CAUSALITY IN STRONG SHEAR FLOWS

It is well known that the standard transport equations violate causality when gradients are large or when temporal variations are rapid. We derive a modified set of transport equations that satisfy causality.'These equations are obtained from the underlying Boltzmann equation. We use a simple model for particle collisions that enables us to derive moment equations nonperturbatively, that is, without making the usual assumption that the distribution function deviates only slightly from its equilibrium value. We also retain time derivatives of various moments and choose closure relations so that the final set of equations is causal. We apply the model to two problems: particle diffusion and viscous transport. In both cases we show that signals propagate at a finite speed and therefore that the formalism obeys causality. When spatial gradients or temporal variations are small, our theory for particle diffusion.and viscous flows reduces to the usual diffusion and Navier-Stokes equations, respectively. However, in the opposite limit of strong gradients the theory produces causal results with finite transport fluxes, whereas the standard theory gives results that are physically unacceptable. We find that when the velocity gradient is large on the scale of a mean free path, the viscous shear stress is suppressed relative to the prediction of the standard diffusion approximation. The shear stress reaches a maximum at a finite value of the shear amplitude and then decreases as the velocity gradient increases. The decrease of the stress in the limit of large shear is qualitatively different from the case of scalar particle diffusion where the diffusive flux asymptotes to a constant value in the limit of large density gradient. In the case of a steady Keplerian accretion disk with hydrodynamic turbulent viscosity, the stress limit translates to an upper bound on the Shakura-Sunyaev alpha-parameter, namely alpha < 0.07. The limit on alpha is much stronger in narrow boundary layers where the velocity shear is larger than in a Keplerian disk.