We prove an estimate of total (viscous plus modelled turbulent) energy dissipation in general eddy viscosity models for shear flows. For general eddy viscosity models, we show that the ratio of the near wall average viscosity to the effective global viscosity is the key parameter. This result is then applied to the 1-equation, URANS model of turbulence for which this ratio depends on the specification of the turbulence length scale. The model, which was derived by Prandtl in 1945, is a component of a 2-equation model derived by Kolmogorov in 1942 and is the core of many unsteady, Reynolds averaged models for prediction of turbulent flows. Away from walls, interpreting an early suggestion of Prandtl, we set \begin{equation*} l=\sqrt{2}k^{+1/2}τ, \hspace{50mm} \end{equation*} where $τ=$ selected time scale. In the near wall region analysis suggests replacing the traditional $l=0.41d$ ($d=$ wall normal distance) with $l=0.41d\sqrt{d/L}$ giving, e.g., \begin{equation*} l=\min \left\{ \sqrt{2}k{}^{+1/2}τ,\text{ }0.41d\sqrt{\frac{d}{L}} \right\} . \hspace{50mm} \end{equation*} This $l(\cdot )$ results in a simpler model with correct near wall asymptotics. Its energy dissipation rate scales no larger than the physically correct $O(U^{3}/L)$, balancing energy input with energy dissipation.
We show for the first time that sustained turbulence is possible at low magnetic Prandtl number for Keplerian flows with no mean magnetic flux. Our results indicate that increasing the vertical domain size is equivalent to increasing the dynamical range between the energy injection scale and the dissipative scale. This has important implications for a large variety of differentially rotating systems with low magnetic Prandtl number such as protostellar disks and laboratory experiments.
Convective turbulence in a Rayleigh Benard system has shown a marked reluctance to exhibit clear scaling in the energy or entropy spectrum. The recent numerical simulation of Pandey, Verma and Mishra has shown significantly better evidence of scaling in the infinite Prandtl number limit. This prompted us to look at this limit analytically. We find that the inevitable presence of sweeping helps give a very good understanding of the results of Pandey et. al. In the presence of rotation, the Rayleigh number dependence of the Nusselt number shows a strong increase in slope in recent experiments an…
We explore the effect of the magnetic Prandtl number Pm on energy and dissipation in fully resolved direct numerical simulations of steady-state, mechanically forced homogeneous magnetohydrodynamic turbulence in the range Pm = 1/32 to 32. We compare the spectra and show that if the simulations are not fully resolved, the steepness of the scaling of the kinetic-to-magnetic dissipation ratio with Pm is overestimated. We also present results of decaying turbulence with helical and nonhelical magnetic fields, where we find nonhelical reverse spectral transfer for Pm < 1. The results of this system…
Liquid metals play a central role in new generation liquid metal cooled nuclear reactors, for which numerical investigations require the use of appropriate thermal turbulence models for low Prandtl number fluids. Given the limitations of traditional modelling approaches and the increasing availability of high-fidelity data for this class of fluids, we propose a Machine Learning strategy for the modelling of the turbulent heat flux. A comprehensive algebraic mathematical structure is derived and physical constraints are imposed to ensure attractive properties promoting applicability, robustness…
We extend the scaling relations of stably stratified turbulence from the geophysical regime of unity Prandtl number to the astrophysical regime of extremely small Prandtl number applicable to stably stratified regions of stars and gas giants. A transition to a new turbulent regime is found to occur when the Prandtl number drops below the inverse of the buoyancy Reynolds number, i.e. $PrRb<1$, which signals a shift of the dominant balance in the buoyancy equation. Application of critical balance arguments then derives new predictions for the anisotropic energy spectrum and dominant balance of t…
The small-scale dynamo is a process by which turbulent kinetic energy is converted into magnetic energy, and thus is expected to depend crucially on the nature of turbulence. In this work, we present a model for the small-scale dynamo that takes into account the slope of the turbulent velocity spectrum v(l) ~ l^theta, where l and v(l) are the size of a turbulent fluctuation and the typical velocity on that scale. The time evolution of the fluctuation component of the magnetic field, i.e., the small-scale field, is described by the Kazantsev equation. We solve this linear differential equation …
A phenomenology of isotropic magnetohydrodynamic turbulence subject to both rotation and applied magnetic field is presented. It is assumed that the triple correlations decay-time is the shortest between the eddy turn-over time and the ones associated to the rotating frequency and Alfvén wave period. For $Pm=1$ it leads to four kinds of piecewise spectra, depending on the four parameters, injection rate of energy, magnetic diffusivity, rotation rate and applied field. With a shell model of MHD turbulence (including rotation and applied magnetic field), spectra for $Pm \le 1$ are presented, tog…
Three-dimensional miscible Rayleigh--Taylor (RT) turbulence at small Atwood number and at Prandtl number one is investigated by means of high resolution direct numerical simulations of the Boussinesq equations. RT turbulence is a paradigmatic time-dependent turbulent system in which the integral scale grows in time following the evolution of the mixing region. In order to fully characterize the statistical properties of the flow, both temporal and spatial behavior of relevant statistical indicators have been analyzed. Scaling of both global quantities ({\it e.g.}, Rayleigh, Nusselt and Reynold…
We analyze the initial, kinematic stage of magnetic field evolution in an isotropic and homogeneous turbulent conducting fluid with a rough velocity field, v(l) ~ l^alpha, alpha<1. We propose that in the limit of small magnetic Prandtl number, i.e. when ohmic resistivity is much larger than viscosity, the smaller the roughness exponent, alpha, the larger the magnetic Reynolds number that is needed to excite magnetic fluctuations. This implies that numerical or experimental investigations of magnetohydrodynamic turbulence with small Prandtl numbers need to achieve extremely high resolution in o…
Using direct numerical simulations of three-dimensional hydromagnetic turbulence, either with helical or non-helical forcing, we show that the ratio of kinetic-to-magnetic energy dissipation always increases with the magnetic Prandtl number, i.e., the ratio of kinematic viscosity to magnetic diffusivity. This dependence can always be approximated by a power law, but the exponent is not the same in all cases. For non-helical turbulence, the exponent is around 1/3, while for helical turbulence it is between 0.6 and 2/3. In the statistically steady state, the rate of the energy conversion from ki…
Turbulent motions enhance the diffusion of large-scale flows and temperature gradients. Such diffusion is often parameterized by coefficients of turbulent viscosity ($ν_{\rm t}$) and turbulent thermal diffusivity ($χ_{\rm t}$) that are analogous to their microscopic counterparts. We compute the turbulent diffusion coefficients by imposing large-scale velocity and temperature gradients on a turbulent flow and measuring the response of the system. We also confirm our results using experiments where the imposed gradients are allowed to decay. To achieve this, we use weakly compressible three-dime…
Compressible hydrodynamic turbulence is studied under the assumption of a polytropic closure. Following Kolmogorov, we derive an exact relation for some two-point correlation functions in the asymptotic limit of a high Reynolds number.
The paper presents a theory of shear-generated turbulence at asymptotically high Reynolds numbers. It is based on an ensemble of dipole vortex tubes taken as quasi-particles and realized in form of rings, hairpins or filament couples of potentially finite length. In a not necesserily planar cross sectional area through a vortex tangle, taken locally orthogonal through each individual tube, the dipoles are moving with the classical dipole velocity. The vortex radius is directly related with Prandtl's classical mixing length. The quasi-particles perform dipol chaos which reminds of molecular cha…
Horizontally extended turbulent convection, termed mesoscale convection in natural systems, remains a challenge to investigate in both experiments and simulations. This is particularly so for very low molecular Prandtl numbers as in stellar convection and in Earth's outer core. The present study reports three-dimensional direct numerical simulations of turbulent Rayleigh-Bénard convection in square boxes of side length $L$ and height $H$ with the aspect ratio $Γ=L/H$ of 25, for Prandtl numbers that span almost 4 orders of magnitude, $10^{-3}\le Pr \le 7$, and Rayleigh numbers $10^5 \le Ra \le …
We use the more intuitive approach due to Kolmogorov (and subsequently, Landau in his text on fluid dynamics) to calculate some third-order structure functions for quasi-geostrophic turbulence for the forward cascade of pseudo-potential enstrophy and the inverse energy cascade in quasi-geostrophic turbulence.
The fully developed turbulent Boussinesq convection is known to form large-scale rolls, often termed the 'large-scale circulation' (LSC). It is an interesting question how such a large-scale flow is created, in particular in systems when the energy input occurs at small scales, when inverse cascade is required in order to transfer energy into the large-scale modes. Here, the small-scale driving is introduced through stochastic, randomly distributed heat source (say radiational). The mean flow equations are derived by means of simplified renormalization group technique, which can be termed 'wea…
The transition to a new turbulent regime in horizontal convection in the case of low Prandtl numbers is analyzed using the Shishkina, Grossmann & Lohse (SGL) theory. The flow driven by the horizontal gradient along a horizontal surface, perpendicular to the acceleration of gravity is shown to transition to turbulence in the plume and the core. This transition to turbulence sets a sequence of heat transfer and momentum transport scalings which are found to follow the SGL prediction for the scaling factors and the prediction of Hughes, Griffith & Mullarney (HGM) for larger forcing amplitudes. Th…
This paper is a synopsis of the recent book A. Boritchev, S. Kuksin, \textit{One-Dimensional Turbulence and the Stochastic Burgers Equation}, AMS Publications, 2021 (to appear). The book is dedicated to the stochastic Burgers equation as a model for 1d turbulence, and the paper discusses its content in relation to the Kolmogorov theory of turbulence.
We use the 2 particle irreducible Schwinger-Keldysh effective action to set up consistent equations for the velocity and pressure correlations of a turbulent flow. We use these equations to derive the Kadanoff-Baym equations describing the relaxation to Kolmogorov turbulence in the absence of mean velocities.
(abridged) Context: Turbulent diffusion of large-scale flows and magnetic fields play major roles in many astrophysical systems. Aims: Our goal is to compute turbulent viscosity and magnetic diffusivity, relevant for diffusing large-scale flows and magnetic fields, respectively, and their ratio, the turbulent magnetic Prandtl number, ${\rm Pm}_{\rm t}$, for isotropically forced homogeneous turbulence. Methods: We use simulations of forced turbulence in fully periodic cubes composed of isothermal gas with an imposed large-scale sinusoidal shear flow. Turbulent viscosity is computed either from …
We investigate sustenance and dependence on magnetic Prandtl number (${\rm Pm}$) for magnetorotational instability (MRI)-driven turbulence in astrophysical Keplerian disks with zero net magnetic flux using standard shearing box simulations. We focus on the turbulence dynamics in Fourier space, capturing specific/noncanonical anisotropy of nonlinear processes due to disk flow shear. This is a new type of nonlinear redistribution of modes over wavevector orientations in Fourier space -- the nonlinear transverse cascade -- which is generic to shear flows and fundamentally different from usual dir…
The Rayleigh (Ra) and Prandtl (Pr) number scaling of the Nusselt number Nu, the Reynolds number Re, the temperature fluctuations, and the kinetic and thermal dissipation rates is studied for (numerical) homogeneous Rayleigh-Benard turbulence, i.e., Rayleigh-Benard turbulence with periodic boundary conditions in all directions and a volume forcing of the temperature field by a mean gradient. This system serves as model system for the bulk of Rayleigh-Benard flow and therefore as model for the so called ``ultimate regime of thermal convection''. With respect to the Ra dependence of Nu and Re we …
An increasing number of numerical simulations and experiments describing the turbulent spectrum of Rayleigh-Taylor (RT) mixing layers came to light over the past few years. Results reported in recent studies allow to rule out a turbulence {\it à la Kolmogorov} as a mechanism acting on a self similar RT turbulent mixing layer. A different mechanism is presented, which complies with both numerical and experimental results and relates RT flow to other buoyant flows.
Experiments and numerical simulations show that quantum turbulence exists in two distinct limiting regimes: Kolmogorov turbulence (which shares with classical turbulence the important property of a cascade of kinetic energy from large eddies to small eddies) and Vinen turbulence (which is more similar to a random flow). In this work, we define a mesoscale helicity for the superfluid, which, tested in numerical experiments, distinguishes the two turbulent regimes, quantifying the amount of nonlocal vortex interactions and the orientation of the vortex lines.
The present-day Universe is highly magnetized, even though the first magnetic seed fields were most probably extremely weak. To explain the growth of the magnetic field strength over many orders of magnitude fast amplification processes need to operate. The most efficient mechanism known today is the small-scale dynamo, which converts turbulent kinetic energy into magnetic energy leading to an exponential growth of the magnetic field. The efficiency of the dynamo depends on the type of turbulence indicated by the slope of the turbulence spectrum v(l) \propto l^{theta}, where v(l) is the eddy v…
As everyone knows who has opened a kitchen faucet, pipe flow is laminar at low flow velocities and turbulent at high flow velocities. At intermediate velocities there is a transition wherein plugs of laminar flow alternate along the pipe with "flashes" of a type of fluctuating, non-laminar flow which remains poorly known. We show experimentally that the fluid friction of flash flow is diagnostic of turbulence. We also show that the statistics of flash flow are in keeping with Kolmogorov's phenomenological theory of turbulence (so that, e.g., the energy spectra of both flash flow and turbulent …
Rotating turbulence is commonly known for being dominated by geostrophic vortices that are invariant along the rotation axis and undergo inverse cascade. Yet, it has recently been shown to sustain fully three-dimensional states with a downscale energy cascade. In this letter, we investigate the statistical properties of three-dimensional rotating turbulence by the means of direct numerical simulations in a triply periodic box where geostrophic vortices are specifically damped. The resulting turbulent flow is an inertial wave turbulence that verifies the Zakharov-Kolmogorov spectrum derived ana…
We discuss two aspects of turbulent Rayleigh-Bénard convection (RBC) on the basis of high-resolution direct numerical simulations in a unique setting; a closed cylindrical cell of aspect ratio of one. First, we present a comprehensive comparison of statistical quantities such as energy dissipation rates and boundary layer thickness scales. Data are used from three simulation run series at Prandtl numbers $Pr$ that cover two orders of magnitude. In contrast to most previous studies in RBC the focus of the present work is on convective turbulence at very low Prandtl numbers including $Pr=0.021$ …
We look at various correlation functions, which include those that involve both the velocity and the vorticity fields, in 2-D isotropic homogeneous decaying turbulence.We adopt the more intuitive approach due to Kolmogorov (and subsequently, Landau in his text on fluid dynamics) and show that how the 2-D turbulence results, obtainable using other methods, may be established in a simpler way.Also, some experimentally verifiable correlation functions in the dissipation range have been derived for the same system.The paper also showcases the inability of the Kolmogorov-Landau approach to get the …
The classical turbulence theory by Kolmogorov is reconsidered using Navier-Stokes' equation generalized to 6D physical-plus-eddy space. Strong pseudo-singularity is shown to reveal itself along the boundary `ridge' line separating the dissipation and inertial sub-ranges surrounding the origin of the eddy space. A speculation is made that this singularity is generated by two dipoles of opposite sign aligned on the common axis. It is supported by the observation that the universal power spectrum calculated rediscovers the Kolmogorov's -5/3 power law as independent of the dimensional approach.
The genesis, growth and rupture of intracranial aneurysm (IA) are open questions in neurovascular medicine until the present moment. The complexity of aneurysm mechanobiology and pathobiology staggeringly combine intertwining biological and physical processes that are tightly connected. Recently, transition to turbulence in IA blood flow is thought to play a central role in IA growth and rupture as it can be directly linked to endothelial dysfunction. However, the problem of turbulence brings unprecedented complications to the topic. We found turbulence in IA to be of non-Kolmogorov type. For …
Sub-Kolmogorov Intermittency and Multifractal Dissipation in Multiphase Turbulence
University of Modena and Reggio Emilia（摩德纳和雷焦艾米利亚大学）；Sorbonne Université（索邦大学）；Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Université Paris-Saclay（巴黎萨克雷大学）；Inria Saclay - Tau team（法国国家信息与自动化研究所萨克莱-Tau团队）
The last stage of evolution toward the stationary Kolmogorov spectrum of hydrodynamic turbulence is studied using the Leith model. This evolution is shown to manifest itself as a reflection wave in the wavenumber space propagating from the largest toward the smallest wavenumbers, and is described by a self-similar solution of a new (third) kind. This stage follows the previously studied stage of an initial explosive propagation of the spectral front from the smallest to the largest wavenumbers reaching arbitrarily large wavenumbers in a finite time, and which was described by a self-similar so…
The spectral slope of strong MHD turbulence has recently been a matter of controversy. While Goldreich-Sridhar model (1995) predicts Kolmogorov's -5/3 slope of turbulence, shallower slopes were often reported by numerical studies. We argue that earlier numerics was affected by driving due to a diffuse locality of energy transfer in MHD case. Our highest-resolution simulation (3072^2x1024) has been able to reach the asymptotic -5/3 regime of the energy slope. Additionally, we found that so-called dynamic alignment, proposed in the model with -3/2 slope, saturates and therefore can not affect as…
We investigate the statistical properties of Rayleigh-Taylor turbulence in a convective cell of high aspect ratio, in which one transverse side is much smaller that the others. We show that the scale of confinement determines the Bolgiano scale of the system, which in the late stage of the evolution is characterized by the Kolmogorov-Obukhov and the Bolgiano-Obukhov phenomenology at small and large scales, respectively. The coexistence of these regimes is associated to a three to two-dimensional transition of the system which occurs when the width of the turbulent mixing layer becomes larger t…
A landmark of out-of-equilibrium physics is Kolmogorov's phenomenological theory of turbulence. However, the past 20 years have provided evidence of a new, universal type of turbulence cascade, which does not abide to Kolmogorov physics. To address this issue, we revise the classical Kolmogorov cascade, by superimposing on it a mechanism of active information exchange between large and small scales. The new theory yields predictions for the dissipation rate, integral length scale and turbulence kinetic energy, as well as a criterion for the transition of the system from the new, to the classic…
We analyze the Prandtl number (Pr) dependence of spectra and fluxes of kinetic energy, as well as the energy injection rates and dissipation rates, of turbulent thermal convection using numerical data. As expected, for a flow with $ \mathrm{Pr} \lessapprox 1 $, the inertial-range kinetic energy flux is constant, and the kinetic energy spectrum is Kolmogorov-like ($ k^{-5/3} $). More importantly, we show that the amplitudes of the kinetic energy fluxes and spectra and those of structure functions increase with the decrease of Pr, thus exhibiting stronger nonlinearity for flows with small Prandt…
We compute the spectra and fluxes of the velocity and temperature fields in Rayleigh-Bénard convection in turbulent regime for a wide range of Prandtl numbers using pseudo-spectral simulations on $512^3$ grids. Our spectral and flux results support the Kolmogorov-Obukhov (KO) scaling for zero Prandtl number and low Prandtl number ($P=0.02$) convection. The KO scaling for the velocity field in zero-Prandtl number and low-Prandtl number convection is because of the weak buoyancy in the inertial range (buoyancy is active only at the very low wavenumbers). We also observe that for intermediate Pra…
The deformation and dynamics of a single droplet in isotropic turbulence is studied using a Lattice Boltzmann diffuse interface model involving exact boundary flow conditions to allow for the creation of an external turbulent flow. We focus on a small, sub-Kolmogorov droplet, whose scale is much smaller than the Kolmogorov length scale of the turbulent flow. The external flow field is obtained via pseudo-spectral simulation data describing the trajectory of a passive tracer in isotropic turbulence. In this way we combine the microscopic scale of the droplet and the macroscopic scale of the tur…
The process of star formation in interstellar molecular clouds is believed to be controlled by driven supersonic magnetohydrodynamic turbulence. We suggest that in the inertial range such turbulence obeys the Kolmogorov law, while in the dissipative range it behaves as Burgers turbulence developing shock singularities. On the base of the She-Leveque analytical model we then predict the velocity power spectrum in the inertial range to be E_k ~ k^{-1.74}. This result reproduces the observational Larson law, <u^2_l>~ l^{0.74...0.76}, [Larson, MNRAS 194 (1981) 809] and agrees well with recent nume…
Solar wind turbulence is dominated by Alfvénic fluctuations but the power spectral exponents somewhat surprisingly evolve toward the Kolmogorov value of -5/3, that of hydrodynamic turbulence. We show that at 1AU the turbulence decomposes linearly into two coexistent components perpendicular and parallel to the local average magnetic field. The first of these is consistent with propagating Alfvén wavepackets and shows the scaling expected of Alfvénic turbulence, namely Irosnikov- Kraichnan. The second shows Kolmogorov scaling which we also find in the number and magnetic energy density, and Poy…
The Rayleigh--Taylor (RT) turbulence is investigated by means of high resolution numerical simulations. The main question addressed here is on whether RT phenomenology can be considered as a manifestation of universality of Navier--Stokes equations with respect to forcing mechanisms. At a theoretical level the situation is far from being firmly established and, indeed, contrasting predictions have been formulated. Our first aim here is to clarify the above controversy through a deep analysis of scaling behavior of relevant statistical observables. The effects of intermittency on the mean field…
It has long been established that turbulence energy spectra scale on the Kolmogorov (1941) variables over a wide range of Reynolds numbers and in vastly different physical systems, depending only on the dissipation rate, the kinematic viscosity and the wavenumber. On the other hand, the analogous study of structure functions in real space is strongly influenced by the Kolmogorov (1962) refined theory, which introduced a dependence on a large length scale Lext, characteristic of the system size. If such a dependence exists it is surprising that it does not show up in the study of wavenumber spe…
We study Kolmogorov flow on a three dimensional, periodic domain with aspect ratios fixed to unity. Using an energy method, we give a concise proof of the linear stability of the laminar flow profile. Since turbulent motion is observed for high enough Reynolds numbers, we expect the domain of attraction of the laminar flow to be bounded by the stable manifolds of simple invariant solutions. We show one such edge state to be an equilibrium with a spatial structure reminiscent of that found in plane Couette flow, with stream wise rolls on the largest spatial scales. When tracking the edge state,…
We use two related non-stationarity functions as measures of the degree of scale-by-scale non-equilibrium in homogeneous isotropic turbulence. The values of these functions indicate significant non-equilibrium at the upper end of the inertial range. Wind tunnel data confirm Lundgren's (2002, 2003) prediction that the two-point separation $r$ where the second and third order structure functions are closest to their Kolmogorov scalings is proportional to the Taylor length scale $λ$, and that both structure functions increasingly distance themselves from their Kolmogorov equilibrium form as $r$ i…
The Weak Turbulence Theory has been applied to waves in thin elastic plates obeying the Föppl-Von Kármán dynamical equations. Subsequent experiments have shown a strong discrepancy between the theoretical predictions and the measurements. Both the dynamical equations and the Weak Turbulence Theory treatment require some restrictive hypotheses. Here a direct numerical simulation of the Föppl-Von Kármán equations is performed and reproduces qualitatively and quantitatively the experimental results when the experimentally measured damping rate of waves $γ_\mathbf{k}= a + bk^2$ is used. This confi…
The turbulent Prandtl number has been calculated in the two-loop approximation of the $\eps$ expansion of the stochastic theory of turbulence. The strikingly small value obtained for the two-loop correction explains the good agreement of the earlier one-loop result with the experiment. This situation is drastically different from other available nontrivial two-loop results, which exhibit corrections of the magnitude of the one-loop term. The reason is traced to the mutual cancellation of additional divergences appearing in two dimensions which have had a major effect on the results of previous…
We consider long simulations of 2D Kolmogorov turbulence body-forced by $\sin4y \ex$ on the torus $(x,y) \in [0,2π]^2$ with the purpose of extracting simple invariant sets or `exact recurrent flows' embedded in this turbulence. Each recurrent flow represents a sustained closed cycle of dynamical processes which underpins the turbulence. These are used to reconstruct the turbulence statistics in the spirit of Periodic Orbit Theory derived for certain types of low dimensional chaos. The approach is found to be reasonably successful at a low value of the forcing where the flow is close to but not…
We investigate the dynamo problem in the limit of small magnetic Prandtl number ($\Pm$) using a shell model of magnetohydrodynamic turbulence. The model is designed to satisfy conservation laws of total energy, cross helicity and magnetic helicity in the limit of inviscid fluid and null magnetic diffusivity. The forcing is chosen to have a constant injection rate of energy and no injection of kinetic helicity nor cross helicity. We find that the value of the critical magnetic Reynolds number ($\Rm$) saturates in the limit of small $\Pm$. Above the dynamo threshold we study the saturated regime…
The idea that chaotic set of quantum vortices can mimic classical turbulence, or at least reproduce many main features, is currently actively being developed. Appreciating significance of the challenging problem of the classical turbulence it can be expressed that the idea to study it in terms of quantized line is indeed very important and may be regarded as a breakthrough. For this reason, this theory should be carefully scrutinized. One of the basic arguments supporting this point of view is the fact that vortex tangle decays at zero temperature, when the apparent mechanism of dissipation (m…
