Unsteady MHD free-convection flows on a porous plate with time-dependent heating in a rotating medium

The three-dimensional unsteady free-convection flows of a viscous fluid near a porous infinite vertical plate in a rotating medium in the presence of a constant transverse magnetic field are investigated under an arbitrary time-dependent heating of the plate. By using the Laplace transform technique, the Green function of the problem is determined and exact solutions are obtained for special cases of the impulsive and the accelerated heating effect for an arbitrary Prandtl number. The thermal influence on skin friction at the plate and the displacement thickness of the boundary layers are discussed.     

Numerical treatment of the unsteady hydromagnetic thermal boundary layer problem

This paper presents a suitable numerical method for the treatment of the unsteady hydromagnetic thermal boundary layer problem for flows past an infinite porous flat plate, the motion of which is governed by a general time-dependent law, under the influence of a transverse externally set magnetic field. The normal velocity of suction/injection at the plate is also assumed to be time-dependent. The results obtained on the basis of numerical approximations seem to compare favourably with earlier results (Pandeet al., 1976; Tokis, 1978). Analytical approximations are given for the cases of a plate (i) generally accelerated and (ii) harmonically oscillating. The direct numerical treatment is obviously advantageous since it allows, handling of cases where the known methods for analytical approximations are not applicable. This problem is closely related to the motions and heat transfer occurring locally on the surfaces of stars.     

Unsteady hydromagnetic free-convection flow with radiative heat transfer in a rotating fluid

We consider the buoyancy-induced flow of an electrically-conducting fluid with radiative heat transfer past a vertical flat plate of infinite length. We assume that the density obeys the simple Boussinesq equation of state while the viscosity and thermal conductivity vary with temperature, that is a compressible fluid. If the temperature of the plate is such that a time-dependent component is superimposed on a constant value, the problem is tackled by asymptotic approximation. The results are compared and contrasted with those of incompressible flow.     

On the Nonlinear Thermal Evolution of Solar Corona Structures

The thermal evolution of structures is investigated for different values of the size of structure. A simplified cooling function and a constant heating mechanism are assumed. Analytical criteria for thermal instability are obtained. It is found that the response of the thermal structure not only depends on the amplitude of the disturbances, but also on whether the disturbance increases or decreases the initial steady temperature. Additionally, the evolution of the structure is examined numerically by using a time-dependent code under the assumption that the inertia terms are small. In particular, the analytical criteria obtained for thermal instability are verified.     

Validity of the isobaric assumption to the solar corona

Many coronal heating mechanisms have been suggested to balance the losses from this tenuous medium by radiation, conduction, and plasma mass flows. A previous paper (Walsh, Bell, and Hood, 1995) considered a time-dependent heating supply where the plasma evolved isobarically along the loop length. The validity of this assumption is investigated by including the inertial terms in the fluid equations making it necessary to track the sound waves propagating in a coronal loop structure due to changes in the heating rate with time. It is found that the temperature changes along the loop are mainly governed by the variations in the heating so that the thermal evolution can be approximated to a high degree by the simple isobaric case. A typical isobaric evolution of the plasma properties is reproduced when the acoustic time scale is short enough. However, the cooling of a hot temperature equilibrium to a cool one creates supersonic flows which are not allowed for in this model.     

Evolution of active galactic nuclei broad-line region clouds: low- and high-ionization lines

The formation of quasar broad-line region (BLR) clouds via thermal instability in the presence of Alfvén heating has been discussed by Gonçalves, Jatenco-Pereira & Opher. In particular, these studies showed the relevance of Alfvén heating in establishing the stability of BLR clouds in the intercloud medium. The present paper shows the results of time-dependent calculations (we use a time-dependent hydrodynamic code) following the evolution of BLR clouds, since their formation from the 10⁷-K intercloud medium. We also calculate the UV and optical line emission associated with the clouds in order to compare with observations. Our results are compared with those of UV and optical monitoring of well-studied AGN, which suggest that the BLR is most probably composed of at least two different regions, each one giving rise to a kind of line variability, since low- and high-ionization lines present different patterns of variability. We discuss the alternative scenario in which lines of different ionization could be formed at the same place but heated/excited by distinct mechanisms, considering the Alfvén heating as the non-radiative mechanism.     

A whole-moon thermal history model of Europa: Impact of hydrothermal circulation and salt transport

A whole-moon numerical model of Europa is developed to simulate its thermal history. The thermal evolution covers three phases: (i) an initial, roughly 0.5 Gyr-long period of radiogenic heating and differentiation, (ii) a long period from 0.5 Gyr to 4 Gyr with continuing radiogenic heating but no tidal dissipative heating (TDH), and (iii) a final period covering the last 0.5 Gyr until the present, during which TDH is active. Hydrothermal plumes develop after the initial period of heating and differentiation and transport heat and salt from Europa’s silicate mantle to its ice shell. We find that, even without TDH, vigorous hydrothermal convection in the rocky mantle can sustain flow in an ocean layer throughout Europa’s history. When TDH becomes active, the ice shell melts quickly to a thickness of about 20 km, leaving an ocean 80 km or more deep. Parameterized convection in the ice shell is non-uniform spatially, changes over time, and is tied to the deeper ocean–mantle dynamics. We also find that the dynamics are affected by salt concentrations. An initially non-uniform salt distribution retards plume penetration, but is homogenized over time by turbulent diffusion and time-dependent flow driven by initial thermal gradients. After homogenization, the uniformly distributed salt concentrations are no longer a major factor in controlling plume transport. Salt transport leads to the formation of a heterogeneous brine layer and salt inclusions at the bottom of the ice shell; the presence of salt in the ice shell could strongly influence convection in that layer.     

Thermal balance of the mid-latitude F2-region at night

The thermal balance of the plasma in the night-time mid-latitude F2-region is examined using solutions of the steady-state O⁺ and electron heat balance equations. The required concentrations and field-aligned velocities are obtained from a simultaneous solution of the time-dependent O⁺ continuity and momentum equations.The results demonstrate the systematic trend for the O⁺ temperature to be 10–20 K greater than the electron temperature during the night at around 300 km, as observed at St. Santin by Bauer and Mazaudier. It is shown that frictional heating between the O⁺ and neutral gases is the cause of the O⁺ temperature being greater than the electron temperature; the greater the importance of frictional heating in the thermal balance the greater is the difference in the O⁺ and electron temperatures. A study is made of the roles played in the thermal balance of the plasma by the thermal conductivity of the O⁺ and electron gases; collisional heat transfer between O⁺ electrons and neutrals; frictional heating between the O⁺ and neutral gases; and advection and convection due to field-aligned O⁺ and electron motions. The results of the study show that, at around 300 km, electron cooling by excitation of the fine structure of the ground state of atomic oxygen plays a major role in the thermal balance of the electrons and, since the temperature of the ions is little affected by this electron cooling process, in determining the difference between the ion and electron temperatures.     

The thermospheric heating efficiency under electron precipitation conditions

A time-dependent model calculation of the interaction between energetic electrons and the thermosphere is performed to evaluate the heating efficiency. All energy channels which contribute to the neutral heating of the atmosphere are considered in detail. Neutral chemical reactions and deactivation of metastables are found to be the major heat sources. It is shown that, below 150km, the heating efficiency is on the order of 60% and nearly independent of altitude. At higher altitude, its value decreases in a manner depending on the atmosphere density and composition. It is concluded that, in the dayside polar cusp, particle heating may be an important source of thermospheric motion.     

The effects of induced magnetic field on unsteady hydromagnetic flows in a rotating medium

Unsteady flows of a viscous incompressible electrically conducting fluid filling the semi-infinite space in contact with an infinite conducting plate in a rotating medium in the presence of a transverse induced magnetic field are investigated under an arbitrary time-dependent forcing effect on the motion of the plate where the plate and the fluid rotate uniformly as a rigid body. By using the Laplace transform technique, the Greens function of the problem is determined and certain asymptotic expansions of the exact solutions are analyzed. the steady-state oscillatory flow problem is solved, and structure of waves and displacement thickness of the boundary layers are discussed for different cases of some natural parameters in the problem. The results are compared with similar flows in the presence of a constant magnetic field.     

Thermal response properties of the Earth's ionospheric plasma

The thermal response of the Earth's ionospheric plasma is calculated for various suddenly applied electron and ion heat sources. The time-dependent coupled electron and ion energy equations are solved by a semi-automatic computational scheme that employs Newton's method for coupled vector systems of non-linear parabolic (second order) partial differential equations in one spatial dimension. First, the electron and composite ion energy equations along a geomagnetic field line are solved with respect to a variety of ionospheric heat sources that include: thermal conduction in the daytime ionosphere; heating by electric fields acting perpendicular to the geomagnetic field line; and heating within a stable auroral red are (SAR-arc). The energy equations are then extended to resolve differential temperature profiles, first for two separate ion species (H⁺, O⁺) and then for four separate ion species (H⁺, He⁺, N⁺, O⁺) in addition to the electron temperature. The electron and individual ion temperatures are calculated for conditions within a night-time SAR-arc excited by heat flowing from the magnetosphere into the ionosphere, and also for typical midlatitude daytime ionospheric conditions. It is shown that in the lower ionosphere all ion species have the same temperature; however, in the topside ionosphere above about 400 km, ion species can display differential temperatures depending upon the balance between thermal conduction, heating by collision with electrons, cooling by collisions with the neutrals, and energy transfer by inter-ion collisions. Both the time evolution and steady-state distribution of such ion temperature differentials are discussed.The results show that below 300km both the electrons and ions respond rapidly (<30s) to variations in direct thermal forcing. Above 600 km the electrons and ions display quite different times to reach steady state, depending on the electron density: when the electron density is low the electrons reach steady state temperatures in 30 s, but typically require 700 s when the density is high; the ions, on the other hand, reach steady state in 700 s when the density is high, and 1500–2500 s when the density is low. Between 300 and 600 km, a variety of thermal structures can exist, depending upon the electron density and the type of thermal forcing; however steady state is generally reached in 200–1000 s.     

Unsteady hydromagnetic free-convection flow with radiative heat transfer in a rotating fluid

We study the unsteady free-convection flow near a moving infinite flat plate in a totating medium by imposing a time-dependent perturbation on a constant plate temperature. The temperatures involved are assumed to be very large so that radiative heat transfer is significant, which renders the problem very nonlinear even on the assumption of a differential approximation for the radiative flux. When the perturbation is small, the transient flow is tackled by the Laplace transform technique. Complete first-order solutions are deduced for an impulsive motion.     

Unsteady hydromagnetic free-convection flow with radiative heat transfer in a rotating fluid of arbitrary optical thickness,III

This paper investigates transient effects on the flow of a thermally-radiating and electrically-conducting compressible gas in a rotating medium bounded by a vertical flat plate. The transience is provoked by a time-dependent perturbation on a constant plate temperature. The problem particularly focusses on an optically thick gas and a gas of arbitrary optical thickness when the difference between the wall and free-stream temperatures is small. Analytical results are possible only for limiting values of time and these results are discussed quatitatively. Indeed the assumption of small temperature difference is more appropriate for plates which are opaque than transparent.     

Time-dependent heating of the solar corona

The problem of how the corona is heated is of central importance in solar physics research. Here it is assumed that the heating occurs in a regular time-dependent manner and the response of the plasma is investigated. If the magnetic field is strong then the dynamics reduces to a one-dimensional problem along the field. In addition if the radiative time in the corona is much longer than the sound travel time then the plasma evolvesisobarically. The frequency with which heat is deposited in the corona is investigated and it is shown that there is a critical frequency above which a hot corona can be maintained and below which the plasma temperature cools to chromospheric values. An evaluation of the isobaric assumption to the solar corona and the implications of time-dependent heating upon the forthcoming SOHO observations are also presented.     

Unsteady MHD free-convection oscillatory flow on a porous plate in a rotating medium

Unsteady free-convection oscillatory flow on a porous plate near an infinite vertical plate in a rotating medium in the presence of a constant transverse magnetic field is investigated under an oscillatory forcing effect on the plate. An exact solution of the problem is determined by using the Laplace transform method. The thermal influence on the skin friction at the plate is determined, and the structure of the thermal waves is presented.     

Photometry of the fast-rotating late-type star W92 in NGC 2264

The effect of a uniform transverse magnetic field on the free-convection flow of an electrically conducting fluid past a uniformly accelerated infinite vertical porous plate is discussed. Finite-difference method has been used to obtain the solution of the governing equations when the Prandtl number is not equal to unity. The velocity profiles have been shown graphically for both cases, cooling and heating of the porous plate. The numerical values of the skin-friction are entered in table and the effects of the various parameter are discussed on the flow field.     

Unsteady magnetohydrodynamic flows in a rotating elasto-viscous fluid

The unsteady flow of an incompressible electrically-conducting and elasto-viscous fluid (Walter's liquidB), filling the semi-infinite space, in contact with an infinite non-conducting plate, in a rotating medium and in the presence of a transverse magnetic field is investigated. An arbitrary time-dependent forcing effect on the motion of the plate is considered and the plate and fluid rotate uniformly as a rigid body. The solution of the problem is obtained with the help of the Laplace transform technique and the analytical expressions for the velocity field as well as for the skin-friction are given.     

On the integral equation method for radiative transfer past a rotating plate in unsteady flow

The transient effect on the flow of a thermally-radiating and electrically-conducting compressible gas in a rotating medium bounded by a vertical flat plate, is studied when the radiative flux satisfies the exact integral expression. The transience is provoked by a time-dependent perturbation on a constant plate temperature. The solution is constructed for the flow near and away from the plate by the Laplace transform method. The results are compared with the recent work of Bestman and Adjepong (1988).     

Derivation of a conservation equation for mean molecular weight for a two-constituent gas within a three-dimensional,time-dependent model of the thermosphere

Systematic circulation systems within the thermosphere create major departures of composition of both major and minor species from diffusive equilibrium. For example, latitudinal gradients in the mixing ratios of major and minor species in recent empirical models of the Earth's thermosphere are inconsistent with changes of the thermal structure alone or with temporal or spatial changes of the turbopause altitude. A conservation equation describing the time rate of change of mean molecular weight is derived for a two-species gas, in the presence of molecular and turbulent diffusion and general global circulation. The equation is fully three-dimensional and time-dependent and is derived from a combination of the general diffusion equation and the time-dependent continuity equation. In the Earth's thermosphere, the two species are [O] the light species and [N₂,O₂] the heavy species and the approach is valid since the time constants of dissociation of [O₂] and recombination of [O] are long compared with typical dynamical time constants. One of the major effects of allowing a wind-driven departure from diffusive equilibrium is that, at the solstice, the pole to pole exospheric temperature difference is increased by more than 50%, while the prevailing summer to winter meridional wind actually decreases. A conservation equation of this kind has general application to any planetary atmosphere which may be considered to be predominantly comprised of two species. Results for a three-dimensional, time-dependent thermospheric model for solstice conditions are presented for the conditions of solar heating only. The model results are compared with previous model results with composition fixed at pressure levels and with empirical temperature and composition models of MSIS.     

Noctilucent cloud formation and the effects of water vapor variability on temperatures in the middle atmosphere

To investigate the occurrence of low temperatures and the formation of noctilucent clouds in the summer mésosphere a one-dimensional time-dependent photochemical-thermal numerical model of the atmosphere between 50 and 120 km has been constructed. The model includes the important chemistry of the hydrogen and oxygen species and transport by eddy and molecular processes. The thermal balance incorporates: heating by solar ultraviolet radiation; transport of chemical potential energy; eddy diffusion and dissipation; molecular conduction; airglow emissions; and infrared cooling by carbon dioxide. A non- LTE parameterization is used to calculate 15 μm band cooling by carbon dioxide. The model self-consistently solves the coupled photochemical and thermal equations as perturbation equations from a reference state assumed to be in equilibrium and is used to consider the effect of variability in water vapor in the lower mesosphere on the temperature in the region of noctilucent cloud formation. It is found that change in water vapor from an equilibrium value of 5 ppm at 50 km to a value of 10 ppm, a variation consistent with observations, can produce a ~ 15 K drop in temperature at 82 km. It is suggested that this process may produce long periods (weeks) of cold temperatures and influence noctilucent cloud formation.