Closure of the Stochastic Dynamic Equations

The Monte Carlo (MC) approach uses a randomly chosen set of points, usually obtained from a random Gaussian distribution, to evaluate the impact of uncertainty in initial conditions for the time integration of a physical system. The Stochastic Dynamic Equation set (SDE) approach begins with an infinite ensemble of initial states represented simply by the variance of the initial conditions. The MC is thus an approximation of the SDE method. However, the SDE has a closure issue. Time derivatives of second moments in the system of equations also require knowledge of third moments. Time derivatives of third moments involve fourth moments, and so on.

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.