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The existence of multiple steady states with the same farfield behavior is discussed for simple 1-D transonic model problems. These multiple solutions all have only entropy satisfying compressive steady shock waves. Only some of these solutions are stable in the time-dependent system and are accessible through physical time-dependent perturbations. This is demonstrated by some elementary explicit solutions in a scalar model problem. However, for a large class of initial data and large C.F.L. numbers, numerical experiments show that implicit schemes can converge to the physically unstable steady states and this phenomenon is analysed. The scalar model is also discussed as a very simple numerical test problem for implicit schemes with rich structure in both the steady state and time-dependent regimes.

A popular contemporary approach in predicting enhanced flame speeds in premixed turbulent combustion involves averaging or closure theories for the G-equation involving both large-scale flows and small-scale turbulence. The G-equation is a Hamilton-Jacobi equation involving advection by an incompressible velocity field and nonlinear dependence on the laminar flame speed; this G-equation has been derived from the complete Navier-Stokes equations under the tacit assumptions that the velocity field varies on only the integral scale and that the ratio of the flame thickness to this integral scale is small. Thus there is a potential source of error in using the averaged G-equation with' turbulent velocities varying on length scales smaller than the integral scale in predicting enhanced flame speeds. Here these issues are discussed ip the simplest context involving velocity fields varying on two scales where a complete theory of nonlinear averaging for predicting enhanced flame speeds without any ad hoc approximations has been developed recently by the authors. The predictions for enhanced flame speeds of this complete averaging theory versus the averaging approach utilizing the G-equation are compared here in the simplest context involving a constant mean flow and a small-scale steady periodic flow where both theories can be solved exactly through analytical formulas. The results of this comparison are summarized briefly as follows: The predictions of enhanced flame speeds through the averaged G-equation always underestimate those computed by complete averaging. Nevertheless, when the transverse component of the mean flow relative to the shear is less than one in magnitude, the agreement between the two approaches is excellent. However, when the transverse component of the mean flow relative to the shear exceeds one in magnitude, the predictions of the enhanced flame speed by the averaged G-equation significantly underestimate those computed through complete nonlinear averaging, and in some cases, by more than an order of magnitude. 0 199.5 American Institute of Physics.

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In this paper, we explore the strong rotation limit of the rotating and stratified Boussinesq equations with periodic boundary conditions when the stratification is order 1 ([Rossby number] Ro = ε, [Froude number] Fr = O(1), as ε → 0). Using the same framework of Embid & Majda (Geophys. Astrophys. Fluid Dyn., vol. 87, 1998, p. 1), we show that the slow dynamics decouples from the fast. Furthermore, we derive equations for the slow dynamics and their conservation laws. The horizontal momentum equations reduce to the two-dimensional Navier–Stokes equations. The equation for the vertically averaged vertical velocity includes a term due to the vertical average of the buoyancy. The buoyancy equation, the only variable to retain its three-dimensionality, is advected by all three two-dimensional slow velocity components. The conservation laws for the slow dynamics include those for the two-dimensional Navier–Stokes equations and a new conserved quantity that describes dynamics between the vertical kinetic energy and the buoyancy. The leading order potential enstrophy is slow while the leading order total energy retains both fast and slow dynamics. We also perform forced numerical simulations of the rotating Boussinesq equations to demonstrate support for three aspects of the theory in the limit Ro → 0: (i) we find the formation and persistence of large-scale columnar Taylor–Proudman flows in the presence of O(1) Froude number; after a spin-up time, (ii) the ratio of the slow total energy to the total energy approaches a constant and that at the smallest Rossby numbers that constant approaches 1 and (iii) the ratio of the slow potential enstrophy to the total potential enstrophy also approaches a constant and that at the lowest Rossby numbers that constant is 1. The results of the numerical simulations indicate that even in the presence of the low wavenumber white noise forcing the dynamics exhibit characteristics of the theory.

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