Is it possible to express the problem of calculating turbulent flame speeds as an eigenvalue problem that is analogous to the laminar flame speed problem? It is argued for grid turbulence that the answer is affirmative, and some benefits of pursuing such a calculation are exploited for the limiting case of a first-order reaction with vanishingly small heat release. The streamwise turbulent transport of reactant occupies a central role in the analysis. The equation governing the ensemble average of this quantity assumes different simplified forms in the limits of small-scale and large-scale turbulence. The criterion which is obtained for separating the small-scale and large-scale régimes differs from that of Damköhler and also from that of Kovasznay and Klimov. In the small-scale régime, turbulence produces a spatially varying diffusivity, the form of which can be ascertained only through an investigation of non-linear equations describing the statistical dynamics of production and decay of the velocity–concentration correlation. In the large-scale régime, which is of greater practical importance, the ensemble average of the streamwise turbulent reactant flux satisfies a linear ordinary differential equation whose solution for the growth and decay of the flux contains effects resembling wrinkling of the laminar flame, increasing of the effective diffusivity and augmentation of the effective reaction rate. An exact solution to the linear eigenvalue problem which arises in the large-scale limit reveals that turbulence enhances mean reactant consumption in the upstream portion of the flame and retards reactant consumption downstream. Formulas are given for the increase in flame speed and the increase in flame thickness that are produced by turbulence in the large-scale limit. Since the equations are relatively tractable in the large-scale limit, it is suggested that further study of these equations may yield improved descriptions of realistic turbulent flames.