oscillate periodically, oscillate chaotically or incite numerical instability leading to unbounded variations. It is interesting to observe the chaotic regime and to contemplate what aspect of non-linearity causes the maintenance of the wild oscillations? Turning on the different forcings gives rise to a richer variety of possible climatic regimes. For example, linear forcing introduces almost imperceptible effects in the short term, but ultimately creates instability whose character is appreciated by the IDL visualization. Although it might be expected that linear forcings can give rise to unbounded climate states, it raises the question of what constitutes an appropriate forcing of the climate? Example 1 - Without Forcing In its original form, the forcing functions F1(t), F2(t) and F3(t) are absent. These external forcings are introduced to study the long term stability of the climate system and are of two types: deterministic or stochastic, according to how well these forcings are known. The external forcing functions used here are variants of these used in the work of Pielke and Zeng (1994). The introduction of memory effects in the model is made possible by the addition of an integral that convolves an exponential kernel function with the response. The result is a system of nonlinear integro-differential equations, described qualitatively by Palmer (1993) and solved here. In their absence, the system allows for the steady state solution given by X=Y=Z=0. When b>1 the system can be shown to possess two additional steady state solutions. As b becomes large, Lorentz showed that the steady state convection becomes unstable. For example, choosing a=10, b=28 and c=8/3, results in phase space trajectories that are aperiodic, wandering about in a chaotic manner . Calculations show that the variables X, Y, and Z have continuous spectra and that the solution is extremely sensitive to the initial conditions.
Example 2 - With Forcing If the forcing functions are turned on, as may be expected, the climatic response can be modified greatly. The forcing functions included in this demonstration can be periodic, aperiodic or random, with corresponding responses shown in figures below. It must be understood that these simulations are only in some sense, a gross characterization of the real climate system. In fact the latter is considerably more complex than can be accounted by this relatively simple model. However, many aspects of the real climate system are captured by these simulations.
Example 3 - Bifurcation Qualitative changes in the dynamical behavior of nonlinear system can occur at a critical value of a nonlinearity parameter. In the case of the Lorentz model, the stable steady solution bifurcates to a limit cycle as parameter 'a' increases through its critical value, as can be seen in the figures below. At the critical value the solution curve branches into two paths that may be stable. Using the IDL program you can examine points of bifurcation, for example, by considering the Lorentz system without forcing. To check the bifurcation points of parameter 'a', set b=28.000 and c=2.660, x0=y0=z0=1.0, dt=0.02 and t=7000. Use the forth order Runge Kutta solver (RK4).
You can observe the next bifurcation point near a=18.45294. You can also set 'a' and 'b' as constant parameters, and then look for the bifurcation in parameter 'c', or you can set the other two parameters constant and examine parameter 'b'. Suggestions for experimenting with the model
References
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