โ Scribed by Stuart D. Jordan
No coin nor oath required. For personal study only.
The low chromosphere now seems definitely to require mechanical heating, and dissipation of initially acoustic waves by shocking is one of the most promising possibilities. Results of recent calculations indicate that the weak shock theory may be applicable here, but discrepancies exist among various applications of this theory, and the explanations offered to date are not completely satisfactory. It is shown here that the different approximations used by different authors to evaluate the mechanical flux integral play an important role in producing these discrepancies, in addition to the already well known effects of the density scale heights and the wave periods. Arguments are presented favoring Ulmschneider's method for evaluation of this flux integral.
There has been disagreement about weak shock theory applied to the solar chromosphere since Osterbrock (1961) claimed that it could predict the mechanical heating in both magnetic (network and plage) and non-magnetic regions. This conclusion has been challenged or, at least, modified many times. Pikel'ner and Livshits (1965) argued that weak shock propagation in magnetic regions (B>~ 10 G, in the chromosphere) would be so strongly affected by non-linear processes that no reliable estimates of the heating could be obtained. Jordan (1970) then took the equations of Osterbrock and, retaining wave periods T of the order of 100 s or more in the non-magnetic regions, showed that, for more current atmospheric models unavailable to Osterbrock, a catastrophically rapid growth of the shock strength occurs.
On the order hand, Ulmschneider (1970, 1971a, b) has concluded that the shock strength grows slowly enough in the non-magnetic chromosphere to preserve the validity of the weak shock theory up to the base of the transition region. Furthermore, his calculations yield an almost constant value for the shock strength throughout this region, and the corresponding mechanical heating does not differ significantly from local net radiation loss estimates. Since these calculations are done for wave periods which are short compared to 100 s (cf. Stein, 1968), Ulmschneider concludes that this accounts for the smaller values of the computed shock strength.
While it is true that shorter periods yield smaller values for the shock strength, it is not obvious, and, in fact, proves to be untrue, that the discrepant results are due mainly to the different periods. The calculations reported here show that the major source of the discrepancy is due to differences in the way the mechanical flux integral is approximated.
To see this, consider the mechanical flux past a point in a fluid due to a periodic wave train of period T:
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