Geostrophic adjustment within a stratified fluid

Abstract

This investigation considers the geostrophically adjusted flow profile generated from a stratified fluid subject to lock-release initial conditions. The system is formed using a three-layer stratified fluid contained within a cuboidal domain. The fluid is initially in solid body rotation with a flat free surface and a step imposed upon the pycnocline. This study uses a combined approach involving analytical, numerical and experimental methods to investigate how asymmetrical initial step locations affect flow strength. An analytical model of the geostrophically adjusted state using a two-layer density profile approximation is developed. This is achieved by decoupling and solving the two-layer shallow water equations by seeking solutions in terms of modal variables. We begin by seeking a solution within a spatially infinite domain with the resulting flow tending towards a baroclinic solution. The model is then generalised to introduce two parallel vertical walls to form an infinite channel. In the channel, we show a system of two jets travelling between the bounding walls at the location of the initial imbalance. Additionally, spectral methods are applied to determine the adjusted state for flow contained within a rectangular domain. In the bounded domain, we analytically predict the formation of two counter-rotating gyre systems. Parameters governing the initial conditions are varied in order to investigate the resulting upper layer flow strength. A linear relationship between the height of imposed lock-release initial conditions and the resulting gyre strength is revealed. Furthermore, we show that the gyres strength exhibits a strength preference regarding the initial condition location. Technical details for our chosen numerical framework (IAMR) as well as our experimental protocols are provided. This includes the methodology used to select appropriate timeaveraging intervals in both cases. The predicted gyre system is observed in the numerical framework and the laboratory experiments. Non-linear behaviour is also observed during the numerical and experimental studies despite achieving a low Rossby number. These non-linear dynamics are absent from the linearised analytic model and manifest in the deformation of the gyre boundaryA quantitative comparison between the gyre dynamics in numerical simulations and laboratory experiments is provided to confirm the good agreement between the two schemes. Finally, the investigation is concluded by presenting a qualitative assessment of how the imposed initial conditions affect the resulting adjusted flow strength. This investigation numerically and experimentally verifies the linear relationship between the height of imposed lock-release initial conditions and the resulting gyre strength revealed. In addition, we explore the relationship between imposed barrier location and the resulting gyre strength. This reveals a preferred barrier location associated with a strength preference in the adjusted state gyres, however, the preferred barrier location differs when comparing numerical and laboratory results