Forty Years of Celebration of Discipline: An Interview with Richard Foster

Wow, Richard, it’s been 40 years since your first book, Celebration of Discipline, was published, and it’s still a best seller (selling over 2 million copies), having been translated into 25 languages! Reflecting on that now, could you say something about your original vision for the book, and how God has blessed its impact over the years

The linear spin-up of a stratified, rotating fluid in a square cylinder

Here we present experimental and theoretical results for how a stratified fluid, initially rotating as a solid body with constant angular velocity, , within a closed cylinder of square cross-section, is spun up when subject to a small, impulsive increase, , in the cylinder’s rotation rate. The fluid’s adjustment to the new state of solid rotation can be characterized by: (a) an inviscid, horizontal starting flow which conserves the vorticity of the initial condition; (b) the eruption of Ekman layer fluid from the perimeter region of the cylinder’s base and lid; (c) horizontal-velocity Rayleigh layers that grow into the interior from the container’s sidewalls; and (d) the formation and decay of columnar vortices in the vertical corner regions. Asymptotic results describe the inviscid starting flow, and the subsequent interior spin-up that occurs due to the combined effects of Ekman suction through the base and lid Ekman layers, and the growth of the sidewall Rayleigh layers. Attention is focused on the flow development over the spin-up time scale , where is the Ekman number. (The spin-up process over the much longer diffusive time scale, , is not considered here.) Experiments were performed using particle imaging velocimetry (PIV) to measure horizontal velocity components at fixed heights within the flow interior and at regular stages during the spin-up period. The velocity data obtained are shown to be in excellent agreement with the asymptotic theory

Multiple Party Accounts: Georgia Law Compared with the Uniform Probate Code

Joint accounts established in financial institutions have become increasingly popular as inexpensive and convenient means of nontestamentary disposition of wealth. Varied and often unsuitable legal theories which have been relied upon to validate such attempts have, however, resulted in inconsistent case results in what should otherwise be a fairly simple area. In their article, Professor Wellman and Mr. Clark explain this disparate treatment and demonstrate the desirability of Article VI, Part 1 of the Uniform Probate Code as a statutory solution for the problems presented

On the formation of axial corner vortices during spin-up in a cylinder of square cross-section

We present experimental and theoretical results for the adjustment of a fluid (homogeneous or linearly stratified), which is initially rotating as a solid body with angular frequency Ω−ΔΩ, to a nonlinear increase ΔΩ in the angular frequency of all bounding surfaces. The fluid is contained in a cylinder of square cross-section which is aligned centrally along the rotation axis, and we focus on the O(Ro−1Ω−1) time scale, where Ro=ΔΩ/Ω is the Rossby number. The flow development is shown to be dominated by unsteady separation of a viscous sidewall layer, leading to an eruption of vorticity that becomes trapped in the four vertical corners of the container. The longer-time evolution on the standard ‘spin-up’ time scale, E−1/2Ω−1 (where E is the associated Ekman number), has been described in detail for this geometry by Foster & Munro (J. Fluid Mech., vol. 712, 2012, pp. 7–40), but only for small changes in the container’s rotation rate (i.e. Ro≪1). In the linear case, for Ro≪E1/2≪1, there is no sidewall separation. In the present investigation we focus on the fully nonlinear problem, Ro=O(1), for which the sidewall viscous layers are Prandtl boundary layers and (somewhat unusually) periodic around the container’s circumference. Some care is required in the corners of the container, but we show that the sidewall boundary layer breaks down (separates) shortly after an impulsive change in rotation rate. These theoretical boundary-layer results are compared with two-dimensional Navier–Stokes results which capture the eruption of vorticity, and these are in turn compared to laboratory observations and data. The experiments show that when the Burger number, S=(N/Ω)2 (where N is the buoyancy frequency), is relatively large – corresponding to a strongly stratified fluid – the flow remains (horizontally) two-dimensional on the O(Ro−1Ω−1) time scale, and good quantitative predictions can be made by a two-dimensional theory. As S was reduced in the experiments, three-dimensional effects were observed to become important in the core of each corner vortex, on this time scale, but only after the breakdown of the sidewall layers