Ekman Model Research Articles

Complex turbulent exchange coefficient in Akerblom–Ekman model

Abstract The turbulent exchange in boundary layer models is usually characterized by a scalar eddy viscosity coefficient assumed to be a positive function of the vertical variable. We introduce a more general form for the turbulence exchange description, which includes two functions that describe the turbulence without any assumption about their positivity. We construct a model of the Akerblom–Ekman type, but with a complex coefficient of turbulent exchange. The basic quality criterion for these models and algorithms is the maximal agreement with meteorological observations. We optimize the agreement between the global meteorological archive of high-resolution wind observations that are provided by World Meteorological Organization (WMO) in Binary Universal Form for the Representation (BUFR). The main result of our work is that agreement between model solutions and observations will be much better if the turbulent exchange coefficient is optimized in the space of all complex-valued functions, and not limited to the cone of real positive functions.

A Study of Wind Stress Effects on the Vertical Eddy Viscosity Coefficient Using the Ekman Model with Data Assimilation

The vertical eddy viscosity coefficient (VEVC) is an important parameter used in ocean dynamics studies to describe the intensity of mixing in the vertical direction and the process of momentum transport. In this paper, an adjoint assimilation method was used to invert the VEVC, based on the Ekman model, with the measured wind field and current data. The main purpose was to study the effect of changes in the background wind field on VEVC, and thus investigate the role of wind stress in the inversion process. The results indicated that the inverse vertical eddy viscosity coefficient increased slightly at the surface layer, reaching its maximum at around 10–12 m, and then decreased monotonically with depth. The maximum VEVC value corresponds to different depths for different wind speed ranges. Additionally, wind steering could affect the VEVC inversion curve, causing it to deviate from the general trend. The kinetic energy ratio increased with depth, peaked at 18–20 m, and then rapidly decreased to nearly zero beyond 24 m. The impact of wind field strength and steering on VEVC was observed in the kinetic energy ratio curve, which confirms the speculative wind stress effect. This study revealed the characteristics and mechanisms of VEVC in coastal waters under different wind conditions, which could provide a reference for further research on physical oceanography.

O СПИРАЛЬНОСТИ ПОГРАНИЧНЫХ СЛОЕВ ЭКМАНОВСКОГО ТИПА

The sensitivity of the results of calculating atmospheric flow helicity in the atmospheric boundary layer with respect to the factor of nonstationarity and details of the eddy exchange description is theoretically investigated. The study shows that estimates often used in the framework of the classical Ekman model can lead to significantly overestimated values of helicity. Appropriate estimates and variants of refined calculations are given.

Scaling Approaches to Quasigeostrophic Theory for Moist, Precipitating Air

Abstract Quasigeostrophic (QG) theory is of fundamental importance in the study of large-scale atmospheric flows. In recent years, there has been growing interest in extending the classical QG plus Ekman friction layer model (QG–Ekman) to systematically include additional physical processes known to significantly contribute to real-life weather phenomena. This paper lays the foundation for combining two of these developments, namely, Smith and Stechmann’s family of precipitating quasigeostrophic (PQG) models on the one hand, and the extension of QG–Ekman for dry air by a strongly diabatic layer (DL) of intermediate height (QG–DL–Ekman) on the other hand. To this end, Smith and Stechmann’s PQG equations for soundproof motions are first corroborated within a general asymptotic modeling framework starting from a full compressible flow model. The derivations show that the PQG model family is naturally embedded in the asymptotic hierarchy of scale-dependent atmospheric flow models introduced by one of the present authors. Particular emphasis is then placed on an asymptotic scaling regime for PQG that accounts for a generic Kessler-type bulk microphysics closure and is compatible with QG–DL–Ekman theory. The detailed derivation of a moist QG–DL–Ekman model is deferred to a future publication.

Experiments on real-life emotions challenge Ekman's model

Ekman's emotions (1992) are defined as universal basic emotions. Over the years, alternative models have emerged (e.g. Greene and Haidt 2002; Barrett 2017) describing emotions as social and linguistic constructions. The variety of models existing today raises the question of whether the abstraction provided by such models is sufficient as a descriptive/predictive tool for representing real-life emotional situations. Our study presents a social inquiry to test whether traditional models are sufficient to capture the complexity of daily life emotions, reported in a textual context. The intent of the study is to establish the human-subject agreement rate in an annotated corpus based on Ekman's theory (Entity-Level Tweets Emotional Analysis) and the human-subject agreement rate when using Ekman's emotions to annotate sentences that don’t respect the Ekman’s model (The Dictionary of Obscure Sorrows). Furthermore, we investigated how much alexithymia can influence the human ability to detect and categorise emotions. On a total sample of 114 subjects, our results show low within subjects agreement rates for both datasets, particularly for subjects with low levels of alexithymia; low levels of agreement with the original annotations; frequent use of emotions based on Ekman model, particularly negative one, in people with high levels of alexithymia.

Atmospheric Boundary Layers over an Oceanic Eddy

Abstract Imagery and numerical modeling show an abundance of submesoscale oceanic eddies in the upper ocean. Large-eddy simulation (LES) is used to elucidate eddy impacts on the atmospheric boundary layer (ABL) forced by winds, convection, and an eddy with varying radius; the maximum azimuthal eddy speed is 1 m s−1. Simulations span the unstable regime −1/L = [0, ∞], where L is the Monin–Obukhov (M–O) stability parameter. A linearized Ekman model and the LES couple ABL winds to an eddy through rough-wall M–O boundary conditions. The eddy currents cause a surface stress anomaly that induces Ekman pumping in a dipole horizontal pattern. The dipole is understood as a consequence of surface winds aligned or opposing surface currents. In free convection a vigorous updraft is found above the eddy center and persists over the ABL depth. Heterogeneity in surface temperature flux is responsible for the full ABL impact. With winds and convection, current stress coupling generates a dipole in surface temperature flux even with constant sea surface temperature. Wind, pressure, and temperature anomalies are sensitive to an eddy under light winds. The eddy impact on ABL secondary circulations is on the order of the convective velocity scale but grows with increasing current speed, decreasing wind, or increasing convection. Flow past an isolated eddy develops a coherent ABL “wake” and secondary circulations for at least five eddy radii downwind. Kinetic energy exchanges by wind work indicate an eddy-killing effect on the oceanic eddy current, but only a spatial rearrangement of the atmospheric wind work.

Influence of Stratification and Bottom Boundary Layer on the Classical Ekman Model

A depth understanding of the different processes of water movements produced by the wind surface stress yields a better description and improvement of the marine food chain and ecosystem. The classical Ekman model proposes a hypothetical ocean, excluding the influence of continents and the Coriolis force. It also assumes infinite depth and a constant vertical eddy viscosity. The current study aims to understand how the vertical velocity profile is affected by the variation of the eddy viscosity coefficient (kz) and the consideration of a finite depth. The study uses an ideal analytical model with the Ekman classical model as a starting point. It has been demonstrated that, for a very stratified profile, when the depth is not considered infinity, the Ekman transport tends to a direction smaller than 80°. It differs from the classical Ekman model, which proposes an approximated angle equal to 90°. Considering the modified model, it was also found that the surface current deviation is smaller than 40°, which differs from the 45° proposed by the classical model. In addition, it was determined that for ocean depths smaller than 180 m, the maximum velocity does not occur at the water surface, as in the classical model, but at deeper levels.

Application of the Trigonometric Polynomial Interpolation for the Estimation of the Vertical Eddy Viscosity Coefficient Based on the Ekman Adjoint Assimilation Model

In this study, a triangular polynomial interpolation (TPI) scheme was developed to estimate the vertical eddy viscosity coefficient (VEVC) on the basis of the Ekman model with adjoint assimilation. In the twin experiments, the advantages and disadvantages of estimating the VEVC using the TPI scheme under different factors are discussed. The results indicated that (1) the TPI scheme proves to be better than the cubic spline interpolation (CSI) and Cressman interpolation (CI) schemes; (2) the inversion results are more sensitive to observations from upper ocean layers than those from lower layers, and the TPI scheme is less likely to be influenced by missing data; (3) for various boundary layer depths, the inversion results of the TPI scheme remain consistent with the given distributions; (4) the inversion results can be influenced considerably by observational errors, and the TPI scheme is more resistant to noise than the CSI and CI schemes; and (5) the inversion accuracy of the TPI scheme can be improved by selecting the temporal wind stress drag coefficients. In practical experiments, the adjoint method with the TPI scheme was developed to estimate the Ekman currents by assimilating the observations from a buoy stationed in the Yellow Sea. The results showed the successful estimation of the VEVC and demonstrated that more precise current velocities can be obtained with this estimation scheme. In summary, this study provides a useful approach for the effective estimation of the VEVC.

Tell Me More: Automating Emojis Classification for Better Accessibility and Emotional Context Recognition

Users of web or chat social networks typically use emojis (e.g., smilies, memes, hearts) to convey in their textual interactions the emotions underlying the context of the communication, aiming for better interpretability, especially for short polysemous phrases. Semantic-based context recognition tools, employed in any chat or social network, can directly comprehend text-based emoticons (i.e., emojis created from a combination of symbols and characters) and translate them into audio information (e.g., text-to-speech readers for individuals with vision impairment). On the other hand, for a comprehensive understanding of the semantic context, image-based emojis require image-recognition algorithms. This study aims to explore and compare different classification methods for pictograms, applied to emojis collected from Internet sources. Each emoji is labeled according to the basic Ekman model of six emotional states. The first step involves extraction of emoji features through convolutional neural networks, which are then used to train conventional supervised machine learning classifiers for purposes of comparison. The second experimental step broadens the comparison to deep learning networks. The results reveal that both the conventional and deep learning classification approaches accomplish the goal effectively, with deep transfer learning exhibiting a highly satisfactory performance, as expected.

Emotion Recognition from Physiological Channels Using Graph Neural Network

In recent years, a number of new research papers have emerged on the application of neural networks in affective computing. One of the newest trends observed is the utilization of graph neural networks (GNNs) to recognize emotions. The study presented in the paper follows this trend. Within the work, GraphSleepNet (a GNN for classifying the stages of sleep) was adjusted for emotion recognition and validated for this purpose. The key assumption of the validation was to analyze its correctness for the Circumplex model to further analyze the solution for emotion recognition in the Ekman modal. The novelty of this research is not only the utilization of a GNN network with GraphSleepNet architecture for emotion recognition, but also the analysis of the potential of emotion recognition based on differential entropy features in the Ekman model with a neutral state and a special focus on continuous emotion recognition during the performance of an activity The GNN was validated against the AMIGOS dataset. The research shows how the use of various modalities influences the correctness of the recognition of basic emotions and the neutral state. Moreover, the correctness of the recognition of basic emotions is validated for two configurations of the GNN. The results show numerous interesting observations for Ekman’s model while the accuracy of the Circumplex model is similar to the baseline methods.

Coupling of Wind and Potential Temperature in an Ekman Model in the Stratified Atmospheric Boundary Layer

Abstract A linear Ekman model in the stratified atmospheric boundary layer (ABL) is proposed based on the steady-state version of the linearized three-dimensional primitive equations with the inclusion of the vertical diffusivity. Due to the inclusion of the potential temperature equation and hydrostatic equation, pressure and potential temperature couple with wind in the proposed model, and thus are not arbitrarily specified variables as in previous studies on the baroclinicity in the Ekman model. The extended thermal wind balance equation and the Ekman potential vorticity equation are derived to describe the coupling. The two equations, along with the equation describing the constraint on potential temperature, are employed to derive the analytical solutions of the proposed Ekman model. Because potential temperature is not a specified variable but part of the solution, the derived analytical solutions have very different forms from those derived in previous studies. The differences illustrate the impact of the inclusion of the potential temperature equation and hydrostatic equation on wind, pressure, and potential temperature in the proposed Ekman model. It is found that the computed wind profiles based on the proposed model can capture some important features of the observed wind profiles.

QG–DL–Ekman: Dynamics of a Diabatic Layer in the Quasi-Geostrophic Framework

AbstractQuasigeostrophic (QG) theory describes the dynamics of synoptic-scale flows in the troposphere that are balanced with respect to both acoustic and internal gravity waves. Within this framework, effects of (turbulent) friction near the ground are usually represented by Ekman layer theory. The troposphere covers roughly the lowest 10 km of the atmosphere while Ekman layer heights are typically just a few hundred meters. However, this two-layer asymptotic theory does not explicitly account for substantial changes of the potential temperature stratification due to diabatic heating associated with cloud formation or with radiative and turbulent heat fluxes which can be significant in about the lowest 3 km and in the middle latitudes. To address this deficiency, this paper extends the classical QG–Ekman layer model by introducing an intermediate dynamically and thermodynamically active layer, called the “diabatic layer” (DL) from here on. The flow in this layer is also in acoustic, hydrostatic, and geostrophic balance but, in contrast to QG flow, variations of potential temperature are not restricted to small deviations from a stable and time-independent background stratification. Instead, within the DL diabatic processes are allowed to affect the leading-order stratification. As a consequence, this layer modifies the pressure field at the top of the Ekman layer, and with it the intensity of Ekman pumping seen by the quasigeostrophic bulk flow. The result is the proposed extended quasigeostrophic three-layer QG–DL–Ekman model for midlatitude dynamics.

A Scheme for Estimating Time-Varying Wind Stress Drag Coefficient in the Ekman Model with Adjoint Assimilation

In this study, the time-varying wind stress drag coefficient in the Ekman model was inverted by the cubic spline interpolation scheme based on the adjoint method. Twin experiments were carried out to investigate the influences of several factors on inversion results, and the conclusions were (1) the inverted distributions with the cubic spline interpolation scheme were in good agreement with the prescribed distributions of the wind stress drag coefficients, and the cubic spline interpolation scheme was superior to direct inversion by the model scheme and Cressman interpolation scheme; (2) the cubic spline interpolation scheme was more advantageous than the Cressman interpolation scheme even if there is moderate noise in the observations. The cubic spline interpolation scheme was further validated in practical experiments where Ekman currents and wind speed derived from mooring data of ocean station Papa were assimilated. The results demonstrated that the variation of the time-varying wind stress drag coefficient with time was similar to that of wind speed with time, and a more accurate inversion result could be obtained by the cubic spline interpolation scheme employing appropriate independent points. Overall, this study provides a potential way for efficient estimation of time-varying wind stress drag coefficient.

Regionalizing the Impacts of Wind‐ and Wave‐Induced Currents on Surface Ocean Dynamics: A Long‐Term Variability Analysis in the Mediterranean Sea

AbstractEffects of wind and waves on the surface dynamics of the Mediterranean Sea are assessed using a modified Ekman model including a Stokes‐Coriolis force in the momentum equation. Using 25 years of observations, we documented intermittent but recurrent episodes during which Ekman and Stokes currents substantially modulate the total mesoscale dynamics by two nonexclusive mechanisms: (a) by providing a vigorous input of momentum (e.g., where regional winds are stronger) and/or (b) by opposing forces to the main direction of the geostrophic component. To properly characterize the occurrence and variability of these dynamical regimes, we perform an objective classification combining self‐organizing maps and wavelet coherence analyses. It allows proposing a new regional classification of the Mediterranean Sea based on the respective contributions of wind, wave, and geostrophic components to the total mesoscale surface dynamics. We found that the effects of wind and waves are more prominent in the northwestern Mediterranean, while the southwestern and eastern basins are mainly dominated by the geostrophic component. The resulting temporal variability patterns show a strong seasonal signal and cycles of 5–6 years in the total kinetic energy arising from both geostrophic and ageostrophic components. Moreover, the whole basin, specially the regions characterized by strong wind‐ and wave‐induced currents, shows a characteristic period of variability at 5 years. This can be related to climate modes of variability. Regional trends in the geostrophic and ageostrophic currents show an intensification of 0.058 ± 1.43 · 10−5 cm/s per year.

Wave-Modified Ekman Current Solutions for the Time-Dependent Vertical Eddy Viscosity

This study aimed to highlight a general lack of clarity regarding the scale of the temporal averaging implicit in Ekman-type models. Under the assumption of time and depth-dependent eddy viscosity, we present an analytical Fourier series solution for a wave-modified Ekman model. The depth dependence of eddy viscosity is based on the K-Profile Parameterization (KPP) scheme. The solution reproduces major characteristics of diurnal variation in ocean velocity and shear. Results show that the time variability in eddy viscosity leads to an enhanced mean current near-surface and a decrease in the effective eddy viscosity, which finally results in an intensified near-surface shear and wakes a low-level jet flow. Rectification values are dominated by the strength of diurnal mixing, and partly due to the nonlinear depth dependence of the eddy viscosity.

The generalized Ekman model for the tropical cyclone boundary layer revisited: Addendum

AbstractMotivated by prior research examining the myth of inertial stability as a radial restoring force in the tropical cyclone boundary layer, we explore factors determining the vertical velocity at the top of the linear vortex boundary layer. Possible applications of these findings to mature tropical cyclone vortices are discussed briefly.

Correction to: The generalized Ekman model for the tropical cyclone boundary layer revisited

Correction to: The generalized Ekman model for the tropical cyclone boundary layer revisited

The generalized Ekman model for the tropical cyclone boundary layer revisited: The myth of inertial stability as a restoring force

AbstractWe revisit the linear boundary‐layer approximation that expresses a generalized Ekman balance and use it to clarify a range of interpretations in the previous literature on the tropical cyclone boundary layer. Some of these interpretations relate to the reasons for inflow in the boundary layer and others relate to the presumed effects of inertial stability on boundary‐layer dynamics. Inertial stability has been invoked, for example, to explain aspects of boundary‐layer behaviour, including the frontogenetic nature of the boundary layer and its relationship to vortex spin‐up. Our analysis exposes the fallacy of invoking inertial stability as a resistance to radial inflow in the boundary layer. The analysis shows also that the nonlinear acceleration terms become comparable to the linear Coriolis acceleration terms in relatively narrow vortices that are inertially stable above the boundary layer. Estimates of the nonlinear accelerations using the linear solutions are expected to underestimate the actual contribution in a nonlinear boundary‐layer model, cautioning against neglecting the nonlinear terms in diagnostic or prognostic models.

A Method on Estimating Time-Varying Vertical Eddy Viscosity for an Ekman Layer Model with Data Assimilation

Temporal vertical eddy viscosity coefficient (VEVC) in an Ekman layer model is estimated using an adjoint method. Twin experiments are carried out to investigate the influences of several factors on inversion results, and the conclusions of twin experiments are 1) the adjoint method is a capable method to estimate different kinds of temporal distributions of VEVCs; 2) the gradient descent algorithm is better than CONMIN and L-BFGS for the present problem, although the posterior two algorithms perform better on convergence efficiency; 3) inversion results are sensitive to initial guesses; 4) the model is applicable to different wind conditions; 5) the inversion result with thick boundary layer depth (BLD) is slightly better than thin BLD; 6) inversion results are more sensitive to observations in upper layers than those in lower layers; 7) inversion results are still acceptable when data noise exists, indicating the method can sustain noise to a certain degree; 8) a regularization method is proved to be useful to improve the results for present problem; and 9) the present method can tolerate the existence of balance errors due to the imperfection of governing equations. The methodology is further validated in practical experiments where Ekman currents are derived from Bermuda Testbed Mooring data and assimilated. Modeled Ekman currents coincide well with observed ones, especially for upper layers. The results demonstrate that the assumptions of depth dependence and time dependence are equally important for VEVCs. The feasibility of the typical Ekman model, the imperfection of Ekman balance equations, and the deficiencies of the present method are discussed. This method provides a potential way to realize the time variations of VEVCs in ocean models.

Influences of Random Surface Waves on the Estimates of Wind Energy Input to the Ekman Layer in the Antarctic Circumpolar Current Region

AbstractSea surface waves significantly affect the wind energy input to the Ekman layer in the upper ocean. In the study, we first incorporated the wave‐induced Coriolis‐Stokes forcing, the reduction of wind stress caused by wave generation, and wave dissipation into the classical Ekman model to investigate the kinetic energy balance in the wave‐affected Ekman layer. Then, both the theoretical steady state solution for the idealized condition and the nonsteady state solution for the realistic ocean were derived. Total energy input to the wave‐affected Ekman layer includes the wind stress energy input and the wave‐induced energy input. Based on the WAVEWATCH III model, the wave spectrum was simulated to represent realistic random directional wave conditions. The wind stress energy input and the wave‐induced energy input to the wave‐affected Ekman layer in the Antarctic Circumpolar Current in the period from 1988 to 2010 were then calculated. The annual mean total energy input in the Antarctic Circumpolar Current region was 402.5 GW and the proportions of the wind stress energy input and the wave‐induced energy input were, respectively, 85% and 15%. Particularly, total energy input in the Antarctic Circumpolar Current in the wave‐affected Ekman layer model was 59.8% lower than that in the classical Ekman model. We conclude that surface waves play a significant role in the wind energy input to the Ekman layer.