This work presents a combined analysis of the spatial distributions of oceanographic fields based on Eulerian and Lagrangian approaches. In the Eulerian approach, the fluid motion is considered as a function of coordinates and time, and the equations of motion are formulated for specific points in space. In contrast, the Lagrangian approach focuses on tracking individual particles in the ocean, representing water masses or other materials, as separate objects moving according to the equations of motion. The variability of oceanographic characteristics in the Northwest Pacific region is examined, specifically analyzing the spatial distributions of oceanographic fields on October 24, 2010. The AMEDA algorithm is used to identify mesoscale eddies. Within the Eulerian approach, it is shown that regions prohibiting the stretching of mesoscale eddies only partially correspond to areas where large-scale eddies are located. Vortical structures and elements of large-scale circulation are well manifested in the distributions of the Lyapunov exponent (Λ-maps) and maps characterizing the trajectories of advected particles (S-maps). Enhanced values of kinetic and available potential energy correspond to the main circulation structures in the region. However, for analyzing the evolution of circulation structures, they are less informative compared to Λ-maps and S-maps, which accumulate information from the previous period. Spatial distributions of relative and potential vorticity are constructed. Positive values of relative vorticity outline the Kuroshio current from the north, while negative values outline it from the south, which is typical of jet currents with a pronounced zonal component. Mesoscale eddies exhibit elevated values of vorticity and a shielding effect (shielded vortices). In the Lagrangian experiment with passive tracer patches, it is demonstrated that the most deformation-resistant patch spent most of the analyzed time interval in a domain where the stretching of eddies is prohibited.

In this paper, a new analysis of the known topographic models of Rossby waves for piecewise exponential topography profiles is performed. A mathematical method is proposed that allows us to find analytically the group velocity and variance. A numerical comparison is made of the relations presented in the study of Buchwald and Adams and the dependencies obtained within the framework of a new analytical approach. Numerical comparative analysis showed that the discrepancy for the phase velocities lies in the range of five percent. For group speeds, the discrepancy reaches nineteen percent for the first mode and decreases for higher mode numbers. We also consider long-wave asymptotics of eigenfunctions. It is established that the long-wave limit for Rossby shelf waves has specifics: the longitudinal wave number tends to zero, and the transverse wave number reaches a certain finite positive constant, which is the greater the higher the mode number. It is shown that in the long-wave limit, Rossby shelf waves transform into shelf topographic currents, while there is a certain self-similarity for the phase and group velocities of shelf currents depending on the value of the topography gradient.

The article discusses various aspects of the interaction of vortices with the barotropic flow. Vortex interaction with a flow results in rotation variants, nutational oscillations, and unlimited stretching of its core. The vortex remains in a localized formation, with the semi-axes of the ellipse experiencing fluctuations near an average value in the first two cases. In the third case, the vortex is significantly elongated, and its shape in the horizontal plane changes as follows: one axis of the ellipse increases, and the other decreases. In this case, the vortex, when viewed from above, stretches into a thread, while remaining ellipsoidal. These vortex formations are called filaments. The latter arise from initially almost circular vortices in the horizontal plane and represent structures with non-zero vorticity elongated in one direction. Here, we aim to study the energy transformation of a vortex during its evolution process, mainly due to changes in its shape by stretching. The energy evolution of a mesoscale vortex located in the Norwegian Sea is analyzed using GLORYS12V1 ocean reanalysis data to verify the theoretical conclusions. During the evolution, the vortex is found to transform from a round shape and becomes elongated, and after three weeks its longitudinal scale becomes 4 times larger than the transverse one. During the transformation of a vortex, the kinetic energy and available potential energy decrease respectively by 3 times and 1.7 times. Concurrently, the total energy of the vortex is found to decrease by 2.3 times. We argue that the stretching of vortices results in a loss of energy as well as its redistribution from mesoscale to submesoscale. The lost part of the energy returns to the flow and results in the occurrence of the reverse energy cascade phenomenon.

In this paper, sub-inertial waves propagating on the Kuril shelf and the oceanic trench are considered. Against the background of a historical review of the beginning of the study of topographic waves and the appearance of relevant terms, a description of the features of wave propagation and the derivation of the main dispersion equations are given. We show that all variants of the topographic solutions presented in the article are basically based on the same dispersion relation: this is the dispersion relation for Rossby topographic waves. Two separate classes of localized solutions have been constructed: one is for the shelf, and the second, in fact, is also for the shelf, but which is commonly called trench waves. We demonstrate that the transverse wave number for trench waves is not independent, as for shelf waves, but is a function of the longitudinal wave number. In other words, Rossby topographic waves are two–dimensional waves, while shelf waves are quasi-one-dimensional solutions. The analytical novelty of the work consists of the fact that we can make crosslinking of trench and shelf waves. This fact was not presented in previous articles on this topic.

This study investigates the stability of wave modes in zonal shear currents on a beta-plane. The main research methods are analytical analysis and numerical modeling, utilizing MATLAB to solve the governing equations and visualize the results. The main result is the identification of stable ranges of wavenumbers for both sinusoidal and varicose modes that manifest within the flow. Numerical analysis reveals a β-destabilizing effect, which leads to the emergence of “β-destabilized” modes. It is shown that the isofrequencies in phase space consist of two branches: one corresponding to “flow waves modified by the β-effect” and the other to “Rossby waves modified by the current.” The distinction between “flow waves modified by the β-effect” (wave number k > 0) and “Rossby waves modified by the current” (wave number k < 0) reflects the dominant factor governing their properties: the β-effect for flow waves and the current for Rossby waves. These waves play a key role in long-distance energy transfer, influencing the redistribution of heat and salinity in the ocean. It is hypothesized that the spatial functions corresponding to the sinusoidal mode represent a pair of monopoles of the same sign, while the varicose mode corresponds to dipoles. A pair of monopoles with opposite signs forms a vortex pair. Studying of the β-destabilizing effect and transverse energy radiation provides deeper insights into the mechanisms of oceanic current dynamics and their impact on global climate processes. This is particularly relevant in global warming, where changes in ocean circulation can have far-reaching consequences for the planet's climate and ecosystems. While specific parameterizations for existing climate model related to β-destabilizing effect are beyond the scope of this study, our results contribute to that fundamental understanding.

A comparative assessment of the contribution of eddies and other dynamic structures to the energy of the Lofoten Basin is carried out. The basis of the study is the data of global ocean reanalysis GLORYS12V1 for 1993–2019. We estimate the total kinetic and potential energy as well as the corresponding contribution of eddy energy for a region bounded by an isobath of 3000 m. We use the method of automatic identification of vortices for the analysis which allows us to calculate the eddy kinetic and potential energy in the automatically selected areas of the eddies. We establish that the potential energy of both cyclones and anticyclones is on average 2–3 times higher than the kinetic energy values, and the energy values of kinetic and potential for anticyclones dominate relative to the energy of cyclones. We also consider the interannual variability and seasonal course of eddy kinetic and potential energy. The seasonal course revealed an increase in both types of energy in the winter months. It is established that the contribution of eddies to the total energy of the basin is small. The contribution of eddy kinetic energy to the total energy of the basin is 7.3%, and the contribution of eddy potential energy is 8.4%. This means that the main contribution to the energy of the basin is made not by mesoscale eddies, but by other dynamic structures i.e. filaments and background flow.