Two-layer Quasigeostrophic Turbulence Research Articles

Application of a Simple Diffusivity Formulation to Examine Regime Transition and Jet–Eddy Energy Partitioning in Quasi-Geostrophic Turbulence

Abstract This study uses a simple diffusivity formulation to examine flow regime transition and jet–eddy energy partitioning in two-layer quasigeostrophic turbulence. Guided by simulations, the formulation is empirically constructed so that the diffusivity is bounded by a f-plane asymptote (Df) in the limit of vanishing β (termed drag-controlled) while reduced to a drag-independent scaling (Dβ) of Lapeyre and Held toward large β (termed β-controlled). Good agreement is found for diffusivities diagnosed from simulations with both quadratic and linear drag and in 2D turbulence. From the formulation, a regime diagram is readily constructed, with Df/Dβ = 1 separating the drag-controlled and β-controlled regimes. The diagram also sets the parameter range where an eddy velocity scaling is applicable. The quantitative representations of eddy variables then enable a reasonably skillful theory for zonal jet speed to be developed from energy balance. It is shown that, using Df/Dβ ≥ 10, a state where eddy statistics are approximately drag insensitive could be identified and interpreted using wave-damping competitions in slowing an inverse cascade. However, contrary to an existing hypothesis, the energy dissipation in such a state is not dominated by zonal jets. A modest revision for a way to maintain balance while keeping eddies drag insensitive is proposed. In the regime diagram, a subspace of zonostrophic condition, defined as jet dissipation surpassing eddy, is further quantified. It is demonstrated that a rough scaling could help interpret how the relative importance of jet and eddy dissipation varies across the parameter space.

Revisiting the Baroclinic Eddy Scalings in Two-Layer, Quasigeostrophic Turbulence: Effects of Partial Barotropization

Abstract In this study, modifications of the Held scaling (Held) are proposed and tested against simulations of f-plane, two-layer quasigeostrophic turbulence. The aim is to better constrain the eddy mixing length and rms barotropic velocity in response to varied quadratic and linear bottom drag. The proposed modifications allow eddies to be partially barotropized, to relax the commonly invoked barotropization approximation, and consider a drag-dependent cascade rate per energy input, to account for the lack of an inertial range. Quantitative comparisons with the vortex gas scaling are also carried out. It is shown that the progressively weakened sensitivity in eddy scales to increased drag strength is mainly a result of eddy partial barotropization. For both drag forms except toward the limit of weak linear drag, accounting for partial barotropization alone leads to good predictions of eddy velocity, although not of mixing length. It also partly resolves the degeneracy of balance constraints for linear drag because partial barotropization acts like scale-dependent damping. Adding a cascade correction, which is interpreted as allowing for changes in spectral room for cascade, further improves the mixing length representation. Overall, the proposed theory can augment the existing scalings by extending the eddy scale predictions to O(1) quadratic drag and has skills generally comparable to the vortex gas scaling for linear drag. However, toward the weak linear drag limit where eddies approach complete barotropization, the proposed theory breaks down but the vortex gas performs well. Potential issues concerning the applicability of vortex gas to this limit are discussed.

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.

Numerical Schemes for Stochastic Backscatter in the Inverse Cascade of Quasigeostrophic Turbulence

Backscatter is the process of energy transfer from small to large scales in turbulence; it is crucially important in the inverse energy cascades of geophysical turbulence, where the net transfer of energy is from small to large scales. One approach to modeling backscatter in underresolved simulations is to add a stochastic forcing term. This study, set in the idealized context of the inverse cascade of two-layer quasigeostrophic turbulence, focuses on the importance of spatial and temporal correlation in numerical stochastic backscatter schemes when used with low-order finite-difference spatial discretizations. A minimal stochastic backscatter scheme is developed as a stripped-down version of stochastic superparameterization [Grooms and Majda, J. Comput. Phys., 271, (2014), pp. 78--98]. This simplified scheme allows detailed numerical analysis of the spatial and temporal correlation structure of the modeled backscatter. Its essential properties include a local formulation amenable to implementation in finite difference codes and nonperiodic domains, and tunable spatial and temporal correlations. Experiments with this scheme in the idealized context of homogeneous two-layer quasigeostrophic turbulence demonstrate the need for stochastic backscatter to be smooth at the coarse grid scale when used with low-order finite-difference schemes in an inverse-cascade regime. In contrast, temporal correlation of the backscatter is much less important for achieving realistic energy spectra. It is expected that the spatial and temporal correlation properties of the simplified backscatter schemes examined here will inform the development of stochastic backscatter schemes in more realistic models.

New perspectives on superparameterization for geophysical turbulence

This is a research expository paper regarding superparameterization, a class of multi-scale numerical methods designed to cope with the intermittent multi-scale effects of inhomogeneous geophysical turbulence where energy often inverse-cascades from the unresolved scales to the large scales through the effects of waves, jets, vortices, and latent heat release from moist processes. Original as well as sparse space–time superparameterization algorithms are discussed for the important case of moist atmospheric convection including the role of multi-scale asymptotic methods in providing self-consistent constraints on superparameterization algorithms and related deterministic and stochastic multi-cloud parameterizations. Test models for the statistical numerical analysis of superparameterization algorithms are discussed both to elucidate the performance of the basic algorithms and to test their potential role in efficient multi-scale data assimilation. The very recent development of grid-free seamless stochastic superparameterization methods for geophysical turbulence appropriate for “eddy-permitting” mesoscale ocean turbulence is presented here including a general formulation and illustrative applications to two-layer quasigeostrophic turbulence, and another difficult test case involving one-dimensional models of dispersive wave turbulence. This last test case has randomly generated solitons as coherent structures which collapse and radiate wave energy back to the larger scales, resulting in strong direct and inverse turbulent energy cascades.

The effect of asymmetric large-scale dissipation on energy and potential enstrophy injection in two-layer quasi-geostrophic turbulence

AbstractIn the Nastrom–Gage spectrum of atmospheric turbulence, we observe a${k}^{\ensuremath{-} 3} $energy spectrum that transitions into a${k}^{\ensuremath{-} 5/ 3} $spectrum, with increasing wavenumber$k$. The transition occurs near a transition wavenumber${k}_{t} $, located near the Rossby deformation wavenumber${k}_{R} $. The Tung–Orlando theory interprets this spectrum as a double downscale cascade of potential enstrophy and energy, from large scales to small scales, in which the downscale potential enstrophy cascade coexists with the downscale energy cascade over the same length scale range. We show that, in a temperature-forced two-layer quasi-geostrophic model, the rates with which potential enstrophy and energy are injected place the transition wavenumber${k}_{t} $near${k}_{R} $. We also show that, if the potential energy dominates the kinetic energy in the forcing range, then the Ekman term suppresses the upscale cascading potential enstrophy more than it suppresses the upscale cascading energy, a behaviour contrary to what occurs in two-dimensional turbulence. As a result, the ratio$\eta / \varepsilon $of injected potential enstrophy over injected energy, in the downscale direction, decreases, thereby tending to decrease the transition wavenumber${k}_{t} $further. Using a random Gaussian forcing model, we reach the same conclusion, under the modelling assumption that the asymmetric Ekman term predominantly suppresses the bottom layer forcing, thereby disregarding a possible entanglement between the Ekman term and the nonlinear interlayer interaction. Based on these results, we argue that the Tung–Orlando theory can account for the approximate coincidence between${k}_{t} $and${k}_{R} $. We also identify certain open questions that require further investigation via numerical simulations.

Cascade Inequalities for Forced–Dissipated Geostrophic Turbulence

AbstractAnalysis of spectral kinetic energy fluxes in satellite altimetry data has demonstrated that an inverse cascade of kinetic energy is ubiquitous in the ocean. In geostrophic turbulence models, a fully developed inverse cascade results in barotropic eddies with large horizontal scales. However, midocean eddies contain substantial energy in the baroclinic mode and in compact horizontal scales (scales comparable to the deformation radius Ld). This paper examines the possibility that relatively strong bottom friction prevents the oceanic cascade from becoming fully developed. The importance of the vertical structure of friction is demonstrated by contrasting numerical simulations of two-layer quasigeostrophic turbulence forced by a baroclinically unstable mean flow and damped by bottom Ekman friction with turbulence damped by vertically symmetric Ekman friction (equal decay rates in the two layers). “Cascade inequalities” derived from the energy and enstrophy equations are used to interpret the numerical results. In the symmetric system, the inequality formally requires a cascade to large-scale barotropic flow, independent of the stratification. The inequality is less strict when friction is in the bottom layer only, especially when stratification is surface intensified. Accordingly, model runs with surface-intensified stratification and relatively strong bottom friction retain substantial small-scale baroclinic energy. Altimetric data show that the symmetric inequality is violated in the low- and midlatitude ocean, again suggesting the potential impact of the “bottomness” of friction on eddies. Inequalities developed for multilayer turbulence suggest that high baroclinic modes in the mean shear also enhance small-scale baroclinic eddy energy. The inequalities motivate a new interpretation of barotropization in weakly damped turbulence. In that limit the barotropic mode dominates the spatial average of kinetic energy density because large values of barotropic density are found throughout the model domain, consistent with the barotropic cascade to large horizontal scales, while baroclinic density is spatially localized.

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.

Baroclinically Unstable Geostrophic Turbulence in the Limits of Strong and Weak Bottom Ekman Friction: Application to Midocean Eddies

Abstract This paper examines the plausibility of mesoscale eddy generation through local baroclinic instability of weak midocean gyre flows. The main tool is a statistically steady, two-layer quasigeostrophic turbulence model driven by an imposed, horizontally homogeneous, vertically sheared mean flow and dissipated by bottom Ekman friction. A wide range of friction strengths is investigated. In the weakly damped limit, flow is nearly barotropic, and the horizontal length scale of barotropic energy increases with decreasing friction, consistent with previous studies. The strongly damped limit, explored here for the first time, is equivalent barotropic (lower-layer velocities are nearly zero) and features an increase in the horizontal scale of potential energy with increasing friction. Current-meter data suggest that midocean eddies lie between the barotropic and equivalent barotropic limits. In accord with this suggestion, the moderately damped regime of the model compares well to observations of eddy amp...

Two-layer quasi-geostrophic turbulence in a simple special case

Abstract In the case of equal layer depths and uniform vertical energy density, the quadratic integral invariants of two-layer rotating flow are close analogs of the corresponding invariants of two-dimensional turbulence. A simple theory based on the invariants and on the selection rules governing triad interactions qualitatively explains the major features of forced equilibrium flow. The general physical picture is very similar to that of Rhines (1977). In the geophysically interesting case. net baroclinic energy is produced at low wavenumbers and moves toward hisher wavenumbers in relatively nonlocal triad interactions which are unhampered by the constraint to conserve enstrophy. The energy converts to barotropic mode and moves back toward low wavenumbers in more local interactions which are similar to those in two-dimensional turbulence. Equilibrium wavenumber spectra are obtainable from a simple Markovian turbulence closure model in which the estimate of turbulent scramhling rate includes a contributi...