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In this paper we analyze crowd turbulence from both classical and quantum perspective. We analyze various crowd waves and collisions using crowd macroscopic wave function. In particular, we will show that nonlinear Schr{o}dinger (NLS) equation is fundamental for quantum turbulence, while its closed-form solutions include shock-waves, solitons and rogue waves, as well as planar de Broglies waves. We start by modeling various crowd flows using classical fluid dynamics, based on Navier-Stokes equations. Then, we model turbulent crowd flows using quantum turbulence in Bose-Einstein condensation, based on modified NLS equation. Keywords: Crowd behavior dynamics, classical and quantum turbulence, shock waves, solitons and rogue waves
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We propose an entropic geometrical model of psycho-physical crowd dynamics (with dissipative crowd kinematics), using Feynman action-amplitude formalism that operates on three synergetic levels: macro, meso and micro. The intent is to explain the dynamics of crowds simultaneously and consistently across these three levels, in order to characterize their geometrical properties particularly with respect to behavior regimes and the state changes between them. Its most natural statistical descriptor is crowd entropy $S$ that satisfies the Prigogines extended second law of thermodynamics, $partial_tSgeq 0$ (for any nonisolated multi-component system). Qualitative similarities and superpositions between individual and crowd configuration manifolds motivate our claim that goal-directed crowd movement operates under entropy conservation, $partial_tS = 0$, while natural crowd dynamics operates under (monotonically) increasing entropy function, $partial_tS > 0$. Between these two distinct topological phases lies a phase transition with a chaotic inter-phase. Both inertial crowd dynamics and its dissipative kinematics represent diffusion processes on the crowd manifold governed by the Ricci flow, with the associated Perelman entropy-action. Keywords: Crowd psycho-physical dynamics, action-amplitude formalism, crowd manifold, Ricci flow, Perelman entropy, topological phase transition
We have obtained a contiguous set of long-slit spectra of a shock wave in the Cygnus Loop to investigate its structure, which is far from the morphology predicted by 1D models. Proper motions from Hubble Space Telescope images combined with the known distance to the Cygnus Loop provide an accurate shock speed. Earlier analyses of shock spectra estimated the shock speed, postshock density, temperature, and elemental abundances. In this paper we determine several more shock parameters: a more accurate shock speed, ram pressure, density, compression ratio, dust destruction efficiency, magnetic field strength, and vorticity in the cooling region. From the derived shock properties we estimate the emissivities of synchrotron emission in the radio and pion decay emission in the gamma rays. Both are consistent with the observations if we assume simple adiabatic compression of ambient cosmic rays as in the van der Laan mechanism. We also find that, although the morphology is far from that predicted by 1D models and the line ratios vary dramatically from point to point, the average spectrum is matched reasonably well by 1D shock models with the shock speed derived from the measured proper motion. A subsequent paper will analyze the development of turbulence in the cooling zone behind the shock.
Supersonic turbulence generates distributions of shock waves. Here, we analyse the shock waves in three-dimensional numerical simulations of uniformly driven supersonic turbulence, with and without magnetohydrodynamics and self-gravity. We can identify the nature of the turbulence by measuring the distribution of the shock strengths. We find that uniformly driven turbulence possesses a power law distribution of fast shocks with the number of shocks inversely proportional to the square root of the shock jump speed. A tail of high speed shocks steeper than Gaussian results from the random superposition of driving waves which decay rapidly. The energy is dissipated by a small range of fast shocks. These results contrast with the exponential distribution and slow shock dissipation associated with decaying turbulence. A strong magnetic field enhances the shock number transverse to the field direction at the expense of parallel shocks. A simulation with self-gravity demonstrates the development of a number of highly dissipative accretion shocks. Finally, we examine the dynamics to demonstrate how the power-law behaviour arises.
Radiative shock waves in the Cygnus Loop and other supernova remnants show different morphologies in [O III] and H{alpha} emission. We use HST spectra and narrowband images to study the development of turbulence in the cooling region behind a shock on the west limb of the Cygnus Loop. We refine our earlier estimates of shock parameters that were based upon ground-based spectra, including ram pressure, vorticity and magnetic field strength. We apply several techniques, including Fourier power spectra and the Rolling Hough Transform, to quantify the shape of the rippled shock front as viewed in different emission lines. We assess the relative importance of thermal instabilities, the thin shell instability, upstream density variations, and upstream magnetic field variations in producing the observed structure.
Turbulence structure resulting from multi-fluid or multi-species, variable-density isotropic turbulence interaction with a Mach 2 shock is studied using turbulence-resolving shock-capturing simulations and Eulerian (grid) and Lagrangian (particle) methods. The complex roles density play in the modification of turbulence by the shock wave are identified. Statistical analyses of the velocity gradient tensor (VGT) show that the density variations significantly change the turbulence structure and flow topology. Specifically, a stronger symmetrization of the joint probability density function (PDF) of second and third invariants of the anisotropic velocity gradient tensor, PDF$(Q^ast, R^ast)$, as well as the PDF of the vortex stretching contribution to the enstrophy equation, are observed in the multi-species case. Furthermore, subsequent to the interaction with the shock, turbulent statistics also acquire a differential distribution in regions having different densities. This results in a nearly symmetrical PDF$(Q^ast, R^ast)$ in heavy fluid regions, while the light fluid regions retain the characteristic tear-drop shape. To understand this behavior and the return to standard turbulence structure as the flow evolves away from the shock, Lagrangian dynamics of the VGT and its invariants are studied by considering particle residence times and conditional particle variables in different flow regions. The pressure Hessian contributions to the VGT invariants transport equations are shown to be not only affected by the shock wave, but also by the density in the multi-fluid case, making them critically important to the flow dynamics and turbulence structure.
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