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We study the linear spatiotemporal stability of an infinite row of equal point vortices under symmetric confinement between parallel walls. These rows of vortices serve to model the secondary instability leading to the merging of consecutive (Kelvin-Helmholtz) vortices in free shear layers, allowing us to study how confinement limits the growth of shear layers through vortex pairings. Using a geometric construction akin to a Legendre transform on the dispersion relation, we compute the growth rate of the instability in different reference frames as a function of the frame velocity with respect to the vortices. This approach is verified and complemented with numerical computations of the linear impulse response, fully characterizing the absolute/convective nature of the instability. Similar to results by Healey on the primary instability of parallel tanh profiles [J. Fluid Mech. 623, 241 (2009), 10.1017/S0022112008005284], we observe a range of confinement in which absolute instability is promoted. For a parallel shear layer with prescribed confinement and mixing length, the threshold for absolute/convective instability of the secondary pairing instability depends on the separation distance between consecutive vortices, which is physically determined by the wavelength selected by the previous (primary or pairing) instability. In the presence of counterflow and moderate to weak confinement, small (large) wavelength of the vortex row leads to absolute (convective) instability. While absolute secondary instabilities in spatially developing flows have been previously related to an abrupt transition to a complex behavior, this secondary pairing instability regenerates the flow with an increased wavelength, eventually leading to a convectively unstable row of vortices. We argue that since the primary instability remains active for large wavelengths, a spatially developing shear layer can directly saturate on the wavelength of such a convectively unstable row, by
Training of the trunk or core muscles for enhanced health, rehabilitation, and athletic performance has received renewed emphasis. Instability resistance exercises have become a popular means of training the core and improving balance. Whether instability resistance training is as, more, or less effective than traditional ground-based resistance training is not fully resolved. The purpose of this review is to address the effectiveness of instability resistance training for athletic, nonathletic, and rehabilitation conditioning. The anatomical core is defined as the axial skeleton and all soft tissues with a proximal attachment on the axial skeleton. Spinal stability is an interaction of passive and active muscle and neural subsystems. Training programs must prepare athletes for a wide variety of postures and external forces, and should include exercises with a destabilizing component. While unstable devices have been shown to be effective in decreasing the incidence of low back pain and increasing the sensory efficiency of soft tissues, they are not recommended as the primary exercises for hypertrophy, absolute strength, or power, especially in trained athletes. For athletes, ground-based free-weight exercises with moderate levels of instability should form the foundation of exercises to train the core musculature. Instability resistance exercises can play an important role in periodization and rehabilitation, and as alternative exercises for the recreationally active individual with less interest or access to ground-based free-weight exercises. Based on the relatively high proportion of type I fibers, the core musculature might respond well to multiple sets with high repetitions (e.g., >15 per set); however, a particular sport may necessitate fewer repetitions.
Recently, a new parallel velocity instability has been found.(J. M. Finn, Phys. Plasmas), 2, 12 (1995) This mode is a tearing mode driven unstable by curvature effects and sound wave coupling in the presence of parallel velocity shear. Under such conditions, linear theory predicts that tearing instabilities will grow even in situations in which the classical tearing mode is stable. This could then be a viable seed mechanism for the neoclassical tearing mode, and hence a non-linear study is of interest. Here, the linear and non-linear stages of this instability are explored using a fully implicit, fully nonlinear 2D reduced resistive MHD code,(L. Chacon et al), ``Implicit, Jacobian-free Newton-Krylov 2D reduced resistive MHD nonlinear solver,'' submitted to J. Comput. Phys. (2000) including viscosity and particle transport effects. The nonlinear implicit time integration is performed using the Newton-Raphson iterative algorithm. Krylov iterative techniques are employed for the required algebraic matrix inversions, implemented Jacobian-free (i.e., without ever forming and storing the Jacobian matrix), and preconditioned with a ``physics-based'' preconditioner. Nonlinear results indicate that, for large total plasma beta and large parallel velocity shear, the instability results in the generation of large poloidal shear flows and large magnetic islands even in regimes when the classical tearing mode is absolutely stable. For small viscosity, the time asymptotic state can be turbulent.
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