Vertical shear-induced resonant triads in Keplerian discs

2019 ◽  
Vol 488 (3) ◽  
pp. 4207-4219 ◽  
Author(s):  
Yuri Shtemler ◽  
Michael Mond

ABSTRACT The vertical-shear instability (VSI) is studied through weakly non-linear analysis of unmagnetized vertically isothermal thin Keplerian discs under small radial temperature gradients. Vertically global and radially local axisymmetric compressible perturbations are considered. The VSI excites three classes of quasi-resonant triads of non-linearly interacting modes characterized by distinct temporal evolution. Most of the triads belong to the two-mode regime of non-linear interaction. Such triads are comprised of one unstable non-linear mode that grows quasi-exponentially, and two other modes that practically decoupled from the former. The latter two modes perform non-linear oscillations around their either linear prototypes (class I) or respective initial values (class II). The rest of the resonant triads belong to class III where all three modes exhibit non-linear oscillations. The proposed model describes an intermediate non-linear stage of the VSI prior to its saturation.

2020 ◽  
Vol 499 (3) ◽  
pp. 3222-3232
Author(s):  
Yuri Shtemler ◽  
Michael Mond

ABSTRACT Clustering of resonant triads that are induced by vertical-shear instability (VSI), driven by the combined effect of the vertical speed shear and small temperature gradients, is studied for vertically isothermal thin unmagnetized Keplerian discs. The authors’ recent study of isolated VSI resonant triads is extended to illustrate their clustering. The coupling conditions for two VSI resonant triads with one common mode are derived and generalized to higher dimension clustering. The clustering of two, three, and four triads connected via one common mode is numerically simulated. The numerical simulations demonstrate the chaotization of non-linear oscillations about the prototypes of the linearly stable modes with a growing cluster’s dimension that is accompanied by a decrease of the characteristic time of chaotization and an increase of the characteristic frequency of perturbations. The chaos associated with the VSI resonant clustering is believed to precede transition to sustainable turbulence in astrophysical discs.


2004 ◽  
Vol 61 (7-12) ◽  
pp. 1055-1071
Author(s):  
N. N. Gerasimova ◽  
V. G. Sinitsin ◽  
Yu. M. Yampolski

2013 ◽  
Vol 194 (3) ◽  
pp. 1920-1940 ◽  
Author(s):  
B. N. Kuvshinov ◽  
T. J. H. Smit ◽  
X. H. Campman

Author(s):  
Stanley M. Yamashiro ◽  
Takahide Kato

A minimal model of cerebral blood flow and respiratory control was developed to describe hypocapnic and hypercapnic responses. Important non-linear properties such as cerebral blood flow changes with arterial partial pressure of carbon dioxide (PaCO2) and associated time dependent circulatory time delays were included. It was also necessary to vary cerebral metabolic rate as a function of PaCO2. The cerebral blood flow model was added to a previously developed respiratory control model to simulate central and peripheral controller dynamics for humans. Model validation was based on previously collected data. The variable time delay due to brain blood flow changes in hypercapnia was an important determinant of predicted instability due to non-linear interaction in addition to linear loop gain considerations. Peripheral chemoreceptor gains above a critical level, but within normal limits, was necessary to produce instability. Instability was observed in recovery from hypercapnia and hypocapnia. The 20 sec breath-hold test appears to be a simple test of brain blood flow mediated instability in hypercapnia. Brain blood flow was predicted to play an important role with non-linear properties. There is an important interaction predicted by the current model between central and peripheral control mechanisms related to instability in hypercapnia recovery. Post hyperventilation breathing pattern can also reveal instability tied to brain blood flow. Previous data collected in patients with chronic obstructive lung disease was closely fitted with the current model and instability predicted. Brain vascular volume was proposed as a potential cause of instability despite cerebral autoregulation promoting constant brain flow.


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