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Register a new userFormation of turbulence via an interaction potential
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Rafail V. Abramov
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In a recent work, we proposed a hypothesis that the turbulence in gases could be produced by particles interacting via a potential - for example, the interatomic potential at short ranges, and the electrostatic potential at long ranges. Here, we examine the proposed mechanics of turbulence formation in a simple model of two particles, which interact solely via a potential. Following the kinetic theory approach, we derive a hierarchy of the velocity moment transport equations, and then truncate it via a novel closure based on the high Reynolds number condition. While standard closures of the velocity moment hierarchy of the Boltzmann equation lead to the compressible Euler and Navier-Stokes systems of equations, our closure leads to a transport equation for the velocity alone, which is driven by the potential forcing. Starting from a large scale laminar shear flow, we numerically simulate the solutions of our velocity transport equation for the electrostatic, gravity, Thomas-Fermi and Lennard-Jones potentials, as well as the Vlasov-type large scale mean field potential. In all studied scenarios, the time-averaged Fourier spectra of the kinetic energy clearly exhibit Kolmogorovs five-thirds power decay rate.
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For a large system of identical particles interacting by means of a potential, we find that a strong large scale flow velocity can induce motions in the inertial range via the potential coupling. This forcing lies in special bundles in the Fourier space, which are formed by pairs of particles. These bundles are not present in the Boltzmann, Euler and Navier-Stokes equations, because they are destroyed by the Bogoliubov-Born-Green-Kirkwood-Yvon formalism. However, measurements of the flow can detect certain bulk effects shared across these bundles, such as the power scaling of the kinetic energy. We estimate the scaling effects produced by two types of potentials: the Thomas-Fermi interatomic potential (as well as its variations, such as the Ziegler-Biersack-Littmark potential), and the electrostatic potential. In the near-viscous inertial range, our estimates yield the inverse five-thirds power decay of the kinetic energy for both the Thomas-Fermi and electrostatic potentials. The electrostatic potential is also predicted to produce the inverse cubic power scaling of the kinetic energy at large inertial scales. Standard laboratory experiments confirm the scaling estimates for both the Thomas-Fermi and electrostatic potentials at near-viscous scales. Surprisingly, the observed kinetic energy spectrum in the Earth atmosphere at large scales behaves as if induced by the electrostatic potential. Given that the Earth atmosphere is not electrostatically neutral, we cautiously suggest a hypothesis that the atmospheric kinetic energy spectra in the inertial range are indeed driven by the large scale flow via the electrostatic potential coupling.
Turbulence in a system of nonlinearly interacting waves is referred to as wave turbulence. It has been known since seminal work by Kolmogorov, that turbulent dynamics is controlled by a directional energy flux through the wavelength scales. We demonstrate that an energy cascade in wave turbulence can be bi-directional, that is, can simultaneously flow towards large and small wavelength scales from the pumping scales at which it is injected. This observation is in sharp contrast to existing experiments and wave turbulence theory where the energy flux only flows in one direction. We demonstrate that the bi-directional energy cascade changes the energy budget in the system and leads to formation of large-scale, large-amplitude waves similar to oceanic rogue waves. To study surface wave turbulence, we took advantage of capillary waves on a free, weakly charged surface of superfluid helium He-II at temperature 1.7K. Although He-II demonstrates non-classical thermomechanical effects and quantized vorticity, waves on its surface are identical to those on a classical Newtonian fluid with extremely low viscosity. The possibility of directly driving a charged surface by an oscillating electric field and the low viscosity of He-II have allowed us to isolate the surface dynamics and study nonlinear surface waves in a range of frequencies much wider than in experiments with classical fluids.
The low wavenumber expansion of the energy spectrum takes the well known form: $ E(k,t) = E_2(t) k^2 + E_4(t) k^4 + ... $, where the coefficients are weighted integrals against the correlation function $C(r,t)$. We show that expressing $E(k,t)$ in terms of the longitudinal correlation function $f(r,t)$ immediately yields $E_2(t)=0$ by cancellation. We verify that the same result is obtained using the correlation function $C(r,t)$, provided only that $f(r,t)$ falls off faster than $r^{-3}$ at large values of $r$. As power-law forms are widely studied for the purpose of establishing bounds, we consider the family of model correlations $f(r,t)=alpha_n(t)r^{-n}$, for positive integer $n$, at large values of the separation $r$. We find that for the special case $n=3$, the relationship connecting $f(r,t)$ and $C(r,t)$ becomes indeterminate, and (exceptionally) $E_2 eq 0$, but that this solution is unphysical in that the viscous term in the K{a}rm{a}n-Howarth equation vanishes. Lastly, we show that $E_4(t)$ is independent of time, without needing to assume the exponential decrease of correlation functions at large distances.
Wave--current interaction (WCI) dynamics energizes and mixes the ocean thermocline by producing a combination of Langmuir circulation, internal waves and turbulent shear flows, which interact over a wide range of time scales. Two complementary approaches exist for approximating different aspects of WCI dynamics. These are the Generalized Lagrangian Mean (GLM) approach and the Gent--McWilliams (GM) approach. Their complementarity is evident in their Kelvin circulation theorems. GLM introduces a wave pseudomomentum per unit mass into its Kelvin circulation integrand, while GM introduces a an additional `bolus velocity to transport its Kelvin circulation loop. The GLM approach models Eulerian momentum, while the GM approach models Lagrangian transport. In principle, both GLM and GM are based on the Euler--Boussinesq (EB) equations for an incompressible, stratified, rotating flow. The differences in their Kelvin theorems arise from differences in how they model the flow map in the Lagrangian for the Hamilton variational principle underlying the EB equations. A recently developed approach for uncertainty quantification in fluid dynamics constrains fluid variational principles to require that Lagrangian trajectories undergo Stochastic Advection by Lie Transport (SALT). Here we introduce stochastic closure strategies for quantifying uncertainty in WCI by adapting the SALT approach to both the GLM and GM approximations of the EB variational principle. In the GLM framework, we introduce a stochastic group velocity for transport of wave properties, relative to the frame of motion of the Lagrangian mean flow velocity and a stochastic pressure contribution from the fluctuating kinetic energy. In the GM framework we introduce a stochastic bolus velocity in addition to the mean drift velocity by imposing the SALT constraint in the GM variational principle.
We are modelling multi-scale, multi-physics uncertainty in wave-current interaction (WCI). To model uncertainty in WCI, we introduce stochasticity into the wave dynamics of two classic models of WCI; namely, the Generalised Lagrangian Mean (GLM) model and the Craik--Leibovich (CL) model. The key idea for the GLM approach is the separation of the Lagrangian (fluid) and Eulerian (wave) degrees of freedom in Hamiltons principle. This is done by coupling an Euler--Poincare {it reduced Lagrangian} for the current flow and a {it phase-space Lagrangian} for the wave field. WCI in the GLM model involves the nonlinear Doppler shift in frequency of the Hamiltonian wave subsystem, which arises because the waves propagate in the frame of motion of the Lagrangian-mean velocity of the current. In contrast, WCI in the CL model arises because the fluid velocity is defined relative to the frame of motion of the Stokes mean drift velocity, which is usually taken to be prescribed, time independent and driven externally. We compare the GLM and CL theories by placing them both into the general framework of a stochastic Hamiltons principle for a 3D Euler--Boussinesq (EB) fluid in a rotating frame. In other examples, we also apply the GLM and CL methods to add wave physics and stochasticity to the familiar 1D and 2D shallow water flow models. The differences in the types of stochasticity which arise for GLM and CL models can be seen by comparing the Kelvin circulation theorems for the two models. The GLM model acquires stochasticity in its Lagrangian transport velocity for the currents and also in its group velocity for the waves. The Kelvin circulation theorem stochastic CL model can accept stochasticity in its both its integrand and in the Lagrangian transport velocity of its circulation loop.
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