Nonlinear modeling of stratified shear instabilities, wave breaking, and wave-topography interactions using vortex method

Abstract

Theoretical studies on linear shear instabilities often use simple velocity and density profiles (e.g., constant, piecewise) for obtaining good qualitative and quantitative predictions of the initial disturbances. Furthermore, such simple profiles provide a minimal model for obtaining a mechanistic understanding of otherwise elusive shear instabilities. However, except a few specific cases, the efficacy of simple profiles has remained limited to the linear stability paradigm. In this work, we have proposed a general framework that can simulate the fully nonlinear evolution of a variety of stratified shear instabilities as well as wave-wave and wave-topography interaction problems having simple piecewise constant and/or linear profiles. To this effect, we have modified the classical vortex method by extending the Birkhoff-Rott equation to multiple interfaces and, furthermore, have incorporated background shear across a density interface. The latter is more subtle and originates from the understanding that Bernoulli’s equation is not just limited to irrotational flows but can be modified to make it applicable for piecewise linear velocity profiles. We have solved diverse problems that can be essentially reduced to the multiple interacting interfaces paradigm, e.g., spilling and plunging breakers, stratified shear instabilities like Holmboe and Taylor-Caulfield, jet flows, and even wave-topography interaction problems like Bragg resonance. Free-slip boundary being a vortex sheet, its effect can also be effectively captured using vortex method. We found that the minimal models capture key nonlinear features, e.g., wave breaking features like cusp formation and roll-ups, which are observed in experiments and/or extensive simulations with smooth, realistic profiles.

Linear temporal stability analysis on the inviscid sheared convective boundary layer flow

“…where U * is the horizontal base flow velocity in the x * direction, θ * is temperature, z * is the vertical coordinate (z * = 0 is at the middle height of the sheared stratified layer), L and the buoyancy is b = γθ, where γ is the thermal expansion coefficient, and the profile of θ is also applicable for b. The typical hyperbolic profiles of U * and θ * in the central sheared stratified layer allow the formation of prominent vortex structures and are commonly used in the previous studies on the pure shear stratified flow [27][28][29][30][31][32][33][34] . The linear profile of θ * at the bottom unsheared stratified layer inherits those usually applied in the linear analysis of the pure thermal convection 35 and the convective boundary layer (CBL) 36,37 .…”

On the opposing roles of the Boussinesq and non‐Boussinesq baroclinic torques in surface gravity wave propagation

“…Recently, there has been a growing body of literature (Holmboe, ; Sakai, ; Baines and Mitsudera, ; Caulfield, ; Harnik et al ., ; Rabinovich et al ., ; Carpenter et al ., ; Guha and Lawrence, ; Heifetz and Mak, ) dealing with stratified shear flow instability that treats the dynamics of density discontinuity surfaces as interfacial vorticity waves .This approach provides a mechanistic rationalization for Taylor–Caulfield and Holmboe instabilities, and also paves the path for efficient vortex‐method‐based computation schemes to simulate the nonlinear evolution, including wave‐breaking ( Bhardwaj and Guha, ). However, while exploring these relatively complex problems, the analysis of surface gravity waves – probably the simplest set‐up of a density discontinuity – in terms of interfacial vorticity waves has been “left behind.” Hence, here we suggest an alternative derivation of surface gravity waves, based on the above‐mentioned approach.…”