Analysis of Periodic Updating for Systems with Multiple Timescales

Abstract

Current meteorological observational networks are capable of observing only a limited number of the dependent variables that describe the state of the atmosphere. For example, the large-scale temperature and horizontal wind are commonly observed, but not the large-scale vertical velocity. In the late 1960s, Charney suggested that any missing dependent variables might be reconstructed from the time history of the fields that are observed; for example, the winds could be reconstructed by continually inserting satellite observations of the temperature into a numerical weather prediction model. (Some modern weather prediction models are essentially still using this technique to reconstruct the missing variables.) Charney's hypothesis is analyzed for systems of equations with and without multiple timescales. In the absence of dissipation, the hypothesis is not correct. However, the addition of dissipation can produce convergence that varies in degree relative to the variables that are inserted and the amount of dissipation. The analysis of the insertion process for the multiple-timescale case proves that less dissipation is required and better rates of convergence are achieved in the case that the slow variables are inserted. The advantage of slow variable insertion is even more apparent when the system is skewed, for example, in the external mode case. An alternative approach that requires no dissipation is suggested.