Search

The North Atlantic Ocean circulation, fueled by winds and surface buoyancy fluxes, carries 1.25$\,$PettaWatts of heat poleward in the subtropics, and plays an important role in regulating global weather and climate patterns. Using a series of simulations with perturbed surface forcing, we study how winds and surface heat flux gradients affect the Atlantic meridional heat transport. We decompose the Atlantic meridional heat transport into contributions from circulation cells at warm and cold temperatures (resembling a subtropical gyre and the dense overturning circulation respectively), and a mixed circulation that contains water masses traversing both these cells. Variations in wind stress initially alter the amount of heat carried by the warm and mixed cells, but on long time scales ($>$10 years), changes in the temperature distribution restore the heat transport to equilibrium. Changes in surface buoyancy forcing control the cold cell's circulation, and its associated meridional heat transport, through high-latitude processes., Comment: 14 pages, 3 figures, submitted to the Geophysical Research Letters

Internal waves propagate on the ocean stratification and carry energy and momentum through the ocean interior. The two most significant sources of these waves in the ocean are surface winds and oscillatory tidal flow across topography. We propose a hybrid of these two mechanisms, in which wind induced oscillations of sea surface and isopycnal heights are rapidly communicated to the seafloor via hydrostatic pressure. In the presence of topography, the resulting oscillatory bottom velocity may then generate internal waves in a similar manner to the barotropic tide. We investigate this mechanism in an idealised numerical isopycnal model of a storm passing over a mid ocean ridge, and perform several perturbation experiments in which ocean and wind properties are varied. Bottom-generated internal waves are identified propagating away from the ridge in the wake of the storm. Estimates of the total wave energy suggest that in the right circumstances these waves could be a significant source of internal wave energy, with a local wind work to wave energy conversion rate of up to 50% of the corresponding conversion to surface generated near-inertial waves in our domain. Our results suggest a need for further investigation in less idealised scenarios to more precisely quantity this novel mechanism of deep ocean wave generation, and how it may affect abyssal mixing., Comment: submitted to the Journal of Physical Oceanography

We describe CATKE, a parameterization for fluxes associated with small-scale or "microscale" ocean turbulent mixing on scales between 1 and 100 meters. CATKE uses a downgradient formulation that depends on a prognostic turbulent kinetic energy (TKE) variable and a diagnostic mixing length scale that includes a dynamic convective adjustment (CA) component. With its dynamic convective mixing length, CATKE predicts not just the depth spanned by convective plumes but also the characteristic convective mixing timescale, an important aspect of turbulent convection not captured by simpler static convective adjustment schemes. As a result, CATKE can describe the competition between convection and other processes such as shear-driven mixing and baroclinic restratification. To calibrate CATKE, we use Ensemble Kalman Inversion to minimize the error between 21 large eddy simulations (LES) and predictions of the LES data by CATKE-parameterized single column simulations at three different vertical resolutions. We find that CATKE makes accurate predictions of both idealized and realistic LES compared to microscale turbulence parameterizations commonly used in climate models., Comment: submitted to J. Adv. Model. Earth Sy

The structure of the Antarctic Slope Current at the continental shelf is crucial in governing the poleward transport of warm water. Canyons on the continental slope may provide a pathway for warm water to cross the slope current and intrude onto the continental shelf underneath ice shelves, which can increase rates of ice shelf melting, leading to reduced buttressing of ice shelves, accelerating glacial flow and hence increased sea level rise. Observations and modelling studies of the Antarctic Slope Current and cross-shelf warm water intrusions are limited, particularly in the East Antarctica region. To explore this topic, an idealised configuration of the Antarctic Slope Current is developed, using an eddy-resolving isopycnal model that emulates the dynamics and topography of the East Antarctic sector. Warm water intrusions via canyons are found to occur in discrete episodes of large onshore flow induced by eddies, even in the absence of any temporal variability in external forcings, demonstrating the intrinsic nature of these intrusions to the slope current system. Canyon width is found to play a key role in modulating cross-shelf exchanges; warm water transport through narrower canyons is more irregular than transport through wider canyons. The intrinsically episodic cross-shelf transport is found to be driven by feedbacks between wind energy input and eddy generation in the Antarctic Slope Current. Improved understanding of the intrinsic variability of warm water intrusions can help guide future observational and modelling studies in the analysis of eddy impacts on Antarctic shelf circulation.

Gyres are central features of large-scale ocean circulation and are involved in transporting tracers such as heat, nutrients, and carbon-dioxide within and across ocean basins. Traditionally, the gyre circulation is thought to be driven by surface winds and quantified via Sverdrup balance, but it has been proposed that surface buoyancy fluxes may also contribute to gyre forcing. Through a series of eddy-permitting global ocean model simulations with perturbed surface forcing, the relative contribution of wind stress and surface heat flux forcing to the large-scale ocean circulation is investigated, focusing on the subtropical gyres. In addition to gyre strength being linearly proportional to wind stress, it is shown that the gyre circulation is strongly impacted by variations in the surface heat flux (specifically, its meridional gradient) through a rearrangement of the ocean's buoyancy structure. On shorter timescales ($\sim$ decade), the gyre circulation anomalies are proportional to the magnitude of the surface heat flux gradient perturbation, with up to $\sim 0.15\,\mathrm{Sv}$ anomaly induced per $\mathrm{W}\,\mathrm{m}^{-2}$ change in the surface heat flux. On timescales longer than a decade, the gyre response to surface buoyancy flux gradient perturbations becomes non-linear as ocean circulation anomalies feed back onto the buoyancy structure induced by the surface buoyancy fluxes. These interactions complicate the development of a buoyancy-driven theory for the gyres to complement the Sverdrup relation. The flux-forced simulations underscore the importance of surface buoyancy forcing in steering the large-scale ocean circulation., Comment: Submitted to the Journal of Physical Oceanography

We report a numerical study of horizontal convection (HC) at Prandtl number $Pr = 1$, with both no-slip and free-slip boundary conditions. We obtain 2D and 3D solutions and determine the relation between the Rayleigh number $Ra$ and the Nusselt number $Nu$. In 2D we vary $Ra$ between $0$ and $10^{14}$. In the range $10^6 \le Ra \le 10^{10}$ the $Nu$-$Ra$ relation is $Nu \sim Ra^{1/5}$. With $Ra$ greater than about $10^{11}$ we find a 2D regime with $Nu \sim Ra^{1/4}$ over three decades, up to the highest 2D $Ra$. In 3D, with maximum $Ra = 10^{11.5}$, we find only $Nu \sim Ra^{1/5}$. These results apply to both free slip and no slip boundary conditions. The $Nu \sim Ra^{1/4}$ regime has a double boundary layer (BL): there is a thin BL with thickness $\sim Ra^{-1/4}$ nested inside a thicker BL with thickness $\sim Ra^{-1/5}$. The $Ra^{-1/4}$ BL thickness, which determines $Nu$, coincides with the Kolmogorov and Batchelor scales of HC. Numerical and theoretical results indicate that 3D HC is qualitatively and quantitatively similar to 2D HC. At the same $Ra$, the 3D $Nu$ exceeds the 2D $Nu$ by less than $20$%, i.e., there is very little 3D enhancement of heat transport. Boundary conditions are more important than dimensionality: the 2D free-slip solutions have larger $Nu$ than 3D no-slip solutions. Using the mechanical energy power integral of HC we show that the mean square vorticity of 3D HC is nearly equal to that of 2D HC at the same $Ra$. Thus vorticity amplification by strain-mediated vortex stretching does not operate in 3D HC., Comment: 32 pages, 12 figures, submitted to the Journal of Fluid Mechanics

Regularising the parameter matrices of neural networks is ubiquitous in training deep models. Typical regularisation approaches suggest initialising weights using small random values, and to penalise weights to promote sparsity. However, these widely used techniques may be less effective in certain scenarios. Here, we study the Koopman autoencoder model which includes an encoder, a Koopman operator layer, and a decoder. These models have been designed and dedicated to tackle physics-related problems with interpretable dynamics and an ability to incorporate physics-related constraints. However, the majority of existing work employs standard regularisation practices. In our work, we take a step toward augmenting Koopman autoencoders with initialisation and penalty schemes tailored for physics-related settings. Specifically, we propose the "eigeninit" initialisation scheme that samples initial Koopman operators from specific eigenvalue distributions. In addition, we suggest the "eigenloss" penalty scheme that penalises the eigenvalues of the Koopman operator during training. We demonstrate the utility of these schemes on two synthetic data sets: a driven pendulum and flow past a cylinder; and two real-world problems: ocean surface temperatures and cyclone wind fields. We find on these datasets that eigenloss and eigeninit improves the convergence rate by up to a factor of 5, and that they reduce the cumulative long-term prediction error by up to a factor of 3. Such a finding points to the utility of incorporating similar schemes as an inductive bias in other physics-related deep learning approaches., Comment: 18 pages

Studies of ocean surface transport often invoke the "Eulerian-mean hypothesis": that wave-agnostic general circulation models neglecting explicit surface waves effects simulate the Eulerian-mean ocean velocity time-averaged over surface wave oscillations. Acceptance of the Eulerian-mean hypothesis motivates reconstructing the total, Lagrangian-mean surface velocity by adding Stokes drift to model output. Here, we show that the Eulerian-mean hypothesis is inconsistent, because wave-agnostic models cannot accurately simulate the Eulerian-mean velocity if Stokes drift is significant compared to the Eulerian-mean or Lagrangian-mean velocity. We conclude that Stokes drift should not be added to ocean general circulation model velocities. We additionally show the viability of the alternative "Lagrangian-mean hypothesis" using a theoretical argument and by comparing a wave-agnostic global ocean simulation with an explicitly wave-averaged simulation. We find that our wave-agnostic model accurately simulates the Lagrangian-mean velocity even though the Stokes drift is significant., Comment: 13 pages, 3 figures

Arctic icebergs, unconstrained sea ice floes, oil slicks, mangrove drifters, lost cargo containers, and other flotsam are known to move at 2-4% of the prevailing wind velocity relative to the water, despite vast differences in the material properties, shapes, and sizes of objects. Here, we revisit the roles of density, aspect ratio, and skin and form drag in determining how an object is driven by winds and water currents. Idealized theoretical considerations show that although substantial differences exist for end members of the parameter space (either very thin or thick and very light or dense objects), most realistic cases of floating objects drift at $\approx$3% of the free-stream wind velocity (measured outside an object's surface boundary layer). This relationship, known as a long-standing rule of thumb for the drift of various types of floating objects, arises from the square root of the ratio of the density of air to that of water. We support our theoretical findings with flume experiments using floating objects with a range of densities and shapes., Comment: accepted at the Journal of Physical Oceanography

Atmosphere and ocean are coupled via air-sea interactions. The atmospheric conditions fuel the ocean circulation and its variability, but the extent to which ocean processes can affect the atmosphere at decadal time scales remains unclear. In particular, such low-frequency variability is difficult to extract from the short observational record, meaning that climate models are the primary tools deployed to resolve this question. Here, we assess how the ocean's intrinsic variability leads to patterns of upper-ocean heat content that vary at decadal time scales. These patterns have the potential to feed back on the atmosphere and thereby affect climate modes of variability, such as El Ni\~no or the Interdecadal Pacific Oscillation. We use the output from a global ocean-sea ice circulation model at three different horizontal resolutions, each driven by the same atmospheric reanalysis. To disentangle the variability of the ocean's direct response to atmospheric forcing from the variability due to intrinsic ocean dynamics, we compare model runs driven with inter-annually varying forcing (1958-2018) and model runs driven with repeat-year forcing. Models with coarse resolution that rely on eddy parameterizations, show (i) significantly reduced variance of the upper-ocean heat content at decadal time scales and (ii) differences in the spatial patterns of low-frequency variability compared with higher resolution models. Climate projections are typically done with general circulation models with coarse-resolution ocean components. Therefore, these biases affect our ability to predict decadal climate modes of variability and, in turn, hinder climate projections. Our results suggest that for improving climate projections, the community should move towards coupled climate models with higher oceanic resolution.

Despite the nonlinear nature of turbulence, there is evidence that part of the energy-transfer mechanisms sustaining wall turbulence can be ascribed to linear processes. The different scenarios stem from linear stability theory and comprise exponential instabilities, neutral modes, transient growth from non-normal operators, and parametric instabilities from temporal mean-flow variations, among others. These mechanisms, each potentially capable of leading to the observed turbulence structure, are rooted in theoretical and conceptual arguments. Whether the flow follows any or a combination of them remains elusive. Here, we evaluate the linear mechanisms responsible for the energy transfer from the streamwise-averaged mean-flow ($\bf U$) to the fluctuating velocities ($\bf u'$). We use cause-and-effect analysis based on interventions. This is achieved by direct numerical simulation of turbulent channel flows at low Reynolds number, in which the energy transfer from $\bf U$ to $\bf u'$ is constrained to preclude a targeted linear mechanism. We show that transient growth is sufficient for sustaining realistic wall turbulence. Self-sustaining turbulence persists when exponential instabilities, neutral modes, and parametric instabilities of the mean flow are suppressed. We further show that a key component of transient growth is the Orr/push-over mechanism induced by spanwise variations of the base flow. Finally, we demonstrate that an ensemble of simulations with various frozen-in-time $\bf U$ arranged so that only transient growth is active, can faithfully represent the energy transfer from $\bf U$ to $\bf u'$ as in realistic turbulence. Our approach provides direct cause-and-effect evaluation of the linear energy-injection mechanisms from $\bf U$ to $\bf u'$ in the fully nonlinear system and simplifies the conceptual model of self-sustaining wall turbulence.

Despite the nonlinear nature of wall turbulence, there is evidence that the energy-injection mechanisms sustaining wall turbulence can be ascribed to linear processes. The different scenarios stem from linear stability theory and comprise exponential instabilities from mean-flow inflection points, transient growth from non-normal operators, and parametric instabilities from temporal mean-flow variations, among others. These mechanisms, each potentially capable of leading to the observed turbulence structure, are rooted in simplified theories and conceptual arguments. Whether the flow follows any or a combination of them remains unclear. In the present study, we devise a collection of numerical experiments in which the Navier-Stokes equations are sensibly modified to quantify the role of the different linear mechanisms. This is achieved by direct numerical simulation of turbulent channel flows with constrained energy extraction from the streamwise-averaged mean-flow. We demonstrate that (i) transient growth alone is not sufficient to sustain wall turbulence and (ii) the flow remains turbulent when the exponential instabilities are suppressed. On the other hand, we show that (iii) transient growth combined with the parametric instability of the time-varying mean-flow is able to sustain turbulence., Comment: arXiv admin note: substantial text overlap with arXiv:1909.05490

We consider the problem of horizontal convection in which non-uniform buoyancy, $b_{\rm s}(x,y)$, is imposed on the top surface of a container and all other surfaces are insulating. Horizontal convection produces a net horizontal flux of buoyancy, $\mathbf{J}$, defined by vertically and temporally averaging the interior horizontal flux of buoyancy. We show that $\overline{\mathbf{J}\cdot\mathbf{\nabla}b_{\rm s}}=-\kappa\langle|\boldsymbol{\nabla}b|^2\rangle$; overbar denotes a space-time average over the top surface, angle brackets denote a volume-time average and $\kappa$ is the molecular diffusivity of buoyancy $b$. This connection between $\mathbf{J}$ and $\kappa\langle|\boldsymbol{\nabla}b|^2\rangle$ justifies the definition of the horizontal-convective Nusselt number, $Nu$, as the ratio of $\kappa \langle|\boldsymbol{\nabla}b|^2\rangle$ to the corresponding quantity produced by molecular diffusion alone. We discuss the advantages of this definition of $Nu$ over other definitions of horizontal-convective Nusselt number currently in use. We investigate transient effects and show that $\kappa \langle|\boldsymbol{\nabla}b|^2\rangle$ equilibrates more rapidly than other global averages, such as the domain averaged kinetic energy and bottom buoyancy. We show that $\kappa\langle|\boldsymbol{\nabla} b|^2\rangle$ is essentially the volume-averaged rate of Boussinesq entropy production within the enclosure. In statistical steady state, the interior entropy production is balanced by a flux of entropy through the top surface. This leads to an equivalent "surface Nusselt number", defined as the surface average of vertical buoyancy flux through the top surface times the imposed surface buoyancy $b_{\rm s}(x,y)$. In experiments it is likely easier to evaluate the surface entropy flux, rather than the volume integral of $|\mathbf{\nabla}b|^2$ demanded by $\kappa\langle|\mathbf{\nabla}b|^2\rangle$., Comment: 16 pages, 7 figures

Despite the nonlinear nature of wall turbulence, there is evidence that the mechanism underlying the energy transfer from the mean flow to the turbulent fluctuations can be ascribed to linear processes. One of the most acclaimed linear instabilities for this energy transfer is the modal growth of perturbations with respect to the streamwise-averaged flow (or streaks). Here, we devise a numerical experiment in which the Navier--Stokes equations are sensibly modified to suppress these modal instabilities. Our results demonstrate that wall turbulence is sustained with realistic mean and fluctuating velocities despite the absence of streak instabilities.

"Eddy saturation" is the regime in which the total time-mean volume transport of an oceanic current is relatively insensitive to the wind stress forcing and is often invoked as a dynamical description of Southern Ocean circulation. We revisit the problem of eddy saturation using a primitive-equations model in an idealized channel setup with bathymetry. We apply only mechanical wind stress forcing; there is no diapycnal mixing or surface buoyancy forcing. Our main aim is to assess the relative importance of two mechanisms for producing eddy saturated states: (i) the commonly invoked baroclinic mechanism that involves the competition of sloping isopycnals and restratification by production of baroclinic eddies, and (ii) the barotropic mechanism, that involves production of eddies through lateral shear instabilities or through the interaction of the barotropic current with bathymetric features. Our results suggest that the barotropic flow-component plays a crucial role in determining the total volume transport., Comment: 13 pages, 4 figures

The mesoscale eddy field plays a key role in the mixing and transport of physical and biological properties and redistribute energy budgets in the ocean. Eddy kinetic energy is commonly defined as the kinetic energy of the time-varying component of the velocity field. However, this definition contains all processes that vary in time, including coherent mesoscale eddies, jets, waves, and large-scale motions. The focus of this paper is on the eddy kinetic energy contained in coherent mesoscale eddies. We present a new method to decompose eddy kinetic energy into oceanic processes. The proposed method uses a new eddy-identification algorithm (TrackEddy). This algorithm is based on the premise that the sea level signature of a coherent eddy can be approximated as a Gaussian feature. The eddy Gaussian signature then allows for the calculation of kinetic energy of the eddy field through the geostrophic approximation. TrackEddy has been validated using synthetic sea surface height data, and then used to investigate trends of eddy kinetic energy in the Southern Ocean using Satellite Sea Surface Height anomaly (AVISO+). We detect an increasing trend of eddy kinetic energy associated with mesoscale eddies in the Southern Ocean. This trend is correlated with an increase of the coherent eddy amplitude and the strengthening of wind stress over the last two decades., Comment: 20 pages, 8 figures; TrackEddy available at https://github.com/josuemtzmo/trackeddy

We study the mechanism of energy injection from the mean flow to the fluctuating velocity necessary to maintain wall turbulence. This process is believed to be correctly represented by the linearized Navier--Stokes equations, and three potential linear mechanisms have been considered, namely, modal instability of the streamwise mean cross-flow $U(y,z,t)$, non-modal transient growth, and non-modal transient growth supported by parametric instability. We have designed three numerical experiments of plane turbulent channel flow with additional forcing terms aiming to neutralize one or various linear mechanisms for energy extraction. From our preliminary experiments, only cases with mean cross-flows capable of supporting modal instabilities were found to sustain turbulence. However, the question whether such a new turbulence complies with the same physical mechanisms as those occurring in actual (unforced) turbulence remains unanswered. On the other hand, cases exclusively supported by transient growth decayed until laminarization., Comment: Corrected typo in author name

Magnetic induction in magnetohydrodynamic fluids at magnetic Reynolds number (Rm) less than~1 has long been known to cause magnetic drag. Here, we show that when $\mathrm{Rm} \gg 1$ and the fluid is in a hydrodynamic-dominated regime in which the magnetic energy is much smaller than the kinetic energy, induction due to a mean shear flow leads to a magnetic eddy viscosity. The magnetic viscosity is derived from simple physical arguments, where a coherent response due to shear flow builds up in the magnetic field until decorrelated by turbulent motion. The dynamic viscosity coefficient is approximately $(B_p^2/2\mu_0) \tau_{\rm corr}$, the poloidal magnetic energy density multiplied by the correlation time. We confirm the magnetic eddy viscosity through numerical simulations of two-dimensional incompressible magnetohydrodynamics. We also consider the three-dimensional case, and in cylindrical or spherical geometry, theoretical considerations similarly point to a nonzero viscosity whenever there is differential rotation. Hence, these results serve as a dynamical generalization of Ferraro's law of isorotation. The magnetic eddy viscosity leads to transport of angular momentum and may be of importance to zonal flows in astrophysical domains such as the interior of some gas giants., Comment: 16 pages, 8 figures

Zonal flows in rotating systems have been previously shown to be suppressed by the imposition of a background magnetic field aligned with the direction of rotation. Understanding the physics behind the suppression may be important in systems found in astrophysical fluid dynamics, such as stellar interiors. However, the mechanism of suppression has not yet been explained. In the idealized setting of a magnetized beta plane, we provide a theoretical explanation that shows how magnetic fluctuations directly counteract the growth of weak zonal flows. Two distinct calculations yield consistent conclusions. The first, which is simpler and more physically transparent, extends the Kelvin-Orr shearing wave to include magnetic fields and shows that weak, long-wavelength shear flow organizes magnetic fluctuations to absorb energy from the mean flow. The second calculation, based on the quasilinear, statistical CE2 framework, is valid for arbitrary wavelength zonal flow and predicts a self-consistent growth rate of the zonal flow. We find that a background magnetic field suppresses zonal flow if the bare Alfven frequency is comparable to or larger than the bare Rossby frequency. However, suppression can occur for even smaller magnetic fields if the resistivity is sufficiently small enough to allow sizable magnetic fluctuations. Our calculations reproduce the $\eta/B_0^2 = \text{const.}$ scaling that describes the boundary of zonation, as found in previous work, and we explicitly link this scaling to the amplitude of magnetic fluctuations., Comment: 17 pages, 7 figures

Zonal jets in a barotropic setup emerge out of homogeneous turbulence through a flow-forming instability of the homogeneous turbulent state (`zonostrophic instability') which occurs as the turbulence intensity increases. This has been demonstrated using the statistical state dynamics (SSD) framework with a closure at second order. Furthermore, it was shown that for small supercriticality the flow-forming instability follows Ginzburg-Landau (G-L) dynamics. Here, the SSD framework is used to study the equilibration of this flow-forming instability for small supercriticality. First, we compare the predictions of the weakly nonlinear G-L dynamics to the fully nonlinear SSD dynamics closed at second order for a wide ranges of parameters. A new branch of jet equilibria is revealed that is not contiguously connected with the G-L branch. This new branch at weak supercriticalities involves jets with larger amplitude compared to the ones of the G-L branch. Furthermore, this new branch continues even for subcritical values with respect to the linear flow-forming instability. Thus, a new nonlinear flow-forming instability out of homogeneous turbulence is revealed. Second, we investigate how both the linear flow-forming instability and the novel nonlinear flow-forming instability are equilibrated. We identify the physical processes underlying the jet equilibration as well as the types of eddies that contribute in each process. Third, we propose a modification of the diffusion coefficient of the G-L dynamics that is able to capture the asymmetric evolution for weak jets at scales other than the marginal scale (side-band instabilities) for the linear flow-forming instability., Comment: 27 pages, 17 figures

"Eddy saturation" refers to a regime in which the total volume transport of an oceanic current is insensitive to the wind stress strength. Baroclinicity is currently believed to be key to the development of an eddy-saturated state. In this paper, it is shown that eddy saturation can also occur in a purely barotropic flow over topography, without baroclinicity. Thus, eddy saturation is a fundamental property of barotropic dynamics above topography. It is demonstrated that the main factor controlling the appearance or not of eddy-saturated states in the barotropic setting is the structure of geostrophic contours, that is the contours of $f/H$ of the ratio of the Coriolis parameter to the ocean's depth. Eddy-saturated states occur when the geostrophic contours are open, that is when the geostrophic contours span the whole zonal extent of the domain. This minimal requirement for eddy-saturated states is demonstrated using numerical integrations of a single-layer quasi-geostrophic flow over two different topographies characterized by either open or closed geostrophic contours with parameter values loosely inspired by the Southern Ocean. In this setting, transient eddies are produced through a barotropic-topographic instability that occurs due tot the interaction of the large-scale zonal flow with the topography. Through the study of this barotropic-topographic instability insight is gained on how eddy-saturated states are established., Comment: 14 pages, 8 figures, submitted to the Journal of Physical Oceanography

Using a one-layer QG model, we study the effect of random monoscale topography on forced beta-plane turbulence. The forcing is a uniform steady wind stress that produces both a uniform large-scale zonal flow $U(t)$ and smaller-scale macroturbulence (both standing and transient eddies). The flow $U(t)$ is retarded by Ekman drag and by the domain-averaged topographic form stress produced by the eddies. The topographic form stress typically balances most of the applied wind stress, while the Ekman drag provides all of the energy dissipation required to balance the wind work. A collection of statistically equilibrated solutions delineates the main flow regimes and the dependence of the time-mean $U$ on the problem parameters and the statistical properties of the topography. If $\beta$ is smaller than the topographic PV gradient then the flow consists of stagnant pools attached to pockets of closed geostrophic contours. The stagnant dead zones are bordered by jets and the flow through the domain is concentrated into a narrow channel of open geostrophic contours. If $\beta$ is comparable to, or larger than, the topographic PV gradient then all geostrophic contours are open and the flow is uniformly distributed throughout the domain. In this case there is an "eddy saturation" regime in which $U$ is insensitive to changes in the wind stress. We show that eddy saturation requires strong transient eddies that act as PV diffusion. This PV diffusion does not alter the energy of the standing eddies, but it does increase the topographic form stress by enhancing the correlation between topographic slope and the standing-eddy pressure field. Last, using bounds based on the energy and enstrophy we show that as the wind stress increases the flow transitions from a regime in which form stress balances the wind stress to a regime in which the form stress is very small and large transport ensues., Comment: 32 pages, 21 figures. Major revisions in the abstract, introduction and discussion and conclusion sections

The perspective of statistical state dynamics (SSD) has recently been applied to the study of mechanisms underlying turbulence in various physical systems. An example implementation of SSD is the second order closure referred to as stochastic structural stability theory (S3T), which has provided insight into the dynamics of wall turbulence and specifically the emergence and maintenance of the roll/streak structure. This closure eliminates nonlinear interactions among the perturbations has been removed, restricting nonlinearity in the dynamics to that of the mean equation and the interaction between the mean and perturbation covariance. Here, this quasi-linear restriction of the dynamics is used to study the structure and dynamics of turbulence in plane Poiseuille flow at moderately high Reynolds numbers in a closely related dynamical system, referred to as the restricted nonlinear (RNL) system. RNL simulations reveal that the essential features of wall-turbulence dynamics are retained. Remarkably, the RNL system spontaneously limits the support of its turbulence to a small set of streamwise Fourier components giving rise to a naturally minimal representation of its turbulence dynamics. Although greatly simplified, this RNL turbulence exhibits natural-looking structures and statistics. Surprisingly, even when further truncation of the perturbation support to a single streamwise component is imposed the RNL system continues to produce self-sustaining turbulent structure and dynamics. RNL turbulence at the Reynolds numbers studied is dominated by the roll/streak structure in the buffer layer and similar very-large-scale structure (VLSM) in the outer layer. Diagnostics of the structure, spectrum and energetics of RNL and DNS turbulence are used to demonstrate that the roll/streak dynamics supporting the turbulence in the buffer and logarithmic layer is essentially similar in RNL and DNS., Comment: published in J. Fluid Mech., 26 pages, 15 figures. arXiv admin note: text overlap with arXiv:1401.7816

Jets coexist with planetary scale waves in the turbulence of planetary atmospheres. The coherent component of these structures arises from cooperative interaction between the coherent structures and the incoherent small-scale turbulence in which they are embedded. It follows that theoretical understanding of the dynamics of jets and planetary scale waves requires adopting the perspective of statistical state dynamics (SSD) which comprises the dynamics of the interaction between coherent and incoherent components in the turbulent state. In this work the S3T implementation of SSD for barotropic beta-plane turbulence is used to develop a theory for the jet/wave coexistence regime by separating the coherent motions consisting of the zonal jets together with a selection of large-scale waves from the smaller scale motions which constitute the incoherent component. It is found that mean flow/turbulence interaction gives rise to jets that coexist with large-scale coherent waves in a synergistic manner. Large-scale waves that would only exist as damped modes in the laminar jet are found to be transformed into exponentially growing waves by interaction with the incoherent small scale turbulence which results in a change in the mode structure allowing the mode to tap the energy of the mean jet. This mechanism of destabilization differs fundamentally and serves to augment the more familiar S3T instabilities in which jets and waves arise from homogeneous turbulence with energy source exclusively from the incoherent eddy field and provides further insight into the cooperative dynamics of the jet/waves coexistence regime in planetary turbulence., Comment: 22 pages, 15 figures

This thesis presents a newly developed theory for the formation and maintenance of eddy-driven jets in planetary turbulence. The novelty is that jet formation and maintenance is studied as a dynamics of the statistics of the flow rather than a dynamics of individual realizations. This is pursued using Stochastic Structural Stability Theory (S3T), which studies the closed dynamics of the first two cumulants of the full statistical state dynamics of the flow after neglecting or parameterizing third and higher-order cumulants. Using this statistical closure, large-scale structure formation is studied in barotropic turbulence on a $\beta$-plane. It is demonstrated that at analytically predicted critical parameter values the homogeneous turbulent state undergoes a bifurcation becoming inhomogeneous with the emergence of large-scale zonal and/or non-zonal flows. The mechanisms by which the turbulent Reynolds stresses organize to reinforce infinitesimal mean flow inhomogeneities, thus leading to this statistical state instability, are studied for various regimes of parameter values. It is also demonstrated that the S3T instabilities equilibrate to turbulent states with finite-amplitude jets. Moreover, the relation between large-scale structure formation through modulational instability and the S3T instability of the homogeneous turbulent state is also investigated and it is shown that the modulational instability results are subsumed by the S3T results. Finally, the study of the S3T stability of inhomogeneous turbulent jet equilibria is presented and the relation with the phenomenon of jet merging is investigated. Although all these phenomena cannot be predicted by analysis of the dynamics of single realizations of the flow, it is demonstrated that the predictions of S3T are reflected in individual realizations of the flow., Comment: Ph.D. thesis, National and Kapodistrian University of Athens, Athens, 2015 (typos fixed; correction in figure 1.11)

Zonal jets and non-zonal large-scale flows are often present in forced-dissipative barotropic turbulence on a beta-plane. The dynamics underlying the formation of both zonal and non-zonal coherent structures is investigated in this work within the statistical framework of Stochastic Structural Stability Theory (S3T). Previous S3T studies have shown that the homogeneous turbulent state undergoes a bifurcation at a critical parameter and becomes inhomogeneous with the emergence of zonal and/or large-scale non-zonal flows and that these statistical predictions of S3T are reflected in direct numerical simulations. In this paper, we study the dynamics underlying the S3T statistical instability of the homogeneous state as a function of parameters. It is shown that for weak planetary vorticity gradient, $\beta$, both zonal jets and non-zonal large-scale structures form from upgradient momentum fluxes due to shearing of the eddies by the emerging infinitesimal flow. For large $\beta$, the dynamics of the S3T instability differs for zonal and non-zonal flows but in both the destabilizing vorticity fluxes decrease with increasing $\beta$. Shearing of the eddies by the mean flow continues to be the mechanism for the emergence of zonal jets while non-zonal large-scale flows emerge from resonant and near resonant triad interactions between the large-scale flow and the stochastically forced eddies. The relation between the formation of large-scale structure through modulational instability and the S3T instability of the homogeneous state is also investigated and it is shown that the modulational instability results are subsumed by the S3T results., Comment: 22 pages, 15 figures, submitted to the Journal of the Atmospheric Sciences

S3T (Stochastic Structural Stability Theory) employs a closure at second order to obtain the dynamics of the statistical mean turbulent state. When S3T is implemented as a coupled set of equations for the streamwise mean and perturbation states, nonlinearity in the dynamics is restricted to interaction between the mean and perturbations. The S3T statistical mean state dynamics can be approximately implemented by similarly restricting the dynamics used in a direct numerical simulation (DNS) of the full Navier-Stokes equations (referred to as the NS system). Although this restricted nonlinear system (referred to as the RNL system) is greatly simplified in its dynamics in comparison to the associated NS, it nevertheless self-sustains a turbulent state in wall-bounded shear flow with structures and dynamics comparable to that in observed turbulence. Moreover, RNL turbulence can be analyzed effectively using theoretical methods developed to study the closely related S3T system. In order to better understand RNL turbulence and its relation to NS turbulence, an extensive comparison is made of diagnostics of structure and dynamics in these systems. Although quantitative differences are found, the results show that turbulence in the RNL system closely parallels that in NS and suggest that the S3T/RNL system provides a promising reduced complexity model for studying turbulence in wall-bounded shear flows., Comment: 18 pages, 14 figures

Stochastic Structural Stability Theory (S3T) provides analytical methods for understanding the emergence and equilibration of jets from the turbulence in planetary atmospheres based on the dynamics of the statistical mean state of the turbulence closed at second order. Predictions for formation and equilibration of turbulent jets made using S3T are critically compared with results of simulations made using the associated quasi-linear and nonlinear models. S3T predicts the observed bifurcation behavior associated with the emergence of jets, their equilibration and their breakdown as a function of parameters. Quantitative differences in bifurcation parameter values between predictions of S3T and results of nonlinear simulations are traced to modification of the eddy spectrum which results from two processes: nonlinear eddy-eddy interactions and formation of discrete non-zonal structures. Remarkably, these non-zonal structures, which substantially modify the turbulence spectrum, are found to arise from S3T instability. Formation as linear instabilities and equilibration at finite amplitude of multiple equilibria for identical parameter values in the form of jets with distinct meridional wavenumbers is verified as is the existence at equilibrium of finite amplitude non-zonal structures in the form of nonlinearly modified Rossby waves. When zonal jets and nonlinearly modified Rossby waves coexist at finite amplitude the jet structure is generally found to dominate even if it is linearly less unstable. The physical reality of the manifold of S3T jets and non-zonal structures is underscored by the existence in nonlinear simulations of jet structure at subcritical S3T parameter values which are identified with stable S3T jet modes excited by turbulent fluctuations., Comment: 18 pages, 18 figures, submitted to the Journal of the Atmospheric Sciences

Highly stratified shear layers are rendered unstable even at high stratifications by Holmboe instabilities when the density stratification is concentrated in a small region of the shear layer. These instabilities may cause mixing in highly stratified environments. However these instabilities occur in limited bands in the parameter space. We perform Generalized Stability analysis of the two dimensional perturbation dynamics of an inviscid Boussinesq stratified shear layer and show that Holmboe instabilities at high Richardson numbers can be excited by their adjoints at amplitudes that are orders of magnitude larger than by introducing initially the unstable mode itself. We also determine the optimal growth that is obtained for parameters for which there is no instability. We find that there is potential for large transient growth regardless of whether the background flow is exponentially stable or not and that the characteristic structure of the Holmboe instability asymptotically emerges as a persistent quasi-mode for parameter values for which the flow is stable., Comment: 21 pages, 22 figures corrected typos, added references, added Appendix A, added discussion for numerical calculations, added discussion for quasi-mode