  Two approaches for closing the turbulence subgrid-scale stress tensor in
terms of matrix exponentials are introduced and compared. The first approach is
based on a formal solution of the stress transport equation in which the
production terms can be integrated exactly in terms of matrix exponentials.
This formal solution of the subgrid-scale stress transport equation is shown to
be useful to explore special cases, such as the response to constant velocity
gradient, but neglecting pressure-strain correlations and diffusion effects.
The second approach is based on an Eulerian-Lagrangian change of variables,
combined with the assumption of isotropy for the conditionally averaged
Lagrangian velocity gradient tensor and with the `Recent Fluid Deformation'
(RFD) approximation. It is shown that both approaches lead to the same basic
closure in which the stress tensor is expressed as the product of the matrix
exponential of the resolved velocity gradient tensor multiplied by its
transpose. Short-time expansions of the matrix exponentials are shown to
provide an eddy-viscosity term and particular quadratic terms, and thus allow a
reinterpretation of traditional eddy-viscosity and nonlinear stress closures.
The basic feasibility of the matrix-exponential closure is illustrated by
implementing it successfully in Large Eddy Simulation of forced isotropic
turbulence. The matrix-exponential closure employs the drastic approximation of
entirely omitting the pressure-strain correlation and other `nonlinear
scrambling' terms. But unlike eddy-viscosity closures, the matrix exponential
approach provides a simple and local closure that can be derived directly from
the stress transport equation with the production term, and using physically
motivated assumptions about Lagrangian decorrelation and upstream isotropy.
