  Resolvent analysis provides a framework to predict coherent spatio-temporal
structures of largest linear energy amplification, through a singular value
decomposition (SVD) of the resolvent operator, obtained by linearizing the
Navier--Stokes equations about a known turbulent mean velocity profile.
Resolvent analysis utilizes a Fourier decomposition in time, which has thus-far
limited its application to statistically-stationary or time-periodic flows.
This work develops a variant of resolvent analysis applicable to time-evolving
flows, and proposes a variant that identifies spatio-temporally sparse
structures, applicable to either stationary or time-varying mean velocity
profiles. Spatio-temporal resolvent analysis is formulated through the
incorporation of the temporal dimension to the numerical domain via a discrete
time-differentiation operator. Sparsity (which manifests in localisation) is
achieved through the addition of an L1-norm penalisation term to the
optimisation associated with the SVD. This modified optimization problem can be
formulated as a nonlinear eigenproblem, and solved via an inverse power method.
We first demonstrate the implementation of the sparse analysis on
statistically-stationary turbulent channel flow, and demonstrate that the
sparse variant can identify aspects of the physics not directly evident from
standard resolvent analysis. This is followed by applying the sparse space-time
formulation on systems that are time-varying: a time-periodic turbulent Stokes
boundary layer, and then a turbulent channel flow with a sudden change in
pressure gradient. We present results demonstrating how the sparsity-promoting
variant can either change the quantitative structure of the leading space-time
modes to increase their sparsity, or identify entirely different linear
amplification mechanisms compared to non-sparse resolvent analysis.
