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A numerical model is implemented to describe fluid dynamic processes associated with mid-latitude small- scale (10 km) upper ocean fronts by using modified state of the art computational fluid dynamics tools. A periodic system was simulated using three different turbulent closures: 1) URANS-Reynolds Stress Model (RSM, seven equation turbulence model), 2) LES-Standard Smagorinsky (SS, algebraic model), and 3) LES-Modified Smagorinsky, introducing a correction for non-isotropic grids (MS). The results show the front developing instabilities and generating submesoscale structures after four days of simulation. A strongly unstable shear flow is found to be confined within the mixed layer with a high Rossby number (Ro > 1) and high vertical velocity zones. The positive (negative) vertical velocity magnitude is found to be approximately O(10-3 ) m/s(O(10-2 ) m/s), one (two) order(s) of magnitude larger than the vertical velocity outside the sub-mesoscale structures, where the magnitude is stable at O(10-4 ) m/s. The latter value is consistent with previous numerical and experimental studies that use coarser grid sizes and therefore do not explicitly calculate the small scale structures. The nonlinear flow introduced by the sub-mesoscale dynamics within the mixed layer and the non-isotropic grid used in the calculations generates a disparity between the predicted horizontal wave-number spectra computed using the RSM model with respect to the linear eddy viscosity model SS. The MS approach improves SS predictions. This improvement is more significant below the mixed layer in the absence of flow nonlinearities. The horizontal spectra predicted with the RSM model fits a slope of -3 for large scale structures and a slope between -2 and -5/3 for turbulent structures smaller than 300 m. This work contributes to the investigation of the physical and methodological aspects for the detailed modelling and understanding of small scale structures in ocean turbulence.

This article discusses the role of geophysical fluid dynamics (GFD) in understanding the natural environment, and in particular the dynamics of atmospheres and oceans on Earth and elsewhere. GFD, as usually understood, is a branch of the geosciences that deals with fluid dynamics and that, by tradition, seeks to extract the bare essence of a phenomenon, omitting detail where possible. The geosciences in general deal with complex interacting systems and in some ways resemble condensed matter physics or aspects of biology, where we seek explanations of phenomena at a higher level than simply directly calculating the interactions of all the constituent parts. That is, we try to develop theories or make simple models of the behaviour of the system as a whole. However, these days in many geophysical systems of interest, we can also obtain information for how the system behaves by almost direct numerical simulation from the governing equations. The numerical model itself then explicitly predicts the emergent phenomena—the Gulf Stream, for example—something that is still usually impossible in biology or condensed matter physics. Such simulations, as manifested, for example, in complicated general circulation models, have in some ways been extremely successful and one may reasonably now ask whether understanding a complex geophysical system is necessary for predicting it. In what follows we discuss such issues and the roles that GFD has played in the past and will play in the future.

Oceanic and atmospheric dynamics are often interpreted through potential vorticity, as this quantity is conserved along the geostrophic flow. However, in addition to potential vorticity, surface buoyancy is a conserved quantity, and this also affects the dynamics. Buoyancy at the ocean surface or at the atmospheric tropopause plays the same role of an active tracer as potential vorticity does since the velocity field can be deduced from these quantities. The surface quasi-geostrophic model has been proposed to explain the dynamics associated with surface buoyancy conservation and seems appealing for both the ocean and the atmosphere. In this review, we present its main characteristics in terms of coherent structures, instabilities and turbulent cascades. Furthermore, this model is mathematically studied for the possible formation of singularities, as it presents some analogies with three-dimensional Euler equations. Finally, we discuss its relevance for the ocean and the atmosphere.

In this work, we consider a Shallow‐Water Quasi Geostrophic equation on the sphere, as a model for global large‐scale atmospheric dynamics. This equation, previously studied by Verkley (2009, https://doi.org/10.1175/2008jas2837.1) and Schubert et al. (2009, https://doi.org/10.3894/james.2009.1.2), possesses a rich geometric structure, called Lie‐Poisson, and admits an infinite number of conserved quantities, called Casimirs. In this paper, we develop a Casimir preserving numerical method for long‐time simulations of this equation. The method develops in two steps: first, we construct an N‐dimensional Lie‐Poisson system that converges to the continuous one in the limit N → ∞; second, we integrate in time the finite‐dimensional system using an isospectral time integrator, developed by Modin and Viviani (2020, https://doi.org/10.1017/jfm.2019.944). We demonstrate the efficacy of this computational method by simulating a flow on the entire sphere for different values of the Lamb parameter. We particularly focus on rotation‐induced effects, such as the formation of jets. In agreement with shallow water models of the atmosphere, we observe the formation of robust latitudinal jets and a decrease in the zonal wind amplitude with latitude. Furthermore, spectra of the kinetic energy are computed as a point of reference for future studies. Plain Language Summary We conducted a study on a model that represents the movements of planetary flows. This model has important physical and mathematical properties that are related to its long‐term behavior, which is essential for understanding geophysical turbulence. In this work, we developed a numerical method for simulation that preserves the key mathematical structure of the model through a two‐step process. We applied our method to simulate global atmospheric flow and investigate the impact of varying strengths of planetary rotation. Our findings demonstrate the expected formation of wind patterns known as zonal jets, where stronger winds occur near the equator and weaker winds near the poles. We also present energy spectra that illustrate the influence of planetary rotation on the transfer of turbulent energy, which aligns with existing theoretical predictions found in literature. These results highlight the potential of our numerical method for studying fundamental problems in geophysical fluid dynamics. Key Points We develop a numerical method preserving Casimirs to simulate balanced shallow water flow on the sphere We perform global high‐resolution simulations while accurately accounting for latitude‐dependent effects Our simulations show the formation of robust zonal jets and provide key insights into quasi‐geostrophic turbulence

The under‐ice fluid dynamics during late winter in many freshwater settings remain poorly understood. One example is how cabbeling, the generation of dense water by mixing water masses on different sides of the temperature of maximum density (Tmd), affects the vertical transport. Using high resolution numerical simulations of the development of a stratified parallel shear flow, we show that when the temperature stratification passes through Tmd, cabbeling modifies the three‐dimensional aspects of the instability with the net effect of forming an emergent, stable stratification after a brief period of strong mixing. This stratification effectively separates the quiescent upper layer from the turbulent one below, thereby limiting transport and mixing between them. We propose a simple model for the vertical fluid flux during the mixing regime, and discuss a potential mechanism responsible for the emergent stratification as well as its sensitivity to the bulk Richardson number. Plain Language Summary In late winter, many lakes in temperate climates remain ice covered while warmer water from adjacent rivers flow into the lake beneath the ice. Due to the fact that freshwater has a temperature of maximum density around four degrees, this sets up a situation in which river inflows can lead to mixing processes in the so‐called cabbeling regime. Cabbeling is the process by which two masses of freshwater mix to form a denser mass. This in turn leads to more mixing as the dense water mass sinks. Using high resolution numerical simulations, we demonstrate that the instability process leads to the creation of two distinct zones in the water column. After an initial burst of activity, mixing between the two regions is greatly reduced. In contrast to warmer climates, the system does not return to a marginally stable state. Since ice covered waters lack a mechanism for mechanical forcing, cabbeling shear instability provides a potentially dominant means to enhance vertical transport for the water column outside a thin near‐surface region. Key Points The nonlinear freshwater equation of state leads to creation of a denser mass of water by mixing lighter masses of water (called cabbeling) Cabbeling generates a large scale instability that effectively entrains fluid in a parallel shear flow transporting heat downwards The cabbeling instability drives a competing process that also strengthens the stratification of the interface

Until recently, the lower to middle cloud region of Venus had been supposed to be dynamically quiet, accommodating nearly steady superrotating westward flow. However, observations of the regions by Akatsuki, the latest Venus orbiter operating since 2015, have revealed a variety of cloud features indicative of vortices and waves. Here we report another, and arguably the most conspicuous, example. Akatsuki's near‐infrared imager IR2 captured gigantic vortices rotating cyclonically on 25 August 2016. By using winds estimated by cloud tracking, the feature is shown to be quantitatively consistent with barotropic instability. The size of the vortices (∼1,000 km) and their spacing (∼2,500 km) are more than several times greater than the vortex‐like features reported previously from the observations of Venus, and they are also greater than the largest barotropic instability observed in the Earth's troposphere. Plain Language Summary Hydrodynamical instabilities play important roles in the general circulation of the Earth and planetary atmospheres. Barotropic instability is a kind of instability that arises from horizontal differences in predominantly parallel horizontal flows. We report herewith the first concrete evidence of its occurrence in the atmosphere of Venus. Before this study, reports are limited to vortex‐like cloud features whose appearance is consistent with this instability, but no analyses of flows have been conducted before. The cloud feature like a vortex street reported in this study has a spatial scale far greater than any of previously reported ones, and our study shows that it is dynamically consistent with barotropic instability. Key Points An event of barotropic instability, whose scale is greater than 1,000 km, was found by the Akatsuki Venus orbiter The discovery reinforces the recent view that the lower to middle cloud regions are dynamically active, promoting further studies

Suppose the observations of Lagrangian trajectories for fluid flow in some physical situation can be modelled sufficiently accurately by a spatially correlated Itô stochastic process (with zero mean) obtained from data which is taken in fixed Eulerian space. Suppose we also want to apply Hamilton’s principle to derive the stochastic fluid equations for this situation. Now, the variational calculus for applying Hamilton’s principle requires the Stratonovich process, so we must transform from Itô noise in the data frame to the equivalent Stratonovich noise. However, the transformation from the Itô process in the data frame to the corresponding Stratonovich process shifts the drift velocity of the transformed Lagrangian fluid trajectory out of the data frame into a non-inertial frame obtained from the Itô correction. The issue is, ‘Will non-inertial forces arising from this transformation of reference frames make a difference in the interpretation of the solution behaviour of the resulting stochastic equations?’ This issue will be resolved by elementary considerations.

Inspired by spatiotemporal observations from satellites of the trajectories of objects drifting near the surface of the ocean in the National Oceanic and Atmospheric Administration’s “Global Drifter Program”, this paper develops data-driven stochastic models of geophysical fluid dynamics (GFD) with non-stationary spatial correlations representing the dynamical behaviour of oceanic currents. Three models are considered. Model 1 from Holm (Proc R Soc A 471:20140963, 2015 ) is reviewed, in which the spatial correlations are time independent. Two new models, called Model 2 and Model 3, introduce two different symmetry breaking mechanisms by which the spatial correlations may be advected by the flow. These models are derived using reduction by symmetry of stochastic variational principles, leading to stochastic Hamiltonian systems, whose momentum maps, conservation laws and Lie–Poisson bracket structures are used in developing the new stochastic Hamiltonian models of GFD.

Hypergravity can be realized by creating a field imposed by centripetal acceleration in a centrifuge apparatus. Such an apparatus is often used to test soil response in geotechnical engineering problems. Here we present the potential usage of a centrifuge apparatus to study various topics in hydrodynamics. The scaling law associated with hydrodynamics is first reviewed, and the advantage of controlling the body force is described. One of the perceived disadvantages in such experiments is the unwanted presence of the Coriolis effect in the centrifuge. However, we propose exploiting this effect to our advantage to study geophysical fluid-dynamic problems that occur particularly in the equatorial region.

Dynamical vectors characterizing instability and applicable as ensemble perturbations for prediction with geophysical fluid dynamical models are analysed. The relationships between covariant Lyapunov vectors (CLVs), orthonormal Lyapunov vectors (OLVs), singular vectors (SVs), Floquet vectors and finite-time normal modes (FTNMs) are examined for periodic and aperiodic systems. In the phase-space of FTNM coefficients, SVs are shown to equate with unit norm FTNMs at critical times. In the long-time limit, when SVs approach OLVs, the Oseledec theorem and the relationships between OLVs and CLVs are used to connect CLVs to FTNMs in this phase-space. The covariant properties of both the CLVs, and the FTNMs, together with their phase-space independence, and the norm independence of global Lyapunov exponents and FTNM growth rates, are used to establish their asymptotic convergence. Conditions on the dynamical systems for the validity of these results, particularly ergodicity, boundedness and non-singular FTNM characteristic matrix and propagator, are documented. The findings are deduced for systems with nondegenerate OLVs, and, as well, with degenerate Lyapunov spectrum as is the rule in the presence of waves such as Rossby waves. Efficient numerical methods for the calculation of leading CLVs are proposed. Norm independent finite-time versions of the Kolmogorov-Sinai entropy production and Kaplan-Yorke dimension are presented.