D. R. Moore

Two-dimensional Rayleigh-Benard convection

Two-dimensional convection in a Boussinesq fluid confined between free boundaries is studied in a series of numerical experiments. Earlier calculations by Fromm and Veronis were limited to a maximum Rayleigh number R 50 times the critical value R , for linear instability. This range is extended to 1000 R c . Convection in water, with a Prandtl number p = 6·8, is systematically investigated, together with other models for Prandtl numbers between 0·01 and infinity. Two different modes of nonlinear behaviour are distinguished. For Prandtl numbers greater than unity there is a viscous regime in which the Nusselt number

Convective instability in a compressible atmosphere. II

N \approx 2(R/R_c)^{\frac{1}{3}}

Nonlinear double-diffusive convection

, independently of p . The heat flux is a maximum for cells whose width is between 1·2 and 1·4 times the layer depth. This regime is found when

The Transfer of Visible Radiation through Clouds

5 \leqslant R/R_c \lesssim p^{\frac{3}{2}}

Axisymmetric convection in a cylinder

. At higher Rayleigh numbers advection of vorticity becomes important and N ∞ R 0·365 . When p = 6·8 the heat flux is a maximum for square cells; steady convection is impossible for wider cells and finite amplitude oscillations appear instead, with periodic fluctuations of temperature and velocity in the layer. For p N ∞ R 0·365 , with a constant of proportionality equal to 1·90 when p [Lt ] 1 and decreasing slowly as p is increased. The physical behaviour in these regimes is analysed and related to astrophysical convection.

Transitions to chaos in two-dimensional double-diffusive convection

The onset of steady convection in a polytropic atmosphere with constant viscosity is studied numerically. (AIP)

The breakdown of steady convection

The two-dimensional motion of a fluid confined between two long horizontal planes, heated and salted from below, is examined. By a combination of perturbation analysis and direct numerical solution of the governing equations, the possible forms of large-amplitude motion are traced out as a function of the four non-dimensional parameters which specify the problem : the thermal Rayleigh number RT, the saline Rayleigh number R,, the Prandtl number u and the ratio of the diffusivities r . A branch of time-dependent asymptotic solutions is found which bifurcates from the linear oscillatory instability point. In general, for fixed u, r and R,, as RT increases three further abrupt transitions in the form of motion are found to take place independent of the initial conditions. At the first transition, a rather simple oscillatory motion changes into a more complicated one with different structure, at the second, the motion becomes aperigdic and, at the third, the only asymptotic solutions are time independent. Disordered motions are thus suppressed by increasing R,. The time-independent solutions exist on a branch which, it is conjectured, bifurcates from the timeindependent linear instability point. They can occur for values of R, less than that at which the third transition point occurs. Hence for some parameter ranges two different solutions exist and a hysteresis effect occurs if solutions obtained by increasing RT and then decreasing RT are followed. The minimum value of RT for which time-independent motion can occur is calculated for fourteen different values of u, r and Rs. This minimum value is generally much less than the critical value of time-independent linear theory and for the larger values of u and R, and the smaller values of r , is less than the critical value of time-dependent linear theory.

Period doubling and chaos in partial differential equations for thermosolutal convection

Abstract The transfer of visible radiation through terrestrial clouds has been calculated by a Monte Carlo computer program using a Henyey-Greenstein phase function which is similar to the true scattering function of water droplet clouds. From the fact that the maximum albedo of thick stratocumulus clouds is 0.7–0.8, it is deduced that 1 − ω0 is of the order of 10−3, where ω0 is the single scattering albedo. This value of 1 − ω0 is ∼104 times larger than expected from pure water. It is argued that the aerosols upon which the cloud droplets condensed are the source of the absorption. A set of simple formulae and tables are presented which give the reflection and transmission of clouds having an arbitrary phase function. For optical depths τc > 10, they are accurate to a few per cent provided that (1 − ω0)/(1 − g) < 10−2, where g = ¯cosθ and where θ is the scattering angle. Over the same range of parameters, the formulae and tables are accurate to about 1% for the Henyey-Greenstein phase function.

Difference methods for time-dependent two-dimensional convection

In three-dimensional BBnard convection regions of rising and sinking fluid are dissimilar. This geometrical effect is studied for axisymmetric convection in a Boussinesq fluid contained in a cylindrical cell with free boundaries. Near the critical Rayleigh number R, the solution is obtained from a perturbation expansion, valid only if both the Reynolds number and the PBclet number are small. For values of the Nusselt number N 1 there is a viscous regime with N M 2(R/Rc)i; when R/R, 2 pb, N increases more rapidly, approximately as R0.4. At high Rayleigh numbers a large isothermal region develops, in which the ratio of vorticity to distance from the axis is nearly constant.

The NRL layered ocean model

The partial differential equations governing two-dimensional thermosolutal convection in a Boussinesq fluid with free boundary conditions have been solved numerically in a regime where oscillatory solutions can be found. A systematic study of the transition from nonlinear periodic oscillations to temporal chaos has revealed sequences of period-doubling bifurcations. Overstability occurs if the ratio of the solutal to the thermal diffusivity tau is less than 1 and the solutal Rayleigh number Rs is sufficiently large. Solutions have been obtained for two representative values of tau. For tau = 0.316, R(s) = 10,000, symmetrical oscillations undergo a bifurcation to asymmetry, followed by a cascade of period-doubling bifurcations leading to aperiodicity, as the thermal Rayleigh number R(T) is increased. At higher values of R(T), the bifurcation sequence is repeated in reverse, restoring simple periodic solutions. As R(T) is further increased more period-doubling cascades, followed by chaos, can be identified. Within the chaotic regions there are narrow periodic windows, and multiple branches of oscillatory solutions coexist. Eventually the oscillatory branch ends and only steady solutions can be found. The development of chaos has been investigated for tau = 0.1 by varying R(T) for several different values of R(s). When R(s) is sufficiently small there are periodic solutions whose period becomes infinite at the end of the oscillatory branch. As R(s) is increased, chaos appears in the neighborhood of these heteroclinic orbits. At higher values of R(s), chaos is found for a broader range in R(T). A truncated fifth-order model suggest that the appearance of chaos is associated with heteroclinic bifurcations.