scholarly journals A statistical mechanics approach to mixing in stratified fluids

2016 ◽  
Vol 810 ◽  
pp. 554-583 ◽  
Author(s):  
A. Venaille ◽  
L. Gostiaux ◽  
J. Sommeria
Keyword(s):  
Potential Energy ◽  
Statistical Mechanics ◽  
Richardson Number ◽  
Degrees Of Freedom ◽  
Coarse Grained ◽  
Mixing Efficiency ◽  
Small Scale ◽  
Stratified Turbulence ◽  
Adiabatic Dynamics ◽  
Boussinesq Fluid

Predicting how much mixing occurs when a given amount of energy is injected into a Boussinesq fluid is a long-standing problem in stratified turbulence. The huge number of degrees of freedom involved in these processes renders extremely difficult a deterministic approach to the problem. Here we present a statistical mechanics approach yielding a prediction for a cumulative, global mixing efficiency as a function of a global Richardson number and the background buoyancy profile. Assuming random evolution through turbulent stirring, the theory predicts that the inviscid, adiabatic dynamics is attracted irreversibly towards an equilibrium state characterised by a smooth, stable buoyancy profile at a coarse-grained level, upon which are fine-scale fluctuations of velocity and buoyancy. The convergence towards a coarse-grained buoyancy profile different from the initial one corresponds to an irreversible increase of potential energy, and the efficiency of mixing is quantified as the ratio of this potential energy increase to the total energy injected into the system. The remaining part of the energy is lost into small-scale fluctuations. We show that for sufficiently large Richardson number, there is equipartition between potential and kinetic energy, provided that the background buoyancy profile is strictly monotonic. This yields a mixing efficiency of 0.25, which provides statistical mechanics support for previous predictions based on phenomenological kinematics arguments. In the general case, the cumulative, global mixing efficiency predicted by the equilibrium theory can be computed using an algorithm based on a maximum entropy production principle. It is shown in particular that the variation of mixing efficiency with the Richardson number strongly depends on the background buoyancy profile. This approach could be useful to the understanding of mixing in stratified turbulence in the limit of large Reynolds and Péclet numbers.

scholarly journals The Impact of a Variable Mixing Efficiency on the Abyssal Overturning

2016 ◽  
Vol 46 (2) ◽  
pp. 663-681 ◽  
Author(s):  
Casimir de Lavergne ◽  
Gurvan Madec ◽  
Julien Le Sommer ◽  
A. J. George Nurser ◽  
Alberto C. Naveira Garabato
Keyword(s):  
Potential Energy ◽  
Internal Waves ◽  
Bottom Water ◽  
Internal Tides ◽  
Mixing Efficiency ◽  
Small Scale ◽  
Antarctic Bottom Water ◽  
Energy Fraction ◽  
Geothermal Heating ◽  
The Impact

AbstractIn studies of ocean mixing, it is generally assumed that small-scale turbulent overturns lose 15%–20% of their energy in eroding the background stratification. Accumulating evidence that this energy fraction, or mixing efficiency Rf, significantly varies depending on flow properties challenges this assumption, however. Here, the authors examine the implications of a varying mixing efficiency for ocean energetics and deep-water mass transformation. Combining current parameterizations of internal wave-driven mixing with a recent model expressing Rf as a function of a turbulence intensity parameter Reb = εν/νN2, the ratio of dissipation εν to stratification N2 and molecular viscosity ν, it is shown that accounting for reduced mixing efficiencies in regions of weak stratification or energetic turbulence (high Reb) strongly limits the ability of breaking internal waves to supply oceanic potential energy and drive abyssal upwelling. Moving from a fixed Rf = 1/6 to a variable efficiency Rf(Reb) causes Antarctic Bottom Water upwelling induced by locally dissipating internal tides and lee waves to fall from 9 to 4 Sverdrups (Sv; 1 Sv ≡ 106 m3 s−1) and the corresponding potential energy source to plunge from 97 to 44 GW. When adding the contribution of remotely dissipating internal tides under idealized distributions of energy dissipation, the total rate of Antarctic Bottom Water upwelling is reduced by about a factor of 2, reaching 5–15 Sv, compared to 10–33 Sv for a fixed efficiency. The results suggest that distributed mixing, overflow-related boundary processes, and geothermal heating are more effective in consuming abyssal waters than topographically enhanced mixing by breaking internal waves. These calculations also point to the importance of accurately constraining Rf(Reb) and including the effect in ocean models.

scholarly journals Understanding mixing efficiency in the oceans: do the nonlinearities of the equation of state for seawater matter?

Ocean Science ◽  
2009 ◽  
Vol 5 (3) ◽  
pp. 271-283 ◽  
Author(s):  
R. Tailleux
Keyword(s):  
Equation Of State ◽  
Potential Energy ◽  
Energy Conversion ◽  
Internal Energy ◽  
Molecular Diffusion ◽  
Rate Of Change ◽  
Gravitational Potential Energy ◽  
Mixing Efficiency ◽  
Buoyancy Frequency ◽  
Boussinesq Fluid

Abstract. There exist two central measures of turbulent mixing in turbulent stratified fluids that are both caused by molecular diffusion: 1) the dissipation rate D(APE) of available potential energy APE; 2) the turbulent rate of change Wr, turbulent of background gravitational potential energy GPEr. So far, these two quantities have often been regarded as the same energy conversion, namely the irreversible conversion of APE into GPEr, owing to the well known exact equality D(APE)=Wr, turbulent for a Boussinesq fluid with a linear equation of state. Recently, however, Tailleux (2009) pointed out that the above equality no longer holds for a thermally-stratified compressible, with the ratio ξ=Wr, turbulent/D(APE) being generally lower than unity and sometimes even negative for water or seawater, and argued that D(APE) and Wr, turbulent actually represent two distinct types of energy conversion, respectively the dissipation of APE into one particular subcomponent of internal energy called the "dead" internal energy IE0, and the conversion between GPEr and a different subcomponent of internal energy called "exergy" IEexergy. In this paper, the behaviour of the ratio ξ is examined for different stratifications having all the same buoyancy frequency N vertical profile, but different vertical profiles of the parameter Υ=α P/(ρCp), where α is the thermal expansion coefficient, P the hydrostatic pressure, ρ the density, and Cp the specific heat capacity at constant pressure, the equation of state being that for seawater for different particular constant values of salinity. It is found that ξ and Wr, turbulent depend critically on the sign and magnitude of dΥ/dz, in contrast with D(APE), which appears largely unaffected by the latter. These results have important consequences for how the mixing efficiency should be defined and measured in practice, which are discussed.

scholarly journals Understanding mixing efficiency in the oceans: do the nonlinearities of the equation of state for seawater matter?

2009 ◽  
Vol 6 (1) ◽  
pp. 371-387
Author(s):  
R. Tailleux
Keyword(s):  
Equation Of State ◽  
Potential Energy ◽  
Linear Equation ◽  
Internal Energy ◽  
Molecular Diffusion ◽  
Rate Of Change ◽  
Gravitational Potential Energy ◽  
Mixing Efficiency ◽  
Nonlinear Character ◽  
Boussinesq Fluid

Abstract. There exist two central measures of turbulent diffusive mixing in turbulent stratified fluids, which are both caused by molecular diffusion: 1) the dissipation rate D(APE) of available potential energy APE; 2) the rate of change Wr,mixing of background gravitational potential energy GPEr. So far, these two quantities have often been regarded as representing the same kind of energy conversion, i.e., the irreversible conversion of APE into GPEr, owing to the well known result that D(APE)≈Wr,mixing in a Boussinesq fluid with a linear equation of state. Here, this idea is challenged by showing that while D(APE) remains largely unaffected by a nonlinear equation of state, Wr,mixing is in contrast strongly affected by the latter. This result is rationalized by using the recent results of Tailleux (2008), which argues that D(APE) represents the dissipation of APE into one particular subcomponent of internal energy called the "dead" internal energy IE0, whereas Wr,mixing represents the conversion between a different subcomponent of internal energy – called the "exergy" IEexergy – and GPEr. It follows that the concept of mixing efficiency, which represents the fraction of the stirring mechanical energy ultimately dissipated by molecular diffusion is related to D(APE), not Wr,mixing, which ensures that it should be largely unaffected by the nonlinear character of the equation of state, and therefore correctly described in the context of a Boussinesq fluid with a linear equation of state. The variations of GPEr, on the other hand, are sensitive to the linear or nonlinear character of the equation of state.

Time-dependent, non-monotonic mixing in stratified turbulent shear flows: implications for oceanographic estimates of buoyancy flux

2013 ◽  
Vol 736 ◽  
pp. 570-593 ◽  
Author(s):  
A. Mashayek ◽  
C. P. Caulfield ◽  
W. R. Peltier
Keyword(s):  
Reynolds Number ◽  
Dissipation Rate ◽  
Isotropic Turbulence ◽  
Buoyancy Flux ◽  
Shear Instability ◽  
Turbulent Shear ◽  
Mixing Efficiency ◽  
Small Scale ◽  
Stratified Turbulence ◽  
Flow Evolution

AbstractWe employ direct numerical simulation to investigate the efficiency of diapycnal mixing by shear-induced turbulence in stably stratified free shear layers for flows with bulk Richardson numbers in the range $0. 12\leq R{i}_{0} \leq 0. 2$ and Reynolds number $Re= 6000$. We show that mixing efficiency depends non-monotonically upon $R{i}_{0} $, peaking in the range 0.14–0.16, which coincides closely with the range in which both the buoyancy flux and the dissipation rate are maximum. By detailed analyses of the energetics of flow evolution and the underlying dynamics, we show that the existence of high mixing efficiency in the range $0. 14\lt R{i}_{0} \lt 0. 16$ is due to the emergence of a large number of small-scale instabilities which do not exist at lower Richardson numbers and are stabilized at high Richardson numbers. As discussed in Mashayek & Peltier (J. Fluid Mech., vol. 725, 2013, pp. 216–261), the existence of such a well-populated ‘zoo’ of secondary instabilities at intermediate Richardson numbers and the subsequent high mixing efficiency is realized only if the Reynolds number is higher than a critical value which is generally higher than that achievable in laboratory settings, as well as that which was achieved in the majority of previous numerical studies of shear-induced stratified turbulence. We furthermore show that the primary assumptions upon which the widely employed Osborn (J. Phys. Oceanogr. vol. 10, 1980, pp. 83–89) formula is based, as well as its counterparts and derivatives, which relate buoyancy flux to dissipation rate through a (constant) flux coefficient ($\Gamma $), fail at higher Richardson numbers provided that the Reynolds number is sufficiently high. Specifically, we show that the assumptions of fully developed, stationary, and isotropic turbulence all break down at high Richardson numbers. We show that the breakdown of these assumptions occurs most prominently at Richardson numbers above that corresponding to the maximum mixing efficiency, a fact that highlights the importance of the non-monotonicity of the dependence of mixing efficiency upon Richardson number, which we establish to be characteristic of stratified shear-induced turbulence. At high $R{i}_{0} $, the lifecycle of the turbulence is composed of a rapidly growing phase followed by a phase of rapid decay. Throughout the lifecycle, there is considerable exchange of energy between the small-scale turbulence and larger coherent structures which survive the various stages of flow evolution. Since shear instability is one of the most prominent mechanisms for turbulent dissipation of energy at scales below hundreds of metres and at various depths of the ocean, our results have important implications for the inference of turbulent diffusivities on the basis of microstructure measurements in the oceanic environment.

scholarly journals Stratified Turbulence and Mixing Efficiency in a Salt Wedge Estuary

2016 ◽  
Vol 46 (6) ◽  
pp. 1769-1783 ◽  
Author(s):  
R. C. Holleman ◽  
W. R. Geyer ◽  
D. K. Ralston
Keyword(s):  
Boundary Layer ◽  
Richardson Number ◽  
Mixing Layer ◽  
High Energy ◽  
Inertial Subrange ◽  
Mixing Efficiency ◽  
Stratified Turbulence ◽  
Free Shear Layer ◽  
Lower Efficiency ◽  
Salt Wedge

AbstractHigh-resolution observations of velocity, salinity, and turbulence quantities were collected in a salt wedge estuary to quantify the efficiency of stratified mixing in a high-energy environment. During the ebb tide, a midwater column layer of strong shear and stratification developed, exhibiting near-critical gradient Richardson numbers and turbulent kinetic energy (TKE) dissipation rates greater than 10−4 m2 s−3, based on inertial subrange spectra. Collocated estimates of scalar variance dissipation from microconductivity sensors were used to estimate buoyancy flux and the flux Richardson number Rif. The majority of the samples were outside the boundary layer, based on the ratio of Ozmidov and boundary length scales, and had a mean Rif = 0.23 ± 0.01 (dissipation flux coefficient Γ = 0.30 ± 0.02) and a median gradient Richardson number Rig = 0.25. The boundary-influenced subset of the data had decreased efficiency, with Rif = 0.17 ± 0.02 (Γ = 0.20 ± 0.03) and median Rig = 0.16. The relationship between Rif and Rig was consistent with a turbulent Prandtl number of 1. Acoustic backscatter imagery revealed coherent braids in the mixing layer during the early ebb and a transition to more homogeneous turbulence in the midebb. A temporal trend in efficiency was also visible, with higher efficiency in the early ebb and lower efficiency in the late ebb when the bottom boundary layer had greater influence on the flow. These findings show that mixing efficiency of turbulence in a continuously forced, energetic, free shear layer can be significantly greater than the broadly cited upper bound from Osborn of 0.15–0.17.

scholarly journals Deep learning of mixing by two ‘atoms’ of stratified turbulence

2019 ◽  
Vol 861 ◽  
Author(s):  
Hesam Salehipour ◽  
W. R. Peltier
Keyword(s):  
Deep Learning ◽  
Ad Hoc ◽  
Training Data ◽  
Global Ocean ◽  
Mixing Efficiency ◽  
Small Scale ◽  
Stratified Turbulence ◽  
Diapycnal Mixing ◽  
Ocean Turbulence ◽  
High Level

Current global ocean models rely on ad hoc parameterizations of diapycnal mixing, in which the efficiency of mixing is globally assumed to be fixed at 20 %, despite increasing evidence that this assumption is questionable. As an ansatz for small-scale ocean turbulence, we may focus on stratified shear flows susceptible to either Kelvin–Helmholtz (KHI) or Holmboe wave (HWI) instability. Recently, an unprecedented volume of data has been generated through direct numerical simulation (DNS) of these flows. In this paper, we describe the application of deep learning methods to the discovery of a generic parameterization of diapycnal mixing using the available DNS dataset. We furthermore demonstrate that the proposed model is far more universal compared to recently published parameterizations. We show that a neural network appropriately trained on KHI- and HWI-induced turbulence is capable of predicting mixing efficiency associated with unseen regions of the parameter space well beyond the range of the training data. Strikingly, the high-level patterns learned based on the KHI and weakly stratified HWI are ‘transferable’ to predict HWI-induced mixing efficiency under much more strongly stratified conditions, suggesting that through the application of appropriate networks, significant universal abstractions of density-stratified turbulent mixing have been recognized.

Diffusion Wavelet Decomposition for Coarse-Graining of Polymer Chains

2015 ◽  
Vol 1753 ◽  
Author(s):  
B. Christopher Rinderspacher ◽  
Jaydeep P. Bardhan ◽  
Ahmed E. Ismail
Keyword(s):  
Degrees Of Freedom ◽  
Large Scale ◽  
Coarse Graining ◽  
Polymer Chains ◽  
Coarse Grained ◽  
Small Scale ◽  
Bonding Structure ◽  
Ring Structures ◽  
Spatial Coordinates ◽  
Molecular Bonding

ABSTRACTHere we present an alternative approach to coarse graining, based on the multiresolution diffusion-wavelet approach to operator compression, which does not require explicit atomistic-to-coarse-grained mappings. Our diffusion-wavelet method takes as input the topology and sparsity of the molecular bonding structure of a system, and returns as output a hierarchical set of degrees of freedom (DoFs) of system-specific coarse-grained variables. Importantly, the hierarchical compression provides a clear framework for modeling at many model scales (levels), beyond the common two-level CG representation. Our results show that the resulting hierarchy separates localized modes, such as a single C-C vibrational mode, from larger-scale motions, e.g., long-range concerted backbone vibrational modes. Our approach correctly captures small-scale chemical features, such as cellulose ring structures, and alkane side chains or CH2 units, as well as large-scale features of the backbone. In particular, the new method’s finest-scale modes describe DoFs similar to united atom models and other chemically-defined CG models. Modes at coarser levels describe increasingly large connected portions of the target polymers. For polyethylene and polystyrene, spatial coordinates and their associated forces were compressed by up to two orders of magnitude. The compression in forces is of particular interest as this allows larger timesteps as well as reducing the number of DoFs.

scholarly journals Nonlinear evolution of linear optimal perturbations of strongly stratified shear layers

2017 ◽  
Vol 825 ◽  
pp. 213-244 ◽  
Author(s):  
A. K. Kaminski ◽  
C. P. Caulfield ◽  
J. R. Taylor
Keyword(s):  
Kinetic Energy ◽  
Dissipation Rate ◽  
Richardson Number ◽  
Initial Perturbation ◽  
Necessary Condition ◽  
Mixing Efficiency ◽  
Small Scale ◽  
Buoyancy Frequency ◽  
Strongly Nonlinear ◽  
Perturbation Energy

The Miles–Howard theorem states that a necessary condition for normal-mode instability in parallel, inviscid, steady stratified shear flows is that the minimum gradient Richardson number, $Ri_{g,min}$, is less than $1/4$ somewhere in the flow. However, the non-normality of the Navier–Stokes and buoyancy equations may allow for substantial perturbation energy growth at finite times. We calculate numerically the linear optimal perturbations which maximize the perturbation energy gain for a stably stratified shear layer consisting of a hyperbolic tangent velocity distribution with characteristic velocity $U_{0}^{\ast }$ and a uniform stratification with constant buoyancy frequency $N_{0}^{\ast }$. We vary the bulk Richardson number $Ri_{b}=N_{0}^{\ast 2}h^{\ast 2}/U_{0}^{\ast 2}$ (corresponding to $Ri_{g,min}$) between 0.20 and 0.50 and the Reynolds numbers $\mathit{Re}=U_{0}^{\ast }h^{\ast }/\unicode[STIX]{x1D708}^{\ast }$ between 1000 and 8000, with the Prandtl number held fixed at $\mathit{Pr}=1$. We find the transient growth of non-normal perturbations may be sufficient to trigger strongly nonlinear effects and breakdown into small-scale structures, thereby leading to enhanced dissipation and non-trivial modification of the background flow even in flows where $Ri_{g,min}>1/4$. We show that the effects of nonlinearity are more significant for flows with higher $\mathit{Re}$, lower $Ri_{b}$ and higher initial perturbation amplitude $E_{0}$. Enhanced kinetic energy dissipation is observed for higher-$Re$ and lower-$Ri_{b}$ flows, and the mixing efficiency, quantified here by $\unicode[STIX]{x1D700}_{p}/(\unicode[STIX]{x1D700}_{p}+\unicode[STIX]{x1D700}_{k})$ where $\unicode[STIX]{x1D700}_{p}$ is the dissipation rate of density variance and $\unicode[STIX]{x1D700}_{k}$ is the dissipation rate of kinetic energy, is found to be approximately 0.35 for the most strongly nonlinear cases.

Driving Torsion Scans with Wavefront Propagation

2020 ◽  
Author(s):  
Yudong Qiu ◽  
Daniel Smith ◽  
Chaya Stern ◽  
mudong feng ◽  
Lee-Ping Wang
Keyword(s):  
Potential Energy ◽  
Force Field ◽  
Degrees Of Freedom ◽  
Computational Cost ◽  
Initial Guess ◽  
Field Development ◽  
Force Field Development ◽  
Wavefront Propagation ◽  
Open Source Software Package

<div>The parameterization of torsional / dihedral angle potential energy terms is a crucial part of developing molecular mechanics force fields.</div><div>Quantum mechanical (QM) methods are often used to provide samples of the potential energy surface (PES) for fitting the empirical parameters in these force field terms.</div><div>To ensure that the sampled molecular configurations are thermodynamically feasible, constrained QM geometry optimizations are typically carried out, which relax the orthogonal degrees of freedom while fixing the target torsion angle(s) on a grid of values.</div><div>However, the quality of results and computational cost are affected by various factors on a non-trivial PES, such as dependence on the chosen scan direction and the lack of efficient approaches to integrate results started from multiple initial guesses.</div><div>In this paper we propose a systematic and versatile workflow called \textit{TorsionDrive} to generate energy-minimized structures on a grid of torsion constraints by means of a recursive wavefront propagation algorithm, which resolves the deficiencies of conventional scanning approaches and generates higher quality QM data for force field development.</div><div>The capabilities of our method are presented for multi-dimensional scans and multiple initial guess structures, and an integration with the MolSSI QCArchive distributed computing ecosystem is described.</div><div>The method is implemented in an open-source software package that is compatible with many QM software packages and energy minimization codes.</div>

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