Mean Dissipation Rate Research Articles

The average dissipation rate of turbulent kinetic energy in the neutral and unstable atmospheric surface layer

The mean rate of dissipation of turbulent kinetic energy is related to the surface fluxes of momentum and heat through the turbulent kinetic energy budget equation. This relationship may be used to estimate surface fluxes from measurements of the dissipation rates. The success of recent applications of the approach has been limited by uncertainties surrounding the functional relationship between the dimensionless dissipation rates and the atmospheric stability parameter. A pair of field experiments was designed and carried out in the atmospheric surface layer to identify this functional relationship over a broad range of neutral and convective flows, covering greater than 3 orders of magnitude in the stability parameter. Mean dissipation rates were computed using Fourier power spectra, second‐order structure functions, and third‐order structure functions. Arguments are presented for the superiority of the third‐order approach. A three‐sublayer conceptual model is invoked to guide the dimensional analysis, and the resulting dissipation rates are shown to scale uniquely in the three sublayers. Near the wall, in the dynamic sublayer, dissipation is significantly less than production, as energy is transported up to the more convective regions, where an equality between dissipation and production is achieved.

Fractal Characterization of Temperature Field in Convective Boundary Layer

In turbulent convection, the mean rate of dissipation of thermal energy follows the law of similarity and may be fractally distributed. It may be used as an indicator of the intensity of turbulence. The more intense is turbulence, the greater the fractal dimension (FD) of mean rate of dissipation of thermal energy. In the atmospheric boundary layer (ABL), many types of structures are observed. One of these structures is plume which occurs frequently. In the plume body temperature field is highly random. In between two plumes we generally observe a zone of tranquility, at least in comparative scale. It is conjectured that a plume has a greater fractal dimension than the interplume zone. In the present paper, a two-dimensional analysis of turbulent convection in the atmospheric boundary layer (ABL) was carried out using local intensities of temperature structures as realized from a sodar echogram. It is argued that the local intensities of the temperature structures are linearly related with the local temperatures. A parameter analogous to the local mean of rate of dissipation of thermal energy has been calculated for a series of vertical columns of the echogram and their FD's have been estimated. It has been visually matched with the echogram. It is observed that plumes have greater FDs than the interplume zone.

Images of dissipation layers to quantify mixing within a turbulent jet

Images were obtained that visualize the structure of the mixture fraction field and the structure of the scalar dissipation layers that exist near the base of a turbulent jet. To characterize the local mixing rate, measured profiles of the mean dissipation rate were compared with a general scaling analysis. In addition, the joint probability density function (£, %) was measured. Some effects of adding coaxial air and swirl are discussed. The especially large values of the mean dissipation rate (up to 30 s1) and the instantaneous dissipation rate (up to 175 s1) that were measured in the base region are due to the large local values of the mean velocity gradient in this region. It was found that the dissipation layers in the base region have a unique structure; the layers tend to be aligned at approximately a 45-deg angle to the flow and thus differ from layers in the far field, which tend to be oriented in an isotropic pattern, as was shown previously. Strong dissipation layers exist at the boundary of the entrained air, and the observed 45-deg alignment in the base region is believed to be due to the ordered vortex pattern in the shear layer. Dissipation layers are typically 0.3 mm thick. Dissipation rates varied in a manner consistent with the general scaling analysis, and the proper scaling constant was measured to be 9300.

Multifractal scaling of the kinetic energy flux in solar wind turbulence

Abstract. The geometrical and scaling properties of the energy flux of the turbulent kinetic energy in the solar wind have been studied. Using present experimental technology in solar wind measurements we cannot directly measure the real volumetric dissipation rate, ε(t), but are constrained to represent it by its surrogate the energy flux near the dissipation range at the proton gyro scale. There is evidence for the multifractal nature of the so defined dissipation field ε(t), a result derived from the scaling exponents of its statistical moments. The generalized dimension Dq has been determined and reveals that the dissipation field has a multifractal structure, which is not compatible with a scale-invariant cascade. The related multifractal spectrum α has been estimated for the first time for MHD turbulence in the solar wind. Its features resemble those obtained for turbulent fluids and other nonlinear multifractal systems. The generalized dimension Dq can for turbulence in high-speed streams be fitted well by the functional dependence of the p-model with a comparatively large parameter p1=0.87, indicating a strongly intermittent multifractal energy cascade. The experimental value for Dp/3 used in the scaling exponent s(p) of the velocity structure function gives an exponent that can describe some of the observations. The scaling exponent μ of the autocorrelation function of ε (t) has also been directly evaluated, being 0.37. Finally, the mean dissipation rate was determined, which could be used in solar wind heating models.

Multifractal scaling of the kinetic energy flux in solar wind turbulence

The geometrical and scaling properties of the energy flux of the turbulent kinetic energy in the solar wind have been studied. Using present experimental technology in solar wind measurements we cannot directly measure the real volumetric dissipation rate, ε(t), but are constrained to represent it by its surrogate the energy flux near the dissipation range at the proton gyro scale. There is evidence for the multifractal nature of the so defined dissipation field ε(t), a result derived from the scaling exponents of its statistical moments. The generalized dimension Dq has been determined and reveals that the dissipation field has a multifractal structure, which is not compatible with a scale-invariant cascade. The related multifractal spectrum α has been estimated for the first time for MHD turbulence in the solar wind. Its features resemble those obtained for turbulent fluids and other nonlinear multifractal systems. The generalized dimension Dq can for turbulence in high-speed streams be fitted well by the functional dependence of the p-model with a comparatively large parameter p1=0.87, indicating a strongly intermittent multifractal energy cascade. The experimental value for Dp/3 used in the scaling exponent s(p) of the velocity structure function gives an exponent that can describe some of the observations. The scaling exponent μ of the autocorrelation function of ε (t) has also been directly evaluated, being 0.37. Finally, the mean dissipation rate was determined, which could be used in solar wind heating models.

On the parameterization of equatorial turbulence: Effect of fine‐scale variations below the range of the diurnal cycle

We seek relationships among turbulence and fine‐scale and large‐scale flow by ensemble averaging the observations taken in the Pacific Equatorial Undercurrent (EUC) at 0°/140°W in April 1987. A combination of fine‐ and microscale sensors resolves vertical wavenumber spectra of shear and temperature from scales of 300 m to the viscous and thermal diffusive cutoffs. We study the depth range 50–350 m, which encompasses high‐shear layers below and above the core of the EUC and the thermostad, but not the diurnal cycle near the surface. Fine‐scale shear dominates over large‐scale shear (related to the slowly varying EUC) at 50–170 m and below 270 m, where large‐scale gradient Proude numbers ( ) drop below 1, while large‐scale shear dominates in the weakly stratified thermostad at 170–270 m, where > 1. We analyze shear fluctuations in different vertical wavenumber bands and pragmatically separate nonturbulent and turbulent fluctuations, the latter being associated with vertical overturning and viscous dissipation. In part of the fine‐scale range, shear spectra fall off approximately in inverse proportion to vertical wavenumber. The shear variance in this wavenumber band stays close to the squared buoyancy frequency independent of large‐scale shear. At a vertical resolution that resolves turbulent overturning, the Kunze et al. (1990) model of shear instability well predicts average dissipation rates below 100 m. Yet instantaneous high‐resolution gradient Froude numbers show virtually no correlation with turbulent dissipation rates. At 20‐m vertical resolution, mean dissipation rates from below 50 m and total rms shear Stot are well correlated as . Fine‐scale shear is essential in this relationship. Large‐scale gradient Froude numbers and large‐scale shear are comparatively poorly correlated with mixing parameters.

The solution to the Tullock rent-seeking game when R>2: Mixed-strategy equilibria and mean dissipation rates

In Tullock's rent-seeking model, the probability a player wins the game depends on expenditures raised to the power R. We show that a symmetric mixed-strategy Nash equilibrium exists when R>2, and that overdissipation of rents does not arise in any Nash equilibrium. We derive a tight lower bound on the level of rent dissipation that arises in a symmetric equilibrium when the strategy space is discrete, and show that full rent dissipation occurs when the strategy space is continuous. Our results are shown to be consistent with recent experimental evidence on the dissipation of rents.

The diurnal cycle of the upper equatorial ocean: Turbulence, fine‐scale shear, and mean shear

Physical processes within the diurnal cycle of the upper ocean in the Pacific Equatorial Undercurrent are investigated based on observations at 0°/140°W of April 1987. The cycle extended below the surface mixed layer into the pycnocline, where nighttime maxima and daytime minima occurred in turbulent dissipation rates ε, fine‐scale shear variance, and in “large‐scale,” low‐frequency zonal shear, buoyancy frequency, and gradient Froude number. The time‐of‐day average e varied by a factor of 40 at 20 m, near the base of the mixed layer. Its night/day ratio decreased with increasing depth, the diurnal signal extending to 45–60 m, compared to nighttime mixed layer depths of 10–25 m. The daily variation of ε extended throughout the layer of high shear and high gradient Froude number Fr, above the Undercurrent core, being cut off where the large‐scale Fr dropped below 1. The diurnal cycle of ε remained unaffected by changes in the large‐scale shear and stratification. The nighttime fine‐scale shear variance in the 5–10 m vertical wavelength band exceeded daytime minima by a factor of 2 on average, the daily variation decaying rapidly with increasing depth. Turbulent overturns were smaller than 5 m, so that we suspect that the fine‐scale shear was related to internal waves. Fine‐scale shear was large enough to be a factor in local shear instability. A zone some 20 m thick below the mixed layer showed weak nighttime maxima in the large‐scale, low‐frequency zonal shear, stratification, and Froude number. We suspect that these variations were a product of turbulent mixing. They could become a discriminative tool in a future comparison of our observations with upper ocean models. The application of such models is encouraged by indications of a local control of the upper ocean heat budget by the surface heat flux. In apparent response to a weaker Undercurrent, stronger stratification, and weaker trade winds, mean and nighttime dissipation rates were lower in 1987 than in November 1984. The night/day ratio, however, was larger in 1987 owing to the absence of daytime high‐ε events.

Experiments revealing small impact of turbulence on the energy budget of the mesosphere and lower thermosphere

We have measured a total of 17 in situ profiles of small‐scale density fluctuations (typical resolution: meters) in the lower thermosphere and upper mesosphere, which are used to derive turbulent parameters, such as the turbulent energy dissipation rate ε, the turbulent diffusion coefficient K, and the mean turbulent velocity wturb. The accuracy of the absolute numbers is unprecedented thanks to the very high spatial resolution and a recently improved data analysis procedure. Concentrating on the 12 flights which were performed during winter conditions at high latitudes (69°N), we find mean energy dissipation rates of 1–2 mW kg−1 in the lower mesosphere (<75 km) and 10–20 mW kg−1 in the upper mesosphere and lower thermosphere (<100 km). The corresponding heating rates are approximately 0.1 and 1 K d−l, respectively. These values are at least 1 order of magnitude smaller than most of the previous measurements and are also significantly smaller than typical values assumed in models. Our observations suggest that the heating effect of turbulence is negligible compared to the most prevailing terms of the heat budget. It can be shown by theoretical considerations involving the turbulent energy budget equation that cooling by turbulent heat conduction is also negligible if ε is small.

Dissipation in random wave groups incident on a beach

Variations in short-wave properties on the time-scale of wave groups provide improtant mechanisms in the generation of long waves. An essential component of models that describe the propagation of wave groups is the description of the slowly varying dissipation due to wave breaking. A model is proposed for the time-varying dissipation rate. As the time-varying dissipation rate cannot be obtained from direct measurements, calibration and verification is performed in an indirect way. The formulation is incorporated in several models of the time-averaged dissipation by combining it with a number of theoretical probability distributions for the wave energy. These models predict measurable quantities such as the mean dissipation, the fraction of breaking waves and some statistical wave height parameters. The parameters in the dissipation formulation are calibrated against some available measurements of the mean wave energy. A verification is then performed for some internal parameters in the models and for other datasets. The calibrated models of the mean dissipation rate in random waves show some improvements in the prediction of internal parameters. The usefulness of these models is also enhanced because the calibration coefficients can be kept constant over a wide range of conditions. The calibrated formulation of the time-varying dissipation rate is quite simple and can be readily used in wave propagation models that take into account variations on the time-scale of wave groups.

The renormalization group, the ?-expansion and derivation of turbulence models

We reformulate the renormalization group (RNG) and theɛ-expansion for derivation of turbulence models. The procedure is developed for the Navier-Stokes equations and the transport equations for the kinetic energyK and energy dissipation rate ℰ. The derivation draws on the works of Yakhot and Orszag (1986) and Smith and Reynolds (1992), and all results are found at low order in the underlying perturbation expansion in powers ofɛ. The sum of the source terms in the ℰ-equation is known to beO(1) due to the balance at leading order ofO(R T 1/2 ) terms. Smith and Reynolds (1992) showed the cancellation of some of theO(R T 1/2 ) terms generated by the RNG procedure. Here we show that including the random-force contribution to ℰ-production results in the cancellation ofall theO(R 1/2 ) terms. We find that two of theO(1) terms in the RNG equation for the mean dissipation rate ℰ have the same form as those in the widely used model -equation. The values of the coefficients of the familiar terms are close to those used in practice. An extra production term is predicted which is small for slow distortions, but important for rapid distortions. Hence, it may be a term that should be added to the model equation. We believe that the present derivation places the model equation on a more solid theoretical basis.

Fluxes across a thermohaline interface

Measurements of velocity and temperature microstructure and hydrography were made with a towed vehicle moving in and around a single interface in a double-diffusive staircase. The interface was traversed 222 times in a saw-tooth pattern over a track 35 km long. The salinity and potential temperature and density in the mixed layers adjacent to the interface were spatially uniform except for one 8 km long anomaly. The rate of dissipation of kinetic energy was uniformly low in the interface and in the mixed layers, except for one section 600 m long where a Kelvin-Helmholtz instability generated turbulence. For the non-turbulent section of the interface, the mean rate of dissipation was 30.2 × 10 −10 W kg −1 in the mixed layers and 9.5 × 10 −10 W kg −1 in the interface. The non-dimensional dissipation rate, ϵ/vN 2, was almost always less than 16 in the interface and therfore, there was no turblent buoyancy flux according to Rohr et al. (1988, Journal of Fluid Mechanics, 195, 77–111). The average double-diffusive flux of buoyancy by heat was 3.6 × 10 −10 W kg −1. Under certain assumptions the ratio of the flux of buoyancy by heat and salt can be estimated to be 0.53 ± 0.10, in good agreement with laboratory and theoretical estimates for salt fingers. The average Cox number was about 8 in the interface, consistent with the theories of Stern (1975, Ocean circulation physics, Academic Press) and Kunze (1987, Journal of Marine Research, 45 533–556), but displayed an inverse dependence on the vertical temperature gradient which was not predicted. As a result, the flux of buoyancy, as well as the individual contributions by heat and salt, were independent of the local mean vertical temperature gradient and the buoyancy frequency. The length of the turbulent section of the interface was only 1.7% of the total length observed. However, the turbulence was intense—the mean rate of dissipation was 2.5 × 10 −8 W kg −1—and may have sufficiently enhanced the flux of heat to increase the net flux ratio to 0.72, which would be consistent with the large-scale changes in layer properties reported by Schmitt (1987 EOS, Transactions of the American Geophysical Union, 68, 57–70) and the O/(10 km) scale changes observed in this study.

Turbulent Couette flow profiles that maximize the efficiency function

A previous study revealed that only maximization of the efficiency function, from among a large class of mean field moments, results in both a logarithmic law and a velocity defect law in turbulent Poiseuille channel flow. The efficiency function, [Escr ], is the product of a drag coefficient and the ratio of the fluctuation and mean dissipation rate integrals. Here, maximum [Escr ] is explored in Couette flow to test its generality as a statistical stability criterion for turbulent shear flows. The optimal flow exhibits a logarithmic law but does not have a velocity defect law. A decreasing velocity defect is predicted for Reynolds numbers up to 30 000. This prediction is shown to be supported by the existing data, which are limited to Reynolds numbers less than 20 000.

“Turbulent mixing” in a salt finger staircase

Towed thermistor chain measurements are examined for patches of “turbulent mixing” occurring within salt finger interfaces in the Caribbean staircase (the Caribbean Sheets and Layers Transects (C‐SALT) experimental area). Patches are identified as regions having short overturning internal waves, resembling Kelvin‐Helmholtz billows, and higher‐wave number, more random fluctuations. For a patch turbulent dissipation rate of ≈ 2×10−8 W kg−1 (based on other C‐SALT measurements and consistent with the observed billow heights of 2–5 m) and an observed patch occurrence of ≈1%, the mean dissipation rate is ≈ 2×10−10 W kg−1. This amount of turbulence would increase the buoyancy flux of heat and salt by 10–20% over fluxes from fingers acting alone and would increase the flux ratio by about 10% to 0.83, close to the value inferred from conductivity‐temperature‐depth data by Schmitt et al. (1987).

Dissipation of turbulent kinetic energy in buoyant free shear flows

A new scaling of the mean rate of turbulent kinetic energy dissipation is proposed for application to free shear flows. The advantage of the proposed scaling is that it permits a direct comparison of the dissipation rate in flows driven by completely different mechanisms (e.g. inertia or buoyancy forces) and of different flow configurations. The same scaling, when applied to pipe flow suggests that the non-dimensional parameter so formed is analogous to a ‘friction factor’ for this motion. Application of this development to the motion of buoyant jets and plumes suggests that the mean rate of dissipation in buoyancy driven plume flows may be twice that for jets and is associated with the larger rate of entrainment in such flows.

Pressure drop and flow rate measurements in a human aortic bifurcation cast for steady and pulsatile flow

Pressure drop and flow rate measurements in a rigid cast of a human aortic bifurcation under both steady and physiological pulsatile flow conditions are reported. Integral momentum and mechanical energy balances are used to calculate impedance, spatially averaged wall shear stress and viscous dissipation rate from the data. In the daughter branches, steady flow impedance is within 30% of the Poiseuille flow prediction, while pulsatile flow impedance is within a factor of 2 of fully developed, oscillatory, straight tube flow theory (Womersley theory). Estimates of wall shear stress are in accord with measurements obtained from velocity profiles. Mean pressure drop and viscous dissipation rate are elevated in pulsatile flow relative to steady flow at the mean flow rate, and the exponents of their Reynolds number dependence are in accord with available theory.

On Tidal Damping in Laplace's Global Ocean

Abstract Laplace's tidal equations are augmented by dissipation in a bottom boundary layer that is intermediate in character between those of Ekman and Stokes. Laplace's tidal equation for a global ocean remains second-order and self-adjoint, but the operator and eigenvalues are complex with imaginary parts are O(E½), where E = ν/2ωh2 (ν is the vertical component of the kinematic eddy viscosity, ω the rotational speed of the Earth, and h the depth of the global ocean). The imaginary part of the eigenvalue is expressed as a quadratic integral of the corresponding Hough function. The Q for a free oscillation is expressed as the ratio of two quadratic integrals that represent the mean energy and dissipation rates. Approximate calculations for the semidiurnal tides (with azimuthal wave number 2) are given.

Mass transfer from small particles suspended in turbulent fluid

Small rigid spherical particles are suspended in fluid, and material is being transferred from the surface of each sphere by convection and diffusion. The fluid is in statistically steady turbulent motion maintained by some stirring device. It is assumed that the Peclet number of the flow around a particle is large compared with unity, so that a concentration boundary layer exists at the particle surface, and that the Reynolds number of the flow around the particle is sufficiently small for the velocity distribution near the particle surface to be given by the Stokes equations.The flow around a particle is a superposition of (a) a streaming flow due to a translational motion of the particle relative to the fluid with a velocity proportional to the density difference, and (b) a flow due to the velocity gradient in the ambient fluid. An expression for the mean transfer rate which is asymptotically exact for large Peclet numbers is obtained in terms of statistical parameters of these two superposed flow fields. As a consequence of the partial suppression of convective transfer by particle rotation, the only relevant parameters are the mean translational velocity of the particle in the direction of the ambient vorticity vector and the mean ambient rate of extension in the direction of the ambient vorticity. The former is shown to be zero in common turbulent flow fields, and an expression for the latter in terms of the mean dissipation rate e is obtained from the equilibrium theory of the small-scale components of the turbulence. The final non-dimensional expression for the transfer rate is 0·55(a2e½/κν½)1/3, where a is the particle radius. This is found to agree well with some previously published sets of data for values of less than 102.

Measurements of dissipation rate and some other characteristics of turbulent plane and circular jets

The mean rate of dissipation ε is measured along the axes of a plane jet and three circular jets, covering good ranges of the jet Reynolds number Ujd/ν and the downstream distance x. The data confirm the universal relations, derived from requirements of self-preservation, between ε and x, for both the circular jet and the plane jet. For either type of jet, the turbulence Reynolds number Rλ and the local Reynolds number Rc (based on local velocity and length scales) are found to be related as Rλ∼2.3Rc1/2.

Spaced antenna VHF radar observations of tropospheric velocities and irregularities

The application of spaced antenna techniques to VHF radar studies of the motions and structure of irregularities in the clear atmosphere is discussed. Observations made with the 53.5 MHz SOUSY‐VHF radar working at vertical incidence and using spaced antennas for reception confirm previous suggestions that echoes from the troposphere have contributions from specular (i.e., partial) reflection, also called ‘Fresnel reflection,’ as well as random scatter. Measurements of the angular width and spatial coherence show the reflecting regions to be thin and quasi‐horizontal in structure. These reflecting regions also are called sheets. Reflected to scattered power ratios are in the range between 0.2 and 10. The angular spectrum of reflecting/scattering structures is investigated by means of a radar interferometer technique. The angular width and the apparent tilt of the sheets is measured to be a few degrees. It has been possible to measure horizontal velocities by the drift method even on those returns where the angular widths are so narrow that the use of conventional Doppler techniques would be impracticable. The drift velocities agree well with balloon measured winds. It is found that stable, reflecting sheets occur primarily in regions of high dynamic stability. In regions of lower dynamic stability, scattering from turbulent irregularities is observed. In this case the mean dissipation rate, diffusion coefficient, and outer scale of the inertial subrange of turbulence are estimated from measured values of correlation time and Richardson number. The estimates are in accord with other turbulence measurements.