17:15 〜 18:45
[MIS21-P06] 非粘性平行シアー流の quasi mode
キーワード：線形安定性解析、臨界レベル、連続スペクトル、ランダウ極
The occurrence of a linear unstable mode in an inviscid parallel shear flow is often interpreted as the interaction between two discrete neutral waves (e.g. Hayashi & Young, 1987 [1]). However, the smooth profile of the basic flow sometimes prevents one from understanding so in a straightforward manner (e.g. Iga, 2013 [2]). This is due to the advent of a continuous spectrum attributed to critical levels, at which the phase velocity of a linear wave is equal to the basic flow velocity. When the gradient of the background vorticity does not vanish at a critical level corresponding to an eigenmode, its eigenfunction always has the logarithmic singularity at the level and the mode can not be discrete. If the gradient of the background vorticity is non-zero throughout the whole space (except for some levels), no discrete neutral mode exists even when a linear unstable mode yields at a certain critical wave number. Iga (2013) added a small window region(s) having a uniform background vorticity to the profile of a basic flow to retrieve such missing discrete eigenvalues. Although Iga (2013)'s method finds plausible discrete eigenmodes, it is unclear whether such a modification of the profile gives a proper understanding of linear waves or not.
''Quasi modes'' may give an alternative approach to expand the interpretation that the interaction between two discrete modes produces an unstable mode. Corresponding to poles discovered from the next Riemann sheet beyond a branch cut lying in a continuous spectrum on the complex plane of the phase velocity, quasi modes behave as exponentially decaying waves and can be regarded as an aggregate of singular eigenmodes of a continuous spectrum (e.g. Briggs et al., 1970[3]; Shrira & Sazonov, 2001 [4]). I suspect that the interaction between a discrete and a quasi mode or between two quasi modes occurs, and want to confirm this hypothesis through this study.
In the present presentation, I utilize the method of spectral deformation (e.g. Crawford & Hislop, 1989[5]; Spencer & Rasband, 1997[6]) to obtain poles of quasi modes from the linearized governing equations. Eigenvalue problems for some parallel shear flows were solved numerically with the Chebyshev collocation method. The below figure shows the results without (with) a spectral deformation in the left (right) panel for the case when the profile of the background flow has a smooth part sandwiched by two regions where the gradients of the background vorticity vanish. The abscissas of each panel represent the nondimensional streamwise wavenumber, and the ordinates are the real part of the dimensionless phase velocity for the upper panels and the imaginary part for the lower ones. In the right panels, the scatter plots are colored just for ease of visibility. I can observe, from the result with the spectral deformation, that the pole of a quasi mode covered with the continuous spectrum exists.
Similar difficulties concerning a continuous spectrum appear in the study of ideal magnetohydrodynamics waves within the Earth's outer core which may cause geomagnetic variations (Nakashima & Yoshida, submitted [7]), addressed by future work.
[References]
[1] Hayashi, Y.-Y., Young, W.R. (1987) J. Fluid Mech., 184, 477-504, doi: 10.1017/S0022112087002982
[2] Iga, K. (2013) J. Fluid Mech., 715, 452-476, doi: 10.1017/jfm.2012.529
[3] Briggs, R.J., Daugherty, J.D., Levy, R.H. (1970) Phys. Fluids, 13, 421–432, doi: 10.1063/1.1692936
[4] Shrira, V.I., Sazonov I.A. (2001) J. Fluid Mech., 446, 133-171, doi: 10.1017/S0022112001005742
[5] Crawford, J.D., Hislop, P.D. (1989) Ann. Phys., 189, 265-317, doi: 10.1016/0003-4916(89)90166-8
[6] Spencer, R.L., Rasband, S.N. (1997) Phys. Plasmas, 4, 53-60, doi: 10.1063/1.872497
[7] Nakashima, R., Yoshida, S., submitted to Geophys. Astrophys. Fluid Dyn., the non-peer reviewed preprint has been posted on arXiv (doi: 10.48550/arXiv.2310.01341) and EarthArXiv (doi: 10.31223/X5Z67T).
''Quasi modes'' may give an alternative approach to expand the interpretation that the interaction between two discrete modes produces an unstable mode. Corresponding to poles discovered from the next Riemann sheet beyond a branch cut lying in a continuous spectrum on the complex plane of the phase velocity, quasi modes behave as exponentially decaying waves and can be regarded as an aggregate of singular eigenmodes of a continuous spectrum (e.g. Briggs et al., 1970[3]; Shrira & Sazonov, 2001 [4]). I suspect that the interaction between a discrete and a quasi mode or between two quasi modes occurs, and want to confirm this hypothesis through this study.
In the present presentation, I utilize the method of spectral deformation (e.g. Crawford & Hislop, 1989[5]; Spencer & Rasband, 1997[6]) to obtain poles of quasi modes from the linearized governing equations. Eigenvalue problems for some parallel shear flows were solved numerically with the Chebyshev collocation method. The below figure shows the results without (with) a spectral deformation in the left (right) panel for the case when the profile of the background flow has a smooth part sandwiched by two regions where the gradients of the background vorticity vanish. The abscissas of each panel represent the nondimensional streamwise wavenumber, and the ordinates are the real part of the dimensionless phase velocity for the upper panels and the imaginary part for the lower ones. In the right panels, the scatter plots are colored just for ease of visibility. I can observe, from the result with the spectral deformation, that the pole of a quasi mode covered with the continuous spectrum exists.
Similar difficulties concerning a continuous spectrum appear in the study of ideal magnetohydrodynamics waves within the Earth's outer core which may cause geomagnetic variations (Nakashima & Yoshida, submitted [7]), addressed by future work.
[References]
[1] Hayashi, Y.-Y., Young, W.R. (1987) J. Fluid Mech., 184, 477-504, doi: 10.1017/S0022112087002982
[2] Iga, K. (2013) J. Fluid Mech., 715, 452-476, doi: 10.1017/jfm.2012.529
[3] Briggs, R.J., Daugherty, J.D., Levy, R.H. (1970) Phys. Fluids, 13, 421–432, doi: 10.1063/1.1692936
[4] Shrira, V.I., Sazonov I.A. (2001) J. Fluid Mech., 446, 133-171, doi: 10.1017/S0022112001005742
[5] Crawford, J.D., Hislop, P.D. (1989) Ann. Phys., 189, 265-317, doi: 10.1016/0003-4916(89)90166-8
[6] Spencer, R.L., Rasband, S.N. (1997) Phys. Plasmas, 4, 53-60, doi: 10.1063/1.872497
[7] Nakashima, R., Yoshida, S., submitted to Geophys. Astrophys. Fluid Dyn., the non-peer reviewed preprint has been posted on arXiv (doi: 10.48550/arXiv.2310.01341) and EarthArXiv (doi: 10.31223/X5Z67T).