Surface Quasi-geostrophic Turbulence Research Articles

Nonequilibrium fluctuations of the direct cascade in surface quasi-geostrophic turbulence

Nonequilibrium fluctuations of the direct cascade in surface quasi-geostrophic turbulence

Characteristics of the Kinetic Energy Spectra in the Subpolar North Atlantic

Abstract The subpolar North Atlantic is known to have rich mesoscale and submesoscale variations; however, their spectral characteristics have not been documented in observations. This study documents the kinetic energy (KE) spectra using acoustic Doppler current profiler measurements that cover both the Iceland Basin and the Irminger Sea. The KE spectrum is partitioned into geostrophically balanced motions and unbalanced motions. The results reveal that balanced motions dominate the KE spectra. The unbalanced motions enhance in spring and fall to flatten the spectra and dominate small-scale (<50 km) energy, though uncertainty is high due to measurement noise and method assumptions. In addition, the dynamical framework that drives the balanced motions undergoes distinct seasonal shifts. In the spring and summer seasons of the Iceland Basin, as well as the summer season of the Irminger Sea, the wavenumber spectra of balanced motions exhibit a slope of approximately −3, consistent with the internal quasigeostrophic turbulence theory. Conversely, in the fall season of the Iceland Basin and the spring and fall seasons of the Irminger Sea, the wavenumber spectra of geostrophic balanced motions have a slope close to −2, consistent with surface quasigeostrophic turbulence theory. Additionally, we have found that the intensity of mesoscale eddies in the spring season can modulate both the slope and intensity of the wavenumber spectra of geostrophic balanced flows.

The buoyancy staircase limit in surface quasigeostrophic turbulence

Surface buoyancy gradients over a quasigeostrophic fluid permit the existence of surface-trapped Rossby waves. The interplay of these Rossby waves with surface quasigeostrophic turbulence results in latitudinally inhomogeneous mixing that, under certain conditions, culminates in a surface buoyancy staircase: a meridional buoyancy profile consisting of mixed zones punctuated by sharp buoyancy gradients, with eastward jets centred at the sharp gradients and weaker westward flows in between. In this article, we investigate the emergence of this buoyancy staircase limit in surface quasigeostrophic turbulence and we examine the dependence of the resulting dynamics on the vertical stratification. Over decreasing stratification ($\mathrm {d}N(z)/\mathrm {d}z\leq 0$, where$N(z)$is the buoyancy frequency), we obtain flows with a longer interaction range (than in uniform stratification) and highly dispersive Rossby waves. In the staircase limit, we find straight jets that are perturbed by eastward propagating along jet waves, similar to two-dimensional barotropic$\beta$-plane turbulence. In contrast, over increasing stratification ($\mathrm {d}N(z)/\mathrm {d}z\geq 0$), we obtain flows with shorter interaction range and weakly dispersive Rossby waves. In the staircase limit, we find sinuous jets with large latitudinal meanders whose shape evolves in time due to the westward propagation of weakly dispersive along jet waves. These along jet waves have larger amplitudes over increasing stratification than over decreasing stratification, and, as a result, the ratio of domain-averaged zonal to meridional speeds is two to three times smaller over increasing stratification than over decreasing stratification. Finally, we find that, for a given Rhines wavenumber, jets over increasing stratification are closer together than jets over decreasing stratification.

Surface Quasigeostrophic Turbulence in Variable Stratification

Abstract Numerical and observational evidence indicates that, in regions where mixed layer instability is active, the surface geostrophic velocity is largely induced by surface buoyancy anomalies. Yet, in these regions, the observed surface kinetic energy spectrum is steeper than predicted by uniformly stratified surface quasigeostrophic theory. By generalizing surface quasigeostrophic theory to account for variable stratification, we show that surface buoyancy anomalies can generate a variety of dynamical regimes depending on the stratification’s vertical structure. Buoyancy anomalies generate longer-range velocity fields over decreasing stratification and shorter-range velocity fields over increasing stratification. As a result, the surface kinetic energy spectrum is steeper over decreasing stratification than over increasing stratification. An exception occurs if the near-surface stratification is much larger than the deep-ocean stratification. In this case, we find an extremely local turbulent regime with surface buoyancy homogenization and a steep surface kinetic energy spectrum, similar to equivalent barotropic turbulence. By applying the variable stratification theory to the wintertime North Atlantic, and assuming that mixed layer instability acts as a narrowband small-scale surface buoyancy forcing, we obtain a predicted surface kinetic energy spectrum between k−4/3 and k−7/3, which is consistent with the observed wintertime k−2 spectrum. We conclude by suggesting a method of measuring the buoyancy frequency’s vertical structure using satellite observations.

Energy Spectra and Vorticity Dynamics in a Two-Layer Shallow Water Ocean Model

Abstract The dynamically adaptive WAVETRISK-OCEAN global model is used to solve one- and two-layer shallow water ocean models of wind-driven western boundary current (WBC) turbulence. When the submesoscale is resolved, both the one-layer simulation and the barotropic mode of the two-layer simulations have an energy spectrum with a power law of −3, while the baroclinic mode has a power law of −5/3 to −2 for a Munk boundary layer. This is consistent with the theoretical prediction for the power laws of the barotropic and baroclinic (buoyancy variance) cascades in surface quasigeostrophic turbulence. The baroclinic mode has about 20% of the energy of the barotropic mode in this case. When a Munk–Stommel boundary layer dominates, both the baroclinic and barotropic modes have a power law of −3. Local energy spectrum analysis reveals that the midlatitude and equatorial jets have different energy spectra and contribute differently to the global energy spectrum. We have therefore shown that adding a single baroclinic mode qualitatively changes WBC turbulence, introducing an energy spectrum component typical of what occurs in stratified three-dimensional ocean flows. This suggests that the first baroclinic mode may be primarily responsible for the submesoscale turbulence energy spectrum of the oceans. Adding more vertical layers, and therefore more baroclinic modes, could strengthen the first baroclinic mode, producing a dual cascade spectrum (−5/3, −3) or (−3, −5/3) similar to that predicted by quasigeostrophic and surface quasigeostrophic models, respectively. Significance Statement This research investigates how wind energy is transferred from the largest ocean scales (thousands of kilometers) to the small turbulence scales (a few kilometers or less). We do this by using an idealized model that includes the simplest representation of density stratification. Our main finding is that this simple model captures an essential feature of the energy transfer process. Future work will compare our results to those obtained using ocean models with more realistic stratifications.

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.

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.

Interpreting Energy and Tracer Spectra of Upper-Ocean Turbulence in the Submesoscale Range (1–200 km)

AbstractSubmesoscale (1–200 km) wavenumber spectra of kinetic and potential energy and tracer variance are obtained from in situ observations in the Gulf Stream region and in the eastern subtropical North Pacific. In the Gulf Stream region, steep kinetic energy spectra at scales between 200 and 20 km are consistent with predictions of interior quasigeostrophic–turbulence theory, both in the mixed layer and in the thermocline. At scales below 20 km, the spectra flatten out, consistent with a growing contribution of internal-wave energy at small scales. In the subtropical North Pacific, the energy spectra are flatter and inconsistent with predictions of interior quasigeostrophic–turbulence theory. The observed spectra and their dependence on depth are also inconsistent with predictions of surface quasigeostrophic–turbulence theory for the observed ocean stratification. It appears that unbalanced motions, most likely internal tides at large scales and the internal-wave continuum at small scales, dominate the energy spectrum throughout the submesoscale range. Spectra of temperature variance along density surfaces, which are not affected by internal tides, are also inconsistent with predictions of geostrophic-turbulence theories. Reasons for this inconsistency could be the injection of energy in the submesoscale range by small-scale baroclinic instabilities or modifications of the spectra by coupling between surface and interior dynamics or by ageostrophic frontal effects.

Functional renormalization-group approach to decaying turbulence

We reconsider the functional renormalization-group (FRG) approach to decaying Burgers turbulence, and extend it to decaying Navier–Stokes and surface quasi-geostrophic turbulence. The method is based on a renormalized small-time expansion, equivalent to a loop expansion, and naturally produces a dissipative anomaly and a cascade after a finite time. We explicitly calculate and analyze the one-loop FRG equations in the zero-viscosity limit as a function of the dimension. For Burgers turbulence they reproduce the FRG equation obtained in the context of random manifolds, extending previous results of one of us. Breakdown of energy conservation due to shocks and the appearance of a direct energy cascade corresponds to failure of dimensional reduction in the context of disordered systems. For Navier–Stokes turbulence in three dimensions, the velocity–velocity correlation function acquires a linear dependence on the distance, ζ2 = 1, in the inertial range, instead of Kolmogorov’s ζ2 = 2/3; however, the possibility remains for corrections at two- or higher-loop order. In two dimensions, we obtain a numerical solution which conserves energy and exhibits an inverse cascade, with explicit analytical results for both large and small distances, in agreement with the scaling proposed by Batchelor. In large dimensions, the one-loop FRG equation for Navier–Stokes turbulence converges to that of Burgers turbulence.

Number of degrees of freedom and energy spectrum of surface quasi-geostrophic turbulence

AbstractWe study both theoretically and numerically surface quasi-geostrophic turbulence regularized by the usual molecular viscosity, with an emphasis on a number of classical predictions. It is found that the system’s number of degrees of freedom $N$, which is defined in terms of local Lyapunov exponents, scales as ${\mathit{Re}}^{3/ 2} $, where $\mathit{Re}$ is the Reynolds number expressible in terms of the viscosity, energy dissipation rate and system’s integral scale. For general power-law energy spectra ${k}^{\ensuremath{-} \ensuremath{\alpha} } $, a comparison of $N$ with the number of dynamically active Fourier modes, i.e. the modes within the energy inertial range, yields $\ensuremath{\alpha} = 5/ 3$. This comparison further renders the scaling ${\mathit{Re}}^{1/ 2} $ for the exponential dissipation rate at the dissipation wavenumber. These results have been predicted on the basis of Kolmogorov’s theory. Our approach thus recovers these classical predictions and is an analytic alternative to the traditional phenomenological method. The implications of the present findings are discussed in conjunction with related results in the literature. Support for the analytic results is provided through a series of direct numerical simulations.

Effects of surface quasi-geostrophic turbulence on phytoplankton competition and coexistence

This paper aims at studying numerically the competition between two mutually exclusive phytoplankton species in a fully-turbulent field consisting of interacting mesoscale and submesoscale structures. A simple NPPZD ecosystem model is embedded in a Surface Quasi-Geostrophic model which is able to reproduce frontogenesis and the associated nutrient vertical pump. The two simulated phytoplankton species differ by their size and their affinity for nutrients. In this study, we rationalize the role played by eddies and filaments in the distribution of the two phytoplankton species. We show that the SQG dynamics are responsible for the coexistence of the two phytoplankton species on a single limiting resource at statistical steady state. In addition, we show that as a result of strong vertical injections, filaments contain 64% of the phytoplankton biomass. The two phytoplankton species coexist in filaments but the large phytoplankton is predominant. By contrast, this latter is completely excluded from eddy cores where only the small phytoplankton develops. Since eddies are coherent structures (unlike filaments) and since their edges are almost impermeable to horizontal transport, the large phytoplankton can barely enter eddies. Therefore, eddies are ecological niches which shelter the small phytoplankton. Finally we show that interactions between eddies such as eddy merger can favor the survival of phytoplankton species within eddies on long time scales in the ocean.

Interacting scales and triad enstrophy transfers in generalized two-dimensional turbulence

The local and nonlocal characteristics of triad enstrophy transfer in the enstrophy inertial range of generalized two-dimensional turbulence, so-called alpha turbulence, are investigated using direct numerical simulations, with a special emphasis on alpha=1 , 2, and 3. The enstrophy transfer via nonlocal triad interactions dominates the transfer dynamics in the enstrophy inertial range, irrespective of alpha . However, the contributions from more local interactions to the total enstrophy transfer increase as alpha decreases. The results are discussed in connection with the local and nonlocal transition of the enstrophy transfer at alpha=2 expected from the phenomenological scaling theory. The specific nature of the enstrophy transfer in surface quasigeostrophic turbulence (alpha=1) is also discussed.

A new proof on net upscale energy cascade in two-dimensional and quasi-geostrophic turbulence

A general proof that more energy flows upscale than downscale in two-dimensional turbulence and barotropic quasi-geostrophic (QG) turbulence is given. A proof is also given that in surface QG turbulence, the reverse is true. Though some of these results are known in restricted cases, the proofs given here are pedagogically simpler, require fewer assumptions and apply to both forced and unforced cases.

Is the subdominant part of the energy spectrum due to downscale energy cascade hidden in quasi-geostrophic turbulence?

In systems governing two-dimensional turbulence, surface quasi-geostrophic turbulence, (more generally $\alpha$-turbulence), two-layer quasi-geostrophic turbulence, etc., there often exist two conservative quadratic quantities, one "energy''-like and one "enstrophy''-like. In a finite inertial range there are in general two spectral fluxes, one associated with each conserved quantity. We derive here an inequality comparing the relative magnitudes of the "energy'' and "enstrophy'' fluxes for finite or infinitesimal dissipations, and for hyper or hypo viscosities. When this inequality is satisfied, as is the case of 2D turbulence,where the energy flux contribution to the energy spectrum is small, the subdominant part will be effectively hidden. In sQG turbulence, it is shown that the opposite is true: the downscale energy flux becomes the dominant contribution to the energy spectrum. A combination of these two behaviors appears to be the case in 2-layer QG turbulence, depending on the baroclinicity of the system.

Diminishing inverse transfer and non-cascading dynamics in surface quasi-geostrophic turbulence

The inverse transfer in the forced-dissipative surface quasi-geostrophic equation is studied, when the natural dissipation operator μ ( − Δ ) 1 / 2 is employed. The nonlinear transfer of this system conserves the two quadratic quantities Ψ 1 = 〈 | ( − Δ ) 1 / 4 ψ | 2 〉 / 2 and Ψ 2 = 〈 | ( − Δ ) 1 / 2 ψ | 2 〉 / 2 (kinetic energy), where ψ is the stream function and 〈 ⋅ 〉 denotes a spatial average. In the limit of infinite domain, the kinetic energy density Ψ 2 remains bounded, for the natural dissipation operator. For the power-law inverse-transfer region, the inverse flux of Ψ 1 diminishes as it proceeds toward sufficiently low wavenumbers, whenever the kinetic energy Ψ 2 remains bounded. This implies that no persistent (non-dissipative) inverse cascade of Ψ 1 to ever-lower wavenumbers is sustainable, as long as the dissipation parameter μ is held fixed. This result does not rule out the possibility that for sufficiently small μ , a finite inverse flux would reach a certain low wavenumber. Numerical results supporting the theoretical predictions are presented.

Large-scale energy spectra in surface quasi-geostrophic turbulence

The large-scale energy spectrum in two-dimensional turbulence governed by the surface quasi-geostrophic (SQG) equation \[\partial_t(-\Delta)^{1/2}\psi+J\big(\psi,(-\Delta)^{1/2}\psi\big)=\mu\Delta\psi+f\] is studied. The nonlinear transfer of this system conserves the two quadratic quantities but dependent on the domain size. Results from numerical simulations confirming the theoretical predictions are presented.

Energy budgets in Charney-Hasegawa-Mima and surface quasigeostrophic turbulence.

We study energy transfer in unbounded Charney-Hasegawa-Mima and surface quasigeostrophic turbulence. The possible inverse-cascading quantities in these systems are, respectively, I identical with integral ( infinity )(0)k(-2)E(k) dk and J identical with integral ( infinity )(0)k(-1)E(k) dk, where E(k) is the kinetic energy spectrum. The supposed direct-cascading quantities for both surface quasigeostrophic and Navier-Stokes turbulence are shown to be bounded. We derive a constraint on E(k) for the surface quasigeostrophic system.

Turbulent diffusion in the geostrophic inverse cascade

Motivated in part by the problem of large-scale lateral turbulent heat transport in the Earth's atmosphere and oceans, and in part by the problem of turbulent transport itself, we seek to better understand the transport of a passive tracer advected by various types of fully developed two-dimensional turbulence. The types of turbulence considered correspond to various relationships between the streamfunction and the advected field. Each type of turbulence considered possesses two quadratic invariants and each can develop an inverse cascade. These cascades can be modified or halted, for example, by friction, a background vorticity gradient or a mean temperature gradient. We focus on three physically realizable cases: classical two-dimensional turbulence, surface quasi-geostrophic turbulence, and shallow-water quasi-geostrophic turbulence at scales large compared to the radius of deformation. In each model we assume that tracer variance is maintained by a large-scale mean tracer gradient while turbulent energy is produced at small scales via random forcing, and dissipated by linear drag. We predict the spectral shapes, eddy scales and equilibrated energies resulting from the inverse cascades, and use the expected velocity and length scales to predict integrated tracer fluxes.When linear drag halts the cascade, the resulting diffusivities are decreasing functions of the drag coefficient, but with different dependences for each case. When β is significant, we find a clear distinction between the tracer mixing scale, which depends on β but is nearly independent of drag, and the energy-containing (or jet) scale, set by a combination of the drag coefficient and β. Our predictions are tested via high- resolution spectral simulations. We find in all cases that the passive scalar is diffused down-gradient with a diffusion coefficient that is well-predicted from estimates of mixing length and velocity scale obtained from turbulence phenomenology.

Surface quasigeostrophic turbulence: The study of an active scalar.

We study the statistical and geometrical properties of the potential temperature (PT) field in the surface quasigeostrophic (SQG) system of equations. In addition to extracting information in a global sense via tools such as the power spectrum, the g-beta spectrum, and the structure functions we explore the local nature of the PT field by means of the wavelet transform method. The primary indication is that an initially smooth PT field becomes rough (within specified scales), though in a qualitatively sparse fashion. Similarly, initially one-dimensional iso-PT contours (i.e., PT level sets) are seen to acquire a fractal nature. Moreover, the dimensions of the iso-PT contours satisfy existing analytical bounds. The expectation that the roughness will manifest itself in the singular nature of the gradient fields is confirmed via the multifractal nature of the dissipation field. Following earlier work on the subject, the singular and oscillatory nature of the gradient field is investigated by examining the scaling of a probability measure and a sign singular measure, respectively. A physically motivated derivation of the relations between the variety of scaling exponents is presented, the aim being to bring out some of the underlying assumptions which seem to have gone unnoticed in previous presentations. Apart from concentrating on specific properties of the SQG system, a broader theme of the paper is a comparison of the diagnostic inertial range properties of the SQG system with both the two- and three-dimensional Euler equations. (c) 2002 American Institute of Physics.