tokens: be measured, then this would provide constraints on the field strength and distribution. We also explored the possibility of non-WKBJ effects relevant to mixed modes of low degree. Our results demonstrate that the form of the quantization condition remains very similar to that in the case where WKBJ is applicable throughout the star. However, in our more general expression, an additional offset g appears, given in equation( 90 ). This is unrelated to magnetic effects and has its origins in the non-WKBJ nature of strongly coupled cavities. Measurements of this may provide information about the structure of the evanescent zone and base of the convective envelope. The effects of rotation have not yet been incorporated and would be the subject of future work. APPENDIX A: TREATMENT OF THE UPPER LAYERS In this appendix, we discuss the convective envelope and evanescent zone (regions P and E), for the purposes of connecting the solutions in these regions to those in the g-mode cavity (treated in Section 4 ), and thereby obtaining the mixed-mode quantization condition. A1 Region P Here we still shall assume N 2 0, making the region isentropic. Equations( 15 )( 16 ) then combine to give where . Making a change of variable , we obtain the DE where The first term in the expression for is expected to dominate, since this corresponds to the squared acoustic wavenumber, which greatly exceeds 1/ r for typical of stochastically driven oscillations in the outer parts of the convection zone. Retaining only the first term in when considering amplitude factors in the WKBJ approximation, the WKBJ solution to ( A2 ) is corresponding to where C 3 and are arbitrary constants. Here r t is the lower turning point where vanishes. We shall assume for simplicity that this exceeds r 0 , where N 2 drops to zero from a positive value further in. Otherwise, the lower limit of the integral should be replaced by r 0 , as ( A1 ) does not apply where N 2 0. The constants C 3 and are in the first instance determined by the requirement that the solution( A4 ) is a continuation of the one that is specified just interior to r e . Asserting the continuity of the solutions multiplying C 1 and C 2 in equation( 34 ), we write The amplitude factors A 1 and A 2 and the phases 1 and 2 in general have to be determined from numerical integrations if the WKBJ approach cannot be used throughout region E, as will be the case for small . A1.1 Incorporating a surface boundary condition The solution representing a mode of oscillation must also satisfy a boundary condition at the surface. Following the usual procedure when constructing WKBJ solutions, we assume that this condition can be met with a solution of the form ( A5 ) by specifying the phase . The value of 0 is determined by the surface boundary condition. Given the above continuations, it is readily found that the condition that the linear combination occurring in equation( 34 ) leads to the correct phase condition specified above in this
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