Questions about outer boundary conditions / asymptotic behaviour of lag_P / eul_P
Posted: Tue Jul 10, 2018 3:26 am
Dear Rich,
dear all,
while computing solar p-mode eigenfunctions and trying to understand the different outer boundary conditions (BCs), I came across a few difficulties.
I computed eigenfunctions for l=2 with gyre 5.1, JCD outer boundary conditions, and Model S, see the enclosed input file.
1.) The first question is whether it is possible to obtain eigenfunctions with the outer boundary condition of a vanishing Eulerian pressure perturbation? I am asking because I observe that eul_P = (P_1 /P) is steeply increasing at the outer boundary (where P_1 is the Eulerian pressure perturbation and eul_P is the output variable from gyre).
2.) After going through the description of the outer BCs implemented in gyre (in the gyre papers and equations.pdf), I am still not sure exactly which conditions from the ones discussed in Christensen-Dalsgaard (2008, "JCD") and Unno (1989, "UNNO") are actually implemented. My guess is that in the JCD condition, the Lagrangian pressure perturbation vanishes (first outer BC mentioned in JCDs paper). In the UNNO condition, I guess that in addition to zero Lagrangian pressure perturbation, also P=\rho=0 is required at the outer boundary (Unno 1989, p. 105, see text above eq. 14.10)? However, in addition, JCD (2008, page 3) discusses matching to an isothermal atmosphere, he also cites Unno (1989), and he mentions also a vacuum outer BC. In Unno (1989), there is also a discussion that does not require P = 0 and a discussion of some other options in section 21.2.
3.) I also observe that lag_P, which is \delta P / P caused by the p mode (\delta P being the Lagrangian pressure perturbation), takes its largest values at the outer boundary, where it steeply increases, see the enclosed figure. For this figure, I normalized the mode to a surface amplitude of 1cm. As I understood the JCD outer boundary condition to be \delta P = 0, I expected \delta P / P to go to zero, too, because P does not vanish in model S at the outer boundary.
What is the reason for this behaviour? Maybe there is a good reason for this which I am missing? Is it possible or does it make sense to request \delta P / P = 0 at the outer boundary? Of course, I am aware that 1/P strongly increases at the surface.
Looking at lag_P*P, which is Lagrangian \delta P caused by the p mode, it looks in principle fine, see the bottom panel in the enclosed figure.
In the plots, you find always two nearly identical curves. One is from the output lag_P from gyre, and the other one I computed from the displacement eigenfunctions to check. I used this way to check the behaviour for the modes that I have from adipls, too, and they look the same.
4) Finally, I tried the other outer BCs available, while everything else was as in the enclosed input file. UNNO gives the same answer as JCD within 1E-4 / 1E-5 in relative numbers. Is that because Model S does not vanish at the outer boundary? Running the code for VACUUM and DZIEM did not work for some reason, I posted a bug report in an additional thread.
Looking forward to any answers. Thanks in advance for any help or advice anyone can give!
Best regards,
Vincent Böning
dear all,
while computing solar p-mode eigenfunctions and trying to understand the different outer boundary conditions (BCs), I came across a few difficulties.
I computed eigenfunctions for l=2 with gyre 5.1, JCD outer boundary conditions, and Model S, see the enclosed input file.
1.) The first question is whether it is possible to obtain eigenfunctions with the outer boundary condition of a vanishing Eulerian pressure perturbation? I am asking because I observe that eul_P = (P_1 /P) is steeply increasing at the outer boundary (where P_1 is the Eulerian pressure perturbation and eul_P is the output variable from gyre).
2.) After going through the description of the outer BCs implemented in gyre (in the gyre papers and equations.pdf), I am still not sure exactly which conditions from the ones discussed in Christensen-Dalsgaard (2008, "JCD") and Unno (1989, "UNNO") are actually implemented. My guess is that in the JCD condition, the Lagrangian pressure perturbation vanishes (first outer BC mentioned in JCDs paper). In the UNNO condition, I guess that in addition to zero Lagrangian pressure perturbation, also P=\rho=0 is required at the outer boundary (Unno 1989, p. 105, see text above eq. 14.10)? However, in addition, JCD (2008, page 3) discusses matching to an isothermal atmosphere, he also cites Unno (1989), and he mentions also a vacuum outer BC. In Unno (1989), there is also a discussion that does not require P = 0 and a discussion of some other options in section 21.2.
3.) I also observe that lag_P, which is \delta P / P caused by the p mode (\delta P being the Lagrangian pressure perturbation), takes its largest values at the outer boundary, where it steeply increases, see the enclosed figure. For this figure, I normalized the mode to a surface amplitude of 1cm. As I understood the JCD outer boundary condition to be \delta P = 0, I expected \delta P / P to go to zero, too, because P does not vanish in model S at the outer boundary.
What is the reason for this behaviour? Maybe there is a good reason for this which I am missing? Is it possible or does it make sense to request \delta P / P = 0 at the outer boundary? Of course, I am aware that 1/P strongly increases at the surface.
Looking at lag_P*P, which is Lagrangian \delta P caused by the p mode, it looks in principle fine, see the bottom panel in the enclosed figure.
In the plots, you find always two nearly identical curves. One is from the output lag_P from gyre, and the other one I computed from the displacement eigenfunctions to check. I used this way to check the behaviour for the modes that I have from adipls, too, and they look the same.
4) Finally, I tried the other outer BCs available, while everything else was as in the enclosed input file. UNNO gives the same answer as JCD within 1E-4 / 1E-5 in relative numbers. Is that because Model S does not vanish at the outer boundary? Running the code for VACUUM and DZIEM did not work for some reason, I posted a bug report in an additional thread.
Looking forward to any answers. Thanks in advance for any help or advice anyone can give!
Best regards,
Vincent Böning