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 $\epsilon = f/N$. Working in regimes characterized by moderate Burger numbers, i.e. $Bu = 1/\epsilon^2 < 1$ 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 $\epsilon=1$, for $\epsilon < 1$ 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 $\epsilon$ 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 $\epsilon > 1$ 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 $\epsilon = 1$. With regard to the vortical modes, for $\epsilon \le 1$, 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 $\epsilon > 1$ 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.Comment: 18 pages, 6 figs. (abbreviated abstract

Stages of dynamics in the Fermi-Pasta-Ulam system as probed by the first Toda integral

We investigate the long term evolution of trajectories in the Fermi-Pasta-Ulam (FPU) system, using as a probe the first non-trivial integral J in the hierarchy of integrals of the corresponding Toda lattice model. To this end we perform simulations of FPU-trajectories for various classes of initial conditions produced by the excitation of isolated modes, packets, as well as `generic' (random) initial data. For initial conditions corresponding to localized energy excitations, J exhibits variations yielding `sigmoid' curves similar to observables used in literature, e.g., the `spectral entropy' or various types of `correlation functions'. However, J(t) is free of fluctuations inherent in such observables, hence it constitutes an ideal observable for probing the timescales involved in the stages of FPU dynamics. We observe two fundamental timescales: i) the `time of stability' (in which, roughly, FPU trajectories behave like Toda), and ii) the `time to equilibrium' (beyond which energy equipartition is reached). Below a specific energy crossover, both times are found to scale exponentially as an inverse power of the specific energy. However, this crossover goes to zero with increasing the degrees of freedom N as εc∼N^(−b), with b∈[1.5,2.5]. For `generic data' initial conditions, instead, J(t) allows to quantify the continuous in time slow diffusion of the FPU trajectories in a direction transverse to the Toda tori

Observations and models of low-mode internal waves in the ocean

The generation of internal gravity waves in the ocean is largely driven by tides, winds, and interaction of currents with the seafloor. Models and observations indicate a global energy supply for the internal wave field of about 1 TW by the conversion of barotropic tides at mid-ocean ridges and abrupt topographic features. Winds acting on the oceanic mixed layer contribute 0.3--1.5 TW, and mesoscale flow over rough topography adds about 0.2 TW. Globally, 1--2 TW are needed to maintain the observed stratification of the deep ocean by diapycnal mixing that results from the breaking of internal waves. Ocean circulation models show significant impact of the spatial distribution of internal wave dissipation and mixing on the ocean state, e.g., thermal structure, stratification, and meridional overturning circulation. Observations indicate that the local ratio of generation and dissipation of internal waves is often below unity, and thus, the energy available for mixing must be redistributed by internal tides and near-inertial waves at low vertical wavenumber that can propagate thousands of kilometers from their source regions. Eddy-permitting global ocean circulation models are able to quantify the different sources of energy input and can also simulate the propagation of the lowest internal wave modes. However, the variation of the internal wave energy flux along its paths by wave--wave interaction, topographic scattering, and refraction by mesoscale features as well as its ultimate fate by dissipation remains to be parameterized