Text of report
The subject if our report is the influence of the structure and
physics of the uppermost superadaibatic layers on the oscillational
spectrum of p-modes. It is generally known, that p-modes represent
acoustic waves trapped in the Sun, and that they are mainly
sensitive to the structure of the upper layers of the solar
convection zone, specifically, to highly sophisticated
superadiabatic layers near the upper boundary of the convection
zone.
It is also well known, that changes in structure or physics in
these uppermost layers cause specific disturbances of
eigenfrequencies, in such way that they are appeared as a function
of frequency only. In our report we will try to use a most simple
representation of the spectrum, plotting the differences between
theoretical (that is models') and observational frequencies against
a frequency. Also, we restrict the set of frequencies in degrees of
modes up to 100 and selected only them which trapped in the
convection zone. It allow us to focused on the effects of the
uppermost layers and ignore possible sophistications from effect of
more deeper layers. Fig.1 shows the typical discrepancies of the
selected modes set. Relatively close collapsing of the points onto
one line reflects the simplicity and adequateness in some respect of
the description deeper, adiabatic layers of the convection zone. The
amplitudes of discrepancies are rather large and consist a chief
part of modern helioseismic discrepancies.
The frequencies plotted on the Fig.1 are calculated for the very
old and simple model without any attempts to adjust it to
observational frequencies. The model was only chosen as a basic
model in our illustrations and obtained with the standard
Mixing-Length Theory of convection and very old opacities in
atmosphere corresponding Cox&Tabor, 1976 tables. Helium abundance is
25 percents, Equation of State is MHD, and entropy in deep
convective layers is close, but not exactly to fitted whole solar
model. However, those parameters are fixed in our further
consideration, so we will focus on the effects of the atmospheric
opacities and structure of the superadiabatic convection region.
We start from well established effects, connected with lack of
exact knowledge of the density profile in solar radiative
atmosphere. Due to this fact, we should completely relay on the
theoretic opacities at low temperature during the model
construction. As it was revealed, that simple calculation of the
opacities may be very approximate and differ from more elaborated
one in several times. It is well known, that changing of assumed
opacities in model calculations leads to quite remarkable changes of
the discrepancy spectrum. On the Fig 2 we have plotted the frequency
discrepancy for models with artificially increased atmospheric
opacities by multiplying by factors two and three correspondingly.
One can see, that increasing of opacities leads to improving of the
agreement between theory and observation. Physically, this effect is
rather simple and can be interpreted as increasing of acoustic
radius of the Sun. But one can mentioned several limitations of such
approach to problem. First, we hardly can assume increasing the
opacities arbitrary, beyond, for example modern elaborated
calculation by Alexsander or Kuruch, which roughly corresponds to
model with tripled opacities on our figure. Secondly, This effect is
rather limited in its appearance in frequencies, and, for example,
cannot explain some small but unremovable divergences of frequencies
for the modes between 2 and 3 mHz. But in sense of amplitude, this
effect is rather prominent.
Next obvious candidate as a reason of the spectral discrepancies,
is the structure of the uppermost superadiabatic region of the solar
convection zone. Under condition of the fixed entropy of the deep
convective layers, the problem of the structure of the
superadiabatic layers is referred as a problem of convection theory.
The possible influence of the convective description on the spectrum
of oscillations broadly considered in a lot of papers, and now we
point out only, that we will concentrated on the horizontally
averaged descriptions the convective layers and moreover, included
in consideration only so-called local convective theories. In our
study, and in papers other authors (for example Ch-D, M.Montero,
Thompson), a lot of convective descriptions have been considered,
but now for sake of demonstration, we show only quite popular Canuto
and Mazzitelly descriptions in couple of variation and in compare
with MLT model.
To check the structure difference it is instructive to consider
profiles in (temperature-density) plane, as it is shown on Fig.3. On
this figure only very narrow stratum, with thickness of couple
hundred kilometers are plotted, but namely this stratum appeared to
be essentially convective sensitive. We supposes, that there a
transition between minimal entropy on the upper convective points
and a limit adiabate of deep convective zone takes place. It is
clear, that MLT predicts rather smooth transition between these two
adiabates, whereas CM-theories corresponds to more sharp transition.
It is also clear, that there is no too much room for other
completely different convective descriptions, and CM-description is
rather close to physically reasonable sharpest transition in
superadiabatic region.
Influence on discrepancy spectrum are shown on next Fig.4. It is
also clear, that sharpening of the convective transition (or
increasing temperature gradient in other words) leads to desirable
effects on the frequencies, but the amplitude of the effects is not
so large as for opacity correction for CM or CGM description. There
other variant CM-theory are plotted to demonstrate there is
possibility to amplify the convective influence, but this variant
corresponds to extremely large variations of the temperature
gradient in this place, that together with tiny details of spectrum
behaviour prevent us to be adherents of this model. But it worth to
mention, that convection variations able in contrast to atmospheric
variations affect the discrepancy behaviour in interval from 2 to 3
mHz.
Third possible, and most intriguing effects is proposed for
explanation of the frequency discrepancies related to inducing
dynamic features in the model. More exactly, we consider the
situation of appearing additional terms in the hydrostatic equation,
like so-called turbulent pressure. The exact nature of such terms
are is so important. We tried to estimate a possible influence on
the spectrum, but not to reproduce theoretically dynamic features of
turbulent convection. We constructed the model included additional
pressure term, chose the expression of turbulent pressure from
CGM96.
The model changing are quite predictable in this case -
gas-component of pressure is dropping down, what is reached by means
of density deficit - fenomena is well known when buoyancy of
magnetic tubes is discussed. This influence is rather strong
(amplitude of turbulent pressure can reach 12 percent of total one),
but very localized - the stratum revealed remarkable turbulent
pressure is in five times shallower, that rather narrow
superadiabatic region. From other side, the behaviour of total
pressure (which is only obey to hydrostatic equation) is rather
similar to CM model (Fig.5). These two factors, in our opinion,
explain rather surprising result in frequencies discrepancy - the
frequencies for model with turbulent pressure are close to the model
without one (Fig.6)
There are two expected ways of influence dynamic convective
effects on the spectrum of oscillations - via model changes and via
changes of physics of oscillation modes. When first is appeared to
be not very significant, we check possible influence of second -
modal effect. Possible changing in hydrostatic equation can be
traced via liniarized oscillations equation and finally appeared as
changing of Gamma_1 - adiabatic compressibility, corresponding to
the fact, that reaction of the plasma on external compression may
differ for the turbulent component of pressure. We recalculate the
theoretical frequencies for so-called reduced Gamma_1 in turbulent
model (follow a receipt of C.Rothental) and got another results,
that an influence on frequencies is not very significant as well
(Fig.7). This result is less paradoxical, then previous, because it
is known, that even simple variation of sound speed (or variation of
Gamma_1) localized in considered layers produce quite small effect
on frequencies, due to evanescent behaviour of the waves in this
region.
We can summarize the situation in next way. The possibility to
improve the coincidence of the spectrum in a frame of local models
and adiabatic oscillations is mostly exhausted by considered
effects. Adjusting of these effects in combinations are able to
reproduce the observational frequencies up to 3 mHz, but anyway
failed to reproduce the spectrum of high frequency range. For
further improving nondiabatic effects and horizontally nonhomohenius
model are appeared to be perspective. The influence of dynamic
features of convections seems to be much less significant then
expected.