Strength through defects: A novel Bayesian approach for the optimization of architected materials.

We use a previously unexplored Bayesian optimization framework, "evolutionary Monte Carlo sampling," to systematically design the arrangement of defects in an architected microlattice to maximize its strain energy density before undergoing catastrophic failure. Our algorithm searches a design space with billions of 4 × 4 × 5 3D lattices, yet it finds the global optimum with only 250 cost function evaluations. Our optimum has a normalized strain energy density 12,464 times greater than its commonly studied defect-free counterpart. Traditional optimization is inefficient for this microlattice because (i) the design space has discrete, qualitative parameter states as input variables, (ii) the cost function is computationally expensive, and (iii) the design space is large. Our proposed framework is useful for architected materials and for many optimization problems in science and elucidates how defects can enhance the mechanical performance of architected materials.

Three-dimensional vortices generated by self-replication in stably stratified rotating shear flows.

A previously unknown instability creates space-filling lattices of 3D vortices in linearly stable, rotating, stratified shear flows. The instability starts from an easily excited critical layer. The layer intensifies by drawing energy from the background shear and rolls up into vortices that excite new critical layers and vortices. The vortices self-similarly replicate to create lattices of turbulent vortices. The vortices persist for all time. This self-replication occurs in stratified Couette flows and in the dead zones of protoplanetary disks where it can destabilize Keplerian flows.

Jupiter's zonal winds: are they bands of homogenized potential vorticity organized as a monotonic staircase?

The east-west striped pattern of clouds in Jupiter's weather layer is accompanied by a zonal flow containing 12 eastward-going jet streams alternating in latitude with westward-going jet streams. Based on theory, simulation and observations of the Earth's oceans and atmosphere, it is conjectured that Jupiter's weather layer is made of bands of constant potential vorticity (PV), where the interfaces between bands are at the latitudes of the maxima of the eastward-going jet streams. It is speculated that the mixing of PV on Jupiter is analogous to the mixing of salt in the ocean by the Phillips effect, which causes the salt density to form a monotonic 'staircase'. It is hypothesized that the PV in Jupiter's weather layer is also a staircase, decreasing from north to south. PV is a function of vorticity, as well as parameters with unknown values, e.g. the vertical stratification and the zonal flow beneath the observable weather layer. Therefore, these hypotheses cannot be tested directly. Using an atmospheric model that contains these unknown parameters, we solved the inverse problem and found values of the unknown parameters (and their uncertainties) that best fit Jovian observations. The unknown parameters influence how the zonal flow interacts with large vortices, e.g. the Great Red Spot (GRS; the largest and longest-lived Jovian vortex, centred at 23° S) and the Oval BA (the second largest vortex, centred at 33° S). Although we found that the PV distribution is approximately piecewise-constant and that the peaks of the eastward-going jet streams are at the latitudes of PV interfaces, there is also a PV interface at 20° S, where there is a westward-going jet stream. We find that the zonal PV is not a monotonic staircase due to the 'backwards' interface at 20° S. We show that this backwards interface is necessary to make the GRS nearly round, and that without that interface, the Red Spot would be highly elongated in the east-west direction and probably unstable.

Selection rules for the nonlinear interaction of internal gravity waves.

Two intersecting beams of internal gravity waves will generically create two wave packets by nonlinear interaction. The frequency of one packet will be the sum and that of the other packet will be the difference of the frequencies of the intersecting beams. In principle, each packet should form an "X" pattern, or "St. Andrew's cross" consisting of four beams outgoing from the point of intersection. Here we derive selection rules and show that most of the expected nonlinear beams are forbidden. These rules can also be applied to the reflection of a beam from a boundary.

Breaking of rotational symmetry in cylindrically bounded 2D electron plasmas and 2D fluids.

Off-axis final states of cylindrically bounded 2D fluids can develop from initially unstable, but cylindrically symmetric, 2D vorticity distributions. Experiments with electrons in a Malmberg-Penning trap, as well as 2D fluid simulations, demonstrated that such states result when the initial vorticity distribution is close to the boundary, while less extended distributions lead to on-axis states. A simple thermodynamic model, maximizing the entropy of a state consisting of a diffuse background surrounding a strong coherent vortex, is shown to quantitatively predict this bifurcation, while conserving circulation, angular momentum, and energy.

Prediction of a global climate change on Jupiter.

Jupiter's atmosphere, as observed in the 1979 Voyager space craft images, is characterized by 12 zonal jet streams and about 80 vortices, the largest of which are the Great Red Spot and three White Ovals that had formed in the 1930s. The Great Red Spot has been observed continuously since 1665 and, given the dynamical similarities between the Great Red Spot and the White Ovals, the disappearance of two White Ovals in 1997-2000 was unexpected. Their longevity and sudden demise has been explained however, by the trapping of anticyclonic vortices in the troughs of Rossby waves, forcing them to merge. Here I propose that the disappearance of the White Ovals was not an isolated event, but part of a recurring climate cycle which will cause most of Jupiter's vortices to disappear within the next decade. In my numerical simulations, the loss of the vortices results in a global temperature change of about 10 K, which destabilizes the atmosphere and thereby leads to the formation of new vortices. After formation, the large vortices are eroded by turbulence over a time of approximately 60 years--consistent with observations of the White Ovals-until they disappear and the cycle begins again.

Jupiter's Great Red Spot and zonal winds as a self-consistent, one-layer, quasigeostrophic flow.

We present the point of view that both the vortices and the east-west zonal winds of Jupiter are confined to the planet's shallow weather layer and that their dynamics is completely described by the weakly dissipated, weakly forced quasigeostrophic (QG) equation. The weather layer is the region just below the tropopause and contains the visible clouds. The forcing mimics the overshoot of fluid from an underlying convection zone. The late-time solutions of the weakly forced and dissipated QG equations appear to be a small subset of the unforced and undissipated equations and are robust attractors. We illustrate QG vortex dynamics and attempt to explain the important features of Jupiter's Great Red Spot and other vortices: their shapes, locations with respect to the extrema of the east-west winds, stagnation points, numbers as a function of latitude, mergers, break-ups, cloud morphologies, internal distributions of vorticity, and signs of rotation with respect to both the planet's rotation and the shear of their surrounding east-west winds. Initial-value calculations in which the weather layer starts at rest produce oscillatory east-west winds. Like the Jovian winds, the winds are east-west asymmetric and have Karman vortex streets located only at the west-going jets. From numerical calculations we present an empirically derived energy criterion that determines whether QG vortices survive in oscillatory zonal flows with nonzero potential vorticity gradients. We show that a recent proof that claims that all QG vortices decay when embedded in oscillatory zonal flows is too restrictive in its assumptions. We show that the asymmetries in the cloud morphologies and numbers of cyclones and anticyclones can be accounted for by a QG model of the Jovian atmosphere, and we compare the QG model with competing models.