Sparsifying the resolvent forcing mode via gradient-based optimisation

We consider the use of sparsity-promoting norms in obtaining localised forcing structures from resolvent analysis. By formulating optimal problem as a Riemannian optimisation, we are able to maximise cost functionals whilst maintaining unit-energy forcing. Taking functional be energy norm driven response results traditional analysis and is solvable by singular value decomposition (SVD). modifying this with $L_1$ -norm, target spatially that provide an efficient amplification response. showcase optimisation procedure on two flows: plane Poiseuille flow at Reynolds number $Re=4000$ , turbulent past NACA 0012 aerofoil $Re=23\,000$ . In both cases, yields sparse modes maintain important features arising SVD order gain energy. These benefits utilising formulation uncover forcings, specifically view using them actuation locations for control.