Closure of the Stochastic Dynamic Equations
In the early years of research using the stochastic dynamic approach, not just to look at uncertainty in initial conditions, but also to ascertain the impact of uncertainty in climate model forcing parameters, the methods of closure were crude. They were patterned after the eddy-damped quasi-normal theory used in turbulence approximations. Now the equations have been properly closed for those systems which are bounded and dissipative – and there are many examples of such systems, including chaotic systems. The methodology uses both the Monte Carlo and SDE approach and relies on the original physics of the system to assist the closure.
The details of the closure are revealed using the Lorenz set of equations that produced his original strange attractor. There are two fundamental issues to deal with in solving the closure. One issue is the initial “explosive randomness” to control over time and the second issue is obtaining the proper form for the moments in the final solution. The explosive randomness is moderate in nonlinear fixed point solutions, but rather severe in chaotic solutions. Examples of the successful closure for both types of solutions are provided.