arXiv subjects

Leslie M. Smith

Publications and source records attributed to Leslie M. Smith.

24 records · Page 2Linked to original sources

arXiv2009

Nonlinear inertia-gravity wave-mode interactions in three dimensional rotating stratified flows

We investigate the dynamics of inertia-gravity wave modes in 3D rotating stratified fluids. We start by deriving a reduced PDE, the GGG model, consisting of only wave-mode interactions. In principle, comparing this model to the full rotating Boussinesq system allows us to gauge the importance of wave-vortical-wave vs. wave-wave-wave interactions in determining the transfer and distribution of wave-mode energy. As in many atmosphere-ocean phenomena we work in a skewed aspect ratio domain (H/L) with Fr = Ro < 1 such that Bu = 1. Our focus is on the equilibration of wave-mode energy and its spectral scaling under the influence of random large-scale forcing. As anticipated, when forcing is applied to all modes, with Fr=Ro ~ 0.05 and H/L=1/5, the wave-mode energy equilibrates and its spectrum scales as a power-law. For the same parameters, when forcing is restricted to only wave modes, the energy fails to equilibrate in both the full system as well as the GGG subsystem at resolutions we can achieve. This cleary demonstrates the importance of the vortical mode in determining the wave-mode energy distribution. Proceeding to the second set of simulations, i.e. for the larger Fr=Ro ~ 0.1 in a less skewed aspect ratio domain with H/L=1/3, we observe that the energy of the GGG subsystem equilibrates. Further, the full system with forcing restricted to wave modes also equilbrates and both yield identical energy spectra. Thus it is clear that the wave-wave-wave interactions play a role in the overall dynamics at moderate Ro, Fr and aspect ratios. Apart from theoretical concerns, these results highlight the difficulty in properly resolving wave-mode interactions when simulating realistic geophysical phenomena.

physics.flu-dyn↗

arXiv2009

Local and Nonlocal Dispersive Turbulence

We consider the evolution of a family of 2D dispersive turbulence models. The members of this family involve the nonlinear advection of a dynamically active scalar field, the locality of the streamfunction-scalar relation is denoted by $α$, with smaller $α$ implying increased locality. The dispersive nature arises via a linear term whose strength is characterized by a parameter $ε$. Setting $0 < ε\le 1$, 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. Straightforward arguments suggest that for small $α$ the scalar field should consist of progressively larger eddies, while for large $α$ the scalar field is expected to have a filamentary structure resulting from a stretch and fold mechanism. Confirming this, we proceed to forced/dissipative dispersive numerical experiments under weakly non-local to local conditions. For $ε\sim 1$, there is quantitative agreement between non-dispersive estimates and observed slopes in the inverse energy transfer regime. On the other hand, forward enstrophy transfer regime always yields slopes that are significantly steeper than the corresponding non-dispersive estimate. Additional simulations show the scaling in the inverse regime to be sensitive to the strength of the dispersive term : specifically, as $ε$ decreases, the inertial-range shortens and we also observe that the slope of the power-law decreases. On the other hand, for the same range of $ε$ values, the forward regime scaling is fairly universal.

physics.flu-dyn↗

arXiv2007

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 and wave modes of a 3D rotating stratified fluid as a function of $ε= f/N$. Working in regimes characterized by moderate Burger numbers, i.e. $Bu = 1/ε^2 < 1$ or $Bu \ge 1$, our results indicate profound change in the character of vortical and wave mode interactions with respect to $Bu = 1$. 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 $ε\le 1$, 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 plays an energetically smaller role in the overall dynamics.

physics.flu-dyn↗

arXiv2006

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, intially 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 time invariant nature of the probability density function (\pdf{}) associated with the normalized vorticity field. Given the rapid decay of energy at this stage, the spectral scaling is transient and fades with the emergence of a smooth, large-scale, very slowly decaying, (almost) vertically sheared horizontal mode with most of its energy in the potential component -- i.e. the Pearson-Linden regime.

physics.flu-dyn↗

arXiv2006

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.

nlin.CD↗

arXiv2002

Stability of Rossby waves in the beta-plane approximation

Floquet theory is used to describe the unstable spectrum at large scales of the beta-plane equation linearized about Rossby waves. Base flows consisting of one to three Rossby wave are considered analytically using continued fractions and the method of multiple scales, while base flow with more than three Rossby waves are studied numerically. It is demonstrated that the mechanism for instability changes from inflectional to triad resonance at an O(1) transition Rhines number Rh, independent of the Reynolds number. For a single Rossby wave base flow, the critical Reynolds number Re^c for instability is found in various limits. In the limits Rh --> infinity and k --> 0, the classical value Re^c = sqrt(2) is recovered. For Rh --> 0 and all orientations of the Rossby wave except zonal and meridional, the base flow is unstable for all Reynolds numbers; a zonal Rossby wave is stable, while a meridional Rossby wave has critical Reynolds number Re^c = sqrt(2). For more isotropic base flows consisting of many Rossby waves (up to forty), the most unstable mode is purely zonal for 2 <= Rh < infinity and is nearly zonal for Rh = 1/2, where the transition Rhines number is again O(1), independent of the Reynolds number and consistent with a change in the mechanism for instability from inflectional to triad resonance.

physics.ao-ph↗