Convective Energy Flux Research Articles

Impacts of Urban Heat and Surface Roughness on Landfalling Tropical Cyclone Intensity: A Case Study Based on TC Victor (1997) in Coastal South China

AbstractThis study investigates the impacts of urban‐induced anthropogenic heat (AH) and surface roughness on Tropical Cyclone (TC) Victor (1997) using the Weather Research and Forecasting Model. TC Victor originated in the South China Sea and made landfall over the Greater Bay Area (GBA) megacity. The storm was characterized by slow movement, a weakly organized structure, and a small size. Three parallel experiments were conducted in the following setups: “Nourban,” where urban areas in the GBA were replaced by croplands, “AH0” (“AH300”), in which the diurnal maximum AH was set to 0 (300 W/m2) in urban locations. The urbanization effect during the landfall period results in a notable reduction in both the Power Dissipation Index and Integrated Kinetic Energy of the storm. A Lagrangian particle dispersion model further demonstrates that from 33 to 17 hr prelandfall, the entrainment of AH‐affected warm airflow into the TC's circulation leads to a decrease in both the convective available potential energy and surface latent heat flux in the western quadrant of the TC, thus weakening the cyclone's intensity. Furthermore, from 12 hr prelandfall to 8 hr postlandfall, there is an intense influx of AH‐affected airflow into the TC's primary circulation, resulting in a decrease in relative humidity and a stronger asymmetrical structure, declining the TC intensity. Concurrently, urban surface roughness contributes to a decrease in postlandfall storm intensity through frictional energy dissipation. This study underscores the critical role of AH and urban roughness in modulating TC intensity for storms with specific characteristics, emphasizing the need for deeper insights into urban‐TC dynamics.

Using Passive Multi-Modal Sensor Data for Thermal Simulation of Urban Surfaces

Abstract. This paper showcases an integrated workflow hinged on passive airborne multi-modal sensor data for the simulation of the thermal behavior of built-up areas with a focus on urban heat islands. The geometry of the underlying parametrized model, or digital twin, is derived from high-resolution nadir and oblique RGB, near-infrared and thermal infrared imagery. The captured bitmaps get photogrammetrically processed into comprehensive surface models, terrain, dense 3D point clouds and true-ortho mosaics. Building geometries are reconstructed from the projected point sets with procedures presupposing outlining, analysis of roof and fac¸ade details, triangulation, and texturing mapping. For thermal simulation, the composition of the ground is determined using supervised machine learning based on a modified multi-modal DeepLab v3+ architecture. Vegetation is retrieved as individual trees and larger tree regions to be added to the meshed terrain. Building materials are assigned from the available visual, infrared and surface planarity information as well as publicly available references. With actual weather data, surface temperatures can be calculated for any period of time by evaluating conductive, convective, radiative and emissive energy fluxes for triangular layers congruent to the faces of the modeled scene. Results on a sample dataset of the Moabit district in Berlin, Germany, showed the ability of the simulator to output surface temperatures of relatively large datasets efficiently. Compared to the thermal infrared images, several insufficiencies in terms of data and model caused occasional deviations between measured and simulated temperatures. For some of these shortcomings, improvement suggestions within future work are presented.

Тепловое нагружение конструкции отработанной ступени ракеты при баллистическом спуске

The results of a numerical calculation of the aerodynamic heating of a spent rocket stage structural elements during its controlled descent along a ballistic trajectory are presented. The influence of the initial trajectory parameters on the specific convective and radiant heat fluxes is determined. To assess the thermal loading of the spent rocket stage structure, the parameter of the total specific energy flux is introduced. It has been established that in the range of initial speeds of the spent rocket stage from 1800 m/s to 2600 m/s, an increase in the initial speed by 200 m/s leads to an increase in the specific convective flow by an average of 11 % for the tail compartment and the fuel tank and by 5 % for the oxidizer tank. The share of the specific radiant energy flux withdrawn from the surface in relation to the specific convective energy flux is from 0,15 to 0,19 for the tail compartment, from 0,12 to 0,15 for the fuel tank and not more than 0,009 for the oxidizer tank. Empirical dependencies are proposed for the preliminary stage of assessing the temperature state of the spent rocket stage structure when moving on a ballistic trajectory. The error of the calculation results according to the proposed dependencies does not exceed 12 %.

Prandtl number dependence of stellar convection: Flow statistics and convective energy transport

Context. The ratio of kinematic viscosity to thermal diffusivity, the Prandtl number, is much smaller than unity in stellar convection zones. Aims. The main goal of this work is to study the statistics of convective flows and energy transport as functions of the Prandtl number. Methods. Three-dimensional numerical simulations of compressible non-rotating hydrodynamic convection in Cartesian geometry are used. The convection zone (CZ) is embedded between two stably stratified layers. The dominant contribution to the diffusion of entropy fluctuations comes in most cases from a subgrid-scale diffusivity whereas the mean radiative energy flux is mediated by a diffusive flux employing Kramers opacity law. Here, we study the statistics and transport properties of up- and downflows separately. Results. The volume-averaged rms velocity increases with decreasing Prandtl number. At the same time, the filling factor of downflows decreases and leads to, on average, stronger downflows at lower Prandtl numbers. This results in a strong dependence of convective overshooting on the Prandtl number. Velocity power spectra do not show marked changes as a function of Prandtl number except near the base of the convective layer where the dominance of vertical flows is more pronounced. At the highest Reynolds numbers, the velocity power spectra are more compatible with the Bolgiano-Obukhov k−11/5 than the Kolmogorov-Obukhov k−5/3 scaling. The horizontally averaged convected energy flux (F̅conv), which is the sum of the enthalpy (F̅enth) and kinetic energy fluxes (F̅kin), is independent of the Prandtl number within the CZ. However, the absolute values of F̅enth and F̅kin increase monotonically with decreasing Prandtl number. Furthermore, F̅enth and F̅kin have opposite signs for downflows and their sum F̅↓conv diminishes with Prandtl number. Thus, the upflows (downflows) are the dominant contribution to the convected flux at low (high) Prandtl numbers. These results are similar to those from Rayleigh-Benárd convection in the low Prandtl number regime where convection is vigorously turbulent but inefficient at transporting energy. Conclusions. The current results indicate a strong dependence of convective overshooting and energy flux on the Prandtl number. Numerical simulations of astrophysical convection often use a Prandtl number of unity because it is numerically convenient. The current results suggest that this can lead to misleading results and that the astrophysically relevant low Prandtl number regime is qualitatively different from the parameter regimes explored in typical contemporary simulations.

Diurnal Cycle of Precipitation Features Observed during DYNAMO

AbstractThis study uses shipborne (R/V Roger Revelle and R/V Mirai) radar, upper-air, ocean, and surface meteorology datasets from the DYNAMO field campaign to investigate the diurnal cycle (DC) of precipitation over the central Indian Ocean related to two distinct Madden–Julian oscillations (MJOs) observed. This study extends earlier studies on the MJO DC by examining the relationship between the DC of convective organization and the local environment and comparing these results on- and off-equator. During the suppressed phase on-equator, the DC of rain rates exhibited two weak maxima at 1500 and 0100 LT, which was largely controlled by the presence of sub-MCS nonlinear precipitation features (PFs). During the active phase on-equator, MCS nonlinear features dominated the rain volume, and the greatest increase in rain rates occurred between 2100 and 0100 LT. This maximum coincided with the maxima in convective available potential energy (CAPE) and sensible heat flux, and the column moistened significantly overnight. Off-equator, the environment was much drier and there was little large-scale upward motion as a result of limited deep convection. The DC of rain rates during the active phase off-equator was most similar to the DC observed during the suppressed phase on-equator, while rainfall off-equator during the suppressed phase did not vary much throughout the day. The DC of MCS nonlinear PFs closely resembled the DC of rainfall during both phases off-equator, and the DC of environmental parameters, including sea surface temperature, CAPE, and latent heat flux, was typically much weaker off-equator compared to on-equator.

Overstable convective modes in rotating early-type stars

ABSTRACT We calculate overstable convective (OsC) modes of 2-, 4-, and $20\hbox{-}{\rm M}_\odot$ main-sequence stars. To compute non-adiabatic OsC modes in the core, we assume $(\nabla \cdot \rm{\boldsymbol {F}}_{\rm C})^\prime =0$ as a prescription for the approximation called frozen-in convection in pulsating stars, where $\rm{\boldsymbol {F}}_{\rm C}$ is the convective energy flux and the prime ′ indicates Eulerian perturbation. We find that the general properties of the OsC modes are roughly the same as those obtained by Lee & Saio, who assumed $\delta (\nabla \cdot \rm{\boldsymbol {F}}_{\rm C})=0$, except that no OsC modes behave like inertial modes when they tend towards complete stabilization with increasing rotation frequency, where δ indicates the Lagrangian perturbation. As the rotation frequency of the stars increases, the OsC modes are stabilized to resonantly excite g modes in the envelope when the core rotates slightly faster than the envelope. The frequency of the OsC modes that excite envelope g modes is approximately given by σ ∼ |mΩc| in the inertial frame and hence σm = −2 ≈ 2σm = −1, where m is the azimuthal wavenumber of the modes and Ωc is the rotation frequency of the core. We find that the modal properties of OsC modes do not strongly depend on the mass of the stars. We discuss angular momentum transport by OsC modes in resonance with envelope g modes in the main-sequence stars. We suggest that angular momentum transfer takes place from the core to the envelope and that the OsC modes may help the stars rotate uniformly and keep the rotation frequency of the core low during their evolution as main-sequence stars.

Comparison between SOLPS-4.3 and the Lengyel Model for ITER baseline neon-seeded plasmas

If correct, the Lengyel model offers a simple and powerful tool to predict the conditions required for detachment onset in future fusion reactors. We assess its validity against a comprehensive SOLPS-4.3 simulation database of ITER baseline (Q = 10) neon-seeded plasmas (Pacher et al 2015 J. Nucl. Mater. 463 591). In absolute terms, the Lengyel Model is found to significantly overpredict the simulated impurity concentration required in the ITER outer divertor for outer target ion flux rollover (by a factor ∼4.3 in this particular case). Importantly though, at detachment onset, and even beyond onset, the Lengyel model does give a remarkably accurate prediction of the scaling interdependencies between the electron density at the outer divertor entrance, the parallel energy flux density at the outer divertor entrance, and the impurity concentration in the outer divertor. However, the generalisation of these two key results to other machines, and in the presence of additional physics not included in these simulations, requires further studies. The analysis techniques described here provide a framework for such studies. Regarding the factor ∼4.3 overprediction of the simulated outer divertor impurity concentration, the main contributors to the disagreement are found to be other energy loss mechanisms besides impurity cooling (primarily neutral losses and radial transport) combined with convective energy fluxes near the target, as well as non-constant electron static pressure due to poloidally variable T i/T e. None of these are included in the Lengyel model. By themselves, these do not strongly influence the scaling interdependencies of the main Lengyel parameters over the explored parameter range. The impurity residence time τ is observed to increase with density, which tends to flatten out the impurity concentration scaling at low density, relative to the Lengyel model (which usually assumes constant τ). In these simulations, however, this flattening out was cancelled by an accumulation of other effects, so that the scaling prediction of the Lengyel model was still well met. A simple physics model is derived for n e τ that matches the simulation data well. Neon is found to migrate from the inner divertor to the outer divertor with increased puffing, thereby increasing the outer divertor neon enrichment. At outer target ion flux rollover, though, the enrichment is approximately independent of the upstream concentration, so that the Lengyel model predicts well the scaling dependency between the upstream impurity concentration and the upstream electron density, both key quantities dictating the operational range of a tokamak.

A numerical study on the pendant superheater of a 1000 MW coal-fired supercritical CO2 boiler

This work focuses on the pendant superheater (PSH) of a 1000 MW coal-fired supercritical CO2 (S-CO2) boiler. Three-dimensional simulation of coal combustion is performed, in order to obtain the heat load distribution on the S-CO2 fluid. The tube lines simplified from the real system are constructed in CAD software. The results show that two high temperature cores of gas emerge at the furnace outlet, which is also two cores of low convective energy flux. The two high temperature cores are dramatically cooled down by the PSH. As a result, the S-CO2 in the central region near the rear wall absorbs more energy. The final heating effect is that the outlet temperature for S-CO2 only has one peak rather than two in X-direction.

Numerical description of axisymmetric blue whirls over liquid-fuel pools

This numerical investigation is focused on determining the structures of blue whirls, recently found to occur in laboratory investigations of fire whirls when the circulation becomes sufficiently large to produce a vortex breakdown that drastically shortens the fire whirl and correspondingly reduces residence times, so that the yellow flames turn blue. The computations address axisymmetric configurations for round pools of liquid fuels flush with and at the center of a larger solid horizontal disc, at the outer edge of which vanes of adjustable angles cause the entrained air to enter with a controllable azimuthal component of velocity. The nondimensionlized conservation equations employed include realistic Lewis numbers with temperature-dependent transport coefficients and a one-step chemical-kinetic approximation that correctly reproduces laminar burning velocities. Buoyancy and radiant energy transport from the flames to the liquid surface are both taken into account, the latter being found to be essential for the blue whirl. Along with the vaporization-equilibrium and energy-conservation boundary conditions at the fuel surface, inflow boundary conditions are provided by a recently developed solution for the boundary-layer flow over the solid disc, while zero-gradient outflow conditions are applied above the whirl. Controlling nondimensional parameters, besides Reynolds, Damköhler, and Froude numbers, are a ratio of radiant to convective energy flux and a ratio of azimuthal to inward radial flow velocity in the boundary layer at the edge of the disc. The computed conditions for the onset of the blue whirl, as well as the computed structure of the whirl itself, bear close resemblance to what was found experimentally.

Sensitivity of Dynamical Downscaling Seasonal Precipitation Forecasts to Convection and Land Surface Parameterization in a High-Resolution Regional Climate Model

Both convection and land surface parameterization influence seasonal precipitation forecasts. In this study, the sensitivity of dynamical downscaling seasonal precipitation forecasts to convection and land surface parameterization was investigated by nesting the Weather Research and Forecasting (WRF) model into the NCEP’s Climate Forecast System version 2 (CFSv2) retrospective forecasts with four convective schemes: Kain–Fritsch (KF), Betts–Miller–Janjic (BMJ), Grell–Freitas (GF), and new simplified Arakawa–Schubert (NSAS) schemes, and two land surface schemes: Noah and simplified Simple Biosphere (SSiB) schemes over the Han River basin. The CFSv2 model biases are reduced when the KF convective scheme is used in the wet summer season. However, negative biases still exist especially when the combination of BMJ and SSiB schemes is used. Compared with CFSv2 reforecasts and other combinations of schemes, the forecast skills of spatial patterns of precipitation anomalies are highest when the combination of KF and Noah schemes is used in summer. In contrast, the combination of BMJ and SSiB schemes shows lowest forecast skills in summer. To understand the causes of the differences in precipitation forecasts using different parameterization schemes, the simulated moisture flux convergence, thermodynamic parameters at different pressure levels, convective available potential energy (CAPE), convective inhibition (CIN), and heat fluxes are compared with the data in the ERA-5 reanalysis dataset. The WRF model-simulated moisture flux convergence is closer to that of the ERA-5 reanalysis compared with that of the CFSv2 reforecasts in summer. The vertical thermodynamic profiles also suggest that the combination of the KF and Noah schemes has caused a more unstable atmosphere, which is crucial for precipitation. In contrast, the combination of BMJ and SSiB schemes shows a less unstable atmospheric environment in summer, which explains the lower forecast skills compared with other schemes. The spatial patterns of CAPE are also improved when using the WRF model, which further enhances the precipitation forecast skills over the Han River basin.

Differential Rotation in Solar-like Convective Envelopes: Influence of Overshoot and Magnetism

Abstract We present a set of four global Eulerian/semi-Lagrangian fluid solver (EULAG) hydrodynamical (HD) and magnetohydrodynamical (MHD) simulations of solar convection, two of which are restricted to the nominal convection zone, and the other two include an underlying stably stratified fluid layer. While all four simulations generate reasonably solar-like latitudinal differential rotation profiles where the equatorial region rotates faster than the polar regions, the rotational isocontours vary significantly among them. In particular, the purely HD simulation with a stable layer alone can break the Taylor–Proudman theorem and produce approximately radially oriented rotational isocontours at medium to high latitudes. We trace this effect to the buildup of a significant latitudinal temperature gradient in the stable fluid immediately beneath the convection zone, which imprints itself on the lower convection zone. It develops naturally in our simulations as a consequence of convective overshoot and rotational influence of rotation on convective energy fluxes. This favors the establishment of a thermal wind balance that allows evading the Taylor–Proudman constraint. A much smaller latitudinal temperature gradient develops in the companion MHD simulation that includes a stable fluid layer, reflecting the tapering of deep convective overshoot that occurs at medium to high latitudes, which is caused by the strong magnetic fields that accumulate across the base of the convection zone. The stable fluid layer also has a profound impact on the large-scale magnetic cycles developing in the two MHD simulations. Even though both simulations operate in the same convective parameter regime, the simulation that includes a stable layer eventually loses cyclicity and transits to a non-solar, steady quadrupolar state.

Low-density, radiatively inefficient rotating-accretion flow on to a black hole

We study low-density axisymmetric accretion flows onto black holes (BHs) with two-dimensional hydrodynamical simulations, adopting the $\alpha$-viscosity prescription. When the gas angular momentum is low enough to form a rotationally supported disk within the Bondi radius ($R_{\rm B}$), we find a global steady accretion solution. The solution consists of a rotational equilibrium distribution at $r\sim R_{\rm B}$, where the density follows $\rho \propto (1+R_{\rm B}/r)^{3/2}$, surrounding a geometrically thick and optically thin accretion disk at the centrifugal radius, where thermal energy generated by viscosity is transported via strong convection. Physical properties of the inner solution agree with those expected in convection-dominated accretion flows (CDAF; $\rho \propto r^{-1/2}$). In the inner CDAF solution, the gas inflow rate decreases towards the center due to convection ($\dot{M}\propto r$), and the net accretion rate (including both inflows and outflows) is strongly suppressed by several orders of magnitude from the Bondi accretion rate $\dot{M}_{\rm B}$ The net accretion rate depends on the viscous strength, following $\dot{M}/\dot{M}_{\rm B}\propto (\alpha/0.01)^{0.6}$. This solution holds for low accretion rates of $\dot{M}_{\rm B}/\dot{M}_{\rm Edd}< 10^{-3}$ having minimal radiation cooling, where $\dot{M}_{\rm Edd}$ is the Eddington rate. In a hot plasma at the bottom ($r<10^{-3}~R_{\rm B}$), thermal conduction would dominate the convective energy flux. Since suppression of the accretion by convection ceases, the final BH feeding rate is found to be $\dot{M}/\dot{M}_{\rm B} \sim 10^{-3}-10^{-2}$. This rate is as low as $\dot{M}/\dot{M}_{\rm Edd} \sim 10^{-7}-10^{-6}$ inferred for SgrA$^*$ and the nuclear BHs in M31 and M87, and can explain the low luminosities in these sources, without invoking any feedback mechanism.

Oscillatory convective modes in red giants: a possible explanation of the long secondary periods

We discuss properties of oscillatory convective modes in low-mass red giants, and compare them with observed properties of the long secondary periods (LSPs) of semi-regular red giant variables. Oscillatory convective modes are very nonadiabatic g$^{-}$ modes and they are present in luminous stars, such as red giants with $\log L/{\rm L}_\odot \ga 3$. Finite amplitudes for these modes are confined to the outermost nonadiabatic layers, where the radiative energy flux is more important than the convective energy flux. The periods of oscillatory convection modes increase with luminosity, and the growth times are comparable to the oscillation periods. The LSPs of red giants in the Large Magellanic Cloud (LMC) are observed to lie on a distinct period-luminosity sequence called sequence D. This sequence D period-luminosity relation is roughly consistent with the predictions for dipole oscillatory convective modes in AGB models if we adopt a mixing length of 1.2 pressure scale height ($\alpha = 1.2$). However, the effective temperature of the red-giant sequence of the LMC is consistent to models with $\alpha=1.9$, which predict periods too short by a factor of two.

Convective radial energy flux due to resonant magnetic perturbations and magnetic curvature at the tokamak plasma edge

With the resonant magnetic perturbations (RMPs) consolidating as an important tool to control the transport barrier relaxation, the mechanism on how they work is still a subject to be clearly understood. In this work, we investigate the equilibrium states in the presence of RMPs for a reduced MHD model using 3D electromagnetic fluid numerical code with a single harmonic RMP (single magnetic island chain) and multiple harmonics RMPs in cylindrical and toroidal geometry. Two different equilibrium states were found in the presence of the RMPs with different characteristics for each of the geometries used. For the cylindrical geometry in the presence of a single RMP, the equilibrium state is characterized by a strong convective radial thermal flux and the generation of a mean poloidal velocity shear. In contrast, for toroidal geometry, the thermal flux is dominated by the magnetic flutter. For multiple RMPs, the high amplitude of the convective flux and poloidal rotation are basically the same in cylindrical geometry, but in toroidal geometry the convective thermal flux and the poloidal rotation appear only with the islands overlapping of the linear coupling between neighbouring poloidal wavenumbers m, m – 1, and m + 1.

Effect of flue gas recirculation during oxy-fuel combustion in a rotary cement kiln

The effect of Flue Gas Recirculation (FGR) during Oxy-Fuel Combustion in a Rotary Cement Kiln was analyzed by using a CFD model applied to coal combustion process. The CFD model is based on 3D-balance equations for mass, species, energy and momentum. Turbulence and radiation model coupled to a chemical kinetic mechanism for pyrolysis processes, gas–solid and gas–gas reactions was included to predicts species and flame temperature distribution, as well as convective and radiation energy fluxes. The model was used to study coal combustion with air and with oxygen for FGR between 30 and 85% as controller parameter for temperature in the process. Flame length effect and heat transfer by convection and radiation to the clinkering process for several recirculation ratios was studied. Theoretical studies predicted a located increase of energy flux and a reduction in flame length with respect to the traditional system which is based on air combustion. The impact of FGR on the oxy-fuel combustion process and different energy scenarios in cement kilns to increase energy efficiency and clinker production were studied and evaluated. Simulation results were in close agreement with experimental data, where the maximum deviation was 7%.

SOUND-SPEED INVERSION OF THE SUN USING A NONLOCAL STATISTICAL CONVECTION THEORY

Helioseismic inversions reveal a major discrepancy in sound speed between the Sun and the standard solar model just below the base of solar convection zone. We demonstrate that this discrepancy is caused by the inherent shortcomings of the local mixing-length theory adopted in the standard solar model. Using a self-consistent nonlocal convection theory, we construct an envelope model of the Sun for sound-speed inversion. Our solar model has a very smooth transition from the convective envelope to the radiative interior; and the convective energy flux changes sign crossing the boundaries of the convection zone. It shows evident improvement over the standard solar model, with a significant reduction in the discrepancy in sound speed between the Sun and local convection models.

Kinetic theory of the turbulent energy pinch in tokamak plasmas

The turbulent energy fluxes, including up-gradient ‘energy pinch’ effects, are derived using the nonlinear bounce-kinetic equation for trapped electrons and the nonlinear gyrokinetic equation for ions in toroidal geometry. The quasi-universal type of inward turbulent equipartition (TEP) energy pinch is recovered for both ions and trapped electrons, with different field dependence coefficients due to toroidal effects. A contribution from the density gradient to an outward convective energy flux is also obtained. The direction of the total energy convection is primarily determined by the competition between the TEP energy pinch and the outward density gradient driven energy convection. The magnetic shear dependence of the electron energy pinch is discussed. The energy pinches can provide possible explanations for some puzzling experimental observations.

Turbulence simulations of barrier relaxations and transport in the presence of magnetic islands at the tokamak edge

The control of transport barrier relaxation oscillations by resonant magnetic perturbations (RMPs) is investigated with three-dimensional turbulence simulations of the tokamak edge. It is shown that single harmonics RMPs (single magnetic island chains) stabilize barrier relaxations. In contrast to the control by multiple harmonics RMPs, these perturbations always lead to a degradation of the energy confinement. The convective energy flux associated with the non-axisymmetric plasma equilibrium in the presence of magnetic islands is found to play a key role in the erosion of the transport barrier that leads to the stabilization of the relaxations. This convective flux is studied numerically and analytically. In particular, it is shown that in the presence of a mean shear flow (generating the transport barrier), this convective flux is more important than the radial flux associated with the parallel diffusion along perturbed field lines.

How to define the boundaries of a convective zone, and how extended is overshooting?

In non-local convection theory, convection extends without limit and therefore an apparent boundary cannot be defined clearly, as in local theory. From the requirement that a similar structure for both local and non-local models has the same depth of convection zone, and taking into account the driving mechanism of turbulent convection, we argue that a proper definition of the boundary of a convective zone should be the place where the convective energy flux (i.e. the correlation of turbulent velocity and temperature) changes its sign. Therefore, it is a convectively unstable region when the flux is positive, and it is a convective overshooting zone when the flux becomes negative. The physical picture of the overshooting zone drawn by the usual non-local mixing-length theory is incorrect. In fact, convection is already subadiabatic (del < del(ad)) long before reaching the unstable boundary, while in the overshooting zone below the convective zone, convection is subadiabatic and superradiative (del(rad) < del < del(ad)). The transition between the adiabatic and radiative temperature gradients is continuous and smooth instead of being a sudden switch. In the unstable zone, the temperature gradient approaches a radiative temperature gradient rather than an adiabatic temperature gradient. We would like to note again that the overshooting distance is different for different physical quantities. In an overshooting zone at deep stellar interiors, the e-folding lengths of turbulent velocity and temperature are about 0.3H(P), whereas that of the velocity-temperature correlation is much shorter, about 0.09H(P). The overshooting distance in the context of stellar evolution, measured by the extent of the mixing of stellar matter, should be more extended. It is estimated to be as large as 0.25-1.7H(P) depending on the evolutionary time-scale. The larger the overshooting distance, the longer the time-scales. This is because of the participation of the extended overshooting tail in the mixing process.

On the boundaries of a convective zone and the extent of overshooting

AbstractIn this work, we will show that a proper definition of the boundary of a convective zone should be the place where the convective energy flux (i.e. the correlation of turbulent velocity and temperature) changes its sign. Therefore, it is convectively unstable region when the flux is positive, and it is convective overshooting zone when the flux becomes negative. In our nonlocal convection theory, convection is already sub-adiabatic (∇ &lt; ∇ad) far before reaching the unstable boundary; while in the overshooting zone below the convective zone, convection is sub-adiabatic and super-radiative (∇rad &lt; ∇ &lt; ∇ad). The transition between the adiabatic temperature gradient and the radiative one is continuous and smooth instead of a sudden switch. In the unstable zone, the temperature gradient is approaching radiative rather than going to adiabatic. The distance of convective overshooting is different for different physical quantities. The overshooting distance in the context of stellar evolution, measured by the extent of mixing of stellar matter, should be more extended than that of other physical quantities. It is estimated as large as 0.25–1.7 Hp depending on the evolutionary timescale.