DOCSLIB.ORG

Atmospheric Waves Chapter 6

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Atmospheric Waves Chapter 6 Atmospheric Waves Chapter 6 Paul A. Ullrich [email protected] Part 5: Mixed Rossby-Gravity Waves Trapped Waves Trapped waves decay exponentially rapidly away from a boundary (such as the equator or a coastline). Trapped waves include: • Equatorial (and coastal) Kelvin waves • Equatorial Rossby (ER) waves • Mixed Rossby-Gravity (MGR) waves at the equator • Equatorial inertia-gravity waves Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves • Shallow water equations, no topography • Coriolis parameter at the equator is approximated by equatorial β-plane: 2⌦ f βy y ⇡ ⇡ a ✓ ◆ Du @Φ βyv + =0 Dt − @x Dv @Φ + βyu + =0 Dt @y DΦ @u @v + Φ + =0 with Φ = gh Dt @x @y ✓ ◆ Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves Linearize shallow water equations about a u = u ,v = v ,h= H + h state at rest with mean height H 0 0 0 @u0 @Φ0 βyv0 + = 0 (1) @t − @x @v0 @Φ0 + βyu0 + = 0 (2) @t @y @Φ0 @u0 @v0 + gH + = 0 (3) with Φ0 = gh0 @t @x @y ✓ ◆ Seek wave solutions of the form (allow amplitudes to vary in y): (u0,v0, Φ0)= uˆ(y), vˆ(y), Φˆ(y) exp(i(kx ⌫t)) − ⇣ ⌘ Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves Yields the system i⌫uˆ βyvˆ + ikΦˆ = 0 (1) − − @Φˆ i⌫vˆ + βyuˆ + = 0 (2) − @y @vˆ i⌫Φˆ + gH ikuˆ + = 0 (3) − @y ✓ ◆ βy k Solve (1) for uˆ : uˆ = i vˆ + Φˆ (4) ⌫ ⌫ Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves Plug (4) into (2) and (3), rearrange terms @Φˆ β2y2 ⌫2 vˆ ikβyΦˆ i⌫ = 0 (5) − − − @y @vˆ k ⌫2 gHk2 Φˆ + i⌫gH βyvˆ = 0 (6) − @y − ⌫ ✓ ◆ Solve (6) for Φ ˆ : i⌫gH @vˆ k Φˆ = βyvˆ (7) −(⌫2 gHk2) @y − ⌫ − ✓ ◆ Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves Plug (7) into (5) and rearrange terms. Yields second- order differential equation for amplitude function @2vˆ ⌫2 k β2y2 + k2 β vˆ = 0 (8) @y2 gH − − ⌫ − gH ✓ ◆ More complicated equation than before since the coefficient in square brackets is not constant (depends on y) Before discussing the solution in detail, let’s frst look at two asymptotic limits when either H→ ∞ or β=0 Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves Asymptotic limit H→∞ Means that the motion is non-divergent, consequence of the continuity equation (3): @u0 @v0 1 @Φ0 + = 0 for H @x @y −gH @t ! !1 ✓ ◆ Equation (8) becomes @2vˆ k + k2 β vˆ = 0 (9) @y2 − − ⌫ Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves Asymptotic limit H→∞ Asymptotic limit H→∞ @2vˆ k + k2 β vˆ = 0 (9) @y2 − − ⌫ Solutions for the amplitude function vˆ in (9) exists of the form: vˆ = v0 exp(i`y) Provided that the frequency ν satisfes the Rossby wave dispersion relationship: βk ⌫ = 2− 2 k + ` It shows that for non-divergent barotropic fow, equatorial dynamics are in no way special Earth’s rotation enters in form of β, not f: Rossby wave Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves Asymptotic limit β=0 All infuence of rotation is eliminated. Due to linearity, solutions of (9) exists of the form vˆ = v0 exp(i`y) Equation (8) reduces to the shallow water gravity model @2vˆ ⌫2 + k2 vˆ = 0 (10) @y2 gH − Non-trivial solutions if frequency ν satisfes ⌫ = gH(k2 + `2) ± Pure gravity wave responsep (eastward and westward) Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves In the general case with 2 2 2 2 @ vˆ ⌫ 2 k β y + k β vˆ = 0 (8) @y2 gH − − ⌫ − gH ✓ ◆ we seek solutions for the meridional distribution of vˆ with boundary condition vˆ 0 for y ! | | !1 For small y the coefficient in square brackets is positive and solutions oscillate in y. For large y the coefficient is negative, solutions either grow or decay. Only the decaying solution satisfes boundary condition. Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves Matsuno (1966) showed that decaying solutions only exist if the constant part of the coefficient in square brackets satisfes the relationship pgH ⌫2 k k2 β =2n +1 n =0, 1, 2,... β gH − − ⌫ ✓ ◆ This is a cubic dispersion relation for frequency ν. It determines the allowed frequencies ν of equatorially trapped waves for the zonal wavenumber k and meridional mode number n. Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves In (8) replace y by the nondimensional meridional coordinate β 1/2 ⇠ = y pgH "✓ ◆ # ⇠2 Then the solution has the form vˆ(⇠)=v H (⇠)exp 0 n − 2 ✓ ◆ th Where v0 is a constant and H n ( ⇠ ) designates the n Hermite polynomial. The frst few Hermite polynomials have the form 2 H0(⇠)=1,H1(⇠)=2⇠,H2(⇠)=4⇠ 2 − Index n corresponds to the number of N-S roots in v. Paul Ullrich Atmospheric Waves March 2014 Hermite Polynomials Meridional (N-S) structure of the amplitude function: ⇠2 vˆ(⇠)=v H (⇠)exp n =0, 1, 2,... 0 n − 2 ✓ ◆ n n 2 d 2 Hermite polynomials: H (⇠)=( 1) exp(⇠ ) exp ⇠− n − d⇠n N vˆ(⇠) 0 ξ vˆ(λ) S n = 0 n = 1 n = 2 n = 5 n = 10 Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves In general, the three ν solutions of pgH ⌫2 k k2 β =2n +1 n =0, 1, 2,... β gH − − ⌫ ✓ ◆ can be interpreted as an equatorially trapped 1) Eastward moving inertia-gravity wave 2) Westward moving inertia-gravity wave 3) Westward moving equatorial Rossby wave However, the case n = 0 must be treated separately: For n = 0 the meridional velocity perturbation has a Gaussian distribution centered at the equator Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves For n = 0 (special case) the dispersion relation factors as ⌫ β ⌫ k + k =0 pgH − − ⌫ pgH ✓ ◆✓ ◆ The root ⌫ = k gH (westward propagating gravity wave) is not permitted, violates− an earlier assumptions when eliminating Φˆ p The two other roots are 1 4β ⌫ = k gH 1+ "2 ± s k2pgH # p Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves The positive root corresponds to an eastward moving equatorial inertia-gravity wave The negative root corresponds to a westward moving wave, which resembles • An equatorial (westward moving) inertia-gravity (IG) wave for long zonal scale (k → 0) • An equatorial Rossby (ER) wave for zonal scales characteristic of synoptic-scale disturbances This mode (n=0, westward moving) is therefore called Mixed Rossby-Gravity (MRG) wave. Paul Ullrich Atmospheric Waves March 2014 Equatorial Mixed Rossby-Gravity Waves Plane view of horizontal velocity and height perturbations in the (n=0) Mixed Rossby-Gravity (MRG) wave, propagates westward H L Equator L H Paul Ullrich Atmospheric Waves March 2014 .
Load more

Recommended publications
  • Fourth-Order Nonlinear Evolution Equations for Surface Gravity Waves in the Presence of a Thin Thermocline
    J. Austral. Math. Soc. Ser. B 39(1997), 214-229 FOURTH-ORDER NONLINEAR EVOLUTION EQUATIONS FOR SURFACE GRAVITY WAVES IN THE PRESENCE OF A THIN THERMOCLINE SUDEBI BHATTACHARYYA1 and K. P. DAS1 (Received 14 August 1995; revised 21 December 1995) Abstract Two coupled nonlinear evolution equations correct to fourth order in wave steepness are derived for a three-dimensional wave packet in the presence of a thin thermocline. These two coupled equations are reduced to a single equation on the assumption that the space variation of the amplitudes takes place along a line making an arbitrary fixed angle with the direction of propagation of the wave. This single equation is used to study the stability of a uniform wave train. Expressions for maximum growth rate of instability and wave number at marginal stability are obtained. Some of the results are shown graphically. It is found that a thin thermocline has a stabilizing influence and the maximum growth rate of instability decreases with the increase of thermocline depth. 1. Introduction There exist a number of papers on nonlinear interaction between surface gravity waves and internal waves. Most of these are concerned with the mechanism of generation of internal waves through nonlinear interaction of surface gravity waves. Coherent three wave interactions of two surface waves and one internal wave have been investigated by Ball [1], Thorpe [22], Watson, West and Cohen [23] and others. Using the theoretical model of Hasselman [12] for incoherent three-wave interaction, Olber and Hertrich [18] have reported a mechanism of generation of internal waves by coupling with surface waves using a three-layer model of the ocean.
    [Show full text]
  • Internal Gravity Waves: from Instabilities to Turbulence Chantal Staquet, Joël Sommeria
    Internal gravity waves: from instabilities to turbulence Chantal Staquet, Joël Sommeria To cite this version: Chantal Staquet, Joël Sommeria. Internal gravity waves: from instabilities to turbulence. Annual Review of Fluid Mechanics, Annual Reviews, 2002, 34, pp.559-593. 10.1146/an- nurev.fluid.34.090601.130953. hal-00264617 HAL Id: hal-00264617 https://hal.archives-ouvertes.fr/hal-00264617 Submitted on 4 Feb 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License INTERNAL GRAVITY WAVES: From Instabilities to Turbulence C. Staquet and J. Sommeria Laboratoire des Ecoulements Geophysiques´ et Industriels, BP 53, 38041 Grenoble Cedex 9, France; e-mail: [email protected], [email protected] Key Words geophysical fluid dynamics, stratified fluids, wave interactions, wave breaking Abstract We review the mechanisms of steepening and breaking for internal gravity waves in a continuous density stratification. After discussing the instability of a plane wave of arbitrary amplitude in an infinite medium at rest, we consider the steep- ening effects of wave reflection on a sloping boundary and propagation in a shear flow. The final process of breaking into small-scale turbulence is then presented.
    [Show full text]
  • The Internal Gravity Wave Spectrum: a New Frontier in Global Ocean Modeling
    The internal gravity wave spectrum: A new frontier in global ocean modeling Brian K. Arbic Department of Earth and Environmental Sciences University of Michigan Supported by funding from: Office of Naval Research (ONR) National Aeronautics and Space Administration (NASA) National Science Foundation (NSF) Brian K. Arbic Internal wave spectrum in global ocean models Collaborators • Naval Research Laboratory Stennis Space Center: Joe Metzger, Jim Richman, Jay Shriver, Alan Wallcraft, Luis Zamudio • University of Southern Mississippi: Maarten Buijsman • University of Michigan: Joseph Ansong, Steve Bassette, Conrad Luecke, Anna Savage • McGill University: David Trossman • Bangor University: Patrick Timko • Norwegian Meteorological Institute: Malte M¨uller • University of Brest and The University of Texas at Austin: Rob Scott • NASA Goddard: Richard Ray • Florida State University: Eric Chassignet • Others including many members of the NSF-funded Climate Process Team led by Jennifer MacKinnon of Scripps Brian K. Arbic Internal wave spectrum in global ocean models Motivation • Breaking internal gravity waves drive most of the mixing in the subsurface ocean. • The internal gravity wave spectrum is just starting to be resolved in global ocean models. • Somewhat analogous to resolution of mesoscale eddies in basin- and global-scale models in 1990s and early 2000s. • Builds on global internal tide modeling, which began with 2004 Arbic et al. and Simmons et al. papers utilizing Hallberg Isopycnal Model (HIM) run with tidal foricng only and employing a horizontally uniform stratification. Brian K. Arbic Internal wave spectrum in global ocean models Motivation continued... • Here we utilize simulations of the HYbrid Coordinate Ocean Model (HYCOM) with both atmospheric and tidal forcing. • Near-inertial waves and tides are put into a model with a realistically varying background stratification.
    [Show full text]
  • Gravity Waves Are Just Waves As Pressure He Was Observing
    FLIGHTOPS Nearly impossible to predict and difficult to detect, this weather phenomenon presents Gravitya hidden risk, especially close to the ground. regory Bean, an experienced pi- National Weather Service observer re- around and land back at Burlington. lot and flight instructor, says he ferred to this as a “gravity wave.” Neither The wind had now swung around to wasn’t expecting any problems the controller nor Bean knew what the 270 degrees at 25 kt. At this point, he while preparing for his flight weather observer was talking about. recalled, his rate of descent became fromG Burlington, Vermont, to Platts- Deciding he could deal with the “alarming.” The VSI now pegged at the burgh, New York, U.S. His Piper Seneca wind, Bean was cleared for takeoff. The bottom of the scale. Despite the control checked out fine. The weather that April takeoff and initial climb were normal difficulty, he landed the aircraft without evening seemed benign. While waiting but he soon encountered turbulence “as further incident. Bean may not have for clearance, he was taken aback by a rough as I have ever encountered.” He known what a gravity wave was, but report of winds gusting to 35 kt from wanted to climb to 2,600 ft. With the he now knew what it could do to an 140 degrees. The air traffic control- vertical speed indicator (VSI) “pegged,” airplane.1 ler commented on rapid changes in he shot up to 3,600 ft. Finally control- Gravity waves are just waves as pressure he was observing. He said the ling the airplane, Bean opted to turn most people normally think of them.
    [Show full text]
  • Gravity Wave and Tidal Influences On
    Ann. Geophys., 26, 3235–3252, 2008 www.ann-geophys.net/26/3235/2008/ Annales © European Geosciences Union 2008 Geophysicae Gravity wave and tidal influences on equatorial spread F based on observations during the Spread F Experiment (SpreadFEx) D. C. Fritts1, S. L. Vadas1, D. M. Riggin1, M. A. Abdu2, I. S. Batista2, H. Takahashi2, A. Medeiros3, F. Kamalabadi4, H.-L. Liu5, B. G. Fejer6, and M. J. Taylor6 1NorthWest Research Associates, CoRA Division, Boulder, CO, USA 2Instituto Nacional de Pesquisas Espaciais (INPE), San Jose dos Campos, Brazil 3Universidade Federal de Campina Grande. Campina Grande. Paraiba. Brazil 4University of Illinois, Champaign, IL, USA 5National Center for Atmospheric research, Boulder, CO, USA 6Utah State University, Logan, UT, USA Received: 15 April 2008 – Revised: 7 August 2008 – Accepted: 7 August 2008 – Published: 21 October 2008 Abstract. The Spread F Experiment, or SpreadFEx, was per- Keywords. Ionosphere (Equatorial ionosphere; Ionosphere- formed from September to November 2005 to define the po- atmosphere interactions; Plasma convection) – Meteorology tential role of neutral atmosphere dynamics, primarily grav- and atmospheric dynamics (Middle atmosphere dynamics; ity waves propagating upward from the lower atmosphere, in Thermospheric dynamics; Waves and tides) seeding equatorial spread F (ESF) and plasma bubbles ex- tending to higher altitudes. A description of the SpreadFEx campaign motivations, goals, instrumentation, and structure, 1 Introduction and an overview of the results presented in this special issue, are provided by Fritts et al. (2008a). The various analyses of The primary goal of the Spread F Experiment (SpreadFEx) neutral atmosphere and ionosphere dynamics and structure was to test the theory that gravity waves (GWs) play a key described in this special issue provide enticing evidence of role in the seeding of equatorial spread F (ESF), Rayleigh- gravity waves arising from deep convection in plasma bub- Taylor instability (RTI), and plasma bubbles extending to ble seeding at the bottomside F layer.
    [Show full text]
  • Inertia-Gravity Wave Generation: a WKB Approach
    Inertia-gravity wave generation: a WKB approach Jonathan Maclean Aspden Doctor of Philosophy University of Edinburgh 2010 Declaration I declare that this thesis was composed by myself and that the work contained therein is my own, except where explicitly stated otherwise in the text. (Jonathan Maclean Aspden) iii iv Abstract The dynamics of the atmosphere and ocean are dominated by slowly evolving, large-scale motions. However, fast, small-scale motions in the form of inertia-gravity waves are ubiquitous. These waves are of great importance for the circulation of the atmosphere and oceans, mainly because of the momentum and energy they transport and because of the mixing they create upon breaking. So far the study of inertia-gravity waves has answered a number of questions about their propagation and dissipation, but many aspects of their generation remain poorly understood. The interactions that take place between the slow motion, termed balanced or vortical motion, and the fast inertia-gravity wave modes provide mechanisms for inertia-gravity wave generation. One of these is the instability of balanced flows to gravity-wave-like perturbations; another is the so-called spontaneous generation in which a slowly evolving solution has a small gravity-wave component intrinsically coupled to it. In this thesis, we derive and study a simple model of inertia-gravity wave generation which considers the evolution of a small-scale, small amplitude perturbation superimposed on a large-scale, possibly time-dependent flow. The assumed spatial-scale separation makes it possible to apply a WKB approach which models the perturbation to the flow as a wavepacket.
    [Show full text]
  • Chapter 16: Waves [Version 1216.1.K]
    Contents 16 Waves 1 16.1Overview...................................... 1 16.2 GravityWavesontheSurfaceofaFluid . ..... 2 16.2.1 DeepWaterWaves ............................ 5 16.2.2 ShallowWaterWaves........................... 5 16.2.3 Capillary Waves and Surface Tension . .... 8 16.2.4 Helioseismology . 12 16.3 Nonlinear Shallow-Water Waves and Solitons . ......... 14 16.3.1 Korteweg-deVries(KdV)Equation . ... 14 16.3.2 Physical Effects in the KdV Equation . ... 16 16.3.3 Single-SolitonSolution . ... 18 16.3.4 Two-SolitonSolution . 19 16.3.5 Solitons in Contemporary Physics . .... 20 16.4 RossbyWavesinaRotatingFluid. .... 22 16.5SoundWaves ................................... 25 16.5.1 WaveEnergy ............................... 27 16.5.2 SoundGeneration............................. 28 16.5.3 T2 Radiation Reaction, Runaway Solutions, and Matched Asymp- toticExpansions ............................. 31 0 Chapter 16 Waves Version 1216.1.K, 7 Sep 2012 Please send comments, suggestions, and errata via email to [email protected] or on paper to Kip Thorne, 350-17 Caltech, Pasadena CA 91125 Box 16.1 Reader’s Guide This chapter relies heavily on Chaps. 13 and 14. • Chap. 17 (compressible flows) relies to some extent on Secs. 16.2, 16.3 and 16.5 of • this chapter. The remaining chapters of this book do not rely significantly on this chapter. • 16.1 Overview In the preceding chapters, we have derived the basic equations of fluid dynamics and devel- oped a variety of techniques to describe stationary flows. We have also demonstrated how, even if there exists a rigorous, stationary solution of these equations for a time-steady flow, instabilities may develop and the amplitude of oscillatory disturbances will grow with time. These unstable modes of an unstable flow can usually be thought of as waves that interact strongly with the flow and extract energy from it.
    [Show full text]
  • Baroclinic Instability, Lecture 19
    19. Baroclinic Instability In two-dimensional barotropic flow, there is an exact relationship between mass 2 streamfunction ψ and the conserved quantity, vorticity (η)given by η = ∇ ψ.The evolution of the conserved variable η in turn depends only on the spatial distribution of η andonthe flow, whichisd erivable fromψ and thus, by inverting the elliptic relation, from η itself. This strongly constrains the flow evolution and allows one to think about the flow by following η around and inverting its distribution to get the flow. In three-dimensional flow, the vorticity is a vector and is not in general con­ served. The appropriate conserved variable is the potential vorticity, but this is not in general invertible to find the flow, unless other constraints are provided. One such constraint is geostrophy, and a simple starting point is the set of quasi-geostrophic equations which yield the conserved and invertible quantity qp, the pseudo-potential vorticity. The same dynamical processes that yield stable and unstable Rossby waves in two-dimensional flow are responsible for waves and instability in three-dimensional baroclinic flow, though unlike the barotropic 2-D case, the three-dimensional dy­ namics depends on at least an approximate balance between the mass and flow fields. 97 Figure 19.1 a. The Eady model Perhaps the simplest example of an instability arising from the interaction of Rossby waves in a baroclinic flow is provided by the Eady Model, named after the British mathematician Eric Eady, who published his results in 1949. The equilibrium flow in Eady’s idealization is illustrated in Figure 19.1.
    [Show full text]
  • On the Transport of Energy in Water Waves
    On The Transport Of Energy in Water Waves Marshall P. Tulin Professor Emeritus, UCSB [email protected] Abstract Theory is developed and utilized for the calculation of the separate transport of kinetic, gravity potential, and surface tension energies within sinusoidal surface waves in water of arbitrary depth. In Section 1 it is shown that each of these three types of energy constituting the wave travel at different speeds, and that the group velocity, cg, is the energy weighted average of these speeds, depth and time averaged in the case of the kinetic energy. It is shown that the time averaged kinetic energy travels at every depth horizontally either with (deep water), or faster than the wave itself, and that the propagation of a sinusoidal wave is made possible by the vertical transport of kinetic energy to the free surface, where it provides the oscillating balance in surface energy just necessary to allow the propagation of the wave. The propagation speed along the surface of the gravity potential energy is null, while the surface tension energy travels forward along the wave surface everywhere at twice the wave velocity, c. The flux of kinetic energy when viewed traveling with a wave provides a pattern of steady flux lines which originate and end on the free surface after making vertical excursions into the wave, even to the bottom, and these are calculated. The pictures produced in this way provide immediate insight into the basic mechanisms of wave motion. In Section 2 the modulated gravity wave is considered in deep water and the balance of terms involved in the propagation of the energy in the wave group is determined; it is shown again that vertical transport of kinetic energy to the surface is fundamental in allowing the propagation of the modulation, and in determining the well known speed of the modulation envelope, cg.
    [Show full text]
  • The Equilibrium Dynamics and Statistics of Gravity–Capillary Waves
    J. Fluid Mech. (2015), vol. 767, pp. 449–466. c Cambridge University Press 2015 449 doi:10.1017/jfm.2014.740 The equilibrium dynamics and statistics of gravity–capillary waves W. Kendall Melville1, † and Alexey V. Fedorov2 1Scripps Institution of Oceanography, University of California San Diego, La Jolla, CA 92093-0213, USA 2Department of Geology and Geophysics, Yale University, 210 Whitney Avenue, PO Box 208109, New Haven, CT 06520, USA (Received 24 December 2013; revised 8 October 2014; accepted 19 December 2014; first published online 18 February 2015) Recent field observations and modelling of breaking surface gravity waves suggest that air-entraining breaking is not sufficiently dissipative of surface gravity waves to balance the dynamics of wind-wave growth and nonlinear interactions with dissipation for the shorter gravity waves of O.10/ cm wavelength. Theories of parasitic capillary waves that form at the crest and forward face of shorter steep gravity waves have shown that the dissipative effects of these waves may be one to two orders of magnitude greater than the viscous dissipation of the underlying gravity waves. Thus the parasitic capillaries may provide the required dissipation of the short wind-generated gravity waves. This has been the subject of speculation and conjecture in the literature. Using the nonlinear theory of Fedorov & Melville (J. Fluid Mech., vol. 354, 1998, pp. 1–42), we show that the dissipation due to the parasitic capillaries is sufficient to balance the wind input to the short gravity waves over some range of wave ages and wave slopes. The range of gravity wave lengths on which these parasitic capillary waves are dynamically significant approximately corresponds to the range of short gravity waves that Cox & Munk (J.
    [Show full text]
  • ATMS 310 Rossby Waves Properties of Waves in the Atmosphere Waves
    ATMS 310 Rossby Waves Properties of Waves in the Atmosphere Waves – Oscillations in field variables that propagate in space and time. There are several aspects of waves that we can use to characterize their nature: 1) Period – The amount of time it takes to complete one oscillation of the wave\ 2) Wavelength (λ) – Distance between two peaks of troughs 3) Amplitude – The distance between the peak and the trough of the wave 4) Phase – Where the wave is in a cycle of amplitude change For a 1-D wave moving in the x-direction, the phase is defined by: φ ),( = −υtkxtx − α (1) 2π where φ is the phase, k is the wave number = , υ = frequency of oscillation (s-1), and λ α = constant determined by the initial conditions. If the observer is moving at the phase υ speed of the wave ( c ≡ ), then the phase of the wave is constant. k For simplification purposes, we will only deal with linear sinusoidal wave motions. Dispersive vs. Non-dispersive Waves When describing the velocity of waves, a distinction must be made between the group velocity and the phase speed. The group velocity is the velocity at which the observable disturbance (energy of the wave) moves with time. The phase speed of the wave (as given above) is how fast the constant phase portion of the wave moves. A dispersive wave is one in which the pattern of the wave changes with time. In dispersive waves, the group velocity is usually different than the phase speed. A non- dispersive wave is one in which the patterns of the wave do not change with time as the wave propagates (“rigid” wave).
    [Show full text]
  • TTL Upwelling Driven by Equatorial Waves
    TTL Upwelling driven by Equatorial Waves L09804 LIUWhat drives the annual cycle in TTL upwelling? ET AL.: START OF WATER VAPOR AND CO TAPE RECORDERS L09804 Liu et al. (2007) Ortland, D. and M. J. Alexander, The residual mean circula8on in the TTL driven by tropical waves, JAS, (in review), 2013. Figure 1. (a) Seasonal variation of 10°N–10°S mean EOS MLS water vapor after dividing by the mean value at each level. (b) Seasonal variation of 10°N–10°S EOS MLS CO after dividing by the mean value at each level. represented in two principal ways. The first is by the area of [8]ConcentrationsofwatervaporandCOnearthe clouds with low infrared brightness temperatures [Gettelman tropical tropopause are available from retrievals of EOS et al.,2002;Massie et al.,2002;Liu et al.,2007].Thesecond MLS measurements [Livesey et al.,2005,2006].Monthly is by the area of radar echoes reaching the tropopause mean water vapor and CO mixing ratios at 146 hPa and [Alcala and Dessler,2002;Liu and Zipser, 2005]. In this 100 hPa in the 10°N–10°Sand10° longitude boxes are study, we examine the areas of clouds that have TRMM calculated from one full year (2005) of version 1.5 MLS Visible and Infrared Scanner (VIRS) 10.8 mmbrightness retrievals. In this work, all MLS data are processed with temperatures colder than 210 K, and the area of 20 dBZ requirements described by Livesey et al. [2005]. Tropopause echoes at 14 km measured by the TRMM Precipitation temperature is averaged from the 2.5° resolution NCEP Radar (PR).
    [Show full text]