Oblique instability of a stratified oscillatory boundary layer

The instability and dynamics of a vertical oscillatory boundary layer in a container filled with a stratified fluid are addressed. Past experiments have shown that when the boundary oscillation frequency is of the same order as the buoyancy frequency, the system is unstable to a herringbone pattern of oblique waves. Prior studies assuming the basic state to be a unidirectional oscillatory shear flow were unable to account for the oblique waves. By accounting for confinement effects present in the experiments, and the ensuing three-dimensional structure of the basic state, we are able to numerically reproduce the experimental observations, opening the door to fully analysing the impacts of stratification on such boundary layers.

The Multi-Scale Layering-Structure of Thermal Microscale Profiles

Thermal microstructure profiling is an established technique for investigating turbulent mixing and stratification in lakes and oceans. However, it provides only quasi-instantaneous, 1-D snapshots. Other approaches to measuring these phenomena exist, but each has logistic and/or quality weaknesses. Hence, turbulent mixing and stratification processes remain greatly under-sampled. This paper contributes to addressing this problem by presenting a novel analysis of thermal microstructure profiles, focusing on their multi-scale stratification structure. Profiles taken in two small lakes using a Self-Contained Automated Micro-Profiler (SCAMP) were analysed. For each profile, buoyancy frequency (N), Thorpe scales (LT), and the coefficient of vertical turbulent diffusivity (KZ) were determined. To characterize the multi-scale stratification, profiles of d2T/dz2 at a spectrum of scales were calculated and the number of turning points in them counted. Plotting these counts against the scale gave pseudo-spectra, which were characterized by the index D of their power law regression lines. Scale-dependent correlations of D with N, LT and KZ were found, and suggest that this approach may be useful for providing alternative estimates of the efficiency of turbulent mixing and measures of longer-term averages of KZ than current methods provide. Testing these potential uses will require comparison of field measurements of D with time-integrated KZ values and numerical simulations.

Asymptotics of the Far Fields of Internal Gravity Waves Excited by a Source of Radial Symmetry

Abstract— The problem of the far field of internal gravity waves generated by a perturbation source of radial symmetry aroused at an initial instant of time is solved. The constant model distribution of the buoyancy frequency is considered and, using the Fourier–Hankel transform, an analytical solution to the problem is obtained in the form of the sum of wave modes. Asymptotics of the solutions that describe the spatial-temporal characteristics of elevation of the isopycnic lines and the vertical and horizontal velocity components far from the perturbation source are obtained. The asymptotics of the components of the wave field are expressed in terms of the square of the Airy function and its derivatives in the neighborhood of the wave fronts of an individual wave mode. The exact and asymptotic results are compared and it is shown that the asymptotic method makes it possible to calculate effectively the far wave fields at times of the order of ten and more of the Brunt–Väisälä periods.

Phase Structure of Internal Gravity Waves in the Ocean with Shear Flows

Purpose. The description of the internal gravity waves dynamics in the ocean with background fields of shear currents is a very difficult problem even in the linear approximation. The mathematical problem describing wave dynamics is reduced to the analysis of a system of partial differential equations; and while taking into account the vertical and horizontal inhomogeneity, this system of equations does not allow separation of the variables. Application of various approximations makes it possible to construct analytical solutions for the model distributions of buoyancy frequency and background shear ocean currents. The work is aimed at studying dynamics of internal gravity waves in the ocean with the arbitrary and model distributions of density and background shear currents. Methods and Results. The paper represents the numerical and analytical solutions describing the main phase characteristics of the internal gravity wave fields in the stratified ocean of finite depth, both for arbitrary and model distributions of the buoyancy frequency and the background shear currents. The currents are considered to be stationary and horizontally homogeneous on the assumption that the scale of the currents' horizontal and temporal variability is much larger than the characteristic lengths and periods of internal gravity waves. Having been used, the Fourier method permitted to obtain integral representations of the solutions under the Miles – Howard stability condition is fulfilled. To solve the vertical spectral problem, proposed is the algorithm for calculating the main dispersion dependences that determine the phase characteristics of the generated wave fields. The calculations for one real distribution of buoyancy frequency and shear flow profile are represented. Transformation of the dispersion surfaces and phase structures of the internal gravitational waves’ fields is studied depending on the generation parameters. To solve the problem analytically, constant distribution of the buoyancy frequency and linear dependences of the background shear current on depth were used. For the model distribution of the buoyancy and shear flow frequencies, the explicit analytical expressions describing the solutions of the vertical spectral problem were derived. The numerical and asymptotic solutions for the characteristic oceanic parameters were compared. Conclusions. The obtained results show that the asymptotic constructions using the model dependences of the buoyancy frequency and the background shear velocities’ distribution, describe the numerical solutions of the vertical spectral problem to a good degree of accuracy. The model representations, having been applied for hydrological parameters, make it possible to describe qualitatively correctly the main characteristics of internal gravity waves in the ocean with the arbitrary background shear currents.

Basic Regularities of Vertical Turbulent Exchange in the Mixed and Stratified Layers of the Black Sea

Purpose. The purpose of the study is to assess the coefficient of vertical turbulent exchange for different layers of the Black Sea basin based on the experimental data on microstructure of the physical fields obtained for the period 2004–2019 in the Black Sea and using the semi-empirical models. Methods and Results. For the upper mixed layer, the turbulent energy dissipation rate ɛ and the exchange coefficient were calculated using the velocity fluctuation spectra based on the Kolmogorov hypotheses on the turbulence spectrum inertial range. In the stratified layers, the turbulence coefficient and the dissipation rate were experimentally determined both from the spectra of the velocity horizontal fluctuations’ gradients and the vertical spectra of temperature fluctuations using the concept of the effective scale of turbulent patches. Depending on the features of the hydrological regime and the prevailing energy contributors to turbulence generation, five layers were identified and described (including their characteristic power dependences of the vertical turbulent diffusion coefficients K on the buoyancy frequency N) using the 1.5D-model of vertical turbulent exchange for the basin under study. For the stratified layers, the 1.5D-model results were comparatively analyzed with those of the other semi-empirical and theoretical models describing the most probable hydrophysical processes in each specific layer; the relations for the vertical turbulent exchange coefficient were obtained depending on the buoyancy frequency. Conclusions. Comparison of the experimental data collected under different hydrometeorological conditions with the simulations resulted from the known turbulence models for the sea upper layer showed that the best agreement between the simulation and measurement data was provided by a multiscale model taking into account three basic mechanisms of turbulence generation: current velocity shear, instability of wave motions, and wave breaking. The turbulent exchange coefficient dependencies on depth are conditioned by the effect of the turbulence dominant source at a given level. In the stratified layers, the exchange coefficient dependence on buoyancy frequency is determined by the hydrophysical processes in each layer; the relations obtained for individual layers indicate intensity of the contributions of vertical advection, internal wave breakings, turbulence diffusion and geothermal flux.

Stability analysis of implicit semi-Lagrangian methods for numerical solution of non-hydrostatic atmospheric dynamics equations

The stability of implicit semi-Lagrangian schemes for time-integration of the non-hydrostatic atmosphere dynamics equations is analyzed in the present paper. The main reason for the instability of the considered class of schemes is the semi-Lagrangian advection of stratified thermodynamic variables coupled to the fixed point iteration method used to solve the implicit in time upstream trajectory computation problem. We identify two types of unstable modes and obtain stability conditions in terms of the scheme parameters. Stabilization of sound modes requires the use of a pressure reference profile and time off-centering. Gravity waves are stable only for an even number of fixed point method iterations. The maximum time step is determined by inverse buoyancy frequency in the case when the reference profile of the potential temperature is not used. Generally, applying time off-centering and reference profile to pressure variable is necessary for stability. Using reference profile for potential temperature and an even number of the iterations allows one to significantly increase the maximum time-step value.

Phase Structure of Internal Gravity Waves in the Ocean with Shear Flows

Purpose. The description of the internal gravity waves dynamics in the ocean with background fields of shear currents is a very difficult problem even in the linear approximation. The mathematical problem describing wave dynamics is reduced to the analysis of a system of partial differential equations; and while taking into account the vertical and horizontal inhomogeneity, this system of equations does not allow separation of the variables. Application of various approximations makes it possible to construct analytical solutions for the model distributions of buoyancy frequency and background shear ocean currents. The work is aimed at studying dynamics of internal gravity waves in the ocean with the arbitrary and model distributions of density and background shear currents. Methods and Results. The paper represents the numerical and analytical solutions describing the main phase characteristics of the internal gravity wave fields in the stratified ocean of finite depth, both for arbitrary and model distributions of the buoyancy frequency and the background shear currents. The currents are considered to be stationary and horizontally homogeneous on the assumption that the scale of the currents' horizontal and temporal variability is much larger than the characteristic lengths and periods of internal gravity waves. Having been used, the Fourier method permitted to obtain integral representations of the solutions under the Miles – Howard stability condition is fulfilled. To solve the vertical spectral problem, proposed is the algorithm for calculating the main dispersion dependences that determine the phase characteristics of the generated wave fields. The calculations for one real distribution of buoyancy frequency and shear flow profile are represented. Transformation of the dispersion surfaces and phase structures of the internal gravitational waves’ fields is studied depending on the generation parameters. To solve the problem analytically, constant distribution of the buoyancy frequency and linear dependences of the background shear current on depth were used. For the model distribution of the buoyancy and shear flow frequencies, the explicit analytical expressions describing the solutions of the vertical spectral problem were derived. The numerical and asymptotic solutions for the characteristic oceanic parameters were compared. Conclusions. The obtained results show that the asymptotic constructions using the model dependences of the buoyancy frequency and the background shear velocities’ distribution, describe the numerical solutions of the vertical spectral problem to a good degree of accuracy. The model representations, having been applied for hydrological parameters, make it possible to describe qualitatively correctly the main characteristics of internal gravity waves in the ocean with the arbitrary background shear currents.

Basic Regularities of Vertical Turbulent Exchange in the Mixed and Stratified Layers of the Black Sea

Purpose. The purpose of the study is to assess the coefficient of vertical turbulent exchange for different layers of the Black Sea basin based on the experimental data on microstructure of the physical fields obtained for the period 2004–2019 in the Black Sea and using the semi-empirical models. Methods and Results. For the upper mixed layer, the turbulent energy dissipation rate ɛ and the exchange coefficient were calculated using the velocity fluctuation spectra based on the Kolmogorov hypotheses on the turbulence spectrum inertial range. In the stratified layers, the turbulence coefficient and the dissipation rate were experimentally determined both from the spectra of the velocity horizontal fluctuations’ gradients and the vertical spectra of temperature fluctuations using the concept of the effective scale of turbulent spots. Depending on the features of the hydrological regime and the prevailing energy contributors to turbulence generation, five layers were identified and described (including their characteristic power dependences of the vertical turbulent diffusion coefficients K on the buoyancy frequency N) using the 1.5D-model of vertical turbulent exchange for the basin under study. For the stratified layers, the 1.5D-model results were comparatively analyzed with those of the other semi-empirical and theoretical models describing the most probable hydrophysical processes in each specific layer; the relations for the vertical turbulent exchange coefficient were obtained depending on the buoyancy frequency. Conclusions. Comparison of the experimental data collected under different hydrometeorological conditions with the simulations resulted from the known turbulence models for the sea upper layer showed that the best agreement between the simulation and measurement data was provided by a multiscale model taking into account three basic mechanisms of turbulence generation: current velocity shear, instability of wave motions, and wave breaking. The turbulent exchange coefficient dependencies on depth are conditioned by the affect of the turbulence dominant source at a given level. In the stratified layers, the exchange coefficient dependence on buoyancy frequency is determined by the hydrophysical processes in each layer; the relations obtained for individual layers indicate intensity of the contributions of vertical advection, internal wave breakings, turbulence diffusion and geothermal flux.

On the Existence of Long-Time Classical Solutions for the 2D Inviscid Boussinesq Equations

In this paper, we consider the Cauchy problem for the 2D inviscid Boussinesq equations with N being the buoyancy frequency. It is proved that for general initial data u 0 ∈ H s with s > 3 , the life span of the classical solutions satisfies T > C ln N 3 / 4 .

The Efficiency of Upward Wave Propagation Near the Tropopause: importance of the form of the refractive index

The connection between the polar stratospheric vortex and the vertical component of the Eliassen-Palm flux in the lower stratosphere and upper troposphere is examined in model level data from the ERA-5 reanalysis. The particular focus of this work is on the conditions that lead to upward wave propagation between the tropopause and the bottom of the vortex near 100 hPa. The ability of four different versions of the index of refraction to capture this wave propagation are evaluated. The original Charney and Drazin index of refraction includes terms ignored by Matsuno that are shown to be critical for understanding upward wave propagation just above the tropopause in both the climatology and during extreme heat flux events. By adding these terms to the Matsuno index of refraction, it is possible to construct a useful tool that describes wave flux immediately above the tropopause and at the same time also describes the role of meridional variations within the stratosphere. It is shown that a stronger tropopause inversion layer tends to restrict upward wave propagation. It is also shown that while only 38% of extreme wave-1 Eliassen-Palm flux vertical component (Fz) at 100hPa events are preceded by extreme Fz at 300hPa, there are almost no extreme events at 100hPa in which the anomaly at 300hPa is of opposite sign or very weak. Overall, wave propagation near the tropopause is sensitive to vertical gradients in buoyancy frequency, and these vertical gradients may not be accurately captured in models or reanalysis products especially with lower vertical resolutions.