During the Early Jurassic, cyst-forming dinoflagellates began a long-term radiation that would portend ecological importance of these taxa in the pelagic plankton community throughout the rest of the Mesozoic era. The factors that contributed to the evolutionary success of dinoflagellates are poorly understood. Here we examine the relationship between oceanographic and climatic conditions during the Hettangian–Toarcian interval in relation to the radiation of dinoflagellates and other organic-walled phytoplankton taxa in the Tethys Ocean. Our analysis is based on two data sets. The first includes δ13Ccarb, δ13Corg, total organic carbon (TOC), and quantitative palynological observations derived from the Mochras Core (Wales, U.K.), which spans the complete Early Jurassic. The second is a coupled Mg/Ca and δ18O record derived from analyses of belemnite calcite obtained from three sections in northern Spain, covering the upper Sinemurian to Toarcian. From these two data sets we reconstructed the influence of sea level, trophism, temperature, and salinity on dinoflagellate cyst abundance and diversity in northwest Europe. Our results suggest that organic-walled phytoplankton (acritarchs, prasinophytes, and dinoflagellates) diversity increased through the Early Jurassic. The radiation coincides with a long-term eustatic rise and overall increase in the areal extent of continental shelves, a factor critical to cyst germination. On shorter timescales, we observed short bursts of dinoflagellate diversification during the late Sinemurian and late Pliensbachian. The former diversification is consistent with the opening of the Hispanic Corridor during the late Sinemurian, which apparently allowed the pioneer dinoflagellate, Liasidium variabile, to invade the Tethys from the Paleo-Pacific. A true radiation pulse during the late Pliensbachian, with predominantly cold-water taxa, occurred during sea level fall, suggesting that climate change was critical to setting the evolutionary tempo. Our belemnite δ18O and Mg/Ca data indicate that late Pliensbachian water masses cooled (ΔT ≈ −6°C) and became more saline (ΔS ≈ +2 psu). Cooling episodes during generally warm and humid Early Jurassic climate conditions would have produced stronger winter monsoon northeast trade winds, resulting in hydrographic instability, increased vertical mixing, and ventilation of bottom waters. During the late Pliensbachian, dinoflagellates replaced green algae, including prasinophytes and acritarchs, as primary producers. By producing benthic resting cysts, dinoflagellates may have been better adapted to oxidized ocean regimes. This hypothesis is supported by palynological data from the early Toarcian ocean anoxic event, which was marked by highly stratified anoxic bottom water overlain by low-salinity, warm surface waters. These conditions were advantageous to green algae, while cyst-producing dinoflagellates temporarily disappeared. Our results suggest that the rise in dinoflagellate diversity later in the Jurassic appears to correspond to deep water ventilation as a result of the opening of the Atlantic seaway, conditions that appear to have simultaneously led to a loss of prasinophyte dominance in the global oceans.

Vortex dipoles in a two-dimensional, inviscid flow are obtained by prescribing the profile function relating the vorticity to the stream function. The profile functions used are smooth, and the solutions obtained have a smooth transition from the exterior flow to the interior of the vortex. The dipoles are nearly elliptical, and this relates this work to the ‘supersmooth’ dipoles obtained recently by Kizner & Khvoles (Regular Chaotic Dyn., vol. 9, 2004, pp. 509–518). The solutions found here are obtained by an iterative method for solving the nonlinear partial differential equation for the stream function. This iterative method is both robust and flexible. Solutions are also obtained in a β-plane, and they are shielded, as has also been found in previous work.

A necessary and sufficient condition for a homogeneous left invariant partial differential operator P on a nilpotent Lie group G to be hypoelliptic is that π(P) be injective in π for every nontrivial irreducible unitary representation π of G. This was conjectured by Rockland in [18], where it was also proved in the case of the Heisenberg group. The necessity of the condition in the general case was proved by Beals [2] and the sufficiency by Helffer and Nourrigat [4]. In this paper we present a microlocal version of this theorem when G is step two nilpotent. The operator may be homogeneous with respect to any family of dilations on G, not just the natural dilations. We may also consider pseudodifferential operators as well as partial differential operators.

A general procedure is presented for computing axisymmetric swirling vortices which are steady with respect to an inviscid flow that is either uniform at infinity or includes shear. We consider cases both with and without a spherical obstacle. Choices of numerical parameters are given which yield vortex rings with swirl, attached vortices with swirl analogous to spherical vortices found by Moffatt, tubes of vorticity extending to infinity and Beltrami flows. When there is a spherical obstacle we have found multiple solutions for each set of parameters. Flows are found by numerically solving the Bragg–Hawthorne equation using a non-Newton-based iterative procedure which is robust in its dependence on an initial guess.

The stability of steady inviscid vortex pairs in equilibrium with a circular cylinder is studied by discretizing equations derived from contour dynamics. There are two families of vortices, one with a pair of counter-rotating vortices standing behind the cylinder, which may be thought of as desingularizing the Föppl point vortices, and the other with the vortices standing directly above and below the cylinder. Vortices in the first family are found to be neutrally stable with respect to symmetric perturbations. When asymmetric perturbations are included, there is a single unstable mode and a single asymptotically stable mode. Vortices above and below the cylinder have two modes of instability, one symmetric and the other asymmetric, and likewise two asymptotically stable modes.

Direct numerical simulations (DNS) are conducted of a model hydrocarbon–nitrogen mixing layer under supercritical conditions. The temporally developing mixing layer configuration is studied using heptane and nitrogen supercritical fluid streams at a pressure of 60 atm as a model system related to practical hydrocarbon-fuel/air systems. An entirely self-consistent cubic Peng–Robinson equation of state is used to describe all thermodynamic mixture variables, including the pressure, internal energy, enthalpy, heat capacity, and speed of sound along with additional terms associated with the generalized heat and mass transport vectors. The Peng–Robinson formulation is based on pure-species reference states accurate to better than 1% relative error through comparisons with highly accurate state equations over the range of variables used in this study (600 [les ] T [les ] 1100 K, 40 [les ] p [les ] 80 atm) and is augmented by an accurate curve fit to the internal energy so as not to require iterative solutions. The DNS results of two-dimensional and three-dimensional layers elucidate the unique thermodynamic and mixing features associated with supercritical conditions. Departures from the perfect gas and ideal mixture conditions are quantified by the compression factor and by the mass diffusion factor, both of which show reductions from the unity value. It is found that the qualitative aspects of the mixing layer may be different according to the specification of the thermal diffusion factors whose value is generally unknown, and the reason for this difference is identified by examining the second-order statistics: the constant Bearman–Kirkwood (BK) thermal diffusion factor excites fluctuations that the constant Irwing–Kirkwood (IK) one does not, and thus enhances overall mixing. Combined with the effect of the mass diffusion factor, constant positive large BK thermal diffusion factors retard diffusional mixing, whereas constant moderate IK factors tend to promote diffusional mixing. Constant positive BK thermal diffusion factors also tend to maintain density gradients, with resulting greater shear and vorticity. These conclusions about IK and BK thermal diffusion factors are species-pair dependent, and therefore are not necessarily universal. Increasing the temperature of the lower stream to approach that of the higher stream results in increased layer growth as measured by the momentum thickness. The three-dimensional mixing layer exhibits slow formation of turbulent small scales, and transition to turbulence does not occur even for a relatively long non-dimensional time when compared to a previous, atmospheric conditions study. The primary reason for this delay is the initial density stratification of the flow, while the formation of strong density gradient regions both in the braid and between-the-braid planes may constitute a secondary reason for the hindering of transition through damping of emerging turbulent eddies.