Statistical mechanics of two-dimensional Euler flows and minimum enstrophy states

A simplified thermodynamic approach of the incompressible 2D Euler equation is considered based on the conservation of energy, circulation and microscopic enstrophy. Statistical equilibrium states are obtained by maximizing the Miller-Robert-Sommeria (MRS) entropy under these sole constraints. The vorticity fluctuations are Gaussian while the mean flow is characterized by a linear $\bar{\omega}-\psi$ relationship. Furthermore, the maximization of entropy at fixed energy, circulation and microscopic enstrophy is equivalent to the minimization of macroscopic enstrophy at fixed energy and circulation. This provides a justification of the minimum enstrophy principle from statistical mechanics when only the microscopic enstrophy is conserved among the infinite class of Casimir constraints. A new class of relaxation equations towards the statistical equilibrium state is derived. These equations can provide an effective description of the dynamics towards equilibrium or serve as numerical algorithms to determine maximum entropy or minimum enstrophy states. We use these relaxation equations to study geometry induced phase transitions in rectangular domains. In particular, we illustrate with the relaxation equations the transition between monopoles and dipoles predicted by Chavanis and Sommeria [J. Fluid. Mech. 314, 267 (1996)]. We take into account stable as well as metastable states and show that metastable states are robust and have negative specific heats. This is the first evidence of negative specific heats in that context. We also argue that saddle points of entropy can be long-lived and play a role in the dynamics because the system may not spontaneously generate the perturbations that destabilize them.Comment: 26 pages, 10 figure

Statistical mechanics of Beltrami flows in axisymmetric geometry: Equilibria and bifurcations

We characterize the thermodynamical equilibrium states of axisymmetric Euler-Beltrami flows. They have the form of coherent structures presenting one or several cells. We find the relevant control parameters and derive the corresponding equations of state. We prove the coexistence of several equilibrium states for a given value of the control parameter like in 2D turbulence [Chavanis and Sommeria, J. Fluid Mech. 314, 267 (1996)]. We explore the stability of these equilibrium states and show that all states are saddle points of entropy and can, in principle, be destabilized by a perturbation with a larger wavenumber, resulting in a structure at the smallest available scale. This mechanism is therefore reminiscent of the 3D Richardson energy cascade towards smaller and smaller scales. Therefore, our system is truly intermediate between 2D turbulence (coherent structures) and 3D turbulence (energy cascade). We further explore numerically the robustness of the equilibrium states with respect to random perturbations using a relaxation algorithm in both canonical and microcanonical ensembles. We show that saddle points of entropy can be very robust and therefore play a role in the dynamics. We evidence differences in the robustness of the solutions in the canonical and microcanonical ensembles. A scenario of bifurcation between two different equilibria (with one or two cells) is proposed and discussed in connection with a recent observation of a turbulent bifurcation in a von Karman experiment [Ravelet et al., Phys. Rev. Lett. 93, 164501 (2004)].Comment: 25 pages; 16 figure

Neuropathic pain after thoracotomy: tracking signs and symptoms before and at monthly intervals following surgery

Background: Because the development of neuropathic symptoms contributes to pain severity and chronification after surgery, their early prediction is important to allow targeted treatment. Objectives: We longitudinally investigated trajectories of signs and symptoms in patients undergoing thoracotomy and assessed whether and at which time they were related to the development of neuropathic pain symptoms six months after surgery. Methods: Presurgical and six, monthly postsurgical assessments included questionnaires for mental and physical well-being (e.g. depression/anxiety, pain catastrophizing, sleep quality, neuropathic pain symptoms), and quantitative sensory testing (QST). Results: QST trajectories indicated nerve impairment of the surgery site with predominant loss of function. Signs of recovery towards the end of the assessment period were observed for some tests. Unsupervised cluster analysis with NPSI scores six months after surgery as clustering variable identified one group with no/low levels of neuropathic symptoms and one with moderate levels. The two groups differed w.r.t. several signs and symptoms already at early time points. Notably, neuropathic pain anywhere in the body differed already preoperatively and sleep impairment differentiated the two groups at all time points. Regression analysis revealed three factors that seemed particularly suited to predicted six months NPSI scores, namely preoperative neuropathic pain symptoms, with contributions from sleep impairment one month after surgery and the presence of dynamic mechanical allodynia three months after surgery. Conclusions: Clinical routine should focus on the individual’s physiological state, including pre-existing neuropathic pain and sleep quality to identify patients early who might be at risk to develop chronic post-surgical neuropathic pain

Statistical mechanics of Fofonoff flows in an oceanic basin

We study the minimization of potential enstrophy at fixed circulation and energy in an oceanic basin with arbitrary topography. For illustration, we consider a rectangular basin and a linear topography h=by which represents either a real bottom topography or the beta-effect appropriate to oceanic situations. Our minimum enstrophy principle is motivated by different arguments of statistical mechanics reviewed in the article. It leads to steady states of the quasigeostrophic (QG) equations characterized by a linear relationship between potential vorticity q and stream function psi. For low values of the energy, we recover Fofonoff flows [J. Mar. Res. 13, 254 (1954)] that display a strong westward jet. For large values of the energy, we obtain geometry induced phase transitions between monopoles and dipoles similar to those found by Chavanis and Sommeria [J. Fluid Mech. 314, 267 (1996)] in the absence of topography. In the presence of topography, we recover and confirm the results obtained by Venaille and Bouchet [Phys. Rev. Lett. 102, 104501 (2009)] using a different formalism. In addition, we introduce relaxation equations towards minimum potential enstrophy states and perform numerical simulations to illustrate the phase transitions in a rectangular oceanic basin with linear topography (or beta-effect).Comment: 26 pages, 28 figure

The Paganica Fault and surface coseismic ruptures caused by the 6 april 2009 earthquake (L’Aquila, central Italy)

On 6 April 2009, at 01:32 GMT, an Mw 6.3 seismic event hit the central Apennines, severely damaging the town of L’Aquila and dozens of neighboring villages and resulting in approximately 300 casualties (Istituto Nazionale di Geofisica e Vulcanologia, http://www.ingv.it; MedNet, http://mednet.rm.ingv.it/proce- dure/events/QRCMT/090406_013322/qrcmt.html). This earth- quake was the strongest in central Italy since the devastating 1915 Fucino event (Mw 7.0). The INGV national seismic net- work located the hypocenter 5 km southwest of L’Aquila, 8–9 km deep. Based on this information and on the seismotectonic framework of the region, earthquake geologists traveled to the field to identify possible surface faulting (Emergeo Working Group 2009a, 2009b). The most convincing evidence of pri- mary surface rupture is along the Paganica fault, the geometry of which is consistent with seismological, synthetic aperture radar (SAR) and GPS data. Investigation of other known nor- mal faults of the area, i.e., the Mt. Pettino, Mt. San Franco, and Mt. Stabiata normal faults suggested that these structures were not activated during the April 6 shock (Emergeo Working Group 2009a, 2009b). In this report, we first describe the seismotectonic frame- work of the area, and then we present the field information that supports the occurrence of surficial displacement on the Paganica fault.Published940-9503.2. Tettonica attivaJCR Journalope

Susceptibility divergence, phase transition and multistability of a highly turbulent closed flow

Using time-series of stereoscopic particle image velocimetry data, we study the response of a turbulent von K\'{a}rm\'{a}n swirling flow to a continuous breaking of its forcing symmetry. Experiments are carried over a wide Reynolds number range, from laminar regime at $Re = 10^{2}$ to highly turbulent regime near $Re = 10^{6}$. We show that the flow symmetry can be quantitatively characterized by two scalars, the global angular momentum $I$ and the mixing layer altitude $z_s$, which are shown to be statistically equivalent. Furthermore, we report that the flow response to small forcing dissymetry is linear, with a slope depending on the Reynolds number: this response coefficient increases non monotonically from small to large Reynolds number and presents a divergence at a critical Reynolds number $Re_c = 40\,000 \pm 5\,000$. This divergence coincides with a change in the statistical properties of the instantaneous flow symmetry $I(t)$: its pdf changes from Gaussian to non-Gaussian with multiple maxima, revealing metastable non-symmetrical states. For symmetric forcing, a peak of fluctuations of $I(t)$ is also observed at $Re_c$: these fluctuations correspond to time-intermittencies between metastable states of the flow which, contrary to the very-long-time-averaged mean flow, spontaneously and dynamically break the system symmetry. We show that these observations can be interpreted in terms of divergence of the susceptibility to symmetry breaking, revealing the existence of a phase transition. An analogy with the ferromagnetic-paramagnetic transition in solid-state physics is presented and discussed.Comment: to appear in Journal of Statistical Mechanic

The Paganica Fault and surface coseismic ruptures caused by the 6 april 2009 earthquake (L’Aquila, central Italy)

On 6 April 2009, at 01:32 GMT, an Mw 6.3 seismic event hit the central Apennines, severely damaging the town of L’Aquila and dozens of neighboring villages and resulting in approximately 300 casualties (Istituto Nazionale di Geofisica e Vulcanologia, http://www.ingv.it; MedNet, http://mednet.rm.ingv.it/proce- dure/events/QRCMT/090406_013322/qrcmt.html). This earth- quake was the strongest in central Italy since the devastating 1915 Fucino event (Mw 7.0). The INGV national seismic net- work located the hypocenter 5 km southwest of L’Aquila, 8–9 km deep. Based on this information and on the seismotectonic framework of the region, earthquake geologists traveled to the field to identify possible surface faulting (Emergeo Working Group 2009a, 2009b). The most convincing evidence of pri- mary surface rupture is along the Paganica fault, the geometry of which is consistent with seismological, synthetic aperture radar (SAR) and GPS data. Investigation of other known nor- mal faults of the area, i.e., the Mt. Pettino, Mt. San Franco, and Mt. Stabiata normal faults suggested that these structures were not activated during the April 6 shock (Emergeo Working Group 2009a, 2009b). In this report, we first describe the seismotectonic frame- work of the area, and then we present the field information that supports the occurrence of surficial displacement on the Paganica fault

Effectiveness of non-animal chondroitin sulfate supplementation in the treatment of moderate knee osteoarthritis in a group of overweight subjects: A randomized, double-blind, placebo-controlled pilot study

Osteoarthritis (OA) is the most common form of arthritis in the world and is characterized by pain, various disabilities and loss of quality of life. Chondroitin sulfate (CS) is recommended as first-line therapy. CS of non-animal origin is of great interest for safety and sustainability reasons. This study aims to investigate the anti-inflammatory effects, anti-pain and ability-enhancement of a short-term supplementation with non-animal CS in overweight subjects with OA. In a randomized, double-blind, placebo-controlled pilot study, 60 overweight adults with symptomatic OA were allocated to consume 600 mg of non-animal CS (n = 30) or a placebo (n = 30) daily for 12 consecutive weeks. The assessment of knee-pain, quality of life, related inflammation markers and body composition was performed at 0, 4 and 12 weeks. The Tegner Lysholm Knee Scoring (TLKS) scale of the experimental group showed a statistically significant increase (+10.64 points; confidence interval (95% confidence interval (CI) 5.57; 15.70; p < 0.01), while the Western Ontario and McMaster Universities Osteoarthritis Index (WOMAC) score decreased (−12.24 points; CI 95% −16.01; −8.38; p < 0.01). The results also showed a decrease in the C-reactive protein (CRP) level (−0.14 mg/dL, CI 95% −0.26; −0.04; p < 0.01) and erythrocyte sedimentation rate (ESR) level (−5.01 mm/h, CI 95% −9.18; −0.84, p < 0.01) as well as the visual analogue scale (VAS) score in both knees. In conclusion, this pilot study demonstrates the effectiveness of non-animal CS supplementation in overweight subjects with knee OA in improving knee function, pain and inflammation markers