Periodic Lateral Boundary Conditions Research Articles

Wind and boundary layers in Rayleigh-Bénard convection. II. Boundary layer character and scaling

The scaling of the kinematic boundary layer thickness lambda(u) and the friction factor C(f) at the top and bottom walls of Rayleigh-Bénard convection is studied by direct numerical simulation (DNS). By a detailed analysis of the friction factor, a new parameterisation for C(f) and lambda(u) is proposed. The simulations were made of an L/H=4 aspect-ratio domain with periodic lateral boundary conditions at Ra=(10(5), 10(6), 10(7), 10(8)) and Pr=1. The continuous spectrum, as well as significant forcing due to Reynolds stresses, clearly indicates a turbulent character of the boundary layer, while viscous effects cannot be neglected, judging from the scaling of classical integral boundary layer parameters with Reynolds number. Using a conceptual wind model, we find that the friction factor C(f) should scale proportionally to the thermal boundary layer thickness as C(f) proportional variant lambda(Theta)/H, while the kinetic boundary layer thickness lambda(u) scales inversely proportionally to the thermal boundary layer thickness and wind Reynolds number lambda(u)/H proportional variant (lambda(Theta)/H)(-1)Re(-1). The predicted trends for C(f) and lambda(u) are in agreement with DNS results.

Treatment of LBCs in 2D simulation of convection over hills

A series of idealized model simulations are analyzed to determine the sensitivity of model results to dierent configurations of the lateral boundary conditions (LBCs) in simulating mesoscale shallow convection over hilly terrain. In the simulations with steady thermal forcing at the model surface, a radiation condition at both boundaries is the best choice under high wind conditions, and the best results are produced when both the normal velocities and the temperature are treated with the radiation scheme in which the phase speed is the same for dierent variables. When the background wind speed is reasonably small, the LBC configuration with either the radiation or the zero gradient condition at both boundaries tends to make the numerical solution unstable. The choice of a constant condition at the inflow boundary and a radiation outflow boundary condition is appropriate in most cases. In the simulations with diurnal thermal forcing at the model surface, dierent LBC schemes are combined together to reduce spurious signals induced by the outflow boundary. A specification inflow boundary condition, in which the velocity fields at the inflow boundary are provided using the time-dependent results of a simulation with periodic LBCs over a flat domain, is tested and the results indicate that the specification condition at the inflow boundary makes it possible to use a smaller model domain to obtain reasonable results. The model horizontal domain length should be greater than a critical length, which depends on the domain depth H and the angle between gravity wave phase lines and the vertical. An estimate of minimum domain length is given by [(H zi)/U ] p N2L2 4 2U2 , where N and U are the background stability and wind speed, respectively, Lx is the typical gravity wavelength scale, and zi is the convective boundary layer (CBL) depth.

Baroclinic instability of a symmetric, rotating, stratified flow: a study of the nonlinear stabilisation mechanisms in the presence of viscosity

Abstract. This paper presents the analysis of symmetric circulations of a rotating baroclinic flow, forced by a steady thermal wind and dissipated by Laplacian friction. The analysis is performed with numerical time-integration. Symmetric flows, vertically bound by horizontal walls and subject to either periodic or vertical wall lateral boundary conditions, are investigated in the region of parameter-space where unstable small amplitude modes evolve into stable stationary nonlinear solutions. The distribution of solutions in parameter-space is analysed up to the threshold of chaotic behaviour and the physical nature of the nonlinear interaction operating on the finite amplitude unstable modes is investigated. In particular, analysis of time-dependent energy-conversions allows understanding of the physical mechanisms operating from the initial phase of linear instability to the finite amplitude stable state. Vertical shear of the basic flow is shown to play a direct role in injecting energy into symmetric flow since the stage of linear growth. Dissipation proves essential not only in limiting the energy of linearly unstable modes, but also in selecting their dominant space-scales in the finite amplitude stage.

Application of a Perturbation Recycling Method in the Large-Eddy Simulation of a Mesoscale Convective Internal Boundary Layer

An arrangement of boundary conditions is described and demonstrated that facilitates the large-eddy simulation (LES) of inhomogeneous boundary layers such as internal boundary layers. In addition to the domain where the internal boundary layer develops, the method requires a section of domain over the upwind surface that is of the order of 10 boundary layer thicknesses and thus similar in size to that needed for the LES of the upwind boundary layer. In addition to periodic lateral, closed upstream, and open downstream boundary conditions, a simple and efficient perturbation recycling technique, which follows from one of Lund, Wu, and Squires, is used to generate a steady supply of fully developed turbulence from the inflow boundary. The arrangement is used to simulate a convective internal boundary layer during a cold-air outbreak over water. Results show the method consistently produces a solution that is homogeneous over the upwind surface and inhomogeneous over the downwind surface, and that the statistics are stationary after spinup. The sensitivity to the placement of the outflow boundary is tested, and examples of instantaneous and mean fields generated by the simulations are shown.

Compressible magnetoconvection in three dimensions: pattern formation in a strongly stratified layer

The interaction between magnetic fields and convection is interesting both because of its astrophysical importance and because the nonlinear Lorentz force leads to an especially rich variety of behaviour. We present several sets of computational results for magnetoconvection in a square box, with periodic lateral boundary conditions, that show transitions from steady convection with an ordered planform through a regime with intermittent bursts to complicated spatiotemporal behaviour. The constraints imposed by the square lattice are relaxed as the aspect ratio is increased. In wide boxes we find a new regime, in which regions with strong fields are separated from regions with vigorous convection. We show also how considerations of symmetry and associated group theory can be used to explain the nature of these transitions and the sequence in which the relevant bifurcations occur.

Three-Dimensional Week-Long Simulations of TOGA COARE Convective Systems Using the MM5 Mesoscale Model

Abstract A three-dimensional nonhydrostatic mesoscale model, the Pennsylvania State University/National Center for Atmospheric Research mesoscale model (MM5), is used to simulate the evolution of convective systems over the intensive flux array (IFA) during the Tropical Ocean Global Atmosphere Coupled Ocean–Atmosphere Response Experiment, during 19–26 December 1992. The model is driven by a time-varying “IFA mean forcing” based on the average advective tendencies of temperature and moisture over the IFA. The domain-averaged horizontal wind is kept close to the observed IFA mean using Newtonian relaxation. Periodic lateral boundary conditions are imposed. Simulations with three horizontal grid spacings, 2, 15, and 60 km, are conducted. With 15- and 60-km resolution, subgrid-scale cumulus convection is parameterized while mesoscale convective organization is explicitly resolved over a (600 km)2 domain. With 2-km resolution, convection is fully resolved over a (210 km)2 domain. Despite their different horizo...

How valid are dynamic models of subduction and convection when plate motions are prescribed?

A detailed comparison between fully dynamic and kinematic plate formulations has been made in models of mantle convection. Plate velocity is computed self-consistently from fully dynamic plate models with temperature- and stress-dependent viscosity and preexisting mobile faults. In fully dynamic models, the flow is driven solely by internal buoyancy, while in kinematic models the flow is driven by a combination of the prescribed surface velocity and internal buoyancy. Only a temperature-dependent viscosity, close to the effective viscosity determined from the fully dynamic models, is used in the kinematic models. The two types of models give very similar temperature structures and slab evolutionary histories when the effective viscosity and surface velocity are nearly identical. In kinematic plate models, the additional work introduced by the prescribed velocity boundary condition is apparently dissipated within the lithosphere and has little influence on the convection under the lithosphere. In models with periodic lateral boundary conditions, slabs sink into the lower mantle at an oblique angle and this contrasts with the vertical sinking which occurs with reflecting boundary conditions. Models show that we can simulate fully dynamic models with kinematic models under either periodic boundary conditions or reflecting boundary conditions.

Fractal Stability Border in Plane Couette Flow

Fachbereich Physik und Institut fu¨r Chemie und Biologie des Meeres,der C.v. Ossietzky Universit¨at, Postfach 25 03, D-26111 Oldenburg, GermanyandFachbereich Physik der Philipps Universt¨at Marburg, D-35032 Marburg, Germany(permanent)(February 5, 2008)We study the dynamics of localised perturbations in plane Couette flow with periodic lateralboundary conditions. For small Reynolds number and small amplitude of the initial state theperturbation decays on a viscous time scale t ∝Re. For Reynolds number larger than about 200,chaotic transients appear with life times longer than the viscous one. Depending on the type of theperturbation isolated initial conditions with infinite life time appear for Reynolds numbers largerthan about 270–320. In this third regime, the life time as a function of Reynolds number andamplitude is fractal. These results suggest that in the transition region the turbulent dynamics ischaracterised by a chaotic repeller rather than an attractor.47.20.Ft,47.20.Ky,47.15.Fe,05.45.+b

Cloud-Resolving Modeling of Tropical Cloud Systems during Phase III of GATE. Part I: Two-Dimensional Experiments

Abstract A formal framework is established for the way in which cloud-resolving numerical models are used to investigate the role of precipitating cloud systems in climate and weather forecasting models. Emphasis is on models with periodic lateral boundary conditions that eliminate unrealistic numerically generated circulations caused by open boundary conditions in long-term simulations. Defined in this formalism is the concept of large-scale forcing and the cloud-environment interactions that are consistent with the periodic boundary conditions. Two-dimensional numerical simulations of the evolution of cloud systems during 1–7 September 1974 in Phase III of the Global Atmospheric Research Program Atlantic Tropical Experiment (GATE) are conducted. Based on the above formalism, a simple technique is used to force an anelastic cloud-resolving model with evolving large-scale horizontal wind field and large-scale forcing for the temperature and moisture obtained from the GATE data. The 7-day period selected i...

Multiple bifurcations in triple convection with non-ideal boundary conditions

We investigate the behaviour associated with a multiple steady-oscillatory bifurcation in a triply diffusive fluid layer. We consider two-dimensional motion confined between two infinite horizontal planes under periodic lateral boundary conditions that confer O(2) symmetry upon the system. Previous investigations with “ideal” (stress-free, im-penetrable) horizontal boundaries have revealed a degenerate cubic-order normal form for the oscillatory bifurcation. This degeneracy does not arise when more realistic horizontal boundary conditions are applied, where at least one of the horizontal planes is rigid, and there is no slip of the fluid at a rigid boundary. For such (non-ideal) problems the normal form coefficients must be computed numerically. We perform these calculations for two triply diffusive fluid systems: heat-KCL-sucrose and KClNaCl-sucrose. Numerical integrations of the normal form equations for the multiple instability compare well with theoretical predictions of their behaviour. For various ranges of parameter values we find stable steady states, mixed steady-standing wave states, travelling waves and modulated travelling waves. We also find that a stable three-frequency flow (already known to exist for the ideal problem) can undergo period-doubling cascades, and can coexist with a branch of stable modulated travelling waves.

Compressible magnetoconvection in three dimensions: planforms and nonlinear behaviour

Convection in a compressible fiuid with an imposed vertical magnetic field is studied numerically in a three-dimensional Cartesian geometry with periodic lateral boundary conditions. Attention is restricted to the mildly nonlinear regime, with parameters chosen first so that convection at onset is steady, and then so that it is oscillatory.Steady convection occurs in the form of two-dimensional rolls when the magnetic field is weak. These rolls can become unstable to a mean horizontal shear flow, which in two dimensions leads to a pulsating wave in which the direction of the mean flow reverses. In three dimensions a new pattern is found in which the alignment of the rolls and the shear flow alternates.If the magnetic field is sufficiently strong, squares or hexagons are stable at the onset of convection. Both the squares and the hexagons have an asymmetrical topology, with upflow in plumes and downflow in sheets. For the squares this involves a resonance between rolls aligned with the box and rolls aligned digonally to the box. The preference for three-dimensional flow when the field is strong is a consequence of the compressibility of the layer- for Boussinesq magnetoconvection rolls are always preferred over squares at onset.In the regime where convection is oscillatory, the preferred planform for moderate fields is found to be alternating rolls - standing waves in both horizontal directions which are out of phase. For stronger fields, both alternating rolls and two-dimensional travelling rolls are stable. As the amplitude of convection is increased, either by dcereasing the magnetic field strength or by increasing the temperature contrast, the regular planform structure seen at onset is soon destroyed by secondary instabilities.