8/21/08. Modeling the General Circulation of the Atmosphere. Topic 4: Equatorial Wave Dynamics. Moisture and Equatorial Waves

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1 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 H E R I C S C I E N C E S Moisture and Equatorial Waves What determines the spectrum of linear disturbances in the tropics? How does condensation affect equatorial waves? Why do current AGCMs tend to simulate equatorial waves poorly? Importance of Equatorial Waves/Variability Equatorial Wave Dynamics Connected to: Tropical weather predictability Monsoon onset Hurricanes El Nino Midlatitude weather Another classic multiscale problem Strategy here: Derive a new simplified model for tropical dynamics Galerkin truncation of equations to two modes Use these to develop some understanding of tropical wave dynamics Test predictions in simplified moist GCM and full GCMs 1

2 Tropical Equations of Motion Tropical Equations of Motion Start with Boussinesq equations (linearized about a basic state density) Equatorial beta-plane Coriolis force is zero at the equator Differential rotation (just north/south of equator) is profoundly important though We ll make a Galerkin truncation of these equations to two modes Boussinesq equations on beta-plane Dimensional Non-dimensional See Frierson, Majda and Pauluis (2004) for more info See Frierson, Majda and Pauluis (2004) for more info Decomposition into Modes Galerkin Truncation to Two Modes Decompose into vertical mean and deviations: And expand into cosine structures: Two vertical modes: vertical mean and first baroclinic mode Barotropic mode Baroclinic mode Temperature equation Vertical structures 2

3 No Barotropic Flow Dispersion Relations for Equatorial Waves Next, assume no vertical mean flow Ū = 0 System has the following: (see Majda 2003 for more details) Kelvin waves (nondispersive eastward propagating waves) Mixed Rossby-gravity wave (Yanai mode) Equatorial Rossby waves Inertia-gravity waves These equations gives a remarkably rich system of waves Non-rotating, 1-D derivation Kelvin wave derivation (waves with v = 0) Frequency Wavenumber Observations of Equatorial Waves Observed Equatorial Waves Wheeler and Kiladis (1999) showed that all these wave types are visible in observations In observations, speeds are significantly slower than predicted There s a simple theory for why though that involves condensation The waves are convectively coupled, i.e., they re seen in the precipitation fields 3

4 Non-rotating, 1-D case Effect of Moisture on Convectively Coupled Equatorial Waves Non-rotating, 1-D case (dimensional) u t = T x T t = s u x + P where s is the dry static stability Gives a wave equation with speed Momentum equation Thermodynamic equation s Non-rotating, 1-D case (dimensional) u t = T x Momentum equation T t = s u x + P Thermodynamic equation But precip is correlated w/ convergence: P = q u x which implies T t = m u x where m = s q is the gross moist stability Phase speed goes as m Alternate Theory for Decreased Phase Speeds Nonlinearities in Precipitation Another theory (alternative to gross moist stability) That second baroclinic mode is important in setting the phase speed Second baroclinic mode dry gravity wave speed is similar to observed speed too Also, second baroclinic mode structures are seen in the waves, in addition to first baroclinic mode structure Main question that we will test with full GCMs Precipitation has a funny nonlinearity though: it can t be negative! P = q u x only when P > 0 (otherwise P = 0) Leads to some interesting dynamics in this model 4

5 Conservation Laws Conservation Laws We added a full moisture budget and parameterized precipitation by Looking for conservation laws in full system: This implies: So, solutions can t ever go to infinity Next, formulate a system with the derivatives of quantities: (This form assumes no barotropic mode) Derivative System Conservation Laws Precipitation Fronts Derivative system has an energy principle too: This energy principle is dissipative as well (not shown) Gradients can t go to infinity Discontinuities can t form However, due to form of precipitation parameterization, no conservation laws for higher derivatives exist So, derivatives can develop jumps in them! Basic variables can develop kinks We used Rankine-Hugoniot jump conditions to solve for speed of precipitation fronts 5

6 Back to Equatorial Wave Dynamics Multiscale Aspects of Moist Processes Question of what determines phase speed of equatorial waves: Let s test in a simplified GCM Waves have scale of a few 1000 km Precipitation/clouds can occur on 1 km or smaller scale How to parameterize precipitation on the larger scale? How (and why) do different parameterizations of precip affect the large scale waves and mean circulation? Hierarchical Modeling Approach Effect of Convective Parameterization Models of increasing complexity to build understanding Isolate particular physical effects Main focus here: Intermediate complexity moist GCM (Frierson et al 2006) Isolate the dynamical impact of water vapor (latent heating when condensation occurs) Interaction between different levels of complexity is key Make connections to simple theories and full GCM simulations How about sensitivity to convection scheme? Simplified Betts-Miller convection scheme Relaxes temperature and humidity to reference profiles Few parameters, easy to interpret 6

7 Effect of convection scheme Hadley Circulation Sensitivity Simplified Betts-Miller convection scheme Relaxes temperature and humidity to reference profiles Few parameters, easy to interpret Instantaneous precipitation Hadley circulation is strongly sensitive to certain parameter changes: Tropical precipitation distribution with identical forcing, moisture content, etc. Not sensitive to others: Changing convective relaxation time by a factor of 8 See Frierson (2007a) for more detail Hadley circulation changes Equatorial Waves in Idealized GCM Circulation is sensitive to aspects of the convection scheme that change the gross moist stability (GMS) of the tropics GMS is efficiency of energy transport of cell Smaller GMS => less efficient Hadley cell, more mass flux required to transport same amount of energy Effect of convective parameterization on the GMS: GMS is larger when convection can easily occur up to high levels GMS is smaller when there s an abrupt trigger for convection Effect on equatorial waves? Kelvin waves dominate the spectrum They can propagate around and around the equator multiple times! See Frierson (2007a) for more detail 7

8 Convectively coupled Kelvin waves Convectively Coupled Kelvin Waves GMS reduction leads to slower convectively coupled waves: Vertical structure is essentially first baroclinic mode though (unlike observations) Temperature and zonal wind GMS = 7 K GMS = 4.5 K GMS = 2.5 K Wavespeed can be tuned to essentially any value in this model See Frierson (2007b) for more detail Equatorial Waves in a Full GCM Vertical structures Experiments with SNU atmospheric GCM Run over observed SSTs, realistic geography Simplified Arakawa-Schubert convection scheme Varying strength of convective trigger In full GCM, the waves show realistic vertical phase tilts (unlike in simplified GCM) Shallow -> deep -> stratiform MJO is also quite realistic in this simulation Wavespeed decreases with stronger moisture trigger Due to smaller GMS, as in simplified GCM Collaboration with Jialin Lin, Daehyun Kim, In-Sik Kang, and Myong-In Lee See Lin et al (2008) and Frierson et al (in prep) for more detail 8

9 Conclusions Other recent work on tropical dynamics In a highly simplified moist GCM, parameterization of convection can affect Strength of Hadley circulation ITCZ precipitation Speed and intensity of convectively coupled waves In a full GCM: Convection also affects speed and intensity of convectively coupled waves Gross moist stability determination is key to these processes Sensitivity of ITCZ to high latitude forcing With Sarah Kang, Isaac Held, and Ming Zhao In simplified GCM and full GCM Effect of surface fluxes on MJO With Adam Sobel, Eric Maloney, and Gilles Bellon Eliminating evaporation-wind feedback severely reduces amplitude of MJO in GFDL and NCAR GCMs Not traditional WISHE idea though nonlinear WISHE 9