Quasi-equilibrium Theory of Small Perturbations to Radiative- Convective Equilibrium States
|
|
- Aubrey Tucker
- 6 years ago
- Views:
Transcription
1 Quasi-equilibrium Theory of Small Perturbations to Radiative- Convective Equilibrium States See CalTech 2005 paper on course web site Free troposphere assumed to have moist adiabatic lapse rate (s* does not vary with height Boundary layer quasi-equilibrium applies
2 Basis of statistical equilibrium physics Dates to Arakawa and Schubert (1974) Analogy to continuum hypothesis: Perturbations must have space scales >> intercloud spacing TKE consumption by convection ~ CAPE generation by large scale Numerical models on the verge of simulating clouds + large-scale waves We further assume convective criticality
3 Implications of the moist adiabatic lapse rate for the structure of tropical disturbances Approximate moist adiabatic condition as that of constant saturation entropy: Lq v * s* = c ln T p Rd ln p T 0 p + 0 T Assume hydrostatic perturbations: φ ' = α ' p
4 Maxwell s relation: α Integrate: α T ' = s*' = s*' s p * p s* ( ) φ φ b xyt Txyt T s ' = '(,, ) + (,, ) *' Only barotropic and first baroclinic mode survive
5 This implies, through the linearized momentum equations, e.g. u φ = + t x that the horizontal velocities may be partitioned similarly: fv ( ) u= u(, xyt,) + Txyt (,,) T u*(, xyt,); b ( ) v= v( xyt,, ) + Txyt (,, ) T v*( xyt,, ). b
6
7 Implications for vertical structure of vertical velocity Integrate: ω u v = + p x y ( p ) p u 0 b vb u* v* ω = ( p0 p) + ( p0 p) T Tdp' +. x y x y
8 At tropopause: ω t ( p p ) u = b + 0 t x v b y This implies that if a rigid lid is imposed at the tropopause, the divergence of the barotropic velocities must vanish and the barotropic components therefore satisfy the barotropic vorticity equation: ηb = Vi ηb, t η kˆ i V + 2Ωsinθ b b
9
10 Feedback of Air Motion on (virtual) Temperature Convection cannot change vertically integrated enthalpy, k = cpt + Lvq The neglecting surface fluxes, radiation, and horizontal advection, t kdp = h ω dp, p Neelin and Held (1987): This function is negative for upward motion
11 Upward motion is associated with column moistening: T q c dp= kdp Lv dp t t t p Ascent leads to cooling Yano and Emanuel, 1991: N ( ) 2 = 1 ε N 2 eff p
12 Prediction: Inviscid, small amplitude perturbations under rigid lid: Shallow water solutions with reduced equivalent depth
13 β Quasi-Linear Plane System, Neglecting Barotropic Mode u s* = ( T ) s T + β yv ru t x v s* = ( T ) s T β yu rv t y s Γ s Q ( ε ) = + pm w t Γ * d d rad m z sb h = C V s * s M w s s t ( ) ( )( ) k 0 b b m
14 u v w + + = x y H 0 Quasi-Equilibrium Assumption: sb t = s * t Gives closure for convective mass flux, M System closed except for specification of Q, s *, s, ε rad 0 m p
15 Additional Approximations: Boundary Layer QE (Raymond, 1995): Neglect s h b, gives simpler expression t for M s * 0 sb M = w+ Ck V sb sm Weak Temperature Approximation (Sobel and Bretherton, 2000): Neglect s * (Over-determined system, ignore momentum equation for irrotational flow) t
16 Important Feedbacks: Wind-Induced Surface Heat Exchange (WISHE) Coupling of surface enthalpy flux to wind perturbations (Neelin et al. 1987, Emanuel, 1987) Moisture-Convection Feedback: Dependence of sm on and/or ε p on s s Cloud-Radiation Feedback: Dependence of Q rad on M or s * sm M * m
17 Ocean-Atmosphere Feedback (e.g. ENSO): Feedback between perturbation surface wind and ocean surface temperature, as represented by s * 0
18 Simple Example: Q rad Ignore perturbations of Ignore fluctuations of ε p Make boundary layer QE approximation Fully linearize surface fluxes: 2 2 V = U + u* V ' = Uu ' V
19 Introduce scalings: First define a merdional scale, L y : Then let 4 Γ 1 d s ε ( ) d p Ly = Ts T H 2 Γm z β x a x y L y a ack V t t u u β L H 2 y y 2 LC y k V ack V β Ly v v s* s* H H T T ( ) s
20 Separate scalings for ocean temperature and lower tropospheric entropy: 1 ε s * p s s s ( ) o b m ε p 0 s m 1 ε ε p p 2 ( ) s s b m ( ) s * s* 0 s m
21 Nondimensional parameters: α R 1 ε ε p ra β L p 2 y ac k U ( s ) 0 * s* ( ) m H V s* s ( WISHE) ( Rayleigh friction) χ 1 ε ac V βl ε p ( s ) 0 * s* 2 p k y H( T ) s T s s ( * ) m 2 ( surface damping) a δ = L y 2 ( zonal geostropy)
22 Nondimensional Equations: u t s = + yv x Ru v t s = δ yu y Rv s u v = + + αu+ s + s χs 0 m t x y
23 Steady System with R = s = 0 : m s x s + αy χy 2 s= y 2 s y 0 Similar to Gill Model, but forcing is directly in terms of SST (s 0 ), not latent heating For SST of the form s = RE G( y) e ikx 0 there are solutions of the form s ikx = RE J( y) e,
24 where 2 2 ik χ y y 1+ ik χu α 2α α 2α J( y) = y e Gu e du 0 Example: 2 G = e by
25 α = 0, k = 2, b= 1.5, χ = 1.5
26 α = 1, k = 2, b= 1.5, χ = 1.5
27 Basic linear wave dynamics on the equatorial β plane Omit damping and WISHE terms from linear nondimensional equations: u t s = + x yv v t s = δ y s u v = + t x y yu
28 Fully equivalent to the shallow water equations on a β plane Eliminate s and u in favor of v: v v v v t t x y x δ + δyv δ = Let v = V( y) e ikx i t ω + = dy δ ω dv ω k k 2 y V 0 2
29 Boundary conditions: V well behaved at y ± Solution in terms of discrete parabolic cylinder functions D n : v= D ( y), n 2 2y 2 where Dn = e 1, 2 y, 4 y 2, provided ω satisfies the dispersion relation ω 2 2 δ k k = 2n + 1 ω
30 There is, in addition, another mode satisfying v=0 everywhere. From first and third linear equations: 2 2 u u = t x Satisfied by u = F x t ( ) Eastward-propagating, nondispersive equatorially trapped Kelvin wave Note that this happens to satisfy derived dispersion relation when n= -1..
31 There are three roots of the general dispersion relation: n = 0: 2 2 ω k k 1= 0 δ ω Factor : ω k 1 ( ω + k ) = 0 δ ω ω = k root not allowed does not satisfy BCs 1 ω = ± + 2 ( ) 2 k k 4δ ( ) Mixed Rossby-Gravity Waves (MRG)
32 For n 1, two well defined limits: 1. ω << k : ω k 2n + 1+ k 2. ω >> k : ω δ k + δ(2n+ 1) Planetary Rossby waves Inertia-gravity waves
33
34 Kelvin wave
35 Mixed Rossby-Gravity
36 Rossby
37 Inertia-gravity
38 Intraseasonal Variability Stochastic excitation of the equatorial waveguide WISHE Moisture-convection feedback Cloud-radiation feedback Ocean interaction
39 Wind-Induced Surface Heat Exchange (WISHE)
40 Add back WISHE term to linear undamped equations: u t s = + x yv v s δ = yu t y s u v = + + t x y α u
41 First look for Kelvin-like modes with v=0: u t s t s x u = + x u u u u = + α t 2 x 2 x Let u ω = = α u e ikx iωt α = k i k :
42 Note: α must be < 0 for ω r > 0 and ω i > 0
43 As k ω i -α/2
44 Effect of Stratosphere (Yano and Emanuel, 1991)
45 Effect of finite convective response time: s 1 u v = + + t 1 ε x y p M eq M t M eq τ c M M ε u v = p ε x y p = α u
46
47
48
49
50 Go back to dimensional, quasilinear QE equations on β plane
51 β Quasi-Linear Plane System, Neglecting Barotropic Mode u s* = ( T ) s T + β yv ru t x v s* = ( T ) s T β yu rv t y s Γ s Q ( ε ) = + pm w t Γ * d d rad m z sb h = C V s * s M w s s t ( ) ( )( ) k 0 b b m
52 u v w + + = x y H 0 Define an equilibrum updraft mass flux from boundary layer QE: M w+ C eq k s 0 * s V b s s b m Relax to equilibrium over a finite time scale: M t and enforce M 0 = M τ eq M convective
53 Numerical solution of β plane quasi-linear equations Nonlinearity retained only in surface fluxes Zonally symmetric SST specified; also symmetric about equator Background easterly wind of 2 ms -1 imposed Convection relaxed to equilibrium over time scale of 3 hours
54
55
56
57
58
59
60 Cloud-Radiative Feedback Set OLR proportional to difference in θ e between boundary layer and mid troposphere (Sandrine Bony)
61
62
63
64 Moisture-Convection Feedback Allow precipitation efficiency to depend on relative humidity Greater heating/upward motion in moister air upward motion moistens air Necessary for tropical cyclones Appears to excite planetary Rossby waves near equator
65
66 Possible effects of ocean response
67
Dynamics of the Zonal-Mean, Time-Mean Tropical Circulation
Dynamics of the Zonal-Mean, Time-Mean Tropical Circulation First consider a hypothetical planet like Earth, but with no continents and no seasons and for which the only friction acting on the atmosphere
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 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 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 information8/21/08. Modeling the General Circulation of the Atmosphere. Topic 4: Equatorial Wave Dynamics. Moisture and Equatorial Waves
Modeling the General Circulation of the Atmosphere. Topic 4: Equatorial Wave Dynamics D A R G A N M. W. F R I E R S O N U N I V E R S I T Y O F W A S H I N G T O N, D E P A R T M E N T O F A T M O S P
More informationRadiative-Convective Instability
Radiative-Convective Instability Kerry Emanuel, Allison Wing, and Emmanuel Vincent Massachusetts Institute of Technology Self-Aggregation of Deep Moist Convection Cloud Clusters Tropical Cyclone Genesis
More informationAdiabatic expansion Isothermal compression Adiabatic compression
Tropical Cyclones: Steady State Physics 1 Energy Production 2 Carnot Theorem: Maximum efficiency results from a pa rticular energy e cycle: Isothermal expansion Adiabatic expansion Isothermal compression
More informationModeling the General Circulation of the Atmosphere. Topic 2: Tropical General Circulation
Modeling the General Circulation of the Atmosphere. Topic 2: Tropical General Circulation DARGAN M. W. FRIERSON UNIVERSITY OF WASHINGTON, DEPARTMENT OF ATMOSPHERIC SCIENCES 1-28-16 Today What determines
More informationParameterizing large-scale dynamics using the weak temperature gradient approximation
Parameterizing large-scale dynamics using the weak temperature gradient approximation Adam Sobel Columbia University NCAR IMAGe Workshop, Nov. 3 2005 In the tropics, our picture of the dynamics should
More information8 Mechanisms for tropical rainfall responses to equatorial
8 Mechanisms for tropical rainfall responses to equatorial heating More reading: 1. Hamouda, M. and Kucharski, F. (2019) Ekman pumping Mechanism driving Precipitation anomalies in Response to Equatorial
More informationLarge-scale disturbances and convection. Željka Fuchs, University of Split
Large-scale disturbances and convection Željka Fuchs, University of Split Huatulco airport Tropical disturbances Tropical cyclones Monsoons Easterly waves Madden-Julian oscillation Convectively
More informationWaVaCS summerschool Autumn 2009 Cargese, Corsica
Introduction Part I WaVaCS summerschool Autumn 2009 Cargese, Corsica Holger Tost Max Planck Institute for Chemistry, Mainz, Germany Introduction Overview What is a parameterisation and why using it? Fundamentals
More informationRadiative-Convective Instability. Kerry Emanuel Massachusetts Institute of Technology
Radiative-Convective Instability Kerry Emanuel Massachusetts Institute of Technology Program Basic radiative-convective equilibrium Macro-instability of the RC state Some consequences Radiative Equilibrium
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 informationEquatorially Trapped Waves in Shallow Water
! Revised November 20, 2012 6:08 PM! 1 Equatorially Trapped Waves in Shallow Water David Randall Introduction Matsuno (1966; Fig. 1) studied the linearized shallow water equations applied to an equatorial
More informationHurricanes: Their physics and relationship to climate. Kerry Emanuel Massachusetts Institute of Technology
Hurricanes: Their physics and relationship to climate Kerry Emanuel Massachusetts Institute of Technology Topics Overview of Tropical Cyclones Tropical Cyclone Physics What have TCs been like in the past,
More informationA new theory for moist convection in statistical equilibrium
A new theory for moist convection in statistical equilibrium A. Parodi(1), K. Emanuel(2) (2) CIMA Research Foundation,Savona, Italy (3) EAPS, MIT, Boston, USA True dynamics: turbulent, moist, non-boussinesq,
More informationThe Tropical Atmosphere: Hurricane Incubator
The Tropical Atmosphere: Hurricane Incubator Images from journals published by the American Meteorological Society are copyright AMS and used with permission. A One-Dimensional Description of the Tropical
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 informationρ x + fv f 'w + F x ρ y fu + F y Fundamental Equation in z coordinate p = ρrt or pα = RT Du uv tanφ Dv Dt + u2 tanφ + vw a a = 1 p Dw Dt u2 + v 2
Fundamental Equation in z coordinate p = ρrt or pα = RT Du uv tanφ + uw Dt a a = 1 p ρ x + fv f 'w + F x Dv Dt + u2 tanφ + vw a a = 1 p ρ y fu + F y Dw Dt u2 + v 2 = 1 p a ρ z g + f 'u + F z Dρ Dt + ρ
More informationTropical Cyclones: Steady State Physics
Tropical Cyclones: Steady State Physics Energy Production Carnot Theorem: Maximum efficiency results from a particular energy cycle: Isothermal expansion Adiabatic expansion Isothermal compression Adiabatic
More informationRadiative-Convective Instability
Radiative-Convective Instability Kerry Emanuel, Allison Wing, and Emmanuel Vincent Massachusetts Institute of Technology Self-Aggregation of Deep Moist Convection Cloud Clusters Tropical Cyclone Genesis
More informationLecture 10a: The Hadley Cell
Lecture 10a: The Hadley Cell Geoff Vallis; notes by Jim Thomas and Geoff J. Stanley June 27 In this short lecture we take a look at the general circulation of the atmosphere, and in particular the Hadley
More informationThe Weak Temperature Gradient Approximation and Balanced Tropical Moisture Waves*
3650 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 58 The Weak Temperature Gradient Approximation and Balanced Tropical Moisture Waves* ADAM H. SOBEL Department of Applied Physics and Applied Mathematics,
More informationMultiple Equilibria in a Cloud Resolving Model: Using the Weak Temperature Gradient Approximation to Understand Self-Aggregation 1
Multiple Equilibria in a Cloud Resolving Model: Using the Weak Temperature Gradient Approximation to Understand Self-Aggregation 1 Sharon Sessions, Satomi Sugaya, David Raymond, Adam Sobel New Mexico Tech
More informationMoisture modes, cloud-radiative feedbacks and the MJO
Moisture modes, cloud-radiative feedbacks and the MJO Adam Sobel Eric Maloney with bits from Shuguang Wang, Jim Benedict; thanks also Gilles Bellon, Dargan Frierson, Daehyun Kim EUCLIPSE summer school
More informationMoisture Modes and the Eastward Propagation of the MJO
JANUARY 2013 S O B E L A N D M A L O N E Y 187 Moisture Modes and the Eastward Propagation of the MJO ADAM SOBEL Department of Applied Physics and Applied Mathematics, and Department of Earth and Environmental
More informationThermodynamics Review [?] Entropy & thermodynamic potentials Hydrostatic equilibrium & buoyancy Stability [dry & moist adiabatic]
Thermodynamics Review [?] Entropy & thermodynamic potentials Hydrostatic equilibrium & buoyancy Stability [dry & moist adiabatic] Entropy 1. (Thermodynamics) a thermodynamic quantity that changes in a
More informationThe Effect of Moist Convection on the Tropospheric Response to Tropical and Subtropical Zonally Asymmetric Torques
DECEMBER 2013 B O O S A N D S H A W 4089 The Effect of Moist Convection on the Tropospheric Response to Tropical and Subtropical Zonally Asymmetric Torques WILLIAM R. BOOS Department of Geology and Geophysics,
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 informationWeak Temperature Gradient Simulations For Different Convective Environments
Weak Temperature Gradient Simulations For Different Convective Environments 1 Benjamin Hatchett and Sharon Sessions 1 2 Division of Atmospheric Science, Desert Research Institute, Reno, Nevada 2 Department
More informationRadiative-Convective Models. The Hydrological Cycle Hadley Circulation. Manabe and Strickler (1964) Course Notes chapter 5.1
Climate Modeling Lecture 8 Radiative-Convective Models Manabe and Strickler (1964) Course Notes chapter 5.1 The Hydrological Cycle Hadley Circulation Prepare for Mid-Term (Friday 9 am) Review Course Notes
More informationThe Wavelength Dependence of the Gross Moist Stability and the Scale Selection in the Instability of Column-Integrated Moist Static Energy
JANUARY 2011 K U A N G 61 The Wavelength Dependence of the Gross Moist Stability and the Scale Selection in the Instability of Column-Integrated Moist Static Energy ZHIMING KUANG Department of Earth and
More informationCHAPTER 4. THE HADLEY CIRCULATION 59 smaller than that in midlatitudes. This is illustrated in Fig. 4.2 which shows the departures from zonal symmetry
Chapter 4 THE HADLEY CIRCULATION The early work on the mean meridional circulation of the tropics was motivated by observations of the trade winds. Halley (1686) and Hadley (1735) concluded that the trade
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 informationRadiative equilibrium Some thermodynamics review Radiative-convective equilibrium. Goal: Develop a 1D description of the [tropical] atmosphere
Radiative equilibrium Some thermodynamics review Radiative-convective equilibrium Goal: Develop a 1D description of the [tropical] atmosphere Vertical temperature profile Total atmospheric mass: ~5.15x10
More informationExcitation of Intraseasonal Variability in the Equatorial Atmosphere by Yanai Wave Groups via WISHE-Induced Convection
210 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 68 Excitation of Intraseasonal Variability in the Equatorial Atmosphere by Yanai Wave Groups via WISHE-Induced Convection AVIV SOLODOCH
More informationQ.1 The most abundant gas in the atmosphere among inert gases is (A) Helium (B) Argon (C) Neon (D) Krypton
Q. 1 Q. 9 carry one mark each & Q. 10 Q. 22 carry two marks each. Q.1 The most abundant gas in the atmosphere among inert gases is (A) Helium (B) Argon (C) Neon (D) Krypton Q.2 The pair of variables that
More information2. Meridional atmospheric structure; heat and water transport. Recall that the most primitive equilibrium climate model can be written
2. Meridional atmospheric structure; heat and water transport The equator-to-pole temperature difference DT was stronger during the last glacial maximum, with polar temperatures down by at least twice
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 informationParameterizing large-scale circulations based on the weak temperature gradient approximation
Parameterizing large-scale circulations based on the weak temperature gradient approximation Bob Plant, Chimene Daleu, Steve Woolnough and thanks to GASS WTG project participants Department of Meteorology,
More informationGoverning Equations and Scaling in the Tropics
Governing Equations and Scaling in the Tropics M 1 ( ) e R ε er Tropical v Midlatitude Meteorology Why is the general circulation and synoptic weather systems in the tropics different to the those in the
More informationLecture 2 ENSO toy models
Lecture 2 ENSO toy models Eli Tziperman 2.3 A heuristic derivation of a delayed oscillator equation Let us consider first a heuristic derivation of an equation for the sea surface temperature in the East
More information5. General Circulation Models
5. General Circulation Models I. 3-D Climate Models (General Circulation Models) To include the full three-dimensional aspect of climate, including the calculation of the dynamical transports, requires
More information2. Conservation laws and basic equations
2. Conservation laws and basic equations Equatorial region is mapped well by cylindrical (Mercator) projection: eastward, northward, upward (local Cartesian) coordinates:,, velocity vector:,,,, material
More informationAssessing the strength of self-aggregation feedbacks from in situ data
Assessing the strength of self-aggregation feedbacks from in situ data Caroline Muller Laboratoire de Météorologie Dynamique Dave Turner NOAA Allison Wing Florida State University Assessing the strength
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 informationAsymptotic Solutions of the Axisymmetric Moist Hadley Circulation in a Model with Two Vertical Modes
Theoretical and Computational Fluid Dynamics manuscript No. (will be inserted by the editor) Samuel P. Burns Adam H. Sobel Lorenzo M. Polvani Asymptotic Solutions of the Axisymmetric Moist Hadley Circulation
More informationImpact of Explicit Atmosphere Ocean Coupling on MJO-Like Coherent Structures in Idealized Aquaplanet Simulations
SEPTEMBER 2006 G R A B O W S K I 2289 Impact of Explicit Atmosphere Ocean Coupling on MJO-Like Coherent Structures in Idealized Aquaplanet Simulations WOJCIECH W. GRABOWSKI National Center for Atmospheric
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 informationAn Introduction to Coupled Models of the Atmosphere Ocean System
An Introduction to Coupled Models of the Atmosphere Ocean System Jonathon S. Wright jswright@tsinghua.edu.cn Atmosphere Ocean Coupling 1. Important to climate on a wide range of time scales Diurnal to
More informationEquatorial Superrotation on Tidally Locked Exoplanets
Equatorial Superrotation on Tidally Locked Exoplanets Adam P. Showman University of Arizona Lorenzo M. Polvani Columbia University Synopsis Most 3D atmospheric circulation models of tidally locked exoplanets
More informationTriggered Convection, Gravity Waves, and the MJO: A Shallow-Water Model
2476 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 70 Triggered Convection, Gravity Waves, and the MJO: A Shallow-Water Model DA YANG AND ANDREW P. INGERSOLL Division of Geological
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 information2.5 Shallow water equations, quasigeostrophic filtering, and filtering of inertia-gravity waves
Chapter. The continuous equations φ=gh Φ=gH φ s =gh s Fig..5: Schematic of the shallow water model, a hydrostatic, incompressible fluid with a rigid bottom h s (x,y), a free surface h(x,y,t), and horizontal
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 informationLecture #3: Gravity Waves in GCMs. Charles McLandress (Banff Summer School 7-13 May 2005)
Lecture #3: Gravity Waves in GCMs Charles McLandress (Banff Summer School 7-13 May 2005) 1 Outline of Lecture 1. Role of GWs in the middle atmosphere 2. Background theory 3. Resolved GWs in GCMs 4. Parameterized
More informationWind Evaporation Feedback and the Axisymmetric Transition to Angular Momentum Conserving Hadley Flow
3758 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 65 Wind Evaporation Feedback and the Axisymmetric Transition to Angular Momentum Conserving Hadley Flow WILLIAM R. BOOS AND KERRY
More informationA Toy Model of the Instability in the Equatorially Trapped Convectively Coupled Waves on the Equatorial Beta Plane
A Toy Model of the Instability in the Equatorially Trapped Convectively Coupled Waves on the Equatorial Beta Plane The Harvard community has made this article openly available. Please share how this access
More informationModeling Large-Scale Atmospheric and Oceanic Flows 2
Modeling Large-Scale Atmospheric and Oceanic Flows V. Zeitlin known s of the Laboratoire de Météorologie Dynamique, Univ. P. and M. Curie, Paris Mathematics of the Oceans, Fields Institute, Toronto, 3
More information) 2 ψ +β ψ. x = 0. (71) ν = uk βk/k 2, (74) c x u = β/k 2. (75)
3 Rossby Waves 3.1 Free Barotropic Rossby Waves The dispersion relation for free barotropic Rossby waves can be derived by linearizing the barotropic vortiticy equation in the form (21). This equation
More information2.1 Effects of a cumulus ensemble upon the large scale temperature and moisture fields by induced subsidence and detrainment
Atmospheric Sciences 6150 Cloud System Modeling 2.1 Effects of a cumulus ensemble upon the large scale temperature and moisture fields by induced subsidence and detrainment Arakawa (1969, 1972), W. Gray
More informationNonlinear shallow-water solutions using the weak temperature gradient approximation
Theor. Comput. Fluid Dyn. (26) DOI.7/s62-6-28-8 ORIGINAL ARTICLE Bo Zhou Adam H. Sobel Nonlinear shallow-water solutions using the weak temperature gradient approximation Received: 6 August 25 / Accepted:
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 informationCHAPTER 5. WAVES AT LOW LATITUDES 73 Figure 5.1: Configuration of a one-layer fluid model with a + fu
Chapter 5 WAVES AT LOW LATITUDES Acharacteristic of the atmosphere is its shallow depth; 99% of the mass lies below a height of 30 km whereas the mean earth radius is 6,380 km. Over this 30 km which extends
More informationChapter 5. Sound Waves and Vortices. 5.1 Sound waves
Chapter 5 Sound Waves and Vortices In this chapter we explore a set of characteristic solutions to the uid equations with the goal of familiarizing the reader with typical behaviors in uid dynamics. Sound
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 informationMoist Hadley circulation: possible role of wave-convection. coupling in aqua-planet experiments. Takeshi Horinouchi
Generated using version 3.1 of the official AMS L A TEX template Moist Hadley circulation: possible role of wave-convection coupling in aqua-planet experiments Takeshi Horinouchi Faculty of Environmental
More informationWhy do GCMs have trouble with the MJO?
Why do GCMs have trouble with the MJO? The Madden-Julian Oscillation West East 200 [hpa] 500 Cool & dry Cool & dry p 700 850 SST Lag Day +20 +15 +10 +5 0-5 -10-15 -20 ~20 days ~10 days ~10-15 days
More informationBoundary layer equilibrium [2005] over tropical oceans
Boundary layer equilibrium [2005] over tropical oceans Alan K. Betts [akbetts@aol.com] Based on: Betts, A.K., 1997: Trade Cumulus: Observations and Modeling. Chapter 4 (pp 99-126) in The Physics and Parameterization
More information3. Midlatitude Storm Tracks and the North Atlantic Oscillation
3. Midlatitude Storm Tracks and the North Atlantic Oscillation Copyright 2006 Emily Shuckburgh, University of Cambridge. Not to be quoted or reproduced without permission. EFS 3/1 Review of key results
More informationIntroduction to Climate ~ Part I ~
2015/11/16 TCC Seminar JMA Introduction to Climate ~ Part I ~ Shuhei MAEDA (MRI/JMA) Climate Research Department Meteorological Research Institute (MRI/JMA) 1 Outline of the lecture 1. Climate System (
More informationA Simple Dynamical Model Capturing the Key Features of Central Pacific El Niño (Supporting Information Appendix)
A Simple Dynamical Model Capturing the Key Features of Central Pacific El Niño (Supporting Information Appendix) Nan Chen and Andrew J. Majda Department of Mathematics, and Center for Atmosphere Ocean
More informationPart-8c Circulation (Cont)
Part-8c Circulation (Cont) Global Circulation Means of Transfering Heat Easterlies /Westerlies Polar Front Planetary Waves Gravity Waves Mars Circulation Giant Planet Atmospheres Zones and Belts Global
More informationCloud Radiative Feedbacks in GCMs : A Challenge for the Simulation of Tropical Climate Variability and Sensitivity
Cloud Radiative Feedbacks in GCMs : A Challenge for the Simulation of Tropical Climate Variability and Sensitivity Sandrine Bony LMD/IPSL, CNRS, UPMC Boite 99, 4 Place Jussieu, 75252 Paris, France bony@lmd.jussieu.fr
More informationCHAPTER 8 NUMERICAL SIMULATIONS OF THE ITCZ OVER THE INDIAN OCEAN AND INDONESIA DURING A NORMAL YEAR AND DURING AN ENSO YEAR
CHAPTER 8 NUMERICAL SIMULATIONS OF THE ITCZ OVER THE INDIAN OCEAN AND INDONESIA DURING A NORMAL YEAR AND DURING AN ENSO YEAR In this chapter, comparisons between the model-produced and analyzed streamlines,
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 informationSimple Mathematical, Dynamical Stochastic Models Capturing the Observed Diversity of the El Niño-Southern Oscillation
Simple Mathematical, Dynamical Stochastic Models Capturing the Observed Diversity of the El Niño-Southern Oscillation Lectures 2 and 3: Background and simple ENSO models 14 september 2014, Courant Institute
More informationROSSBY WAVE PROPAGATION
ROSSBY WAVE PROPAGATION (PHH lecture 4) The presence of a gradient of PV (or q.-g. p.v.) allows slow wave motions generally called Rossby waves These waves arise through the Rossby restoration mechanism,
More informationResponse of convection to relative SST: Cloud-resolving simulations in 2D and 3D
1 2 Response of convection to relative SST: Cloud-resolving simulations in 2D and 3D S. Wang, 1 and A. H. Sobel 2 -------------- Shuguang Wang, Department of Applied Physics and Applied Mathematics, Columbia
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 informationESCI 344 Tropical Meteorology Lesson 11 Tropical Cyclones: Formation, Maintenance, and Intensification
ESCI 344 Tropical Meteorology Lesson 11 Tropical Cyclones: Formation, Maintenance, and Intensification References: A Global View of Tropical Cyclones, Elsberry (ed.) Global Perspectives on Tropical Cylones:
More information4. Atmospheric transport. Daniel J. Jacob, Atmospheric Chemistry, Harvard University, Spring 2017
4. Atmospheric transport Daniel J. Jacob, Atmospheric Chemistry, Harvard University, Spring 2017 Forces in the atmosphere: Gravity g Pressure-gradient ap = ( 1/ ρ ) dp / dx for x-direction (also y, z directions)
More informationThe Structure of Precipitation Fronts for Finite Relaxation Time
Theoretical and Computational Fluid Dynamics manuscript No. (will be inserted by the editor) Samuel N. Stechmann Andrew J. Majda The Structure of Precipitation Fronts for Finite Relaxation Time This Draft:
More informationContents. Parti Fundamentals. 1. Introduction. 2. The Coriolis Force. Preface Preface of the First Edition
Foreword Preface Preface of the First Edition xiii xv xvii Parti Fundamentals 1. Introduction 1.1 Objective 3 1.2 Importance of Geophysical Fluid Dynamics 4 1.3 Distinguishing Attributes of Geophysical
More informationTropical Meridional Circulations: The Hadley Cell
Tropical Meridional Circulations: The Hadley Cell Introduction Throughout much of the previous sections, we have alluded to but not fully described the mean meridional overturning circulation of the tropics
More informationScienceDirect. Numerical modeling of atmospheric water content and probability evaluation. Part I
Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 70 ( 2014 ) 321 329 12th International Conference on Computing and Control for the Water Industry, CCWI2013 Numerical modeling
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 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 informationESCI 485 Air/Sea Interaction Lesson 1 Stresses and Fluxes Dr. DeCaria
ESCI 485 Air/Sea Interaction Lesson 1 Stresses and Fluxes Dr DeCaria References: An Introduction to Dynamic Meteorology, Holton MOMENTUM EQUATIONS The momentum equations governing the ocean or atmosphere
More informationBalanced Dynamics and Convection in the Tropical Troposphere
Balanced Dynamics and Convection in the Tropical Troposphere David J. Raymond, Željka Fuchs, Saška Gjorgjievska, and Sharon L. Sessions Physics Department and Geophysical Research Center New Mexico Tech
More informationarxiv: v1 [physics.ao-ph] 18 Jul 2007
GEOPHYSICAL RESEARCH LETTERS, VOL.???, XXXX, DOI:10.1029/, arxiv:07.2750v1 [physics.ao-ph] 18 Jul 2007 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Multiple Equilibria in a Single-Column Model of the Tropical Atmosphere
More informationLARGE SCALE DYNAMICS OF PRECIPITATION FRONTS IN THE TROPICAL ATMOSPHERE: A NOVEL RELAXATION LIMIT
COMM. MATH. SCI. Vol., No., pp. 591 c International Press LARGE SCALE DYNAMICS OF PRECIPITATION FRONTS IN THE TROPICAL ATMOSPHERE: A NOVEL RELAXATION LIMIT DARGAN M. W. FRIERSON, ANDREW J. MAJDA, AND OLIVIER
More informationPrototype Instabilities
Prototype Instabilities David Randall Introduction Broadly speaking, a growing atmospheric disturbance can draw its kinetic energy from two possible sources: the kinetic and available potential energies
More informationMultiple equilibria in a cloud resolving model using the weak temperature gradient approximation
Click Here for Full Article JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115,, doi:10.1029/2009jd013376, 2010 Multiple equilibria in a cloud resolving model using the weak temperature gradient approximation Sharon
More informationDynamics of the Extratropical Response to Tropical Heating
Regional and Local Climate Modeling and Analysis Research Group R e L o C l i m Dynamics of the Extratropical Response to Tropical Heating (1) Wegener Center for Climate and Global Change (WegCenter) and
More informationEffects of moisture feedback in a frictional coupled Kelvin Rossby wave model and implication in the Madden Julian oscillation dynamics
Clim Dyn DOI 10.1007/s00382-016-3090-y Effects of moisture feedback in a frictional coupled Kelvin Rossby wave model and implication in the Madden Julian oscillation dynamics Fei Liu 1 Bin Wang 2 Received:
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 informationUnderstanding the local and global impacts of model physics changes
ECMWF Annual Seminar 2008 MJR 1 Understanding the local and global impacts of model physics changes Mark Rodwell Work with Thomas Jung 4 September 2008 Thanks to: MJR 2 Motivation The real world and GCMs
More information