The evolution of a front in turbulent thermal wind balance. Part 2. Numerical simulations

In Crowe &amp; Taylor (J. Fluid Mech., vol. 850, 2018, pp. 179–211) we described a theory for the evolution of density fronts in a rotating reference frame subject to strong vertical mixing using an asymptotic expansion in small Rossby number, $Ro$. We found that the front reaches a balanced state where vertical diffusion is balanced by horizontal advection in the buoyancy equation. The depth-averaged buoyancy obeys a nonlinear diffusion equation which admits a similarity solution corresponding to horizontal spreading of the front. Here we use numerical simulations of the full momentum and buoyancy equations to investigate this problem for a wide range of Rossby and Ekman numbers. We examine the accuracy of our asymptotic solution and find that many aspects of the solution are valid for $Ro=O(1)$. However, the asymptotic solution departs from the numerical simulations for small Ekman numbers where the dominant balance in the momentum equation changes. We trace the source of this discrepancy to a depth-independent geostrophic flow that develops on both sides of the front and we develop a modification to the theory described in Crowe &amp; Taylor (2018) to account for this geostrophic flow. The refined theory closely matches the numerical simulations, even for $Ro=O(1)$. Finally, we develop a new scaling for the intense vertical velocity that can develop in thin bands at the edges of the front.</jats:p

Baroclinic instability with a simple model for vertical mixing

AbstractHere, we examine baroclinic instability in the presence of vertical mixing in an idealized setting. Specifically, we use a simple model for vertical mixing of momentum and buoyancy and expand the buoyancy and vorticity in a series for small Rossby numbers. A flow in subinertial mixed layer (SML) balance (see the study by Young in 1994) exhibits a normal mode linear instability, which is studied here using linear stability analysis and numerical simulations. The most unstable modes grow by converting potential energy associated with the basic state into kinetic energy of the growing perturbations. However, unlike the inviscid Eady problem, the dominant energy balance is between the buoyancy flux and the energy dissipated by vertical mixing. Vertical mixing reduces the growth rate and changes the orientation of the most unstable modes with respect to the front. By comparing with numerical simulations, we find that the predicted scale of the most unstable mode matches the simulations for small Rossby numbers while the growth rate and orientation agree for a broader range of parameters. A stability analysis of a basic state in SML balance using the inviscid QG equations shows that the angle of the unstable modes is controlled by the orientation of the SML flow, while stratification associated with an advection/diffusion balance controls the size of growing perturbations for small Ekman numbers and/or large Rossby numbers. These results imply that baroclinic instability can be inhibited by small-scale turbulence when the Ekman number is sufficiently large and might explain the lack of submesoscale eddies in observations and numerical models of the ocean surface mixed layer during summer.</jats:p

The evolution of a front in turbulent thermal wind balance. Part 1. Theory

Here, we examine the influence of small-scale turbulence on the evolution of fronts in the ocean and atmosphere. Specifically, we consider the evolution of an initially balanced density front subject to an imposed viscosity and diffusivity as a simple analogue for small-scale turbulence. At late times, the dominant balance is found to be the quasisteady turbulent thermal wind balance with time evolution due to an advection–diffusion balance in the buoyancy equation. We use the leading-order balance to determine analytical similarity solutions for the spreading of a front and find that the spreading rate is maximum for an intermediate value of the Ekman number, with the spreading resulting from shear dispersion associated with the cross-front flow and vertical diffusion of density. In response to shear dispersion, the front evolves towards a density profile that is nearly linear in the cross-front coordinate. At the edges of the frontal zone, the density field develops large curvature, and these regions are associated with narrow bands of intense vertical velocity

An Analysis of Selected Characteristics in Beginning Business Students: Young Adult Inmates and Male College Freshmen

A thesis presented to the faculty of the Department of Business Education at Morehead State University in partial fulfillment of the requirements for the Degree of Master of Business Education by Richard Godfrey Crowe, Jr. in May of 1970

The evolution of a front in turbulent thermal wind balance. Part 1. Theory

Here, we examine the influence of small-scale turbulence on the evolution of fronts in the ocean and atmosphere. Specifically, we consider the evolution of an initially balanced density front subject to an imposed viscosity and diffusivity as a simple analogue for small-scale turbulence. At late times, the dominant balance is found to be the quasisteady turbulent thermal wind balance with time evolution due to an advection–diffusion balance in the buoyancy equation. We use the leading-order balance to determine analytical similarity solutions for the spreading of a front and find that the spreading rate is maximum for an intermediate value of the Ekman number, with the spreading resulting from shear dispersion associated with the cross-front flow and vertical diffusion of density. In response to shear dispersion, the front evolves towards a density profile that is nearly linear in the cross-front coordinate. At the edges of the frontal zone, the density field develops large curvature, and these regions are associated with narrow bands of intense vertical velocity.</jats:p