scholarly journals Comments on “A Combined Derivation of the Integrated and Vertically Resolved, Coupled Wave–Current Equations”

2017 ◽  
Vol 47 (9) ◽  
pp. 2377-2385 ◽  
Author(s):  
Fabrice Ardhuin ◽  
Nobuhiro Suzuki ◽  
James C. McWilliams ◽  
Hidenori Aiki
Keyword(s):  
Vertical Integration ◽  
Order Term ◽  
Momentum Equation ◽  
Total Momentum ◽  
Leading Order ◽  
Mean Flow ◽  
Wave Forcing ◽  
The Mean ◽  
New Formulation ◽  
Wave Momentum

AbstractSeveral equivalent equations for the evolution of the wave-averaged current momentum have been proposed, implemented, and used. In contrast, the equation for the total momentum, which is the sum of the current and wave momenta, has not been widely used because it requires a less practical wave forcing. In an update on previous derivations, Mellor proposed a new formulation of the wave forcing for the total momentum equation. Here, the authors show that this derivation misses a leading-order term that has a zero depth-integrated value. Corrected for this omission, the wave forcing is equivalent to that in the first paper by Mellor. When this wave forcing effect on the currents is approximated it leads to an inconsistency. This study finally repeats and clarifies that the vertical integration of several various forms of the current-only momentum equations are consistent with the known depth-integrated equations for the mean flow momentum obtained by subtracting the wave momentum equation from the total momentum equation. Several other claims in prior Mellor manuscripts are discussed.

Wave-induced mean flows in rotating shallow water with uniform potential vorticity

2018 ◽  
Vol 839 ◽  
pp. 408-429 ◽  
Author(s):  
Jim Thomas ◽  
Oliver Bühler ◽  
K. Shafer Smith
Keyword(s):  
Shallow Water ◽  
Potential Vorticity ◽  
Rotation Rate ◽  
Standing Waves ◽  
Leading Order ◽  
Mean Flow ◽  
Wave Fields ◽  
The Mean ◽  
Rotating Shallow Water ◽  
Wave Induced

Theoretical and numerical computations of the wave-induced mean flow in rotating shallow water with uniform potential vorticity are presented, with an eye towards applications in small-scale oceanography where potential-vorticity anomalies are often weak compared to the waves. The asymptotic computations are based on small-amplitude expansions and time averaging over the fast wave scale to define the mean flow. Importantly, we do not assume that the mean flow is balanced, i.e. we compute the full mean-flow response at leading order. Particular attention is paid to the concept of modified diagnostic relations, which link the leading-order Lagrangian-mean velocity field to certain wave properties known from the linear solution. Both steady and unsteady wave fields are considered, with specific examples that include propagating wavepackets and monochromatic standing waves. Very good agreement between the theoretical predictions and direct numerical simulations of the nonlinear system is demonstrated. In particular, we extend previous studies by considering the impact of unsteady wave fields on the mean flow, and by considering the total kinetic energy of the mean flow as a function of the rotation rate. Notably, monochromatic standing waves provide an explicit counterexample to the often observed tendency of the mean flow to decrease monotonically with the background rotation rate.

scholarly journals Perturbation Solution for Pulsatile Flow of a Non-Newtonian Fluid in a Rock Fracture: A Logarithmic Model

Water ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 1341
Author(s):  
Irene Daprà ◽  
Giambattista Scarpi
Keyword(s):  
Power Series ◽  
Newtonian Fluid ◽  
Constant Pressure ◽  
Rock Fracture ◽  
Momentum Equation ◽  
Constitutive Law ◽  
Mean Flow ◽  
Logarithmic Model ◽  
The Past ◽  
The Mean

The purpose of this work is to study the motion of a non-Newtonian fluid in a rock fracture, generated by a constant pressure gradient to which a pulsating component is superposed. The momentum equation is faced analytically by adopting a logarithmic constitutive law; the velocity is expressed as a power series of the amplitude of the pulsating component, up to the second order, easily usable for numerical calculations. The results obtained are compared with those provided in the past by the authors, using a three-parameter Williamson model. The comparison highlights that the value of the mean flow rate in a period differs by less than 10% even if the velocity profiles look quite different.

Turbulent planar wakes under pressure gradient conditions

2017 ◽  
Vol 830 ◽  
Author(s):  
Sina Shamsoddin ◽  
Fernando Porté-Agel
Keyword(s):  
Pressure Gradient ◽  
Maximum Velocity ◽  
Wind Farms ◽  
Parameter Tuning ◽  
Momentum Equation ◽  
Mean Flow ◽  
Self Similarity ◽  
High Lift ◽  
Velocity Deficit ◽  
The Mean

Accurate prediction of the spatial evolution of turbulent wake flows under pressure gradient conditions is required in some engineering applications such as the design of high-lift devices and wind farms over topography. In this paper, we aim to develop an analytical model to predict the evolution of a turbulent planar wake under an arbitrary pressure gradient condition. The model is based on the cross-stream integration of the streamwise momentum equation and uses the self-similarity of the mean flow. We have also made an experimentally supported assumption that the ratio of the maximum velocity deficit to the wake width is independent of the imposed pressure gradient. The asymptotic response of the wake to the pressure gradient is also investigated. After its derivation, the model is successfully validated against experimental data by comparing the evolution of the wake width and maximum velocity deficit. The inputs of the model are the imposed pressure gradient and the wake width under zero pressure gradient. The model does not require any parameter tuning and is deemed to be practical, computationally fast, accurate enough, and therefore useful for the scientific and engineering communities.

scholarly journals Stratospheric Wave–Mean Flow Feedbacks and Sudden Stratospheric Warmings in a Simple Model Forced by Upward Wave Activity Flux

2014 ◽  
Vol 71 (11) ◽  
pp. 4055-4071 ◽  
Author(s):  
Jeremiah P. Sjoberg ◽  
Thomas Birner
Keyword(s):  
Wave Propagation ◽  
Inversion Layer ◽  
Wave Activity ◽  
Mean Flow ◽  
Stable States ◽  
Mass Model ◽  
Incoming Wave ◽  
Wave Forcing ◽  
The Mean ◽  
Wave Activity Flux

Abstract A classic result of studying stratospheric wave–mean flow interactions presented by Holton and Mass is that, for constant incoming wave forcing (at a notional tropopause), a vacillating stratospheric response may ensue. Simple models, such as the Holton–Mass model, typically prescribe the incoming wave forcing in terms of geopotential perturbation, which is not a proxy for upward wave activity flux. Here, the authors reformulate the Holton–Mass model such that incoming upward wave activity flux is prescribed. The Holton–Mass model contains a positive wave–mean flow feedback whereby wave forcing decelerates the mean flow, allowing enhanced wave propagation, which then further decelerates the mean flow, etc., until the mean flow no longer supports wave propagation. By specifying incoming wave activity flux, this feedback is constrained to the model interior. Bistability—where the zonal wind may exist at one of two distinct steady states for a given incoming wave forcing—is maintained in this reformulated model. The model is perturbed with transient pulses of upward wave activity flux to produce transitions between the two stable states. A minimum of integrated incoming wave activity flux necessary to force these sudden stratospheric warming–like transitions exists for pulses with time scales on the order of 10 days, arising from a wave time scale internal to the model at which forcing produces the strongest mean-flow response. The authors examine how the tropopause affects the internal feedback for this model setup and find that the tropopause inversion layer may potentially provide an important source of wave activity in the lower stratosphere.

scholarly journals Reformulation of quasi-linear theory

1972 ◽  
Vol 8 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Allan N. Kaufman
Keyword(s):  
Diffusion Equation ◽  
Linear Theory ◽  
Total Momentum ◽  
New Formulation ◽  
Wave Momentum

A new formulation of quasi-linear theory is presented, which allows for only resonant diffusion, caused by both growing and damped waves. Nonresonant terms do not appear in the diffusion equation, but contribute to wave momentum and energy, and ensure conservation of total momentum and energy.

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 Role of Finite-Amplitude Rossby Waves and Nonconservative Processes in Downward Migration of Extratropical Flow Anomalies

2018 ◽  
Vol 75 (5) ◽  
pp. 1385-1401 ◽  
Author(s):  
Sandro W. Lubis ◽  
Clare S. Y. Huang ◽  
Noboru Nakamura ◽  
Nour-Eddine Omrani ◽  
Martin Jucker
Keyword(s):  
Finite Amplitude ◽  
Wave Activity ◽  
Mean Flow ◽  
Amplitude Wave ◽  
Wave Forcing ◽  
The Mean ◽  
Finite Amplitude Wave Activity ◽  
Tropospheric Wind ◽  
Stratospheric Variability

There is growing evidence that stratospheric variability exerts a noticeable imprint on tropospheric weather and climate. Despite clear evidence of these impacts, the principal mechanism whereby stratospheric variability influences tropospheric circulation has remained elusive. Here, the authors introduce a novel approach, based on the theory of finite-amplitude wave activity, for quantifying the role of adiabatic and nonconservative effects on the mean flow that shape the downward coupling from the stratosphere to the troposphere during stratospheric vortex weakening (SVW) events. The advantage of using this theory is that eddy effects (at finite amplitude) on the mean flow can be more readily distinguished from nonconservative effects. The results show (in confirmation of previous work) that the downward migration of extratropical wind anomalies is largely attributable to dynamical adjustments induced by fluctuating finite-amplitude wave forcing. The nonconservative effects, on the other hand, contribute to maintaining the downward signals in the recovery stage within the stratosphere, hinting at the importance of mixing and diabatic heating. The analysis further indicates that variations in stratospheric finite-amplitude wave forcing are too weak to account for the attendant changes and shapes in the tropospheric flow. It is suggested that the indirect effect of tropospheric finite-amplitude wave activity through the residual displacements is needed to amplify and prolong the tropospheric wind responses over several weeks. The results also reveal that the local tropospheric wave activity over the North Pacific and North Atlantic sectors plays a significant role in shaping the high-latitude tropospheric wind response to SVW events.

scholarly journals Comments on “The Three-Dimensional Current and Surface Wave Equations”

2008 ◽  
Vol 38 (6) ◽  
pp. 1340-1350 ◽  
Author(s):  
Fabrice Ardhuin ◽  
Alastair D. Jenkins ◽  
Konstadinos A. Belibassakis
Keyword(s):  
Momentum Flux ◽  
Large Scale ◽  
Wave Equations ◽  
Three Dimensional ◽  
Wave Theory ◽  
Correction Method ◽  
Vertical Flux ◽  
Mean Flow ◽  
Wave Forcing ◽  
The Mean

Abstract The lowest order sigma-transformed momentum equation given by Mellor takes into account a phase-averaged wave forcing based on Airy wave theory. This equation is shown to be generally inconsistent because of inadequate approximations of the wave motion. Indeed the evaluation of the vertical flux of momentum requires an estimation of the pressure p and coordinate transformation function s to first order in parameters that define the large-scale evolution of the wave field, such as the bottom slope. Unfortunately, there is no analytical expression for p and s at that order. A numerical correction method is thus proposed and verified. Alternative coordinate transforms that allow a separation of wave and mean flow momenta do not suffer from this inconsistency nor do they require a numerical estimation of the wave forcing. Indeed, the problematic vertical flux is part of the wave momentum flux, thus distinct from the mean flow momentum flux, and not directly relevant to the mean flow evolution.

On the diffusion of momentum and mass by internal gravity waves

1976 ◽  
Vol 77 (4) ◽  
pp. 789-823 ◽  
Author(s):  
Peter Mtfller
Keyword(s):  
Wave Field ◽  
Gravity Waves ◽  
General Circulation ◽  
Viscosity Coefficient ◽  
Internal Gravity Waves ◽  
Momentum Equation ◽  
Momentum Balance ◽  
Mean Flow ◽  
Vertical Diffusion ◽  
The Mean

The interaction between short internal gravity waves and a larger-scale mean flow in the ocean is analysed in the Wkbj approximation. The wave field determines the radiation-stress term in the momentum equation of the mean flow and a similar term in the buoyancy equation. The mean flow affects the propagation characteristics of the wave field. This cross-coupling is treated as a small perturbation. When relaxation effects within the wave field are considered, the mean flow induces a modulation of the wave field which is a linear functional of the spatial gradients of the mean current velocity. The effect that this modulation itself has on the mean flow can be reduced to the addition of diffusion terms to the equations for the mass and momentum balance of the mean flow. However, there is no vertical diffusion of mass and other passive properties. The diffusion coefficients depend on the frequency spectrum and the relaxation time of the internal-wave field and can be evaluated analytically. The vertical viscosity coefficient is found to be vv [ape ] 4 x 103cm2/s and exceeds values typically used in models of the general circulation by at least two orders of magnitude.

Sign in / Sign up

Export Citation Format

Share Document