Noether-type discrete conserved quantities arising from a finite element approximation of a variational problem

In this work, we prove a weak Noether-type Theorem for a class of variational problems that admit broken extremals. We use this result to prove discrete Noether-type conservation laws for a conforming finite element discretisation of a model elliptic problem. In addition, we study how well the finite element scheme satisfies the continuous conservation laws arising from the application of Noether’s first theorem (1918). We summarise extensive numerical tests, illustrating the conservation of the discrete Noether law using the p-Laplacian as an example and derive a geometric-based adaptive algorithm where an appropriate Noether quantity is the goal functional.

Rossby wave propagation on potential vorticity fronts with finite width

The horizontal gradient of potential vorticity (PV) across the tropopause typically declines with lead time in global numerical weather forecasts and tends towards a steady value dependent on model resolution. This paper examines how spreading the tropopause PV contrast over a broader frontal zone affects the propagation of Rossby waves. The approach taken is to analyse Rossby waves on a PV front of finite width in a simple single-layer model. The dispersion relation for linear Rossby waves on a PV front of infinitesimal width is well known; here an approximate correction is derived for the case of a finite width front, valid in the limit that the front is narrow compared to the zonal wavelength. Broadening the front causes a decrease in both the jet speed and the ability of waves to propagate upstream. The contribution of these changes to Rossby wave phase speeds cancel at leading order. At second order the decrease in jet speed dominates, meaning phase speeds are slower on broader PV fronts. This asymptotic phase speed result is shown to hold for a wide class of single-layer dynamics with a varying range of PV inversion operators. The phase speed dependence on frontal width is verified by numerical simulations and also shown to be robust at finite wave amplitude, and estimates are made for the error in Rossby wave propagation speeds due to the PV gradient error present in numerical weather forecast models.

Followers are not followed: observed group interactions modulate subsequent social attention

We asked whether previous observations of group interactions modulate subsequent social attention episodes. Participants first completed a learning phase with two conditions. In the ‘leader’ condition one of three identities turned her gaze first, followed by the two other faces. In the ‘follower’ condition, one of the identities turned her gaze after the two other faces had first shifted their gaze. Thus, participants observed that some individuals were consistently ‘leaders’ and others ‘followers’ of others’ attention. In the test phase, the faces of ‘leaders’ and ‘followers’ were presented in a gaze cueing paradigm. Remarkably, the ‘followers’ did not elicit gaze cueing. Our data demonstrate that individuals who do not guide group attention in exploring the environment are ineffective social attention directors in later encounters. Thus, the role played in previous group social attention interactions modulates the relative weight assigned to others’ gaze: we ignore the gaze of group followers.