Numerical Computation of Instabilities and Internal Waves from In Situ Measurements via the Viscous Taylor–Goldstein Problem

Abstract

We explore numerical methods for the stability analysis of stratified, parallel shear flows considering the effects of small-scale turbulence represented by eddy viscosity and diffusivity. The result is an extension of the classical Taylor–Goldstein problem applicable to oceanic and atmospheric flows. Solutions with imaginary frequency describe shear and convective instabilities, whereas those with real frequency represent internal gravity waves. Application to large observational datasets can involve considerable computation and therefore requires a compromise between speed and accuracy. We compare several numerical methods to identify optimal approaches to various problems.

The butterfly effect and the transition to turbulence in a stratified shear layer

“…To allow the subharmonic mode to develop, the streamwise periodicity interval accommodates two wavelengths of the fastest-growing KH mode based on linear stability analysis (Lian, Smyth & Liu 2020). The spanwise periodicity interval is sufficient for the development of 3DSIs (e.g.…”

The coupled dynamics of internal waves and hairpin vortices in stratified plane Poiseuille flow

“…Solutions are sought to the viscous Taylor-Goldstein equations (Liu et al 2012;Lian, Smyth & Liu 2020) which are derived from the linearised momentum, continuity and buoyancy transport equations of § 3.4 by assuming perturbations take the form…”

“…where λ = σ − iω is the complex growth rate, v( y) and b( y) represent complex vertical structure functions and it is assumed the wave vector is aligned with the x direction (k y = 0). Following Lian et al (2020), and ensuring dimensional consistency with the full 3-D simulations, the dimensionless viscous Taylor-Goldstein equations are…”

“…Inclusion of eddy viscosity (not shown) actually does not affect the obtained dispersion relation and only has a minor smoothing effect on the structure functions at the critical levels, where ω = Uk x . Equations (3.12) and (3.13) can be reconstructed as a generalised differential eigenvalue problem, and solved numerically using finite differencing for given profiles of U( y) and B( y), and wavenumber k x (see Lian et al (2020) for further details). The eigenvalue problem is solved on a uniform grid consisting of 401 points over a wavenumber range of k x = 0.5-80.…”

“…where λ = σ − iω is the complex growth rate, v( y) and b( y) represent complex vertical structure functions and it is assumed the wave vector is aligned with the x direction (k y = 0). Following Lian et al (2020), and ensuring dimensional consistency with the full 3-D simulations, the dimensionless viscous Taylor-Goldstein equations are with viscous and diffusive operators (3.15) where A h and A v are horizontal and vertical viscosities, and K h and K v are horizontal and vertical diffusivities. Although in principle this form of the equations allows for the modelling of turbulence through a turbulent diffusion closure, here, only molecular diffusion is allowed, such that…”