Penetrative convection in rotating fluids: A model for the base of stellar convection zones

Abstract

To model penetrative convection at the base of a stellar convection zone we consider two plane parallel, co-rotating Boussinesq layers coupled at their fluid interface. The system is such that the upper layer is unstable to convection while the lower is stable. Following the method of Kondo and Unno (1982, 1983) we calculate critical Rayleigh numbers R_c for a wide class of parameters. Here, R_c is typically much less than in the case of a single layer, although the scaling R_c～T^2/3 as T → ∞ still holds, where T is the usual Taylor number. With parameters relevant to the Sun the helicity profile is discontinuous at the interface, and dominated by a large peak in a thin boundary layer beneath the convecting region. In reality the distribution is continuous, but the sharp transition associated with a rapid decline in the effective viscosity in the overshoot region is approximated by a discontinuity here. This source of helicity and its relation to an alpha effect in a mean-field dynamo is especially relevant since it is a generally held view that the overshoot region is the location of magnetic field generation in the Sun.     

The effect of stratification and compressibility on anelastic convection in a rotating plane layer

We study the effect of stratification and compressibility on the threshold of convection and the heat transfer by developed convection in the nonlinear regime in the presence of strong background rotation. We consider fluids both with constant thermal conductivity and constant thermal diffusivity. The fluid is confined between two horizontal planes with both boundaries being impermeable and stress-free. An asymptotic analysis is performed in the limits of weak compressibility of the medium and rapid rotation (τ^?1/12???|θ|???1, where τ is the Taylor number and θ is the dimensionless temperature jump across the fluid layer). We find that the properties of compressible convection differ significantly in the two cases considered. Analytically, the correction to the characteristic Rayleigh number resulting from small compressibility of the medium is positive in the case of constant thermal conductivity of the fluid and negative for constant thermal diffusivity. These results are compared with numerical solutions for arbitrary stratification. Furthermore, by generalizing the nonlinear theory of Julien and Knobloch [Fully nonlinear three-dimensional convection in a rapidly rotating layer. Phys. Fluids 1999, 11, 1469–1483] to include the effects of compressibility, we study the Nusselt number in both cases. In the weakly nonlinear regime we report an increase of efficiency of the heat transfer with the compressibility for fluids with constant thermal diffusivity, whereas if the conductivity is constant, the heat transfer by a compressible medium is more efficient than in the Boussinesq case only if the specific heat ratio γ is larger than two.     

The catastrophe structure of thermohaline convection in a two-dimensional fluid model and a comparison with low-order box models

Abstract

We impose a surface forcing on the 2D, Boussinesq, thermohaline equations in a rectangular domain, in the form of equatorially symmetric cosine distributions of salinity flux and temperature. This system may be seen as an idealization of the ocean thermohaline circulation on the global scale over intervals of centuries or millenia. Multiple steady states are found numerically. They reflect the competition between the opposite signs of the temperature and salinity-driven equatorially symmetric circulations. There are also pole-to-pole, equatorially asymmetric circulations. In the control space of the temperature and salinity-flux forcing amplitudes, these equilibria form two cusp catastrophes, and transitions between stable equilibria occur through several distinct bifurcations. These catastrophes can be reproduced in simple box models connecting stirred reservoirs through capillary pipes. This steady-state analysis may provide a framework for a better understanding of climatic transitions between different stable regimes of the ocean-atmosphere system.     

Convection in rotating annular channels heated from below. Part 2. Transitions from steady flow to turbulence

We report the results of fully three-dimensional numerical simulations of nonlinear convection in a Boussinesq fluid in an annular channel rotating about a vertical axis with lateral no-slip or stress-free sidewalls, stress-free top and bottom, uniformly heated from below, a problem first studied by Davies-Jones and Gilman (1971 Davies-Jones, RP and Gilman, PA. 1971. Convection in a rotating annulus uniformly heated from below.. J. Fluid Mech., 46: 65–81.  [Google Scholar]) and Gilman (1973 Gilman, PA. 1973. Convection in a rotating annulus uniformly heated from below. Part 2. Nonlinear results. J. Fluid Mech., 57: 381–400.  [Google Scholar]). A substantial range of the Rayleigh number R (R_c≤R≤O(100 R_c)), where R_c denotes the critical value at the onset of convection) is considered. It is found that the wall-localized convection mode, unaffected by the velocity boundary condition imposed on the sidewalls, is nonlinearly robust. Both directions of travelling waves, one propagating against the sense of rotation near the outer sidewall and the other propagating in the same sense as the rotation in the vicinity of the inner sidewall, are always present in the nonlinear solutions. In contrast to nonlinear convection in a rotating Bénard layer, neither convection rolls nor the Küpper–Lortz instability can exist in a rotating annular channel because of the effect of the sidewalls. It is the nonlinear interaction between the wall-localized modes and the internal mode that plays an essential role in determining the nonlinear properties of convection in a rotating annular channel. Our studies reveal systematically the various nonlinear phenomena, from steady travelling waves trapped in the vicinities of the sidewalls to convective turbulence exhibiting columnar structure.     

Instability of flows near the onset of convection in a rotating layer with stress-free horizontal boundaries

We investigate instability of convective flows of simple structure (rolls, standing and travelling waves) in a rotating layer with stress-free horizontal boundaries near the onset of convection. We show that the flows are always unstable to perturbations, which are linear combinations of large-scale modes and short-scale modes, whose wave numbers are close to those of the perturbed flows. Depending on asymptotic relations of small parameters α (the difference between the wave number of perturbed flows and the critical wave number for the onset of convection) and ε (ε² being the overcriticality and the perturbed flow amplitude being O(ε)), either small-angle or Eckhaus instability is prevailing. In the case of small-angle instability for rolls the largest growth rate scales as ε^8/5, in agreement with results of Cox and Matthews (Cox, S.M. and Matthews, P.C., Instability of rotating convection. J. Fluid. Mech., 2000, 403, 153–172) obtained for rolls with k = k _c . For waves, the largest growth rate is of the order ε^4/3. In the case of Eckhaus instability the growth rate is of the order of α².     

Convection in rotating annular channels heated from below. Part 1. Linear stability and weakly nonlinear mean flows

Convection in a Boussinesq fluid in an annular channel rotating about a vertical axis with lateral rigid sidewalls, stress-free top and bottom, uniformly heated from below is investigated. The sidewalls are assumed to be either perfectly insulating or conducting. Three different types of convection are identified when the channel is rotating sufficiently fast: (i) global oscillatory convection preferred for small Prandtl numbers in channels with intermediate or large aspect ratios (width to height ratio), (ii) wall-localized oscillatory convection representing the most unstable mode for moderate or large Prandtl numbers in channels with intermediate or large aspect ratios and (iii) global stationary convection preferred in channels with sufficiently small aspect ratios regardless of the size of the Prandtl number. The corresponding weakly nonlinear problem describing differential rotation and meridional circulation is also examined, showing that geostrophic, multiple-peaked (two prograde and two retrograde) differential rotation can be maintained by the Reynolds stresses in wall-localized convective eddies in a rapidly rotating channel.     

Stratified convection with multiple states

A simplified box model of the cooling of a salt-stratified ocean is analyzed analytically and numerically. A large isothermal basin of salt water has a layer of fresh water at the surface. Beside this is a small basin, cooled from above and connected to the large basin by horizontal tubes at the top, middle and bottom. For small cooling rate, fresh water enters the small basin, is cooled and leaves through the middle tube. For greater cooling rate, the fresh water leaves the small basin through the middle and bottom tube. If the top tube is smaller than the deeper tubes and the fresh water layer is sufficiently shallow, flow in the middle tube reverses at a critical cooling rate. In this case, a mixture of salt and fresh water is cooled and leaves the bottom tube. Increased cooling produces much greater flow rate; consequentially temperature increases rather than decreases in the small basin. A relaxation heat flow condition results in multiple equilibria. One of the stable modes has fresh surface water descending in the small basin and flowing out through the middle and bottom tube. The other has a greater rate of flow of both fresh and salty water (through the middle tube) into the basin with the flow of mixed salty water out of the bottom tube. Implications for deep convection in the ocean are discussed.     

Water Vapor and Cloud Radiative Forcings over the Pacific Ocean Simulated by the LASG/IAP AGCM: Sensitivity to Convection Schemes

Characteristics of the total clear-sky greenhouse effect (GA) and cloud radiative forcings (CRFs), along with the radiative-related water vapor and cloud properties simulated by the Spectral Atmospheric Model developed by LASGIAP (SAMIL) are evaluated. Impacts of the convection scheme on the simulation of CRFs are discussed by using two AMIP (Atmospheric Model Inter-comparison Project) type simulations employing different convection schemes: the new Zhang-McFarlane (NZH) and Tiedtke (TDK) convection schemes. It shows that both the climatological GA and its response to El Nio warming are simulated well, both in terms of spatial pattern and magnitude. The impact of the convection scheme on GA is not significant. The climatological longwave CRF (LWCRF) and its response to El Nio warming are simulated well, but with a prominently weaker magnitude. The simulation of the climatology (response) of LWCRF in the NZH (TDK) run is slightly more realistic than in the TDK (NZH) simulation, indicating significant impacts of the convection scheme. The shortwave CRF (SWCRF) shows large biases in both spatial pattern and magnitude, and the results from the TDK run are better than those from the NZH run. A spuriously excessive negative climatological SWCRF over the southeastern Pacific and an insufficient response of SWCRF to El Nio warming over the tropical Pacific are seen in the NZH run. These two biases are alleviated in the TDK run, since it produces vigorous convection, which is related to the low threshold for convection to take place. Also, impacts of the convection scheme on the cloud profile are discussed.     

热带海温异常对东南亚夏季风爆发的影响

通过季风指数Im定义了能表征东南亚地区降水实况的东南亚夏季风指数,根据东南亚夏季风指数测算出东南亚夏季风爆发的平均时间为5月7日.利用东南亚夏季风指数分析热带海温场及垂直速度场的变化后发现,在东南亚夏季风爆发的前期秋、冬季节,中东太平洋地区以及中西印度洋地区的冷海温有利于东南亚地区夏季风的提前爆发.当中东太平洋地区是冷(暖)海温时,对应着纬向的Walker环流及季风环流圈强(弱),东南亚地区的对流也强(弱),则东南亚地区夏季风爆发早(迟).