Dynamics of fingering convection II: The formation of thermohaline staircases

Regions of the ocean's thermocline unstable to salt fingering are often observed to host thermohaline staircases, stacks of deep well-mixed convective layers separated by thin stably-stratified interfaces. Decades after their discovery, however, their origin remains controversial. In this paper we use 3D direct numerical simulations to shed light on the problem. We study the evolution of an analogous double-diffusive system, starting from an initial statistically homogeneous fingering state and find that it spontaneously transforms into a layered state. By analysing our results in the light of the mean-field theory developed in Paper I, a clear picture of the sequence of events resulting in the staircase formation emerges. A collective instability of homogeneous fingering convection first excites a field of gravity waves, with a well-defined vertical wavelength. However, the waves saturate early through regular but localized breaking events, and are not directly responsible for the formation of the staircase. Meanwhile, slower-growing, horizontally invariant but vertically quasi-periodic gamma-modes are also excited and grow according to the gamma-instability mechanism. Our results suggest that the nonlinear interaction between these various mean-field modes of instability leads to the selection of one particular gamma-mode as the staircase progenitor. Upon reaching a critical amplitude, this progenitor overturns into a fully-formed staircase. We conclude by extending the results of our simulations to real oceanic parameter values, and find that the progenitor gamma-mode is expected to grow on a timescale of a few hours, and leads to the formation of a thermohaline staircase in about one day with an initial spacing of the order of one to two metres.Comment: 18 pages, 9 figures, associated mpeg file at http://earth.uni-muenster.de/~stellma/movie_small.mp4, submitted to JF

Shallow current of viscous fluid flowing between diverging or converging walls

We investigate the shallow flow of viscous fluid into and out of a channel whose gap width increases as a power-law ($x^n$), where $x$ is the downstream axis. The fluid flows slowly, while injected at a rate in the form of $t^\alpha$, where $t$ is time and $\alpha$ is a constant. The invading fluid has higher viscosity than the ambient fluid, thus avoiding Saffman-Taylor instability. Similarity solutions of the first kind for the outflow problem are found using approximations of lubrication theory. Zheng et al [2014] studied the deep-channel case and found divergent behaviour of the similarity variable as $n\rightarrow1$ and $n\rightarrow3$, when fluid flows into and out of the channel respectively. No divergence is found in the shallow case presented here. The characteristic equilibration time for the numerically simulated constant-volume flow to converge to the similarity solution is calculated assuming inverse dependence on the ratio disagreement between the current front using the method of lines (MOL). The inverse power dependence between equilibration time and ratio disagreement is found for channels of different powers. A similarity solution of the second kind for the inflow problem is found using the phase plane formalism and the bisection method. An exponential decay relationship is found between $n$ and the degree $\delta$ of the similarity variable $xt^{-\delta}$, which does not show any divergent behaviour for large $n$. An asymptotic behaviour is found for $\delta$ that approaches $1/2$ as $n\rightarrow\infty$

Neural correlates of early deliberate emotion regulation: Young children\u27s responses to interpersonal scaffolding.

Deliberate emotion regulation, the ability to willfully modulate emotional experiences, is shaped through interpersonal scaffolding and forecasts later functioning in multiple domains. However, nascent deliberate emotion regulation in early childhood is poorly understood due to a paucity of studies that simulate interpersonal scaffolding of this skill and measure its occurrence in multiple modalities. Our goal was to identify neural and behavioral components of early deliberate emotion regulation to identify patterns of competent and deficient responses. A novel probe was developed to assess deliberate emotion regulation in young children. Sixty children (age 4-6 years) were randomly assigned to deliberate emotion regulation or control conditions. Children completed a frustration task while lateral prefrontal cortex (LPFC) activation was recorded via functional near-infrared spectroscopy (fNIRS). Facial expressions were video recorded and children self-rated their emotions. Parents rated their child\u27s temperamental emotion regulation. Deliberate emotion regulation interpersonal scaffolding predicted a significant increase in frustration-related LPFC activation not seen in controls. Better temperamental emotion regulation predicted larger LPFC activation increases post- scaffolding among children who engaged in deliberate emotion regulation interpersonal scaffolding. A capacity to increase LPFC activation in response to interpersonal scaffolding may be a crucial neural correlate of early deliberate emotion regulation

Topographic controls on gravity currents in porous media

We present a theoretical and experimental study of the propagation of gravity currents in porous media with variations in the topography over which they flow, motivated in part by the sequestration of carbon dioxide in saline aquifers. We consider cases where the height of the topography slopes upwards in the direction of the flow and is proportional to the nth power of the horizontal distance from a line or point source of a constant volumetric flux. In two-dimensional cases with n>1/2, the current evolves from a self-similar form at early times, when the effects of variations in topography are negligible, towards a late-time regime that has an approximately horizontal upper surface and whose evolution is dictated entirely by the geometry of the topography. For n<1/2, the transition between these flow regimes is reversed. We compare our theoretical results in the case n=1 with data from a series of laboratory experiments in which viscous glycerine is injected into an inclined Hele-Shaw cell, obtaining good agreement between the theoretical results and the experimental data. In the case of axisymmetric topography, all topographic exponents n>0 result in a transition from an early-time similarity solution towards a topographically controlled regime that has an approximately horizontal free surface. We also analyse the evolution over topography that can vary with different curvatures and topographic exponents between the two horizontal dimensions, finding that the flow transitions towards a horizontally topped regime at a rate which depends strongly on the ratio of the curvatures along the principle axes. Finally, we apply our mathematical solutions to the geophysical setting at the Sleipner field, concluding that topographic influence is unlikely to explain the observed non-axisymmetric flow

Stokes at 200: A celebration of the remarkable achievements of Sir George Gabriel Stokes two hundred years after his birth

Sir George Gabriel Stokes PRS was for 30 years an inimitable Secretary of the Royal Society and its President from 1885 to 1890. Two hundred years after his birth, Stokes is a towering figure in physics and applied mathematics; fluids, asymptotics, optics, acoustics among many other fields. At the Stokes200 meeting, held at Pembroke College, Cambridge from 15-18th September 2019, an invited audience of about 100 discussed the state of the art in all the modern research fields that have sprung from his work in physics and mathematics, along with the history of how we have got from Stokes' contributions to where we are now. This theme issue is based on work presented at the Stokes200 meeting. In bringing together people whose work today is based upon Stokes' own, we aim to emphasize his influence and legacy at 200 to the community as a whole. This article is part of the theme issue 'Stokes at 200 (Part 1)'

Stokes at 200 (part 2)

We present the second half of the papers from the Stokes200 symposium celebrating the bicentenary of George Gabriel Stokes. This article is part of the theme issue 'Stokes at 200 (part 2)'

On the p-length of some finite p-soluble groups

The main aim of this paper is to give structural information of a finite group of minimal order belonging to a subgroup-closed class of finite groups and whose p-length is greater than 1, p a prime number. Alternative proofs and improvements of recent results about the influence of minimal p-subgroups on the p-nilpotence and p-length of a finite group arise as consequences of our study