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We consider a 2-layer quasi-geostrophic ocean model where the upper layer is forced by a steady Kolmogorov wind stress in a periodic channel domain, which allows to mathematically study the nonlinear development of the resulting flow. The model supports a steady parallel shear flow as a response to the wind stress. As the maximal velocity of the shear flow (equivalently the maximal amplitude of the wind forcing) exceeds a critical threshold, the zonal jet destabilizes due to baroclinic instability and we numerically demonstrate that a first transition occurs. We obtain reduced equations of the system using the formalism of dynamic transition theory and establish two scenarios which completely describe this first transition. The generic scenario is that two modes become critical and a Hopf bifurcation occurs as a result. Under an appropriate set of parameters describing midlatitude oceanic flows, we show that this first transition is continuous: a supercritical Hopf bifurcation occurs and a stable time periodic solution bifurcates. We also investigate the case of double Hopf bifurcations which occur when four modes of the linear stability problem simultaneously destabilize the zonal jet. In this case we prove that, in the relevant parameter regime, the flow exhibits a continuous transition accompanied by a bifurcated attractor homeomorphic to $S^3$. The topological structure of this attractor is analyzed in detail and is shown to depend on the system parameters. In particular, this attractor contains (stable or unstable) time-periodic solutions and a quasi-periodic solution., Comment: 20 pages, 12 figures, 2 tables

Flow over a surface can be stratified by imposing a fixed mean vertical temperature (density) gradient profile throughout or via cooling at the surface. These distinct mechanisms can act simultaneously to establish a stable stratification in a flow. Here, we perform a series of direct numerical simulations of open-channel flows to study adaptation of a neutrally stratified turbulent flow under the combined or independent action of the aforementioned mechanisms. We force the fully developed flow with a constant mass flow rate. This flow forcing technique enables us to keep the bulk Reynolds number constant throughout our investigation and avoid complications arising from the acceleration of the bulk flow when a constant pressure gradient approach were to be adopted to force the flow instead. When both stratification mechanisms are active, the dimensionless stratification perturbation number emerges as an external flow control parameter, in addition to the Reynolds, Froude, and Prandtl numbers. We demonstrate that significant deviations from the Monin-Obukhov similarity formulation are possible when both types of stratification mechanisms are active within an otherwise weakly stable flow, even when the flux Richardson number is well below 0.2. An extended version of the similarity theory due to Zilitinkevich and Calanca shows promise in predicting the dimensionless shear for cases where both types of stratification mechanisms are active, but the extended theory is less accurate for gradients of scalar. The degree of deviation from neutral dimensionless shear as a function of the vertical coordinate emerges as a qualitative measure of the strength of stable stratification for all the cases investigated in this study., Comment: revised manuscript submitted to the Journal of the Atmospheric Sciences

The Cosmic Ray Albedo Neutron Decay (CRAND) is believed to be the principal mechanism for the formation of inner proton radiation belt -- at least for relatively higher energy particles. We implement this mechanism in a Monte Carlo simulation procedure to calculate the trapped proton radiation at the low Earth orbits, through event-by-event interaction of the cosmic ray particles in the Earth's atmosphere and their transportation in the magnetosphere. We consider the generation of protons from subsequent decay of the secondary neutrons from the cosmic ray interaction in the atmosphere and their transport (and/or trapping) in the geomagnetic field. We address the computational challenges for this type of calculations and develop an optimized algorithm to minimize the computation time. We consider a full 3D description of the Earth's atmospheric and magnetic-field configurations using the latest available models. We present the spatial and phase-space distribution of the trapped protons considering the adiabatic invariants and other parameters at the low Earth orbits. We compare the simulation results with the trapped proton flux measurements made by PAMELA experiment at low Earth orbit and explain certain features observed by the measurement., 28 pages, 11 figures, Accepted for publication in Advances in Space Research

Eddy saturation describes the nonlinear mechanism in geophysical flows whereby, when average conditions are considered, direct forcing of the zonal flow increases the eddy kinetic energy, while the energy associated with the zonal flow does not increase. We present a minimal baroclinic model that exhibits complete eddy saturation. Starting from Phillips’ classical quasi-geostrophic two-level model on the beta channel of the mid-latitudes, we derive a reduced order model comprising of six ordinary differential equations including parameterised eddies. This model features two physically realisable steady state solutions, one a purely zonal flow and one where, additionally, finite eddy motions are present. As the baroclinic forcing in the form of diabatic heating is increased, the zonal solution loses stability and the eddy solution becomes attracting. After this bifurcation, the zonal components of the solution are independent of the baroclinic forcing, and the excess of heat in the low latitudes is efficiently transported northwards by finite eddies, in the spirit of baroclinic adjustment.

Volcanic jet flows in explosive eruptions emit radio frequency signatures, indicative of their fluid dynamic and electrostatic conditions. The emissions originate from sparks supported by an electric field built up by the ejected charged volcanic particles. When shock-defined, low-pressure regions confine the sparks, the signatures may be limited to high-frequency content corresponding to the early components of the avalanche-streamer-leader hierarchy. Here, we image sparks and a standing shock together in a transient supersonic jet of micro-diamonds entrained in argon. Fluid dynamic and kinetic simulations of the experiment demonstrate that the observed sparks originate upstream of the standing shock. The sparks are initiated in the rarefaction region, and cut off at the shock, which would limit their radio frequency emissions to a tell-tale high-frequency regime. We show that sparks transmit an impression of the explosive flow, and open the way for novel instrumentation to diagnose currently inaccessible explosive phenomena., Comment: 9 pages, 6 figures

On average once every four years, the Tropical Pacific warms considerably during events called El Niño, leading to weather disruptions over many regions on Earth. Recent machine-learning approaches to El Niño prediction, in particular Convolutional Neural Networks (CNNs), have shown a surprisingly high skill at relatively long lead times. In an attempt to understand this high skill, we here use data from distorted physics simulations with an intermediate complexity El Niño model to determine what aspects of El Niño physics are represented in a specific CNN-based classification method. We find that the CNN can adequately correct for distortions in the ocean adjustment processes, but that the machine-learning method has far more trouble to deal with distortions in upwelling feedback strength., 17 pages, 12 figures

Wind tunnel experiments were carried out on four urban morphologies: two tall canopies with uniform-height and two super-tall canopies with a large variation in element heights (where the maximum element height is more than double the average canopy height, $h_{max}$=2.5 $h_{avg}$). {The average canopy height and packing density were fixed across the surfaces to $h_{avg} = 80$ mm, and $\lambda_{p} = 0.44$, respectively.} A combination of laser doppler anemometry and direct drag measurements were used to calculate and scale the mean velocity profiles {within the boundary layer depth, $\delta$}. In the uniform-height experiment, the high packing density resulted in a `skimming flow' regime with very little flow penetration into the canopy. This led to a surprisingly shallow roughness sublayer ($z\approx1.15h_{avg}$), and a well-defined inertial sublayer above it. {In the heterogeneous-height canopies, despite the same packing density and average height, the flow features were significantly different.} {The height heterogeneity enhanced mixing thus encouraging deep flow penetration into the canopy. A deeper roughness sublayer was found to exist and extend up to just above the tallest element height (corresponding to $z/h_{avg} = 2.85$)}, which was found to be the dominant lengthscale controlling the flow behaviour. {Results points toward the existence of an inertial sublayer for all surfaces considered herein despite the severity of the surface roughness ($\delta/h_{avg} = 3 - 6.25$)}. This contrasts with previous literature., Comment: 25 pages, 12 figures. Revised submission to Boundary-Layer Meteorology

Ocean swell plays an important role in the transport of energy across the ocean, yet its evolution is still not well understood. In the late 1960s, the nonlinear Schr{\"o}dinger (NLS) equation was derived as a model for the propagation of ocean swell over large distances. More recently, a number of dissipative generalizations of the NLS equation based on a simple dissipation assumption have been proposed. These models have been shown to accurately model wave evolution in the laboratory setting, but their validity in modeling ocean swell has not previously been examined. We study the efficacy of the NLS equation and four of its generalizations in modeling the evolution of swell in the ocean. The dissipative generalizations perform significantly better than conservative models and are overall reasonable models for swell amplitudes, indicating dissipation is an important physical effect in ocean swell evolution. The nonlinear models did not out-perform their linearizations, indicating linear models may be sufficient in modeling ocean swell evolution.

The Earth's N2-dominated atmosphere is a very special feature. Firstly, N2 as main gas is unique on the terrestrial planets in the inner solar system and gives a hint for tectonic activity. Studying the origins of atmospheric nitrogen and its stability provides insights into the uniqueness of the Earth's habitat. Secondly, the coexistence of N2 and O2 within an atmosphere is unequaled in the entire solar system. Such a combination is strongly linked to the existence of aerobic lifeforms. The availability of nitrogen on the surface, in the ocean, and within the atmosphere can enable or prevent the habitability of a terrestrial planet, since nitrogen is vitally required by all known lifeforms. In the present work, the different origins of atmospheric nitrogen, the stability of nitrogen dominated atmospheres, and the development of early Earth's atmospheric N2 are discussed. We show why N2-O2-atmospheres constitute a biomarker not only for any lifeforms but for aerobic lifeforms, which was the first major step that led to higher developed life on Earth.

We describe a simple classroom demonstration of a fluid-dynamic instability. The demonstration requires only a bucket of water, a piece of string and some used tealeaves or coffee grounds. We argue that the mechanism for the instability, at least in its later stages, is two-dimensional barotropic (shear-flow) instability and we present evidence in support of this. We show results of an equivalent basic two-dimensional numerical non-linear model, which simulates behavior comparable to that observed in the bucket demonstration. Modified simulations show that the instability does not depend on the curvature of the domain, but rather on the velocity profile., 22 pages, 8 figures. This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in American Journal of Physics 88, Issue 12, 1041 (2020) and may be found at https://doi.org/10.1119/10.0002438

The Nobel Prize in Physics 2021 was awarded to Syukuro Manabe, Klaus Hasselmann, and Giorgio Parisi for their “groundbreaking contributions to our understanding of complex systems,” including major advances in the understanding of our climate and climate change. In this Perspective article, we review their key contributions and discuss their relevance in relation to the present understanding of our climate. We conclude by outlining some promising research directions and open questions in climate science. All authors received funding from the European Union’s Horizon 2020 research and innovation program under Marie Sklodowska Curie Grant Agreement No. 813844. C.M. also acknowledges funding from the Spanish Ministerio de Ciencia, Innovación y Universidades (No. PGC2018-099443-B-I00) and the ICREA ACADEMIA program of Generalitat de Catalunya. J.K. was supported by the Russian Ministry of Science and Education (Agreement No. 075-15-2020-808).

Traditional general circulation models, or GCMs -- i.e. 3D dynamical models with unresolved terms represented in equations with tunable parameters -- have been a mainstay of climate research for several decades, and some of the pioneering studies have recently been recognized by a Nobel prize in Physics. Yet, there is considerable debate around their continuing role in the future. Frequently mentioned as limitations of GCMs are the structural error and uncertainty across models with different representations of unresolved scales; and the fact that the models are tuned to reproduce certain aspects of the observed Earth. We consider these shortcomings in the context of a future generation of models that may address these issues through substantially higher resolution and detail, or through the use of machine learning techniques to match them better to observations, theory, and process models. It is our contention that calibration, far from being a weakness of models, is an essential element in the simulation of complex systems, and contributes to our understanding of their inner workings. Models can be calibrated to reveal both fine-scale detail, or the global response to external perturbations. New methods enable us to articulate and improve the connections between the different levels of abstract representation of climate processes, and our understanding resides in an entire hierarchy of models where GCMs will continue to play a central role for the foreseeable future.., 9 pages, 5 figures, 93 citations

The Cassini and Juno probes have revealed large coherent cyclonic vortices in the polar regions of Saturn and Jupiter, a dramatic contrast from the east-west banded jet structure seen at lower latitudes. Debate has centered on whether the jets are shallow, or extend to greater depths in the planetary envelope. Recent experiments and observations have demonstrated the relevance of deep convection models to a successful explanation of jet structure and cyclonic coherent vortices away from the polar regions have been simulated recently including an additional stratified shallow layer. Here we present new convective models able to produce long-lived polar vortices. Using simulation parameters relevant for giant planet atmospheres we find flow regimes that are in agreement with geostrophic turbulence (GT) theory in rotating convection for the formation of large scale coherent structures via an upscale energy transfer fully three-dimensional. Our simulations generate polar characteristics qualitatively similar to those seen by Juno and Cassini: they match the structure of cyclonic vortices seen on Jupiter; or can account for the existence of a strong polar vortex extending downwards to lower latitudes with a marked spiral morphology and the hexagonal pattern seen on Saturn. Our findings indicate that these vortices can be generated deep in the planetary interior. A transition differentiating these two polar flows regimes is described, interpreted in terms of different force balances and compared with previous shallow atmospheric models which characterised polar vortex dynamics in giant planets. In addition, the heat transport properties are investigated confirming recent scaling laws obtained in the context of reduced models of GT., Comment: 18 pages, 13 figures and 3 tables

We use a database of direct numerical simulations to evaluate parametrizations for energy dissipation rate in stably stratified flows. We show that shear-based formulations are more appropriate for stable boundary layers than commonly used buoyancy-based formulations. As part of the derivations, we explore several length scales of turbulence and investigate their dependence on local stability.

We present a publicly available library of model atmospheres with radiative-convective equilibrium Pressure-Temperature ($P$-$T$) profiles fully consistent with equilibrium chemical abundances, and the corresponding emission and transmission spectrum with R$\sim$5000 at 0.2 $\mu$m decreasing to R$\sim$35 at 30 $\mu$m, for 89 hot Jupiter exoplanets, for four re-circulation factors, six metallicities and six C/O ratios. We find the choice of condensation process (local/rainout) alters the $P$-$T$ profile and thereby the spectrum substantially, potentially detectable by JWST. We find H$^-$ opacity can contribute to form a strong temperature inversion in ultra-hot Jupiters for C/O ratios $\geq$ 1 and can make transmission spectra features flat in the optical, alongside altering the entire emission spectra. We highlight how adopting different model choices such as thermal ionisation, opacities, line-wing profiles and the methodology of varying the C/O ratio, effects the $P$-$T$ structure and the spectrum. We show the role of Fe opacity to form primary/secondary inversion in the atmosphere. We use WASP-17b and WASP-121b as test cases to demonstrate the effect of grid parameters across their full range, while highlighting some important findings, concerning the overall atmospheric structure, chemical transition regimes and their observables. Finally, we apply this library to the current transmission and emission spectra observations of WASP-121b, which shows H$_2$O and tentative evidence for VO at the limb, and H$_2$O emission feature indicative of inversion on the dayside, with very low energy redistribution, thereby demonstrating the applicability of library for planning and interpreting observations of transmission and emission spectrum., Comment: 26 pages, 19 figures in the main paper. 13 pages, 6 figures, 3 tables in the supplementary material attached with the main paper here. Accepted for Publication in MNRAS. Full grid of model P-T profiles, chemical abundances, transmission and emission spectra, contribution functions are available here, https://drive.google.com/drive/folders/1zCCe6HICuK2nLgnYJFal7W4lyunjU4JE

A demanding challenge in atmospheric research is the night-time characterization of aerosols using passive techniques, that is, by extracting information from scattered light that has not been emitted by the observer. Satellite observations of artificial night-time lights have been used to retrieve some basic integral parameters, like the aerosol optical depth. However, a thorough analysis of the scattering processes allows one to obtain substantially more detailed information on aerosol properties. In this Letter we demonstrate a practicable approach for determining the aerosol particle size number distribution function in the air column, based on the measurement of the angular radiance distribution of the scattered light emitted by night-time lights of cities and towns, recorded from low Earth orbit. The method is self-calibrating and does not require the knowledge of the absolute city emissions. The input radiance data are readily available from several spaceborne platforms, like the VIIRS-DNB radiometer onboard the Suomi-NPP satellite., Author prepared version of a manuscript accepted for publication in Monthly Notices of the Royal Astronomical Society (Letter) 5 pages, 5 figures

The combined effectiveness of model reduction and the quasilinear approximation for the reproduction of the low-order statistics of oceanic surface boundary-layer turbulence is investigated. Idealized horizontally homogeneous problems of surface-forced thermal convection and Langmuir turbulence are studied in detail. Model reduction is achieved with a Galerkin projection of the governing equations onto an subset of modes determined by proper orthogonal decomposition. For less than 0.2% of the modes retained, the reduced quasilinear model is able to reproduce vertical profiles of horizontal mean fields as well as certain energetically important second-order turbulent transport statistics and energies to within 30% error. For intermediate sizes of the basis truncation some statistics approach those found in the fully nonlinear simulations. Thus basis reduction can actually improve upon the accuracy of quasilinear dynamics. A test model with a small total number of modes demonstrates the non-monotonic convergence toward the correct statistics as the size of the basis is increased., 24 pages, 14 figures

Employing the finite element and computational fluid dynamics methods, we have determined the conditions for the fragmentation of space bodies or preservation of their integrity when they penetrate into the Earth's atmosphere. The origin of forces contributing to the fragmentation of space iron bodies during the passage through the dense layers of the planetary atmosphere has been studied. It was shown that the irregular shape of the surface can produce transverse aerodynamic forces capable of causing deformation stress in the body exceeding the tensile strength threshold of iron., Comment: 9 pages, 14 figures

Nighttime monitoring of the aerosol content of the lower atmosphere is a challenging task, because appropriate reference natural light sources are lacking. Here we show that the anthropogenic night sky brightness due to city lights can be successfully used for estimating the aerosol optical depth of arbitrarily thick atmospheric layers. This method requires measuring the zenith night sky brightness with two detectors located at the limiting layer altitudes. Combined with an estimate of the overall atmospheric optical depth (available from ground-based measurements or specific satellite products), the ratio of these radiances provides a direct estimate of the differential aerosol optical depth of the air column between these two altitudes. These measurements can be made with single-channel low-cost radiance detectors widely used by the light pollution research community., 5 1/2 pages, 3 figures. Accepted for publication (2020 November 03) in Monthly Notices of the Royal Astronomical Society Letters

A statistical indicator for dynamic stability known as the $\Upsilon$ indicator is used to gauge the stability and hence detect approaching tipping points of simulation data from a reduced 5-box model of the North-Atlantic Meridional Overturning Circulation (AMOC) exposed to a time dependent hosing function. The hosing function simulates the influx of fresh water due to the melting of the Greenland ice sheet and increased precipitation in the North Atlantic. The $\Upsilon$ indicator is designed to detect changes in the memory properties of the dynamics, and is based on fitting ARMA (auto-regressive moving-average) models in a sliding window approach to time series data. An increase in memory properties is interpreted as a sign of dynamical instability. The performance of the indicator is tested on time series subject to different types of tipping, namely bifurcation-induced, noise-induced and rate-induced tipping. The numerical analysis show that the indicator indeed responds to the different types of induced instabilities. Finally, the indicator is applied to two AMOC time series from a full complexity Earth systems model (CESM2). Compared with the doubling CO$_2$ scenario, the quadrupling CO$_2$ scenario results in stronger dynamical instability of the AMOC during its weakening phase., Comment: The following article has been submitted to Chaos (AIP-Publishing). After it is published, it will be found at https://publishing.aip.org/resources/librarians/products/journals/

\citet{turner1957} proposed that dry thermals entrain because of buoyancy (via a constraint which requires an increase in the radius $a$). This however, runs counter to the scaling arguments commonly used to derive the entrainment rate, which rely on either the self-similarity of \citet{scorer1957} or the turbulent entrainment hypothesis of \citet{morton1956}. The assumption of turbulence-driven entrainment was investigated by \citet{lecoanet2018}, who found that the entrainment efficiency $e$ varies by less than $20\%$ between laminar (Re = 630) and turbulent (Re = 6300) thermals. This motivated us to utilize Turner's argument of buoyancy-controlled entrainment in addition to the thermal's vertical momentum equation to build a model for thermal dynamics which does not invoke turbulence or self-similarity. We derive simple expressions for the thermals' kinematic properties and their fractional entrainment rate $\epsilon$ and find close quantitative agreement with the values in direct numerical simulations. In particular, our expression for entrainment rate is consistent with the parameterization $\epsilon \sim B/w^2$, for Archimedean buoyancy $B$ and vertical velocity $w$. We also directly validate the role of buoyancy-driven entrainment by running simulations where gravity is turned off midway through a thermal's rise. The entrainment efficiency $e$ is observed to drop to less than 1/3 of its original value in both the laminar and turbulent cases when $g=0$, affirming the central role of buoyancy in entrainment in dry thermals., Comment: 12 pages, 8 figures

Internal redox reactions may irreversibly alter the mantle composition and volatile inventory of terrestrial and super-Earth exoplanets and affect the prospects for atmospheric observations. The global efficacy of these mechanisms, however, hinges on the transfer of reduced iron from the molten silicate mantle to the metal core. Scaling analysis indicates that turbulent diffusion in the internal magma oceans of sub-Neptunes can kinetically entrain liquid iron droplets and quench core formation. This suggests that the chemical equilibration between core, mantle, and atmosphere may be energetically limited by convective overturn in the magma flow. Hence, molten super-Earths possibly retain a compositional memory of their accretion path. Redox control by magma ocean circulation is positively correlated with planetary heat flow, internal gravity, and planet size. The presence and speciation of remanent atmospheres, surface mineralogy, and core mass fraction of atmosphere-stripped exoplanets may thus constrain magma ocean dynamics., Comment: Accepted for publication in ApJL; 7 pages, 2 figures; summaries available at http://bit.ly/Lichtenberg21blog (blog) and http://bit.ly/Lichtenberg21ApJLvideo (video)

We examine the dynamics of low-frequency gravito-inertial waves (GIWs) in differentially rotating deformed radiation zones in stars and planets by generalising the traditional approximation of rotation (TAR). The TAR treatment was built on the assumptions that the star is spherical and uniformly rotating. However, it has been generalised in our previous work by including the effects of the centrifugal deformation using a non-perturbative approach in the uniformly rotating case. We aim to carry out a new generalisation of the TAR treatment to account for the differential rotation and the strong centrifugal deformation simultaneously. We generalise our previous work by taking into account the differential rotation in the derivation of our complete analytical formalism that allows the study of the dynamics of GIWs in differentially and rapidly rotating stars. We derived the complete set of equations that generalises the TAR, simultaneously taking the full centrifugal acceleration and the differential rotation into account. Within the validity domain of the TAR, we derived a generalised Laplace tidal equation for the horizontal eigenfunctions and asymptotic wave periods of the GIWs, which can be used to probe the structure and dynamics of differentially rotating deformed stars with asteroseismology. A new generalisation of the TAR, which simultaneously takes into account the differential rotation and the centrifugal acceleration in a non-perturbative way, was derived. This generalisation allowed us to study the detectability and the signature of the differential rotation on GIWs in rapidly rotating deformed stars and planets. We found that the effects of the differential rotation in early-type deformed stars on GIWs is theoretically largely detectable in modern space photometry using observations from $\textit{Kepler}$ and TESS., 13 pages, 11 figures, 1 table, abstract shortened for arXiv. Accepted for publication in A&A. arXiv admin note: substantial text overlap with arXiv:2104.09302

We study the impact of applying stochastic forcing to the Ghil-Sellers energy balance climate model in the form of a fluctuating solar irradiance. Through numerical simulations, we explore the noise-induced transitions between the competing warm and snowball climate states. We consider multiplicative stochastic forcing driven by Gaussian and $α$-stable Lévy - $α\in(0,2)$ - noise laws, examine the statistics of transition times, and estimate most probable transition paths. While the Gaussian noise case has been carefully studied in a plethora of investigations on metastable systems, much less is known about the Lévy case, especially in the case of high- and infinite-dimensional systems. In the weak noise limit, the expected residence time in each metastable state scales in a fundamentally different way in the Gaussian vs. Lévy noise case with respect to the intensity of the noise. In the former case, the classical Kramers-like exponential law is recovered. In the latter case, power laws are found, with the exponent equal to $-α$, in apparent agreement with rigorous results obtained for additive noise in a related - yet different - reaction-diffusion equation as well as in simpler models. This can be better understood by treating the Lévy noise as a compound Poisson process. The transition paths are studied in a projection of the state space and remarkable differences are observed between the two different types of noise. The snowball-to-warm and the warm-to-snowball most probable transition path cross at the single unstable edge state on the basin boundary. In the case of Lévy noise, the most probable transition paths in the two directions are wholly separated, as transitions apparently take place via the closest basin boundary region to the outgoing attractor. This property can be better elucidated by considering singular perturbations to the solar irradiance., 36 pages, 8 figures

We consider a model of a circular lenticular vortex immersed into a deep and vertically stratified viscous fluid in the presence of gravity and rotation. The vortex is assumed to be baroclinic with a Gaussian profile of angular velocity both in the radial and axial directions. Assuming the base state to be in a cyclogeostrophic balance, we derive linearized equations of motion and seek for their solution in a geometric optics approximation to find amplitude transport equations that yield a comprehensive dispersion relation. Applying algebraic Bilharz criterion to the latter, we establish that stability conditions are reduced to three inequalities that define stability domain in the space of parameters. The main destabilization mechanism is either stationary or oscillatory axisymmetric instability depending on the Schmidt number ($Sc$), vortex Rossby number and the difference between the radial and axial density gradients as well as the difference between the epicyclic and vertical oscillation frequencies. We discover that the boundaries of the regions of stationary and oscillatory axisymmetric instabilities meet at a codimension-2 point, forming a singularity of the neutral stability curve. We give an exhaustive classification of the geometry of the stability boundary, depending on the values of the Schmidt number. Although we demonstrate that the centrifugally stable (unstable) Gaussian lens can be destabilized (stabilized) by the differential diffusion of mass and momentum and that destabilization can happen even in the limit of vanishing diffusion, we also describe explicitly a set of parameters in which the Gaussian lens is stable for all $Sc>0$.

The theory of low-frequency internal gravity waves (IGWs) is readdressed in the stable stratified weakly ionized Earth's ionosphere. The formation of dipolar vortex structures and their dynamical evolution, as well as, the emergence of chaos in the wave-wave interactions are studied both in presence and absence of the Pedersen conductivity. The latter is shown to inhibit the formation of solitary vortices and the onset of chaos., Comment: 6 pages, 4 figures; Submitted for consideration to IEEE Transactions on Plasma Science (Special Issue on Plenary, Invited, and Selected Minicourse Papers from ICOPS2021)

Driven by growing momentum in two-dimensional geophysical flow modeling, this paper introduces a general family of "thermal" rotating shallow-water models. The models are capable of accommodating thermodynamic processes, such as those acting in the ocean mixed layer, by allowing buoyancy to vary in horizontal position and time as well as with depth, in a polynomial fashion up to an arbitrary degree. Moreover, the models admit Euler--Poincare variational formulations and possess Lie--Poisson Hamiltonian structure. Such a geometric property provides solid fundamental support to the theories described with consequences for numerical implementation and the construction of unresolved motion parametrizations. In particular, it is found that stratification halts the development of small-scale filament rollups recently observed in a popular model, which, having vertically homogeneous density, represents a special case of the models presented here., Revised; to appear in Physics of Fluids

We study the thermodynamic equilibrium spectra of the Charney–Hasegawa–Mima (CHM) equation in its weakly nonlinear limit. In this limit, the equation has three adiabatic invariants, in contrast to the two invariants of the 2D Euler or Gross–Pitaevskii equations, which are examples for comparison. We explore how the third invariant considerably enriches the variety of equilibrium spectra that the CHM system can access. In particular we characterise the singular limits of these spectra in which condensates occur, i.e. a single Fourier mode (or pair of modes) accumulate(s) a macroscopic fraction of the total invariants. We show that these equilibrium condensates provide a simple explanation for the characteristic structures observed in CHM systems of finite size: highly anisotropic zonal flows, large-scale isotropic vortices, and vortices at small scale. We show how these condensates are associated with combinations of negative thermodynamic potentials (e.g. temperature).

We use an idealized three-dimensional general circulation model to study condensible-rich atmospheres with an ineffective cold trap on slowly rotating tidally locked terrestrial planets. In particular, we show the climate dynamics in a thin and temperate atmosphere with condensible water vapor. The similarities between our thin and temperate atmosphere and the warm and thick atmosphere approaching the water vapor runaway greenhouse in previous works are discussed, including the reversal of the thermal emission between the day and night hemispheres. Different from the transit spectroscopy of water vapor that depends on the absolute amount of atmospheric water vapor, the contrast between the dayside and nightside thermal emission provides information regarding the relative ratio of water vapor to the background atmosphere as well as the atmospheric pressure near the substellar tropopause and the emission level on the nightside on potentially habitable worlds., 15 pages, 4 figures

The Traditional Approximation of Rotation (TAR) is a treatment of the hydrodynamic equations of rotating and stably stratified fluids in which the action of the Coriolis acceleration along the direction of the entropy and chemical stratifications is neglected because it is weak in comparison with the buoyancy force. The dependent variables in the equations for the dynamics of gravito-inertial waves (GIWs) then become separable into radial and horizontal parts as in the non-rotating case. The TAR is built on the assumptions that the star is spherical (i.e. its centrifugal deformation is neglected) and uniformly rotating. We study the feasibility of carrying out a generalisation of the TAR to account for the centrifugal acceleration in the case of strongly deformed uniformly and rapidly rotating stars (and planets), and to identify the validity domain of this approximation. We built analytically a complete formalism that allows the study of the dynamics of GIWs in spheroidal coordinates which take into account the flattening of rapidly rotating stars by assuming the hierarchies of frequencies adopted within the TAR in the spherical case and by deriving a generalised Laplace tidal equation for the horizontal eigenfunctions of the GIWs and their asymptotic wave periods, which can be used to probe the structure and dynamics of rotating deformed stars with asteroseismology. Using 2D ESTER stellar models, we determine the validity domain of the generalised TAR as a function of the rotation rate of the star normalised by its critical angular velocity and its pseudo-radius. This generalisation allows us to study the signature of the centrifugal effects on GIWs in rapidly rotating deformed stars. We found that the effects of the centrifugal acceleration in rapidly rotating early-type stars on GIWs are theoretically detectable in modern space photometry using observations from Kepler., Comment: 17 pages, 14 figures, 1 table, abstract shortened for arXiv. Accepted for publication in A&A

Small non-spherical particles settling in a quiescent fluid tend to orient so that their broad side faces down, because this is a stable fixed point of their angular dynamics at small particle Reynolds number. Turbulence randomises the orientations to some extent, and this affects the reflection patterns of polarised light from turbulent clouds containing ice crystals. An overdamped theory predicts that turbulence-induced fluctuations of the orientation are very small when the settling number Sv (a dimensionless measure of the settling speed) is large. At small Sv, by contrast, the overdamped theory predicts that turbulence randomises the orientations. This overdamped theory neglects the effect of particle inertia. Therefore we consider here how particle inertia affects the orientation of small crystals settling in turbulent air. We find that it can significantly increase the orientation variance, even when the Stokes number St (a dimensionless measure of particle inertia) is quite small. We identify different asymptotic parameter regimes where the tilt-angle variance is proportional to different inverse powers of Sv. We estimate parameter values for ice crystals in turbulent clouds and show that they cover several of the identified regimes. The theory predicts how the degree of alignment depends on particle size, shape and turbulence intensity, and that the strong horizontal alignment of small crystals is only possible when the turbulent energy dissipation is weak, of the order of $1\,$cm$^2$/s$^3$ or less., Comment: 16 pages, 5 figures, 2 tables, supplemental material

Data assimilation (DA) aims at optimally merging observational data and model outputs to create a coherent statistical and dynamical picture of the system under investigation. Indeed, DA aims at minimizing the effect of observational and model error, and at distilling the correct ingredients of its dynamics. DA is of critical importance for the analysis of systems featuring sensitive dependence on the initial conditions, as chaos wins over any finitely accurate knowledge of the state of the system, even in absence of model error. Clearly, the skill of DA is guided by the properties of dynamical system under investigation, as merging optimally observational data and model outputs is harder when strong instabilities are present. In this paper we reverse the usual angle on the problem and show that it is indeed possible to use the skill of DA to infer some basic properties of the tangent space of the system, which may be hard to compute in very high-dimensional systems. Here, we focus our attention on the first Lyapunov exponent and the Kolmogorov-Sinai entropy, and perform numerical experiments on the Vissio-Lucarini 2020 model, a recently proposed generalisation of the Lorenz 1996 model that is able to describe in a simple yet meaningful way the interplay between dynamical and thermodynamical variables., 23 pages, 10 figures

The influence of forward speed on stochastic free-surface crossing, in a Gaussian wave field, is investigated. The case of a material point moving with a constant forward speed is considered; the wave field is assumed stationary in time, and homogeneous in space. The focus is on up-crossing events, which are defined as the material point crossing the free surface, into the water domain. The effect of the Doppler shift (induced by the forward speed) on the up-crossing frequency, and the related conditional joint distribution of wave kinematic variables is analytically investigated. Some general trends are illustrated through different examples, where three kinds of wave direction distribution are considered: unidirectional, short-crested anisotropic, and isotropic. The way the developed approach may be used in the context of slamming on marine structures is briefly discussed., Accepted for publication in Ocean Engineering

The impact of the El Ni\~no-Southern Oscillation (ENSO) on the extratropics is investigated in an idealized, reduced-order model that has a tropical and an extratropical module. Unidirectional ENSO forcing is used to mimick the atmospheric bridge between the tropics and the extratropics. The variability of the coupled ocean-atmosphere extratropical module is then investigated through the analysis of its pullback attractors (PBAs). This analysis focuses on two types of ENSO forcing generated by the tropical module, one periodic and the other aperiodic. For a substantial range of the ENSO forcing, two chaotic PBAs are found to coexist for the same set of parameter values. Different types of extratropical low-frequency variability are associated with either PBA over the parameter ranges explored. For periodic ENSO forcing, the coexisting PBAs exhibit only weak nonlinear instability. For chaotic forcing, though, they are quite unstable and certain extratropical perturbations induce transitions between the two PBAs. These distinct stability properties may have profound consequences for extratropical climate predictions: in particular, ensemble averaging may no longer help isolate the low-frequency variability signal., Comment: 35 pages, 17 figures, manuscript submitted to JAMES

We present turbulent Schmidt number ($$ Sc_{\text{t}} $$) estimations above three-dimensional urban canopies, where $$ Sc_{\text{t}} $$ is a property of the flow defined as the ratio of the eddy diffusivity of momentum ($$ K_{M} $$) to the eddy diffusivity of mass ($$ D_{t} $$). Despite the fact that $$ Sc_{t} $$ modelling is of great interest, inter alia, for pollutant dispersion simulations conducted via computational fluid dynamics, no universal value is known. Simultaneous measurements of fluid velocity and mass concentration are carried out in a water channel for three staggered arrays of cubical obstacles corresponding to isolated flow, wake-interference, and skimming-flow regimes. A passive tracer is released from a continuous point source located at a height $$ z = 1.67H $$ where H is the obstacle height. The results show an increase of $$ Sc_{\text{t}} $$ with height above the canopy for all three arrays, with values at $$ z = 2H $$ ($$ Sc_{t} \approx 0.6 $$) about double compared to that at $$ z = H $$. The observed $$ Sc_{t} $$ agrees well with that modelled by using a simple formulation for $$ Sc_{t} $$ based on expressions for $$ K_{M} $$ and $$ D_{t} $$ published in previous studies. Comparisons with other $$ Sc_{t} $$ models found in the literature are also presented and discussed.

In this work we present the first parameterizations of the global occurrence rate and chemical influence of Blue Jets, a type of transient luminous event taking place in the stratospheric region above thunderclouds. These parameterizations are directly coupled with five different lightning parameterizations implemented in the Whole Atmosphere Community Climate Model (WACCM4). We have obtained a maximum Blue Jet global occurrence rate of about 0.9 BJ per minute. The geographical occurrence of Blue Jets is closely related to the chosen lightning parameterization. Some previously developed local chemical models of Blue Jets predicted an important influence onto the stratospheric concentration of N2O, NOx, and O3. We have used these results together with our global implementations of Blue Jets in WACCM4 to estimate their global chemical influence in the atmosphere. According to our results, Blue Jets can inject about 3.8 Tg N2O-N/year and 0.07 Tg NO-N/year near the stratosphere, where N2O-N and NO-N stand for the mass of nitrogen atoms in N2O and NO molecules, respectively. These production rates of N2O and NOx could have a direct impact on, for example, the acidity of rainwater or the greenhouse effect. We have found that Blue Jets could also slightly contribute to the depletion of stratospheric ozone. In particular, we have estimated that the maximum difference in the concentration of O3 at 30 km of altitude between simulations with and without Blue Jets can be about −5% in equatorial and polar regions. ©2019. American Geophysical Union. All Rights Reserved., This work was supported by the Spanish Ministry of Science and Innovation, MINECO under projects and ESP2017-86263-C4-4-R and by the EU through the H2020 Science and Innovation with Thunderstorms (SAINT) project (Ref. 722337) and the FEDER program. Authors F.J.P.I and F.J.G.V acknowledge financial support from the State Agency for Research of the Spanish MCIU through the >Center of Excellence Severo Ochoa> award for the Instituto de Astrofisica de Andalucia(SEV-2017-0709). The National Center for Atmospheric Research is sponsored by the National Science Foundation. The CESM project is supported by the National Science Foundation and the Office of Science (BER) of the U.S. Department of Energy. Computing resources were provided by the Climate Simulation Laboratory at NCAR's Computational and Information Systems Laboratory (CISL), sponsored by the National Science Foundation and other agencies. F. J. P.-I. acknowledges a PhD research contract, code BES-2014-069567. F. J. G.-V. acknowledges support from the Spanish Ministry of Education and Culture under the Salvador de Madariaga program PRX17/00078.

We calculate the rate of ocean waves energy dissipation due to whitecapping by numerical simulation of deterministic phase resolving model for dynamics of ocean surface. Two independent numerical experiments are performed. First, we solve the $3D$ Hamiltonian equation that includes three- and four-wave interactions. This model is valid for moderate values of surface steepness only, $\mu < 0.09$. Then we solve the exact Euler equation for non-stationary potential flow of an ideal fluid with a free surface in $2D$ geometry. We use the conformal mapping of domain filled with fluid onto the lower half-plane. This model is applicable for arbitrary high levels of steepness. The results of both experiments are close. The whitecapping is the threshold process that takes place if the average steepness $\mu > \mu_{cr} \simeq 0.055$. The rate of energy dissipation grows dramatically with increasing of steepness. Comparison of our results with dissipation functions used in the operational models of wave forecasting shows that these models overestimate the rate of wave dissipation by order of magnitude for typical values of steepness., Comment: 6 pages, 2 figures

We present an empirical model for nitric oxide NO in the mesosphere ($\approx$60--90 km) derived from SCIAMACHY (SCanning Imaging Absorption spectroMeter for Atmospheric CHartoghraphY) limb scan data. This work complements and extends the NOEM (Nitric Oxide Empirical Model; Marsh et al., 2004) and SANOMA (SMR Acquired Nitric Oxide Model Atmosphere; Kiviranta et al., 2018) empirical models in the lower thermosphere. The regression ansatz builds on the heritage of studies by Hendrickx et al. (2017) and the superposed epoch analysis by Sinnhuber et al. (2016) which estimate NO production from particle precipitation. Our model relates the daily (longitudinally) averaged NO number densities from SCIAMACHY (Bender et al., 2017a, b) as a function of geomagnetic latitude to the solar Lyman-alpha and the geomagnetic AE (auroral electrojet) indices. We use a non-linear regression model, incorporating a finite and seasonally varying lifetime for the geomagnetically induced NO. We estimate the parameters by finding the maximum posterior probability and calculate the parameter uncertainties using Markov chain Monte Carlo sampling. In addition to providing an estimate of the NO content in the mesosphere, the regression coefficients indicate regions where certain processes dominate., Comment: 13 pages, 6 figures; published in Atmos. Chem. Phys

We derive exact analytical expressions for flow configurations that optimize the instantaneous growth rate of energy in the linear Eady problem, along with the associated growth rates. These optimal perturbations are relevant linear stability analysis, but, more importantly, they are relevant for understanding the energetics of fully nonlinear baroclinic turbulence. The optimal perturbations and their growth rates are independent of the Richardson number. The growth rates of the optimal perturbations grow linearly as the horizontal wavelength of the perturbation decreases. Perturbation energy growth at large scales is driven by extraction of potential energy from the mean flow, while at small scales it is driven by extraction of kinetic energy from the mean shear. We also analyze the effect of spatial discretization on the optimal perturbations and their growth rates. A second order energy-conserving discretization on the Arakawa B grid generally has too-weak growth rates at small scales and is less accurate than two second order discretizations on the Arakawa C grid. The two C grid discretizations, one that conserves energy and another that conserves both energy and enstrophy, yield very similar optimal perturbation growth rates that are significantly more accurate than the B grid discretization at small scales., 19 pages, 3 figures

The representation of cloud processes in weather and climate models is crucial for their feedback on atmospheric flows. Since there is no general macroscopic theory of clouds, the parameterization of clouds in corresponding simulation software depends crucially on the underlying modeling assumptions. In this study we present a new model of inter-mediate complexity (a one-and-a-half moment scheme) for warm clouds, which is derived from physical principles. Our model consists of a system of differential-algebraic equations which allows for supersaturation and comprises intrinsic automated droplet activation due to a coupling of the droplet mass- and number concentrations tailored to this problem. For the numerical solution of this system we recommend a semi-implicit integration scheme, with efficient solvers for the implicit parts. The new model shows encouraging numerical results when compared with alternative cloud parameterizations, and it is well suited to investigate model uncertainties and to quantify predictability of weather events in moist atmospheric regimes., Comment: 37 pages, 12 figures

Clouds are important components of the atmosphere. Since it is usually not possible to treat them as ensembles of huge numbers of particles, parameterizations on the basis of averaged quantities (mass and/or number concentration) must be derived. Since no first-principles derivations of such averaged schemes are available today, many alternative approximating schemes of cloud processes exist. Most of these come in the form of nonlinear differential equations. It is unclear whether these different cloud schemes behave similarly under controlled local conditions, and much less so when they are embedded dynamically in a full atmospheric flow model. We use mathematical methods from the theory of dynamical systems and asymptotic analysis to compare two operational cloud schemes and one research scheme qualitatively in a simplified context in which the moist dynamics is reduced to a system of ODEs. It turns out that these schemes behave qualitatively differently on shorter time scales, whereas at least their long time behavior is similar under certain conditions. These results show that the quality of computational forecasts of moist atmospheric flows will generally depend strongly on the formulation of the cloud schemes used., 29 pages, 6 figures

During a solar eclipse the solar irradiance reaching the top-of-atmosphere (TOA) is reduced in the Moon shadow. The solar irradiance is commonly measured by Earth observation satellites before the start of the solar eclipse and is not corrected for this reduction, which results in a decrease of the computed TOA reflectances. Consequently, air quality products that are derived from TOA reflectance spectra, such as the ultraviolet (UV) Absorbing Aerosol Index (AAI), are distorted or undefined in the shadow of the Moon. The availability of air quality satellite data in the penumbral and antumbral shadow during solar eclipses, however, is of particular interest to users studying the atmospheric response to solar eclipses. Given the time and location of a point on the Earth's surface, we explain how to compute the obscuration during a solar eclipse taking into account wavelength-dependent solar limb darkening. With the calculated obscuration fractions, we restore the TOA reflectances and the AAI in the penumbral shadow during the annular solar eclipses on 26 December 2019 and 21 June 2020 measured by the TROPOMI/S5P instrument. In the corrected products, the signature of the Moon shadow disappeared, but only if wavelength-dependent solar limb darkening is taken into account. We conclude that the correction method of this paper can be used to detect real AAI rising phenomena during a solar eclipse and has the potential to restore any other product that is derived from TOA reflectance spectra. This would resolve the solar eclipse anomalies in satellite air quality measurements and would allow for studying the effect of the eclipse obscuration on the composition of the Earth's atmosphere from space., Accepted for publication in Atmospheric Chemistry and Physics

Much attention has been given to the climate dynamics and habitable boundaries of synchronously rotating planets around low mass stars. However, other rotational states are possible, including spin–orbit resonant configurations, particularly when higher eccentricity orbits can be maintained in a system. Additionally, the oscillating strain as a planet moves from periastron to apoastron results in friction and tidal heating, which can be an important energy source. Here, we simulate the climate of ocean-covered planets near the inner edge of the habitable zone around M to solar stars with the NASA GISS ROCKE-3D general circulation model, and leverage the planetary evolution software package, VPLanet, to calculate tidal heating rates for Earth-sized planets orbiting 2600 and 3000 K stars. This study is the first to use a 3D general circulation model that implements tidal heating to investigate habitability for multiple resonant states. We find that for reference experiments without tidal heating, the resonant state has little impact on the radial position of the inner edge because for a given stellar flux, higher-order states tend to be warmer than synchronous rotators, but for a given temperature, have drier upper atmospheres. However, when strong tidal heating is present, the rotational component implies a strong dependence of habitable conditions on the system evolution and rotational state. Since tidal and stellar heating both decrease rapidly with orbital distance, this results in a compact orbital width separating temperate and uninhabitable climates. We summarize these results and also compare ROCKE-3D to previously published simulations of the inner edge.

We review a testable, the axion quark nugget (AQN) model outside of the standard WIMP paradigm. The model was originally invented to explain the observed similarity between the dark and the visible components, $\Omega_{\rm DM}\approx \Omega_{\rm visible}$ in a natural way as both types of matter are formed during the same QCD transition and proportional to the same dimensional fundamental parameter of the system, $\Lambda_{\rm QCD}$. In this framework the baryogenesis is actually a charge segregation (rather than charge generation) process which is operational due to the $\cal{CP}$-odd axion field,while the global baryon number of the Universe remains zero. The nuggets and anti-nuggets are strongly interacting but macroscopically large objects with approximately nuclear density. We overview several specific recent applications of this framework. First, we discuss the "solar corona mystery" when the so-called nanoflares are identified with the AQN annihilation events in corona. Secondly, we review a proposal that the recently observed by the Telescope Array puzzling events is a result of the annihilation events of the AQNs under thunderstorm. Finally, we overview a broadband strategy which could lead to the discovery the AQN-induced axions representing the heart of the construction., Comment: invited brief review to be published in MPLA

In this contribution I present results achieved recently in the field of the OT forecast that push further the limit of the accuracy of the OT forecasts and open to new perspectives in this field., Comment: Comments: 2 figure, OSA Optical Sensors and Sensing Congress, Invited Talk, Vancouver, Canada, 22 to 26 June 2020, Propagation Through and Characterisation of Atmospheric and Oceanic Phenomena (pcAOP) and Adaptive Optics (AO)

The energy and flux budget (EFB) closure theory for a passive scalar (non-buoyant and non-inertial particles or gaseous admixtures) is developed for stably stratified turbulence. The physical background of the EFB turbulence closures is based on the budget equations for the turbulent kinetic and potential energies and turbulent fluxes of momentum and buoyancy, as well as the turbulent flux of particles. The EFB turbulence closure is designed for stratified geophysical flows from neutral to very stable stratification and it implies that turbulence is maintained by the velocity shear at any stratification. In a steady-state, expressions for the turbulent flux of passive scalar and the anisotropic non-symmetric turbulent diffusion tensor are derived, and universal flux Richardson number dependencies of the components of this tensor are obtained. The diagonal component in the vertical direction of the turbulent diffusion tensor is suppressed by strong stratification, while the diagonal components in the horizontal directions are not suppressed, and they are dominant in comparison with the other components of turbulent diffusion tensor. This implies that any initially created strongly inhomogeneous particle cloud is evolved into a thin pancake in horizontal plane with very slow increase of its thickness in the vertical direction. The turbulent Schmidt number increases linearly with the gradient Richardson number. Considering the applications of these results to the atmospheric boundary-layer turbulence, the theoretical relationships are derived which allow to determine the turbulent diffusion tensor as a function of the vertical coordinate measured in the units of the local Obukhov length scale. The obtained relations are potentially useful in modelling applications of particle dispersion in the atmospheric boundary-layer turbulence and free atmosphere turbulence., 18 pages, revtex4-2.cls, revised paper

The evolution of a single raindrop falling below a cloud is governed by fluid dynamics and thermodynamics fundamentally transferable to planetary atmospheres beyond modern Earth's. Here, we show how three properties that characterize falling raindrops -- raindrop shape, terminal velocity, and evaporation rate -- can be calculated as a function of raindrop size in any planetary atmosphere. We demonstrate that these simple, interrelated characteristics tightly bound the possible size range of raindrops in a given atmosphere, independently of poorly understood growth mechanisms. Starting from the equations governing raindrop falling and evaporation, we demonstrate that raindrop ability to vertically transport latent heat and condensible mass can be well captured by a new dimensionless number. Our results have implications for precipitation efficiency, convective storm dynamics, and rainfall rates, which are properties of interest for understanding planetary radiative balance and (in the case of terrestrial planets) rainfall-driven surface erosion., Comment: submitted to JGR: Planets; 40 pages, 8 figures, 3 tables, 4 appendices; supporting information with 9 pages, 8 figures, 1 table; associated code at https://github.com/kaitlyn-loftus/rainprops

This paper presents a novel method for obtaining the probability wave of breaking (Pb) of deep water, dominant wind‐sea waves (that is, waves made of the energy within ±30% of the peak wave frequency) derived from Gaussian wave field theory. For a given input wave spectrum we demonstrate how it is possible to derive a joint probability density function between wave phase speed (c) and horizontal orbital velocity at wave crest (u) from which a model for Pb can be obtained. A non‐linear kinematic wave breaking criterion consistent with the Gaussian framework is further proposed. Our model would allow, therefore, for application of the classical wave breaking criterion (that is, wave breaking occurs if u/c > 1) in spectral wave models which, to the authors’ knowledge, has not been done to date. Our results show that the proposed theoretical model has errors in the same order of magnitude as six other historical models when assessed using three field datasets. With optimization of the proposed model's single free parameter, it can become the best performing model for specific datasets. Although our results are promising, additional, more complete wave breaking datasets collected in the field are needed to comprehensively assess the present model, especially in regards to the dependence on phenomena such as direct wind forcing, long wave modulation and wave directionality. Plain Language Summary Waves will break if the speed of the water particles on the wave crest is greater than the speed of the wave itself, causing the wave crest to overtake the front part of the wave, leading to wave breaking. Precisely simulating real ocean waves requires, therefore, a particle‐by‐particle description of the water motion, which is too expensive for the current computers to handle in real‐world applications. Instead, wave models describe waves by means of their statistical properties, that is, averaged over a large number of waves. In this paper, we present a mathematical formulation that allows to calculate the combined probability between the speed of particles on the wave crest and the wave speed based only on statistical properties. From these combined probabilities, we model the probability of wave breaking. Our results indicate that our model performed relatively well when compared to six other models using three historical datasets. Because of a lack of observed data to assess our model, we recommend that future research should focus on collecting more wave breaking data measured in the field. Future advances on this line of research could lead, for example, to improvements on operational weather forecast models.

We study the two-dimensional rotating shallow-water model describing Earth's oceanic layers. It is formally analogue to a Schr\"odinger equation where the tools from topological insulators are relevant. Once regularized at small scale by an odd-viscous term, such a model has a well-defined bulk topological index. However, in presence of a sharp boundary, the number of edge modes depends on the boundary condition, showing an explicit violation of the bulk-edge correspondence. We study a continuous family of boundary conditions with a rich phase diagram, and explain the origin of this mismatch. Our approach relies on scattering theory and Levinson's theorem. The latter does not apply at infinite momentum because of the analytic structure of the scattering amplitude there, ultimately responsible for the violation., Comment: 26 pages, 5 figures

The size distribution of marine microplastics provides a fundamental data source for understanding the dispersal, break down, and biotic impacts of the microplastics in the ocean. The observed size distribution at the sea surface generally shows, from large to small sizes, a gradual increase followed by a rapid decrease. This decrease has led to the hypothesis that the smallest fragments are selectively removed by sinking or biological uptake. Here we propose a new model of size distribution without any removal of material from the system. The model uses an analogy with black-body radiation and the resultant size distribution is analogous to Planck's law. In this model, the original large plastic piece is broken into smaller pieces once by the application of "energy" or work by waves or other processes, under two assumptions, one that fragmentation into smaller pieces requires larger energy and the other that the probability distribution of the "energy" follows the Boltzmann distribution. Our formula well reproduces observed size distributions over wide size ranges from micro- to mesoplastics. According to this model, the smallest fragments are fewer because large "energy" required to produce such small fragments occurs more rarely., 17 pages, 5 figures, and Supporting information