The influence of the thermal diffusivity of the lower boundary on eddy motion in convection

The paper presents new concepts and results for the eddy structure of turbulent convection in a horizontal fluid layer of depth h which lies above a solid base with thickness hb. The fluid parameters are the kinematic viscosity ν, the thermal diffusivity κ, which is taken to be comparable with ν, the density ρ, the specific heat cp and the expansion parameter β. The thermal diffusivity of the solid is κb. The results are an extension of the more commonly studied cases, where a constant heat flux or constant temperature is applied at the interface between the fluid and the base. The buoyancy forces induce eddy motions with a typical velocity w∗∼(gβFθh)1/3 where ρcpFθ is the average heat flux and Fθ the covariance of the fluctuations of the temperature and of the vertical velocity. At moderate Reynolds numbers (Re=w∗h/ν), say less than about 103, an order-of-magnitude analysis shows that for the case of high diffusivity of the base (i.e. κb≫κ) elongated ‘plumes’ form at the surface and extend to the top of the fluid layer. When the base diffusivity is low (i.e. κb≤κ) the surface cools below the developing ‘plume’ and either the plume breaks up into elongated puffs or, if κb≪κ, horizontal pressure gradients form so that only small-scale puffs can form near the surface. At very high Reynolds numbers, approximately greater than 104, the surface boundary layer below each puff/plume is highly turbulent with a local logarithmic velocity and temperature profile. An approximate analysis indicates for this case that there is insufficient buoyancy flux from the base, irrespective of its diffusivity, to maintain plumes, because of the high turbulent heat transfer. So puffs dominate high-Reynolds-number thermal convection as numerical simulations and field experiments demonstrate. However, when the surface heat flux is uniform, for example as a result of radiant heat transfer or by forcing with a constant heat flux below a very thin conducting base, plumes are the dominant form of eddy motion, as is commonly observed. In the numerical solutions presented here, where Re∼3×102 and the slab thickness hb=h, it is shown that the spatial scales of eddy structures in the fluid layer close to the surface become significantly smaller as κb/κ is reduced from 100 to 0.1. At the same time in the core of the convective layer the change in the autocorrelation and spatial correlation function indicates that there is a transition from long-duration plumes into shorter-duration and smaller-length-scale elongated puffs. The simulations show that the largest temperature fluctuations near the surface occur when a constant heat flux is applied at the bottom of the fluid layer. The smallest temperature fluctuations are associated with the constant-temperature boundary condition. The finite base diffusivity cases lie in between these limits, with the largest fluctuations occurring when the thermal diffusivity of the base is small. The hypothesis introduced above has been tested qualitatively in a laboratory set-up when the effective diffusivity of the base was varied. The flow structure was observed as it changed from being characterized by nearly steady plumes, into unsteady plumes and finally into puffs when the thickness of the conducting base was first increased and then the diffusivity was decreased

Effects of rotation and sloping terrain on fronts of density current fronts

The initial stage of the adjustment of a gravity current to the effects of rotation with angular velocity f/2 is analysed using a short time analysis where Coriolis forces are initiated in an inviscid von Kármán–Benjamin gravity current front at tF=0. It is shown how, on a time-scale of order 1/f, as a result of ageostrophic dynamics, the slope and front speed UF are much reduced from their initial values, while the transverse anticyclonic velocity parallel to the front increases from zero to O(NH0), where N=g′/H0−−−−−√ is the buoyancy frequency, and g′=gΔρ/ρ0 is the reduced acceleration due to gravity. Here ρ0 is the density and Δρ and H0 are the density difference and initial height of the current. Extending the steady-state theory to account for the effect of the slope σ on the bottom boundary shows that, without rotation, UF has a maximum value for σ=\upi/6, while with rotation, UF tends to zero on any slope. For the asymptotic stage when ftF≫1, the theory of unsteady waves on the current is reviewed using nonlinear shallow-water equations and the van der Pol averaging method. Their motions naturally split into a ‘balanced’ component satisfying the Margules geostrophic relation and an equally large ‘unbalanced’ component, in which there is horizontal divergence and ageostrophic vorticity. The latter is responsible for nonlinear oscillations in the current on a time scale f−1, which have been observed in the atmosphere and field experiments. Their magnitude is mainly determined by the initial potential energy in relation to that of the current and is proportional to the ratio \it Bu−−−−−√=LR/R0, where LR=NH0/f is the Rossby deformation radius and R0 is the initial radius. The effect of slope friction also prevents the formation of a steady front. From the analysis it is concluded that a weak mean radial flow must be driven by the ageostrophic oscillations, preventing the mean front speed UF from halting sharply at ftF∼1. Depending on the initial value of LR/R0, physical arguments show that UF decreases slowly in proportion to (ftF)−1/2, i.e. UF/UF0=F(ftF,\it Bu). Thus the front only tends to the geostrophic asymptotic state of zero radial velocity very slowly (i.e. as ftF→∞) for finite values of LR/R0. However, as LR/R0→0, it reaches this state when ftF∼1. This analysis of the overall nonlinear behaviour of the gravity current is consistent with two two-dimensional non-hydrostatic (Navier–Stokes) and axisymmetric hydrostatic (shallow-water) Eulerian numerical simulations of the varying form of the rotating gravity current. When the effect of surface friction is considered, it is found that the mean movement of the front is significantly slowed. Furthermore, the oscillations with angular frequency f and the slow growth of the radius, when ftF≥1, are consistent with recent experiments

Assessment of Planetary Boundary-Layer Schemes in the Weather Research and Forecasting Mesoscale Model Using MATERHORN Field Data

The study was aimed at understanding the deficiencies of numerical mesoscale models by comparing predictions with a new high-resolution meteorological dataset collected during the Mountain Terrain Atmospheric Modelling and Observations (MATERHORN) Program. The simulations focussed on the stable boundary layer (SBL), the predictions of which continue to be challenging. High resolution numerical simulations (0.5-km horizontal grid size) were conducted to investigate the efficacy of six planetary boundary-layer (PBL) parametrizations available in the advanced research version of the Weather Research and Forecasting model. One of the commonly used PBL schemes was modified to include eddy diffusivities that account for enhanced momentum transport compared to heat transport in the SBL, representing internal wave dynamics. All of the tested PBL schemes, including the modified scheme, showed a positive surface temperature bias. None of the PBL schemes was found to be superior in predicting the vertical wind and temperature profiles over the lowest 500 m, however two of the schemes appeared superior in capturing the lower PBL structure. The lowest model layers appear to have a significant impact on the predictions aloft. Regions of sporadic flow interactions delineated by the MATERHORN observations were poorly predicted, given such interactions are not represented in typical PBL schemes