A buoyant flow structure in a magnetic field: Quasi-steady states and linear-nonlinear transitions

The confined evolution of a buoyant blob of fluid subject to a vertical magnetic field is investigated in the limit of low magnetic Reynolds number. When the applied magnetic field is strong, the rise velocity of the blob is small. As the vorticity diffuses along the magnetic field lines, a quasi-steady state characterised by a balance between the work done by buoyancy and Ohmic dissipation is eventually reached at time tqs(L2/δ2)τ, where L is the axial dimension of the fluid domain, δ is the radius of the buoyant blob and τ is the magnetic damping time. However, when the applied magnetic field is weak or the axial length is sufficiently large compared to the blob size, the growth of axial velocity eventually makes the advection of vorticity significant. The typical time for the attainment of this nonlinear phase is , where N0 is the magnetic interaction parameter at time t=τ. The order-of-magnitude estimates for the timescales tqs and tnl are verified by computational experiments that capture both the linear and nonlinear phases

Tricks with the lorenz curve

This note develops, for the Gini coefficient of inequality, a very simple generalization that directly incorporates judgments on ‘relative inter-group inequality aversion' by making the inequality measure sensitive to the skewness of the Lorenz curve. The resulting family of inequality indices can be seen as complements to the Gini coefficient: some members of the family reflect ‘left-leaning', and others ‘right-leaning', distributional values relative to the ‘centrist' position assumed by Gini.Lorenz Curve, Gini coefficient, skewness

Confinement of rotating convection by a laterally varying magnetic field

Spherical shell dynamo models based on rotating convection show that the flow within the tangent cylinder is dominated by an off-axis plume that extends from the inner core boundary to high latitudes and drifts westward. Earlier studies explained the formation of such a plume in terms of the effect of a uniform axial magnetic field that significantly increases the lengthscale of convection in a rotating plane layer. However, rapidly rotating dynamo simulations show that the magnetic field within the tangent cylinder has severe lateral inhomogeneities that may influence the onset of an isolated plume. Increasing the rotation rate in our dynamo simulations (by decreasing the Ekman number $E$) produces progressively thinner plumes that appear to seek out the location where the field is strongest. Motivated by this result, we examine the linear onset of convection in a rapidly rotating fluid layer subject to a laterally varying axial magnetic field. A cartesian geometry is chosen where the finite dimensions $(x,z)$ mimic $(\phi,z)$ in cylindrical coordinates. The lateral inhomogeneity of the field gives rise to a unique mode of instability where convection is entirely confined to the peak-field region. The localization of the flow by the magnetic field occurs even when the field strength (measured by the Elsasser number $\varLambda$) is small and viscosity controls the smallest lengthscale of convection. The lowest Rayleigh number at which an isolated plume appears within the tangent cylinder in spherical shell dynamo simulations agrees closely with the viscous-mode Rayleigh number in the plane layer linear magnetoconvection model. The localized excitation of viscous-mode convection by a laterally varying magnetic field provides a mechanism for the formation of isolated plumes within Earth's tangent cylinder.Comment: 12 figures, 3 table

Online detection of temporal communities in evolving networks by estrangement confinement

Temporal communities result from a consistent partitioning of nodes across multiple snapshots of an evolving complex network that can help uncover how dense clusters in a network emerge, combine, split and decay with time. Current methods for finding communities in a single snapshot are not straightforwardly generalizable to finding temporal communities since the quality functions used for finding static communities have highly degenerate landscapes, and the eventual partition chosen among the many partitions of similar quality is highly sensitive to small changes in the network. To reliably detect temporal communities we need not only to find a good community partition in a given snapshot but also ensure that it bears some similarity to the partition(s) found in immediately preceding snapshots. We present a new measure of partition distance called "estrangement" motivated by the inertia of inter-node relationships which, when incorporated into the measurement of partition quality, facilitates the detection of meaningful temporal communities. Specifically, we propose the estrangement confinement method, which postulates that neighboring nodes in a community prefer to continue to share community affiliation as the network evolves. Constraining estrangement enables us to find meaningful temporal communities at various degrees of temporal smoothness in diverse real-world datasets. Specifically, we study the evolution of voting behavior of senators in the United States Congress, the evolution of proximity in human mobility datasets, and the detection of evolving communities in synthetic networks that are otherwise hard to find. Estrangement confinement thus provides a principled approach to uncovering temporal communities in evolving networks

Seismic Sounding of Convection in the Sun

Our Sun, primarily composed of ionized hydrogen and helium, has a surface temperature of 5777~K and a radius $R_\odot \approx 696,000$ km. In the outer $R_\odot/3$, energy transport is accomplished primarily by convection. Using typical convective velocities $u\sim100\,\rm{m\,s^{-1}}$ and kinematic viscosities of order $10^{-4}$ m$^{2}$s$^{-1}$, we obtain a Reynolds number $Re \sim 10^{14}$. Convection is thus turbulent, causing a vast range of scales to be excited. The Prandtl number, $Pr$, of the convecting fluid is very low, of order $10^{-7}$\,--\,$10^{-4}$, so that the Rayleigh number ($\sim Re^2 Pr$) is on the order of $10^{21}\,-\,10^{24}$. Solar convection thus lies in extraordinary regime of dynamical parameters, highly untypical of fluid flows on Earth. Convective processes in the Sun drive global fluid circulations and magnetic fields, which in turn affect its visible outer layers ("solar activity") and, more broadly, the heliosphere ("space weather"). The precise determination of the depth of solar convection zone, departures from adiabaticity of the temperature gradient, and the internal rotation rate as a function of latitude and depth are among the seminal contributions of helioseismology towards understanding convection in the Sun. Contemporary helioseismology, which is focused on inferring the properties of three-dimensional convective features, suggests that transport velocities are substantially smaller than theoretical predictions. Furthermore, helioseismology provides important constraints on the anisotropic Reynolds stresses that control the global dynamics of the solar convection zone. This review discusses the state of our understanding of convection in the Sun, with a focus on helioseismic diagnostics. We present our considerations with the interests of fluid dynamicists in mind.Comment: 29 pages, 12 figures, in review, Annual Reviews of Fluid Mechanic