Temperature fluctuations in the solar photosphere

The general problem of interpreting granulation data, in particular Edmonds' r.m.s. intensity fluctuation distribution against heliocentric angle , is discussed.A method is developed for investigating a variety of models of inhomogeneous departures from radiative equilibrium using two dimensional solutions of the equation of radiative transfer, and theoretical r.m.s. intensity fluctuation distributions are computed. It is found that only a very narrow range of models yields distributions which exhibit the essential features of Edmonds' distribution (a center-of-disk value of 14 % and a maximum value of 20.5 % at a heliocentric angle of 53°). The feature of these models is a maximum in the temperature fluctuations of about 660 K r.m.s., which represents a temperature difference between hot and cold regions of 2000 K, at a depth of about 250 km below ₅₀₀₀= 0.03. Below this the temperature fluctuations decrease rapidly in the next 70 km.These results are interpreted in terms of convective and radiative transport of energy. Velocities of the order of 8 km/sec are deduced in the essentially convective regime near 320 km, decreasing through 4 km/sec near the temperature fluctuation maximum to negligible values in the radiative region above 200 km.These features are shown to be consistent with modern theoretical and laboratory studies of convection in incompressible fluids. Further, these studies indicate that a second temperature fluctuation should occur at the bottom of a convective layer. For this reason, further photospheric models are studied in which, below the region of small temperature fluctuations near 320 km, the fluctuations increase sharply. For one of these models a theoretical intensity r.m.s. distribution is obtained which closely fits not only the maximum at = 53° in Edmonds' observed distribution but also the initial decrease and smaller minimum near 24°.Of the National Bureau of Standards and the University of Colorado.     

The Evershed motion in sunspots

The Evershed motion is postulated as a steady, laminar flow of material along a limiting field line which separates the umbral magnetic field from the penumbral. Assuming that the Evershed flow starts from the spot-base with a velocity which is adequate to carry the convective flux at that level, the velocity at the surface comes out to be of the order of 1 km/sec, in good agreement with the observed Evershed velocity.Supported in part by the National Science Foundation [GP-5391] and the Office of Naval Research [Nonr-220 (47)].     

A model of the sunspot umbra

Equations governing the structure of the umbra of a single spot have been integrated on the spot-axis. It is shown that a consistent umbral model can be obtained only for a narrow range of the electron pressures at the spot surface. The spot center is found to be at a depth of about 400 km below the normal solar surface. The reduced energy flux observed at the surface is assumed to be due to the effect of the spot-magnetic field on subphotospheric convection and an empirical factor is introduced to take into account the reduction in the convective energy flux. With an inferred expression for as a function of the internal and magnetic-energy density it is shown that in a consistent model the physical variables in the spot approach their ambient values at about 2330 km below the undisturbed solar surface and at the same depth the energy flux approaches the normal solar value and the magnetic interference with convection vanishes. The present investigation is a refinement of an earlier paper by Chitre (1963).Supported in part by the Office of Naval Research [Nonr-220(47)] and the National Science Foundation [GP-5391].     

Polarized radiation diagnostics of magnetohydrodynamic models of the solar atmosphere

Solar magnetic elements and their dynamical interaction with the convective surface layers of the Sun are numerically simulated. Radiation transfer in the photosphere is taken into account. A simulation run over 18.5 minutes real time shows that the granular flow is capable of moving and bending a magnetic flux sheet (the magnetic element). At times it becomes inclined by up to 30° with respect to the vertical around the level ₅₀₀₀ = 1 and it moves horizontally with a maximal velocity of 4 km/s. Shock waves form outside and within the magnetic flux sheet. The latter cause a distinctive signature in a time series of synthetic Stokes V-profiles. Such shock events occur with a mean frequency of about 2.5 minutes. A time resolution of at least 10 seconds in Stokes V recordings is needed to reveal an individual shock event by observation.The National Center for Atmospheric Research is sponsored by the National Science Foundation     

Convective cloud heights as a diagnostic for methane environment on Titan

The appearance of convective clouds in Titan’s troposphere has been documented from ground-based observation for more than a decade. Cloud tops have been reported between 14 and 25 km. Higher resolution Cassini data have shown smaller portions of the cloud system can reach up to 42 km. We use the Titan Regional Atmospheric Modeling System (TRAMS) to explore environments which allow convective clouds to reach the tropopause. In general, cloud tops remain below 30 km, but for environments where the surface humidity of methane is greater than 50%, a small portion at the center of the cloud rises briefly to higher altitudes; for ?65% humidity, the cloud top reaches nearly to the tropopause (∼40 km). A number of other parameters also have noticeable affects on cloud top such as nucleation critical saturation, haze abundance, and collisional growth of cloud particles.     

The greenhouse of Titan

Both non-gray radiative equilibrium and gray convective equilibrium calculations for Titan indicate that the discrepancy between the equilibrium temperature of an atmosphereless Titan and the observed infrared temperatures can be explained by a massive molecular hydrogen greenhouse effect. The convective calculations indicate a probable minimum optical depth of 14, corresponding to many tens of km-atm of H₂, and total pressures of ～0.1 bar. The tropopause is several hundred km above the Titanian surface and at a temperature of about 90°K. Methane condensation is likely at this level. Such an atmosphere is unstable against atmospheric blow-off unless typical mesosphere scale heights are < 25km, an unlikely situation. Blow-off can also be circumvented by exospheric temperatures near the freezing point of hydrogen. It is considered more plausible that the present atmosphere is in equilibrium between outgassing and blow-off of the one hand and accretion from protons trapped in a hypothetical Saturnian magnetic field on the other; or exhibits uncompensated blow-off of outgassing products. To maintain the present blow-off rate without compensation for all of geological time requires an outgassing equivalent to the volatilization of a few km of subsurface ices. Photo-dissociation of these volatilized ices produces the observed high abundance of H₂ as well as large quantities of complex organic chromophores which may explain the reddish coloration of the Titanian cloud deck. An extensive circum-Titanian hydrogen corona is postulated. Surface temperatures as high as 200°K are not excluded. Because of its high temperatures and pressures and the probable large abundance of organic compounds, Titan is a prime target for spacecraft exploration in the outer solar system.     

Convective energy transport in stellar atmospheres

The motion of convective cells in an environment which changes rapidly with depth is examined. In such an environment a cell may move through regions with different levels of ionization and with associated differences in heat capacity. The energy equation is cast in a manner which is independent of the history of these cells. The convective flux at a given level of the atmosphere is written as an average over an ensemble of cells originating at a range of other levels. A procedure for correcting the temperature gradient for these non-local effects is described and results for a model solar atmosphere are given. The principal results are: (1) The rms velocity varies smoothly and is non-zero well into the photosphere (e.g.,v _rsm=1.4 km/sec at =0.2); (2) Convective overshoot reduces the radiative flux to 60% and 90% of the total at =2.5 and 0.2 respectively; and (3) The interior adiabat of the convective envelope is less sensitive to the assumed value of the average cell size than in the usual treatment of convection.Supported in part by the National Science Foundation [GP-9433, GP-9114], the Office of Naval Research [Nonr-220(47)], and Air Force Grant AG-AFOSR-171-67.     

The photolytic stability of the Martian atmosphere

Several recent suggestions for stabilizing the Martian atmosphere against photolysis have proved untenable. However, downward convective transport as well as a low altitude (0–35 km) aerosol, which catalyzes two-body molecular recombination reactions, can bring about such stability. The ‘effective’ convection velocity and ‘average’ two-body reaction rate coefficients required by observed abundances are evaluated. The computed profiles of CO and O at high altitude are shown to agree well with observations.     

Nonlinear simulations of solar rotation effects in supergranules

Nonlinear calculations for the three-dimensional and time dependent convective flow in a plane parallel layer of fluid are carried out with parameter values appropriate for supergranules on the Sun. A rotation vector is used which is tilted from the vertical to represent various latitudes. For the incompressible fluid used in this model the solar rotation produces turning motions sufficient to completely twist a fluid column in about one day. It is suggested that this effect will be greatly enhanced in a compressible fluid. The tilted rotation vector produces anisotropies and systematic Reynolds stresses which drive mean flows. The resulting flows produce a rotation rate which increases inward and a meridional circulation with poleward flow along the outer surface.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Density and stress distribution in the moon

A model is presented for the lateral variations of density within the Moon. The model gives rise to a gravitational potential which is equal to the observed potential at the lunar surface, moreover, it minimizes the total shear-strain energy of the Moon. The model exhibits lateral variations of about ±0.25 g cc^–1 within 50 km depth. The variations, however, reduce to ±0.06 and ±0.008 g cc^–1 within layers at 50–135 and 135–235 km respectively, and they become negligible below this region. The associated stress differences are found to be about 50 bar within 600 km depth, having their maximum values of about 90 bars at a depth of about 250 km. On the basis of these stress differences a strength of about 100 bar is concluded for the upper 400 km of the lunar interior for the last 3.3 b.y.Paper dedicated to Professor Harold C. Urey on the occasion of his 80th birthday on 29 April, 1973.The Lunar Science Institute is operated by the Universities Space Research Association under Contract No. NSR 09-051-001 with the National Aeronautics and Space Administration. This paper is Lunar Science Institute Contribution No. 117.     

Modeling the morphological diversity of impact craters on icy satellites

Impact craters on icy satellites display a wide range of morphologies, some of which have no counterpart on rocky bodies. Numerical simulation studies have struggled to reproduce the diversity of features, such as central pits and transitions in crater depth with increasing diameter, observed on the icy Galilean satellites. The transitions in crater depth (at diameters of about 26 and 150 km on Ganymede and Callisto) have been interpreted as reflecting subsurface structure. Using the CTH shock physics code, we model the formation of craters with diameters between 400 m and about 200 km on Ganymede using different subsurface temperature profiles. Our calculations include recent improvements in the model equation of state for H₂O and quasi-static strength parameters for ice. We find that the shock-induced formation of dense high-pressure polymorphs (ices VI and VII) creates a gap in the crater excavation flow, which we call discontinuous excavation. For craters larger than about 20 km, discontinuous excavation concentrates a hot plug of material (>270 K and mostly on the melting curve) in the center of the crater floor. The size and occurrence of the hot plug are in good agreement with the observed characteristics of central pit craters, and we propose that a genetic link exists between them. We also derive depth versus diameter curves for different internal temperature profiles. In a 120 K isothermal crust, calculated craters larger than about 30 km diameter are deeper than observed and do not reproduce the transition at about 26 km diameter. Calculated crater depths are shallower and in good agreement with observations between about 30 and 150 km diameter using a warm thermal gradient representing a convective interior. Hence, the depth-to-diameter transition at about 26 km reflects thermal weakening of ice. Finally, simulation results generally support the hypothesis that the anomalous interior morphologies for craters larger than 100 km are related to the presence of a subsurface ocean.     

Umbral boundaries,convection, and the depth of sunspots

The boundary between the umbra and penumbra of a sunspot is consistently observed to be very sharp, on the order of 500 km. Approximating the sunspot as a static region in a homogeneous medium with a radiative surface, temperature distributions resulting from a variety of convective motions exterior to the sunspot are calculated. The calculations suggest that, for the exterior convection to produce the observed boundary, the maximum depth of the region of inhibited convection below a sunspot umbra is on the order of 10³ km.     

Onset of Convection,Heat Flow and Thickness of the Europa‘s ice Shell

The observational evidence given by Galileo spacecraft about Europa supports an icy rigid layer of several kilometers over another ductile layer of ice in convection, which floats over an internal ocean of liquid water. Before the onset of convection, heat is transmitted into the crust by conduction. The heat flow analysis in the potentially convective layer gives values higher than those obtained previously by tidal dissipation models, and suggests that the ice may be limited to a thin layer of ∼4 km total thickness. This revised version was published online in July 2006 with corrections to the Cover Date.     

The nature of the sunspot phenomenon

This paper points out the basic relation between the conversion of thermal energy into convective fluid motion (Alfvén waves when a strong vertical magnetic field is present) and the convective transport of thermal energy. It is shown that heat transport necessarily accompanies convective driving of fluid motion. Convective motions restricted to a layer whose thickness is a small fraction of the local scale height can divert no more than the same fraction of the energy into Alfvén waves. But if the convecting layer extends over many scale heights, then the convective forces may convert more energy into fluid motion than they transport. Hence the creation of a cool sunspot requires convection extending coherently over several scale heights, at least 500 km. This requirement is basically just the familiar thermodynamic efficiency of an ideal heat engine. The calculations establish that convection need not be much less efficient than the ideal.     

Heat flow, lenticulae spacing, and possibility of convection in the ice shell of europa

Two opposing models to explain the geological features observed on Europa’s surface have been proposed. The thin-shell model states that the ice shell is only a few kilometers thick, transfers heat by conduction only, and can become locally thinner until it exposes an underlying ocean on the satellite’s surface. According to the thick-shell model, the ice shell may be several tens of kilometers thick and have a lower convective layer, above which there is a cold stagnant lid that dissipates heat by conduction. Whichever the case, from magnetic data there is strong support for the presence of a layer of salty liquid water under the ice. The present study was performed to examine whether the possibility of convection is theoretically consistent with surface heat flows of ∼100-200 mW m⁻², deduced from a thin brittle lithosphere, and with the typical spacing of 15-23 km proposed for the features usually known as lenticulae. It was obtained that under Europa’s ice shell conditions convection could occur and also account for high heat flows due to tidal heating of the convective (nearly isothermal) interior, but only if the dominant water ice rheology is superplastic flow (with activation energy of 49 kJ mol⁻¹; this is the rheology thought dominant in the warm interior of the ice shell). In this case the ice shell would be ∼15-50 km thick. Furthermore, in this scenario explaining the origin of the lenticulae related to convective processes requires ice grain size close to 1 mm and ice thickness around 15-20 km.     

The structure of a sunspot

Typical intensity profiles across a sunspot at several heliocentric angles are selected from recent observations of the Wilson Effect. In addition, the profile of the mean intensity at the surface of the spot is inferred from these observed profiles.With these data, the transfer equation is solved for the two-dimensional source function distribution within the sunspot for several models of the opacity distribution. For an opacity model in which unit optical depth in the umbra occurs at least 700 km below unit optical depth in the mean photosphere, it is possible to reproduce qualitatively all the features of the observed profiles.Although no assumption is made about the extent of the umbra below the surface, these solutions clearly show that, at a depth of 700 km below unit optical depth in the photosphere, the diameter of the umbral region, which is 10800 km at the surface, has increased to about 12000 km. Thus the shape of the umbral region below the surface is part of an inverted cone of semi-vertical angle approximately 45°. The run of gas pressure and density in the umbra is computed for the model and compared with the corresponding photospheric values.Of the National Bureau of Standards and University of Colorado.     

On convection in ice I shells of outer Solar System bodies, with detailed application to Callisto

It has been argued that the dominant non-Newtonian creep mechanisms of water ice make the ice shell above Callisto's ocean, and by inference all radiogenically heated ice I shells in the outer Solar System, stable against solid-state convective overturn. Conductive heat transport and internal melting (oceans) are therefore predicted to be, or have been, widespread among midsize and larger icy satellites and Kuiper Belt objects. Alternatively, at low stresses (where non-Newtonian viscosities can be arbitrarily large), convective instabilities may arise in the diffusional creep regime for arbitrarily small temperature perturbations. For Callisto, ice viscosities are low enough that convection is expected over most of geologic time above the internal liquid layer for plausible ice grain sizes (?a few mm); the alternative for early Callisto, a conducting shell over a very deep ocean (>450 km), is not compatible with Callisto's present partially differentiated state. Moreover, if convection is occurring today, the stagnant lid would be quite thick (∼100 km) and compatible with the lack of active geology. Nevertheless, Callisto's steady-state heat flow may have fallen below the convective minimum for its ice I shell late in Solar System history. In this case convection ends, the ice shell melts back at its base, and the internal ocean widens considerably. The presence of such an ocean, of order 200 km thick, is compatible with Callisto's moment-of-inertia, but its formation would have caused an ∼0.25% radial expansion. The tectonic effects of such a late, slow expansion are not observed, so convection likely persists in Callisto, possibly subcritically. Ganymede, due to its greater size, rock fraction and full differentiation, has a substantially higher heat flow than Callisto and has not reached this tectonic end state. Titan, if differentiated, and Triton should be more similar to Ganymede in this regard. Pluto, like Callisto, may be near the tipping point for convective shutdown, but uncertainties in its size and rock fraction prevent a more definitive assessment.     

The speeds of coronal mass ejection events

The outward speeds of mass ejection events observed with the white light coronagraph experiment on Skylab varied over a range extending from less than 100 km s^–1 to greater than 1200 km s^–1. For all events the average speed within the field of view of the experiment (1.75 to 6 solar radii) was 470 km s^–1. Typically, flare associated events (Importance 1 or greater) traveled faster (775 km s^–1) than events associated with eruptive prominences (330 km s^–1); no flare associated event had a speed less than 360 km s^–1, and only one eruptive prominence associated event had a speed greater than 600 km s^–1. Speeds versus height profiles for a limited number of events indicate that the leading edges of the ejecta move outward with constant or increasing speeds.Metric wavelength type II and IV radio bursts are associated only with events moving faster than about 400 km s^–1; all but two events moving faster than 500 km ^–1 produced either a type II or IV radio burst or both. This suggests that the characteristic speed with which MHD signals propagate in the lower (1.1 to 3 solar radii) corona, where metric wavelength bursts are generated, is about 400 to 500 km s^–1. The fact that the fastest mass ejection events are almost always associated with flares and with metric wavelength type II and IV radio bursts explains why major shock wave disturbances in the solar wind at 1 AU are most often associated with these forms of solar activity rather than with eruptive prominences.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

The moon's thermal state and an interpretation of the lunar electrical conductivity distribution

Heat convection, being a more general theory than conduction theory, compels one to give reasons for using the latter theory as the basis of thermal evolution studies. Such reasons are supplied by considerations of material rheology.The specific case of the thermal regime of the Moon is first considered as a steady state problem. It is demonstrated that no plausible creep resistance of lunar material and heat generation is compatible with a purely conductive theory of lunar thermal evolution. The most plausible, steady state models give convective cores extending to within 200–300 km of the surface. The radial temperature gradients in such cores is virtually confined to a thermal boundary layer but averages to about a tenth of degree/km. The (steady) central temperature for the most plausible lunar rheologies lie between 600–1000°C. Such models are compatible with the first interpretations of lunar magnetometry. The case for considering the lunar thermal state as such a quasi-static state is discussed.It is also predicted that in very local zones the viscous dissipation of the general circulation produces much higher temperatures. Chemical differentiation and seismicity would have their origin in such low viscosity zones.     

Dynamical aspects of the venus 4-day circulation

Evidence for the 4-day retrograde zonal circulation of the upper Venus atmosphere is summarized. The ‘moving flame’ phenomenon, convective instability to a mean shear and tidal forcing are discussed as possible dynamical explanations for the 4-day rotation. Tidal forcing seems feasible only if momentum diffusion is molecular in nature. Convective instability to a mean shear, although it can account for the magnitude of the 4-day circulation, must be supplemented by another mechanism, the ‘moving flame’ say, to explain the direction of the zonal motion. However, numerical computations indicate that the ‘moving flame’ by itself can account for both the magnitude and direction of the 4-day rotation. It appears that the stable stratification above altitudes of about 60 km is an essential factor in the ‘moving flame’ mechanism for generating the retrograde atmospheric rotation.