Leslie M. Smith

Transfer of energy to two-dimensional large scales in forced, rotating three-dimensional turbulence

Forced turbulence in a rotating frame is studied using numerical simulations in a triply periodic box. The random forcing is three dimensional and localized about an intermediate wavenumber kf. The results show that energy is transferred to scales larger than the forcing scale when the rotation rate is large enough. The scaling of the energy spectrum approaches E(k)∝k−3 for k<kf. Almost all of the energy for k<kf lies in the two-dimensional (2D) plane perpendicular to the rotation z-axis, and thus the large-scale motions are quasi-2D with E(k)≈E(kh,kz=0), where kh and kz are, respectively, the horizontal and vertical components of the wavevector. The large scales consist of cyclonic vortices. Possible mechanisms responsible for the two-dimensionalization are discussed. The development of the 2D spectrum E(kh,kz=0)∝kh−3 is analogous to the dynamics of β-plane turbulence leading to the Rhines spectrum E(ky,kx=0)∝ky−5.

Generation of slow large scales in forced rotating stratified turbulence

Numerical simulations are used to study homogeneous, forced turbulence in three-dimensional rotating, stably stratified flow in the Boussinesq approximation, where the rotation axis and gravity are both in the z ˆ-direction. Energy is injected through a three-dimensional isotropic white-noise forcing localized at small scales. The parameter range studied corresponds to Froude numbers smaller than an O (1) critical value, below which energy is transferred to scales larger than the forcing scales. The values of the ratio N / f range from ≈1/2 to ∞, where N is the Brunt–Vaisala frequency and f is twice the rotation rate. For strongly stratified flows ( N / f [Gt ]1), the slow large scales generated by the fast small-scale forcing consist of vertically sheared horizontal flow. Quasi-geostrophic dynamics dominate, at large scales, only when 1/2 [les ] N / f [les ] 2, which is the range where resonant triad interactions cannot occur.

On near resonances and symmetry breaking in forced rotating flows at moderate Rossby number

Numerical simulations are used to study a series of reduced models of homogeneous, rotating flow at moderate Rossby numbers Ro 0.1, for which both numerical and physical experiments show the generation of quasi-two-dimensional vortices and symmetry breaking in favour of cyclones. A random force at intermediate scales injects energy at a constant average rate. The nonlinear term of reduced models is restricted to include only a subset of triad interactions in Fourier space. Reduced models of near-resonant, non-resonant and near two-dimensional triad interactions are considered. Only the model of near resonances reproduces all of the important characteristics of the full simulations: (i) efficient energy transfer from three-dimensional forced modes to two-dimensional large-scale modes, (ii) large-scale energy spectra scaling approximately as k -3 h , where k h is the wavenumber in the plane perpendicular to the axis of rotation, and (iii) strong cyclone/anticyclone asymmetry in favour of cyclones. Non-resonances, defined as the complement to near resonances, act to reduce the energy transfer to large scales.

Vortical and wave modes in 3D rotating stratified flows: random large-scale forcing

Utilizing an eigenfunction decomposition, we study the growth and spectra of energy in the vortical (geostrophic) and wave (ageostrophic) modes of a three-dimensional (3D) rotating stratified fluid as a function of ε = f/N, where f is the Coriolis parameter and N is the Brunt–Vaisala frequency. Throughout, we employ a random large-scale forcing in a unit aspect ratio domain and set these parameters such that the Froude and Rossby numbers are roughly comparable and much less than unity. Working in regimes characterized by moderate Burger numbers, i.e. Bu = 1/ε2 < 1 or Bu ≥ 1, our results indicate profound change in the character of vortical and wave mode interactions with respect to Bu = 1. Indeed, previous analytical work concerning the qualitatively different nature of these interactions has been in limiting conditions of rotation or stratification domination (i.e. when Bu ≪ 1 or Bu ≫ 1, respectively). As with the reference state of ε = 1, for ε < 1 the wave mode energy saturates quite quickly and the ensuing forward cascade continues to act as an efficient means of dissipating ageostrophic energy. Further, these saturated spectra steepen as ε decreases: we see a shift from k −1 to k −5/3 scaling for k f < k < k d (where k f and k d are the forcing and dissipation scales, respectively). On the other hand, when ε > 1 the wave mode energy never saturates and comes to dominate the total energy in the system. In fact, in a sense the wave modes behave in an asymmetric manner about ε = 1. With regard to the vortical modes, for ε ≤ 1, the signatures of 3D quasigeostrophy are clearly evident. Specifically, we see a k −3 scaling for k f < k < k d and, in accord with an inverse transfer of energy, the vortical mode energy never saturates but rather increases for all k < k f . In contrast, for ε > 1 and increasing, the vortical modes contain a progressively smaller fraction of the total energy indicating that the 3D quasigeostrophic subsystem, though always present, plays an energetically smaller role in the overall dynamics. Combining the vortical and wave modes, the total energy for k > k f and ε ≤ 1 shows a transition as k increases wherein the vortical modes contain a large portion of the energy at large scales, while the wave modes dominate at smaller scales. There is no such transition when ε > 1 and the wave modes dominate the total energy for all k > k f .

Local and nonlocal dispersive turbulence

We consider the evolution of a family of two-dimensional (2D) dispersive turbulence models. The members of this family involve the nonlinear advection of a dynamically active scalar field, and as per convention, the locality of the streamfunction-scalar relation is denoted by α, with smaller α implying increased locality (α=1 gives traditional 2D dynamics). The dispersive nature arises via a linear term whose strength, after nondimensionalization, is characterized by a parameter ϵ. Setting 0<ϵ≤1, we investigate the interplay of advection and dispersion for differing degrees of locality. Specifically, we study the forward (inverse) transfer of enstrophy (energy) under large-scale (small-scale) random forcing along with the geometry of the scalar field. Straightforward arguments suggest that for small α the scalar field should consist of progressively larger isotropic eddies, while for large α the scalar field is expected to have a filamentary structure resulting from a stretch and fold mechanism, much like...

New intermediate models for rotating shallow water and an investigation of the preference for anticyclones

New intermediate models for the rotating shallow water (RSW) equations are derived by considering the nonlinear interactions between subsets of the eigenmodes for the linearized equations. It is well-known that the two-dimensional quasi-geostrophic (QG) equation results when the nonlinear interactions are restricted to include only the vortical eigenmodes. Continuing past QG in a non-perturbative manner, the new models result by including subsets of interactions which include inertial-gravity wave (IG) modes. The such simplest model adds nonlinear interactions between one IG mode and two vortical modes. In sharp contrast to QG, the latter model behaves similar to the full RSW equations for decay from balanced initial conditions as well as unbalanced random initial conditions with divergence-free velocity. Quantitative agreement is observed for statistics that measure structure size, intermittency and cyclone/anticyclone asymmetry. In particular, dominance of anticyclones is observed for Rossby numbers Ro in the range 0.1 < Ro < 1 (away from the QG parameter regime Ro → 0). A hierarchy of models is explored to determine the effects of wave-vortical and wave-wave interactions on statistics such as the skewness of vorticity in decaying turbulence. Possible advantages over previously derived intermediate models include (i) the non-perturbative nature of the new models (not restricting them a priori to any particular parameter regime) and (ii) insight into the physical and mathematical consequences of vortical-wave interactions.

A priori tests of one-equation LES modeling of rotating turbulence

A priori tests of subgrid-scale (SGS) models are performed using results of 1283 direct numerical simulations for forced isotropic (Reλ = 100) and rotating turbulence (0.1 < Ro ω3 < 0.4). A range of SGS models is tested varying from algebraic, gradient, and scale similarity, to one-equation viscosity and non-viscosity dynamic structure models. Anisotropy and material frame indifference (MFI) requirements for SGS models in rotating systems are reviewed and used to help construct new models based on the dynamic structure approach. The models are evaluated primarily using correlation and regression coefficients of individual components of the SGS tensor, components of the divergence of the SGS stresses, and the kinetic energy transfer term between large and small scales. For all measures examined, the MFI-consistent dynamic structure models perform significantly better, especially for rotating turbulence.

On the Two-point Correlation of Potential Vorticity in Rotating and Stratified Turbulence

A framework is developed to describe the two-point statistics of potential vorticity in rotating and stratified turbulence as described by the Boussinesq equations. The Karman-Howarth equation for the dynamics of the two-point correlation function of potential vorticity reveals the possibility of inertial-range dynamics in certain regimes in the Rossby, Froude, Prandtl and Reynolds number parameters. For the case of large Rossby and Froude numbers, and for the case of quasi-geostrophic dynamics, a linear scaling law with 2/3 prefactor is derived for the third-order mixed correlation between potential vorticity and velocity, a result that is analogous to the Kolmogorov 4/5-law for the third-order velocity structure function in turbulence theory.

A-posteriori tests of one-equation LES modeling of rotating turbulence

Eight subgrid-scale (SGS) models were evaluated using two flow configurations: homogeneous decaying turbulence, and rotating turbulence forced at large or intermediate scales. Testing was performed for the first configuration through a systematic comparison between direct numerical simulation results and large eddy simulation results of many characteristics, including resolved kinetic energy, SGS energy production, molecular dissipation, and kinetic energy spectrum. The new models, which are based on dynamic structure model and satisfy the consistency of material frame indifference with the SGS stress, showed more accurate results than traditional models. In the forced testing, the new models were better able to capture essential features of rotating turbulence, including cyclonic/anti-cyclonic asymmetry, quasi-2D at large scales, and reverse kinetic energy transfer from small to large scales.

Self-similarity in decaying two-dimensional stably stratified adjustment

The evolution of large-scale density perturbations is studied in a stably stratified, two-dimensional flow governed by the Boussinesq equations. As is known, initially smooth density (or temperature) profiles develop into fronts in the very early stages of evolution. This results in a frontally dominated k−1 potential energy spectrum. The fronts, initially characterized by a relatively simple geometry, spontaneously develop into severely distorted sheets that possess structure at very fine scales, and thus there is a transfer of energy from large to small scales. It is shown here that this process culminates in the establishment of a k−5∕3 kinetic energy spectrum, although its scaling extends over a shorter range as compared to the k−1 scaling of the potential energy spectrum. The establishment of the kinetic energy scaling signals the onset of enstrophy decay, which proceeds in a mildly modulated exponential manner and possesses a novel self-similarity. Specifically, the self-similarity is seen in the ti...