  Resolvent analysis is a powerful tool for modeling and analyzing turbulent
flows and in particular provides an approximation of coherent flow structures.
Despite recent algorithmic advances, computing resolvent modes for flows with
more than one inhomogeneous spatial coordinate remains computationally
expensive. In this two-part paper, we show how efficient and accurate
approximations of resolvent modes can be obtained using a well-posed spatial
marching method for flows that contain a slowly varying direction. In this
first part of the paper, we derive a well-posed and convergent one-way equation
describing the downstream-traveling waves supported by the linearized
Navier-Stokes equations. Integrating these equations, which requires
significantly less CPU and memory resources than a direct solution of the
linearized Navier-Stokes equations, approximates the action of the resolvent
operator on a forcing vector. This capability is leveraged in part 2 of the
paper to compute approximate resolvent modes. The method is validated and
demonstrated using the examples of a simple acoustics problem and a supersonic
turbulent jet.
