Since the advent of extensive ultraviolet observations of cool stars, it has been clear that the stellar activity observed is not directly correlated with the star's position on the HR diagram (Basri and Linsky 1979, Stencel et al. 1980). Observations of an important chromospheric diagnostic, the MgII resonance lines, led to the conclusion that stellar chromospheric activity had only a weak dependence on spectral type, and exhibited large variations within a given spectral type. Because of the strong observed correlation of solar activity with magnetic fields, the field is thought to be a natural candidate for the extra parameter which predicts the level of activity. Unfortunately, it is quite difficult to measure magnetic fields directly in most cool stars. Another method with which to examine correlations between magnetic field and stellar activity indirectly is the hypothesis that magnetic fluxes are directly related to a combination of the convective and rotational parameters of a star through its generation in a magnetic dynamo. The α-ω dynamo theory (Parker, 1979) predicts a direct correlation between differential rotational velocities and field generated. Durney and Robinson (1982) predict basically a linear dependence of the emergent flux on the angular velocity of the star. One might therefore expect that in stars with the same fundamental stellar parameters, the amount of activity observed would depend on the rotational velocities. This is difficult to test because most cool stars are slow rotators and only a few rotational velocities are known.

For a sample of 58 late type stars we analyse the relations between the soft X-ray flux density Fx, the Ca II H and K line-core flux density FH+K, and parameters determining the global stellar structure. By analysing the soft X-ray spectra from 15 stars we determine the coronal temperatures T and specific emission measures per unit area ζ. We discuss the dependence of T on B-V, Fx and stellar radius R. The diagram of the specific emission measure ζ against the temperature T is interpreted in terms of a coronal model consisting of static loops. Also, a search for time variations in the X-ray flux has been performed.

Three topics of current interest are treated in this review:

1. a) The success and shortcomings of dynamo models for the solar cycle are explained, and oscillator models discussed briefly.

2. b) The intermittent (flux tube) nature of magnetic fields in convection zones leads to new conjectures about stellar dynamos. Arguments are given that the dynamo may operate in the overshoot region below a convective envelope. Mean-field theory for intermittent fields is illustrated.

3. c) I review nonlinear dynamo models and some attempts to interpret observational results concerning late-type active stars.

Dolginov (1977) pointed out that a non spherical distribution of molecular weight μ can drive a Biermann type “battery” mechanism in a star and so generate a toroidal magnetic field. We consider uniformly rotating axisymmetric models of this type, containing a ‘primeval’ poloidal field, in which the currents maintaining the toroidal field flow along poloidal field lines, thus ensuring that the configuration is torque free. When the molecular weight distribution is chosen so as to produce a meridional circulation which exactly cancels the Eddington-Sweet circulation, then battery fields of order 1000 gauss can be maintained against ohmic decay in models appropriate to Ap stars near the main sequence with period of order 5 days.

Calculations by Drobyshevski and Yuferev (1974) suggest that magnetic flux is ‘topologically pumped’ to the bottom of a convecting Benard layer as the electrical conductivity is increased. This is shown to be mistaken: however, compressible flows where motion is mainly around fieldlines can confine flux near their base.

I see my assignment in this talk as being to focus on the interaction between the magnetic fields produced by dynamo action and the dynamics of the fluid flow which drives this dynamo, and to make some connections to the solar and stellar dynamo problems. To do this really requires we start with a fluid dynamical model that satisfies relevant laws of fluid dynamics, and in which the flow can actually respond to the induced magnetic field. Thus the so-called kinematic dynamo models are not enough for our purposes, and we must address the full MHD dynamo problem in a self-consistent way. For example, we do not allow ourselves the license to vary independently the convection and differential rotation, as is commonly done in kinematic dynamo calculations, because the laws of physics do not allow that. We have been attempting to do self consistent MHD dynamo modeling at Boulder for the past several years, starting from a nonlinear fluid dynamical model for convection in a rotating spherical shell. This model we believe is physically complete in itself, with a minimum of ad hoc assumptions. It is much simpler than the real sun, but contains a lot of the physics we consider most relevant to the solar and stellar dynamo problem. I like to view the model more as an analog to the solar or a stellar convection zone, rather than as an approximation—much as a laboratory rat or monkey is used as an analog to a human being in many medical experiments. Each exists and is physically complete, and much, though not all, of their biochemistries are the same or are closely related.

Regeneration of magnetic fields by dynamo effect in the presence of turbulence is of interest both for astro-geophysical applications and because it may produce unwanted effects in large scale fast breader reactors. In such machines magnetic Reynolds numbers RM of the order of 100 can be reached, possibly in excess of the critical value for dynamo action.

A computational model, based on diffusion, differential rotation, and meridional circulation, has been developed to simulate the transport of magnetic flux on the Sun. Using Kitt Peak magnetograms as input, we have determined a best-fit diffusion constant by comparing the computed and observed fields at later times. Our value of 730 ± 250 km2/s is consistent with Leighton's (1964) estimate of 770–1540 km2/s and is significantly larger than Mosher's (1977) estimate of 200–400 km2/s. This suggests that diffusion may be fast enough to account for the observed polar magnetic field reversal without requiring a significant assist from meridional currents.

I discuss the connection between the rotation rate of late-type dwarf stars and their level of surface “activity” (as deduced from visible, UV, and X-ray observations) from both theoretical and observational perspectives.

Previous work on the concentration of magnetic field by cellular convection in a Boussinesq fluid is extended to a perfect gas, so that the gas pressure is reduced in the presence of a strong magnetic field. Attention is focussed on 2-dimensional flows and on the case of small layer depths, so that the pressure is effectively uniform in field free regions. It is shown that if the field is sufficiently strong, flux sheets with significantly reduced pressures and densities may form. Criteria are established which measure the relative importance of this effect to the more familiar ‘magnetic drag’ which acts to prevent the concentration of field by generating counter vorticity.

We have constructed a simple parametrized mean field dynamo model that includes the dynamical interaction between the magnetic field and differential rotation. This system of seven coupled nonlinear ordinary differential equations has finite amplitude oscillatory solutions (corresponding to Parker's dynamo waves) when the dynamo number D>1. We have studied two regimes. In the first, the velocity shear is reduced by the Lorentz force and there are stable periodic solutions for all D>1. In the second there is a transition from strictly periodic oscillations to aperiodic (chaotic) behaviour as D is increased. This simple example shows that nonlinear hydromagnetic dynamos can produce aperiodic cycles, with Maunder minima, as observed in the sun and other late-type stars.

In this review based largely on observations with the IUE and Einstein satellites, I will summarize the different roles that magnetic fields play in controlling the structure and energy balance in the chromospheres and transition regions of late-type stars. Solar observations clearly show that magnetic flux tubes are the dominant structural element in the solar atmosphere, but the rotational modulation of plages (structures that are bright in ultraviolet emission lines) that overlie dark starspots provide strong evidence that magnetic flux tubes are the dominant structural elements in late-type stellar atmospheres as well. The wide range of radiative loss rates (and thus heating rates) observed in chromospheric and transition region emission lines also provides evidence for the importance of magnetic fields, but it is not yet clear whether the most active stars can be understood in terms of a large fractional coverage by solar-like magnetic flux tubes or whether brighter flux tubes are needed. I propose that the existence of a boundary between solar-like stars and those with little or no hot plasma, as well as the different types of G-K giants and supergiants, can be understood in terms of the fractional surface coverage by closed magnetic structures. Transition region downflows, the chromospheric heating mechanism, and the relative heating rates at different layers can be simply explained by the control of the energy balance by magnetic fields. Finally, I will intercompare models computed for active and quiet regions on the Sun with similar models computed for active and quiet stars, that is stars with intrinsically bright or weak emission lines.

Calculations are presented for the inhomogeneous magnetic field structure above a stellar photosphere which has magnetic flux tubes located at the downdraughts of its supergranulation pattern. Regions can be delineated where the ambient magnetic energy density is large or small compared with the thermal energy density derived from a model atmosphere. This enables the relative importance of magnetic versus non-magnetic heating mechanisms to be assessed. For the quiet Sun, over half the chromospheric emission must be supplied non-magnetically, whilst the network and active regions require a magnetic supply. For other late-type stars, a simple working rule suggests that when the magnetic field is strong enough to be directly observable, the chromosphere will be magnetically dominated.

We review the origins and uses of scaling laws in studies of stellar outer atmospheres, with particular emphasis on the properties of coronal loops. The evidence for a fundamental structuring of the Solar corona is reviewed and a discussion of thermodynamic scaling laws is presented. In order to intercompare different theories for coronal formation and heating, it is necessary to recast the theories in terms of observable quantities. As an example, we present a discussion of magnetic field-related heating and scaling laws which can be obtained relating coronal pressure, temperature and magnetic field strength; available data are shown to be consistent with scaling laws obtained in this way. However, some parameters of the theory must be treated as adjustable at the present time and it is necessary to examine data from other stars in order to determine whether these are true parameters in coronal heating. We examine some of the difficulties involved in using unresolved stellar data when dealing with loop atmospheres, by first treating the Sun as an unresolved source. Using the detailed observations now available we examine the limits of applicability of single-loop models. The possibilities and limits of stellar data are then discussed.

The 6 cm radiation of solar active regions marks the legs of dipolar loops which have their footpoints in lower-lying sunspots. The temperatures T ≈ 106K and longitudinal magnetic field strengths Hℓ ≈ 600 Gauss at heights h ≈ 4 × 109 cm above sunspot umbrae. The circularly polarized emission at 6 cm delineates the magnetic structure above sunspot penumbrae. The 20 cm radiation of solar active regions delineates the ubiquitous coronal loops previously detected at X-ray wavelengths. We infer semilengths L ≈ 5 × 109cm, maximum electron temperatures Te(max) ≈ 3 × 106K, emission measures and electron densities Ne ≈ 109cm−3 for the 20 cm bremsstrahlung. Future V.L.A. observations at 20 cm may be used to determine the magnetic field strength of coronal loops. Changes in temperature and magnetic structure before and during solar bursts are briefly discussed.

Traditionally stellar sizes and shapes have been determined by analyzing the optical light curves of eclipsing binaries. Now it is possible to extend our knowledge to structures above the photosphere by making similar observations at X-ray, UV, and radio wavelengths. We report the first such observations, together with their geometrical and physical interpretation, which allow us to determine coronal and chromospheric properties of the stars in the RS CVn binary AR Lac. In this paper we discuss the properties of the system, Einstein observations which provide a reasonably good X-ray light curve, IUE data which enable us to tie the inferred coronal features to the surfaces, and radio observations which allow us to separate geometric and intrinsic variations by acting as a flare monitor. We also discuss the construction of simple coronal model which, when applied to the X-ray light curve, yields the sizes and shapes of the coronae in this system. In another paper (Paper II: Walter, Gibson, and Basri, 1982), the geometrical properties of the coronae are discussed and used to determine the physical conditions within.

The last ten years of observations have shown beyond doubt the fundamental role played by the magnetic field in the heating of stellar atmospheres. After the recognition of the extreme inhomogeneity of the solar corona, two basic new trends have appeared in the theoretical literature on the coronal heating problem. One is the adoption of a global point of view that stresses the connection of the properties of the upper layers to those of the underlying ones. In this way a general framework is provided, capable of accomodating many possible heating mechanisms that need not to be specified at this stage. The second novelty is the explicit inclusion in the theory of the inhomogeneous nature of the stellar envelopes, as a result of the presence of magnetic fields. The present status of knowledge on the subject as determined by the above evolution of the theoretical approach will be reviewed.

The high sensitivity and angular resolving power of the VLA permits observations for the first time of the steady (quiescent) microwave emission from stellar coronae as well as detailed studies of stellar flares. Gary and Linsky (1981) first detected steady 6 cm emission from χ1 Ori (GO V) and UV Cet (M5.6e V), and Topka and Marsh (1982) subsequently detected quiescent emission from both components of EQ Peg (dM3.5e + dM4.5e).

The solar-coronal complex is the only astrophysical system whose X-ray structure has to some degree been spatially resolved. The most important lesson that we have learned from these relatively high-resolution observations is that the corona is highly structured, comprising a variety of closed, loop-like regions of enhanced radiative output. Furthermore, coronal radiation loops are now known to be spatially coincident with magnetic loops which confine the radiating plasma.

A theory for the thermal stability of hot coronal loops is presented, which is based on the resonant electrodynamic heating theory of Ionson (1982) and the evaporation/condensation scenario of Krall and Antiochos (1980). The theory predicts that gradual changes in the length of a loop or in its magnetic field strength can trigger catastrophic changes in the X-ray visibility of the loop, without the need for a change in the magnetic field topology.

A natural explanation is thereby given for the observations of X-ray brightenings in loops and loop evacuations with coronal rain.