Convection in a rotating annulus having sidewalls of height-dependent thermal conductance

Abstract

Thermal convection in a vertically-mounted, rotating annulus of a particular design proposed by Davies and Walin (1977) is investigated. The annulus used in the present study differs from the conventional type in some important aspects: the sidewalls are finitely conducting, and the thermal conductance of the sidewalls is height-dependent. The theoretical model due to Davies and Walin is briefly recounted. The present study aims to verify the theoretical model; we have acquired numerical solutions to the governing Navier-Stokes equations. The numerical results are supportive of the theoretical contentions. The near-linear dependence of the isothermal slope on the parameter D, which is a function of Ω and ΔT, is corroborated within reasonable limits. New data on the vertical and radial structures of the meridional and azimuthal flows are presented. The numerical results also confirm that the shape of the sidewall thickness has a substantial influence on the meridional flow patterns. In the bulk of the interior flow field, the dominant azimuthal flow field and the temperature field are linked by the thermal wind relation.     

Flow of a stratified fluid bounded by walls of finite thermal conductance

Abstract

A study is made of the behavior of a thermally stratified fluid in a container when the non-horizontal boundaries have finite thermal conductance. The theory of Rahm and Walin is briefly recounted. Numerical solutions to the Navier-Stokes equations for a Boussinesq fluid in a cylinder, adopting a Newtonian heat flux condition at the vertical sidewall, are presented. Results on the details of flow and temperature fields are given over ranges of the Rayleigh number Ra, the container aspect ratio H, and the sidewall conductance S. As S increases, the isotherms in the meridional plane are horizontal at small radii but they diverge at large radii. This creates temperature nonuniformilies in the horizontal direction, and convective motions result. The salient features of the interior temperature profiles are captured by the theoretical model. The velocity field is characterized by two oppositely-directed circulations. As Ra or S varies, the qualitative circulation patterns remain substantially unchanged, but the magnitudes of the convective flows differ by large amounts. The effects of the externally-imposed parameters on the flow and temperature structures are examined.     

Flows within the Ekman layer during spin-up of a thermally stratified fluid

Abstract

Finite-difference numerical solutions were obtained to present the flow and temperature field details within the transient Ekman layer during spin-up of a thermally stratified fluid in a cylinder. This complements the earlier studies on stratified spin-up which examined the flows in the interior core region. As the stratification increases, the following changes in the flow field are noticeable. The radial velocity in the Ekman layer decreases in magnitude. The azimuthal flows adjust smoothly from the interior region to the endwall boundary, and the Ekman layer in the azimuthal flow field fades. Vertical motions are inhibited, resulting in a weakened Ekman pumping. The axial vorticity field behaves similarly to the azimuthal flows. The temperature deviation from the equilibrium profile decreases, and the heat transfer flux from the endwall to the fluid decreases. The thickness of the thermal layer is larger than the velocity layer thickness. Illustrative comparisons of the relative sizes of the terms in the governing equations are conducted in order to assess the stratification effect in the adjustment process of the fluid.     

Flow of a double-diffusive stratified fluid in a differentially-rotating cylinder

Abstract

A study has been made of a basic state of axisymmetric flow, at large rotational Reynolds numbers, in a double-diffusive stratified fluid contained in a vertically-mounted, differentially-rotating cylindrical cavity. The aim is to describe the qualitative characteristics of the flow of a fluid, the density of which is stratified by two diffusive effects, i.e., temperature and salinity gradients. Attention is confined to situations in which the temperature and salinity gradients make opposing contributions to the overall density profile, the undisturbed stratification being gravitationally stable. Finite difference numerical solutions of the governing Navier-Stokes equations have been obtained using the Boussinesq approximation. The results are presented in a way that illustrates the explicit effects of double-diffusivity when the cavity aspect ratio, height/radius, is O(1). The principal non-dimensional parameters characterizing the flow field are identified. In the interior core, the primary dynamic balance is between the horizontal density gradient and the vertical shear of the prevailing azimuthal velocity. The effective stratification is seen to decrease as the double-diffusivity increases, even if the overall stratification parameter, St, is held constant. The solute field contains a very thin boundary layer structure at large Lewis numbers. The effective stratification increases with the Prandtl number. Results have been derived for extreme values of the cavity aspect ratio. For small cavity aspect ratios, the dominant dynamic ingredients are viscous diffusion and rotation. For large aspect ratios, the bulk of the flow field is determined by the rotating sidewall. In this case, the direct influence of the double-diffusivity is minor.     

Comment on rotating hydraulics

Abstract

Numerical solutions of the axisymmetric flows during the relatively early phase of spin-up from rest of a stratified fluid in a cylinder are presented. Detailed results are given for a cylinder of aspect ratio of O(l) and for a minute Ekman number, showing axisymmetric spin-up for three values of the stratification parameter. As the stratification increases, the meridional circulation is confined to a region closer to the Ekman layers. An axisymmetric shear wave propagates radially inward from the sidewall, but, unlike the strictly vertical front for a homogeneous fluid, the interface which separates rotating from nonrotating fluid is bow-shaped. For a stratified fluid, the axial vorticity distribution is nonuniform both in the vertical and in the radial directions. With increasing stratification, diffusive vorticity production near the sidewall is more pronounced. Axisymmetric flows in the early phase of spin-up of a stratified fluid are controlled by both the inviscid dynamic effect and the viscous diffusion effect. At a location close to the Ekman layers, the inviscid effect outweighs the viscous effect, in much the same way as in a homogeneous fluid. However, at a location close to mid-depth, the viscous diffusion effect, enhanced by substantial flow gradients in that region, is dominant. This points to the necessity of including the direct effect of viscous diffusion in the interior in formulating an analytical model of stratified spin-up problems.     

Buoyant boundary layer flows- an analogy to the Ekman spiral

Abstract

Flow details inside the buoyant boundary layer in the heat-up process of a contained, stably stratified, fluid are presented. Numerical solutions were obtained for the heatup problem in a cylinder considered by Sakurai and Matsuda (1972). By plotting the scaled vertical velocity W versus the scaled temperature θ as functions of the normal distance from the sidewall, the precise shape of the buoyant layer spiral is constructed. The analogy between this spiral and the Ekman spiral in rotating fluids is apparent. As the Rayleigh number Ra increases, the magnitude of the scaled vertical velocity increases substantially, but the scaled temperature does not vary appreciably. The buoyant layer thickness is determined by measuring the zero-crossing normal distance for the vertical velocity. The buoyant layer suction increases significantly as Ra increases. The effects of vertical level and of time on the qualitative behavior of buoyant layer flows are found to be small. The buoyant layer flows decay over the heat-up time scale t _n ; t _h characterizes the time span over which the overall adjustment process in the inviscid interior region is accomplished. This work clarifies that the analogy between heat-up and spin-up, which has been known to exist in the main body of inviscid fluid, applies equally well to the boundary layer regions.     

The criterion of dynamic stability of zonal motion including the vertical shear

Summary Many writers treated on the problem of dynamic instability of westerly flow due to the excessive horizontal shear, and the present author discusses the corresponding dynamic instability due to the vertical shear. The critical vertical shear in indifferent stratification is given by the condition — the meridional component of absolute vorticity vanishes, — and is an approximate negative valueof 10^–4 sec ^–1 in middle latitude. However the critical vertical shear in normal stable stratification is a fairly large negative value of 2 sec^–1. It might be emphasized that the problem of this study differs fromRichardson's criterion of turbulence, for the present author discusses the condition under which the zonal flow is dynamically stable, whileRichardson expressed the condition under which the turbulence will decrease.     

Experimental investigation of stratorotational instability using a thermally stratified system: instability,waves and associated momentum flux

Stratorotational instability (SRI) has been proposed as a mechanism for outward angular momentum transport in Keplerian accretion disks. A particular designed Taylor–Couette laboratory experiment with axial stratification is suitable for studying the instability. Bottom endplate is cooled and top endplate is heated to achieve axial stratification. Due to constructive constraints, endplates are visually unamenable and quantitative measurement techniques in the co-rotating frame can only be done by looking through the outer cylinder. For this purpose, we built a co-rotating mini-PIV (Particle Image Velocimetry) system with a camera having a tilted viewing angle regarding the horizontal laser sheet. The aim of this study is (i) to quantify the uncertainty of the mini-PIV together with the used calibration technique and (ii) to compare experimental findings on SRI with theoretical predictions.

We perform measurements of the azimuthal and radial component of the velocity in axial stably stratified Taylor–Couette flows, consider velocity profiles and do frequency-filtering and flow decomposition. The absolute error of the mini-PIV system is 2% and we realised that stratified Taylor–Couette flows have smaller Ekman endwall effects than homogeneous ones. Still, Ekman pumping has an impact of the flow and might be responsible for differences between the data and theoretical models ignoring the endwalls. Here we focus on the flow structure during transition to SRI, the drift rate of SRI modes and the radial momentum flux as a function of the Reynolds number. Whereas the structure in form of trapped boundary Kelvin modes and the drift rate corresponds well with earlier predictions, the momentum flux shows a nonlinear dependency with respect to the Reynolds number. Away from the region of transition, theoretical models show a linear relationship. Several possible reasons for the mismatch between the experimental and theoretical models are discussed. Most important, we experimentally demonstrated that in the Rayleigh stable flow regime the SRI can provide a significant amount of outward momentum flux which makes this instability interesting in the context of accretion disks and also of atmospheric vortices where rotation and stratification also play a significant role.     

Form-drag instability in a barotropic and a baroclinic atmosphere

We discuss the form-drag instability for a quasi-geostrophic channel flow. We first study the characteristics of this instability in a barotropic flow, considering in detail the influence of the meridional scale and discussing which structure of the perturbation zonal flow must be chosen in order to describe properly this instability.We then consider a continuous quasi-geostrophic channel model in which the topography enters only through the bottom boundary condition, and we discuss how in this case the effects of the form-drag are felt by the mean zonal flow through the ageostrophic mean meridional circulation. Because the meridional structure of the perturbation zonal flow cannot simply be extended from the barotropic to the continuous case, we show how to modify it properly.We then study the baroclinic model in the particular case of constant (in the vertical) basic-state zonal flow and show how this case closely resembles the barotropic, demonstrating the barotropic nature of the form-drag instability.Symbols _t is the partial derivative with respect tot. - _x is the partial derivative with respect tox. - _y is the partial derivative with respect toy. - represents the geostrophic stream function. - u is the eastward component of the geostrophic wind. - v is the northward component of the geostrophic wind. - u _a is the eastward component of the ageostrophic wind. - v _a is the northward component of the ageostrophic wind. - w is the vertical component of the wind. - f is the Coriolis parameter=2 sin f _o+y. - f _o is the Coriolis parameter evaluated at mid-latitude. - N is the Brunt-Vaisala frequency. - [A] is the zonal (x) average ofA at constantp andy. - <A> is the horizontal (x andy) average ofA at constantp     

The degree of metamorphism of organic matter in sedimentary rocks as a paleo-geothermometer,applied to the Upper Rhine Graben

The subsidence of sedimentary layers implies increasing temperature downwards within the sedimentary column, so that the degree of coalification of organic matter increases continually. Apart from temperature, the slowly reacting chemical compounds of the organic matter strongly depend on time, too.It is shown that the coal rank is proportional to the integral of temperature and time of burial (t) for the Tertiary sedimentary rocks of the Upper Rhine Graben. This relationship is used to calculate paleogeothermal gradients (gradT) for some boreholes in the Upper Rhine Graben, from which the rate of burial during geological history (z(t)) is known. The degree of coalification is measured by its mean optical reflectivity (R _m), so that the relationship between coalification and geothermal history isR _m ² gradT z(t) dt.The results show high heat flow during Lower Tertiary and a decrease during Upper Tertiary at some locations of the Upper Rhine Graben. The recent high heat flow is not detectable in coalification. The young thermal anomaly is perhaps caused by ascending pore fluid and/or by heat conduction from a heat source in the lower crust.     

Thermal and magnetic instabilities in a rapidly rotating fluid sphere

Abstract

A linear analysis is used to study the stability of a rapidly rotating, electrically-conducting, self-gravitating fluid sphere of radius r ₀, containing a uniform distribution of heat sources and under the influence of an azimuthal magnetic field whose strength is proportional to the distance from the rotation axis. The Lorentz force is of a magnitude comparable with that of the Coriolis force and so convective motions are fully three-dimensional, filling the entire sphere. We are primarily interested in the limit where the ratio q of the thermal diffusivity κ to the magnetic diffusivity η is much smaller than unity since this is possibly of the greatest geophysical relevance.

Thermal convection sets in when the temperature gradient exceeds some critical value as measured by the modified Rayleigh number R_c. The critical temperature gradient is smallest (R_c reaches a minimum) when the magnetic field strength parameter Λ ? 1. [R_c and Λ are defined in (2.3).] The instability takes the form of a very slow wave with frequency of order κ/r ² ₀ and its direction of propagation changes from eastward to westward as Λ increases through Λ _c ? 4.

When the fluid is sufficiently stably stratified and when Λ > Λ_m ? 22 a new mode of instability sets in. It is magnetically driven but requires some stratification before the energy stored in the magnetic field can be released. The instability takes the form of an eastward propagating wave with azimuthal wavenumber m = 1.     

Estimating the internal pressure gradient errors in a σ-coordinate ocean model for the Nordic Seas

Internal pressure gradient estimation is problematic in σ-coordinate ocean models and models based on more generalised topography following coordinate systems. Artificial pressure gradients in these models may create artificial flow. In recent literature, several methods for reducing the errors in the estimated internal pressure gradients are suggested. A basin with a bell-shaped seamount in the middle has often been applied as a test case. To supplement the findings from these more idealised experiments, the internal pressure gradient errors in a σ-coordinate ocean model for the Nordic Seas are discussed in the present paper. Three methods for estimating internal pressure gradients are applied in these experiments. The sensitivity of the results to the subtraction of background stratification and to the horizontal viscosity are also investigated. For the extended Nordic Seas case, basin scale modes dominate after a few days of simulation. The errors in the transports across some sections may be larger than 1 Sv (1 Sv = 10⁶ m³ s^− 1) in these studies with 16-km grid resolution. The order of magnitude of the errors in the transports of Atlantic water into the Nordic Seas is approximately 0.5 Sv or between 5 and 10 % of recent transport estimates based on measurements. The results do not indicate that the errors are generally reduced if the background stratification is subtracted when estimating internal pressure gradients in terrain following models. However, the results from the experiments initialised with the background stratification show that the erroneous flows may be reduced considerably by using more recent techniques for estimating internal pressure gradients, especially for higher values of horizontal viscosity.     

Electromagnetic fields of a vertical magnetic dipole above a laterally inhomogeneous thin layer of conductive overburden

An analytical formulation is developed for the resultant electromagnetic field of an oscillating vertical magnetic dipole located over a thin conductive sheet of infinite extent. The sheet is characterized by a conductivity-thickness product or conductance d that may be a function of the horizontal coordinates. The system of integral equations arising in the general formulation is simplified greatly when azimuthal symmetry prevails. Numerical results for a Gaussian variation of d in the radial direction are presented for the case of a symmetrically located source. These results are for the fields at the level of the source dipole over the conductive sheet. It is shown that the quadrature response of the sheet is enhanced when there is rapid variation of the conductance. The null in the resultant wave tilt is also found to be shifted toward the direction of increasing conductance.     

Numerical investigation of the anisotropic hydraulic conductivity behavior in heterogeneous porous media

The behavior of the mean equivalent hydraulic conductivity normal and parallel to stratification (K₁, and K₂, respectively) is studied here through Monte Carlo simulations of three-dimensional, steady-state flow in statistically anisotropic, bounded, and heterogeneous media. For water flow normal to stratification in strongly heterogeneous porous media (²_Y=3) the value of K₁ is not unique; it ranges from an arithmetic to a geometric, and finally, to a harmonic mean behavior depending on field dimensions, and medium anisotropy. For a fixed anisotropy ratio and variance of Y = ln K, the larger the distance, in the direction perpendicular to stratification, over which water flow takes place, the faster the rate at which, K_H, behavior is approached. However, even for large anisotropy ratios, harmonic mean behavior appears to be a good approximation only for aquifer thickness L₁ that is large enough to allow stratified flow to occur. For small aquifer thickness (L₁/₁<8, where ₁ is the integral scale normal to stratification) the limiting behavior, for large anisotropy ratios, appears to be, instead, that of two-dimensional flow, i.e., water flows primarily parallel to the planes of stratification. When the aquifer thickness is very small compared to the horizontal dimensions (and with relative similar integral scales in the three directions) a behavior resembling arithmetic mean conditions is exhibited, i.e., water flow takes place through heterogeneous, vertical, soil volumes. The geostatistical expressions of Desbarats (1992a) for upscaling hydraulic conductivity values were utilized and closed form empirical relations were developed for the main components of the upscaled hydraulic conductivity tensor.     

Magnetic tracers,a probe of the solar convective layers

Abstract

A meridional circulation of sunspots has been measured through the digital analysis of the Meudon spectroheliograms from 1978 to 1983. Old and young sunspots follow a zonal meridional circulation, in several bands of latitude, in which two adjacent bands have opposite motions. This meridional circulation pattern is time-dependent. Using the H _α filaments as magnetic field tracers, a large-scale magnetic pattern has been found that was also obtained independently by direct measurement of the magnetic field (Hoeksema, 1988).

The coincidence of a large-scale magnetic pattern with a zonal meridional circulation suggests the existence of azimuthal rolls below the surface, and these azimuthal rolls can explain a number of properties of the solar cycle. New rolls occur with increasing proximity to the Equator, thereby indicating the direction of propagation of the dynamo wave. The occurrence of rolls is very favorable to the emergence of the magnetic regions. The rolls also influence the magnetic complexity of the active regions. They modulate the surface rotation through the Coriolis force, which accelerates or decelerates the fluid particles. They therefore offer a plausible explanation of the torsional oscillation pattern.

There are a number of problems raised by such an unexpected circulation pattern: for example, the coexistence of axisymmeric rolls with hypothetical giant cells, the location of the dynamo source below or within the convective zone, and the coupling of the radiative interior and the convective layers. To resolve these important issues, continuous observational studies are needed of the manifestation of solar activity, as well as of radius and luminosity variations. So, we have aimed our paper at an audience of theoreticians in the hope that they take up the challenges we describe.     

Characterization of volcanic thermal anomalies by means of sub-pixel temperature distribution analysis

The simultaneous solution of the Planck equation (involving the widely used “dual-band” technique) using two shortwave infrared (SWIR) bands allows for an estimate of the fractional area of the hottest part of an active lava flow (f _h) and the background temperature of the cooler crust (T _c). The use of a high spectral and spatial resolution imaging spectrometer with a wide dynamic range of 15 bits (DAIS 7915) in the wavelength range from 0.501 to 12.67 μm resulted in the identification of crustal temperature and fractional areas for an intra-crater hot spot at Mount Etna, Italy. This study indicates the existence of a relationship between these T _c and f _h extracted from DAIS and Landsat TM data. When the dual band equation system is performed on a lava flow, a logarithmic distribution is obtained from a plot of the fractional area of the hottest temperature vs. the temperature of the cooler crust. An entirely different distribution is obtained over active degassing vents, where increases in T _c occur without any increase in f _h. This result indicates that we can use scatter plots of T _c vs. fh to discriminate between different types of volcanic activity, in this case between degassing vents and lava flows, using satellite thermal data.     

Non-linear marginal convection in a rotating magnetic system

Abstract

An inviscid, electrically conducting fluid is contained between two rigid horizontal planes and bounded laterally by two vertical walls. The fluid is permeated by a strong uniform horizontal magnetic field aligned with the side wall boundaries and the entire system rotates rapidly about a vertical axis. The ratio of the magnitudes of the Lorentz and Coriolis forces is characterized by the Elsasser number, A, and the ratio of the thermal and magnetic diffusivities, q. By heating the fluid from below and cooling from above the system becomes unstable to small perturbations when the adverse density gradient as measured by the Rayleigh number, R, is sufficiently large.

With the viscosity ignored the geostrophic velocity, U, which is aligned with the applied magnetic field, is independent of the coordinate parallel to the rotation axis but is an arbitrary function of the horizontal cross-stream coordinate. At the onset of instability the value of U taken ensures that Taylor's condition is met. Specifically the Lorentz force, which results from marginal convection must not cause any acceleration of the geostrophic flow. It is found that the critical Rayleigh number characterising the onset of instability is generally close to the corresponding value for the usual linear problem, in which Taylor's condition is ignored and U is chosen to vanish. Significant differences can occur when q is small owing to a complicated flow structure. There is a central interior region in which the local magnetic Reynolds number, R_m , based on U is small of order q and on exterior region in which R_m is of order unity.     

Convection in a rotating magnetic system and taylor's constraint part II,numerical results

Abstract

Results are presented of a numerical study of marginal convection of electrically conducting fluid, permeated by a strong azimuthal magnetic field, contained in a circular cylinder rotating rapidly about its vertical axis of symmetry. To this basic state is added a geostrophic flow U_G (s), constant on geostrophic cylinders radius s. Its magnitude is fixed by requiring that the Lorentz forces induced by the convecting mode satisfy Taylor's condition. The nonlinear mathematical problem describing the system was developed in an earlier paper (Skinner and Soward, 1988) and the predictions made there are confirmed here. In particular, for small values of the Roberts number q which measures the ratio of the thermal to magnetic diffusivities, two distinct regions can be recognised within the fluid with the outer region moving rapidly compared to the inner. Otherwise, conditions for the onset of instability via the Taylor state (U_G 0) do not differ significantly from those appropriate to the static (U_G = 0) basic state. The possible disruption of the Taylor states by shear flow instabilities is discussed briefly.