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With the aim of assessing internal wave-driven mixing in the ocean, we develop a new technique for numerical simulations of stratified turbulence. Since the spatial scales of energetic ocean currents and internal gravity waves are typically much larger than that of turbulence, fully incorporating both in a model would require a high computational cost and is therefore out of our scope. Alternatively, we cut out a small domain periodically distorted by an unresolved large-scale flow field and locally simulate the energy cascade to the smallest scales. This technique enables us to concentrate the computational resources on resolving the breaking process of small-scale waves and the resulting turbulence while properly incorporating energy supply from large-scale flow components. This time, we demonstrate the results of two typical problems: the parametric subharmonic instability (PSI) of a plane internal gravity wave and the ageostrophic anticyclonic instability (AAI) of an elliptic vortex.

For the PSI case, we simulate the enhancement of a striped pattern of subharmonic internal waves in a tilted and oscillating coordinate system. When the disturbance amplitude grows sufficiently large, secondary instabilities arise and produce much smaller-scale fluctuations. Passing through these two stages, wave energy is transferred into turbulence energy and will be eventually dissipated. Different from the conventional scenarios of vertical shear-induced instabilities, a large part of turbulent potential energy is supplied from the outer wave and directly used for mixing. The mixing coefficient, defined as the dissipation rate of available potential energy divided by that of kinetic energy, is as large as 0.5 and tends to increase with the outer wave Froude number.

In the simulations of AAI, we rotate a model domain following elliptic streamlines of the gradient wind balance to assess the parametric excitation of inertia-gravity waves. From a series of experiments, we identify two different scenarios of wave breaking conditioned on the magnitude of the instability growth rate scaled by the buoyancy frequency. When this parameter is large, the primary wave amplitude excited by AAI quickly goes far beyond the overturning threshold and directly breaks. The resulting state is thus strongly nonlinear turbulence. If the instability parameter is small, weak wave–wave interactions begin to redistribute energy across frequency space before the primary wave reaches a breaking limit. Then, after a sufficiently long time, the system approaches a Garrett–Munk-like stationary spectrum, in which wave breaking occurs at finer vertical scales. In the two wave-breaking scenarios, the energy dissipation rates exhibit distinct scaling properties, similar to D’Asaro and Lien’s wave–turbulence transition model.

References and collaborators:
- Onuki, Yohei, Sylvain Joubaud, and Thierry Dauxois. "Simulating turbulent mixing caused by local instability of internal gravity waves." Journal of Fluid Mechanics 915 (2021): A77.
- Onuki, Yohei, Sylvain Joubaud, and Thierry Dauxois. "Breaking of internal waves parametrically excited by ageostrophic anticyclonic instability." Journal of Physical Oceanography 53.6 (2023): 1591-1613.

Baroclinic instability is a phenomenon in rotating stratified systems, and is known to play an important role in the Earth's atmosphere and ocean. The Eady problem is a text book example of baroclinic instability widely invoked for various applications (along with the Charney and Phillips problem). In the first part we recap some of the features of the standard geostrophic Eady problem (e.g., as described in most standard GFD textbooks), such as the linear instability characteristics, method of solution by analytical means, instability mechanism in terms of Counter-propagating Rossby Waves (CRWs), relations to baroclinic lifecycles, and its parameterisation in atmospheric and/or oceanic systems (e.g., works of Green, Gent-McWilliams and/or Greatbach-Lamb schemes). In the second part, we proceed to highlight and/or clarify some features of a modified Eady linear instability problem in the presence of a slope, namely (i) links of the Eady problem with quantum mechanics under the umbrella of non-Hermitian PT symmetric operators, (ii) a revised view of the CRW mechanism in the present sloped system that is more complete than existing attempts, with a rephrasing of the problem in terms of a dynamical system with a CRW basis, and (iii) analysis of the instability characteristics in terms of the GEOMETRIC framework, with some links to ocean eddy parameterisations. If time at the end, more speculative links of non-Hermitian PT symmetric operators, CRW basis and the more general shear instability problem will be provided.

 Batchelor's self-similarity and Kraichnan's inertial range work on two-dimensional turbulence are unsuccessful due to the formation of coherent vortices which generate spatial hierarchical structures with time. In particular, the vortices create spatial intermittency and non-Gaussianity, and mechanisms for inverse energy cascade and direct enstrophy transfer are still open to probe. Via numerical simulations and self-similar vortex theory, we quantified the vortex populations and found the energy spectrum at high wave numbers follows a steeper slope than that predicted by the Batchelor and Kraichnan theories. Also, we discuss the reasons for the decay of enstrophy, which is due to the debris that is created by the vortex collisions.

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Mesoscale Eddies are rotating structures found in the ocean with size of 50 to 500 km with a lifespan of 10 to 100 days. Eddies can be detected from Sea Surface Height (SSH) data. These circulations play a major role in transportation of heat, salt, carbon and nutrients throughout the ocean and are also associated with complex atmosphere-ocean interactions. That is why the detection of the eddies is extremely important to understand the big picture of the climate. Various, physics based methods were used for eddies detection but dependent upon the threshold values and human expertise which leads to the false detection. Therefore, advance techniques of artificial intelligence are needed to overcome these challenges. Various Deep Learning based works were done over the South China Sea and South Atlantic Ocean but Bay of Bengal is less explored. In this study we trained deep learning models, U-Net and Attention U-Net, for detecting the mesoscale ocean eddies over Bay of Bengal from daily Sea Level Anomaly (SLA) data. These are supervised Deep Learning models renowned for semantic segmentation tasks. The pyEddyTracker and openEddy softwares were used to create two ground truth datasets. On both datasets, Attention U-Net performed slightly better than U-Net. After that very small sized eddies detected by the Attention U-Net model were filtered. After filtering, the Attention U-Net model achieved a Mean Intersection Over Union score of 0.81 and 0.84 for dataset developed using pyEddyTracker and openEddy respectively. Intersection Over Union is defined as the ratio of area of the intersection between the actual and predicted segmentation mask to their area of their union. Deep Learning models are labeled as “black box”. Hence, to explain our model we employed SEG-GRAD-CAM based technique. As ground truth is not perfect, the future focus of our work is to create a near perfect ground truth dataset to enhance the performance of Deep Learning based Eddy Detection methods.

Planning and adapting to future coastal ocean conditions requires accurate coastal ocean predictions of the nutrient, pollutant, heat and sediment transport. As coastal ocean turbulence is often driven by relatively small scale processes such as high-frequency internal waves and submesocale eddies that are not captured in most regional ocean models turbulence parameterisation is challenging Using a combination of process-based field campaigns and long-term monitoring data we have characterised the relationship between diapycnal mixing and diverse external forcings and differing flow regimes on the Australian northwest shelf. We show that the overall diapycnal mixing is dominated by relatively rare but energetic mixing events. We observe that the semi-diurnal barotropic tide, the spring-neap tidal variability, and the seasonal variability in stratification all affect the magnitude of diapycnal mixing and its vertical distribution.

Internal gravity waves pervade the oceans, profoundly shaping their dynamics. Their interactions with eddies and other waves govern energy transfers and can lead to wave breaking, and density mixing, thus influencing large-scale mean flows. Despite their significance, the relative importance of wave-mean flow interactions vis-à-vis wave-wave interactions remains elusive. We investigate internal gravity wave-mean flow interactions with the novel numerical Internal Wave Energy Model (IWEM) based on the six-dimensional radiative transfer equation. We simulate wave interactions with local coherent mesoscale eddies, to find a wave energy loss at the eddy rim akin to critical layer behavior. We investigate internal gravity wave-wave interactions by numerically evaluating the kinetic equation derived from weak interaction assumptions. Our findings unveil a predominantly forward energy cascade from wave-wave interactions

• Eddy viscosity is employed throughout the majority of numerical fluid dynamical models, and has been the subject of a vigorous body of research spanning a variety of disciplines. It has long been recognized that the proper description of eddy viscosity uses tensor mathematics, but in practice it is almost always employed as a scalar due to uncertainty about how to constrain the extra degrees of freedom and physical properties of its tensorial form. This talk will introduce techniques from outside the realm of geophysical fluid dynamics that allow us to consider the eddy viscosity tensor using its eigenvalues and eigenvectors, establishing a new framework by which tensorial eddy viscosity can be tested. This is made possible by a careful analysis of an operation called tensor unrolling, which casts the eigenvalue problem for a fourth-order tensor into a more familiar matrix-vector form, whereby it becomes far easier to understand and manipulate. New constraints are established for the eddy viscosity coefficients that are guaranteed to result in energy dissipation, backscatter, or a combination of both. Finally, I will propose a testing protocol by which tensorial eddy viscosity can be systematically evaluated across a wide range of fluid regimes.

The Indian Ocean (IO) coastline which houses a large population from the continents of Africa, Asia and Australia is vulnerable to a plethora of climatic hazards that are brought on by sea-level rise. The global mean sea level has risen at a rate of ~3.6 mm/yr over the last two decades and is projected to increase by more than 1m by the end of this century. A thorough assessment of the dynamics of the regional sea-level change is vital for effective policymaking to mitigate natural calamities associated with the rising sea levels. We use a suit of 27 models from phase six of the coupled model intercomparison project (CMIP6) simulations to study their representation of dynamic sea level (DSL) and the factors that influence DSL variability in the basin. We show that the multi-model mean DSL exhibits a good correlation with observation with few notable biases consistent across the models. There is a positive bias in the DSL across the basin with a west to east gradient and a pronounced bias in the Antactic circumploar current region. In the case of variability, most of the models underestimate the variability across the basin except the eastern equatorial IO. The poor representation of the equatorial winds in most models produces an Indian Ocean Dipole (IOD) like bias and results in the misrepresentation of climatic modes. Our analysis suggests that a finer horizontal resolution of the ocean component alone cannot guarantee a better representation of the DSL but requires proper representation of wind fields as well. A subset of best performing models among the ensemble is selected to have a more representative estimate of DSL change in the Indian Ocean. The Arabian Sea is expected to experience higher sea level rise (~35 cm), compared to the Bay of Bengal and the southern tropical Indian Ocean under a high emission scenario by the end of 2100. This research aims to gain better insights on the DSL evolution and its future projections in the Indian Ocean and to investigate the model deficiencies associated with the same.

Turbulence reconstruction poses significant challenges in a wide range of fields, including geophysics, astronomy, and even the natural and social sciences. The complexity of these challenges is largely due to the non-trivial geometrical and statistical properties observed over decades of time and spatial scales. Recent advances in machine learning, such as generative adversarial networks (GANs), have shown notable advantages over classical methods in addressing these challenges[1,2]. In addition, the success of generative diffusion models (DMs), particularly in computer vision, has opened up new avenues for tackling turbulence problems. These models use Markovian processes that progressively add and remove noise scale by scale, which naturally aligns with the multiscale nature of turbulence. In this presentation we discuss a conditional DM tailored for turbulence reconstruction tasks. The inherent stochasticity of DM provides a probabilistic set of predictions based on known measurements [3]. We validate DM on a rotating turbulence setup, a representative challenge in geophysical applications where spatial gaps are present in 2D observed snapshots. The effectiveness of DM is compared with both a GAN and an equation-based data assimilation method, nudging. Through systematic comparison, DM demonstrates superior performance in both pointwise reconstruction and statistical metrics. This approach could be instrumental in a range of physical applications, from astrophysics to particle tracking. It provides a robust tool for uncertainty quantification and risk assessment, and has the potential to address complex turbulent systems across different spatial and temporal scales. This research was supported from the European Research Council (ERC) grant 882340 and from the MUR - FARE grant R2045J8XAW.
[1] Li et al., J. Fluid Mech. 971 (2023)
[2] Buzzicotti et al., Phys. Rev. Fluids 6(5) (2021)
[3] Li et al., Atmosphere 15(1) (2024)

Surface currents modify wind driven Ekman pumping in the ocean both by modifying the stress itself and by modifying the relationship between the stress and the Ekman transport. The former effect results in a strong mesoscale structure in the wind stress curl, such as is evident from scatterometer data. This mesoscale forcing is anti-correlated with surface vorticity and thus produces a strong damping effect on ocean eddies and currents. Recent work, however, suggests that this damping effect is over-represented in common parameterizations of the air-sea wind stress. The latter effect is referred to as nonlinear Ekman dynamics. These dynamics take the stress as given and add advective terms to the linear balance. Specifically, cross terms involving the Ekman and non-Ekman components of the flow are added to the linear Ekman balance. This is known to produce small scale (e.g., submesoscale) structures in the pumping velocity.

Here, we first review both the ocean surface velocity dependence in formulations of the wind stress and how it relates to damping of geostrophic currents and eddies. We then propose a new formulation in which the surface velocity is not the velocity of an upper ocean grid cell, but rather is the velocity of a deformable rigid lid assumed to be lying between the atmosphere and ocean. The same boundary layer turbulence theory which gives the stress just above the boundary is also assumed to apply below, and matching the two stresses determines the surface velocity (and hence the stress). This formulation leads to a reduction in the damping of geostrophic eddies and currents. Next, we consider so-called nonlinear Ekman theory in submesoscale permitting simulations of wind driven ocean channel flow. A slowly varying part of near surface model vertical velocity is shown to agree well with the theory. When high frequency forcing is added, inertia-gravity waves interact with the slowly varying Ekman flow to produce a fast-time-scale analog to the Ekman velocity. Finally, we speculate that interactions between this fast Ekman flow and the waves themselves feed back on to the slowly varying Ekman-like flow. That is, we suggest that waves and (non-linear) Ekman dynamics can help to explain the rich structure seen in near-surface ocean vertical velocity fields.

Recent efforts in building data-driven surrogates for weather forecasting applications have received a lot of attention and garnered noticeable success. These autoregressive data-driven models yield significantly competitive short-term forecasting performance (often outperforming traditional numerical weather models) at a fraction of the computational cost of numerical models. However, these data-driven models do not remain stable when time-integrated for a long time. Such a long time-integration would provide (1) a method to seamlessly scale a weather model to a climate model and (2) gathering insights into the statistics of that climate system, e.g., the extreme events, owing to the cheap cost of generating multiple ensembles. While many studies have reported this instability, especially for data-driven models of turbulent flow, a causal mechanism for this instability is not clear. Most efforts to obtain stability are ad-hoc and empirical. In this work, we use a canonical quasi-geostrophic model to present a causal mechanism for this instability through the lens of a phenomenon called “spectral bias” in deep learning theory. We would show how spectral bias compounds the error in small scales which eventually, through inverse cascade, intensifies the errors in the large scales. Furthermore, we would show how the compounding of error would increase the intensity of meridional heat flux and momentum flux leading to an unstable and unphysical flow. We further perform rigorous theoretical eigen analysis to provide a framework to identify unstable autoregressive models and quantitatively predict the compounding error growth. In order to mitigate this issue, we provide a rigorous architecture agnostic (shown on different neural networks and neural operators) a three-pronged approach in which (1) a higher-order integrator is constructed within the model architecture, (2) a spectral regularizer is introduced to reduce the effect of spectral bias, and (3) a self-supervised optimization strategy is introduced, wherein the deepest layers are updated during autoregressive inference to correct the Fourier spectrum of the small scales. Finally, we would apply our approach to a data-driven model trained on ERA5 data as well as ocean reanalysis data and show optimistic results in both short-term performance and long-term stability opening new frontiers towards seamless weather-to-climate models.

Consider a system described by a multi-dimensional state vector X. The evolution of x is governed by a set of equations in the form of dx/dt=F(X(t)). x is a component of X. F(X(t)), the differential forcing of x, is a deterministic function of X. The solution of such a system often exhibits randomness, where the solution at one time is independent of the solution at another time. This study investigates the mechanism responsible for such randomness. We do so by exploring the integral forcing of x, G_T (t), a definite integral of F over the time span extending from t to t+T, which links the solution at two times, t and t+T.

We show that, for a system in equilibrium, G_T (t) can be expressed as G_T (t)=c_T+d_T x(t)+f_T (t), which consists of (apart from constant c_T) a dissipating component with strength d_T and a fluctuating component f_T (t), in line with the fluctuation-dissipation theorem that for a system in equilibrium, anything that generates fluctuations must also damp the fluctuations. Moreover, for a sufficiently large value of T, G_T (t) emerges as a unified forcing, characterized by d_T=-1 and a white-noise like fluctuating component f_T (t). The evolution of x from t to t+T, which is described by x(t+T)=x(t)+G_T (t) nominally, is then described by x(t+T)=c_T+f_T (t). This evolution is random, since x(t+T) is independent of x(t). This evolution is also irreversible, since the dissipating component of G_T (t) cancels with x(t) little by little and eventually completely by the time when G_T (t) emerges and generates x(t+T). The unified forcing results from interactions of x(t) with other components of X(t) that are completed during the forward integration over the time span [t,t+T], which cannot be included in the differential forcing F. In general, randomness and irreversibility are inherent features of a multi-dimensional physical system in equilibrium.

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