Formal Introduction to Scale Separation: Band-Pass Filtering

Abstract

The idea of scale separation introduced in the preceding chapter will now be formalized on the mathematical level, to show how to handle the equations and derive the subgrid models. The representation of the filtering as a convolution product¹ is first presented in the ideal case of a filter of uniform cutoff length over an infinite domain (Sect. 2.1). Fundamental properties of filters and their approximation via differential operators is presented. Extensions to the cases of a bounded domain and a filter of variable cutoff length are then discussed (Sect. 2.2). The chapter is closed by discussing a few properties of the Eulerian time-domain filters (Sect. 2.3).