Quasigeostrophic Turbulence Research Articles

Near-inertial waves and geostrophic turbulence

The two-mode quasigeostrophic model and associated turbulence phenomenology is a holy grail of geophysical fluid dynamics. An illustration of how wind-generated near-inertial waves in the ocean modify the quasigeostrophic turbulence phenomenology is presented.

Role of Surface-Layer Coherent Eddies in Potential Vorticity Transport in Quasigeostrophic Turbulence Driven by Eastward Shear

The transport by materially coherent surface-layer eddies was studied in a two-layer quasigeostrophic model driven by eastward mean shear. The coherent eddies were identified by closed contours of the Lagrangian-averaged vorticity deviation obtained from Lagrangian particles advected by the flow. Attention was restricted to eastward mean flows, but a wide range of flow regimes with different bottom friction strengths, layer thickness ratios, and background potential vorticity (PV) gradients were otherwise considered. It was found that coherent eddies become more prevalent and longer-lasting as the strength of bottom drag increases and the stratification becomes more surface-intensified. The number of coherent eddies is minimal when the shear-induced PV gradient is 10–20 times the planetary PV gradient and increases for both larger and smaller values of the planetary PV gradient. These coherent eddies, with an average core radius close to the deformation radius, propagate meridionally with a preference for cyclones to propagate poleward and anticyclones to propagate equatorward. The meridional propagation preference of the coherent eddies gives rise to a systematic upgradient PV transport, which is in the opposite direction as the background PV transport and not captured by standard Lagrangian diffusivity estimates. The upgradient PV transport by coherent eddy cores is less than 15% of the total PV transport, but the PV transport by the periphery flow induced by the PV inside coherent eddies is significant and downgradient. These results clarify the distinct roles of the trapping and stirring effect of coherent eddies in PV transport in geophysical turbulence.

On energy exchanges between eddies and the mean flow in quasigeostrophic turbulence

We study the term in the eddy energy budget of continuously stratified quasigeostrophic turbulence that is responsible for energy extraction by eddies from the background mean flow. This term is a quadratic form, and we derive Euler–Lagrange equations describing its eigenfunctions and eigenvalues, the former being orthogonal in the energy inner product and the latter being real. The eigenvalues correspond to the instantaneous energy growth rate of the associated eigenfunction. We find analytical solutions in the Eady problem. We formulate a spectral method for computing eigenfunctions and eigenvalues, and compute solutions in Phillips-type and Charney-type problems. In all problems, instantaneous growth is possible at all horizontal scales in both inviscid problems and in problems with linear Ekman friction. We conjecture that transient growth at small scales is matched by linear transfer to decaying modes with the same horizontal structure, and we provide simulations supporting the plausibility of this hypothesis. In Charney-type problems, where the linear problem has exponentially growing modes at small scales, we expect net energy extraction from the mean flow to be unavoidable, with an associated nonlinear transfer of energy to dissipation.

Spectra of atmospheric water in precipitating quasi-geostrophic turbulence

Atmospheric water has a complex behaviour partly due to the influence of precipitation. Consequently, it is challenging to explain properties of water such as the scale-dependence of its variance, for which a range of spectral exponents has been identified in observational data. Here, a precipitating quasi-geostrophic (PQG) model is explored as a possible prototype for contributing to understanding of water spectra, in an idealised setting. Geostrophic turbulence is examined in numerical simulations, where precipitation is included to explore its effect on the water spectrum, but where phase changes are neglected to allow corresponding theoretical analysis. The water spectral exponent is seen to range from approximately −1.4 to approximately −5 depending on the rainfall speed parameter, , which indicates a significant influence of precipitation on the water spectrum. The limiting values of this range are explained through asymptotic analyses for large and small values of . To obtain this theoretical understanding of the model, a key observation is that water can be written as a linear combination of two other tracers (equivalent potential temperature and a moist variable M), which themselves have theoretically tractable spectra. These two other tracers are linked to distinct modes of the PQG equations–the vortical mode and a moist mode – and the analysis here highlights the usefulness of wave or mode decompositions for understanding water in a saturated domain.

Nonintrusive reduced order modeling framework for quasigeostrophic turbulence.

In this study, we present a nonintrusive reduced order modeling (ROM) framework for large-scale quasistationary systems. The framework proposed herein exploits the time series prediction capability of long short-term memory (LSTM) recurrent neural network architecture such that (1) in the training phase, the LSTM model is trained on the modal coefficients extracted from the high-resolution data snapshots using proper orthogonal decomposition (POD) transform, and (2) in the testing phase, the trained model predicts the modal coefficients for the total time recursively based on the initial time history. Hence, no prior information about the underlying governing equations is required to generate the ROM. To illustrate the predictive performance of the proposed framework, the mean flow fields and time series response of the field values are reconstructed from the predicted modal coefficients by using an inverse POD transform. As a representative benchmark test case, we consider a two-dimensional quasigeostrophic ocean circulation model which, in general, displays an enormous range of fluctuating spatial and temporal scales. We first demonstrate that the conventional Galerkin projection-based reduced order modeling of such systems requires a high number of POD modes to obtain a stable flow physics. In addition, ROM-Galerkin projection (ROM-GP) does not seem to capture the intermittent bursts appearing in the dynamics of the first few most energetic modes. However, the proposed nonintrusive ROM framework based on LSTM (ROM-LSTM) yields a stable solution even for a small number of POD modes. We also observe that the ROM-LSTM model is able to capture quasiperiodic intermittent bursts accurately, and yields a stable and accurate mean flow dynamics using the time history of a few previous time states, denoted as the lookback time window in this paper. We show several features of ROM-LSTM framework such as significantly higher accuracy than ROM-GP, and faster performance using larger time step size. Throughout the paper, we demonstrate our findings in terms of time series evolution of the field values and mean flow patterns, which suggest that the proposed fully nonintrusive ROM framework is robust and capable of predicting chaotic nonlinear fluid flows in an extremely efficient way compared to the conventional projection-based ROM framework.

Geophysical turbulence dominated by inertia–gravity waves

Recent evidence from both oceanic observations and global-scale ocean model simulations indicate the existence of regions where low-mode internal tidal energy dominates over that of the geostrophic balanced flow. Inspired by these findings, we examine the effect of the first vertical mode inertia–gravity waves on the dynamics of balanced flow using an idealized model obtained by truncating the hydrostatic Boussinesq equations on to the barotropic and the first baroclinic mode. On investigating the wave–balance turbulence phenomenology using freely evolving numerical simulations, we find that the waves continuously transfer energy to the balanced flow in regimes where the balanced-to-wave energy ratio is small, thereby generating small-scale features in the balanced fields. We examine the detailed energy transfer pathways in wave-dominated flows and thereby develop a generalized small Rossby number geophysical turbulence phenomenology, with the two-mode (barotropic and one baroclinic mode) quasi-geostrophic turbulence phenomenology being a subset of it. The present work therefore shows that inertia–gravity waves would form an integral part of the geophysical turbulence phenomenology in regions where balanced flow is weaker than gravity waves.

Relative particle dispersion in two-dimensional and quasi-geostrophic turbulence

In this paper, phenomenological developments are used to explore relative particle dispersion (RPD) in two-dimensional (2D) and quasi-geostrophic (QG) fully-developed turbulence (FDT). The role played by the 2D and QG FDT cascade physics underlying this process is given special attention. Prevalence of spatial intermittency effects in 2D FDT enstrophy cascade, however small, is shown to lead to a structural change in the RPD growth law, more specifically the development of power-law scaling of RPD; this corroborates the difficulty in observing Lin’s (1972) exponential scaling law in laboratory experiments (Jullien (2003)). QG effects are found to lead to an enhanced RPD in the baroclinic regime of the energy cascade, which seems to be traceable to a negative eddy-viscosity characterizing the latter regime. They are also found to lead to particle clumping in the baroclinic regime of the enstrophy cascade (this aspect appears to be associated with the tendency of divorticity sheets to occur near the vortex nulls where particle clumping is known, as per numerical simulations and laboratory experiments, to be favored to occur). These results are developed from the established scaling relations for 2D and QG FDT and are validated further via alternative dimensional/scaling developments for 2D and QG FDT similar to the one given for 3D FDT by Batchelor and Townsend (1956). The feasibility of spatial intermittency effects is underscored via the nonlinear scaling dependence of RPD on the enstrophy (or energy) dissipation rate.

A Hybrid Approach for Model Order Reduction of Barotropic Quasi-Geostrophic Turbulence

We put forth a robust reduced-order modeling approach for near real-time prediction of mesoscale flows. In our hybrid-modeling framework, we combine physics-based projection methods with neural network closures to account for truncated modes. We introduce a weighting parameter between the Galerkin projection and extreme learning machine models and explore its effectiveness, accuracy and generalizability. To illustrate the success of the proposed modeling paradigm, we predict both the mean flow pattern and the time series response of a single-layer quasi-geostrophic ocean model, which is a simplified prototype for wind-driven general circulation models. We demonstrate that our approach yields significant improvements over both the standard Galerkin projection and fully non-intrusive neural network methods with a negligible computational overhead.

Ensemble Spread Grows More Rapidly in Higher-Resolution Simulations of Deep Convection

Idealized ensemble simulations of mesoscale convective systems (MCSs) with horizontal grid spacings of 1, 1.4, and 2 km are used to analyze the influence of numerical resolution on the rate of growth of ensemble spread in convection-resolving numerical models. The ensembles are initialized with random phases of 91-km-wavelength moisture perturbations that are captured with essentially identical accuracy at all resolutions. The rate of growth of ensemble variance is shown to systematically increase at higher resolution. The largest horizontal wavelength at which the perturbation kinetic energy (KE′) grows to at least 50% of the background kinetic energy spectrum is also shown to grow more rapidly at higher resolution. The mechanism by which the presence of smaller scales accelerates the upscale growth of KE′ is clear-cut in the smooth-saturation Lorenz–Rotunno–Snyder (ssLRS) model of homogeneous surface quasigeostrophic turbulence. Comparing the growth of KE′ from the MCS ensemble simulations to that in the ssLRS model suggests interactions between perturbations at small scales, where KE′ is not yet completely saturated, and somewhat larger scales, where KE′ is clearly unsaturated, are responsible for the faster growth rate of ensemble variance at finer resolution. These results provide some empirical justification for the use of deep-convection-related stochastic parameterization schemes to reduce the problem of underdispersion in coarser-resolution ensemble prediction systems.

Log-Normal Turbulence Dissipation in Global Ocean Models.

Data from turbulent numerical simulations of the global ocean demonstrate that the dissipation of kinetic energy obeys a nearly log-normal distribution even at large horizontal scales O(10 km). As the horizontal scales of resolved turbulence are larger than the ocean is deep, the Kolmogorov-Yaglom theory for intermittency in 3D homogeneous, isotropic turbulence cannot apply; instead, the down-scale potential enstrophy cascade of quasigeostrophic turbulence should. Yet, energy dissipation obeys approximate log-normality-robustly across depths, seasons, regions, and subgrid schemes. The distribution parameters, skewness and kurtosis, show small systematic departures from log-normality with depth and subgrid friction schemes. Log-normality suggests that a few high-dissipation locations dominate the integrated energy and enstrophy budgets, which should be taken into account when making inferences from simplified models and inferring global energy budgets from sparse observations.

Random forcing of geostrophic motion in rotating stratified turbulence

Random forcing of geostrophic motion is a common approach in idealized simulations of rotating stratified turbulence. Such forcing represents the injection of energy into large-scale balanced motion, and the resulting breakdown of quasi-geostrophic turbulence into inertia–gravity waves and stratified turbulence can shed light on the turbulent cascade processes of the atmospheric mesoscale. White noise forcing is commonly employed, which excites all frequencies equally, including frequencies much higher than the natural frequencies of large-scale vortices. In this paper, the effects of these high frequencies in the forcing are investigated. Geostrophic motion is randomly forced with red noise over a range of decorrelation time scales τ, from a few time steps to twice the large-scale vortex time scale. It is found that short τ (i.e., nearly white noise) results in about 46% more gravity wave energy than longer τ, despite the fact that waves are not directly forced. We argue that this effect is due to wave–vortex interactions, through which the high frequencies in the forcing are able to excite waves at their natural frequencies. It is concluded that white noise forcing should be avoided, even if it is only applied to the geostrophic motion, when a careful investigation of spontaneous wave generation is needed.

Solutions of quasi-geostrophic turbulence in multi-layered configurations

We consider quasi-geostrophic (QG) models in two- and three-layers that are useful in theoretical studies of planetary atmospheres and oceans. In these models, the streamfunctions are given by (1+2) partial differential systems of evolution equations. A two-layer QG model, in a simplified version, is dependent exclusively on the Rossby radius of deformation. However, the f-plane QG point vortex model contains factors such as the density, thickness of each layer, the Coriolis parameter, and the constant of gravitational acceleration, and this two-layered model admits a lesser number of Lie point symmetries, as compared to the simplified model. Finally, we study a three-layer oceanography QG model of special interest, which includes asymmetric wind curl forcing or Ekman pumping, that drives double-gyre ocean circulation. In three-layers, we obtain solutions pertaining to the wind-driven doublegyre ocean flow for a range of physically relevant features, such as lateral friction and the analogue parameters of the f-plane QG model. Zero-order invariants are used to reduce the partial differential systems to ordinary differential systems. We determine conservation laws for these QG systems via multiplier methods.

Wavenumber-frequency analysis of single-layer shallow-water beta-plane quasi-geostrophic turbulence

We numerically investigate single-layer shallow-water beta-plane quasi-geostrophic turbulence in a doubly periodic domain with emphasis on wavenumber-frequency spectra. We conduct a broad parameter sweep, varying the deformation radius (Ld), the narrow-band forcing wavenumber (kf), and the meridional gradient of the Coriolis parameter (β). Out of 54 simulations we present ten in detail spanning slowly propagating vortices to strong jets. We define a nondimensional parameter γβ in terms of β, Ld, and the energy injection rate. The moderately low γβ case is characterized by westward propagating coherent vortices and zonal wavenumber-frequency spectra dominated by a nondispersive line (NDL) corresponding to uniform propagation at or near the long-wave Rossby speed. The moderately high γβ case is characterized by jets, and the NDL persists even when there are no coherent vortices. The jets have large meridional excursions (meanders) that propagate westward nearly uniformly at a speed slower than the long-wave Rossby speed. Also at moderately high γβ, a second dispersion relation appears, roughly corresponding to linear waves on a zonal potential vorticity (PV) staircase. At very high γβ, during the slow evolution to a PV staircase, the structure of the linear waves is altered by the small perturbations to a constant potential vorticity gradient. A simple model treating the small perturbation as a sinusoid accurately predicts the meridional wavenumber-frequency spectra in the very high γβ simulations.

Impact of Horizontal Resolution (1/12° to 1/50°) on Gulf Stream Separation, Penetration, and Variability

AbstractThe impact of horizontal resolution (1/12° to 1/50°; 6 to 1.5 km at midlatitudes) on Gulf Stream separation, penetration, and variability is quantified in a series of identical North Atlantic experiments. The questions the authors seek to address are twofold: 1) Is the realism of the modeled solution increased as resolution is increased? 2) How robust is the modeled mesoscale and submesoscale eddy activity as a function of grid spacing and how representative is it of interior quasigeostrophic (QG) or surface quasigeostrophic (SQG) turbulence? This study shows that (i) the representation of Gulf Stream penetration and associated recirculating gyres shifts from unrealistic to realistic when the resolution is increased to 1/50° and when the nonlinear effects of the submesoscale eddies intensifies the midlatitude jet and increases its penetration eastward, (ii) the penetration into the deep ocean drastically increases with resolution and closely resembles the observations, and (iii) surface power spectra in the 70–250-km mesoscale range are independent of the horizontal resolution and of the latitude and are representative of 2D QG and SQG turbulence.

The Role of Rough Topography in Mediating Impacts of Bottom Drag in Eddying Ocean Circulation Models.

Motivated by the substantial sensitivity of eddies in two-layer quasi-geostrophic (QG) turbulence models to the strength of bottom drag, this study explores the sensitivity of eddies in more realistic ocean general circulation model (OGCM) simulations to bottom drag strength. The OGCM results are interpreted using previous results from horizontally homogeneous, two-layer, flat-bottom, f-plane, doubly periodic QG turbulence simulations and new results from two-layer β-plane QG turbulence simulations run in a basin geometry with both flat and rough bottoms. Baroclinicity in all of the simulations varies greatly with drag strength, with weak drag corresponding to more barotropic flow and strong drag corresponding to more baroclinic flow. The sensitivity of the baroclinicity in the QG basin simulations to bottom drag is considerably reduced, however, when rough topography is used in lieu of a flat bottom. Rough topography reduces the sensitivity of the eddy kinetic energy amplitude and horizontal length scales in the QG basin simulations to bottom drag to an even greater degree. The OGCM simulation behavior is qualitatively similar to that in the QG rough bottom basin simulations in that baroclinicity is more sensitive to bottom drag strength than are eddy amplitudes or horizontal length scales. Rough topography therefore appears to mediate the sensitivity of eddies in models to the strength of bottom drag. The sensitivity of eddies to parameterized topographic internal lee wave drag, which has recently been introduced into some OGCMs, is also briefly discussed. Wave drag acts like a strong bottom drag in that it increases the baroclinicity of the flow, without strongly affecting eddy horizontal length scales.

A novel dynamic framework for subgrid scale parametrization of mesoscale eddies in quasigeostrophic turbulent flows

This paper puts forth a modular dynamic subgrid scale modeling framework for large eddy simulation of quasigeostrophic turbulence based on minimizing the errors between structural and functional subgrid scale models. The approximate deconvolution (AD) procedure is used to estimate the free modeling parameters for the eddy viscosity coefficient parameterized in space and time using the Smagorinsky and Leith models. The novel idea here is to estimate the modeling parameters using the AD method rather than the traditionally used test filtering approach. First, a-priori and a-posteriori analyses are presented for solving a canonical homogeneous isotropic decaying turbulence problem. The proposed model is then applied to a wind-driven quasigeostrophic four-gyre ocean circulation problem, which is a standard prototype of more realistic ocean dynamics. Results show that the proposed model captures the quasi-stationary ocean dynamics and provides the time averaged four-gyre circulation patterns. Taking locally resolved flow characteristics into account, the model dynamically provides higher eddy viscosity values for lower resolutions.

Evaluation of scale-aware subgrid mesoscale eddy models in a global eddy-rich model

Two parameterizations for horizontal mixing of momentum and tracers by subgrid mesoscale eddies are implemented in a high-resolution global ocean model. These parameterizations follow on the techniques of large eddy simulation (LES). The theory underlying one parameterization (2D Leith due to Leith, 1996) is that of enstrophy cascades in two-dimensional turbulence, while the other (QG Leith) is designed for potential enstrophy cascades in quasi-geostrophic turbulence. Simulations using each of these parameterizations are compared with a control simulation using standard biharmonic horizontal mixing.Simulations using the 2D Leith and QG Leith parameterizations are more realistic than those using biharmonic mixing. In particular, the 2D Leith and QG Leith simulations have more energy in resolved mesoscale eddies, have a spectral slope more consistent with turbulence theory (an inertial enstrophy or potential enstrophy cascade), have bottom drag and vertical viscosity as the primary sinks of energy instead of lateral friction, and have isoneutral parameterized mesoscale tracer transport. The parameterization choice also affects mass transports, but the impact varies regionally in magnitude and sign.

A deformation-based parametrization of ocean mesoscale eddy reynolds stresses

Ocean mesoscale eddies strongly affect the strength and variability of large-scale ocean jets such as the Gulf Stream and Kuroshio Extension. Their spatial scales are too small to be fully resolved in many current climate models and hence their effects on the large-scale circulation need to be parametrized. Here we propose a parametrization of mesoscale eddy momentum fluxes based on large-scale flow deformation. The parametrization is argued to be suitable for use in eddy-permitting ocean general circulation models, and is motivated by an analogy between turbulence in Newtonian fluids (such as water) and laminar flow in non-Newtonian fluids. A primitive-equations model in an idealised double-gyre configuration at eddy-resolving horizontal resolution is used to diagnose the relationship between the proposed closure and the eddy fluxes resolved by the model. Favourable correlations suggest the closure could provide an appropriate deterministic parametrization of mesoscale eddies. The relationship between the closure and different representations of the Reynolds stress tensor is also described. The parametrized forcing possesses the key quasi-geostrophic turbulence properties of energy conservation and enstrophy dissipation, and allows for upgradient fluxes leading to the sharpening of vorticity gradients. The implementation of the closure for eddy-permitting ocean models requires only velocity derivatives and a single parameter that scales with model resolution.

A scale-aware subgrid model for quasi-geostrophic turbulence

A scale-aware subgrid model for quasi-geostrophic turbulence

Low-Dimensional Reduced-Order Models for Statistical Response and Uncertainty Quantification: Two-Layer Baroclinic Turbulence

Abstract Accurate uncertainty quantification for the mean and variance about forced responses to general external perturbations in the climate system is an important subject in understanding Earth’s atmosphere and ocean in climate change science. A low-dimensional reduced-order method is developed for uncertainty quantification and capturing the statistical sensitivity in the principal model directions with largest variability and in various regimes in two-layer quasigeostrophic turbulence. Typical dynamical regimes tested here include the homogeneous flow in the high latitudes and the anisotropic meandering jets in the low latitudes and/or midlatitudes. The idea in the reduced-order method is from a self-consistent mathematical framework for general systems with quadratic nonlinearity, where crucial high-order statistics are approximated by a systematic model calibration procedure. Model efficiency is improved through additional damping and noise corrections to replace the expensive energy-conserving nonlinear interactions. Model errors due to the imperfect nonlinear approximation are corrected by tuning the model parameters using linear response theory with an information metric in a training phase before prediction. Here a statistical energy principle is adopted to introduce a global scaling factor in characterizing the higher-order moments in a consistent way to improve model sensitivity. The reduced-order model displays uniformly high prediction skill for the mean and variance response to general forcing for both homogeneous flow and anisotropic zonal jets in the first 102 dominant low-wavenumber modes, where only about 0.15% of the total spectral modes are resolved, compared with the full model resolution of 2562 horizontal modes.