Geophysical Turbulence Research Articles

Predictability in a Model of Geophysical Turbulence

AbstractThe nature of predictability is examined in a numerical model relevant to the midlatitude atmosphere and oceans. The approach followed is novel and uses new theoretical tools from information theory, namely entropy functionals, as measures of information content and their application to finite ensembles. Particular attention is paid here to the practical application of these methods to the problem of ensemble prediction in dynamical systems with state spaces of high dimensionality. In this case, typically only an estimate of the prediction probability distribution function is available at coarse resolution. A methodology for estimating the information loss implied by this limited knowledge is introduced and applied to the practical problem of measuring prediction information content in a model able to generate geophysical turbulence. The application studied here generates such turbulence through the mechanism of baroclinic instability via an imposed and constant mean vertical shear. In traditional studies in this area, considerable attention has been paid to variations in ensemble spread as the major determinant of how predictability may change as prediction initial conditions vary. The analysis here reveals that such a scenario neglects the important role of the so-called ensemble signal, which is related to the difference in the first moments of the prediction and climatological distributions. It is found, in fact, that this quantity is often a strong control over variations in predictability of the large-scale barotropic flow. An initial investigation of the role of non-Gaussian effects shows that for the univariate large-scale barotropic case, they are only of minor importance to variations in predictability.

Towards mesh adaptivity for geophysical turbulence: continuous mapping approach

AbstractLooking forward towards mesh adaptivity for simulating turbulent atmospheric/oceanic flows, we are pursuing advanced algorithms for evaluating vector differential operators cast in time‐dependentcurvilinear co‐ordinates. In this paper, we review our effort to date with the development of a deformable‐co‐ordinates multi‐scale anelastic model designed from the bottom‐up relying on strengths ofnon‐oscillatory transport methods. We have shown in earlier works that effective multi‐scale adaptive numerical models for high‐Reynolds‐number meteorological flows can be designed that dispense with rigorous evaluation of the more cumbersome of the vector differential operators, such as the curl or the strain rate. These operators are nonetheless important for budget analyses of the model results, estimating physical uncertainties, driving the mesh adaptivity itself, and extending the model's applicability beyond standard meteorological situations. Here, we discuss selected extensions of the generic explicitly inviscid approach. Copyright © 2005 John Wiley & Sons, Ltd.

Statistical mechanics of geophysical turbulence: application to jovian flows and Jupiter’s great red spot

We propose a parameterization of 2D geophysical turbulence in the form of a relaxation equation similar to a generalized Fokker–Planck equation [P.H. Chavanis, Phys. Rev. E 68 (2003) 036108]. This equation conserves circulation and energy and increases a generalized entropy functional determined by a prior vorticity distribution fixed by small-scale forcing [R. Ellis, K. Haven, B. Turkington, Nonlinearity 15 (2002) 239]. We discuss applications of this formalism to jovian atmosphere and Jupiter’s great red spot. We show that, in the limit of small Rossby radius where the interaction becomes short-range, our relaxation equation becomes similar to the Cahn–Hilliard equation describing phase ordering kinetics. This strengthens the analogy between the jet structure of the great red spot and a “domain wall”. Our relaxation equation can also serve as a numerical algorithm to construct arbitrary nonlinearly dynamically stable stationary solutions of the 2D Euler equation. These solutions can represent jets and vortices that emerge in 2D turbulent flows as a result of violent relaxation. Due to incomplete relaxation, the statistical prediction may fail and the system can settle on a stationary solution of the 2D Euler equation which is not the most mixed state. In that case, it can be useful to construct more general nonlinearly dynamically stable stationary solutions of the 2D Euler equation in an attempt to reproduce observed phenomena.

Simulation of two‐dimensional turbulent flows in a rotating annulus

AbstractRotating water tank experiments have been used to study fundamental processes of atmospheric and geophysical turbulence in a controlled laboratory setting. When these tanks are undergoing strong rotation the forced turbulent flow becomes highly two dimensional along the axis of rotation. An efficient numerical method has been developed for simulating the forced quasi‐geostrophic equations in an annular geometry to model current laboratory experiments. The algorithm employs a spectral method with Fourier series and Chebyshev polynomials as basis functions. The algorithm has been implemented on a parallel architecture to allow modelling of a wide range of spatial scales over long integration times. This paper describes the derivation of the model equations, numerical method, testing and performance of the algorithm. Results provide reasonable agreement with the experimental data, indicating that such computations can be used as a predictive tool to design future experiments. Copyright © 2004 John Wiley & Sons, Ltd.

Comments on "A generic length-scale equation for geophysical turbulence models" by L. Umlauf and H. Burchard

Comments on "A generic length-scale equation for geophysical turbulence models" by L. Umlauf and H. Burchard

Reply to: "Comments on 'A generic length-scale equation for geophysical turbulence models'" by L. Kantha and S. Carniel

Reply to: "Comments on 'A generic length-scale equation for geophysical turbulence models'" by L. Kantha and S. Carniel

Counter-Rotating Quasigeostrophic Ellipsoidal Vortex Pair

In geophysical flows, coherent vortex structures persist for long time and their interactions dominate the dynamics of geophysical turbulence. Miyazaki et al. derived a Hamiltonian dynamical system describing the interactions of N ellipsoidal vortices, where each coherent vortex was modeled by an ellipsoid of uniform potential vorticity embedded in a locally uniform 3D shear field. In this paper, the interaction of counter-rotating vortex pair (dipole) is investigated. According to the ellipsoidal moment model, the inclination from the vertical axis approaches π/2 and one of the principal axis of the ellipsoid grows exponentially with time, if two slender vortices are placed within a critical distance initially. Direct numerical simulations, based on the Contour Advective Semi-Lagrangian algorithm, are performed in order to assess the validity of the model. The CASL computaion tells that the vortices keep the dipole structure quite robustly. They survive from tilting down by emittimg thin filaments from their top and bottom, even if they are placed very close.

Three-dimensional instability of isolated vortices

We study the three-dimensional stability of the family of vortices introduced by Carton and McWilliams [Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence, edited by Nikhoul and Jamart (Elsevier, New York, 1989)] describing isolated vortices. For these vortices, the circulation vanishes outside their core over a distance depending on a single parameter, the steepness α. We proceed to the direct numerical simulation of the linear impulse response to obtain both temporal and spatio-temporal instability results. In the temporal instability framework, growth rates are calculated as a function of the axial wavenumber k and the azimuthal wavenumber m. The stability analysis is performed at a Reynolds number of Re=667. It is shown that the most unstable mode is the axisymmetric mode m=0, regardless of the steepness parameter in the investigated range. When the steepness α is increased the band of unstable azimuthal modes widens, i.e., larger m are destabilized. The study of the spatio-temporal spreading of the wave packet shows that the m=2 mode is always the fastest traveling mode, for all studied values of the steepness parameter.

A generic length-scale equation for geophysical turbulence models

A generalization of a class of differential length-scale equations typically used in second-order turbulence models for oceanic flows is suggested. Commonly used models, like the κ-e model and the Mellor-Yamada model, can be recovered as special cases of this generic model, and thus can be rationally compared. In addition, a method is proposed that yields a generalized framework for the calibration of the most frequently used class of differential length-scale equations. The generic model, calibrated with this method, exhibits a greater range of applicability than any of the traditional models. Stratified flows, plane mixing layers, and turbulence introduced by breaking surface waves are considered besides some classical test cases.

Vortex statistics from Eulerian and Lagrangian time series.

Coherent vortices are an important component of the dynamics of geophysical turbulence, but direct estimates of the properties of the vortex population from measured data is usually difficult. Motivated by this problem, we propose a new method for determining the statistical properties of coherent vortices in two-dimensional turbulence based on a small number of Lagrangian and Eulerian time series. The method provides reliable estimates of the mean vortex size and vortex number density.

Numerical Validation of Quasigeostrophic Ellipsoidal Vortex Model

In geophysical flows, coherent vortex structures persist for long time and their interactions dominate the dynamics of geophysical turbulence. Meacham et al. obtained a series of exact unsteady solution of the quasigeostrophic equation, which represents a uniform ellipsoidal vortex patch embedded in a uniform 3D shear field. Miyazaki et al. derived a Hamiltonian dynamical system describing the interactions of N ellipsoidal vortices, where each coherent vortex was modeled by an ellipsoid of uniform potential vorticity. In this paper, direct numerical simulations based on a Contour Advective Semi-Lagrangian algorithm (CASL) are performed in order to assess the validity of the Hamiltonian model. First, the instability of a tilted spheroid is investigated. A prolate spheroid becomes unstable against the third Legendre mode when the aspect ratio is less than 0.44 and the inclination angle is larger than 0.48. Weakly unstable flatter spheroidal vortices emit thin filaments from their top and bottom, whereas str...

VLES modelling of geophysical fluids with nonoscillatory forward‐in‐time schemes

AbstractAtmospheres and oceans are mostly incompressible and turbulent. The experience of the meteorological community with modelling such flows is based primarily on centred in time and space (CTS) methods. Our experience applying non‐oscillatory forward in time methods (NFT) to a range of flows has revealed some unexpected benefits specifically in the area of modelling geophysical turbulence where broad span of scales, density stratification, planetary rotation, inhomogeneity of the lower boundary, etc., make explicit modelling of subgrid‐scale motions particularly challenging. It turns out that in the absence or insufficiency of a proper subgrid‐scale model, NFT methods supply their own, implicit, turbulence models that are quite effective in assuring quality simulations of high‐Reynolds number flows. Since such simulations abandon rigorous notion of the large‐eddy simulation approach, and merely aim at computing explicitly large coherent eddies resolvable on the grid, they are referred to as very‐large‐eddy simulations (VLES). In this paper we will describe advantages of the NFT approach and illustrate them with an example of gravity‐wave‐breaking induced turbulence in a deep atmosphere. On the philosophical side, we challenge a common misconception that NFT methods are overly diffusive and therefore inadequate for high Reynolds number flow simulations. Copyright © 2002 John Wiley & Sons, Ltd.

Numerical simulation of geophysical turbulence

The differences between simulations of geophysical flows and flows of direct interest to engineers are discussed in some detail. A large number of direct numerical and large eddy simulations of turbulent stratified flows that are mainly of interest in environmental fluid mechanics and geophysics that have been made over that last ten years are described. These include homogeneous stratified shear flow, oscillating grid flow (with and without a thermocline or inversion), stratified channel flow and Langmuir circulations. The ability of simulations to provide physical insights and information that are useful in constructing parameterizations for these flows is demonstrated.

Anisotropy of turbulence in stably stratified mixing layers

Direct numerical simulations of turbulence resulting from Kelvin–Helmholtz instability in stably stratified shear flow are used to study sources of anisotropy in various spectral ranges. The set of simulations includes various values of the initial Richardson and Reynolds numbers, as well as Prandtl numbers ranging from 1 to 7. We demonstrate that small-scale anisotropy is determined almost entirely by the spectral separation between the small scales and the larger scales on which background shear and stratification act, as quantified by the buoyancy Reynolds number. Extrapolation of our results suggests that the dissipation range becomes isotropic at buoyancy Reynolds numbers of order 105, although we cannot rule out the possibility that small-scale anisotropy persists at arbitrarily high Reynolds numbers, as some investigators have suggested. Correlation-coefficient spectra reveal the existence of anisotropic flux reversals in the dissipation subrange whose magnitude decreases with increasing Reynolds number. The scalar concentration field tends to be more anisotropic than the velocity field. Estimates of the dissipation rates of kinetic energy and scalar variance based on the assumption of isotropy are shown to be accurate for buoyancy Reynolds numbers greater than O(102). Such estimates are therefore reliable for use in the interpretation of most geophysical turbulence data, but may give misleading results when applied to smaller-scale flows.

Vertical turbulence structure and second-moment budgets in convection with rotation: A large-eddy simulation study

The structure of the instantaneous flow fields and turbulence statistics and the second-order moment budgets in convection affected by rotation are analysed using a large-eddy simulation (LES) dataset. Three archetypes of convective flows driven by the surface buoyancy flux are generated. One is the reference case of the non-rotating convective boundary layer (CBL) growing into a quiescent stably stratified fluid. The other two are CBLs affected by rotation. In the geophysical turbulence context, these non-rotating and rotating CBLs mimic an early stage and a mature stage, respectively, of the vertical mixing phase of open-ocean deep convection. Instantaneous flow structures reveal strong localization of the buoyancy anomalies and the n on-hydrostatic pressure anomalies near the surface in rotating CBLs and their dilution as one moves towards the CBL outer edge. These anomalies are associated with the localized cyclonic vortices which are the centres of intense vertical motions. Most of the cyclones never reach the outer edge of the CBL. Increasing rotation results in less mixing, reducing the entrapment flux at the CBL outer edge and maintaining a negative buoyancy gradient throughout the CBL. The effect of counter-gradient transport, which occurs in free convection, is largely reduced. The vertical-velocity variance and the layer-averaged turbulence kinetic energy are reduced by rotation, while the variances of buoyancy and pressure are enhanced. The vertical velocity and buoyancy fields are positively skewed in both rotating and free convection. The buoyancy skewness is considerably larger in the bulk of the rotating CBL than in the non-rotating CBL, reflecting strong localization of positive buoyancy anomalies. The pressure transport term in the turbulence kinetic-energy budget becomes more important as the rotation rate increases, whereas the contribution of the third-order transport term is reduced. All terms in the buoyancy variance budget grow in amplitude as the rotation rate increases. The mean-gradient term and the turbulent transport term are both gains that are offset by a loss to dissipation in the bulk of the CBL. This is different from free convection where the buoyancy variance budget in mid CBL is maintained mainly by turbulent transport since the mean-gradient term is small there. The budget of the vertical buoyancy flux in convection with rotation is strongly dominated by the pressure-gradient/buoyancy covariance and the buoyancy production terms. Evaluation of closures for the turbulence energy dissipation against the LES data supports the idea of imposing a limitation on the dissipation length scale due to the background rotation. This limitation is required to account for the reduced turbulence energy in convection with rotation. A similar limitation on the length scale for the dissipation of buoyancy variance is not found to be important. Analysis of parametrized budgets of the third-order moments reveals the dominance of the direct effects of buoyancy. These effects are enhanced with increasing rotation rate. They must be included in parametrizations for the third-order moments. The conventional down-gradient approximations neglecting the buoyancy effects would greatly underestimate the turbulent transport.

Vertical turbulence structure and second‐moment budgets in convection with rotation: A large‐eddy simulation study

AbstractThe structure of the instantaneous flow fields and turbulence statistics and the second‐order moment budgets in convection affected by rotation are analysed using a large‐eddy simulation (LES) dataset. Three archetypes of convective flows driven by the surface buoyancy flux are generated. One is the reference case of the non‐rotating convective boundary layer (CBL) growing into a quiescent stably stratified fluid. The other two are CBLs affected by rotation. In the geophysical turbulence context, these non‐rotating and rotating CBLs mimic an early stage and a mature stage, respectively, of the vertical mixing phase of open‐ocean deep convection.Instantaneous flow structures reveal strong localization of the buoyancy anomalies and the n on‐hydrostatic pressure anomalies near the surface in rotating CBLs and their dilution as one moves towards the CBL outer edge. These anomalies are associated with the localized cyclonic vortices which are the centres of intense vertical motions. Most of the cyclones never reach the outer edge of the CBL.Increasing rotation results in less mixing, reducing the entrapment flux at the CBL outer edge and maintaining a negative buoyancy gradient throughout the CBL. The effect of counter‐gradient transport, which occurs in free convection, is largely reduced. The vertical‐velocity variance and the layer‐averaged turbulence kinetic energy are reduced by rotation, while the variances of buoyancy and pressure are enhanced. The vertical velocity and buoyancy fields are positively skewed in both rotating and free convection. The buoyancy skewness is considerably larger in the bulk of the rotating CBL than in the non‐rotating CBL, reflecting strong localization of positive buoyancy anomalies.The pressure transport term in the turbulence kinetic‐energy budget becomes more important as the rotation rate increases, whereas the contribution of the third‐order transport term is reduced. All terms in the buoyancy variance budget grow in amplitude as the rotation rate increases. The mean‐gradient term and the turbulent transport term are both gains that are offset by a loss to dissipation in the bulk of the CBL. This is different from free convection where the buoyancy variance budget in mid CBL is maintained mainly by turbulent transport since the mean‐gradient term is small there. The budget of the vertical buoyancy flux in convection with rotation is strongly dominated by the pressure‐gradient/buoyancy covariance and the buoyancy production terms.Evaluation of closures for the turbulence energy dissipation against the LES data supports the idea of imposing a limitation on the dissipation length scale due to the background rotation. This limitation is required to account for the reduced turbulence energy in convection with rotation. A similar limitation on the length scale for the dissipation of buoyancy variance is not found to be important. Analysis of parametrized budgets of the third‐order moments reveals the dominance of the direct effects of buoyancy. These effects are enhanced with increasing rotation rate. They must be included in parametrizations for the third‐order moments. The conventional down‐gradient approximations neglecting the buoyancy effects would greatly underestimate the turbulent transport.

Some problems and issues in geophysical turbulence

Velocity fluctuations in the large scales of the atmosphere's meso-scale have turbulent characteristics of random fluctuations and a scale-size distribution near k −5/3 (Gage, 1979. J. Atmos. Sci. 36, 1950–1954; Lilly, 1983. J. Atmos. Sci. 40, 749–761; Lilly et al., 1998. Theoret. Comput. Fluid Dyn. 11, 139–153). Explanations of this motion field have ranged from inverse cascading quasi-geostrophic (i.e. quasi-two-dimensional) turbulence, to gravity waves (VanZandt, 1982. Geophys. Res. Lett. 9, 575–578). We describe efforts to relate observational spectra to various theories ranging from quasi-geostrophic turbulence to gravity waves. We note that at the larger scales quasi-geostrophic theory may suffice, but at smaller scales, a quasi-geostrophic explanation becomes untenable because the importance of rotation becomes progressively weaker as scales of the flow becomes smaller (the Rossby number approaches unity). We then discuss numerical simulations designed to discriminate between alternative explanations of the flow. Several simulations are reviewed, starting with those of Herring and Métais (1989. J. Fluid Mech. 202, 97–115), and finally those described by Lilly et al. (1998. Theoret. Comput. Fluid Dyn. 11, 139–153).

Intermittency of two-dimensional decaying electron magnetohydrodynamic turbulence

The intermittent nature of energy dissipation in two-dimensional electron-magnetohydrodynamic turbulence is investigated by means of high resolution direct numerical simulations. It is found that, when the main contribution to the energy is given by the magnetic field, dissipation is mostly concentrated on one-dimensional filaments. As a consequence, the multifractal spectrum has a simple form which can be approximately described in terms of a bifractal model. @S1063-651X~99!01803-6# PACS number~s!: 47.27.Eq The statistical theory of three-dimensional fully developed hydrodynamic turbulence relies on one outstanding issue: the nonlinear transfer of energy from large to small scales @1,2#. The energy flux is constant over the intermediate scales of the inertial range, but does not need to be homogeneous in space. Moreover, experiments in fluid turbulence indicate that the self-similarity of the energy dissipation distribution is broken by the presence of small-scale structures in the flow. Recent direct numerical simulations revealed the presence of filaments and other dissipative structures candidate as physical sources of intermittency @3#. It is therefore interesting to look for two-dimensional turbulent systems sharing these same features. Actually many of them exhibit a reversed energy flux, from the small scales to the larger ones, as is the case of two-dimensional ~2D! Navier-Stokes turbulence @4‐7#, Hasegawa-Mima turbulence @8#, or its geophysical counterpart equivalent barotropic turbulence @9#. In this framework 2D electronmagnetohydrodynamic ~EMHD! turbulence deserves special attention, beyond its modeling applications, since it has been shown to display, for the freely decaying case, a forward

Two-dimensional turbulence properties of the ECMWF reanalyses

A spherical harmonic analysis is made of the stationary and transient rotational motions during14 winter and 15 summer seasons from the reanalyses of the European Centre for Medium-Range Weather Forecasts (ECMWF) for the northern hemisphere. Vertically-integrated kineticenergy, enstrophy, rotational non-linear interactions and the baroclinic source term are diagnosedas a function of total wavenumber n. The contributions to the non-linear transfers fromtriads involving wavenumber and frequency bands of meterological relevance are mapped. Theinter-seasonal and intra-seasonal variability are computed. The non-linear energy and enstrophytendencies and fluxes are examined and compared to existing geophysical turbulence theories.The transient divergent kinetic energy is less than the rotational energy. The spectral energyslope seen in the range n ˜ 10—40 is roughly - 2.5 ˜ - 2.6. Based on the variability of the slopeon seasonal and 10-day time scales, this slope is significantly different than - 3. There is noindication of the – 5/3 mesoscale energy regime seen in observations. A broad enstrophy dissipationregimes is seen for n > 40. Non-linear terms transfer transient energy from a band centeredat n ˜ 15 to one at n ˜ 7, with the latter predominantly associated with non-zonal (zonal wavenumberm0) flow. Non-linear terms transfer mean energy from n=7 to the mean zonal flowm=0, n=3 and n=5. The non-linear transfer of transient energy is quite variable, with about 16% of 10-day periods yielding a tendency twice the mean, and 16% showing no upscaletendency whatever. This variability is greatly reduced when interactions involving only synopticscales (n ˜ 10—40) are retained. The latter set of interactions are associated mostly with triadsinvolving both high and low frequencies, with associated periods in the 1–9 day and 11—90range, respectively. Non-local planetary wave advective interactions play an important rôle inthe downscale transfer of enstrophy. More local interactions involving synoptic scales dominatethe non-linear energy transfers. The main seasonal effects are a weakening in summer of thetotal energy and shifting to higher wave number of the peak, and a distinct shift to smallerscales in the transition between the large-scale and synoptic-scale regimes in the energy budget.The predominant time scale for non-linear maintenance of the planetary waves (about 9 days)is roughly the same as that of the baroclinic support of larger synoptic scale waves. The timescale of baroclinic conversion which maintains the smaller synoptic waves (n ˜ 20—40) is shorter(about 4 days).

Order and Resolution for Computational Ocean Dynamics

An ocean flow that has all its scales resolved on a model grid can be more efficiently calculated to within a required accuracy by using high-order numerics than by grid refinement with low-order numerics. The differencing order must be at least as great as the space–time dimensionality D of the model to ensure that grid refinement reduces truncation error at least as quickly as computational cost increases. Ocean flows often have variability on a wide range of scales that cannot all be resolved on any practical grid. In such circumstances the distribution of variability among the scales determines whether grid refinement or increased order results in the greatest accuracy per unit computational cost. A model that simulates the −5/3 power law of the inertial subrange of three-dimensional turbulence would most efficiently exploit low-order numerics for all terms. The spectra of different terms in the equations of motion can be different and can therefore require different orders of accuracy for efficient computation. Modeling geophysical turbulence with a power law of −3 would require high-order numerics for the advective terms but low-order numerics would be sufficient for other terms. Output from several ocean models are observed to have spectra that are sufficiently red to justify using high-order numerics for all terms. In the case of one relatively simple ocean modeling problem the author demonstrates that leading-order terms dominate the truncation error.