Sparsity-promoting methods for automated simplification of linearized operators in fluid mechanics

Linear energy amplification mechanisms can play in important role in even highly nonlinear, turbulent flows. In many cases, mechanisms leading to large energy amplification arise due to the properties and interactions of a limited set of terms within the linearized equations. This work develops and applies a method to identify the minimal set of such terms that preserve the energy amplification properties of the original equations. This is done through the use of sparsity-promoting techniques, formulating an optimization problem that penalizes both the difference between the amplification properties of the true and simplified dynamics, and the number of retained terms in the simplified equations. The simplified operators that are identified through this procedure typically have very similar pseudospectral properties as the original equations, but with only the most important active terms present. We demonstrate the use of this method on both incompressible and compressible parallel shear flows, and relate the results to known amplification mechanisms. We additionally discuss the potential applicability of this method for a wider class of fluid flows.