On Derivation and Interpretation of Kuo Eliassen Equation
|
|
- Elizabeth Norman
- 6 years ago
- Views:
Transcription
1 1 On Derivation and Interpretation of Kuo Eliassen Equation Jun-Ichi Yano 1 1 GAME/CNRM, Météo-France and CNRS, Toulouse Cedex, France Manuscript submitted 22 September 2010 The Kuo Eliassen equation provides a balance condition for both tropical cyclone like vortex systems as well as zonally symmetric meridional circulations. This condition is examined with the former application more in mind. The condition is derived more pedagogically based on the bounded derivative method. Some physical interpretations as well as basic mathematical remarks on this condition are provided. Analogy with quasi geostrophic system is also remarked. 1. Introduction Balanced vortex dynamics on the f plane is originally introduced by Eliassen (1952), and is investigated by Shapiro and Willoughby (1982), Schubert and Hack (1982), and many others. Schubert and McNoldy (2010, SM10 hereinafter) develop a physical interpretation of this system. The present note, being inspired by SM10, attempts to provide further interpretations on this system from a more basic level. In considering this problem, a balance condition (Eq of SM10) arises from a consistency of the system. A similar balance condition can be derived in considering zonally symmetric meridional circulations, as originally considered by Kuo (1956). For this reason, this balance condition is sometimes called Kuo Eliassen equation (e.g., Krishnamurti et al. 1994, Chang 1996). Two version are, of course, different in details mainly due to difference of horizontal scales of the two types of circulations. The present note considers a version for the balanced vortex dynamics as considered by SM10. By presenting three different types of analytical solutions, SM10 presents interpretations of Rossby length and depth arise in Kuo Eliassen equation. The goal of the present notes is, in turn, to examine the derivation of this balance equation and then offer further interpretations of this equation under further approximations. The next section examines the derivation of Kuo Eliassen equation from a point of view of the bounded derivative method (Kreiss 1980, Browning et al. 1980). Further interpretations of this equation is attempted from a more basic level than SM10 in Sec. 3. Some mathematical issues are discussed separately in Sec. 4 in order to better understand exact analytical solutions provided by SM10. Finally, Sec. 5 points out analogy between the Kuo Eliassen equation and the quasi geostrophic model. 2. Derivation of Kuo Eliassen Equation Kuo Eliassen equation is derived by SM10 as their Eq. (2.10). Starting point for deriving this equation is a balance vortex dynamics equation system given by their Eq. (2.1) (2.5). In order to obtain insights on the derivation of this equation, I recover neglected weak local time tendencies in their equations for azimuthal (Eq. 2.1) and vertical (Eq. 2.3) velocities with their smallness indicated by a parameter ǫ. Thus, the system in concern is given by ǫ u t f av = φ r, (1a) v t + ζ au + w v z = 0 ǫ w t + φ z = g θ 0 θ, (1b) (1c) (ru) rr + (ρw) = 0, (1d) ρz θ t + uθ r + wθ z = θ. (1e) To whom correspondence should be addressed. Jun-Ichi Yano, CNRM, Météo-France, 42 av Coriolis, Toulouse Cedex, France. jun-ichi.yano@zmaw.de.
2 2 Yano Here, f a = f +v/r is an absolute Coriolis parameter felt by the radial wind v, and ζ a = f + (rv)/rr is the absolute vorticity, which also works as an effective Croiolis parameter for the azimuthal wind u. Note otherwise the identical notations as in SM10 are adopted here. This set (Eq. 1.a e) is obtained by partially recovering O(ǫ) terms from Eq. (1.1) (1.4) of Shapiro and Willoughby (1982). The parameter ǫ can be considered as a measure of the ratio of the radial and vertical velocities to the azimuthal velocity. The purpose here is to indicate the main terms neglected in the balanced vortex dynamics: acceleration of azimuthal and vertical velocities. Then we ask a question what conditions ensure the neglect of those terms. More terms of O(ǫ) from the original full system may be recovered, but only with an expense of making the following derivative more involved. Only the essential terms are retained in order to make the point as lucid as possible. A series of balance conditions can be obtained by writing down a linear tendency equation for the vorticity, ζ ϕ = u/z w/r, in the azimuthal direction. Note that an arising question is naturally linear by already neglecting the advection terms in Eq. (1a) and (1c). For this purpose, we take z derivative and r derivative on Eq. (1a) and (1c), respectively, and take a difference: ǫ ζ ϕ t = f av z g θ 0 θ r. It is important to note that this linear vorticity tendency equation is satisfied even after taking time derivatives any number of times. Thus, by taking time derivatives for n times on the above, we obtain a series of the conditions: ǫ n+1 ζ ϕ n = tn+1 t n (f av z g θ θ 0 r ) for n = 0, 1,. This series states that the right hand side must remain always small enough in order that the acceleration (as defined as any order of time derivatives) of vorticity (left hand side) is well bounded to O(ǫ) as required for the balanced dynamics. The condition reduces in the limit ǫ 0 to: n av t n(f z g θ ) = 0. (2) θ 0 r The method for posing a series of constraints (2) with n = 0, 1, is called the bounded derivative in the context of the model initialization problem (Kreiss 1980, Browning et al. 1980). For the present purpose, suffices it to pose the condition (2) for n = 0 and 1. With n = 0 in Eq. (2), we recover the thermal wind balance given as the second equality in Eq. (2.8) of SM10. With n = 1, we recover Kuo Eliassen equation given by (2.10) of SM10. This condition states that the acceleration of the azimuthal Coriolis force (the first term) must balance with that of buoyancy (the second term) in order that the system is always balanced (or bounded to O(ǫ)). The acceleration rate of the Coriolis force, f a v, is evaluated as f a v t = (f a + v r )v t = (f a+ v r )(ζ au+w v z ) ˆf 2 u (3a) Here, the last approximate equality is obtained by neglecting the advection term. An effective Coriolis parameter ˆf is introduced by ˆf 2 = (f a + v r )ζ a (cf., Vigh and Schubert 2009). Note that ˆf reduces to the Coriolis parameter itself, f, in linear limit. Here, Eq. (1a) says that the azimuthal wind u is accelerated by the Coriolis force f a v with a rate characterized by the absolute Coriolis parameter f a. According to Eq. (3a), the Coriolis force is, in turn, accelerated by a rate characterized by another effective Coriolis factor, f a + v/r. As the whole, the azimuthal wind u is doubly accelerated by a rate proportional the square of the effective Coriolis parameter, ˆf: 2 u t 2 ˆf 2 u This balance leads to an oscillation tendency of the azimuthal wind, u, with the inertial frequency ˆf f. On the other hand, the acceleration rate of buoyancy is estimated by the thermodynamic equation (1e) as g θ θ 0 t g θ θ z w g θ θ = N 2 w g θ θ. (3b) The approximate equality above is again obtained by neglecting the horizontal advection. As the first term in the middle expression shows, the buoyancy acceleration is primarily dictated by the vertical heat advection. That leads to an oscillation tendency of the vertical velocity, w, by the Vaisala frequency, N = (gθ/θz) 1/2 : 2 w t 2 N2 w Substitution of Eq. (3a, b) into Eq. (2) with n = 1 leads to z ( ˆf 2 u) r (N2 w) g θ θ 0 r. JAMES-D
3 Kuo Eliassen Equation 3 Consequently, in order to maintain the balance condition, horizontal acceleration tendency with the inertial frequency ˆf f (first term) must balance with vertical acceleration tendency with the Vaisala frequency N. By further substituting the definition of the streamfucntion, ψ, given by Eq. (2.9) of SM10 into the above, we obtain an approximate form of the Kuo Eliassen equation: [ r N 2 ρ (1 r r r) + z ˆf 2 ρ z ]ψ g θ 0 θ r. (4) Eq. (4) agrees with Eq. (2.10) of SM10 when the definitions of the coefficients, A = N 2 /ρ and C = ˆf 2 /ρ, as given by their Eq. (2.8), are substituted, and also setting B 0. It is interesting to note that neglect of advection has an equivalent effect as neglecting the baroclinicity of the system. 3. Interpretation of Kuo Eliassen Equation Interpretation of Kuo Eliassen equation is further prompted by assuming the coefficients, A and C, are constants, as assumed by SM10, as well as neglecting the curvature effects in the radial direction, by setting (1/r)(/r)r /r. As a result, Eq. (4) reduces to [ 1ˆf2 2 r N 2 2 z 2]ψ 1 (N ˆf) 2 g θ 0 θ r. (5) In order to obtain a simple interpretation of Eq. (5), let us assume that the secondary circulation is approximately represented by single horizontal and vertical wavenumbers, k and m, respectively. Let us also concentrate on the homogeneous problem by setting θ = 0 by following the main focus in SM10. Then Eq. (5) furthermore reduces to: [( kˆf ) 2 + ( m N )2 ]ψ 0 The equation has a simple interpretation that the horizontal phase velocity c H = ˆf/k due to the inertial oscillation must match with the vertical phase velocity c V = N/m due to the buoyancy oscillation to the order of magnitude, i.e., c H c V (6) or ˆf N k m. (7) When Eq. (7) is solved for the horizontal wavenumber, k for a given vertical wavenumber, m, we obtain an inverse horizontal wavenumber: k 1 1 m (Ṋ f ), (8a) which is the Rossby length (or Rossby radius of deformation). Alternatively, when Eq. (7) is solved for the vertical wavenumber, m for a given horizontal wavenumber, k, we obtain an inverse vertical wavenumber: m 1 1 k ( ˆf N ), (8b) which is the Rossby depth. Consequently, Rossby length and depth (Eq. 8a, b) are interpreted as scales that the matching of the two phase velocities is satisfied (Eq. 6) for given vertical and horizontal scales, so that the secondary circulation can be maintained in a manner consistent with the balance condition for the vortical motion. Here, many readers may wonder why I did not put Eq. (6) instead more precisely as c H ±ic V. The purpose above is merely to show that we can understand the meaning of Rossby length and depth by a very simple order of magnitude argument. For such a purpose, even the question of whether c H and c V are real or imaginary does not count. These mathematically more subtle issues are discussed in the next section in order to better understand the analytical solutions presented by SM10 from a wider perspective. 4. Mathematical Remarks From mathematical point of view, Eq. (7) suggests when either the horizontal coordinate is stretched by a rate, N/ ˆf, or alternatively the vertical coordinate is stretched by a rate, ˆf/N, Eq. (5) reduces to a Poisson problem: [ r 2 z 2]ψ g θ ˆα2 θ 0 r. with the definition of a constant ˆα depending on which coordinate is actually stretched. The star implies stretched coordinates. The stretching factor Γ = N/ ˆf plays a major role in analytical solutions of SM10. See their Eq. (3.11) and (4.5) especially. For better elucidation, here, again, let us seek a solution with local wavenumbers, k and m for a situation
4 4 Yano without diabatic heating, i.e., θ = 0. The above equation, then reduces to: k 2 + m 2 = 0. (9) The result (9) means that when a real wavenumber, k, is assumed in the horizontal direction, a vertical structure of the solution is evanescent (i.e., m 2 < 0), and vice versa. It is important to note that this mathematical structure of the problem is also reflected on the full problem with the cylindrical coordinates as considered by SM10. In Sec. 3, SM10 seek a solution under a Fourier expansion in the vertical direction. This corresponds to the case in Eq. (9) that considering a real wavenumber, m, in the vertical direction. Thus, the solution in the horizontal direction must be evanescent. It is reflected on the fact that SM10 found a radial structure described by modified Bessel functions. Those functions are essentially evanescent in their behavior (cf., Olver 1974). On the other hand, in Sec. 4, SM10 seek a solution assuming a Bessel function form in the radial direction. Being equivalent of assuming a wave form in the radial direction, the vertical structure of the solution become evanescent as expected from Eq. (9), and as found in their solution (Eq. 4.6). The relation (9) suggests that the secondary circulation can locally be wave like either in the radial or the vertical direction only. Consistency of this conclusion with the full solution with the cylindrical coordinates in SM10 furthermore suggests that an overall character of the secondary circulation solution can well be inferred without explicitly taking into account the curvature effects of the system. 5. Analogy with the quasi geostrophic system Finally, it would be helpful to recall that a similar issue with Rossby length and depth is also found in the quasi geostrophic system. The quasi geostrophic system is dictated by a conservation law of a potential vorticity q: ( t + v H H )q = 0 with the horizontal velocity v H = (u x, u y ) under geostrophy: u x = 1 φ f y, u y = 1 f φ x with the Cartesian coordinates (x, y). Here, the potential vorticity, q, is defined by q = 1 f Hφ + f ρ z ( ρ N 2 φ) (10) z under the notations of SM10. A similarity between Eq. (10) and Eq. (4) is hard to miss. By assuming ρ/n 2 constant, Eq. (10) further reduces to: q = f( 1 f 2 H + 1 N 2 2 z 2)φ, which can be more directly compared with Eq. (5). Note that the horizontal Laplacian H above plays the same role as 2 /r 2 in Eq. (5). Thus, the behavior of the quasi geostrophic system can be understood, to some extent, in analogy with that of the Kuo Eliassen equation discussed above. Important role of the Rossby length, 1/m(N/f), in the quasi geostrophic dynamics is hard to overemphasize (cf., Pedlosky 1987). This knowledge would also help to further understand the Kuo Eliassen equation. References Browning, G., A. Kasahara, H O. Kreiss, 1980: Initialization of the primitive equations by the bounded derivative method. J. Atmos. Sci., 37, Chang, E. K. M., 1996: Mean meridional circulation driven by eddy forcings of different timescales. J. Atmos. Sci., 53, Eliassen, A., 1952: Slow thermally or frictionally controlled meridional circulations in a circular vortex. Astrophysica Norvegica, 5, Kreiss H. O., 1980: Problems with different time scales for partial-differential equations. Comm. Pure Appl. Math., 33, Krishnamurti T. N., H. S. Bedi, D. Oosterhof, V. Hardiker, 1994: The formation of Hurricane-Frederic of Mon. Wea. Rev., 122, Kuo, H-L., 1956: Forced and free meridional circulations in the atmosphere. J. Meteor., 13, Olver, F. W. J., 1974: Asymptotics and Special Functions, Academic Press, 572pp. Pedlosky, J., 1987: Geophysical Fluid Dynamics. 710pp, Springer Verlag. 2nd Ed., Schubert, W. H., and J. J. Hack, 1982: Inertial Stability and Tropical Cyclone Development. J. Atmos. Sci., 39, Schubert, W. H., and B. D. McNoldy, 2010: Application of the concepts of Rossby length and Rossby depth to tropical cyclone dynamics. accepted to J. Adv. Model. Earth Syst. JAMES-D
5 Kuo Eliassen Equation 5 Shapiro, L. J., and H. E. Willoughby, 1982: The response of balance hurricane to local sources of heat and momentum. J. Atmos. Sci., 39, Vigh, J. L., and W. H. Schubert, 2009: Rapid development of the tropical cyclone warm core. J. Atmos. Sci., 66,
( u,v). For simplicity, the density is considered to be a constant, denoted by ρ 0
! Revised Friday, April 19, 2013! 1 Inertial Stability and Instability David Randall Introduction Inertial stability and instability are relevant to the atmosphere and ocean, and also in other contexts
More informationQuasi-geostrophic ocean models
Quasi-geostrophic ocean models March 19, 2002 1 Introduction The starting point for theoretical and numerical study of the three dimensional large-scale circulation of the atmosphere and ocean is a vorticity
More informationChapter 9. Geostrophy, Quasi-Geostrophy and the Potential Vorticity Equation
Chapter 9 Geostrophy, Quasi-Geostrophy and the Potential Vorticity Equation 9.1 Geostrophy and scaling. We examined in the last chapter some consequences of the dynamical balances for low frequency, nearly
More informationChapter 2. Quasi-Geostrophic Theory: Formulation (review) ε =U f o L <<1, β = 2Ω cosθ o R. 2.1 Introduction
Chapter 2. Quasi-Geostrophic Theory: Formulation (review) 2.1 Introduction For most of the course we will be concerned with instabilities that an be analyzed by the quasi-geostrophic equations. These are
More informationNonlinear baroclinic dynamics of surface cyclones crossing a zonal jet
Nonlinear baroclinic dynamics of surface cyclones crossing a zonal jet Jean-Baptiste GILET, Matthieu Plu and Gwendal Rivière CNRM/GAME (Météo-France, CNRS) 3rd THORPEX International Science Symposium Monterey,
More information= vorticity dilution + tilting horizontal vortices + microscopic solenoid
4.4 Vorticity Eq 4.4.1 Cartesian Coordinates Because ζ = ˆk V, gives D(ζ + f) x minus [v momentum eq. in Cartesian Coordinates] y [u momentum eq. in Cartesian Coordinates] = vorticity dilution + tilting
More informationModel equations for planetary and synoptic scale atmospheric motions associated with different background stratification
Model equations for planetary and synoptic scale atmospheric motions associated with different background stratification Stamen Dolaptchiev & Rupert Klein Potsdam Institute for Climate Impact Research
More information1/18/2011. Conservation of Momentum Conservation of Mass Conservation of Energy Scaling Analysis ESS227 Prof. Jin-Yi Yu
Lecture 2: Basic Conservation Laws Conservation Law of Momentum Newton s 2 nd Law of Momentum = absolute velocity viewed in an inertial system = rate of change of Ua following the motion in an inertial
More information( ) (9.1.1) Chapter 9. Geostrophy, Quasi-Geostrophy and the Potential Vorticity Equation. 9.1 Geostrophy and scaling.
Chapter 9 Geostrophy, Quasi-Geostrophy and the Potential Vorticity Equation 9.1 Geostrophy and scaling. We examined in the last chapter some consequences of the dynamical balances for low frequency, nearly
More informationDynamics Rotating Tank
Institute for Atmospheric and Climate Science - IACETH Atmospheric Physics Lab Work Dynamics Rotating Tank Large scale flows on different latitudes of the rotating Earth Abstract The large scale atmospheric
More informationGFD 2012 Lecture 1: Dynamics of Coherent Structures and their Impact on Transport and Predictability
GFD 2012 Lecture 1: Dynamics of Coherent Structures and their Impact on Transport and Predictability Jeffrey B. Weiss; notes by Duncan Hewitt and Pedram Hassanzadeh June 18, 2012 1 Introduction 1.1 What
More informationFundamentals of Atmospheric Modelling
M.Sc. in Computational Science Fundamentals of Atmospheric Modelling Peter Lynch, Met Éireann Mathematical Computation Laboratory (Opp. Room 30) Dept. of Maths. Physics, UCD, Belfield. January April, 2004.
More informationOn the Motion of a Typhoon (I)*
On the Motion of a Typhoon (I)* By S. Syono Geophysical Institute, Tokyo University (Manuscript received 2 November 1955) Abstract Solving barotropic vorticity equation, the motion of a disturbance of
More informationEffects of Convective Heating on Movement and Vertical Coupling of Tropical Cyclones: A Numerical Study*
3639 Effects of Convective Heating on Movement and Vertical Coupling of Tropical Cyclones: A Numerical Study* LIGUANG WU ANDBIN WANG Department of Meteorology, School of Ocean and Earth Science and Technology,
More informationEliassen-Palm Theory
Eliassen-Palm Theory David Painemal MPO611 April 2007 I. Introduction The separation of the flow into its zonal average and the deviations therefrom has been a dominant paradigm for analyses of the general
More informationConservation of Mass Conservation of Energy Scaling Analysis. ESS227 Prof. Jin-Yi Yu
Lecture 2: Basic Conservation Laws Conservation of Momentum Conservation of Mass Conservation of Energy Scaling Analysis Conservation Law of Momentum Newton s 2 nd Law of Momentum = absolute velocity viewed
More informationBy convention, C > 0 for counterclockwise flow, hence the contour must be counterclockwise.
Chapter 4 4.1 The Circulation Theorem Circulation is a measure of rotation. It is calculated for a closed contour by taking the line integral of the velocity component tangent to the contour evaluated
More informationThe General Circulation of the Atmosphere: A Numerical Experiment
The General Circulation of the Atmosphere: A Numerical Experiment Norman A. Phillips (1956) Presentation by Lukas Strebel and Fabian Thüring Goal of the Model Numerically predict the mean state of the
More informationATMOSPHERIC AND OCEANIC FLUID DYNAMICS
ATMOSPHERIC AND OCEANIC FLUID DYNAMICS Fundamentals and Large-scale Circulation G E O F F R E Y K. V A L L I S Princeton University, New Jersey CAMBRIDGE UNIVERSITY PRESS An asterisk indicates more advanced
More informationMeasurement of Rotation. Circulation. Example. Lecture 4: Circulation and Vorticity 1/31/2017
Lecture 4: Circulation and Vorticity Measurement of Rotation Circulation Bjerknes Circulation Theorem Vorticity Potential Vorticity Conservation of Potential Vorticity Circulation and vorticity are the
More informationGoals of this Chapter
Waves in the Atmosphere and Oceans Restoring Force Conservation of potential temperature in the presence of positive static stability internal gravity waves Conservation of potential vorticity in the presence
More informationCan a Simple Two-Layer Model Capture the Structure of Easterly Waves?
Can a Simple Two-Layer Model Capture the Structure of Easterly Waves? Cheryl L. Lacotta 1 Introduction Most tropical storms in the Atlantic, and even many in the eastern Pacific, are due to disturbances
More informationTransformed Eulerian Mean
Chapter 15 Transformed Eulerian Mean In the last few lectures we introduced some fundamental ideas on 1) the properties of turbulent flows in rotating stratified environments, like the ocean and the atmosphere,
More information1/25/2010. Circulation and vorticity are the two primary
Lecture 4: Circulation and Vorticity Measurement of Rotation Circulation Bjerknes Circulation Theorem Vorticity Potential Vorticity Conservation of Potential Vorticity Circulation and vorticity are the
More informationDependence of tropical cyclone intensification on the Coriolis parameter
May 2012 Li et al. 1 Dependence of tropical cyclone intensification on the Coriolis parameter Tim Li a n d Xu ya n g Ge Department of Meteorology and IPRC, University of Hawaii, Honolulu, Hawaii Melinda
More informationInfluences of Asymmetric Flow Features on Hurricane Evolution
Influences of Asymmetric Flow Features on Hurricane Evolution Lloyd J. Shapiro Meteorological Institute University of Munich Theresienstr. 37, 80333 Munich, Germany phone +49 (89) 2180-4327 fax +49 (89)
More informationPV Thinking. What is PV thinking?
PV Thinking CH 06 What is PV thinking? Two main approaches to solving fluid flow problems:. We can integrate the momentum, continuity and thermodynamic equations (the primitive equations) directly.. In
More informationConference Proceedings Paper Vortex Motion State of the Dry Atmosphere with Nonzero Velocity Divergence
1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 1 3 4 5 6 7 8 9 30 31 3 33 34 35 36 37 38 39 40 41 Conference Proceedings Paper Vortex Motion State of the Dry Atmosphere with Nonzero Velocity Divergence
More informationIn two-dimensional barotropic flow, there is an exact relationship between mass
19. Baroclinic Instability In two-dimensional barotropic flow, there is an exact relationship between mass streamfunction ψ and the conserved quantity, vorticity (η) given by η = 2 ψ.the evolution of the
More informationBALANCED FLOW: EXAMPLES (PHH lecture 3) Potential Vorticity in the real atmosphere. Potential temperature θ. Rossby Ertel potential vorticity
BALANCED FLOW: EXAMPLES (PHH lecture 3) Potential Vorticity in the real atmosphere Need to introduce a new measure of the buoyancy Potential temperature θ In a compressible fluid, the relevant measure
More informationModeling the atmosphere of Jupiter
Modeling the atmosphere of Jupiter Bruce Turkington UMass Amherst Collaborators: Richard S. Ellis (UMass Professor) Andrew Majda (NYU Professor) Mark DiBattista (NYU Postdoc) Kyle Haven (UMass PhD Student)
More informationDynamics and Kinematics
Geophysics Fluid Dynamics () Syllabus Course Time Lectures: Tu, Th 09:30-10:50 Discussion: 3315 Croul Hall Text Book J. R. Holton, "An introduction to Dynamic Meteorology", Academic Press (Ch. 1, 2, 3,
More informationGeophysics Fluid Dynamics (ESS228)
Geophysics Fluid Dynamics (ESS228) Course Time Lectures: Tu, Th 09:30-10:50 Discussion: 3315 Croul Hall Text Book J. R. Holton, "An introduction to Dynamic Meteorology", Academic Press (Ch. 1, 2, 3, 4,
More informationBaroclinic Rossby waves in the ocean: normal modes, phase speeds and instability
Baroclinic Rossby waves in the ocean: normal modes, phase speeds and instability J. H. LaCasce, University of Oslo J. Pedlosky, Woods Hole Oceanographic Institution P. E. Isachsen, Norwegian Meteorological
More informationGravity Waves. Lecture 5: Waves in Atmosphere. Waves in the Atmosphere and Oceans. Internal Gravity (Buoyancy) Waves 2/9/2017
Lecture 5: Waves in Atmosphere Perturbation Method Properties of Wave Shallow Water Model Gravity Waves Rossby Waves Waves in the Atmosphere and Oceans Restoring Force Conservation of potential temperature
More informationTropical Cyclone Intensification
Tropical Cyclone Intensification Theories for tropical cyclone intensification and structure CISK (Charney and Eliassen 1964) Cooperative Intensification Theory (Ooyama 1969). WISHE (Emanuel 1986, Holton
More informationEddy PV Fluxes in a One Dimensional Model of Quasi-Geostrophic Turbulence
Eddy PV Fluxes in a One Dimensional Model of Quasi-Geostrophic Turbulence Christos M.Mitas Introduction. Motivation Understanding eddy transport of heat and momentum is crucial to developing closure schemes
More information2. Baroclinic Instability and Midlatitude Dynamics
2. Baroclinic Instability and Midlatitude Dynamics Midlatitude Jet Stream Climatology (Atlantic and Pacific) Copyright 26 Emily Shuckburgh, University of Cambridge. Not to be quoted or reproduced without
More informationwhere p oo is a reference level constant pressure (often 10 5 Pa). Since θ is conserved for adiabatic motions, a prognostic temperature equation is:
1 Appendix C Useful Equations Purposes: Provide foundation equations and sketch some derivations. These equations are used as starting places for discussions in various parts of the book. C.1. Thermodynamic
More informationVortex Rossby Waves and Hurricane Evolution in the Presence of Convection and Potential Vorticity and Hurricane Motion
LONG-TERM GOALS/OBJECTIVES Vortex Rossby Waves and Hurricane Evolution in the Presence of Convection and Potential Vorticity and Hurricane Motion Michael T. Montgomery Department of Atmospheric Science
More informationSatellite Applications to Hurricane Intensity Forecasting
Satellite Applications to Hurricane Intensity Forecasting Christopher J. Slocum - CSU Kate D. Musgrave, Louie D. Grasso, and Galina Chirokova - CIRA/CSU Mark DeMaria and John Knaff - NOAA/NESDIS Center
More informationEART164: PLANETARY ATMOSPHERES
EART164: PLANETARY ATMOSPHERES Francis Nimmo Last Week Radiative Transfer Black body radiation, Planck function, Wien s law Absorption, emission, opacity, optical depth Intensity, flux Radiative diffusion,
More informationCirculation and Vorticity
Circulation and Vorticity Example: Rotation in the atmosphere water vapor satellite animation Circulation a macroscopic measure of rotation for a finite area of a fluid Vorticity a microscopic measure
More informationQuasi-geostrophic system
Quasi-eostrophic system (or, why we love elliptic equations for QGPV) Charney s QG the motion of lare-scale atmospheric disturbances is overned by Laws of conservation of potential temperature and potential
More informationEAS372 Open Book Final Exam 11 April, 2013
EAS372 Open Book Final Exam 11 April, 2013 Professor: J.D. Wilson Time available: 2 hours Value: 30% Please check the Terminology, Equations and Data section before beginning your responses. Answer all
More informationTOWARDS A BETTER UNDERSTANDING OF AND ABILITY TO FORECAST THE WIND FIELD EXPANSION DURING THE EXTRATROPICAL TRANSITION PROCESS
P1.17 TOWARDS A BETTER UNDERSTANDING OF AND ABILITY TO FORECAST THE WIND FIELD EXPANSION DURING THE EXTRATROPICAL TRANSITION PROCESS Clark Evans* and Robert E. Hart Florida State University Department
More informationLecture 12: Angular Momentum and the Hadley Circulation
Lecture 12: Angular Momentum and the Hadley Circulation September 30, 2003 We learnt last time that there is a planetary radiative drive net warming in the tropics, cooling over the pole which induces
More informationRotating stratified turbulence in the Earth s atmosphere
Rotating stratified turbulence in the Earth s atmosphere Peter Haynes, Centre for Atmospheric Science, DAMTP, University of Cambridge. Outline 1. Introduction 2. Momentum transport in the atmosphere 3.
More informationBalanced Flow Geostrophic, Inertial, Gradient, and Cyclostrophic Flow
Balanced Flow Geostrophic, Inertial, Gradient, and Cyclostrophic Flow The types of atmospheric flows describe here have the following characteristics: 1) Steady state (meaning that the flows do not change
More information6 Two-layer shallow water theory.
6 Two-layer shallow water theory. Wewillnowgoontolookatashallowwatersystemthathastwolayersofdifferent density. This is the next level of complexity and a simple starting point for understanding the behaviour
More informationDiagnosis of a Quasi-Geostrophic 2-Layer Model Aaron Adams, David Zermeño, Eunsil Jung, Hosmay Lopez, Ronald Gordon, Ting-Chi Wu
Diagnosis of a Quasi-Geostrophic 2-Layer Model Aaron Adams, David Zermeño, Eunsil Jung, Hosmay Lopez, Ronald Gordon, Ting-Chi Wu Introduction For this project we use a simple two layer model, which is
More informationAtmospheric dynamics and meteorology
Atmospheric dynamics and meteorology B. Legras, http://www.lmd.ens.fr/legras III Frontogenesis (pre requisite: quasi-geostrophic equation, baroclinic instability in the Eady and Phillips models ) Recommended
More informationThe Hadley Circulation and the Weak Temperature Gradient Approximation
1744 JOURNAL OF THE ATMOSPHERIC SCIENCES The Hadley Circulation and the Weak Temperature Gradient Approximation L. M. POLVANI AND A. H. SOBEL Department of Applied Physics and Applied Mathematics, and
More informationUsing simplified vorticity equation,* by assumption 1 above: *Metr 430 handout on Circulation and Vorticity. Equations (4) and (5) on that handout
Rossby Wave Equation A. Assumptions 1. Non-divergence 2. Initially, zonal flow, or nearly zonal flow in which u>>v>>w. 3. Initial westerly wind is geostrophic and does not vary along the x-axis and equations
More informationHurricanes are intense vortical (rotational) storms that develop over the tropical oceans in regions of very warm surface water.
Hurricanes: Observations and Dynamics Houze Section 10.1. Holton Section 9.7. Emanuel, K. A., 1988: Toward a general theory of hurricanes. American Scientist, 76, 371-379 (web link). http://ww2010.atmos.uiuc.edu/(gh)/guides/mtr/hurr/home.rxml
More informationd v 2 v = d v d t i n where "in" and "rot" denote the inertial (absolute) and rotating frames. Equation of motion F =
Governing equations of fluid dynamics under the influence of Earth rotation (Navier-Stokes Equations in rotating frame) Recap: From kinematic consideration, d v i n d t i n = d v rot d t r o t 2 v rot
More informationCoriolis force in Geophysics: an elementary introduction and examples
Coriolis force in Geophysics: an elementary introduction and examples F. Vandenbrouck, L. Berthier, and F. Gheusi Laboratoire de Physique de la Matière Condensée, Collège de France, 11 place M. Berthelot,
More informationarxiv:physics/ v1 [physics.ed-ph] 10 May 2000
Coriolis force in Geophysics: an elementary introduction and examples F. Vandenbrouck, L. Berthier, and F. Gheusi Laboratoire de Physique de la Matière Condensée, Collège de France, 11 place M. Berthelot,
More informationSAMPLE CHAPTERS UNESCO EOLSS WAVES IN THE OCEANS. Wolfgang Fennel Institut für Ostseeforschung Warnemünde (IOW) an der Universität Rostock,Germany
WAVES IN THE OCEANS Wolfgang Fennel Institut für Ostseeforschung Warnemünde (IOW) an der Universität Rostock,Germany Keywords: Wind waves, dispersion, internal waves, inertial oscillations, inertial waves,
More information7 Balanced Motion. 7.1 Return of the...scale analysis for hydrostatic balance! CSU ATS601 Fall 2015
7 Balanced Motion We previously discussed the concept of balance earlier, in the context of hydrostatic balance. Recall that the balanced condition means no accelerations (balance of forces). That is,
More information1/27/2010. With this method, all filed variables are separated into. from the basic state: Assumptions 1: : the basic state variables must
Lecture 5: Waves in Atmosphere Perturbation Method With this method, all filed variables are separated into two parts: (a) a basic state part and (b) a deviation from the basic state: Perturbation Method
More informationLecture 8. Lecture 1. Wind-driven gyres. Ekman transport and Ekman pumping in a typical ocean basin. VEk
Lecture 8 Lecture 1 Wind-driven gyres Ekman transport and Ekman pumping in a typical ocean basin. VEk wek > 0 VEk wek < 0 VEk 1 8.1 Vorticity and circulation The vorticity of a parcel is a measure of its
More informationLecture 14. Equations of Motion Currents With Friction Sverdrup, Stommel, and Munk Solutions Remember that Ekman's solution for wind-induced transport
Lecture 14. Equations of Motion Currents With Friction Sverdrup, Stommel, and Munk Solutions Remember that Ekman's solution for wind-induced transport is which can also be written as (14.1) i.e., #Q x,y
More informationFluid Dynamics Exercises and questions for the course
Fluid Dynamics Exercises and questions for the course January 15, 2014 A two dimensional flow field characterised by the following velocity components in polar coordinates is called a free vortex: u r
More informationDiabatically Induced Secondary Flows in Tropical Cyclones. Part I: Quasi-Steady Forcing
VOLUME 137 M O N T H L Y W E A T H E R R E V I E W MARCH 2009 Diabatically Induced Secondary Flows in Tropical Cyclones. Part I: Quasi-Steady Forcing ANGELINE G. PENDERGRASS Department of Atmospheric Sciences,
More informationA Note on the Barotropic Instability of the Tropical Easterly Current
April 1969 Tsuyoshi Nitta and M. Yanai 127 A Note on the Barotropic Instability of the Tropical Easterly Current By Tsuyoshi Nitta and M. Yanai Geophysical Institute, Tokyo University, Tokyo (Manuscript
More informationarxiv: v1 [physics.ao-ph] 15 May 2017
Elliptic Transverse Circulation Equations for Balanced Models in a Generalized Vertical Coordinate arxiv:1705.05460v1 [physics.ao-ph] 15 May 2017 1. Introduction Wayne H. Schubert Department of Atmospheric
More informationProceedings Vortex Motion State of the Dry Atmosphere with Nonzero Velocity Divergence
Proceedings Vortex Motion State of the Dry Atmosphere with Nonzero Velocity Divergence Robert Zakinyan, Arthur Zakinyan *, Roman Ryzhkov and Julia Semenova Department of General and Theoretical Physics,
More informationFour ways of inferring the MMC. 1. direct measurement of [v] 2. vorticity balance. 3. total energy balance
Four ways of inferring the MMC 1. direct measurement of [v] 2. vorticity balance 3. total energy balance 4. eliminating time derivatives in governing equations Four ways of inferring the MMC 1. direct
More informationWind-driven Western Boundary Ocean Currents in Terran and Superterran Exoplanets
Wind-driven Western Boundary Ocean Currents in Terran and Superterran Exoplanets By Edwin Alfonso-Sosa, Ph.D. Ocean Physics Education, Inc. 10-Jul-2014 Introduction Simple models of oceanic general circulation
More informationNon-hydrostatic sound-proof equations of motion for gravity-dominated compressible fl ows. Thomas Dubos
Non-hydrostatic sound-proof of for gravity-dominated compressible fl ows Thomas Dubos Laboratoire de Météorologie Dynamique/IPSL, École Polytechnique, Palaiseau, France Fabrice Voitus Centre National de
More informationt tendency advection convergence twisting baroclinicity
RELATIVE VORTICITY EQUATION Newton s law in a rotating frame in z-coordinate (frictionless): U + U U = 2Ω U Φ α p U + U U 2 + ( U) U = 2Ω U Φ α p Applying to both sides, and noting ω U and using identities
More informationJet Formation in the Equatorial Oceans Through Barotropic and Inertial Instabilities. Mark Fruman
p. 1/24 Jet Formation in the Equatorial Oceans Through Barotropic and Inertial Instabilities Mark Fruman Bach Lien Hua, Richard Schopp, Marc d Orgeville, Claire Ménesguen LPO IFREMER, Brest, France IAU
More informationTheory and Computation of Wavenumber-2 Vortex Rossby Wave Instabilities in Hurricane-like Vortices
Theory and Computation of Wavenumber-2 Vortex Rossby Wave Instabilities in Hurricane-like Vortices TOY 2008 Christopher Jeffery & Nicole Jeffery (cjeffery@lanl.gov) Los Alamos National Laboratory, Los
More informationNonlinear Balance on an Equatorial Beta Plane
Nonlinear Balance on an Equatorial Beta Plane David J. Raymond Physics Department and Geophysical Research Center New Mexico Tech Socorro, NM 87801 April 26, 2009 Summary Extension of the nonlinear balance
More informationControl Volume. Dynamics and Kinematics. Basic Conservation Laws. Lecture 1: Introduction and Review 1/24/2017
Lecture 1: Introduction and Review Dynamics and Kinematics Kinematics: The term kinematics means motion. Kinematics is the study of motion without regard for the cause. Dynamics: On the other hand, dynamics
More informationLecture 1: Introduction and Review
Lecture 1: Introduction and Review Review of fundamental mathematical tools Fundamental and apparent forces Dynamics and Kinematics Kinematics: The term kinematics means motion. Kinematics is the study
More informationInternal boundary layers in the ocean circulation
Internal boundary layers in the ocean circulation Lecture 9 by Andrew Wells We have so far considered boundary layers adjacent to physical boundaries. However, it is also possible to find boundary layers
More informationSynoptic Meteorology II: Potential Vorticity Inversion and Anomaly Structure April 2015
Synoptic Meteorology II: Potential Vorticity Inversion and Anomaly Structure 14-16 April 2015 Readings: Sections 4.2 and 4.4 of Midlatitude Synoptic Meteorology. Potential Vorticity Inversion Introduction
More informationPAPER 333 FLUID DYNAMICS OF CLIMATE
MATHEMATICAL TRIPOS Part III Wednesday, 1 June, 2016 1:30 pm to 4:30 pm Draft 21 June, 2016 PAPER 333 FLUID DYNAMICS OF CLIMATE Attempt no more than THREE questions. There are FOUR questions in total.
More informationChapter 5. Shallow Water Equations. 5.1 Derivation of shallow water equations
Chapter 5 Shallow Water Equations So far we have concentrated on the dynamics of small-scale disturbances in the atmosphere and ocean with relatively simple background flows. In these analyses we have
More informationNWP Equations (Adapted from UCAR/COMET Online Modules)
NWP Equations (Adapted from UCAR/COMET Online Modules) Certain physical laws of motion and conservation of energy (for example, Newton's Second Law of Motion and the First Law of Thermodynamics) govern
More informationChapter 1. Introduction
Chapter 1. Introduction In this class, we will examine atmospheric phenomena that occurs at the mesoscale, including some boundary layer processes, convective storms, and hurricanes. We will emphasize
More information10 Shallow Water Models
10 Shallow Water Models So far, we have studied the effects due to rotation and stratification in isolation. We then looked at the effects of rotation in a barotropic model, but what about if we add stratification
More informationIntroduction to Mesoscale Meteorology
Introduction to Mesoscale Meteorology Overview Scale Definitions Synoptic Synoptic derived from Greek synoptikos meaning general view of the whole. Also has grown to imply at the same time or simultaneous.
More informationAtmospheric Dynamics: lecture 2
Atmospheric Dynamics: lecture 2 Topics Some aspects of advection and the Coriolis-effect (1.7) Composition of the atmosphere (figure 1.6) Equation of state (1.8&1.9) Water vapour in the atmosphere (1.10)
More information1/3/2011. This course discusses the physical laws that govern atmosphere/ocean motions.
Lecture 1: Introduction and Review Dynamics and Kinematics Kinematics: The term kinematics means motion. Kinematics is the study of motion without regard for the cause. Dynamics: On the other hand, dynamics
More information196 7 atmospheric oscillations:
196 7 atmospheric oscillations: 7.4 INTERNAL GRAVITY (BUOYANCY) WAVES We now consider the nature of gravity wave propagation in the atmosphere. Atmospheric gravity waves can only exist when the atmosphere
More informationBells and whistles of convection parameterization
Bells and whistles of convection parameterization Article Accepted Version Yano, J. I., Machulskaya, E., Bechtold, P. and Plant, R. S. (2013) Bells and whistles of convection parameterization. Bulletin
More informationEAS372 Open Book Final Exam 11 April, 2013
EAS372 Open Book Final Exam 11 April, 2013 Professor: J.D. Wilson Time available: 2 hours Value: 30% Please check the Terminology, Equations and Data section before beginning your responses. Answer all
More informationHadley Cell Dynamics in a Primitive Equation Model. Part II: Nonaxisymmetric Flow
2859 Hadley Cell Dynamics in a Primitive Equation Model. Part II: Nonaxisymmetric Flow HYUN-KYUNG KIM AND SUKYOUNG LEE Department of Meteorology, The Pennsylvania State University, University Park, Pennsylvania
More informationEffective Depth of Ekman Layer.
5.5: Ekman Pumping Effective Depth of Ekman Layer. 2 Effective Depth of Ekman Layer. Defining γ = f/2k, we derived the solution u = u g (1 e γz cos γz) v = u g e γz sin γz corresponding to the Ekman spiral.
More informationIntroduction to Isentropic Coordinates: a new view of mean meridional & eddy circulations. Cristiana Stan
Introduction to Isentropic Coordinates: a new view of mean meridional & eddy circulations Cristiana Stan School and Conference on the General Circulation of the Atmosphere and Oceans: a Modern Perspective
More informationNote that Rossby waves are tranverse waves, that is the particles move perpendicular to the direction of propagation. f up, down (clockwise)
Ocean 423 Rossby waves 1 Rossby waves: Restoring force is the north-south gradient of background potential vorticity (f/h). That gradient can be due to either the variation in f with latitude, or to a
More informationAtmospheric Dynamics Fall 2008
Atmospheric Dynamics Fall 2008 AT601, the first semester of Atmospheric Dynamics, is based on the course notes available over the web and on the highly recommended texts listed below. The course notes
More informationPUBLICATIONS. Journal of Advances in Modeling Earth Systems
PUBLICATIONS Journal of Advances in Modeling Earth Systems RESEARCH ARTICLE./3MS99 Key Points: Structure of wind field and secondary circulation depends on vortex structure 3-D model is governed by vertical
More informationThe general circulation: midlatitude storms
The general circulation: midlatitude storms Motivation for this class Provide understanding basic motions of the atmosphere: Ability to diagnose individual weather systems, and predict how they will change
More informationTraveling planetary-scale Rossby waves in the winter stratosphere: The role of tropospheric baroclinic instability
GEOPHYSICAL RESEARCH LETTERS, VOL.???, XXXX, DOI:.29/, 1 2 Traveling planetary-scale Rossby waves in the winter stratosphere: The role of tropospheric baroclinic instability Daniela I.V. Domeisen, 1 R.
More informationThe Equations of Motion in a Rotating Coordinate System. Chapter 3
The Equations of Motion in a Rotating Coordinate System Chapter 3 Since the earth is rotating about its axis and since it is convenient to adopt a frame of reference fixed in the earth, we need to study
More informationHouze sections 7.4, 8.3, 8.5, Refer back to equations in Section 2.3 when necessary.
Thunderstorm Dynamics Houze sections 7.4, 8.3, 8.5, Refer back to equations in Section.3 when necessary. Bluestein Vol. II section 3.4.6. Review article "Dynamics of Tornadic Thunderstorms" by Klemp handout.
More information