Envelope Convection, Surface Magnetism, and Spots in A and Late B-type Stars

At or near the surface of the star where abundant elements have ionization thresholds, bumps in the heat capacity and opacity do indeed result in convective layers. We call these HCZ, He iCZ, He iiCZ, and FeCZ according to the species ionized there. We use the Modules for Experiments in Stellar Evolution (release 11342) to evolve nonrotating stellar models with masses between 0.9 and 25 $\,{M}_{\odot }$ from the pre-main sequence to close to the end of core H burning⁵ (Paxton et al. 2011, 2013, 2015, 2018, 2019). The models have an initial metallicity of Z = 0.02 with a mixture taken from Asplund et al. (2005). Convective regions are calculated using the mixing length theory (MLT) in the Henyey et al. (1965) formulation with ${\alpha }_{\mathrm{MLT}}$ = 1.6, though we do check in Section 7.2 how our results depend on the ${\alpha }_{\mathrm{MLT}}$ parameter. The boundaries of convective regions are determined using the Schwarzschild criterion. An exponentially decaying overshoot above and below convective regions is accounted for with the parameter ${\alpha }_{\mathrm{ov}}$ = 0.014 (Herwig 2000; Paxton et al. 2011). Because we are just focusing on the convective properties of A and late B-type stars, we do not include the effect of stellar winds, which are basically negligible for the main-sequence evolution of these stars

The location of convection in solar-metallicity main-sequence stars between 0.9 and 25 $\,{M}_{\odot }$ is illustrated in Figure 2. In stars above about 1.5 $\,{M}_{\odot }$, there is no thick convective envelope, just various thin convective layers, each occurring at a particular temperature. At low temperatures, the ionization of H and He i together produce one surface convective layer. Moving from lower to higher masses, each convective layer disappears as the photospheric temperature rises above each ionization temperature. This manifests itself in the locations on the Hertzsprung–Russell (HR) diagram of the boundaries separating stars with different numbers of convective layers, as seen in Figure 3. In our 1D models, surface convection completely disappears around a temperature of 10,000–11,000 K, with hotter models showing only subsurface convection zones.

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Solar-metallicity stars more massive than about 7 $\,{M}_{\odot }$ have an iron-ionization convective layer, the FeCZ (Cantiello et al. 2009). In contrast to the other envelope convection zones (HCZ, He iCZ, and He iiCZ), which are always present when their relevant ionization temperatures occur inside the star, the FeCZ occurs only in more massive stars, even though the ionization transition zone at 1.5 × 10⁵ K is present in all stars. This can be explained as follows. The left-hand side of Equation (3) is essentially the same function of temperature in all stars because opacity and therefore also ∇_κ depend almost exclusively on temperature and only very weakly on pressure, and because (as discussed above) ∇_F − ∇_g ≈ 0. The difference between intermediate-mass stars and massive stars is on the right-hand side—in more massive stars, the radiation pressure P_rad = (a/3)T⁴ accounts for a greater fraction of the total pressure P, and the specific heat capacities are correspondingly greater. To be more precise, in the absence of ionization effects, the value of ∇_ad goes from 2/5 in the limit of a monatomic ideal gas toward 1/4 as radiation pressure becomes dominant. And from Equation (1), we see that the quantity gT⁴/FP is also the same function of temperature in all stars, which can be rearranged to give the radiative pressure fraction as ${P}_{\mathrm{rad}}/P\propto {T}_{\mathrm{eff}}^{4}/{g}_{\mathrm{surf}}\propto {\text{}}L/M$, where the latter is a measure of proximity to the Eddington limit. Plotting stellar models of fixed composition on an HR diagram, the threshold for the existence of the FeCZ is indeed a line of constant ${T}_{\mathrm{eff}}^{4}/{g}_{\mathrm{surf}}$, which is approximately horizontal—convection is present for a luminosity above about 10^3.2$\,{L}_{\odot }$ at Z = 0.02 (Figure 11 in Cantiello et al. 2009).

Note that all of the subsurface convective layers are around 1–1.5 pressure scale heights in thickness. The reason for this can be seen in Figure 1: the opacity peaks are around 0.15 dex wide in temperature, and in pressure they have a width a factor 1/∇ greater than this. Given that ∇ ≈ 2/5, this corresponds to a factor of e in pressure or a little more.

As stars move along the main sequence, convective layers can appear, disappear, merge, and demerge. In Figures 4 and 5, we illustrate the evolution of these layers in a stellar model with 2.4 $\,{M}_{\odot }$. Note that due to the very low densities in the outer layers (ρ < 10⁻⁷ g cm⁻³), these convective regions contain very little mass. As such, they are completely invisible in the usual Kippenhahn plot showing the evolution of the internal structure in Lagrangian mass coordinate (see, e.g., Figure 4).

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In terms of the convective kinetic energy density, the most energetic of the layers is the deepest one, which between about 1.5 and 7 $\,{M}_{\odot }$ is the He iiCZ. In Table 1, we show the properties of the He iiCZ and the overlying radiative layer (ignoring the presence of the weaker convective layers) in 2, 2.4, and 4 $\,{M}_{\odot }$ models during core H burning.

Table 1.  Properties of the Outer Layers in Nonrotating 2 $\,{M}_{\odot }$, 2.4 $\,{M}_{\odot }$, and 4 $\,{M}_{\odot }$ Main-sequence Models with Solar Metallicity

Mini	R⋆	Xca	RHe iib	ΔRHe iic	$\bar{{H}_{{\rm{P}}}}$ d	${v}_{{\rm{c}}}$ e	$\bar{{v}_{{\rm{c}}}}$ f	$\mathrm{log}\bar{\rho }$ g	$\mathrm{log}{M}_{\mathrm{He}{\rm{II}}}$ h	$\mathrm{log}{M}_{\mathrm{top}}$ i	τturnj	τtopk	$\mathrm{log}\dot{M}$
(${M}_{\odot }$)	(${R}_{\odot }$)		(${R}_{\odot }$)	(${R}_{\odot }$)	(${R}_{\odot }$)	(km s−1)	(km s−1)	(g cm−3)	(${M}_{\odot }$)	(${M}_{\odot }$)	(hr)	(yr)	(${M}_{\odot }\,{\mathrm{yr}}^{-1}$)
2.0	1.91	0.5	1.886	0.007	0.006	1.76	0.75	−7.13	−8.38	−8.766	2.89	312	−11.26
2.4	2.36	0.47	2.330	0.009	0.007	1.07	0.42	−7.29	−8.263	−8.653	6.41	401	−11.26
4.0	3.04	0.5	3.020	0.009	0.007	0.16	0.06	−7.68	−8.401	−8.826	48.38	21	−10.14

Notes.

^aCore hydrogen fraction. ^bRadial coordinate of the top of the He iiCZ. ^cRadial extension of the He iiCZ. ^dPressure scale height at top/bottom of the He iiCZ. ^eMaximum of the convective velocity inside the He iiCZ. ^fAverage convective velocity inside the He iiCZ. ^gAverage density in the He iiCZ. ^hMass contained in the convective region. ⁱMass in the radiative layer between the stellar surface and the upper boundary of the convective zone. ^jConvective turnover time, ${\tau }_{\mathrm{turn}}:= 2{H}_{{\rm{P}}}/\langle {v}_{{\rm{c}}}\rangle$. ^kTime it takes to remove the material in the radiative layer, ${\tau }_{\mathrm{top}}:= {\rm{\Delta }}{M}_{\mathrm{top}}/\dot{M}$.

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In the He iiCZ, the transport of energy by convective motions is relatively inefficient: radiation dominates and transports more than 95% of the total flux. This is because the density is very low and the mean free path of photons correspondingly long. In this situation, convection is significantly superadiabatic, and the gradient ${\rm{\nabla }}\equiv d\,\mathrm{ln}\,T/d\,\mathrm{ln}\,P$ is explicitly calculated from the mixing length equations (e.g., Kippenhahn & Weigert 1990).

The convection in the He iCZ and photospheric HCZ is even weaker than that in the He iiCZ. The calculations presented so far have ignored the role of viscosity, as is common practice in stellar modeling, using simply the Schwarzschild criterion appropriate for convection at high Reynolds number. In a viscous fluid, the instability condition can be expressed in terms of the so-called Rayleigh number Ra as

$$\begin{eqnarray}&&\mathrm{Ra}\approx \displaystyle \frac{({\rm{\nabla }}-{{\rm{\nabla }}}_{\mathrm{ad}}){L}^{3}g}{\chi \nu }\gt {\mathrm{Ra}}_{\mathrm{crit}}\approx {10}^{3},\end{eqnarray} \tag{ 4 }$$

where again ∇ and ∇_ad are the actual and adiabatic temperature gradients, χ and ν are the thermal and kinetic diffusivities, g is gravity, and L is some relevant length scale in the vertical direction, presumably either the vertical extent of the convective zone or the scale height if that is smaller. The value of Ra_crit can be determined experimentally but varies by a factor of 2 or 3 depending on the boundary conditions. Note that the presence of rotation also alters the value of Ra_crit. We calculated the Rayleigh number in each convective layer in a 2.4 $\,{M}_{\odot }$ model with an effective temperature T_eff = 9560 K (we chose this specific model to reproduce the conditions of Vega, a well-known A-type star). Using the numbers at the location in each layer of peak convective energy density $\rho {v}_{{\rm{c}}}^{2}/2$, we find Ra ≈ 10², 10³, and 10⁷ in the HCZ, He iCZ, and He iiCZ, respectively. The reason for the difference in Ra between the layers is largely a consequence of the density: at lower density, the mean free path of the photons, which carry both heat and momentum, is greater, and both thermal and kinetic diffusivity are higher. Indeed, at the photosphere, the mean free path is comparable to the scale height. In addition, the scale height used in the numerator in Equation (4) is smaller closer to the surface.

It is not well understood how stellar convection should look at low Rayleigh numbers; there should perhaps be some kind of viscous, nonturbulent motion, similar to what is observed in laboratory experiments. Another uncertainty is our limited inability to predict the temperature gradient and convective velocities accurately; we currently use MLT. Although standard in stellar evolution modeling, we see from laboratory experiments and numerical simulations, as well as the experience of glider pilots, that MLT is based on an inaccurate physical picture. In reality, rising and falling fluid parcels move past each other rather than mixing, surviving over many scale heights; heating and cooling at the boundaries is crucial (see, e.g., Öpik 1950 for an attempt to capture the horizontal exchange of matter between up- and downstreams in a 1D theory). We use the standard MLT in our modeling; the numbers should therefore be treated as the result of a dimensional analysis, and their dependence on the mixing length free parameter ${\alpha }_{\mathrm{MLT}}$ should neither worry nor surprise us. In the literature, a number of works have attempted to simulate these convective regions using multidimensional hydro simulations, although mostly for cool A stars with surface temperatures below 8500 K (see, e.g., Kupka & Montgomery 2002; Trampedach 2004; Kochukhov et al. 2007; Freytag et al. 2012; Kupka & Muthsam 2017). For the more inefficient convective regions arising in hotter A and early B-type stars, the situation still needs to be clarified. This could be done with the aid of radiation hydro simulations, similar to those of Jiang et al. (2015) but for much lower values of the stellar luminosity.

Observationally, the transition from convective to radiative surfaces seems to occur at surface temperatures between 9000 and 10,000 K, depending on the proxy. This is marginally consistent with the results shown in Figures 2 and 3, where surface convection is calculated using the MLT disappears at around 10,000–11,000 K. Landstreet et al. (2009) find microturbulent velocities in the photospheres of stars to be approximately 2 $\,\mathrm{km}\,{{\rm{s}}}^{-1}$ in stars cooler than 10,000 K, while above that temperature there is an upper limit of around 1 $\,\mathrm{km}\,{{\rm{s}}}^{-1}$. The microturbulent velocity is interpreted as convective line broadening, but also internal gravity waves coming from the He iiCZ could contribute (Cantiello et al. 2009). Chromospheric activity indicators and large line-profile asymmetries are also present in stars cooler than about 8250 K (log T_eff < 3.92; Simon et al. 2002; Landstreet et al. 2009), a temperature that coincides pretty well with the transition to largely inefficient surface H convection (see Figures 6 and 7).

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In an astrophysical plasma, a dynamo is a configuration of the flow which is able to generate a magnetic field and sustain it against ohmic dissipation. Depending on the scale of the resulting magnetic field with respect to the scale of the kinetic energy injection, dynamos can be divided into small and large scales. In large-scale dynamos, the field has correlation length bigger than the forcing scale in the flow, while small-scale dynamos result in magnetic fields with correlation scale of order of or smaller than the forcing scale. It is generally considered that large-scale dynamos require fast rotation, characterized by a low Rossby number (the ratio of the rotation period to convective turnover time), as well as perhaps differential rotation (see, e.g., Brandenburg & Subramanian 2005 for a review).

In principle, the He iiCZ could host a small-scale dynamo. Moreover, most intermediate-mass stars rotate rapidly, with rotation periods of 12 hr or 1 day, which corresponds to a rotational period of the order of the convective turnover timescale (Rossby number is in the range 1–10; see, e.g., Figure 11), so it may be possible to build magnetic structure on a length scale larger than that of the convective motion. Assuming the dynamo produces a magnetic energy density equal to the convective kinetic energy density, we calculate magnetic field strengths up to a few hundred Gauss inside the He iiCZ—see Figure 6.

This assumption is supported by numerical simulations showing dynamo excitation by convection in the presence of rotation and shear, which also show magnetic fields reaching equipartition on scales larger than the scale of convection (Käpylä et al. 2008; Cantiello et al. 2010).

Hence, the nature of the dynamo action in the He iiCZ could depend on parameters like the stellar rotation and the shear profile. The dynamo may also be affected by a fossil or failed-fossil large-scale magnetic field penetrating upwards from the radiative zone below (see Section 7). In Figure 8, we show the expected maximum magnetic field inside any envelope convection region, assuming equipartition of magnetic and kinetic energies:

$$\begin{eqnarray}&&\displaystyle \frac{{B}_{\mathrm{eq}}^{2}}{8\pi }=\displaystyle \frac{1}{2}\,\rho \,{v}_{{\rm{c}}}^{2},\end{eqnarray} \tag{ 5 }$$

where we adopted the maximum value of ρ${v}_{{\rm{c}}}$² inside the convective zones as computed in the nonrotating models. Due to the higher power dependency on the velocity, the location of the maximum of ρ${v}_{{\rm{c}}}$² always roughly corresponds to the maximum of ${v}_{{\rm{c}}}$. Values of magnetic fields calculated using the average convective velocity and density typically differ by less than 30%. For models in the range ≃1.5–5 $\,{M}_{\odot }$ and for temperatures above $\mathrm{log}{T}_{\mathrm{eff}}$ ≃ 3.85, the He iiCZ is the convective zone hosting the strongest equipartition magnetic fields. The HCZ contributes with the strongest fields, but only for the cooler models in the lower corner of the plot.

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Now that we have established that a substantial magnetic field can in principle be generated by dynamo action in the He iiCZ, we would like to know how it could rise through the overlying radiative layer and reach the photosphere. In Cantiello & Braithwaite (2011), we compared various mechanisms to bring magnetic field upwards. Upwards advection by mass loss and ohmic diffusion are both slow, and convective overshoot is unlikely to bridge the gap between the He iiCZ and the stellar photosphere (see, e.g., Figure 5), but buoyancy can bring the field to the surface on the dynamic (Alfvén) timescale.

A magnetic field provides pressure without contributing to density, so a magnetic feature that is in pressure equilibrium with its surroundings must have a lower temperature in order to have the same density and avoid rising buoyantly; the result is inwards diffusion of heat. In this context, the mean free path of photons is large, and heat diffusion keeps magnetic features at roughly the same temperature as their surroundings. Consequently, it rises at a speed limited by aerodynamic drag, which works out to be

$$\begin{eqnarray}&&{v}_{\mathrm{drag}}\sim {v}_{{\rm{A}}}{\left(\displaystyle \frac{l}{{H}_{{\rm{P}}}}\right)}^{1/2}\sim \displaystyle \frac{{c}_{{\rm{s}}}}{{\beta }^{1/2}}{\left(\displaystyle \frac{l}{{H}_{{\rm{P}}}}\right)}^{1/2},\end{eqnarray} \tag{ 6 }$$

where l is the size of the magnetic feature, c_s is the sound speed, and β is the ratio between gas pressure and magnetic pressure. Note that this buoyant rise happens at essentially the same speed as the adiabatic magnetic buoyancy instability (Newcomb 1961; Parker 1966), which may also be relevant here. For a magnetic feature with l ∼ ${H}_{{\rm{P}}}$ and β ≈ 500, the time taken to rise one scale height is of order 5 hr in our Vega model.

The difference between the field strengths at the top and bottom of the radiative layer depends on its geometry. In a self-contained magnetic feature (a "blob" or "plasmoid"), the field strength scales as B ∝ ρ^2/3 as it rises. Given that P ∝ ρ^4/3 in the radiative layer (approximately, because ∇ ≈ 1/4), we see that β remains constant during the rise. Such plasmoids might, however, be difficult to detach from the convective layer. An alternative geometry would be a horizontal flux tube, where the scaling is B ∝ ρ. More realistic is a sunspot-like arch, where the central section of the tube rises to the surface while its ends are still in the convective layer, allowing plasma to flow downwards along the tube. Because the temperatures inside and outside the tube are essentially the same, the pressure scale heights are also roughly equal inside and outside.

This means that the plasma β is equal along the length of the tube: we have the same scaling with density B ∝ ρ^2/3 as with the plasmoid scenario. In Figure 9, we show the expected amplitude of surface magnetic fields, calculated assuming the buoyant rise of the equipartition fields shown in Figure 8 (with scaling B ∝ ρ^2/3 as the feature rises).

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Note that there could be other ways for a magnetic field to escape the subsurface convection region and reach the stellar surface. For example, Warnecke & Brandenburg (2010) and Warnecke et al. (2011) studied the magnetic flux produced by a turbulent dynamo in Cartesian and spherical geometries, respectively. They found that magnetic flux can rise above the turbulent region without the need for magnetic buoyancy. This appears to be related to the release of magnetic tension, which leads to the relaxation and emergence of the field. In this case, the magnetic field at the surface could be larger than in the case of magnetic buoyancy. Therefore, the values of surface magnetic fields reported in Figure 9 should be taken as lower estimates (but see also the discussion on the dependency of the equipartition magnetic fields on ${\alpha }_{\mathrm{MLT}}$ in Section 7.2). Even in the absence of a dynamic process bringing the dynamo-generated magnetic fields to the surface, it is important to notice that the radiative layer separating the He iiCZ from the surface contains very little mass. Even the very weak stellar winds of A-type stars can easily remove this mass on a timescale of a few hundred years or so (see Table 1). Therefore, the material present at the surface of an A star has been recently stirred by turbulent convection. Magnetic fields produced by a convective dynamo tend to decay rapidly (on an Alfvén timescale) when they are not constantly regenerated, but in the presence of the rapid rotation typical of A stars, this process can be delayed substantially by the stabilizing effect of the Coriolis force (Braithwaite & Cantiello 2013). Hence, the stellar plasma might still be substantially magnetized by the time the layers are revealed to the surface by stellar winds.

Once a magnetic feature reaches the surface of the star, it would be useful to estimate its lifetime τ_spot. As a lower limit for τ_spot, we could take the Alfvén-crossing time, or equivalently, the time a feature of size l takes to cross the photosphere while rising with a velocity v_drag. For l ∼ ${H}_{{\rm{P}}}$, this gives a timescale of the order of a few hours. Unlike in more massive stars, the upper limit to the lifetime from the removal of the spot by the loss into the wind of the gas it is threading is very long, because the mass-loss rate is very small. A more relevant upper limit is probably set by the dynamo properties. Magnetic features should probably persist for at least the smallest imaginable dynamo timescale, the convective turnover time, which is less than a day. However, with the aid of rotational effects, greater dynamo timescales could be present. If large-scale magnetic fields are produced, these could be stable on much longer timescales, akin to the long timescales of field reversals observed in some stars (∼yr). A coexistence of short-lived and long-lived magnetic features is also possible and required to explain the recent spot evolution observations of Petit et al. (2017).

As far as the direct detection of the magnetic field from spectropolarimetric observations of the Zeeman effect is concerned, the most readily measured quantity is the mean longitudinal field, i.e., the disk-averaged line-of-sight component. The detectability depends on the field strength and geometry. Unfortunately, it is not obvious what the geometry might be. On the entire surface of the Sun, we see magnetic structure on the granulation scale: a small-scale local dynamo located near the surface. If the A-star dynamo looks like this and the field everywhere is able to escape upwards, the entire photosphere will be covered in a randomly oriented magnetic field with a length scale comparable to the scale height in the He ii zone, H_c. Assuming $\sqrt{N}$ statistics in the disk-averaging, the observed mean longitudinal field would be

$$\begin{eqnarray}&&{B}_{\mathrm{long}}\sim \displaystyle \frac{{H}_{{\rm{c}}}}{{R}_{* }}{B}_{\mathrm{surf}},\end{eqnarray} \tag{ 7 }$$

where B_surf is the surface field strength as calculated in the previous section. In our Vega model, B_surf ∼ 5 Gauss and B_long ∼ B_surf/400, which is completely unobservable with current technology and also incompatible with the measured value longitudinal field of 0.6 ± 0.3 G (Lignières et al. 2009). Actually, even $\sqrt{N}$ statistics seems optimistic because it assumes the polarity of a region is random and independent of its neighbors. This is unlikely to be the case, especially in light of the constraint ${\boldsymbol{\nabla }}\cdot {\boldsymbol{B}}=0$. If there were a net mean radial field over some large part of the surface, this flux would need to be brought in from surrounding regions, via the thin convective layer.

Perhaps, though the length scale at the surface is larger than H_c, as magnetic features expand as they rise toward the stellar surface; their size would be greater by a factor (ρ_c/ρ_ph)^1/3, where ρ_c and ρ_ph are the density in the convective zone and at the photosphere, respectively, assuming isotropic expansion. Unfortunately, this is still insufficient to explain the observations.

It is possible, however, for dynamos to produce structure larger than the convective scale. In terms of observational evidence, we see that fully convective stars generally have strong dipolar fields, but that stars with a convective envelope such as the Sun have weaker dipole fields and that most of the energy is at smaller scales (see, e.g., Morin et al. 2010; Gregory et al. 2012). Intuitively, this is understandable in terms of the difficulty in different parts of a convective shell "communicating" with each other to produce coherent large-scale structures. Probably the thinner the shell is, the more serious this problem. Differential rotation probably helps though to connect different parts of a convective shell, probably most effectively in the azimuthal direction. Sunspots are produced by some process around the base of the convective envelope; the active regions we see at the surface are no larger than the scale height at that depth. With only a very thin convective layer to work with, it is difficult to imagine producing large active regions or spots on an A star (e.g., Giampapa & Rosner 1984).

Apart from the Zeeman effect, a magnetic field has various potentially observable indirect effects. For instance, in general, a magnetic field affects the temperature of a radiative photosphere. The photosphere is at a constant optical depth, but lower inside magnetic features because of the contribution of magnetic pressure. Because temperature is a function of height, magnetic spots appear hotter and brighter than their surroundings.⁶ The temperature difference is given by Cantiello & Braithwaite (2011),

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}T}{T}\approx \displaystyle \frac{{{\rm{\nabla }}}_{\mathrm{rad}}}{\beta },\end{eqnarray} \tag{ 8 }$$

which in Vega is only ∼10 K if T = 10⁴ K and β = 500, corresponding to a difference in flux density of less than 1%. If the spot is larger than the photospheric scale height, transport of heat between outside and inside the spot is less effective, and temperature is no longer purely a function of height, so the photospheric temperature difference is likely smaller. Note that even if the surface of some of these stars is convective (see, e.g., Figure 3), the energy transport will still be dominated by radiation (Figure 6). Therefore, the dominant contribution to the emergent flux is the aforementioned local dip in the photosphere, resulting in a brighter feature akin to a facula. While the effect might just be observable on more massive stars, it might be too small to explain the observations of Böhm et al. (2015) of Vega, who find photometric variability which they model as a flux density difference of 5 × 10⁻⁴ in a large spot with a radius equal to 0.3 R_*. Perhaps the observations could also be reproduced with a large number of smaller and somewhat stronger spots (Petit et al. 2017). Rotational modulation of surface structures akin to stellar spots has been reported in a large number of B and A stars observed by Kepler and TESS (Balona 2011; Balona et al. 2019; Pedersen et al. 2019). In Figure 10, we show the location of possible rotational variable stars in Balona et al. (2019), together with the predicted surface magnetic fields coming from subsurface convective regions. While it is difficult to make firm conclusions, it is interesting to note a dearth of candidates for rotational variables in the region corresponding to the minimum predicted surface magnetic fields. In the near future, more observations coming from TESS should help to further explore this correlation.

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There is some evidence of flares in Am/A stars (Balona 2013). Although normally one can invoke an otherwise undetected low-mass companion to explain flares (and also X-rays; see, e.g., Pedersen et al. 2017), it has been claimed that there are at least a few cases where the flare energy is probably too large for this to be plausible (Balona 2015). Explaining these energetic flares with a subsurface magnetic dynamo and associated coronal activity is also not easy.

In the Sun, flares are believed to be produced by magnetic reconnection in strongly magnetized regions (sunspots) having scales L ∼ 0.01 $\,{R}_{\odot }$ and B ∼ kG. In the case of very efficient energy conversion, flares can have energies as high as E_flare ≈ B²L³ ∼ 10³² erg. In the case of an A star like Vega, where the magnetic field is about two orders of magnitude smaller, one needs magnetic field concentrations on scales larger than the stellar diameter to explain the observed maximum energy E_flare ∼ 10³⁶ erg (Balona 2012, 2013), which is clearly not possible. Moreover, far-ultraviolet observations of A stars have shown that chromospheric activity seems to disappear for surface temperatures above 8250 K (Simon et al. 2002).

This said, given the typical flare recurrence time τ of 10–100 days, one can still accommodate them via a buildup of magnetic energy within the He iiCZ or in the radiative layer above it. For a typical A star, the convective luminosity L_c (the rate at which energy passes through the convective motion, of order ρv_c³ per unit area) is a fraction 10⁻²–10⁻³ of the total stellar luminosity (see Figure 6). Again, assuming the equipartition of kinetic energy density with magnetic energy density and the ability of magnetic fields to store a small fraction f of the integrated convective luminosity during the recurrence time, one can power flares with energy as large as

$$\begin{eqnarray}&&{E}_{\mathrm{flare}}\approx f\,\tau {L}_{{\rm{c}}},\end{eqnarray} \tag{ 9 }$$

which for a typical A-star only requires f ∼ 10⁻³ or so to reach 10³⁶ erg. Of course, this would imply that the flares in A stars are produced in a very different way than in Sun-like stars, something that needs further observational and theoretical studies.

Some fraction of A stars, the Ap stars, have large-scale, steady fields of 200 G–30 kG (see, e.g., Mathys 2009, for a review). These are fossil fields, anchored deep in the stellar interior. There is a one-to-one correlation (Aurière et al. 2007) between these strong magnetic fields and chemical peculiarities at the surface caused by gravitational settling and radiative levitation, processes that are usually washed out by convection and surface turbulence in other A stars. This strongly suggests that a magnetic field can suppress at least some of the convection.

A sufficiently strong field does indeed suppress convection, as in sunspots, forcing an increase in temperature gradient so that the entire energy flux is transported radiatively. However, it is not obvious where the field-strength threshold should be. One might naïvely expect convection to be suppressed by a field of greater energy density than the kinetic energy in the convective motion. However, the work of Gough & Tayler (1966), Moss & Tayler (1969), and Mestel (1970) suggests that in that case, the temperature gradient simply steepens further above the adiabatic gradient until convection resumes, and that to suppress convection completely, a field at equipartition with the thermal energy, rather than the convection kinetic energy, is required. However, these studies consider a situation where the energy flux is entirely convective, such as is approximately the case in the bulk of the solar convective zone. In ABO-star subsurface convective layers, the situation is different in that only a small fraction (∼5% or less) of the stellar heat flux is carried by convection, and that the temperature gradient is already significantly above adiabatic. It may be then that the threshold in this context is intermediate between convective and thermal equipartition.

An important clue comes from recent observations. Sundqvist (2014) looked at a sample of O stars with and without detected magnetic fields, finding that while one star with a 20 kG field (NGC 1624-2) lacks measurable macroturbulence, the other stars in the sample, which have fields up to 2.5 kG, all display macroturbulent velocities of at least 20 km s⁻¹. This result is consistent with the threshold for suppression of convection being in equipartition with thermal energy, and probably incompatible with the threshold being in equipartition with the convective energy, as equipartition field strengths for stars in this sample are about 1–2 kG (Cantiello & Braithwaite 2011). A larger sample including stars with fields between 2.5 and 20 kG should shed more light on the situation in the future.

In A stars, the field-strength thresholds for suppression of convection can simply be taken from the lower panel of Figure 6, equipartition with either convection or pressure. In the convective equipartition hypothesis, all subsurface convection should be suppressed in Ap stars. If, however, equipartition with thermal pressure is required, the threshold for He ii convection suppression is around 3 kG, so that convection would be present in the weaker-field Ap stars, and even He i convection may be present in less massive Ap stars at the bottom of the field-strength distribution. This suppression would have consequences on observables connected to this convection, e.g., microturbulence, and chemical abundances, as mentioned above.

Apart from Ap/Bp stars, it is likely that the rest of the A and late-B population harbor weaker large-scale fields in their interiors. A convective dynamo during the pre-main sequence will leave behind a magnetic field that decays on the main sequence on a dynamic timescale given in terms of the Alfvén timescale and the star's rotation by ${\tau }_{{\rm{A}}}^{2}{\rm{\Omega }}$, which goes to infinity as the field strength goes to zero. The field can therefore never disappear completely because the weaker it gets, the more slowly it decays. In Braithwaite & Cantiello (2013), we predicted, by equating this decay timescale to the stellar age, internal field strengths in Vega and Sirius of around 20 and 7 G, respectively, with somewhat weaker fields at the surface. Unable to suppress convection, these fields would coexist with it. That a large-scale field could be screened somehow below the convective layer, as suggested for instance by Gough & McIntyre (1998), seems impossible (Braithwaite & Spruit 2017).

In the case of efficient convection, the velocities calculated by the MLT have a weak dependence on the unknown mixing length parameter ${v}_{{\rm{c}}}$ ∝ ${\alpha }_{\mathrm{MLT}}$^1/3 (see the Appendix). In the case of the (sub)surface convection zones studied here, convection is very inefficient. The small energy excess content carried by the convective element is lost before it can be advected because the thermal timescale of the convective element is comparable to the dynamical timescale of the convective motion. This is quantified by the Peclet number, which measures the ratio between the thermal and the dynamical timescale (see, e.g., Maeder 2009). The convective zones studied in this work all have small Peclet numbers. In the case of inefficient (nonadiabatic) convection, the MLT has to solve a cubic equation, and the resulting dependence of the convective velocities on the uncertain ${\alpha }_{\mathrm{MLT}}$ parameter is much steeper, ${v}_{{\rm{c}}}$ ∝ ${\alpha }_{\mathrm{MLT}}$³ (Appendix). It is difficult to say what exact value of the ${\alpha }_{\mathrm{MLT}}$ parameter is needed to reproduce the properties of stellar convection in these regions. When compared to 2D and 3D numerical simulations, a range of values between 1 and 2 is usually discussed (see e.g., Kupka & Muthsam 2017), so that the values of velocity and resulting equipartition magnetic fields calculated in this work should be considered only as order of magnitude estimates. In Figure 12, we show how some of the relevant properties in our Vega model depend on the choice of ${\alpha }_{\mathrm{MLT}}$ parameters.

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