Weak Gradient Research Articles

A weak Galerkin finite element method for Allen–Cahn equation with a nonuniform two‐step backward differentiation formula scheme

AbstractIn this article, we consider a weak Galerkin finite element method and a nonuniform two‐step backward differentiation formula scheme for solving the Allen–Cahn equation. A discrete weak gradient operator on discontinuous piecewise polynomials is used in the numerical scheme. It is well‐known that the Allen–Cahn equations have a nonlinear stability property, that is, the free‐energy functional decreases with respect to time. A corresponding energy stability analysis of the discretized system is developed. Optimal order of L2‐norm error estimates are derived. Numerical experiments are preformed to support the theoretical results.

Meteotsunamis in the northern Baltic Sea and their relation to synoptic patterns

Low-tidal coastal regions, such as the Baltic Sea, are known to be particularly vulnerable to exceptional high-frequency sea level oscillations such as meteotsunamis. Possibilities of studying sub-hourly sea level variations have recently improved, owing to advancement in temporal resolution of tide gauge observations. In this work, we study high-frequency (period <6 h) sea level oscillations – strongest of which we consider to be meteotsunamis – on the coast of Finland, in the northern Baltic Sea, using quality-checked 1-min observations collected between 2004 and 2015 at 13 tide gauge stations. The intensity of the oscillations varies substantially between stations due to local coastal morphologies. The most intense oscillations predominantly occur in late summer and autumn, although the seasonality may differ between sub-regions. Measured atmospheric data and reanalysis products related to the strongest events reveal two distinct types of atmospheric processes and governing synoptic patterns that are mostly associated with warmer and colder period of year. Consequently, meteotsunamis are classified as summer-type or winter-type events. Most of the summer-type events are caused by surface atmospheric pressure jumps associated with mesoscale convective systems, which are advancing northward over the sea and are embedded into a mid-troposphere jet overtopping an inflow of warm low-troposphere air. At the surface, weak air pressure gradients due to a high-pressure area to the east and a low-pressure area to the west of the Baltic are usually found during summer-type events. The winter-type events, on the contrary, are mostly related to cold fronts and strong northerly-northwesterly-westerly winds at the surface layer. Contrary to summer-type events, surface atmospheric pressure jumps are not necessarily detected during the strongest winter-type events. Deep lows and extratropical cyclones are commonly centered to the north of the Baltic Sea and, at the mid-troposphere level, there is a pronounced westerly jet stream. A hypothesis about the generation mechanism of intense high-frequency sea level oscillations is given: Proudman resonance appears to be the main driver of summer-type events, whereas the main driver of winter-type events is less clear.

Constructing the Milky Way Stellar Halo in the Galactic Center by Direct Orbit Integration

The halo stars on highly-radial orbits should inevitably pass the center regions of the Milky Way. Under the assumption that the stellar halo is in “dynamical equilibrium” and is axisymmetric, we integrate the orbits of ∼10,000 halo K giants at 5 ≤ r ≤ 50 kpc cross-matched from LAMOST DR5 and Gaia DR3. By carefully considering the selection function, we construct the stellar halo distribution at the entire regions of r ≤ 50 kpc. We find that a double-broken power-law function well describes the stellar halo’s density distribution with shallower slopes in the inner regions and the two breaks at r = 10 kpc and r = 25 kpc, respectively. The stellar halo becomes flatter from outer to inner regions but has q ∼ 0.5 at r ≲ 5 kpc. The stellar halo becomes isotropic with a slight prograde rotation in the inner 5 kpc, and reaches velocity dispersions of ∼250 km s−1. We get a weak negative metallicity gradient of −0.005 dex kpc−1 at 5 ≤ r ≤ 50 kpc, while there is an excess of relative metal-rich stars with [Fe/H] > −1 in the inner 10 kpc. The halo interlopers at r ≤ 5 kpc from integration of our sample has a mass of ∼1.2 × 108 M ⊙ (∼4.7×107 M ⊙ at [Fe/H] < 1.5), which can explain 50–100% of the metal-poor stars with [Fe/H] < −1.5 directly observed in the Galactic central regions.

Environmental filtering and mass effect are two important processes driving lake benthic diatoms: Results of a DNA metabarcoding study in a large lake.

Environmental filtering is often found to dominate assembly rules in diatoms. These microalgae are diverse, especially at subspecies level, and tend to exhibit a niche phylogenetic conservatism. Therefore, other rules, such as competition or mass effects, should be detectable when environmental gradients and dispersal barriers are limited. We used metabarcoding to analyse benthic littoral diatom communities in 153 sites in a large lake (Geneva) exhibiting weak geographical barriers and weak environmental gradients outside river estuaries. We assessed assembly rules using variance partitioning, phylogenetic and source tracking analyses. No phylogenetic over-dispersion of communities, indicative of exclusive competition, was detected. Instead, we found these communities to be phylogenetically over-clustered, indicating environmental filtering, which was even stronger near river estuaries where environmental gradients are stronger. Finally, using a Bayesian method (SourceTracker), we found that rivers flowing into the lake bring communities that settle, especially in sites close to estuaries. Rivers with the highest discharges are primarily responsible for immigration, explaining 27% of lake composition. Therefore, despite favourable conditions to observe other rules, our results support that diatom communities are prominently assembled by environmental filtering and immigration processes, in particular from rivers. However, this does not exclude that other assembly rules may be at play at a finer spatial, temporal and/or phylogenetic scale.

Early Paleogene continental temperature patterns and gradients over eastern Eurasia

Early Paleogene latitudinal continental temperature gradients along the Pacific coast of Eurasia are studied in time and space using the Coexistence Approach, for the first time applied on an extensive regional palaeobotanical record. The palaeobotanical data used in this reconstruction are compiled from literature resources on a total of 110 reasonably well-dated floras, including 79 palynofloras, 30 leaf floras and one carpoflora covering the early Paleocene (Danian) to early Eocene (Ypresian), i.e., a time-span of ca. 25 myr, in total. The palaeobotanical records originate from terrestrial deposits of 73 localities situated along the Pacific coast of Eurasia, including the Far East of Russia, Eastern Siberia, China, and Japan. Our results reveal very weak latitudinal temperature gradients during the early Paleogene. Nevertheless, based on mean values of mean annual temperature and cold month mean temperature, two different regional climatic zones can be distinguished in the Paleocene representing in each case the cooler and warmer (subtropical) part of warm temperate climate of the Koeppen-Geiger system. In the early Eocene, the gradient became more clearly pronounced and in addition, a cool temperate zone can be distinguished. The presence of mangroves in our early Eocene records, already known from previous studies, is largely in line with our climate reconstruction and possible can be related to hyperthermal events such as the PETM and ETM.

Fingering Convection in the Stably Stratified Layers of Planetary Cores

AbstractStably stratified layers may be present at the top of the electrically conducting fluid layers of many planets either because the temperature gradient is locally subadiabatic or because a stable composition gradient is maintained by the segregation of chemical elements. Here we study the double‐diffusive processes taking place in such a stable layer, considering the case of Mercury's core where the temperature gradient is stable but the composition gradient is unstable over a 800 km‐thick layer. The large difference in the molecular diffusivities leads to the development of buoyancy‐driven instabilities that drive radial flows known as fingering convection. We model fingering convection using hydrodynamical simulations in a rotating spherical shell and varying the rotation rate and the stratification strength. For small Rayleigh numbers (i.e., weak background temperature and composition gradients), fingering convection takes the form of columnar flows aligned with the rotation axis and with an azimuthal size comparable with the layer thickness. For larger Rayleigh numbers, the flows retain a columnar structure but the azimuthal size is drastically reduced leading to thin sheet‐like structures that are elongated in the meridional direction. The azimuthal size decreases when the thermal stratification increases, following closely the scaling law expected from the linear planar theory (Stern, 1960, https://doi.org/10.1111/j.2153-3490.1960.tb01295.x). We find that the radial flows always remain laminar with local Reynolds number of order 1–10. Equatorially symmetric zonal flows form due to latitudinal variations of the axisymmetric composition. The zonal velocity exceeds the non‐axisymmetric velocities at the largest Rayleigh numbers. We discuss plausible implications for planetary magnetic fields.

Robust weak Galerkin finite element solvers for Stokes flow based on a lifting operator

This paper presents novel finite element solvers for Stokes flow that are pressure-robust due to the use of a lifting operator. Specifically, weak Galerkin (WG) finite element schemes are developed for the Stokes problem on quadrilateral and hexahedral meshes. Local Arbogast-Correa or Arbogast-Tao spaces are utilized for construction of discrete weak gradients. The lifting operator lifts WG test functions into H ( div ) -subspaces and removes pressure dependence of velocity errors. The pressure robustness of these solvers is validated theoretically and illustrated numerically. Comparison with the non-robust classical Taylor-Hood ( Q 2 , Q 1 ) solver is presented.

The amphibian magnetic sense(s).

Sensitivity to the earth's magnetic field is the least understood of the major sensory systems, despite being virtually ubiquitous in animals and of widespread interest to investigators in a wide range of fields from behavioral ecology to quantum physics. Although research on the use of magnetic cues by migratory birds, fish, and sea turtles is more widely known, much of our current understanding of the functional properties of vertebrate magnetoreception has come from research on amphibians. Studies of amphibians established the presence of a light-dependent magnetic compass, a second non-light-dependent mechanism involving particles of magnetite and/or maghemite, and an interaction between these two magnetoreception mechanisms that underlies the "map" component of homing. Simulated magnetic displacement experiments demonstrated the use of a high-resolution magnetic map for short-range homing to breeding ponds requiring a sampling strategy to detect weak spatial gradients in the magnetic field despite daily temporal variation at least an order of magnitude greater. Overall, reliance on a magnetic map for short-range homing places greater demands on the underlying sensory detection, processing, and memory mechanisms than comparable mechanisms used by long-distance migrants. Moreover, unlike sea turtles and migratory birds, amphibians are exceptionally well suited to serve as model organisms in which to characterize the molecular and biophysical mechanisms underlying the light-dependent 'quantum compass'.

The income-mortality paradox among first- and second-generation immigrants in Sweden

Abstract Background This study investigates mortality differences by income among first- and second-generation immigrants and the native ancestral population in Sweden. Despite immigrants’ various vulnerabilities and the exhaustive evidence for a persistent inverse relationship between income and mortality in general Western populations, previous studies from outside Sweden demonstrated surprisingly weak income gradients in mortality among first-generation immigrants. Examining these associations among second-generation immigrants may help to understand this paradox. Methods Swedish register data from 2002 to 2016 were used to study the association between individual income rank positions and all-cause mortality. The study population was restricted to ages 25-64 years. Based on ‘relative indices of inequality’ (RII) derived from Poisson regressions, we measured mortality differentials between the least and most deprived income rank positions stratified by nativity group and sex. Correspondingly, we assessed absolute differences in mortality between the most and least deprived by using ‘slope indices of inequality’ (SII). Results Largest inequalities in mortality by relative income rank positions (based on RII) were found for the Swedish native ancestral population that showed on average a nearly doubled mortality risk for least compared to most deprived rank position. Immigrants disclosed weak or even nullified associations between relative income rank and all-cause mortality. Mortality inequalities by income among second-generation immigrants were substantially higher relative to first-generation immigrants but somewhat lower compared to the native ancestral population. These patterns were consistent between males and females, and confirmed by the use of SII. Conclusions Distinct exposures to inequality structures but also first-generation immigrants’ ‘mortality advantage’ likely contribute to the considerable mortality differences by income between the studied nativity groups. Key messages • Despite their social and economic vulnerabilities, first-generation immigrants in Sweden disclose weak associations between relative income rank and all-cause mortality. • Second-generation immigrants in Sweden show notably higher magnitudes in income-related mortality compared with first-generation immigrants.

Spectroscopy of nanoantenna-covered Cu2O : Towards enhancing quadrupole transitions in Rydberg excitons

Plasmonic antennas channel incoming light into a confined near field that comes along with strong field gradients at the edges. Transverse field gradients are capable of driving quadrupole transitions in a more effective way than the weak longitudinal field gradient in dipolar light. Cuprous oxide Rydberg excitons with principal quantum number $n$ between 5 and 15 are several tens to a few hundreds of nanometers in diameter and can have different orbital angular momentum quantum numbers. Usually, only $P$- and $F$-excitons are electric dipole allowed, while $S$- and $D$-excitons remain dark. Here, we use 30-nm-wide and 60- to 110-nm-long plasmonic aluminum nanoantennas deposited on the cuprous oxide crystal surface to create transverse field gradients in similar spatial dimensions as the size of the Rydberg excitons. This way, light field and matter wave function overlap spatially and quadrupole transitions of $S$excitons are enhanced. The relative enhancement of the $S$-exciton oscillator strength with respect to the $P$-exciton oscillator accounts for a factor of up to 1.4.

Turbulent wakes in a non-uniformly stratified environment

This study explores the behavior of turbulent wakes generated by a sphere propagating with constant speed in a non-uniformly stratified fluid. The investigation is based on a series of high-resolution direct numerical simulations in which the background stratification is systematically varied. We consider one linear and three nonlinear density profiles and discover that even modest, spatially localized non-uniformities of stratification can profoundly influence the wake dynamics, structure, and evolution. The analysis of microstructure signatures shows that wakes in non-uniformly stratified fluids tend to be more spread horizontally, and internal waves are much stronger than in linear stratification. Simulations performed with Gaussian perturbations are characterized by a vertically asymmetric energy distribution, which is attributed to internal wave reflections from low-gradient regions. Using microstructure decay rates, we estimate the effective persistence period of wakes, showing that it substantially increases with the increasing Froude number. We also find that wakes persist much longer in high-gradient profiles, whereas weak local gradients can substantially reduce the wake longevity.

Fluid mechanism analysis on the interaction between natural laminar flow nacelle and wingbody on a transonic aircraft

A transonic natural laminar flow (NLF) nacelle, which is a streamlined fairing used to contain a turbofan engine and mounted under the wing of civil aircraft, can reduce friction drag. Because the fluid mechanism of the interaction between an NLF nacelle and a wingbody is not clear, laminar flow at high Reynolds numbers in the transonic regime is maintained difficultly. In this work, such interaction on a civil aircraft is investigated. Three NLF nacelles with different pressure distribution characteristics and a baseline nacelle with the turbulent flow are examined. These are installed under the wing of a widebody aircraft to investigate the fluid mechanism between a natural laminar flow nacelle and wingbody. The results show that the influence of the wingbody on the fluid characteristics of the nacelle should be considered in the NLF nacelle design. A well-designed isolated NLF nacelle is different from the one that considers the effect of the wingbody. A favorable pressure gradient in the front part of the nacelle is a key factor in drag reduction. An installed NLF nacelle owning a large pressure peak with a weak favorable pressure gradient and a strong shock wave in front of the nacelle is recommended to be applied in civil aircraft.

Quantifying data information content to resolve seabed structure in geoacoustic inversion

This paper considers the importance of quantitative model selection and general parameterizations in estimating and interpreting seabed profiles in geoacoustic inversion, with application to data collected on the New England Mud Patch. In particular, the seabed structure that can be resolved depends on the information content of the acoustic data set under consideration, which varies with a number of factors, including the physics of the seabed acoustic interaction, frequency content of the data, and measurement and theory errors. Quantitative model selection applied to general parameterizations ensures the inclusion of seabed structure that is reliably sensed by the data while avoiding spurious structure. Data sets considered here include ship noise, modal dispersion, and wide-angle reflection coefficients. In each case, seabed models consistent with the data information content are estimated as part of the inversion using trans-dimensional and/or Bernstein-polynomial parameterizations. Results for all data sets indicate a low sound-speed mud layer over higher-speed sand; however, the ability to resolve structure within the mud layer, such as a transition to higher speeds near the mud base and possibly a weak positive gradient in the upper mud, depends on the information content of the various data sets. [Supported by the Office of Naval Research.]

Understanding Changes in the Tropical Circulation under Global Warming Using a Cloud-Resolving Model and a Conceptual Model

Abstract A cloud-resolving model (CRM) is used to investigate how a prototype tropical circulation driven by a sea surface temperature (SST) contrast changes in a warmer climate. The CRM is used to simulate a region of the atmosphere with a positive SST anomaly, and the large-scale circulation in this region is represented using the weak temperature gradient (WTG) and damped gravity wave (DGW) parameterizations, where the large-scale vertical velocity within the domain is related to the deviation of the simulated density profile from a reference profile representative of the tropical mean state. The behavior of the circulation in response to an increase in SST of both the domain and reference state (i.e., uniform warming) is examined. While the vertical velocity shows an increase in its maximum strength with warming, its value in the lower to midtroposphere decreases. Since the water vapor concentration is largest in the lower troposphere, this leads to a dynamic weakening of precipitation under warming. To understand these results, a simple model for the thermodynamic structure of a convecting atmosphere based on a bulk entraining plume is employed. The model uses a fixed entrainment rate and the relative humidity profiles from the CRM to predict the temperature profiles of the domain and reference state. The vertical velocity profiles calculated from these predicted temperature profiles reproduce important aspects of those simulated with the CRM. This simple modeling framework reveals that the effect of entrainment is crucial to understanding the dynamic response of precipitation to warming, providing a stepping stone to understanding the factors driving changes to the tropical precipitation distribution in a future warmer climate.

A weak Galerkin method for elasticity interface problems

This article introduces a weak Galerkin (WG) finite element method for linear elasticity interface problems on general polygonal/polyhedral partitions. The discrete weak divergence and gradient operators are discretized as polynomials and computed by solving inexpensive local problems on each element. The developed WG method has been proved to be stable and accurate with optimal order error estimates in the discrete H1 norm. Some numerical experiments are conducted to verify the efficiency and accuracy of the proposed WG method, in addition, and its uniform convergence independent of the jump of the interface coefficients and the incompressibility.

A first-order condition for the independence on p of weak gradients

It is well known that on arbitrary metric measure spaces, the notion of minimal p-weak upper gradient may depend on p.In this paper we investigate how a first-order condition of the metric-measure structure, that we call Bounded Interpolation Property, guarantees that in fact such dependence is not present.We also show that the Bounded Interpolation Property is stable for pointed measure Gromov Hausdorff convergence and holds on a large class of spaces satisfying curvature dimension conditions.

Dynamics of the Atlantic Marine Intertropical Convergence Zone

AbstractA generalized ω‐equation is used to identify the contributions from different processes that force upward motions in the Atlantic Marine ITCZ (AMI) from a numerical mesoscale simulation of June 2010. This ω‐equation separates the diabatic heating contributions, which lie at the core of the Weak Temperature Gradient (WTG) framework, from the dynamical terms. Three layers of atmosphere are found with different balance. In the Marine Atmospheric Boundary‐Layer (MABL), the upward motions in the AMI are induced by the frontogenesis and buoyancy components, which are regulated by the ageostrophic adjustment due to the presence of thermal‐wind imbalance. The balance of these three processes well captures the variability of the vertical velocity and the associated precipitation, meaning that boundary‐layer processes play a central role in the AMI dynamics. In the layer [600–2,000 m], a zone of strong vertical wind‐shear just above the MABL, the upward motions are induced by the ageostrophic adjustment and radiative components, which are counteracted by evaporation of convective precipitation. Above 2,000 m the ascending motions are driven by the deep convection heating, as expected by the WTG framework, and more surprisingly by the ageostrophic adjustment term within the Tropical Easterly Jet. Thanks to the use of the ω‐equation, these results extend the current WTG framework to the boundary layer, where it is not expected to hold. In the free troposphere, the WTG framework only accounts for half of the AMI ascent, the other half being forced by the dynamical terms.

The Basic Equations under Weak Temperature Gradient Balance: Formulation, Scaling, and Types of Convectively Coupled Motions

Abstract The weak temperature gradient (WTG) approximation is extended to the basic equations on a rotating plane. The circulation is decomposed into a diabatic component that satisfies WTG balance exactly and a deviation from this balance. Scale analysis of the decomposed basic equations reveals a spectrum of motions, including unbalanced inertio-gravity waves and several systems that are in approximate WTG balance. The balanced systems include equatorial moisture modes with features reminiscent of the MJO, off-equatorial moisture modes that resemble tropical depression disturbances, “mixed systems” in which temperature and moisture play comparable roles in their thermodynamics, and moist quasigeostrophic motions. In the balanced systems the deviation from WTG balance is quasi nondivergent, in nonlinear balance, and evolves in accordance to the vorticity equation. The evolution of the strictly balanced WTG circulation is in turn described by the divergence equation. WTG balance restricts the flow to evolve in the horizontal plane by making the isobars impermeable to vorticity and divergence, even in the presence of diabatically driven vertical motions. The vorticity and divergence equations form a closed system of equations when the irrotational circulation is in WTG balance and the nondivergent circulation is in nonlinear balance. The resulting “WTG equations” may elucidate how interactions between diabatic processes and the horizontal circulation shape slowly evolving tropical motions. Significance Statement Many gaps in our understanding of tropical weather systems still exist and there are still many opportunities to improve their forecasting. We seek to further our understanding of the tropics by extending a framework known as the “weak temperature gradient approximation” to all of the equations for atmospheric flow. Doing this reveals a variety of motions whose scales are similar to observed tropical weather systems. We also show that two equations describe the evolution of slow systems: one that describes tropical thunderstorms and one for the rotating horizontal winds. The two equations may help us understand the dynamics of slowly evolving tropical systems.

A Non-Uniform Planar Coil in Electromagnetic Vibration Energy Harvesting for Enhanced Output Power

Planar electromagnetic vibration energy harvester (EVEH) is a promising technology for compact, low form factor self-powered wireless devices. However, planar EVEH has the limitation of non-fully utilized magnetic flux gradient and low output power due to its structure constraint and weak magnetic flux gradient capturing capability. Planar coil is the key component in harvesting magnetic flux gradient and determining the output electromotive force (EMF) voltage and output power. A novel non-uniform distributed planar (NUDP) coil is proposed to break the current limitation of output power for the planar coil. The boost in output power is realized with tight inner windings and loose outer windings. The breakpoint is determined based on power increasing rate for each winding. The optimal winding radius for both inner and outer coils is investigated using discrete non-linear constraint heuristic optimization method. The designed NUDP coil-based EVEH achieves peak-to-peak output voltage of 305 mV and output power of 1.163 mW at 64.3 Hz resonant frequency and 0.05 g acceleration. It realizes 16.3% higher output power and 19.4% higher output voltage than uniform distributed planar (UDP) coil under the same spring and magnet. The power density of the proposed prototype reaches 9.616 mW/cm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> g <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , which is 15.025 times higher than the state-of-the-art EVEHs with the planar coil.

Commutators of local bilinear maximal operator with Sobolev symbols

In the present paper, we introduce the commutators of local bilinear maximal operator, local bilinear maximal commutators and their fractional variants. Under the conditions that the symbol function belongs to the first-order Sobolev spaces, the bounds for the above commutators on the Sobolev spaces are established. These results are based on several new pointwise estimates for the weak gradients of the above commutators.