Parametrizations of subgrid-scale mountains are commonly used in numerical weather prediction and climate models. They try to represent quite separate processes; namely, the enhancement of the turbulent drag by orography, gravity-wave drag, and the effects of low-level flow blocking. Among the gravity wave schemes, some of them distinguish between upward-propagating waves and trapped lee waves. This article makes use of a recent theoretical methodology to propose a formalism that includes all these effects. This theory handles enhanced turbulent drag in the neutral case, gravity waves in the stratified case, and trapped lee waves in the transition. Mountain drag associated with all these processes is estimated analytically, as well as the fraction of the drag that stays within the boundary layer instead of being radiated in the far field. Although the theory used is adapted to gentle hills with small slope, we also try to evaluate the blocked layer depth by combining the sheltering effects that dominate when stratification is small and the blocking effects when stratification is strong.

We address the question of separating the ocean’s deterministic response to time-dependent forcing from its intrinsic chaotic variability. Ideally, one could compute the ensemble mean directly without performing numerous realizations, but this requires knowledge or closure of the second-order statistics — the classical turbulent-closure problem, here recast for a non-equilibrium, geophysical setting. Building on the ideas of nonlinear midlatitude ocean adjustment, we examine this problem using idealized quasi-geostrophic (QG) double-gyre ensembles subjected to episodic temporal variations in wind forcing. Our objective here is not to develop a subgrid parameterization of unresolved eddies, but rather to construct and test prognostic equations for the ensemble mean itself, using the simplest possible closure assumptions. We find that the performance of ensemble mean closures is highly dependent on the spatiotemporal structure of the forcing. Under slowly varying forcing, approximate closures reproduce the mean evolution reasonably well; under rapidly varying, near-zero-mean forcing, the simplest ensemble-mean closures fail, even at the level of basin-averaged total energy and enstrophy. In both regimes, the ensemble-mean response is not simply the accumulated imprint of the applied forcing, but instead appears as a continuing, non-equilibrated dialogue between the mean and eddy fields.

Increased upper-ocean stratification is an unavoidable consequence of global warming and will strongly impact the structure of ocean currents. Using a high-resolution ocean model, we show that intensification of stratification leads to the loss of coherence of the Gulf Stream Extension, replacing its steady eastward path with vigorous, chaotic meanders. This regime shift persists independently of changes in the Atlantic Meridional Overturning Circulation and surface wind forcing. Enhanced meandering under intensified stratification also proves to be a robust feature across both idealized and realistic ocean models that resolve mesoscale eddies, but is not captured by coarse-resolution models that parameterize eddies. The presented findings therefore highlight the need for improved representations of oceanic turbulence in climate projections.

Parameterizations of subgrid scale mountains are commonly used in large scale numerical weather prediction and climate models. They try to represent quite separate processes: the enhancement of the turbulent drag by orography, gravity waves and low level flow blocking. Among the gravity waves some schemes eventually separate between the upward propagating waves and the trapped lee waves. Using a recent theoretical methodology that addresses the interaction of stratified boundary layers with mountains, a theory that handles the transition from neutral to stratified dynamics and trapped waves, we propose a formalism that can include all these effects. As in most parameterizations it separates the flow between a linear part and a blocked part. Here the linear part handles enhanced turbulent drag in the neutral case and gravity waves in the stratified case, trapped lee waves in the transition. In this presentation we evaluate the mountain drag associated to all these processes as well as the fraction of the drag that stays within the boundary layer instead of being radiated in the far field. We also try to evaluate the blocked part by combining the sheltering effects that dominate when stratification is small and the blocking effects that dominate when stratification is large.

The representation of turbulent fluxes during oceanic convective events is important to capture the evolution of the oceanic mixed layer. To improve the accuracy of turbulent fluxes, we examine the possibility of adding a non-gradient component in their expression in addition to the usual downgradient part. To do so, we extend the $k-\varepsilon$ algebraic second-moment closure by relaxing the assumption on the equilibrium of the temperature variance $\overline{\theta’^2}$. With this additional transport equation for the temperature variance, we obtain a $k - \varepsilon - \overline{\theta’^2}$ model (the “$k \varepsilon t$” model) which includes a non-gradient term for the temperature flux. We validate this new model against Large Eddy Simulations (LES) in both wind-forced and buoyancy-driven regimes. In both cases, we find that the vertical profile of temperature is well captured by the $k \varepsilon t$ model. Particularly, for the buoyancy-driven regime, the non-gradient term increases the portion of the mixed layer that is stably stratified. This is an improvement since this portion is too small with the $k - \varepsilon$ parameterization. Finally, a comparison of the non-gradient term with the KPP non-local term gives insights for refining the KPP’s ad hoc shape polynomial.

Under wind forcing, the oceanic mixed layer deepens over time. Scaling laws exist to predict the deepening rate, but either neglect Earth's rotation or do not take into account the vertical structure of the mixed layer. This study aims to address these limitations by investigating the long-term dynamics of a wind-driven mixed layer in a rotating, linearly stratified fluid. We derive an analytical scaling law for the mixed layer depth h as a function of time t, \[ h \propto \frac{u*}{\sqrt{N_0f}} (tf)<^>{1/4} \] h proportional to u & lowast;N0f(tf)1/4 with $ N_0 $ N0 the initial stratification, f the rotation parameter and $ u_* $ u & lowast; the friction velocity. This law extends classical theories by considering rotational effects and is consistent with recent Large Eddy Simulations. We show that mixing, mostly concentrated within the entrainment layer, is primarily controlled by the stationary component of the velocity and that inertial oscillations have little effect on entrainment.

The reconstruction of deep ocean currents is a major challenge in data assimilation due to the scarcity of interior data. In this work, we present a proof of concept for deep ocean flow reconstruction using a Physics-Informed Neural Network (PINN), a machine learning approach that offers an alternative to traditional data assimilation methods. We introduce an efficient algorithm called StrAssPINN (for Stratified Assimilation PINNs), which assigns a separate network to each layer of the ocean model while allowing them to interact during training. The neural network takes spatiotemporal coordinates as input and predicts the velocity field at those points. Using a SIREN architecture (a multilayer perceptron with sine activation functions), which has proven effective in various contexts, the network is trained using both available observational data and dynamical priors enforced at several collocation points. We apply this method to pseudo-observed ocean data generated from a 3-layer quasi-geostrophic model, where the pseudo-observations include surface-level data akin to SWOT observations of sea surface height, interior data similar to ARGO floats, and a limited number of deep ARGO-like measurements in the lower layers. Our approach successfully reconstructs ocean flows in both the interior and surface layers, demonstrating a strong ability to resolve key ocean mesoscale features, including vortex rings, eastward jets associated with potential vorticity fronts, and smoother Rossby waves. This work serves as a prelude to applying StrAssPINN to real-world observational data.

The subtropical mode water in the North Atlantic, often referred to as ‘eighteen-degree water’ (EDW), has been investigated based on observational and theoretical studies. We here discuss the mechanism of EDW by using an ensemble-based approach which offers the advantage of separating the eddy field from the mean flow without making implicit assumptions on the temporal or spatial scales of the eddies. We employ an ensemble of North Atlantic Ocean simulations partially coupled with the atmosphere at mesoscale permitting resolution (1/12°), and determine EDW as a pool of the Ertel potential vorticity (PV) lower than the surroundings. Our results suggest that the maintenance of EDW can be explained by the down-gradient eddy PV fluxes balancing the mean flow: the low PV in the formation region is transported by the eddy fluxes to the pool and mixes with the surrounding high PV.

This study examines the role of stratification in the formation and persistence of eastward jets (like the Gulf Stream and the Kuroshio). Using a wind-driven, 2-layer quasigeostrophic model in a double-gyre configuration, we construct a phase diagram to classify flow regimes. The parameter space is defined by a criticality parameter , which controls the emergence of baroclinic instability, and the ratio of layer depths S, which describes the surface intensification of stratification. Eastward jets detaching from the western boundary are observed when S GG 1 and ;1, representing a regime transition from a vortex-dominated western boundary current to a zonostrophic regime characterized by multiple eastward jets. Remarkably, these surface-intensified patterns emerge without considering bottom friction. The emergence of the coherent eastward jet is further addressed with complementary 1.5-layer simulations and explained through both linear stability analysis and turbulence phenomenology. In particular, we show that coherent eastward jets emerge when the western boundary layer is stable and find that the asymmetry in the baroclinic instability of eastward and westward flows plays a central role in the persistence of eastward jets, while contributing to the disintegration of westward jets.

A theory for flow over gentle hills using a mixing-length turbulence closure is developed to describe the transition from turbulent orographic form drag to gravity wave drag. It confirms that the first is associated with downstream sheltering, and the second with upstream blocking and strong downslope winds. It shows that the altitude at which the incident flow needs to be taken to calculate the drag is the inner layer scale at which dissipation equilibrates disturbance advection. It also shows that the parameter that controls the transition, here a Richardson number, compares the mountain length with the altitude of the turning points above which the upward-propagating gravity waves become evanescent. Our solutions are also used to show that the downslope winds penetrate well into the inner layer and that a good fraction of the drag is deposited in the inner layer: all of it in the neutral case, a large fraction in the intermediate cases when there are trapped lee waves, and even in stable situations without trapping part of the gravity wave drag is eroded in the inner layer. Some discussion on how to combine neutral and stratified effects in the parametrization of subgrid scale orography in large-scale models is given. A theory for the interaction between a boundary layer and a low mountain is derived. The incident wind considered (U0$$ {U}_0 $$, left panel) presents a logarithmic profile near the surface. The theory describes the transition from neutral to stratified flows, and the systems of mountain waves (upward propagating and trapped, see right panel) that develop during the transition. The theory also reproduces the transition from downstream sheltering to downslope winds (zoom) as stratification increases. The mountain drag and Reynolds stress profiles are also discussed.image

The exploration of a two-dimensional wind-driven ocean model with no-slip boundaries reveals the existence of a turbulent asymptotic regime where energy dissipation becomes independent of fluid viscosity. This asymptotic flow represents an out-of-equilibrium state, characterized by a vigorous two-dimensional vortex gas superimposed onto a western-intensified gyre. The properties of the vortex gas are elucidated through scaling analysis for detached Prandtl boundary layers, providing a rationalization for the observed anomalous dissipation. The asymptotic regime demonstrates that boundary instabilities alone can be strong enough to evacuate wind-injected energy from the large-scale oceanic circulation.

We examine the ocean energy cycle where the eddies are defined about the ensemble mean of a partially air-sea coupled, eddy-rich ensemble simulation of the North Atlantic. The decomposition about the ensemble mean leads to a parameter-free definition of eddies, which is interpreted as the expression of oceanic chaos. Using the ensemble framework, we define the reservoirs of mean and eddy kinetic energy (MKE and EKE respectively) and mean total dynamic enthalpy (MTDE). We opt for the usage of dynamic enthalpy (DE) as a proxy for potential energy due to its dynamically consistent relation to hydrostatic pressure in Boussinesq fluids and non-reliance on any reference stratification. The curious result that emerges is that the potential energy reservoir cannot be decomposed into its mean and eddy components, and the eddy flux of DE can be absorbed into the EKE budget as pressure work.We find from the energy cycle that while baroclinic instability, associated with a positive vertical eddy buoyancy flux, tends to peak around February, EKE takes its maximum around September in the wind-driven gyre. Interestingly, the energy input from MKE to EKE, a process sometimes associated with barotropic processes, becomes larger than the vertical eddy buoyancy flux towards the summer and autumn. Our results question the common notion that the inverse energy cascade of winter-time EKE energized by baroclinic instability within the mixed layer is solely responsible for the summer-to-autumn peak in EKE, and suggest that the non-local eddy transport of DE and local transfer of energy from MKE to EKE could also contribute to the seasonal EKE maxima.

The exploration of a two-dimensional wind-driven ocean model with no-slip boundaries reveals the existence of a turbulent asymptotic regime where energy dissipation becomes independent of fluid viscosity. This asymptotic flow represents an out-of-equilibrium state, characterized by a vigorous two-dimensional vortex gas superimposed onto a westernintensified gyre. The properties of the vortex gas are elucidated through scaling analysis for detached Prandtl boundary layers, providing a rationalization for the observed anomalous dissipation. The asymptotic regime demonstrates that boundary instabilities alone can be strong enough to evacuate wind-injected energy from the large-scale oceanic circulation.

Abstract The ocean surface mixed layer plays a crucial role as an entry or exit point for heat, salt, momentum, and nutrients from the surface to the deep ocean. In this study, we introduce a framework to assess the evolution of the mixed layer depth (MLD) for realistic forcings and preconditioning conditions. Our approach involves a physically‐based parameter space defined by three dimensionless numbers: λs representing the relative contribution of the buoyancy flux and the wind stress at the air‐sea interface, Rh the Richardson number which characterizes the stability of the water column relative to the wind shear, and f/Nh which characterizes the importance of the Earth's rotation (ratio of the Coriolis frequency f and the pycnocline stratification Nh). Four MLD evolution regimes (“restratification,” “stable,” “deepening,” and “strong deepening”) are defined based on the values of the normalized temporal evolution of the MLD. We evaluate the 3D parameter space in the context of 1D simulations and we find that considering only the two dimensions (λs, Rh) is the best choice of 2D projection of this 3D parameter space. We then demonstrate the utility of this two‐dimensional λs − Rh parameter space to compare 3D realistic ocean simulations: we discuss the impact of the horizontal resolution (1°, 1/12°, or 1/60°) and the Gent‐McWilliams parameterization on MLD evolution regimes. Finally, a proof of concept of using observational data as a truth indicates how the parameter space could be used for model calibration.

Air-sea fl uxes are the main drivers of ocean circulation, yet their representation in ocean-only models remains challenging. While a zeroth-order formulation accounting only for the state of the atmosphere is well adopted by the community, surface ocean feedback has gained attention over the last decades. In this paper, we focus on thermodynamical indirect feedback of surface ocean currents, which completes the " eddy killing" " effect induced by the mechanical feedback. In this study, we quantify both the mechanical and thermodynamical contributions in the context of idealized, coupled quasigeostrophic simulations through sensitivity experiments on wind stress formulation. As compared to eddy killing which impacts kinetic energy levels, the indirect thermodynamical feedback induces significant fi cant changes in potential energy levels. The thermodynamical feedback also enhances by + 27% the potential-to-kinetic turbulent energy conversion induced by relative wind stress formulation, as well as significant fi cant changes in both forward and inverse cascades of potential energy (PE). That is, accounting for ocean surface currents in the computation of wind stress significantly fi cantly changes transfers of PE from the mean to the turbulent fl ow. These changes are mostly controlled by a reduced upscale energy fl ux rather than a more vigorous downscale fl ux, a process in line with results obtained for kinetic energy fl uxes associated with the eddy killing effect.

13 The mixed layer plays a crucial role as an entry or exit point for heat, salt, momen14 tum and nutrients from the surface to the deep ocean. We propose here a generic frame15 work to evaluate the mixed layer depth (MLD) dynamics across a wide range of forcings 16 and preconditioning conditions. To do so, we propose to use a physically-based param17 eter space formed by two dimensionless numbers: λs the relative contribution of the buoy18 ancy flux and the wind in the surface layer, and the Richardson number Rh which char19 acterizes the stability of the water column at the mixed layer base. Four MLD dynam20 ics regimes (”restratification”, ”stable”, ”deepening” and ”strong deepening”) are de21 fined based on the values of the normalized temporal evolution of the MLD. We show 22 the robustness and the predictive skill of the parameter space in a context of 1D sim23 ulations by showing that to a given (λs, Rh) corresponds a predictable MLD dynamics 24 regime. Finally, we present two applications showing how the parameter space can be 25 used with 3D ocean realistic models. We discuss the impact of the horizontal resolution 26 (1°, 1/12° or 1/60°) and the Gent McWilliams parameterization on the MLD dynamics 27 regimes. 28 Plain Language Summary 29 Vertical mixing of water in the ocean occurs when cold air temperatures create dense 30 cold water at the surface that tends to sink in the ocean or when a strong wind induces 31 turbulence at the ocean surface. These processes mix heat and salt and create a layer 32 at the top of the ocean that has a uniform temperature and salinity and that is called 33 the ”mixed layer”. This mixed layer plays a fundamental role in the Earth climate sys34 tem, and the representation of its dynamics in ocean models hence needs to be assessed. 35 For this purpose, we propose to map the mixed layer dynamics in a two dimensional space 36 where the first axis is related to the wind and the surface heat flux, and the second axis 37 to the stability of the water column. We show that this tool is robust and we give some 38 examples of application on how to use it in the context of realistic ocean models. 39

A wavelet-based method is re-introduced in an oceanographic and spectral context to estimate wavenumber spectrum and spectral flux of kinetic energy and enstrophy. We apply this to a numerical simulation of idealized, doubly-periodic quasi-geostrophic flows, i.e.~the flow is constrained by the Coriolis force and vertical stratification. The double periodicity allows for a straightforward Fourier analysis as the baseline method. Our wavelet spectra agree well with the canonical Fourier approach but with the additional strengths of negating the necessity for the data to be periodic and being able to extract local anisotropies in the flow. Caution is warranted, however, when computing higher-order quantities, such as spectral flux.

The anisotropic mesoscale eddy transport tensor is diagnosed using passive tracers advected in both an idealized 101-member mesoscale-resolving quasi-geostrophic (QG) double-gyre ensemble, and a realistic 24-member eddying ($1/12^\circ$) ensemble of the North Atlantic. We assert that the Reynold's decomposition along the ensemble dimension, rather than the spatial or temporal dimension, allows us to capture the intrinsic spatiotemporal variability of the mean flow and eddies.The tensor exhibits good performance in reconstructing the eddy fluxes of passive tracers, here defined as fluctuations about the ensemble thickness-weighted averaged (TWA) mean.However, the inability of the tensor to reconstruct eddy fluxes of QG potential vorticity, which encapsulates the eddy-mean flow interaction, and other active tracers raises the question: To what extent can the diagnosed tensor be applied to inform the parametrization of mesoscale dynamics?