| description abstract | The stochastic dynamic equations, as investigated in part I of this two-part study, can be applied to any time-dependent set of differential equations which are, at most, nonlinear quadratic. In this study, they are used to explore various aspects of the question of atmospheric predictability. The growth of uncertainty due to ill-defined initial conditions in the nonlinear advection field is viewed by considering a simple barotropic model. A wave number is defined to be ?unpredictable? when the ?uncertain? energy associated with that wave becomes as large as the ?certain? energy associated with it. The predictability of wave number 12 is used as a reference point and as an arbitrary minimum requirement for useful synoptic forecasts. It is found that, based upon the average root-mean-square vector error in the wind field today, such a wave number has a predictability value of about 1.5 days. If this error could be reduced by a factor of 4 (i.e., down to 1 m/s), this value would be approximately 5 days. Using a stochastic barotropic model with 2,015 degrees of freedom, it is found that any initial energy spectra for the certain or uncertain eddy kinetic energy will give essentially the same predictability values. This is because the complete nonlinearity is accounted for in the stochastic dynamic equation set and the dynamics of the two-dimensional fluid tend to drive any initial spectrum into approximately a ? 3 power law in some averaged sense?as expected from theory. It is shown that the rather pessimistic predictability values, based solely upon error growth due to uncertain advection and instability processes, are considerably lengthened (at least in the largest scales) when additional forcing and dissipation terms are included in the mathematical models. However, such additional forces can never be simulated perfectly and the qualitative effect of these imperfections is shown by calculations with a simple baroclinic model having heating and friction. Based upon arguments presented, the author speculates that in 10 yr the projected uncertainties in the physics and the uncertainties arising from the computational wave number cutoff will still restrict the predictability of wave number 12 to within 5?7 days. It is shown how the eventual application of the stochastic dynamic equations to more complicated models can replace such speculation with more concrete evidence. The globally averaged value of predictability considered above is very general and it is shown how the utility of the stochastic dynamic set can provide more meaningful information to the user. Only one aspect of this utility is shown (the growth of the phase error of a wave with time), but the stochastic set of equations gives the ?believability? of each variable at every point in the space-time domain. | |
