In this paper, we study the forcing of baroclinic critical levels, which arise in stratified fluids with horizontal shear flow along the surfaces where the phase speed of a wave relative to the mean flow matches a natural internal wavespeed. Linear theory predicts the baroclinic critical layer dynamics is similar to that of a classical critical layer, characterized by the secular growth of flow perturbations over a region of decreasing width. By using matched asymptotic expansions, we construct a nonlinear baroclinic critical layer theory to study how the flow perturbation evolves once they enter the nonlinear regime. A key feature of the theory is that, because the location of the baroclinic critical layer is determined by the streamwise wavenumber, the nonlinear dynamics filters out harmonics and the modification to the mean flow controls the evolution. At late times, we show that the vorticity begins to focus into yet smaller regions whose width decreases exponentially with time, and that the addition of dissipative effects can arrest this focussing to create a drifting coherent structure. Jet-like defects in the mean horizontal velocity are the main outcome of the critical-layer dynamics. † Email address for correspondence: chenwang@math.ubc.ca arXiv:1909.04620v2 [physics.flu-dyn]

A second-order instability analysis has been performed for sinuous disturbances on two-dimensional planar viscous sheets moving in a stationary gas medium using a perturbation technique. The solutions of second-order interface disturbances have been derived for both temporal instability and spatial instability. It has been found that the second-order interface deformation of the fundamental sinuous wave is varicose or dilational, causing disintegration and resulting in ligaments which are interspaced by half a wavelength. The interface deformation has been presented; the breakup time for temporal instability and breakup length for spatial instability have been calculated. An increase in Weber number and gas-to-liquid density ratio extensively increases both the temporal or spatial growth rate and the second-order initial disturbance amplitude, resulting in a shorter breakup time or length, and a more distorted surface deformation. Under normal conditions, viscosity has a stabilizing effect on the firstorder temporal or spatial growth rate, but it plays a dual role in the second-order disturbance amplitude. The overall effect of viscosity is minor and complicated. In the typical condition, in which the Weber number is 400 and the gas-to-liquid density ratio is 0.001, viscosity has a weak stabilizing effect when the Reynolds number is larger than 150 or smaller than 10; when the Reynolds number is between 150 and 10, viscosity has a weak destabilizing effect.

A second-order weakly nonlinear analysis has been made of the temporal instability for the linear sinuous mode of two-dimensional planar viscoelastic liquid sheets moving in an inviscid gas. The convected Jeffreys models including the corotational Jeffreys model, Oldroyd A model, and the Oldroyd B model are considered as the rheology model of the viscoelastic fluid of the sheet. The solution for the secondorder gas-to-liquid interface displacement has been derived, and the temporal evolution leading to the breakup has been shown. The second-order interface displacement of the linear sinuous mode is varicose, which causes the sheet to fragment into ligaments. First-order constitutive relations of the three rheology models become identical after linearization, so the linear instability results are also the same. For the second-order weakly nonlinear instability, the second-order constitutive relation varies among the corotational Jeffreys model, Oldroyd A model, and the Oldroyd B model, but although they have different disturbance pressures, their disturbance velocities and interface displacements are the same, and therefore, the sheets of the corotational Jeffreys fluid, Oldroyd A fluid, and the Oldroyd B fluid have the same instability behavior characterized by the wave profile and breakup time. The reason for the identical instability behavior is that the effect of different codeformations of the corotational frame, covariant frame, and the contravariant frame is counteracted by the corresponding change in the second-order disturbance pressure, leaving no influence on the second-order velocity. At wavenumbers with maximum instabilities, an increase in the elasticity, or a reduction of the deformation retardation time, leads to a larger linear temporal growth rate, greater second-order disturbance amplitude, and shorter breakup time, thereby enhancing instability. The mechanism of linear instability has been examined using an energy approach, which shows that the main cause of instability is the aerodynamic force. C 2015 AIP Publishing LLC.

Strato-rotational instability (SRI) is normally interpreted as the resonant interactions between normal modes of the internal or Kelvin variety in three-dimensional settings in which the stratification and rotation are orthogonal to both the background flow and its shear. Using a combination of asymptotic analysis and numerical solution of the linear eigenvalue problem for plane Couette flow, it is shown that such resonant interactions can be destroyed by certain singular critical levels. These levels are not classical critical levels, where the phase speed $c$ of a normal mode matches the mean flow speed $U$, but are a different type of singularity where $(c-U)$ matches a characteristic gravity-wave speed $\pm N/k$, based on the buoyancy frequency $N$ and streamwise horizontal wavenumber $k$. Instead, it is shown that a variant of SRI can occur due to the coupling of a Kelvin or internal wave to such ‘baroclinic’ critical levels. Two characteristic situations are identified and explored, and the conservation law for pseudo-momentum is used to rationalize the physical mechanism of instability. The critical level coupling removes the requirement for resonance near specific wavenumbers $k$, resulting in an extensive continuous band of unstable modes.

Abstract

In this paper, the instability of shallow-water shear flow with a sheared parallel magnetic field is studied. Waves propagating in such magnetic shear flows encounter critical levels where the phase velocity relative to the basic flow, $c-U(y)$ , matches the Alfvén wave velocities $\pm B(y)/\sqrt {\mu \rho }$ , based on the local magnetic field $B(y)$ , the magnetic permeability $\mu$ , and the mass density of the fluid $\rho$ . It is shown that when the two critical levels are close to each other, the critical layer can generate an instability. The instability problem is solved, combining asymptotic solutions at large wavenumbers and numerical solutions, and the mechanism of instability explained using the conservation of momentum. For the shallow-water magnetohydrodynamic system, the paper gives the general form of the local differential equation governing such coalescing critical layers for any generic field and flow profiles, and determines precisely how the magnetic field modifies the purely hydrodynamic stability criterion based on the potential vorticity gradient in the critical layer. The curvature of the magnetic field profile, or equivalently the electric current gradient $J' = - B''/\mu$ in the critical layer, is found to play a complementary role in the instability.

The nonlinear temporal instability of gas-surrounded planar liquid sheets, whose linear instability contains both sinuous and varicose modes, is studied. Both the weakly nonlinear analysis using a second-order perturbation expansion and the numerical simulation using a boundary integral method have been applied. Their comparison shows that the weakly nonlinear analysis can precisely predict the shapes of sheets for most of the time of disturbance evolution and qualitatively explain the instability mechanism when sheets break up. Both the first harmonics of the linear sinuous mode and linear varicose mode are varicose; they contribute to the breakup of sheets, but the first harmonic generated by the coupling between the linear sinuous and varicose modes is sinuous; it plays an important role in modulating the wave profile. The instability with various initial phase differences between the upper and lower interfaces is examined. Except for the varicose initial disturbance, the linear sinuous mode dominates in the shapes of sheets when their amplitudes grow large. Within the second-order analysis, the major modes that can cause the breakup include the linear varicose mode, the first harmonic of the linear sinuous mode and the first harmonic of the linear varicose mode. The effects of various flow parameters have been investigated. At relatively large wavenumbers where approximate analytical and numerical results agree well when sheets break up, increasing the wavenumber reduces the wave amplitude. Reducing the initial disturbance amplitude makes the first harmonic of the linear sinuous mode the dominant mode in causing the breakup. Increasing the Weber number or gas-to-liquid density ratio significantly reduces breakup time and enhances instability.

This paper studies the instability of two-dimensional magnetohydrodynamic systems on a sphere using analytical methods. The underlying flow consists of a zonal differential rotation and a toroidal magnetic field is present. Semicircle rules that prescribe the possible domain of the wave velocity in the complex plane for general flow and field profiles are derived. The paper then sets out an analytical study of the ‘clamshell instability’, which features field lines on the two hemispheres tilting in opposite directions (Cally, Sol. Phys., vol. 199, 2001, pp. 231–249). An asymptotic solution for the instability problem is derived for the limit of weak shear of the zonal flow, via the method of matched asymptotic expansions. It is shown that when the zonal flow is solid body rotation, there exists a neutral mode that tilts the magnetic field lines, referred to as the ‘tilting mode’. A weak shear of the zonal flow excites the critical layer of the tilting mode, which reverses the tilting direction to form the clamshell pattern and induces the instability. The asymptotic solution provides insights into properties of the instability for a range of flow and field profiles. A remarkable feature is that the magnetic field affects the instability only through its local behaviour in the critical layer.