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Dynamics of the ZonalZonal-Mean, Mean, Time-Mean Tropical Circulation TimeCirculation 1 First consider a hypothetical yp planet like Earth, but with no continents and no seasons and for which the only friction acting on the atmosphere is at the surfface. This planet has an exact nonlinear equilibrium solution for the flow of the atmosphere, characterized by 1. Every column is in radiative-convective equilibrium, 2. Wind vanishes at planet’s surface 3. Horizontal pressure gradients balanced by 3 Coriolis accelerations 2 Hydrostatic balance: p g z I pressure coordinat In di tes: , p where where 1 specific volume, gz geopotential 3 Geostrophic balance on a sphere: u tan 2 sin urel a a p 2 rel Can be C b phrased h d iin tterms off absolute b l t angular l momentum per unit mass: M a cos a cos u M a cos sin 2 3 a cos 2 4 2 4 4 Eliminate using hydrostatic equation: 1 tan M T s * 2 2 a cos p p p s* p 2 (Thermal wind equation) Take s* s *( ) (convective neutrality) Integrate upward from surface (p=p0), taking g u=0 at surface: 5 s * 2 2 2 M a cos a cos cot Ts T ss * 2 u 2a cos u cot Ts T 0. 2 2 2 Ts T s * u 2 sin y Implies strongest west-east winds where entropy gradient is strongest, weighted toward equator 6 Two potential problems with this solution: 1. Not enough angular momentum available for required west-east wind, 2 Equilibrium solution may be unstable 2. 7 How much angular momentum is need ded? d? At the tropopause, s * 2 2 2 M a cos a cos cot Ts Tt 2 2 2 How much angular momentum is available? M eq a 2 s * Ts Tt 2 4 2 a 2 tan 1 cos 4 cos 2 8 More generally, we require that angular momentum decrease away from the equator: 1 M 2 0 sin 1 2 s * 4 a cos cos cot Ts Tt 0. sin 2 2 3 Let y sec 2 Ts Tt s * 1 y 2 2 0 yy a y 3 9 What happens when this condition i viol is i lated? d? Overturning circulation must develop Acts to drive actual entropy distribution back toward criticality Postulate constant angular momentum at tropopause: M t a 10 2 Integrate thermal wind equations down from tropopause: t s * M a a cos cott T Tt 2 2 4 2 2 T Tt s * a 1 u cos 3 2 cos 2a sin Note: N t u att surfface allways < 0 ffor supercriti iticall entropy distribution 11 12 Critical entropy distribution: Ts Tt s * 1 1 2 1 2 2 a yy 2 y Ts Tt Ts Tt * 1 1 s* 2 2 seq 1 1 y 2 2 a a y 2 Ts Tt * 1 2 1 2 2 seq 1 cos 2 cos a 2 13 Violation results in large-scale overturning circulation, known as the Hadley Circulation, that transports heat poleward and drives surface entropy gradient back toward its critical value From "The General Circulation of the Tropical Atmosphere and Interactions with Extratropical Latitudes - Vol. 1" by MIT Press. Courtesy of MIT Press. Used with permission. 14 15 Simulations by Held and Hou (1980) 16 17 Zonal wind at tropopause 18 The cross-equatorial Hadley Circullatiion Ci 19 Numerical simulation using a 2-D primitive equation equat o model ode with t pa parameterized a ete ed co convection ect o (Pauluis, 2004) Imposed sea surface temperature 20 21 MIT OpenCourseWare http://ocw.mit.edu 12.811 Tropical Meteorology Spring 2011 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 22