Simple Convection Research Articles

Treatment of on/off switches in the adjoint method: FDDA experiments with a simple model

The purpose of this study is to illustrate the feasibility of using the adjoint method in the data assimilation procedure, where the assimilation model involves on/off switches associated with physical processes. Although the assimilation model used in this study is very simple, it has the on/off switches that are typical in more sophisticated and realistic assimilation models. The calculus of variations is used to confirm that variational FDDA using the adjoint method allows the equations of the assimilation model to have a finite number of first-order discontinuous points. These points represent the on/off switches for which the Jacobian matrix of the model equation may not exist. In practice, when the on/off switches are involved, the gradient of the functional can be obtained by assuming that an infinitesimally small initial perturbation does not change the time interval within which the switching takes place. The time when the corresponding switching in the adjoint equations is performed is the same as that in the forward integration of the model equations, i.e., the switching time is determined by the basic state. Numerical experiments are then conducted using the aforementioned ideas in a simple one-dimensional convection model involving on/off switches associated with latent heat release. The implication of the numerical experiments is that there is no difficulty, in theory, in treating the on/off switches in the adjoint of the assimilation model. Rather, the success of the adjoint method is dependent on the numerical aspects of the implementation, such as the proper scaling of the variables. DOI: 10.1034/j.1600-0870.1993.00014.x

Simulation of supersonic and hypersonic flows

A simple convection algorithm for simulation of time-dependent supersonic and hypersonic flows of a perfect but viscous gas is described. The algorithm is based on conservation and convection of mass, momentum and energy in a grid of rectangular cells. Examples are given for starting flow in a shock tube and oblique shocks generated by a wedge at Mach numbers up to 30·4. Good comparisons are achieved with well-known perfect gas flows.

Control of Nutrient Mixing and Uptake by Irrigation Frequency and Relative Humidity

AbstractThe distribution of nutrients and water between mobile and immobile pores should influence nutrient uptake. The distribution can be regulated through control of the water‐filled pore space. This research was conducted to determine the effect of varying soil‐water content and water uptake upon nutrient uptake. Corn (Zea mays L.) was grown in a growth chamber for 2 wk at 35 or 55% relative humidity (RH). Three soils [Boelus LS, 5% slope (sandy over loamy, mixed, mesic Udic Haplustoll); Boelus LS, 2% slope; and Plano Soil (fine‐silty, mixed, thermic Typic Haplustolf)] were watered to field capacity (θmax). Plants on each soil were allowed to extract water to one of three minimal levels (θmin1, θmin2, θmin3) before rewatering. After harvest, P and K content and other root and leaf parameters were determined. The values of θmin were chosen so that, for each soil, the three values ensured no low‐water stress. Effective diffusion coefficients were determined for the three Increased Omin for a soil required for frequent watering, which led to greater mixing of solutes between pores. At 55% RH, no water treatment significantly affected P and K flux, despite significant differences in diffusion coefficients. At 35% RH, however, phosphate flux to roots increased as θmin increased. The significant increase of phosphate flux with more frequent watering at low RH suggests that plant uptake is affected by soil physical processes other than simple diffusion and convection to individual roots. More frequent watering results in greater mixing of solute between pores containing mobile and immobile water and, consequently, greater uptake.

Thermal stress tectonics on the satellites of Saturn and Uranus

Most of the major Saturnian and Uranian satellites have undergone some form of extensional tectonism and/or resurfacing early in their histories. We examine the stresses created by the early thermal evolution of these satellites and consider the geologic implications of these stresses. Starting from an initial accretional temperature profile, we calculate the satellites' temperature profiles as they evolve by conduction. Solid state convection, when it occurs, is handled using a simple parameterized convection scheme. The effects of radiogenic heating and release of heat of phase change are included. The thermal stresses built up in the satellites are then calculated, assuming the satellites to have a Maxwell viscoelastic rheology. The stress calculation takes into account both the effects of thermal stresses and of shifts in the position of the ice I/ice II phase boundary in some satellites. If accretion takes place in a cold, gas‐free environment, the stresses that build up in the smaller satellites (Mimas, Miranda, and Enceladus) are minor. In the larger satellites, however, stresses are large enough to cause fracturing and extensional tectonism under these circumstances. The stresses result largely from rapid cooling contraction of a thin outer shell; the ice II → ice I phase transition also contributes significantly for the largest satellites. If accretion occurs in a warm gaseous nebula that dissipates rapidly, stresses large enough to cause failure are attained in all of the satellites. In all cases, the material immediately underlying the zone of failure is the warmest in the satellite and hence may be buoyant and mobile enough to be extruded to the surface through fractures created by the extension. We conclude that thermal stresses following accretion led to a substantial early episode of extensional tectonism and perhaps also associated resurfacing on many of the Saturnian and Uranian satellites. If evidence of this early episode of extensional tectonism has survived subsequent impacts, then the process may have been responsible for much of the early tectonism and resurfacing observed on these satellites. If not, then it may only be responsible for the very early global resurfacing that some of the satellites are inferred to have undergone.

Convective effects on chemical waves. 2. Simple convection in the iodate-arsenous acid system

Convective effects on chemical waves. 2. Simple convection in the iodate-arsenous acid system

Convective effects on chemical waves. 1. Mechanisms and stability criteria

We consider the effects of convection on chemical waves in which density gradients result from the exothermicity as well as from the isothermal volume change of the reaction. If the signs of the enthalpy (ΔH) and volume (ΔV) of the reaction are opposite, then simple convection will occur, and the increase in the front propagation velocity will be equal to the velocity of the convective fluid flow. If the signs are the same, then simple convection will not occur. Instead multicomponent convection may be present, even though the overall density gradient may appear to be stable. The stability conditions for both mechanisms as functions of ΔH and ΔV and of the direction of front propagation (ascending, descending or horizontal) are analyzed

A simple forced convection film boiling model

A simple analysis of the steady laminar forced convection film boiling flow over a horizontal flat plate is presented. A novel analogy drawn between the mobile liquid-vapor interface in forced convection film boiling and a moving surface in a flowing single-phase fluid enables a simple analysis of this complex flow. As a result, easy to apply closed-form expressions to predict theoretical wall heat transfer and skin friction are obtained, circumventing the need for previous complex analyses.

Fluxes through diffusive staircases: A new formulation

This paper deals with two aspects of the flux law for diffusive convection: (1) the dependence on density ratio, and (2) the dependence on ΔT. Empirical formulations of the dependence of temperature and salinity fluxes on density ratio are developed using published measurements. At high density ratios (Rρ > 4) our temperature fluxes agree with those given by Marmorino and Caldwell's (1976) formulation, but at low density ratios (Rρ < 2.5) our temperature fluxes are smaller by half. However, it is suggested that any formulation of the 4/3 flux law should be used with caution. A simple convection theory implies that fluxes are not proportional to ΔT4/3, as the 4/3 law states, but rather are proportional to ΔT raised to an power closer to 5/4. The theory accurately matches measurements of velocity and heat flux in thermal convection. There are insufficient data to test (or even fully formulate) the theory for diffusive convection, but if the thermal convection results carry over to the diffusive case, then the 4/3 flux law overestimates oceanic fluxes by up to about 30%.

Coupling of mantle temperature anomalies and the flow pattern in the core: interpretation based on simple convection calculations

The association of some relatively stationary features of the magnetic field at the core-mantle boundary (CMB) with temperature anomalies in the deep mantle inferred from seismic tomography has led to the speculation that flow in the core is locked to thermal anomalies in the mantle, with downwellings (upwellings) in the core lying below cold (hot) anomalies in the lower mantle (Bloxham and Gubbins). The relatively large temperature anomalies in the mantle act to control the heat flux out of the core, with high heat flux into colder than average mantle and low (perhaps negative) heat flux into regions of hotter than average mantle. We use a simple two-dimensional, internally heated convection model in a Cartesian geometry with varying thermal boundary conditions to investigate this problem. The relationship between upwellings (downwellings) and maxima (minima) in surface heat flux depends upon the thermal boundary condition, with maximum heat flux sometimes associated with upwellings, sometimes downwellings, and sometimes neither. However, for moderate Rayleigh numbers, large aspect ratio geometries and large variations in heat flux, the Bloxham and Gubbins hypothesis is supported.

Salinity Variations in Submarine Hydrothermal Systems by Layered Double-Diffusive Convection

Various mechanisms have been proposed to explain the salinity variations in vent fluids of seafloor geothermal systems. New experiments reacting diabase and evolved seawater were carried out to reproduce earlier published observations of Cl depletions attributed to formation of an ephemeral Cl-bearing mineral. The absence of any Cl depletions in the present study suggests that the formation of Cl-bearing minerals is not sufficiently widespread to account for the observed salinity variations in the vent fluids. A re-evaluation of both field and laboratory evidence has led to a new model for subseafloor circulation that accounts for salinity variations as well as other chemical and mineralogic observations. In place of a simple single-pass convection system, we propose that the seafloor systems consist of two vertically nested convection cells in which a brine layer at depth heats and drives an overlying seawater cell. Such layering of salinities, a process known in fluid mechanics as double-diffusive conve...

Viscosity structure implied by mantle convection

The total mechanical dissipation by convective motion in the mantle is well constrained by thermodynamic arguments to ∼ 5.6 × 10 12 W. By adopting a simple convection model we can use the observed mean plate speed υ = 3.25 cm y −1 to estimate the mean mantle creep rate, ϵ ̇ ≈ 7.2 × 10 −16 s −1 , which implies a mean effective stress σ ≈ 8.5 × 10 6 Pa and a corresponding viscosity υ ⋍ 1.2 × 10 22 Pa s. This viscosity estimate is biased low by the nature of the implied averaging, but for better estimates we must make assumptions about the internal convective pattern. If we assume that the convection is self-sustaining at all depths, and therefore that the depth distribution of dissipation is the same as the depth distribution of generated power, and that the convective pattern is self-adjusted to make stress independent of depth, then we can specify the mantle viscosity profile. These assumptions imply quasi-steady convection. The fact that viscosities inferred from convection are higher than values from isostatic rebound most plausibly arises as a difference between steady state and transient creep.

Solar sources of interplanetary southward Bz events responsible for major magnetic storms (1978‐1979)

Tsurutani et al. [1988] analyzed the 10 intense interplanetary southward Bz events that led to major magnetic storms (Dst<−100 nT) during the 500‐day interval, August 16, 1978, through December 28, 1979. In this paper we report solar sources of the major storms and discuss their interplanetary effects. Seven of the sources of the 10 major storms are flares and 3 are prominence eruptions. The ratio is consistent with frequency of these two types of events occurring on the Sun during this period. The flares range from M1 to X2 in peak soft X ray emissions. However, the largest of the 10 storms (Dst = −220 nT) as well as the two strongest shocks (M>3.0) are associated with prominence eruptions. For three of the five southward Bz events in which the driver gases are the causes of the intense southward field leading to magnetic storms, the photospheric fields of the solar sources have no dominant southward component, indicating the driver gas fields do not always result from a simple outward convection of solar magnetic fields. Finally we compare the solar events and their resulting interplanetary shocks and find that the standard solar parameters do not correlate with the strengths of the resulting shocks at 1 AU. The implications are discussed.

Transient climatic response to increasing CO<sub>2</sub> concentration: some dynamical scenarios

The global surface temperature record of the last 130 years displays a stage of systematic increase on which is superimposed a small-scale variability, followed by a stage of marked slowing down. At first sight, this appears to be surprising, in view of the ongoing increase of CO 2 concentration in the atmosphere. In this work, the mechanisms that enable a dynamical system to produce such a peculiar response to an increasing control parameter are studied. The main ideas are illustrated, successively, on a linearized energy balance model subject to white or red noise, on nonlinear energy balance models admitting several stable solutions, and on a simple thermal convection model giving rise to nonperiodic (chaotic) solutions. It is shown that the action of stochastic perturbations (especially those having a relatively long correlation time), the sudden switching in the vicinity of a point of marginal stability, or finally chaotic dynamics, give rise to responses that match the temperature record. It is suggested that the variance of climatic observables is much more sensitive to the increase of a control parameter compared to the mean. DOI: 10.1111/j.1600-0870.1988.tb00330.x

Heat transport by parallel-roll convection in a rectangular container

Heat-transport measurements are reported for thermal convection in a rectangular box of aspect’ ratio 10 x 5. Results are presented for Rayleigh numbers up to 35Rc, Prandtl numbers between 2 and 20, and wavenumbers between 0.6 and 1.0kc, where Rc and kc are the critical Rayleigh number and wavenumber for the onset of convection in a layer of infinite lateral extent. The measurements are in good agreement with a phenomenological model which combines the calculations of Nusselt number, as a function of Rayleigh number and roll wavenumber for two-dimensional convection in an infinite layer, with a nonlinear amplitude-equation model developed to account for sidewell attenuation. The appearance of bimodal convection increases the heat transport above that expected for simple parallel-roll convection.

Thermal evolution of a differentiated Ganymede and implications for surface features

Thermal evolution models are presented for Ganymede, assuming a mostly differentiated initial state of a water ocean overlying a rock layer. The only heat sources are assumed to be primordial heat (provided by accretion) and the long-lived radiogenic heat sources in the rock component. As Ganymede cools, the ocean thins, and two ice layers develop, one above composed of ice I, and the other below composed of high-pressure polymorphs of ice. Subsolidus convection proceeds separately in each ice layer, its transport of heat calculated using a simple parameterized convection scheme and the most recent data on ice rheology. The model requires that the average entropy of the deep ice layer exceeds that of the ice I layer. If the residual ocean separating these layers becomes thin enough, then a Rayleigh-Taylor-like (“diapiric”) instability may ensue, driven by the greater entropy of the deeper ice and merging the two ice mantles into a single convective layer. This instability is not predicted by linear analysis but occurs for plausible finite amplitude perturbations associated with large Rayleigh number convection. The resulting warm ice diapirs may lead to a dramatic “heat pulse” at the surface and to fracturing of the lithosphere, and may be directly or indirectly responsible for resurfacing and grooved terrain formation on Ganymede. The timing of this event depends rather sensitively on poorly known rheological parameters, but could be consistent with chronologies deduced from estimated cratering rates. Irrespective of the occurrence or importance of the heat pulse, we find that lithospheric fracturing requires rapid stress loading (on a time scale ⪅10^4 years). Such a time scale can be realized by warm ice diapirism, but not directly by gradual global expansion. In the absence of any quantitative and self-consistent model for the resurfacing of Ganymede by liquid water, we favor resurfacing by warm ice flows, which we demonstrate to be physically possible, a plausible consequence of our models, compatible with existing observations, and a hypothesis testable by Galileo. We discuss core formation as an alternative driver for resurfacing, and conclude that it is less attractive. We also consider anew the puzzle of why Callisto differs so greatly from Ganymede, offering several possible explanations. The models presented do not provide a compelling explanation for all aspects of Ganymedean geological evolution, since we have identified several potential problems, most notably the apparently extended period of grooved terrain formation (several hundred million years), which is difficult to reconcile with the heat pulse phenomenon.

Heat flow observations on the Bermuda Rise and thermal models of midplate swells

The Bermuda Rise is a broad topographic swell which is apparent in both residual depth and geoid anomaly maps of the western North Atlantic. The magnitudes of the depth and geoid anomalies associated with the Bermuda Rise are similar to the anomalies associated with other swells surrounding recent volcanic islands (e.g., Hawaii), suggesting that despite the lack of recent volcanism on Bermuda, the rise has a similar origin to other midplate swells. Results are reported from 171 new heat flow measurements at seven carefully selected sites on the Bermuda Rise and the surrounding seafloor. Off the Bermuda Rise the basement depths are generally shallower and the heat flow higher than either the plate or boundary layer models predict, with the measured heat flow apparently reaching a uniform value of about 50 mW m−2 on 120 m.y. old crust. On the Bermuda Rise the heat flow is significantly higher (57.4±2.6 mW m−2) than off the swell (49.5±1.7 mW m−2). The magnitude of the anomalous heat flux (8–10 mW m−2) is comparable to that previously found along the older portion of the Hawaiian Swell near Midway. The existence of higher heat flow on both the Hawaiian Swell and Bermuda Rise indicates that these features fundamentally have a thermal origin. The differences in the shape, uplift, and subsidence histories of the Hawaiian Swell and Bermuda Rise can be quantitatively explained by the different absolute velocities of the Pacific and North American plates moving across a distributed heat source in the underlying mantle. Two‐dimensional numerical convection models indicate that the observed depth, geoid, and heat flow anomalies are consistent with simple convection models in which the lower part of the thermally defined plate acts as the upper thermal boundary layer of the convection.

Fluctuation-dissipation theorem and intrinsic stochasticity of climate : Nicolis, C., J.P. Boon and G. Nicolis, 1985. Nuovo Cim., (C)8(3): 223–241

S n m m a r y . IrL climate dynamics the effect of internally generated fluctuations is usually described by augmenting the balance equations through the addition of random ]orees. In this paper the properties of these forces arc investigated. A ]luctuation-dissipation theorem relating their covariance matrix to the phenomenological coefficients such as eddy diffusivity is proposed. The theorem is subsequently used to identify the statistical properties of the climatic variables themselves, and thus to characterize climatic variability from the standpoint of the statistical theory of irreversible proeesscs. Applications to a simple thermal convection problem and to a zonally averaged energy-balance model arc presented; the possibility of experimental verification is discussed.

Observations of Complex Oscillations in a Closed Thermosyphon

Observations have been made of thermal oscillations in a slightly inclined closed thermosyphon. The thermosyphon is made up of two isothermal tubes, capped at the outer ends, and joined along their axes by an insulating section. The tube assembly is filled with liquid and inclined slightly with respect to the vertical. The lower isothermal section is hotter than the upper one and convection is driven across the insulating region. Between the applied temperature difference at which simple steady convection occurs, and that required for persistent turbulent motions, there is a wide range over which thermal oscillations are observed. These oscillations reflect quasi-periodic flow as well as a type of periodic chaos.

Fluctuation-dissipation theorem and intrinsic stochasticity of climate

In climate dynamics the effect of internally generated fluctuations is usually described by augmenting the balance equations through the addition ofrandom forces. In this paper the properties of these forces are investigated. Afluctuation-dissipation theorem relating their covariance matrix to the phenomenological coefficients such as eddy diffusivity is proposed. The theorem is subsequently used to identify the statistical properties of the climatic variables themselves, and thus to characterize climatic variability from the standpoint of the statistical theory of irreversible processes. Applications to a simple thermal convection problem and to a zonally averaged energy-balance model are presented; the possibility of experimental verification is discussed.

A New Look at Sound Generation by Blade/Vortex Interaction

As a preliminary attempt to understand the dynamics of blade/vortex interaction, the two-dimensional problem of a rectilinear vortex filament interacting with a Joukowski airfoil is analyzed in both the lifting and nonlifting cases. The vortex velocity components could be obtained analytically and integrated to determine the vortex trajectory. With this information, the aeroacoustic low-frequency Green’s function approach could then be employed to calculate the sound produced during the encounter. The results indicate that the vortex path deviates considerably from simple convection due to the presence of the airfoil and that a reasonably sharp sound pulse is radiated during the interaction whose fundamental frequency is critically dependent upon whether the vortex passes above or below the airfoil. Determination of this gross parameter of the interaction is shown to be highly nonlinearly dependent upon airfoil circulation, vortex circulation, and initial position.