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SUMMARY:S.M.Churilov\, I.G. Shukhman (Institute of Solar-Terrestrial Physi
 cs\, Irkutsk)
DTSTART:20230126T110000Z
DTEND:20230126T120000Z
DTSTAMP:20260423T005852Z
UID:mmandim/50
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/mmandim/50/"
 >Critical layer and weakly nonlinear evolution of unstable quasi-monochrom
 atic perturbations in shear flows</a>\nby S.M.Churilov\, I.G. Shukhman (In
 stitute of Solar-Terrestrial Physics\, Irkutsk) as part of Mathematical mo
 dels and integration methods\n\n\nAbstract\nBy Howard’s semicircle theor
 em\, in a plane-parallel shear flow\, the real part of the phase velocity 
 $c$ of an unstable perturbation $\\sim f(z)\\exp[ik(x-ct)]$ is between the
  minimum and maximum of the flow velocity $V_x = U(z)$ and coincides with 
 $U$ on a critical level $z=z_c$ so that $\\mathrm{Re} \\\, c = U(z_c)$. In
  a narrow neighborhood of this level\, — so called critical layer (CL)\,
  — liquid particles are in phase resonance with the wave and intensively
  interact with it. In the framework of an idealized statement of the probl
 em taking no account of dissipation (viscosity)\, unsteadiness\, and nonli
 nearity\, the perturbation eigenfunction $f(z)$ is singular on the critica
 l level. Taking into consideration any one of these factors makes the solu
 tion regular\, but the relative magnitude of the perturbation inside the C
 L remains large. For this reason\, it is the CL that makes the leading-ord
 er contribution into nonlinear interactions\, and this fact simplifies the
  study of a weakly nonlinear evolution of an unstable perturbation.\n\nEac
 h of these factors specifies the length scale associated with it\, namely\
 ,\n\n(i) viscous $L_ν = (k^3 \\mathrm{Re})^{-1/3} = O(ν^{1/3})$\,\n\n(ii
 ) unsteady $L_t = |(kU'_c A)^{-1} d|A|/dt| = O(γ)$\,\n\n(iii) nonlinear $
 L_N \\sim |A/U'_c|^δ$\,\n\nwhere $\\mathrm{Re}$ is Reynolds number\, $A(t
 )$ is the perturbation amplitude\, $δ$ depends on the behavior of $f(z)$ 
 in the neighborhood of the critical level\, and the prime denotes the deri
 vative in $z$. The greatest of these scales determines not only the width 
 of the CL\, but also the behavior of the solution inside it. Therefore\, i
 t is appropriate to distinguish between viscous\, unsteady\, and nonlinear
  CLs\, taking into account that the CL kind may change in the process of e
 volution in accordance with the scale ratio.\n\nAs a result\, only a limit
 ed number of basic scenarios of evolution do exist. The realization of one
  scenario or another\, or some sequence of them depends mainly on the degr
 ee of supercriticality of the basic unstable flow and on the nature of the
  singularity at the critical level.\n
LOCATION:https://researchseminars.org/talk/mmandim/50/
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