Theoretical results on water waves almost always start by assuming irrotationality of the flow in order to simplify the formulation. In this work, we investigate the well-foundedness of this hypothesis via numerical simulations of the free surface Navier-Stokes equations. We show that, in the presence of a non-flat bathymetry, either angular or smooth, a gravity wave of finite amplitude can shed vortex pairs from the bottom boundary layer into the bulk of the flow. As these eddies approach the free surface, they modify the shape of the wave. It is found that this perturbation does not vanish as the Reynolds number (Re) is increased. The vanishing viscosity limit of water waves is, therefore, singular when no-slip boundary conditions are enforced on the bottom.

Numerical experiments of dynamo action designed to understand the generation of Earth's magnetic field produce different regime branches identified within bifurcation diagrams. Notable are distinct branches where the resultant magnetic field is either weak or strong. Weak-field solutions are identified by the prominent role of viscosity (and/or inertia) on the motion, whereas the magnetic field has a leading-order effect on the flow in strong-field solutions. We demonstrate the persistence of the strong-field branch, preserving the expected force balance of Earth's core, and provide scaling laws governing its onset as parameters move toward values appropriate for the Geodynamo. We introduce a new output parameter, based on dynamically important parts of rotational and magnetic forces, that captures expected O(1) values of strong-field solutions throughout input parameter space. This new measure of the field strength and our bounds on scaling laws can guide future studies in locating strong-field dynamos in parameter space.

Convection is the main heat transport mechanism in the Earth's liquid core and is thought to power the dynamo that generates the geomagnetic field. Core convection is strongly constrained by rotation while being turbulent. Given the difficulty in modelling these conditions, some key properties of core convection are still debated, including the dominant energy-carrying lengthscale. Different regimes of rapidly-rotating, unmagnetised, turbulent convection exist depending on the importance of viscous and inertial forces in the dynamics, and hence different theoretical predictions for the dominant flow lengthscale have been proposed. Here we study the transition from viscously-dominated to inertia-dominated regimes using numerical simulations in spherical and planar geometries. We find that the cross-over occurs when the inertial lengthscale approximately equals the viscous lengthscale. This suggests that core convection in the absence of magnetic fields is dominated by the inertial scale, which is hundred times larger than the viscous scale.

The problem of the geodynamo is simple to formulate (Why does the Earth possess a magnetic field?), yet it proves surprisingly hard to address. As with most geophysical flows, the fluid flow of molten iron in the Earth's core is strongly influenced by the Coriolis effect. Because the liquid is electrically conducting, it is also strongly influenced by the Lorentz force. The balance is unusual in that, whereas each of these effects considered separately tends to impede the flow, the magnetic field in the Earth's core relaxes the effect of the rapid rotation and allows the development of a large-scale flow in the core that in turn regenerates the field. This review covers some recent developments regarding the interplay between rotation and magnetic fields and how it affects the flow in the Earth's core.

Asymptotic solutions are investigated for the travelling wave consisting of infectives I ( x − c t ) propagating at speed c into a region of uninfected susceptibles S = S + , on the basis that S + is large. In the moving frame, three domains are identified. In the narrow leading frontal region, the infectives terminate relatively abruptly. Conditions ahead (increasing x ) of the front control the speed c of the front advance. In the trailing region (decreasing x ), the number of infectives decay relatively slowly. Our asymptotic development focuses on the dependence of I on S in the central region. Then, the apparently simple problem is complicated by the presence of both algebraic and logarithmic dependencies. Still, we can construct an asymptotic expansion to a high order of accuracy that embeds the trailing region solution. A proper solution in the frontal region is numerical, but here the central region solution works well too. We also investigated numerically the evolution from an initial state to a travelling wave. Following the decay of transients, the speed adopted by the wave is fast, though the slowest of those admissible. The asymptotic solutions are compared with the numerical solutions and display excellent agreement.

Most theoretical and numerical results regarding water waves rely on the irrotationality of the flow in order to simplify significantly the mathematical formulation. This assumption is rarely discussed in details. In this work we investigate the well-foundedness of this important hypothesis using numerical simulations of the free-surface Navier-Stokes equation, using a scheme introduced in Riquier and Dormy (2024). We show that, in the presence of an irregular bottom, a gravity wave of non-negligible height can effectively destabilise the bottom boundary layer, stripping off vortices into the main flow. As a vortex approaches the surface the solution of the Navier-Stokes flow in the limit of vanishing viscosity is shown to differ from the inviscid Euler solution.

We present a rigorous mathematical analysis of the modeling of inviscid water waves. The free-surface is described as a parametrized curve. We introduce a numerically stable algorithm which accounts for its evolution with time. The method is shown to converge using approximate solutions, such as Stokes waves and Green-Naghdi solitary waves. It is finally tested on a wave breaking problem, for which an odd-even coupling suffices to achieve numerical convergence up to the splash without the need for additional filtering.

The low-Reynolds-number Stokes flow driven by rotation of two parallel cylinders of equal unit radius is investigated by both analytical and numerical techniques. In Part I, the case of counterrotating cylinders is considered. A numerical (finite-element) solution is obtained by enclosing the system in an outer cylinder of radius R-0 >> 1, on which the no-slip condition is imposed. A model problem with the same symmetries is first solved exactly, and the limit of validity of the Stokes approximation is determined; this model has some relevance for ciliary propulsion. For the two-cylinder problem, attention is focused on the small-gap situation epsilon << 1. An exact analytic solution is obtained in the contact limit epsilon=0, and a net force F(c )acting on the pair of cylinders in this contact limit is identified; this contributes to the torque that each cylinder experiences about its axis. The far-field torque doublet ("torquelet") is also identified. Part II treats the case of corotating cylinders, for which again a finite-element numerical solution is obtained for R-0 >> 1. The theory of Watson [Mathematika 42, 105 (1995)] is elucidated and shown to agree well with the numerical solution. In contrast to the counterrotating case, inertia effects are negligible throughout the fluid domain, however large, provided Re << 1. In the concluding section, the main results for both cases are summarized, and the situation when the fluid is unbounded (R-0=infinity) is discussed. If the cylinders are free to move (while rotating about their axes), in the counterrotating case they will then translate relative to the fluid at infinity with constant velocity, the drag force exactly compensating the self-induced force due to the counterrotation. In the corotating case, if the cylinders are free to move, then they will rotate as a pair relative to the fluid at infinity and the net torque on the cylinder pair is zero; the flow relative to the fluid at infinity is identified as a "radial quadrupole." If, however, the cylinder axes are held fixed, then the Stokes flow in the counterrotating case extends only for a distance r similar to Re(-1)log[Re-1] from the cylinders, and it is argued that the cylinders then experience a (dimensionless) force F-y similar to 1/log[Re(-1)log[Re-1]]; in the corotating case, the cylinder pair experiences a (dimensionless) torque T, which tends to 17.2587 as epsilon down arrow 0; this torque is associated with a vortex-type flow similar to r(-1) that is established in the far field. Situations that can be described by the condition epsilon<0 are treated for both counter- and corotating cases in the Supplemental Material.

We investigate the ocean wave field under Hurricane SAM (2021). Whilst measurements of waves under Tropical Cyclones (TCs) are rare, an unusually large number of quality in situ and remote measurements are available in that case. First, we highlight the good consistency between the wave spectra provided by the Surface Waves Investigation and Monitoring (SWIM) instrument onboard the China-France Oceanography Satellite (CFOSAT), the in situ spectra measured by National Data Buoy Center (NDBC) buoys, and a saildrone. The impact of strong rains on SWIM spectra is then further investigated. We show that whereas the rain definitely affects the normalized radar cross section, both the innovative technology (beam rotating scanning geometry) and the post-processing processes applied to retrieve the 2D wave spectra ensure a good quality of the resulting wave spectra, even in heavy rain conditions. On this basis, the satellite, airborne and in situ observations are confronted to the analytical model proposed by Kudryavtsev et al (2015). We show that a trapped wave mechanism may be invoked to explain the large significant wave height observed in the right front quadrant of Hurricane SAM.

We introduce a numerical strategy to study the evolution of 2D water waves in the presence of a plunging jet. The free-surface Navier-Stokes solution is obtained with a finite but small viscosity. We observe the formation of a surface boundary layer where the vorticity is localised. We highlight convergence to the inviscid solution. The effects of dissipation on the development of a singularity at the tip of the wave is also investigated by characterising the vorticity boundary layer appearing near the interface.

In this paper, the classical problem of two-dimensional flow in a cylindrical domain, driven by a nonuniform tangential velocity imposed at the boundary, is reconsidered in straightforward manner. When the boundary velocity is a pure rotation Omega plus a small perturbation eta Omega f(theta) and when the Reynolds number based on Omega is large (Re >> 1 ), this flow is of "Prandtl-Batchelor" type, namely, a flow of uniform vorticity omega(c )in a core region inside a viscous boundary layer of thickness O(Re)(-1/2) . The O (eta(2)) contribution to omega(c )is determined here by asymptotic analysis up to O (Re-1) . The result is in good agreement with numerical computation for Re greater than or similar to 400 .

We investigate the mechanism for eye formation in hurricane-like vortices, using a formulation adapted from Oruba, Davidson, and Dormy. Numerical simulations are performed using an axisymmetric model of dry rotating Rayleigh-B & eacute;nard convection under the Boussinesq approximation. The fluxes of heat and momentum at the sea surface are described using the bulk aerodynamic formula. A simplified fi ed model for radiative cooling is also implemented. We fi nd that the mechanism for eye formation introduced in Oruba et al., relying on vorticity stripping from the boundary layer, is robust in dry hurricane-like vortices. Furthermore, with these boundary conditions, the structure of the flow is closer to the flow of actual tropical cyclones. The applicability of this mechanism to the moist case however remains uncertain and deserves further study. Finally, energy budgets, obtained either by a heat engine approach or by a direct estimation of the work of buoyancy forces, are investigated. They provide estimations of the surface wind speed as a function of the controlling parameters. SIGNIFICANCE STATEMENT: Tropical cyclones (also known as hurricanes or typhoons, depending on their location) are life-threatening and devastating atmospheric vortices. Their impact worsens with sea level rise and increasing coastal population. Here, we explore, using idealized models, the physics behind the formation of an eye (the quiet and typically clear region at the center of these storms). We then investigate the controlling parameters for the amplitude of the strongest winds in the model, using energy constraints, and compare them to what is observed in actual tropical cyclones.

Numerical simulations of the geodynamo (and other planetary dynamos) have made significant progress in recent years. As computing power has advanced, some new models claim to be ever more appropriate for understanding Earth's core dynamics. One measure of the success of such models is the ability to replicate the expected balance between forces operating within the core; Coriolis and Lorentz forces are predicted to be most important. The picture is complicated for an incompressible flow by the existence of the pressure gradient force which renders the gradient parts of all other forces dynamically unimportant. This can confuse the situation, especially when the scale dependence of forces is considered. In this work we investigate force balances through the alternative approach of eliminating gradient parts of each force to form `solenoidal force balances'. We perform a length-scale-dependent analysis for several spherical simulations and find that removal of gradient parts offers an alternative picture of the force balance compared with looking at traditional forces alone. Solenoidal force balances provide some agreement with the results of previous studies but also significant differences. They offer a cleaner overall picture of the dynamics and introduce differences at smaller scales. This has implications for geodynamo models purporting to have reached Earth-like regimes: in order to achieve a meaningful comparison of forces, only the solenoidal part of forces should be considered.

The net impact of aircraft contrails on global climate change is a matter of controversy today. Among the many parameters potentially influencing this issue, the role played by the aircraft wake has received only little attention so far. Yet the interaction between the engine exhaust jets causing these contrails and the aircraft wake can lead to modifications in the altitude of the contrails on the order of hundreds of meters. This change in altitude heavily influences the net impact of contrails to global climate change, since it affects the ambient temperature and thereby the ice content and the radiative properties of these contrails. The wake entrainment supporting these effects basically depends on the relative positioning of the jet with respect to the tip vortices and on the buoyant forces associated with atmospheric stratification. Here we focus on these two parameters by running a large number of two-dimensional simulations of the flow from the aftermath of the jet turbulent diffusion, vortex roll-up, and initial ice formation up to the vortex destabilization stage, for a range of values of the vortex based normalization of the Brunt-Vaisala frequency and the jet to wing span ratio. The very near wake dynamics are not simulated and instead replaced by an analytical description for the vortex, the jet, and the ice plume. Ice water content is determined from the offset to ice saturation, given prescribed ambient conditions. The jet lateral spacing is considered in the range from fuselage to wing tip. The potential radiative impact of the early wake is calculated using the total extinction induced by the ice plume. The results are indicative of the impact of older contrail cirrus clouds, the largest proportion in the whole contrail radiative impact. The parametric mapping (stratification, jet spacing) highlights the important role played by the jet position on the opacity of early contrails, for regular stratification levels. In particular, a jet located closer to the wing tip results in contrails located at lower altitudes and reduced optical thickness, suggesting that jet positioning could be an interesting mean of contrail mitigation.

Abstract This study focuses on ocean waves impacting the Moorea Island in French Polynesia, where coral reefs play an essential role in the biodiversity and protection of habitations. We investigate how the innovative Surface Waves Investigation and Monitoring (SWIM) instrument of the Chinese‐French Oceanography SATellite satellite enables to document on a multi‐annual basis, the spectral properties of ocean waves reaching the coasts of the Moorea Island. Our analysis is based on comparisons with in situ measurements (wave gauges deployed on the outer slope of the coral reef), and with other satellite observations (altimeter, Synthetic Aperture Radar [SAR]). Accounting for local masking effects, we show that SWIM provides relevant information on short swell or wind waves, which is missed by the SAR observations, in particular in high sea‐state conditions, owing to the dominant propagation direction being close to the azimuth. We, nevertheless, also find that wave properties in low sea‐state conditions are better documented by SAR than by SWIM. Such results are important to accurately measure and predict the wave conditions which fragilize the coral reefs and to evaluate the impact of extreme events on tropical islands and coral reefs.

We consider axisymmetric Boussinesq convection in a shallow cylinder of radius $L$ and depth $H (\ll L)$ , which rotates with angular velocity $\varOmega$ about its axis of symmetry aligned to the vertical. Constant heat flux boundary conditions, top and bottom, are adopted, for which the onset of instability occurs on a long horizontal length scale provided that $\varOmega$ is sufficiently small. We investigate the nonlinear development by well-established two-scale asymptotic expansion methods. Comparisons of the results with the direct numerical simulations (DNS) of the primitive governing equations are good at sufficiently large Prandtl number $\sigma$ . As $\sigma$ is reduced, the finite amplitude range of applicability of the asymptotics reduces in concert. Though the large meridional convective cell, predicted by the DNS, is approximated adequately by the asymptotics, the azimuthal flow fails almost catastrophically, because of significant angular momentum transport at small $\sigma$ , exacerbated by the cylindrical geometry. To appraise the situation, we propose hybrid methods that build on the meridional streamfunction $\psi$ derived from the asymptotics. With $\psi$ given, we solve the now linear azimuthal equation of motion for the azimuthal velocity $v$ by DNS. Our ‘hybrid’ methods enable us to explain features of the flow at large Rayleigh number, found previously by Oruba et al. ( J. Fluid Mech. , vol. 812, 2017, pp. 890–904).

The focus of this study is to investigate primary and secondary bifurcations to weakly nonlinear flows (weak branch) in convective rotating spheres in a regime where only strongly nonlinear oscillatory suband supercritical flows (strong branch) were previously found in [E. J. Kaplan, N. Schaeffer, J. Vidal, and P. Cardin, Phys. Rev. Lett. 119, 094501 (2017)]. The relevant regime corresponds to low Prandtl and Ekman numbers, indicating a predominance of Coriolis forces and thermal diffusion in the system. We provide the bifurcation diagrams for rotating waves (RWs) computed by means of continuation methods and the corresponding stability analysis of these periodic flows to detect secondary bifurcations giving rise to quasiperiodic modulated rotating waves (MRWs). Additional direct numerical simulations (DNS) are performed for the analysis of these quasiperiodic flows for which Poincaré sections and kinetic energy spectra are presented. The diffusion time scales are investigated as well. Our study reveals very large initial transients (more than 30 diffusion time units) for the nonlinear saturation of solutions on the weak branch, either RWs or MRWs, when DNS are employed. In addition, we demonstrate that MRWs have multimodal nature involving resonant triads. The modes can be located in the bulk of the fluid or attached to the outer sphere and exhibit multicellular structures. The different resonant modes forming the nonlinear quasiperiodic flows can be predicted with the stability analysis of RWs, close to the Hopf bifurcation point, by analyzing the leading unstable Floquet eigenmode.

In a previous paper, Oruba et al. (J. Fluid Mech., vol. 818, 2017, pp. 205–240) considered the ‘primary’ quasi-steady geostrophic (QG) motion of a constant density fluid of viscosity $\unicode[STIX]{x1D708}$ that occurs during linear spin-down in a cylindrical container of radius $r^{\dagger }=L$ and height $z^{\dagger }=H$, rotating rapidly (angular velocity $\unicode[STIX]{x1D6FA}$) about its axis of symmetry subject to mixed rigid and stress-free boundary conditions for the case $L=H$. Here, Direct numerical simulation at large $L=10H$ and Ekman numbers $E=\unicode[STIX]{x1D708}/H^{2}\unicode[STIX]{x1D6FA}$ in the range $=10^{-3}{-}10^{-7}$ reveals inertial wave activity on the spin-down time scale $E^{-1/2}\unicode[STIX]{x1D6FA}^{-1}$. Our analytic study, based on $E\ll 1$, builds on the results of Greenspan & Howard (J. Fluid Mech., vol. 17, 1963, pp. 385–404) for an infinite plane layer $L\rightarrow \infty$. In addition to QG spin-down, they identify a ‘secondary’ set of quasi-maximum frequency $\unicode[STIX]{x1D714}^{\dagger }\rightarrow 2\unicode[STIX]{x1D6FA}$ (MF) inertial waves, which is a manifestation of the transient Ekman layer, decaying algebraically $\propto 1/\surd \,t^{\dagger }$. Here, we acknowledge that the blocking of the meridional parts of both the primary-QG and the secondary-MF spin-down flows by the lateral boundary $r^{\dagger }=L$ provides a trigger for other inertial waves. As we only investigate the response to the primary QG-trigger, we call the model ‘reduced’ and for that only inertial waves with frequencies $\unicode[STIX]{x1D714}^{\dagger }<2\unicode[STIX]{x1D6FA}$ are triggered. We explain the ensuing organised inertial wave structure via an analytic study of the thin disc limit $L\gg H$ restricted to the region $L-r^{\dagger }=O(H)$ far from the axis, where we make a Cartesian approximation of the cylindrical geometry. Other than identifying a small scale fan structure emanating from the corner $[r^{\dagger },z^{\dagger }]=[L,0]$, we show that inertial waves, on the gap length scale $H$, radiated (wave energy flux) away from the outer boundary $r^{\dagger }=L$ (but propagating with a phase velocity towards it) reach a distance determined by the mode with the fastest group velocity.

This article describes the first results obtained from the Surface Waves Investigation and Monitoring (SWIM) instrument carried by the China France Oceanography Satellite (CFOSAT), which was launched on October 29, 2018. SWIM is a Ku-band radar with a near-nadir scanning beam geometry. It was designed to measure the spectral properties of surface ocean waves. First, the good behavior of the instrument is illustrated. It is then shown that the nadir products (significant wave height, normalized radar cross section, and wind speed) exhibit an accuracy similar to standard altimeter missions, thanks to a new retracking algorithm, which compensates a lower sampling rate compared to standard altimetry missions. The off-nadir beam observations are analyzed in detail. The normalized radar cross section varies with incidence and wind speed as expected from previous studies presented in the literature. We illustrate that, in order to retrieve the wave spectra from the radar backscattering fluctuations, it is crucial to apply a speckle correction derived from the observations. Directional spectra of ocean waves and their mean parameters are then compared to wave model data at the global scale and to in situ data from a selection of case studies. The good efficiency of SWIM to provide the spectral properties of ocean waves in the wavelength range [70-500 m] is illustrated. The main limitations are discussed, and the perspectives to improve the data quality are presented.