scholarly journals Momentum Balance and Eliassen–Palm Flux on Moist Isentropic Surfaces

2016 ◽  
Vol 73 (3) ◽  
pp. 1293-1314 ◽  
Author(s):  
Ray Yamada ◽  
Olivier Pauluis
Keyword(s):  
Momentum Equation ◽  
Momentum Balance ◽  
Similar Increase ◽  
Moist Air ◽  
Momentum Exchange ◽  
Weak Formulation ◽  
Horizontal Structure ◽  
Vertical Coordinate ◽  
Momentum Budget ◽  
Conditional Averaging

Abstract Previous formulations for the zonally averaged momentum budget and Eliassen–Palm (EP) flux diagnostics do not adequately account for moist dynamics, since air parcels are not differentiated by their moisture content when averages are taken. The difficulty in formulating the momentum budget in moist coordinates lies in the fact that they are generally not invertible with height. Here, a conditional-averaging approach is used to derive a weak formulation of the momentum budget and EP flux in terms of a general vertical coordinate that is not assumed to be invertible. The generalized equation reduces to the typical mass-weighted zonal-mean momentum equation for invertible vertical coordinates. The weak formulation is applied here to study the momentum budget on moist isentropes. Recent studies have shown that the meridional mass transport in the midlatitudes is twice as strong on moist isentropes as on dry isentropes. It is shown here that this implies a similar increase in the EP flux between the dry and moist frameworks. Physically, the increase in momentum exchange is tied to an enhancement of the form drag associated with the horizontal structure of midlatitude eddies, where the poleward flow of moist air is located in regions of strong eastward pressure gradient.

scholarly journals The Tidally Averaged Momentum Balance in a Partially and Periodically Stratified Estuary

2010 ◽  
Vol 40 (11) ◽  
pp. 2418-2434 ◽  
Author(s):  
Mark T. Stacey ◽  
Matthew L. Brennan ◽  
Jon R. Burau ◽  
Stephen G. Monismith
Keyword(s):  
Water Column ◽  
Time Scale ◽  
Eddy Viscosity ◽  
Momentum Equation ◽  
Momentum Balance ◽  
Exchange Flow ◽  
Bed Stress ◽  
Turbulent Momentum ◽  
Stratified Estuary ◽  
Estuarine Exchange

Abstract Observations of turbulent stresses and mean velocities over an entire spring–neap cycle are used to evaluate the dynamics of tidally averaged flows in a partially stratified estuarine channel. In a depth-averaged sense, the net flow in this channel is up estuary due to interaction of tidal forcing with the geometry of the larger basin. The depth-variable tidally averaged flow has the form of an estuarine exchange flow (downstream at the surface, upstream at depth) and varies in response to the neap–spring transition. The weakening of the tidally averaged exchange during the spring tides appears to be a result of decreased stratification on the tidal time scale rather than changes in bed stress. The dynamics of the estuarine exchange flow are defined by a balance between the vertical divergence of the tidally averaged turbulent stress and the tidally averaged pressure gradient in the lower water column. In the upper water column, tidal stresses are important contributors, particularly during the neap tides. The usefulness of an effective eddy viscosity in the tidally averaged momentum equation is explored, and it is seen that the effective eddy viscosity on the subtidal time scale would need to be negative to close the momentum balance. This is due to the dominant contribution of tidally varying turbulent momentum fluxes, which have no specific relation to the subtidal circulation. Using a water column model, the validity of an effective eddy viscosity is explored; for periodically stratified water columns, a negative effective viscosity is required.

scholarly journals A finite-difference convective model for Jupiter's equatorial jet

2006 ◽  
Vol 2 (S239) ◽  
pp. 230-232 ◽  
Author(s):  
Kwing L. Chan
Keyword(s):  
Reynolds Number ◽  
Numerical Model ◽  
Finite Difference ◽  
Spherical Shell ◽  
Reynolds Stress ◽  
Momentum Equation ◽  
Momentum Balance ◽  
Zonal Momentum Balance ◽  
Zonal Momentum ◽  
Convective Model

AbstractWe present results of a numerical model for studying the dynamics of Jupiter's equatorial jet. The computed domain is a piece of spherical shell around the equator. The bulk of the region is convective, with a thin radiative layer at the top. The shell is spinning fast, with a Coriolis number = ΩL/V on the order of 50. A prominent super-rotating equatorial jet is generated, and secondary alternating jets appear in the higher latitudes. The roles of terms in the zonal momentum equation are analyzed. Since both the Reynolds number and the Taylor number are large, the viscous terms are small. The zonal momentum balance is primarily between the Coriolis and the Reynolds stress terms.

Shared dynamical features of smooth- and rough-wall boundary-layer turbulence

2016 ◽  
Vol 792 ◽  
pp. 435-469 ◽  
Author(s):  
R. L. Ebner ◽  
Faraz Mehdi ◽  
J. C. Klewicki
Keyword(s):  
Structural Features ◽  
Momentum Equation ◽  
Rough Wall ◽  
Momentum Balance ◽  
Normal Velocity ◽  
Advective Transport ◽  
Vorticity Field ◽  
Wall Flows ◽  
The Mean ◽  
Spanwise Vorticity

The structure of smooth- and rough-wall turbulent boundary layers is investigated using existing data and newly acquired measurements derived from a four element spanwise vorticity sensor. Scaling behaviours and structural features are interpreted using the mean momentum equation based framework described for smooth-wall flows by Klewicki (J. Fluid Mech., vol. 718, 2013, pp. 596–621), and its extension to rough-wall flows by Mehdiet al.(J. Fluid Mech., vol. 731, 2013, pp. 682–712). This framework holds potential relative to identifying and characterizing universal attributes shared by smooth- and rough-wall flows. As prescribed by the theory, the present analyses show that a number of statistical features evidence invariance when normalized using the characteristic length associated with the wall-normal transition to inertial leading-order mean dynamics. On the inertial domain, the spatial size of the advective transport contributions to the mean momentum balance attain approximate proportionality with this length over significant ranges of roughness and Reynolds number. The present results support the hypothesis of Mehdiet al., that outer-layer similarity is, in general, only approximately satisfied in rough-wall flows. This is because roughness almost invariably leaves some imprint on the vorticity field; stemming from the process by which roughness influences (generally augments) the near-wall three-dimensionalization of the vorticity field. The present results further indicate that the violation of outer similarity over regularly spaced spanwise oriented bar roughness correlates with the absence of scale separation between the motions associated with the wall-normal velocity and spanwise vorticity on the inertial domain.

Mean momentum balance in moderately favourable pressure gradient turbulent boundary layers

2008 ◽  
Vol 617 ◽  
pp. 107-140 ◽  
Author(s):  
M. METZGER ◽  
A. LYONS ◽  
P. FIFE
Keyword(s):  
Pressure Gradient ◽  
Boundary Layers ◽  
Outer Region ◽  
Present Theory ◽  
Momentum Equation ◽  
Turbulent Boundary Layers ◽  
Momentum Balance ◽  
Physical Layers ◽  
Multi Scale ◽  
The Mean

Moderately favourable pressure gradient turbulent boundary layers are investigated within a theoretical framework based on the unintegrated two-dimensional mean momentum equation. The present theory stems from an observed exchange of balance between terms in the mean momentum equation across different regions of the boundary layer. This exchange of balance leads to the identification of distinct physical layers, unambiguously defined by the predominant mean dynamics active in each layer. Scaling domains congruent with the physical layers are obtained from a multi-scale analysis of the mean momentum equation. Scaling behaviours predicted by the present theory are evaluated using direct measurements of all of the terms in the mean momentum balance for the case of a sink-flow pressure gradient generated in a wind tunnel with a long development length. Measurements also captured the evolution of the turbulent boundary layers from a non-equilibrium state near the wind tunnel entrance towards an equilibrium state further downstream. Salient features of the present multi-scale theory were reproduced in all the experimental data. Under equilibrium conditions, a universal function was found to describe the decay of the Reynolds stress profile in the outer region of the boundary layer. Non-equilibrium effects appeared to be manifest primarily in the outer region, whereas differences in the inner region were attributed solely to Reynolds number effects.

scholarly journals Equations of Atmospheric Motion in Non-Eulerian Vertical Coordinates: Vector-Invariant Form and Quasi-Hamiltonian Formulation

2014 ◽  
Vol 142 (10) ◽  
pp. 3860-3880 ◽  
Author(s):  
Thomas Dubos ◽  
Marine Tort
Keyword(s):  
Hamiltonian Formulation ◽  
Point Of View ◽  
Momentum Balance ◽  
Hamiltonian Form ◽  
Conservation Of Energy ◽  
Vertical Coordinate ◽  
Vertical Displacements ◽  
Energy Conserving ◽  
Atmospheric Motion ◽  
Hydrostatic Balance

Abstract The curl form of equations of inviscid atmospheric motion in general non-Eulerian coordinates is obtained. Narrowing down to a general vertical coordinate, a quasi-Hamiltonian form is then obtained in a Lagrangian, isentropic, mass-based or z-based vertical coordinate. In non-Lagrangian vertical coordinates, the conservation of energy by the vertical transport terms results from the invariance of energy under the vertical relabeling of fluid parcels. A complete or partial separation between the horizontal and vertical dynamics is achieved, except in the Eulerian case. The horizontal–vertical separation is especially helpful for (quasi-)hydrostatic systems characterized by vanishing vertical momentum. Indeed for such systems vertical momentum balance reduces to a simple statement: total energy is stationary with respect to adiabatic vertical displacements of fluid parcels. From this point of view the purpose of (quasi-)hydrostatic balance is to determine the vertical positions of fluid parcels, for which no evolution equation is readily available. This physically appealing formulation significantly extends previous work. The general formalism is exemplified for the fully compressible Euler equations in a Lagrangian vertical coordinate and a Cartesian (x, z) slice geometry, and the deep-atmosphere quasi-hydrostatic equations in latitude–longitude horizontal coordinates. The latter case, in particular, illuminates how the apparent intricacy of the time-dependent metric terms and of the additional forces can be absorbed into a proper choice of prognostic variables. In both cases it is shown how the quasi-Hamiltonian form leads straightforwardly to the conservation of energy using only integration by parts. Relationships with previous work and implications for stability analysis and the derivation of approximate sets of equations and energy-conserving numerical schemes are discussed.

scholarly journals Comments on “The Depth-Dependent Current and Wave Interaction Equations: A Revision”

2011 ◽  
Vol 41 (10) ◽  
pp. 2008-2012 ◽  
Author(s):  
Anne-Claire Bennis ◽  
Fabrice Ardhuin
Keyword(s):  
Surf Zone ◽  
Three Dimensional ◽  
Wave Interaction ◽  
Momentum Equation ◽  
Momentum Balance ◽  
Averaging Operators ◽  
Three Dimensional Modeling ◽  
Dimensional Modeling ◽  
Sloping Bottom

Abstract Equations for the wave-averaged three-dimensional momentum equations have been published in this journal. It appears that these equations are not consistent with the known depth-integrated momentum balance, especially over a sloping bottom. These equations should thus be considered with caution, because they can produce erroneous flows, particularly outside of the surf zone. It is suggested that the inconsistency in the equations may arise from the different averaging operators applied to the different terms of the momentum equation. It is concluded that other forms of the momentum equations, expressed in terms of the quasi-Eulerian velocity, are better suited for three-dimensional modeling of wave–current interactions.

scholarly journals Modelling of snow entrainment and deposition in powder-snow avalanches

1998 ◽  
Vol 26 ◽  
pp. 253-258 ◽  
Author(s):  
Dieter Issler
Keyword(s):  
Kinetic Energy ◽  
Three Dimensional ◽  
Momentum Balance ◽  
Model Parameters ◽  
Momentum Exchange ◽  
Erosion And Deposition ◽  
Practical Applications ◽  
Suspension Layer ◽  
Dense Flow ◽  
Avalanche Formation

Following Norem’s description of powder-snow avalanche formation and structure, we propose a mathematical model that consists of a suspension layer and a so-called saltation layer. The latter is only a few meters deep and is modelled by depth-averaged mass and momentum balances. In the suspension layer, the mass and momentum balance equations for the mixture are supplemented by the snow mass balance and the transport equations for turbulent kinetic energy and dissipation. Mass and momentum exchange between the two layers is determined by particle settling, turbulent diffusion against the concentration gradient and aerodynamic shear forces. The net erosion or deposition rate is a function of the kinetic energy of the impacting particles. The saltation layer reacts on the suspension layer in that saltating particles extract momentum from the air flow. The preliminary estimates of the model parameters can be refined by means of saltation-trajectory simulations. Three-dimensional simulations with a simplified model have clearly shown the importance of snow erosion and deposition in practical applications. This approach is well suited for coupling to a dense-flow avalanche model.

scholarly journals Momentum Balance of the Wind-Driven and Meridional Overturning Circulation

2011 ◽  
Vol 41 (5) ◽  
pp. 960-978 ◽  
Author(s):  
David P. Marshall ◽  
Helen R. Pillar
Keyword(s):  
Ocean Circulation ◽  
Large Scale ◽  
Boundary Layer Separation ◽  
Momentum Equation ◽  
Meridional Overturning Circulation ◽  
Momentum Balance ◽  
Overturning Circulation ◽  
Potential Applications ◽  
Rotational Momentum ◽  
Meridional Overturning

Abstract When a force is applied to the ocean, fluid parcels are accelerated both locally, by the applied force, and nonlocally, by the pressure gradient forces established to maintain continuity and satisfy the kinematic boundary condition. The net acceleration can be represented through a “rotational force” in the rotational component of the momentum equation. This approach elucidates the correspondence between momentum and vorticity descriptions of the large-scale ocean circulation: if two terms balance pointwise in the rotational momentum equation, then the equivalent two terms balance pointwise in the vorticity equation. The utility of the approach is illustrated for three classical problems: barotropic Rossby waves, wind-driven circulation in a homogeneous basin, and the meridional overturning circulation in an interhemispheric basin. In the hydrostatic limit, it is shown that the rotational forces further decompose into depth-integrated forces that drive the wind-driven gyres and overturning forces that are confined to the basin boundaries and drive the overturning circulation. Potential applications of the approach to diagnosing the output of ocean circulation models, alternative and more accurate formulations of numerical ocean models, the dynamics of boundary layer separation, and eddy forcing of the large-scale ocean circulation are discussed.

A physical model of the turbulent boundary layer consonant with mean momentum balance structure

2007 ◽  
Vol 365 (1852) ◽  
pp. 823-840 ◽  
Author(s):  
Joe Klewicki ◽  
Paul Fife ◽  
Tie Wei ◽  
Pat McMurtry
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Physical Model ◽  
Momentum Equation ◽  
Momentum Balance ◽  
Dynamical Structure ◽  
Mean Flow ◽  
Structure And Dynamics ◽  
The Mean ◽  
Balance Structure

Recent studies by the present authors have empirically and analytically explored the properties and scaling behaviours of the Reynolds averaged momentum equation as applied to wall-bounded flows. The results from these efforts have yielded new perspectives regarding mean flow structure and dynamics, and thus provide a context for describing flow physics. A physical model of the turbulent boundary layer is constructed such that it is consonant with the dynamical structure of the mean momentum balance, while embracing independent experimental results relating, for example, to the statistical properties of the vorticity field and the coherent motions known to exist. For comparison, the prevalent, well-established, physical model of the boundary layer is briefly reviewed. The differences and similarities between the present and the established models are clarified and their implications discussed.

scholarly journals INTERACTIONS OF WAVES WITH IDEALIZED HIGH-RELIEF BOTTOM ROUGHNESS

2018 ◽  
pp. 57 ◽  
Author(s):  
Xiao Yu ◽  
Johanna H. Rosman ◽  
James L. Hench
Keyword(s):  
Vertical Direction ◽  
Momentum Equation ◽  
Momentum Balance ◽  
Spatial Averaging ◽  
Oscillating Flows ◽  
Horizontal Length ◽  
Large Eddy ◽  
Dispersive Stress ◽  
Infinite Array ◽  
Stress Term

Interactions between waves and high-relief bottom roughness were investigated using Large Eddy Simulations of oscillatory flow over an infinite array of regularly spaced hemispheres. Simulation results were analyzed using a spatially- and phase-averaged momentum balance to provide insight into how flow-topography interactions affect wave-driven oscillating flows. Phase-averaging was applied first, and then spatial averaging was applied over volumes with horizontal length scales greater than the size of a single solid obstacle but fine enough in the vertical direction that the vertical structure of the dynamics was resolved. Spatial averaging of the momentum equation results in terms that represent drag and inertial forces, and a dispersive stress term that represents a vertical momentum flux induced by the spatial heterogeneity of the phase-averaged flow. These new terms require parameterization in coastal ocean wave and circulation models.

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