ATS 421/521 Climate Modeling Spring 2015 Lecture 9 Hadley Circulation (Held and Hou, 1980) General Circulation Models (tetbook chapter 3.2.3; course notes chapter 5.3) The Primitive Equations (tetbook chapter 3.3.1) Surface Processes Parameterizations Grids and Resolution May 1, 2015
Hadley Circulation Course notes chapter 5.2 Simple Model by Held and Hou (1980) based on conservation of angular momentum in a two layer model Angular Momentum M=ru no friction friction zonal velocity
Assume heating: At equator: (assume no zonal flow) At latitude Φ: Φ=30 N: um = 110 m/s
Calculate width of Hadley Cell: Vertical shear: Thermal wind balance: 3000 km or 30 Width is in good agreement with observations but circulation is much too slow.
Lindzen and Hou (1988) showed that the circulation is improved by considering the seasonal cycle annual mean conditions lead to weak heating seasonal heating is much stronger if maimum heating is shifted slightly away from equator
Hadley Circulation and Zonal Wind easterlies westerlies Reanalysis GCM Schmittner et al. (2011)
ERA40 1959-2002 Meridional Water Vapor Flu positive = northward F WV = 360 =0 y y'=90s E P d dy'
Let s follow an air parcel along the Hadley cell. 1) We start at the surface in the ITCZ. Over the ocean the air is saturated (RH = 100%) qsat tropics subtropics condensation precipitation evaporation ocean EQ 30 N
Let s follow an air parcel along the Hadley cell. 2) It raises to higher altitudes, loosing water through condensation (RH = 100%) clouds form, precipitation (rainfall) occurs tropics subtropics condensation precipitation evaporation ocean
Let s follow an air parcel along the Hadley cell. 3) At high elevation it is cold and the air is dry (low specific humidity, but still high relative humidity) tropics subtropics condensation precipitation evaporation ocean
Let s follow an air parcel along the Hadley cell. 4) As the air moves toward the subtropics it stays cold and dry tropics subtropics condensation precipitation evaporation ocean
Let s follow an air parcel along the Hadley cell. 5) As the air sinks in the subtropics its temperature increases but the amount of water vapor does not increase. Thus, the relative humidity decreases. tropics subtropics condensation precipitation evaporation ocean
Let s follow an air parcel along the Hadley cell. 6) When the air arrives at the surface it has warmed but now it has a very low relative humidity. tropics subtropics condensation precipitation evaporation ocean
Let s follow an air parcel along the Hadley cell. 7) On its way back towards the equator evaporation leads to an increase its water vapor content and relative humidity. tropics subtropics condensation precipitation evaporation ocean
Let s follow an air parcel along the Hadley cell. 8) Back at the ITCZ the air is is saturated again. tropics subtropics condensation precipitation evaporation ocean
Primitive Equations Momentum Conservation: Newton s second law Velocity Total (Lagrangian) derivative Force per mass Navier-Stokes Equations: Total derivative:
On rotating sphere Coriolis and centripetal forces: zonal velocity meridional velocity vertical velocity
Mass Conservation: Continuity Equation Incompressible Fluid Energy Conservation: First Law of Thermodynamics Adiabatic Epansion/ Compression Diabatic Processes (e.g. radiation, latent heat release) Potential Temperature = Temperature of fluid parcel if adiabatically brought to surface gas constant dry air heat capacity const. pressure
Conservation of Water Vapor: (In Ocean analogous equation for salinity) If RH > 80% then precipitation Equation of State: Air: Sea Water: Primitive Equations = 5.17-5.23 (5.24) 7 equations with 7 unknowns (u,v,w,ρ,p,t,q)
Surface Processes Empirical Relations (Bulk Formulae) Momentum (wind stress): Sensible heat flu: Moisture flu: Cs are transfer/drag coefficients
Equations are solved on a three-dimensional grid covering the Earth. In each grid bo the primitive equations as well as other equations are solved. Flues between neighboring boes are calculated and used to update the tendencies for the net time step. Typically ocean and atmosphere models have different grid bo sizes (resolution), which requires a coupler to interpolate/average between the two grids.
Parameterizations required due to large grid bo sizes Reynolds Decomposition: grid bo mean deviation from mean mean flow eddy flues Sub-grid scale eddy flues have to be parameterized. E.g. in ocean models they are often treated as diffusion:
Finite Differences Horizontal Staggered Grids Convergence of meridians at poles leads to small grid spacings Δ, which restricts the time step.
Spectral Models Spherical Harmonics l=4 Frequently used for atmospheric models No problem due to convergence of meridians at pole Advection is accurate But: not positive definite Not used for ocean models due to zonal boundaries at continents http://www.maths.nottingham.ac.uk/personal/pcm/sphere/sphere.html
Orography at different spectral resolutions T42 ~ 250 km T21 ~ 500 km Schmittner et al. in press J. Climate
Finite Element Models Cubed Sphere Geodesic Kurihata Washington et al. 2009 Phil. Trans. R. Soc.
Finite Element Models An eample of a global climate model geodesic grid with a color-coded plot of the observed sea-surface temperature distribution. The continents are depicted in white. This grid has 10,242 cells, each of which is roughly 240 km across. Twelve of the cells are pentagons; the rest are heagons. Source: Randall, D. A. et. al., Climate modeling with spherical geodesic grids, Computing in Science and Engineering, 4, 5, 32-41. http://cnls.lanl.gov/~petersen/
Vertical Grids depth-coordinate σ-coordinate isopycnal-coordinate http://www.oc.nps.edu/nom/modeling/vertical_grids.html
Resolution 500 km (T21) 300 km (T42) 150 km (T85) 75 km Washington et al. 2009 Phil. Trans. R. Soc.