The aim of this appendix is to explain in a detailed way how to obtain the boundary condition .
A.1. Set of equations
The full set of linearized ideal MHD equations in the case of adiabatic oscillations is
where the quantities , , denote the Eulerian perturbations of the pressure, density and the magnetic field, and , denote the Lagrangian perturbation of the pressure and the density. The temporal dependence is assumed to be . The displacement and the Lagrangian perturbation of pressure are given by Eq. 3 and Eq. 4. As we consider high frequencies, we adopt the Cowling approximation, i.e. we neglected the perturbation of the gravitational potential, .
In order to simplify this system of equations, we have made several assumptions. First, we assumed large radial wave numbers, therefore,
where X is any perturbed quantity. More, we suppose that horizontal wave numbers are small compared with radial wave numbers, ie
We also assumed . The result for the Lorentz force with these approximations is given by Eq. 6-8. After simplifications thanks to the previous assumptions and a little algebra, we get the following system
A.7 is the equation of continuity, A.8 the equation of motion on , A.10 the equation of motion on , A.12 the equation of motion on . A.9, A.11 have been introduced to get a system of differential equations of the first order.
The following variables have been introduced
with defined by Eq. 1.
V and are defined by
We also introduced the dimensionless frequency by
with the range of frequencies we consider in this paper [1200,2300] , it corresponds to a range of [10,19] for .
We see that the system Eq. A.7-A.12 is singular at the pole and the equator. These two limit cases needs a special treatment (see Appendix B)
The upper and lower boundaries of the magnetic layer depend, as expected, on the value of the photospheric magnetic field . For the top of this layer, we required that tends to vacuum field () as tends to zero. Therefore, we can put the right hand side of Eq. A.10 and Eq. A.12 equal to zero. This requires us to extend the atmosphere sufficiently high to have low density and then verify . At the base of the magnetic layer we need two conditions. The first one corresponds to a negligible magnetic pressure compared with the gas pressure; for this we require . The second condition concerns Alfvénic waves. We require that their wavelengths are much shorter than the pressure scale height, i.e. . We give in Table 1 several values of at the top and the base of the magnetic layer.
Table 1. Limits of the magnetic boundary layer for different values of the magnetic fields, and in each case, the value of the two parameters and describing the importance of the magnetic field.
The general solution vector of this system can be written as a linear combination of six linearly independent solutions. However, we assume that the star is an isolated system, i.e. we reject inward propagating waves which could come from infinity. Hence, this reduces our solutions to three inside the magnetic layer
There are two steps in the numerical integration. The first one consists of finding the three peculiar solutions for each radial position x in the magnetic layer. To do this, we start from the top of the layer with analytical peculiar solutions which are found thanks to a local analysis. We then integrate them numerically, with a method of Runge-Kutta, through the magnetic layer to the base (). The next step of the numerical integration is to find the coefficients of the linear combination . This is done at the base of the boundary layer adding another condition. Since at one has , we assumed a full decoupling between p-modes and Alfvénic modes beneath the magnetic layer. We add also the condition that there are only inwards propagative Alfvénic waves under the magnetic layer (see also Roberts & Soward 1983). Then the solution at the base of the magnetic layer has the following form
The Alfvénic solutions are obtained with a WKBJ method. The components of at the fit only depend on the two variables and . (see below). Therefore, the unknown quantities are and which are found matching the solutions Eq. A.18 and Eq. A.19. This leads to a linear homogeneous system of equations which has non trivial solutions if the secular determinant is zero. This condition gives the relation between and at , that is to say the boundary condition .
A.2. External boundary conditions
We adopted here a local analysis of the system assuming that the coefficients of the system Eq. A.7-A.12 are constant (we assume to be in an isothermal atmosphere). The solutions have the form . This gives a relation of dispersion
One gets three groups of solutions. As said before, we keep only outward propagating or decreasing waves in the atmosphere. The solution may be written as
The three remaining solutions correspond to three different waves numbers of Eq. A.20.
being the dimensionless non-magnetic acoustic cut-off frequency and is given by
The corresponding vector solution can be written as followed
This gives a second vector (with )
For the last vector corresponding to the wave number
one can choose with (with )
Next, each component of the vector S is integrated by a method of Runge-Kutta of the fourth order with adapted step size. The result is the values , and .
A.3. Inner boundary conditions
To obtain the expression for these coefficients, we require new conditions at the base of the magnetic layer. In this part of the star, the ratio is very small. This means that the Alfvénic speed is negligible compared with the sound speed. Since the two perturbations of B and p, say the Alfvénic waves and pressure waves are propagating with very different speeds, it appears a decoupling between the two. Mathematically, this decoupling appears in the system Eq. A.7- A.12 because of the small value of . Then, a solution of the system Eq. A.7-A.12 can be written as
The subscript "p" refers to the p-mode solution, whereas the subscript "A" corresponds to the Alfvénic solution. Since, the effect of the magnetic field is very small below the magnetic layer, we can again use a perturbative approach to find the solutions.
A.3.1. p-modes solution
where we neglected the terms due to since they are small compared with and as we can see on Eq. A.30. This system corresponds to the system of equations in the case of non-magnetic and adiabatic radial oscillations (in the Cowling approximation).
Then, we can write the vector solution for p-modes as function of and .
A.3.2. Alfvénic modes
This corresponds to the first order of perturbations in . We suppose for Alfvénic waves that and also . The system governing the Alfvénic waves is
To solve this system we use a WKBJ approach with the form
k obeys the following relation of dispersion
We keep only solutions which propagate inwards. One has, then, two kinds of solutions
Then, at the base of the magnetic layer the inner boundary condition write as
The solutions with the subscripts 1, 2, 3 are the values at after integration in the boundary layer, and the solutions with the subscript "A" are given by Eq. A.42 and A.43. Requiring that the determinant of the matrix in Eq. A.45 is zero, one obtains the function .
The results presented in the previous section are not valid at the pole () and the equator () because of the singular behaviour of the equations. For these two points we need to find two systems of equations.
B.1. The pole
At the pole the Lorentz force vanishes. Therefore, the system of equations reduces to the second order Eq. A.30-A.32 (i.e. the system for radial p-modes in the non-magnetic and adiabatic case)
B.2. The equator
At the equator the Lorentz Eq. 6-8 gets a simpler expression
It can be shown that the system at the equator reduces to the second order
The formula Eq. 33 provides a good test for numerical calculations. Once the numerical solution is found the eigen vectors are injected in the following formulas Eq. C.1-C.2 which leads to another evaluation of the frequencies. We neglect the terms corresponding to B since the eigen frequencies corresponds to the oscillations below , i.e. where the dynamical effect of B is negligible compared with the effect of gas pressure.
The relative differences between the numerical solutions of and with the frequencies calculated with Eq. C.1 and Eq. C.2 are given in Fig. C.1. We see that the consistency of the calculation are better than for the real part and better than 0.01 for the imaginary part, for the mode and .
© European Southern Observatory (ESO) 2000
Online publication: March 28, 2000