Convection-driven zonal flows in the major planets

The major planets produce heat flux from their interiors that is comparable to the radiative flux they receive from the sun. The dynamics of convection flows carrying the heat flux are discussed, and the dominating effect of the Coriolis force is demonstrated. The characteristic high-velocity jets in the atmospheres of Jupiter and Saturn can be explained on the basis of Reynolds stresses generated by the fluctuating convective motions. A simple annulus model, which elucidates the more complex mathematical analysis of the spherical case given in an earlier paper (Busse, 1983), is considered in detail. Various aspects of the observational evidence are discussed in relation to the model.     

Was there ice along the shore of the Sea of Galilee during the last 12,000?—Reply to a comment by Prange et al. (2007) and a comment by Friedman (2007)

Prange et al. (2007) question our reconstruction of the Sea of Galilee (Lake Kinneret) paleoclimate and argue that a “careful” analysis of the paleoclimatic analysis leads to much smaller cooling events than we have considered. By and large, their “careful” analysis is based on correlating the paleotemperatures of the Lake with those of the northern Red-Sea that (geographically) is much closer to the Lake than the two Mediterranean cores that we used. Ironically, their argument contradicts Friedman’s (2007) statements (the second comment on our original paper), which are based on still-closer cores and support our larger cooling choices. This issue alone would be enough to dismiss the uniqueness of PAL argument but there is another issue with their work that we wish to comment on. In support of their own small cooling argument, PAL present winter correlation maps that indeed show a stronger correlation of the northern Red-Sea SST to the lake SST than the correlation of Mediterranean SST with the lake SST. This seemingly correct correlation argument of PAL is totally false (for both daily and millennial time scales) because it has no climatological basis. On the daily time scale, all the storms that reach the Lake originate in the Mediterranean Sea (to the west of the lake), not the Red-Sea (which lies 700 km south of the lake). Also, although the lake and the Red-Sea are only 700 kilometers apart, their climates are very different because they are subject to two totally different air masses. While the climate of the Red-Sea region is desert-like, the climate of the region surrounding the lake is a typical wet Mediterranean climate. Seasonal correlation maps (and even monthly maps) such as those presented by the authors filter out the storms that control the winter climate in the lake region because these storms occur on a daily scale. With this filtering, all that one is left with is the low frequency first baroclinic mode, which merely reflects the Rossby radius scale (measured from the lake). On the millennial time scale, cold events in the lake regions (from an earlier period) have been attributed to Bond cycles and Heinrich events both of which are global and not local processes. As such, they are probably forced by variability in the solar radiation rather than a local process implied by PAL. Overall, all that the PAL correlation shows for both daily and millennial time scales is that changes in the temperature in the Red-Sea occur at the same time as they do in the Lake. But this does not say anything about the dynamics in question and does not imply that it is better to use records from the Red-Sea (which does not lie within the path of the zonal winds reaching the Lake). Neglecting this issue (as proposed by PAL) distorts the physics and reminds us of the classical statistical example for the limitations involved in the interpretation of correlation—the incidence of lung cancer is strongly correlated with the incidence of carrying matches in ones pocket even though the matches do not cause the cancer and the cancer does not force one to carry matches.     

Is There a Paleolimnological Explanation for ‘Walking on Water’ in the Sea of Galilee?

Lake Kinneret (the Sea of Galilee) is a small freshwater lake (148 km² and a mean depth of 20 m) situated in northern Israel. Throughout recent history there have been no known records of a total ice formation on its top. Furthermore, given that convection requires an initial cooling of the entire lake down to 4 °C, it is difficult to imagine how such a low-latitude lake, presently subject to two-digit temperatures during the winter, could ever freeze. Lake Kinneret is, however, unique in the sense that there are dense (warm and salty) springs along its western shore. The dynamics of the regions adjacent to these springs are investigated using a one-dimensional nonlinear analytical ice model, a paleoceanographic record of the sea surface temperature of the Mediterranean Sea, and a statistical model. We show that, because the water directly above the plume created by the salty springs does not convect when it is cooled down to 4 °C, freezing of the region directly above the salty springs was possible during periods when the climate in the region was somewhat cooler than it is today. We refer to this localized freezing situation as ‘springs ice’. The analytical ice-model involves a slowly varying approach where the ice is part of a thin fresh and cold layer floating on top of the salty and warm spring water below. During the ice formation process, the ice is cooled by the atmosphere above and warmed by the spring water below. The plumes created by the springs have a length scale of 30 m, and it is argued that, during the Younger Dryas when the air temperature in the region was probably 7 °C or more cooler than today, ‘springs ice’ (thick enough to support human weight) was formed once every 27 years or less. During the cold events 1500 and 2500 years ago (when the atmospheric temperature was 3 °C or more lower than today) springs ice occurred about once in 160 years or less. Since the duration of these cold events is of the same order as the springs ice recurrence time, there is a substantial chance that at least one springs ice occurred during these cooler periods. With today's climate, the likelihood of a springs ice is virtually zero (i.e., once in more than 10,000 years). One set of those springs associated with the freezing is situated in Tabgha, an area where many archeological features associated with Jesus Christ have been found. On this basis, it is proposed that the unusual local freezing process might have provided an origin to the story that Christ walked on water. Since the springs ice is relatively small, a person standing or walking on it may appear to an observer situated some distance away to be ‘walking on water’. This is particularly true if it rained after the ice was formed (because rain smoothes out the ice’s surface). Whether this happened or not is an issue for religion scholars, archeologists, anthropologists, and believers to decide on. As natural scientists, we merely point out that unique freezing processes probably happened in that region several times during the last 12,000 years.     

A multi-model study of the restratification phase in an idealized convection basin

The representation of baroclinic instability in numerical models depends strongly upon the model physics and significant differences may be found depending on the vertical discretization of the governing dynamical equations. This dependency is explored in the context of the restratification of an idealized convective basin with no external forcing. A comparison is made between an isopycnic model including a mixed layer (the Miami Isopycnic Coordinate Ocean Model, MICOM), its adiabatic version (MICOM-ADIAB) in which the mixed layer physics are removed and the convective layer is described by a deep adiabatic layer outcropping at the surface instead of a thick dense mixed layer, and a z-coordinate model (OPA model).In the absence of a buoyancy source at the surface, the mixed layer geometry in MICOM prevents almost any retreat of this layer. As a result, lateral heat exchanges in the upper layers are limited while mass transfers across the outer boundary of the deep convective mixed layer result in an unrealistic outward spreading of this layer. Such a widespread deep mixed layer maintains a low level of baroclinic instability, and therefore limits lateral heat exchanges in the upper layers over most of the model domain. The behavior of the adiabatic isopycnic model and z-coordinate model is by far more satisfactory although contrasted features can be observed between the two simulations. In MICOM-ADIAB, the more baroclinic dynamics introduce a stronger contrast between the surface and the dense waters in the eddy kinetic energy and heat flux distributions. Better preservation of the density contrasts around the dense water patch maintains more persistent baroclinic instability, essentially associated with the process of dense water spreading. The OPA simulation shows a faster growth of the eddy kinetic energy in the early stages of the restratification which is attributed to more efficient baroclinic instability and leads to the most rapid buoyancy restoring in the convective area among the three simulations. Dense water spreading and warm surface capping occur on fairly similar time scales in MICOM-ADIAB although the former is more persistent that the latter. In this model, heat is mainly transported by anticyclonic eddies in the dense layer while both cyclonic and anticyclonic eddies are involved in the upper layers. In OPA, heat is mainly brought into the convective zone through the export of cold water trapped in cyclonic eddies with a strong barotropic structure. Probably the most interesting difference between the z-coordinate and the adiabatic isopycnic model is found in the temperature distribution ultimately produced by the restratification process. OPA generates a spurious volume of intermediate water which is not seen in MICOM-ADIAB where the volume of the dense water is preserved.     

The Influence of the Prandtl Number on the Style of Vigorous Thermal Convection

The effect of the Prandtl number on convection in a planar three-dimensional geometry is investigated in this study. We have employed a numerical scheme to integrate the governing equations. Differently from previous studies we have chosen stress-free boundaries. Experiments have been performed at a Rayleigh number of Ra = 10 6 for Prandtl numbers (Pr) ranging from 0.025 to 100. We have further conducted one experiment in the limiting case of infinite Prandtl number. Despite the differences in the geometry and the boundary conditions, as compared to other studies, we find a similar transition in the dynamics of the flow when the Prandtl number is increased. While the velocity and the temperature structure show diffusive character at low Pr, sharp thermal boundary layers form at high Pr. The heat transport efficiency increases with Pr until a transition value is reached, from there on Nu behaves almost asymptotically. The transition can not be caused by a change in hierarchies between velocity and thermal boundary layers, as suggested in other studies. Due to the stress-free boundaries, a velocity boundary layer does not exist. We observe that the toroidal part of the flow is strong at low Pr and looses its strength with increasing Pr, thus it is likely to be responsible for the transition. In a further chapter we demonstrate that due to the neglect of the toroidal part in two-dimensional calculations at low Pr results are obtained which are misleading, even in a qualitative sense. Infinite Pr results from 2D calculations closely resemble the dynamics of fully 3D flows.     

A Simple Mixing Length Formulation for the Eddy-Diffusivity Parameterization of Dry Convection

In this paper a simple mixing length formulation for the eddy-diffusivityparameterization of dry convection is suggested. The new formulation relates the mixinglength to the square root of the turbulent kinetic energy (e) and a time scale ( ):l = e. To close the parameterization the time scale is calculated as a functionof the boundary-layer height (h) and the convective velocity scale (w_*), h/w_*. Thesimpler approach of a constant time scale is also studied. The simulation of a case of dry atmosphericconvection with a one-dimensional boundary-layer model shows that the model with the new formulationreproduces quite well the main properties of the convective boundary layer. In particular,the entrainment is realistically represented by the new mixing length, which has the advantage of naturallydecreasing with the turbulent kinetic energy. Sensitivity studies to the surface flux and the lapserate, in the context of a simplified situation, show the robustness of the new formulation.     

On thermonuclear convection: I shellular instability

Abstract

As the sun evolves, a sharp compositional peak of He ³ builds up in the core. Nuclear reactions involving He ³ are very temperature sensitive, as a result, this He ³ layer is susceptible to thermal instability. The small horizontal wavenumber g-modes have large time scales, comparable to the thermal time scale. Using a two-layer model, we find that such “shellular modes” are the most unstable. As a result of nuclear heating, these modes may be excited in the solar core in a shallow layer confined to the He ³ zone. A possible effect of such shellular convection on the solar neutrino problem is discussed. In this paper we discuss the linear theory; the nonlinear effects will be treated in a subsequent paper.     

地幔对流的实验研究:非立柱状幔柱和地幔涡旋

地幔对流的物理模拟实验结果表明 ,在地幔介质和温度非均匀分布的复杂条件下 ,热卷流 (地幔柱 )往往由立柱状转变为非立柱状 (含斜柱状、涡旋状等 )。在忽略科里奥利力的情况下 ,板块的下插和滞积下沉、岩石圈根的存在以及地幔介质粘度的非均匀分布等都可能构成不同形状的障碍 -导流体 ,导致地幔的涡旋运动。软流圈中的水平涡旋环带属于对数螺线型 ,环带旋转半径及线速度逐渐减小 ,最终在旋转中心处下沉 ,而旋转角速度大致保持恒定     

Convective Instability of a Mushy Layer - I: Uniform Permeability

Mushy layers arise and are significant in a number of geophysical contexts, including freezing of sea ice, solidification of magma chambers and inner-core solidification. A mushy layer is a region of solid and liquid in phase equilibrium which commonly forms between the liquid and solid regions of a solidifying system composed of two or more constituents. We consider the convective instability of a plane mushy layer which advances steadily upwards as heat is withdrawn at a uniform rate from the bottom of a eutectic binary alloy. The solid which forms is assumed to be composed entirely of the denser constituent, making the residual liquid within the mush compositionally buoyant and thus prone to convective motion. In this article we focus on the large-scale mush mode of instability, arguing that the 'boundary-layer' mode is not amenable to the standard stability analysis, because convective motions occur on that scale for any non-zero value of the Rayleigh number. We quantify the minimum critical Rayleigh number and determine the structure of the convective modes of motion within the mush and the associated deflections of the mush-melt and mush-solid boundaries. This study of convective perturbations differs from previous analyses in two ways; the inhibition of motion and deformation of the mush-melt interface by the stable stratification of the overlying melt is properly quantified and deformation of the mush-solid interface is permitted and quantified. We find that the mush-melt interface is almost unaffected by convection while significant deformation of the mush-solid interface occurs. We show that each of these effects causes significant (unit-order) changes in the predicted critical Rayleigh number. The marginal modes depend on three dimensionless parameters: a scaled eutectic temperature, τ _e (which characterizes the eutectic temperature relative to the depression of the liquidus), a scaled superheat, τ_∞ (which measures the amount by which the temperature of the incoming melt exceeds the liquidus temperature) and the Stefan number, S (which measures the latent heat of crystallization). To survey parameter space, we focus on seven cases, a standard case having S = τ_∞ = τ _e = 1, and six others in which one of the parameters is either large or small compared with unity: a nearly pure case (τ _e = 100; having little of the light constituent), the large superheat limit (τ_∞→ ∞), a case of large latent heat (S = 100), the near eutectic limit (τ _e → 0), a case of small superheat (τ_∞ = 0.01) and the case of zero latent heat (S = 0). The critical Rayleigh number and the associated wavelength of the convection pattern are determined in each case. The eigenvector for each case is presented in terms of the streamlines and the isolines of the perturbation temperature and solid fraction.     

Convection in rotating annular channels heated from below. Part 3: Experimental boundary conditions

Results are presented from both linear stability analysis and numerical simulations of three-dimensional nonlinear convection in a Boussinesq fluid in an annular channel, under experimental boundary conditions, rotating about a vertical axis uniformly heated from below. The focus is placed on the Prandtl number Pr = 7.0, representing liquid water at room temperature. The linear analysis shows that, when the aspect ratio is sufficiently small, there exists only one stationary mode that occupies the whole fluid container. When the aspect ratio is moderate or large, however, there exist three different linear solutions: (i) the outer sidewall-localized traveling wave propagating against the sense of rotation; (ii) the inner sidewall-localized traveling wave propagating in the same sense as rotation; and (iii) both the counter-traveling waves occurring simultaneously. Guided by the result of the linear stability analysis, fully three-dimensional simulations are then performed for a channel with a moderate aspect ratio. It is found that neither the prograde nor the retrograde mode is physically realizable near threshold and beyond. The dynamics of nonlinear convection in a rotating channel are chiefly characterized by the interaction between the sidewall-localized waves and the interior convection cells/rolls, producing an interesting and unusual nonlinear phenomenon. In order to compare with the classical Rayleigh–Bénard problem without vertical sidewalls, we also study linear and nonlinear convection at exactly the same parameters but in an infinitely extended layer with periodic horizontal conditions. This reveals that both the linear instability and nonlinear convection in a rotating channel are characteristically different from those in a rotating layer with periodic horizontal conditions.