Vladimir Zeitlin

New evidence of surprising robustness of solitary-wave solutions of the Serre-Green-Naghdi (SGN) equations is presented on the basis of high-resolution numerical simulations conducted using a novel well-balanced finite-volume method. SGN solitons exhibit a striking resemblance with their celebrated Korteweg-deVries (KdV) counterparts. Co-moving solitons are shown to exit intact from double and triple collisions with a remarkably small wave-wake residual. The counter-propagating solitons experiencing frontal collisions and solitons hitting a wall, non-existing in KdV case configurations, are shown to also recover, but with a much larger than in co-moving case residual, confirming with higher precision the results known in the literature. Multiple SGN solitons emerging from localized initial conditions are exhibited, and it is demonstrated that SGN solitons survive hitting localized topographic obstacles, and generate secondary solitons when they encounter a rising escarpment.

Exact steady eastward-moving vortex-dipole solutions, the equatorial modons, are constructed in the asymptotic limit of low divergence and small temperature variations in the thermal rotating shallow water (TRSW) model on the equatorial beta-plane. This regime is known to be relevant for the tropical atmosphere. The model itself is a generalization, allowing for horizontal temperature gradients, of the classical rotating shallow water model. The asymptotic modons can carry a temperature anomaly and exist also on the inhomogeneous temperature background. The modon configurations are then used to initialize numerical simulations, in order to check whether such coherent structures can exist in the full TRSW model. The results show that this is, indeed, the case. The parameter regimes and limitations on the structure of the temperature anomaly inside, in order for the modons to persist, are established. It is also shown that the modons keep their coherence even while evolving on the background of meridionally inhomogeneous temperature fields, or while interacting with sharp temperature fronts. A general scenario of disaggregation of the modons, if they are out of the stability domain, is exhibited and analysed.

A Lagrangian approach to both hydrostatic non-dispersive in the short-wave range and non-hydrostatic dispersive rotating shallow-water magnetohydrodynamics is developed, and used to analyse weakly and fully nonlinear waves described by the model. Hyperbolic structure in the non-dispersive case is displayed and Riemann invariants are constructed. Characteristic equations are used to establish criteria of breaking and formation of shocks by magneto-gravity waves, and conditions of the appearance of contact discontinuities in Alfvén waves. As in the case of non-magnetic rotating shallow water, rotation cannot prevent breaking. The Lagrangian equations of the model are reduced to a single partial differential ‘master’ equation, which is used to analyse the propagation of weakly nonlinear waves of both families, with or without weak rotation, and with or without weak short-wave dispersion. Corresponding modulation equations are constructed and their main properties sketched. The same master equation is used to obtain fully nonlinear finite-amplitude wave solutions in particular cases of no short-wave dispersion or no rotation.

A new class of vortices has been observed in the stratosphere following extreme wildfires (Canada 2017, Australia 2020) and volcanic eruptions (Raikoke 2019). These vortices are long‐lived mesoscale anticyclones (hundreds to 1,000 km in diameter) trapping plumes of aerosols and combustion/volcanic compounds. Owing to their unusual composition, these anticyclonically trapped plumes (ATPs) are associated with a significant radiative heating, which fuels their ascent through the stratosphere. This article investigates the dynamics of ATPs using two complementary approaches: analytically, in a potential vorticity (PV) perspective, and through idealised numerical simulations with the Weather Research and Forecasting (WRF) model. In both cases, we consider the vortical flow forced by a heating Lagrangian tracer. By reformulating the problem in the potential radius–potential temperature coordinate system introduced for tropical cyclones, we first clarify that ATP formation is concomitant with the injection of air into the stratosphere at extratropical latitudes. Then, we derive a set of simplified one‐dimensional equations describing the subsequent evolution of the flow after the injection. The equation obtained for the tracer is a variant of the classical Burgers' equation. In qualitative agreement with the three‐dimensional WRF simulations, this theoretical model predicts that ATPs develop an upper tracer front associated with sustained near‐zero anticyclonic PV, followed by a smooth tracer tail of cyclonic PV. Radiative relaxation of the temperature perturbations induced by the anticyclone and the presence of an initial PV anomaly tend to stabilise ATPs during their ascent. Finally, we note that the theory predicts a similar relationship between the plume and anticyclonic PV for cooled ATPs, which is supported by three‐dimensional simulations and may apply to the 2022 Hunga volcanic plume.

We develop a new second-order flux globalization based path-conservative central-upwind (PCCU) scheme for rotating shallow water magnetohydrodynamic equations. The new scheme is designed not only to maintain the divergence-free constraint of the magnetic field at the discrete level but also to satisfy the well-balanced (WB) property by exactly preserving some physically relevant steady states of the underlying system. The locally divergence-free constraint of the magnetic field is enforced by following the method recently introduced in [A. Chertock, A. Kurganov, M. Redle, and K. Wu, ArXiv preprint (2022), arXiv:2212.02682]: we consider a Godunov-Powell modified version of the studied system, introduce additional equations by spatially differentiating the magnetic field equations, and modify the reconstruction procedures for magnetic field variables. The WB property is ensured by implementing a flux globalization approach within the PCCU scheme, leading to a method capable of preserving both still- and moving-water equilibria exactly. In addition to provably achieving both the WB and divergence-free properties, the new method is implemented on an unstaggered grid and does not require any (approximate) Riemann problem solvers. The performance of the proposed method is demonstrated in several numerical experiments that confirm the lack of spurious oscillations, robustness, and high resolution of the obtained results.

A detailed investigation of linear instabilities of double-eyewall hurricane-like vortices with double maxima (“walls”) of azimuthal velocity and vorticity around the central minimum (“eye”) and of their nonlinear saturation is carried out in the framework of the moist-convective rotating shallow water model. It is shown that developing barotropic instability leads to inward displacement and gradual disappearance of the inner wall and to the formation of a nontrivial tripolar quasi-stationary structure, provided the second, outer, wall of the initial vortex is high enough. The effects of moist convection lead to the enhancement of the outer wall. In comparison, under the influence of the same instability, vortices with an inner wall stronger than the outer one evolve toward monopolar single-eyewall structures.

A class of exact solutions of the magnetohydrodynamic quasi-geostrophic equations (MQG), which result from rotating shallow water magnetohydrodynamics in the limit of small Rossby and magnetic Rossby numbers is constructed analytically. These solutions are magnetic modons, steady-moving dipolar vortices, which are generalizations of the well-known quasi-geostrophic modons. It is shown that various configurations of magnetic modons are possible: with or without external magnetic field, and with or without internal magnetic field trapped inside the dipole. By using the modon solutions as initial conditions for direct numerical simulations of the MQG equations, it is shown that they remain coherent for a long time, running over about a hundred deformation radii without change of form, provided the external and internal magnetic fields are not too strong, and even if a small-amplitude noise is added to initial conditions.

A simple two-layer model, the moist-convective rotating shallow water, which allows for low-cost high-resolution numerical simulations of the dynamics of the moist atmosphere in the presence of topography, is used to identify and understand dynamical processes governing the evolution of easterly waves propagating on the background of a low-latitude easterly jet crossing a land-sea boundary, a setup crudely representing the African Easterly Jet over the West-African plateau and the Atlantic ocean. We perform a thorough linear stability analysis and identify the unstable modes of the jet, which we use then for initialisation of fully nonlinear numerical simulations. In this way, we determine nonlinear evolution of unstable perturbations of the jet, both in the “dry” and moist-convective environments and highlight essential differences between the two cases. We identify a mechanism of formation of intense lower-layer cyclonic vortices at the northern flank of the jet and determine the influence of the land-sea contrast upon this process.

The adjustment of equatorial pressure anomalies is studied using high-resolution numerical simulations within one- and two-layer shallow-water systems on a rotating sphere, which include a simple self-consistent parameterization of the dynamical effects of condensation (the so-called moist-convective shallow-water equations). A systematic generation of localized, moving eastward along the equator cyclonic pairs, termed the equatorial modons, is observed in the one-layer model for sufficiently intense initial perturbations. The modons are strongly enhanced by the moist convection. In the two-layer model with a relatively small depth ratio, baroclinic equatorial modons are also observed. In both the one-layer model and the baroclinic mode of the two-layer model, the emergence of a transient Gill pattern preceding the generation of modons is demonstrated. It is found that the breaking of circumnavigating moist Kelvin waves, generated in the adjustment process, induces extensive easterly jets, which weaken the eastward nonlinear advection tendency of the modons. Finally, an inductive generation of the modons over oceanic warm-pools, modelled as local humidity maxima, is discovered, and explained on the basis of conservation of moist potential vorticity.