Total Energy Dissipation Rate Research Articles

Impact of parameterized lee wave drag on the energy budget of an eddying global ocean model

The impact of parameterized topographic internal lee wave drag on the input and output terms in the total mechanical energy budget of a hybrid coordinate high-resolution global ocean general circulation model forced by winds and air-sea buoyancy fluxes is examined here. Wave drag, which parameterizes the generation of internal lee waves arising from geostrophic flow impinging upon rough topography, is included in the prognostic model, ensuring that abyssal currents and stratification in the model are affected by the wave drag.An inline mechanical (kinetic plus gravitational potential) energy budget including four dissipative terms (parameterized topographic internal lee wave drag, quadratic bottom boundary layer drag, vertical eddy viscosity, and horizontal eddy viscosity) demonstrates that wave drag dissipates less energy in the model than a diagnostic (offline) estimate would suggest, due to reductions in both the abyssal currents and stratification. The equator experiences the largest reduction in energy dissipation associated with wave drag in inline versus offline estimates. Quadratic bottom drag is the energy sink most affected globally by the presence of wave drag in the model; other energy sinks are substantially affected locally, but not in their global integrals. It is suggested that wave drag cannot be mimicked by artificially increasing the quadratic bottom drag because the energy dissipation rates associated with bottom drag are not spatially correlated with those associated with wave drag where the latter are small. Additionally, in contrast to bottom drag, wave drag is a non-local energy sink.All four aforementioned dissipative terms contribute substantially to the total energy dissipation rate of about one terawatt. The partial time derivative of potential energy (non-zero since the isopycnal depths have a long adjustment time), the surface advective fluxes of potential energy, the rate of change of potential energy due to diffusive mass fluxes, and the conversion between internal energy and potential energy also play a non-negligible role in the total mechanical energy budget. Reasons for the <10% total mechanical energy budget imbalance are discussed.

STATISTICAL ANALYSIS OF CURRENT SHEETS IN THREE-DIMENSIONAL MAGNETOHYDRODYNAMIC TURBULENCE

We develop a framework for studying the statistical properties of current sheets in numerical simulations of 3D magnetohydrodynamic (MHD) turbulence. We describe an algorithm that identifies current sheets in a simulation snapshot and then determines their geometrical properties (including length, width, and thickness) and intensities (peak current density and total energy dissipation rate). We then apply this procedure to simulations of reduced MHD turbulence and perform a statistical analysis on the obtained population of current sheets. We evaluate the role of reconnection by separately studying the populations of current sheets which contain magnetic X-points and those which do not. We find that the statistical properties of the two populations are different in general. We compare the scaling of these properties to phenomenological predictions obtained for the inertial range of MHD turbulence. Finally, we test whether the reconnecting current sheets are consistent with the Sweet-Parker model.

Search for the optimality signature of river network development

Whether the evolution of natural river networks pursues a certain optimal state has been a most intriguing and fundamental question. There have been many optimality hypotheses proposed but it has yet to be proved which of these best serves as a quantitative signature of river network development. Here, this fundamental question is investigated for the five hypotheses of "minimum total energy expenditure," "minimum total energy dissipation rate," "minimum total stream power," "minimum global energy expenditure rate," and "minimum topological energy." Using simple example landscapes, I examined whether any of these hypotheses pursues both the treelike river network formation and the concave stream longitudinal profile, the two characteristic patterns of natural landscapes. It is found that none of these hypotheses captures both patterns under the steady-state condition where the balance between tectonic uplift and sediment loss is satisfied. These findings are further verified through simulations of landscapes that satisfy given optimality criteria using an optimization method.

Wave forcing over an intertidal mussel bed

The Mosselwad project studies the stability and opportunities for restoration of mussel beds in the Wadden Sea. In this context we seek to predict mussel bed stability with respect to hydrodynamic forcing. To make accurate predictions with models, field experiments are needed to determine relevant processes and to establish representative estimates for model parameters. These parameter values were determined by a six week campaign on a relatively young mussel bed in the Wadden Sea. During this period wave height, period, propagation velocity, dissipation and flow velocities were measured. From these measurements values for the total rate of energy dissipation and the rate of wave energy dissipation were estimated. Results show a large increase in measured bed shear stress over the mussel bed compared with the uncovered parts of the intertidal flat. This is caused by the large roughness of the mussels. The turbulent kinetic energy was high above the covered parts of the bed. From the dissipation rate of TKE values for the corresponding bed shear stress and roughness height were estimated. These estimates were subsequently applied to calibrate a wave model which was used to determine the spatial distribution of the wave forcing. Model results show that the bed shear stress decreases over the mussel bed but increases behind the bed. Furthermore, a model study on the situation before the mussel bed settled shows that a minimum in wave forcing coincides with the seaward edge of the bed. This suggests that the bed is currently located in the optimal location with respect to wave forcing.

Vortex events in Euler and Navier–Stokes simulations with smooth initial conditions

AbstractWe present high-resolution numerical simulations of the Euler and Navier–Stokes equations for a pair of colliding dipoles. We study the possible approach to a finite-time singularity for the Euler equations, and contrast it with the formation of developed turbulence for the Navier–Stokes equations. We present numerical evidence that seems to suggest the existence of a blow-up of the inviscid velocity field at a finite time (${t}_{s} $) with scaling $\vert u\vert _{\infty } \ensuremath{\sim} \mathop{ ({t}_{s} \ensuremath{-} t)}\nolimits ^{\ensuremath{-} 1/ 2} $, $\vert \omega \vert _{\infty } \ensuremath{\sim} \mathop{ ({t}_{s} \ensuremath{-} t)}\nolimits ^{\ensuremath{-} 1} $. This blow-up is associated with the formation of a ${k}^{\ensuremath{-} 3} $ spectral range, at least for the finite range of wavenumbers that are resolved by our computation. In the evolution toward ${t}_{s} $, the total enstrophy is observed to increase at a slower rate, $\Omega \ensuremath{\sim} \mathop{ ({t}_{s} \ensuremath{-} t)}\nolimits ^{\ensuremath{-} 3/ 4} $, than would naively be expected given the behaviour of the maximum vorticity, ${\omega }_{\infty } \ensuremath{\sim} \mathop{ ({t}_{s} \ensuremath{-} t)}\nolimits ^{\ensuremath{-} 1} $. This indicates that the blow-up would be concentrated in narrow regions of the flow field. We show that these regions have sheet-like structure. Viscous simulations, performed at various $\mathit{Re}$, support the conclusion that any non-zero viscosity prevents blow-up in finite time and results in the formation of a dissipative exponential range in a time interval around the estimated inviscid ${t}_{s} $. In this case the total enstrophy saturates, and the energy spectrum becomes less steep, approaching ${k}^{\ensuremath{-} 5/ 3} $. The simulations show that the peak value of the enstrophy scales as ${\mathit{Re}}^{3/ 2} $, which is in accord with Kolmogorov phenomenology. During the short time interval leading to the formation of an inertial range, the total energy dissipation rate shows a clear tendency to become independent of $\mathit{Re}$, supporting the validity of Kolmogorov’s law of finite energy dissipation. At later times the kinetic energy shows a ${t}^{\ensuremath{-} 1. 2} $ decay for all $\mathit{Re}$, in agreement with experimental results for grid turbulence. Visualization of the vortical structures associated with the stages of vorticity amplification and saturation show that, prior to ${t}_{s} $, large-scale and the small-scale vortical structures are well separated. This suggests that, during this stage, the energy transfer mechanism is non-local both in wavenumber and in physical space. On the other hand, as the spectrum becomes shallower and a ${k}^{\ensuremath{-} 5/ 3} $ range appears, the energy-containing eddies and the small-scale vortices tend to be concentrated in the same regions, and structures with a wide range of sizes are observed, suggesting that the formation of an inertial range is accompanied by transfer of energy that is local in both physical and spectral space.

EXPERIMENTAL AND NUMERICAL STUDIES ON WAVE TRANSFORMATION OVER ARTIFICIAL REEFS

A laboratory measurement on the flow field, turbulence and wave energy of spilling breakers over artificial reefs is presented. Instantaneous velocity fields of propagating breaking waves on artificial reefs were measured using Particle Image Velocimeter (PIV) and Bubble Image Velocimeter (BIV). Variations of water surface elevation were observed by using Charge Coupled Device (CCD) cameras with horizontal posture. The experimental results showed that the initial bubble velocity in the aerated region is faster than phase speed with a factor of 1.26. The velocity profiles are identical to the shallow water theory. It is found that a low flow velocity exists due to an opposite but equal onshore and offshore velocity. Significant turbulent kinetic energy and turbulent Reynolds stress are produced by breaking waves in the front of aerated region, then move offshore and decay. The calculated total energy dissipation rate was compared to that based on a bore approximation.

Accuracy of perturbation theory for nonadiabatic effects in adsorbate-surface dynamics

An independent-electron formalism is developed to describe the energetic distributions of hot electrons and holes excited in the interaction between an adsorbate and a metal surface. The formalism encompasses both a fully nonadiabatic treatment and a perturbation expansion in the adsorbate velocity that can be taken to arbitrary order. The widely used electronic friction and forced oscillator models are shown to be approximations of the second order perturbation result. A simple tight binding model of an atomic adsorbate interacting with a metal surface is used to demonstrate the formalism. It is shown that many orders (&amp;gt;10) of perturbation theory are required for quantitative agreement with fully nonadiabatic calculations of the electron and hole distribution functions. However, lower order approximations can provide a useful, semiquantitative picture of the distribution functions, and they are in good agreement with nonperturbative results for the total rate of nonadiabatic energy dissipation.

Scale-Up Factor for Mean Drop Diameter in Batch Rotor-Stator Mixers with Internal Circulation

The aim of the present study is to propose a scale-up factor for the mean drop diameter in internally circulated batch rotor–stator mixers. The total energy dissipation rate (εt) is used as the scale-up factor for the mean drop diameter. εt is calculated from the power number, flow number, rotational speed, number of rotor blades, and number of stator holes. Because it is difficult to measure the circulation flow rate of an internally circulated mixer directly, the flow rate of a production-scale mixer is estimated by measuring the time required for complete mixing—termed “mixing time”—and taking into account the power number of the production- and pilot-scale mixers and the flow number of the pilot-scale mixer. Our experimental results indicate that εt could influence the mean drop diameter. Theoretical verification suggests that εt includes information on the mixer configuration (rotor diameter, stator hole diameter, stator wall thickness, stator opening ratio, and gap width) and manufacturing conditions (operating time: t; rotational speed: N; and total product volume: V). Our results show that a decrease in the mean drop diameter is proportional to t, N4, and V−1. Further, we estimate the mean drop diameter of the production-scale mixer by performing a model product experiment with a pilot-scale mixer. The above results also indicate that the scale-up criteria for the mean drop diameter can be determined on the basis of εt and in terms of the operating time and not necessarily in terms of the geometric similarities between mixer configurations, a constant rotor tip speed, or a constant clearance between the rotor and stator.

Turbulence and energy dissipations of surf-zone spilling breakers

A laboratory study on the turbulence and wave energy dissipations of spilling breakers in a surf zone is presented. Instantaneous velocity fields of propagating breaking waves on a 1/20 slope were measured using Particle Image Velocimetry (PIV). Due to the large region of the evolving wave breaking generated turbulent flow, seven PIV fields of view (FOVs) were mosaicked to form a continuous flow field in the surf zone. Mean and turbulence quantities were extracted by ensemble averaging 25 repeated instantaneous measurements at each FOV. New results for distribution and evolution of turbulent kinetic energy, mean flow energy, and total energy across the surf zone were obtained from analyzing the data. The turbulence dissipation rate was estimated based on several different approaches. It was found that the vertical distribution of the turbulence dissipation rate decays exponentially from the crest level to the bottom. The resulting energy budget and energy flux were also calculated. The calculated total energy dissipation rate was compared to that based on a bore approximation. It was found that the ratio of turbulence dissipation rate to total energy dissipation rate was about 0.01 in the outer surf zone and increased to about 0.1 after the breaking waves transformed into developed turbulent bores in the inner surf zone.

Lifetime of the thunderstorm electric energy in the global atmospheric circuit and thunderstorm energy characteristics

The lifetime of electric energy in the atmosphere is introduced and investigated as is the total electric energy of the atmosphere related to the total mean rate of electric energy dissipation. This lifetime, as determined from general estimations and convenient analytical expressions, turns out to be very small – from about 10 to about 100 s, depending on the assumptions on the control parameters of principal sources in the global electric circuit. In particular the energy lifetime is less than the relaxation time of the “global condenser” and field relaxation time near the ground surface. It is explained by the high dissipative rate of the electric energy in the atmosphere, taking into account that the regions mainly contributing to the total energy and its dissipative rate are connected to the altitudes of active parts of electrified (thunderstorm) clouds in the atmosphere with exponentially increasing conductivity.

Air entrainment and associated energy dissipation in steady and unsteady plunging jets at free surface

A theoretical formulation that describes energy transformation and dissipation caused by air bubble entrainment in water is proposed, in which the potential energy increase, induced by air bubble displacement and thermodynamical work done on air bubbles, are considered. Laboratory experiments were carried out with measurements of void fraction distributions under vertical steady plunging jets and plunging breakers of water waves. The energy dissipation rate has been estimated from measured data of void fractions and compared with the total energy dissipation rate. The ratios of energy dissipation rate caused by bubble entrainment to total dissipation rate have been found to be 4%–8% for vertical circular plunging jets and 36%–40% for plunging breakers.

Structure of Optimal Transport Networks Subject to a Global Constraint

The structure of pipe networks minimizing the total energy dissipation rate is studied analytically. Among all the possible pipe networks that can be built with a given total pipe volume (or pipe lateral surface area), the network which minimizes the dissipation rate is shown to be loopless. Furthermore, such an optimal network is shown to contain at most N-2 nodes in addition to the N sources plus sinks that it connects. These results are valid whether the possible locations for the additional nodes are chosen freely or from a set of nodes (such as points of a grid). Applications of these results to various physical situations and to the efficient computation of optimal pipe networks are also discussed.

Determination of the Segregation Index to Sense the Mixing Quality of Pilot- and Production-Scale Microstructured Mixers

The performance of the microstructured mixer Star-Laminator300 ( StarLam300) with stacked star-shaped foils has been examined experimentally by the use of the adapted Villermaux/Dushman parallel-competitive iodide-iodate reaction system in continuous processes. The mixing efficiency was quantitatively determined by the segregation index derived from the UV absorption. The mixing performance of the microstructured mixer is analysed in terms of the total volume flow and turbulent energy dissipation rate. The influence of the geometry of the injection nozzles and the mixing chamber and the variation of foil number is also presented and discussed. The latter touches upon the numbering-up concept of microstructured devices. Since mixing efficiencies much better as for conventional equipment at throughputs up to 320 l h −1 were found (with pressure-loss curves showing further potential to 1000 l h −1), this microstructured mixer proved potential to act as process-intensification (PI) equipment.

Energy transfer and intermittency in four-dimensional turbulence

The energy transfer and small scale intermittency in decaying turbulence in four dimensions are studied by direct numerical simulation and by spectral theory. It is found that (1) a 1∕2 law, −(1∕2)ε¯r, in four dimensional (4D) for the longitudinal third-order structure function holds, (2) the energy transfer in 4D is more efficient than in three dimensional (3D), (3) the Kolmogorov constant in 4D is K4=1.28 which is smaller than K3=1.72 in 3D, (4) the velocity gradient intermittency is stronger than in 3D, while (5) the total-energy dissipation rate in 4D is less intermittent than in 3D. The conflicting trends in (4) and (5) are explained by the changes in the balance between the convective and pressure terms as the dimension increases.

Estimates of Energy Dissipation Rates in the Three-Dimensional Deep Ocean Internal Wave Field

Using the “Eikonal Approach” (Henyey et al., 1986), we estimate energy dissipation rates in the three-dimensional Garrett-Munk internal wave field. The total energy dissipation rate within the undisturbed GM internal wave field is found to be 4.34 × 10−9 W kg−1. This corresponds to a diapycnal diffusivity of about 0.3 × 10−4 m2s−1, which is less than the value 10−4 m2s−1 required to sustain the global ocean overturning circulation. Only when the high vertical wavenumber, near-inertial current shear is enhanced can diapycnal diffusivity reach ∼10−4 m2s−1. It follows that the energy supplied at low vertical wavenumbers and low frequencies is efficiently transferred to high vertical wavenumbers and near-inertial frequencies in the mixing hotspots in the real ocean.

Drag Reduction in Bubbly Taylor-Couette Turbulence

In Taylor-Couette flow the total energy dissipation rate and therefore the drag can be determined by measuring the torque on the system. We do so for Reynolds numbers between Re=7 x 10(4) and Re=10(6) after having injected (i) small bubbles (R=1 mm) up to a volume concentration of alpha=5% and (ii) buoyant particles (rhop/rhol=0.14) of comparable volume concentration. In case (i) we observe a crossover from little drag reduction at smaller Re to strong drag reduction up to 20% at Re=10(6). In case (ii) we observe at most little drag reduction throughout. Several theoretical models for bubbly drag reduction are discussed in view of our findings.

The behaviour of quiescent accretion discs in dwarf novae

We propose a simple explanation for the constant mean brightness observed between outbursts in dwarf novae. Secular changes in the total energy dissipation rate of the accretion disc brought about by variations in surface density, temperature and disc radius can be regulated by the gradual cooling of a small, critically-stable hot inner region. The hypothesis is supported by two-dimensional time-dependent numerical models of dwarf nova accretion discs.

Bounds on dissipation in magnetohydrodynamic Couette and Hartmann shear flows

Shear flow with an applied cross-stream magnetic field is studied using dissipative incompressible magnetohydrodynamics. The study incorporates exact solutions, the energy stability method, and exact bounds on the total energy dissipation rate. Two physical configurations are examined: magnetic Couette flow and Hartmann flow, the latter being Poiseuille flow with the existence of a perpendicular magnetic field. Explicit expressions are derived for energy stability regions in the parameter space and these expressions are compared with numerically obtained results. For large enough Reynolds numbers the energy dissipation rate is shown to be bounded by a function of the magnetic Prandtl number. The bounds obtained on the dissipation rate are compared with experimental results.

A physical basis for calculating instrument spatial weighting functions in homogeneous systems

Starting with a streamline‐based argument, the theoretical basis for a spatial weighting function in a steady, homogenous, and isotropic flow system is developed for both compressible and incompressible flow. A physical interpretation of spatial weighting functions in terms of the ratio of steady state energy dissipation rate per unit volume of porous medium to total energy dissipation rate over the entire flow domain is formulated. Finally, applications of the spatial weighting functions are presented based on calculated gas minipermeameter mass flux fields for the conventional surface probe and a new small drill hole probe configuration. Spatial weighting function distributions indicate that with diverging flow field instruments, such as the gas minipermeameter, porous medium volumes in the inlet vicinity are heavily weighted, with volumes near the seal boundaries shown to be extremely important. The technique described allows one to quantify the size and shape of the subvolume of a domain contributing to a hydraulic conductivity or intrinsic permeability measurement. For heterogeneous systems it is suggested that the spatial weighting function for a given instrument will be heterogeneity dependent and therefore nonunique. However, extending the weighting function concept to heterogeneous and/or transient systems is an important research priority.

The Scaling Law of Quasi-Geostrophic Turbulence with Weak Energy Dissipation

When a problem without viscosity is considered as an ideal case, a small dissipation term is necessary in many numerical calculations. Quasi-geostrophic turbulence, which is described by the barotropic version of quasi-geostrophic potential vorticity equation, (also called Charney-Hasegawa-Mima equation as an equivalent) is one of these situations: a dissipation term must be added in order to prevent the energy from piling up in the large wavenumber region. Consequently, the total energy decreases gradually, which should be conserved in an ideal situation. In this article, the total energy dissipation rate in quasi-geostrophic turbulence is estimated, based on the assumption that the energy dissipated in this system is equal to the energy transported to the large wavenumber region. This estimation complements the dynamic scaling laws for the quasi-geostrophic turbulence developed by Watanabe et al. (1998): the parameter expressing the total energy dissipation, which was determined from the result of the numerical calculation in their study, is derived from the parameters of the setting of the numerical calculation. This estimation also suggests the means to determine the appropriate artificial hyperviscosity coefficient used in numerical simulations.