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[MIS12-P03] Revisit of the stability of inviscid horizontal shear flows with the method of spectral deformation
Keywords:critical layers, continuous spectra, spectral deformation, Landau damping, pseudomomentum
Critical layers and their concomitant continuous spectra often appear in the studies of inviscid fluid dynamics and ideal magnetohydrodynamics. Because the spectrum can cover and hide discrete eigenvalues on which we want to focus, a method that enables one to extract embedded discrete eigenvalues from a continuous spectrum is required. Here, I attempt to utilize the method of spectral deformation (Crawford & Hislop, 1989[1]) for retrieving the lost eigenvalues. The change of the independent variable in the approach can indirectly move the positions of critical layers from points on the real axis of the coordinate to ones on the complex plane, hence a deformed continuous spectrum. Some papers (e.g. Spencer & Rasband, 1997[2]) demonstrated that the method is effective for various problems accompanied by such continuous spectra. According to them, the extracted “eigenvalues” (or the phase velocities) are not real but complex, being equivalent to the Landau damping in a broad sense.
The present presentation is devoted to a reinvestigation into the stability of horizontal shear flows in an inviscid fluid. If the profile U(y), where y is the cross-stream direction, of the basic flow is piecewise linear, the interaction between two neutral interfacial Rossby waves localized at the discontinuities of the gradient of U(y) can yield unstable modes (e.g. Iga, 2013[3]). Even for smooth profiles in which the discontinuities of the gradient in the piecewise profiles are locally replaced by curved lines, the physical interpretation based on the resonance of neutral waves remains valid (e.g. Carpenter & Guha, 2019[4]). However, smoother profiles such as U(y)=tanh(y) bring a continuous spectrum that makes a pair of resonating neutral waves undetectable (e.g. Iga, 2013[3]). To retrieve the covered eigenvalues, I solved the eigenvalue problem for the tanh-shaped profile with the Chebyshev collocation method into which the method of spectral deformation is incorporated. The resulting embedded “eigenmodes” possess phase velocities that are pure imaginary numbers, as shown in the below figure. Since my results are inconsistent with those of Iga (2013)[3], further discussion in terms of pseudomomentum is inevitable.
[Reference]
[1] Crawford, J.D., Hislop, P.D. (1989) Ann. Phys., 189, 265-317, doi: 10.1016/0003-4916(89)90166-8
[2] Spencer, R.L., Rasband, S.N. (1997) Phys. Plasmas, 4, 53-60, doi: 10.1063/1.872497
[3] Iga, K. (2013) J. Fluid Mech., 715, 452-476, doi: 10.1017/jfm.2012.529
[4] Carpenter, J.R., Guha, A. (2019) Phys. Fluids, 31, 081701, doi: 10.1063/1.5116633
The present presentation is devoted to a reinvestigation into the stability of horizontal shear flows in an inviscid fluid. If the profile U(y), where y is the cross-stream direction, of the basic flow is piecewise linear, the interaction between two neutral interfacial Rossby waves localized at the discontinuities of the gradient of U(y) can yield unstable modes (e.g. Iga, 2013[3]). Even for smooth profiles in which the discontinuities of the gradient in the piecewise profiles are locally replaced by curved lines, the physical interpretation based on the resonance of neutral waves remains valid (e.g. Carpenter & Guha, 2019[4]). However, smoother profiles such as U(y)=tanh(y) bring a continuous spectrum that makes a pair of resonating neutral waves undetectable (e.g. Iga, 2013[3]). To retrieve the covered eigenvalues, I solved the eigenvalue problem for the tanh-shaped profile with the Chebyshev collocation method into which the method of spectral deformation is incorporated. The resulting embedded “eigenmodes” possess phase velocities that are pure imaginary numbers, as shown in the below figure. Since my results are inconsistent with those of Iga (2013)[3], further discussion in terms of pseudomomentum is inevitable.
[Reference]
[1] Crawford, J.D., Hislop, P.D. (1989) Ann. Phys., 189, 265-317, doi: 10.1016/0003-4916(89)90166-8
[2] Spencer, R.L., Rasband, S.N. (1997) Phys. Plasmas, 4, 53-60, doi: 10.1063/1.872497
[3] Iga, K. (2013) J. Fluid Mech., 715, 452-476, doi: 10.1017/jfm.2012.529
[4] Carpenter, J.R., Guha, A. (2019) Phys. Fluids, 31, 081701, doi: 10.1063/1.5116633