Hadley Cell Observations Summary (小结) evew Temperature field:the equator-pole temperature gradient is much smaller than the RE temperature gradient. Wind fields:meridional winds strongest at tropopause and surface; vertical velocity strongest at mid-level of the troposphere. Jets(zonal winds):strong subtropical jet at upper level with its maximum in the latitudes at the edge or just poleward of the descending branch of the Hadley cell;surface winds-easterlies near the equator and westerlies in the extratropics. Strong seasonal variations:in summer or winter,Hadley cell always appears as a strong single cell across the equator with the ascending branch in the tropics of the summer hemisphere. 授课教师：张洋2
授课教师：张洋 2 Hadley Cell - Observations n Summary （⼩结） n Temperature field: the equator-pole temperature gradient is much smaller than the RE temperature gradient. n Wind fields: meridional winds strongest at tropopause and surface; vertical velocity strongest at mid-level of the troposphere. n Jets (zonal winds): strong subtropical jet at upper level with its maximum in the latitudes at the edge or just poleward of the descending branch of the Hadley cell; surface winds-easterlies near the equator and westerlies in the extratropics. n Strong seasonal variations: in summer or winter, Hadley cell always appears as a strong single cell across the equator with the ascending branch in the tropics of the summer hemisphere. Review
&会 Hadley Cell Theories Held-Hou model (1980) MARCI 1980 ISAAC M.HELD AND 515 Nonlinear Axially Symmetric Circulations in a Nearly Inviscid Atmosphere IsAAC M.HELD ARTHUR Y.Hou Center for Earth and Plaxetary Physics.Harvard University,Cambridge,MA 02138 (Manuscript received 23 July 1979,in final form 16 October 1979) ABSTRACT on a sphere is the polew rving.The cory predicts the lotal po et in the zonal wir nd,and 1.Introduction atmospheres (e.g..Dickinson,1971:Leovy,1964) The importance of mixing induced by large-scale the meridional is determined baroclinic or barotropic instabilities for the general by the parameterized small-scale frictional stresses circulation of the atmosphere can best be a ppreciated in the zonal momentum equation.Detailed analyses by artificially suppressing these instabilities and of such models do not promise to be very fruitful examining the circulation which develops in their as long as theories for small-scale momentum mixing absence.This is most easily accomplished in the are themselves not very well developed. idealized case for which radiative foreing and the Schneider and Lindzen have recently computed lower boundary condition are both axially symmetric some axisymmetric flows forced by small-scale (independent of longitude).The flow of interest in fluxes of heat and momentum that do bear some this case is the large-scale axisymmetric flow resemblance to the observed cireulation (Schneider consistent with radiative forcing and whatever small- and Lindzen,1977;Schneider,1977).Using simple scale mixing is still present in the atmosphere theories for moist convective as well as boundary after the large-scale instabilitics have been suppressed. and radiative fluxes,Schneider obtains a Hadley Such axisymmetric circulations have not received cell which terminates abruptly at more or less the as much attention in the meteorological literature right latitude.a very strong subtropical jet at the as one might expect,given what would appear to be poleward boundary of the Hadley cell,strong trade their natural position as first approximations to the winds in the tropics,and a shallow Ferrel cell and general circulation.Reasons for this neglect are not surface westerlies poleward ofthe trades.Nakamura Isaac M.Held hard to find.It is the accepted wisdom that large (1978)describes an effectively axisymmetric calcula- scale zonally asymmetric baroclinic instabilities are tion (with heating and frictional formulations differ- 0z 8z J义个作寸大PJ八/十 3
n Held-Hou model (1980) 授课教师：张洋 3 Hadley Cell - Theories Isaac M. Held Review
&会 Hadley Cell Theories Held-Hou model (1980) Make assumptions: (Vallis,2006) Z=H Tropopause Angular momentum conserving flow the circulation is steady; Large zonal flow aloft the upper branch conserves Mass flux V angular momentum;surface zonal Warm Cool winds are weak; ascent descent the circulation is in thermal wind balance. Mass flux -V Frictional return flow Z=0,Ground Weak zonal flow at surface Equator Latitude Subtropics 授课教师：张洋 4
授课教师：张洋 4 Hadley Cell - Theories i i i i i i i i Angular momentum conserving flow Equator Subtropics Latitude Warm ascent Cool descent Tropopause Frictional return flow Weak zonal flow at surface Ground Large zonal flow aloft Fig. 11.4 A simple model of the Hadley Cell. Rising air near the equator moves polewards near the tropopause, descending in the subtropics and returning near the surface. The polewards moving air conserves its axial angular momentum, leading to a zonal flow that increases away from the equator. By the thermal wind relation the temperature of the air falls as it moves poleward, and to satisfy the thermodynamic budget it sinks in the subtropics. The return flow at the surface is frictionally retarded and small. From Vallis (2006) From Vallis (2006) (Vallis, 2006) n Held-Hou model (1980) Make assumptions: n the circulation is steady; n the upper branch conserves angular momentum; surface zonal winds are weak; n the circulation is in thermal wind balance. Review Mass flux V Mass flux -V Z = 0, Z = H
&会岛 Hadley Cell (review) Theories Review Held-Hou model (1980) Meet the model(diagram) y Conservation of angular momentum Angular momentum V conserving (axisymmetric) Thermal wind balance Distribution of temperature aCOSΦ Steady flow Latitude extent of Hadley Cell V Strength of Hadley Cell a Y Weak zonal flow at surface Distribution of upper westerly (due to friction) Distribution of surface winds Zonal flow is balanced (thermal wind relation) 受课教师：张洋 5
授课教师：张洋 5 Hadley Cell (review) - Theories n Held-Hou model (1980) Meet the model (diagram) Conservation of angular momentum Thermal wind balance Distribution of temperature Latitude extent of Hadley Cell Strength of Hadley Cell Distribution of upper westerly Distribution of surface winds ⌦ a cos a Angular momentum conserving (axisymmetric) Weak zonal flow at surface (due to friction) Zonal flow is balanced (thermal wind relation) Steady flow Review
&会岛 Held-Hou model -Angular momentum Review The absolute angular momentum per unit mass is M=(2 a cos o+u)acosΦ a is the radius of the earth Due to earth's Deviation from the solid rotation solid rotation D 1ap Dt M=-p∂入 a cos oFx acoSΦ In an axisymmetric flow([M]=M) coiF] In an inviscid (frictionless),axisymmetric flow,the angular momentum is conserved 授课教师：张洋6
M = (⌦a cos + u)a cos D Dt M = 1 ⇢ @p @ + a cos F D Dt[M] = a cos [F] 授课教师：张洋 6 Held-Hou model -Angular momentum n The absolute angular momentum per unit mass is a is the radius of the earth Due to earth’s solid rotation Deviation from the solid rotation ⌦ a cos a In an axisymmetric flow ([M]=M) In an inviscid (frictionless), axisymmetric flow, the angular momentum is conserved. Review
&会品 Held-Hou model -Angular momentum Review [M=(acosφ+[u)acosφ At the equator,as the parcels rise from the surface,where the flow is weak,we assume that the zonal flow is zero there. 2 in-o [u=2a 三UM cos o acoSΦ Then,what is the UM at 10,20,30 degree? Answers:14,57,134 m/s,respectively Combined with the weak surface flow,this indicates strong vertical shear of the zonal wind. 授课教师：张洋7
M = (⌦a cos + u)a cos [u] = ⌦a sin2 cos ⌘ UM 授课教师：张洋 7 Held-Hou model -Angular momentum n At the equator, as the parcels rise from the surface, where the flow is weak, we assume that the zonal flow is zero there. ⌦ a cos a [ ] [ ] Then, what is the UM at 10, 20, 30 degree? Answers: 14, 57, 134 m/s, respectively Combined with the weak surface flow, this indicates strong vertical shear of the zonal wind. Review
&会易 Held-Hou model -Temperature distribution Review Angular momentum: sin2o 4=acos中 UM Thermal wind relation: fu(D)-uo1+tan°2(H)-2o1= gH∂g a日。ab ©(0)-Θ（Φ） 22a2sin40 o 2gH cos2 o Conservation of angular momentum and the maintenance of thermal wind completely determine the variation of temperature within the Hadley Cell 授课教师：张洋8
[u] = ⌦a sin2 cos ⌘ UM f[u(H) u(0)] + tan a [u2(H) u2(0)] = gH a⇥o @⇥˜ @ ⇥˜ (0) ⇥˜ () ⇥o = ⌦2a2 2gH sin4 cos2 授课教师：张洋 8 Held-Hou model -Temperature distribution n Thermal wind relation: n Angular momentum: Conservation of angular momentum and the maintenance of thermal wind completely determine the variation of temperature within the Hadley Cell ! Review
&会 Held-Hou model -Extent of Hadley Cell Review ΦH ΦH 白cosφd0= 白cos od0 0 Radiative equilibrium temperature 百FROM EQ.l2) 日(，=1- anA(mo+a(后-司 △H fractional temperature difference between equator and pole 百（中H)=日EH) △u fractional temperature difference between ground and top Psecond Legendre polynomial,()1) Vertical average: ⊙eo,习=1-号△aPB(sin0) 2 EQUATOR POLE LATITUDE 授课教师：张洋 9
授课教师：张洋 9 Held-Hou model -Extent of Hadley Cell ⇥˜ (H) = ⇥˜ E(H) Z H 0 ⇥˜ cos d = Z H 0 ⇥˜ E cos d H - fractional temperature di↵erence between equator and pole ⇥E(, z) ⇥o = 1 2 3 HP2(sin ) + v( z H 1 2 ) P2 - second Legendre polynomial, P2(x) = 1 2 (3x2 1) ⇥˜ E(, z) ⇥o = 1 2 3 HP2(sin ) n Radiative equilibrium temperature Vertical average: v - fractional temperature di↵erence between ground and top Review
&会 Held-Hou model -Extent of Hadley Cell Review Assume small,sinp~Φ ΦH ΦH 白cos odΦ= 白cos od0 0 (0)E(0) 5gH△ 1822a2 百[FROM EQ.l2) 1/2 d(pH)=ΘH) 2e /2 set R then,φH= R EQUATOR POLE LATITUDE 授课教师：张洋 10
Assume small , sin ⇠ 授课教师：张洋 10 Held-Hou model -Extent of Hadley Cell ⇥˜ (H) = ⇥˜ E(H) Z H 0 ⇥˜ cos d = Z H 0 ⇥˜ E cos d H = ✓5 3 gHH ⌦2a2 ◆1/2 set R = gHH ⌦2a2 , then, H = ✓5 3 R ◆1/2 ⇥˜ (0) ⇥o ⇡ ⇥˜ E(0) ⇥o 5 18 gH2 H ⌦2a2 Review
&会 Held-Hou model -Strength of Hadley Cell Thermodynamic equation at equator and steady state: DΘ ΘE-Θ ∂0、ΘE-Θ Dt T W02 T If the static stability is mostly determined by the forcing instead of meridional circulation: 1∂日 △v Θ。∂z H H ΘE-Θ 5g△2HH2 ≈ Θo△yT 18a2T22△V Using mass continuity: gH032△2 H Characteristic overturning time a23T△V Td w scale can be estimated. 授课教师：张洋11
授课教师：张洋 11 Held-Hou model -Strength of Hadley Cell 1 ⇥o @⇥ @z ⇡ V H n If the static stability is mostly determined by the forcing instead of meridional circulation: n Thermodynamic equation at equator and steady state: w ⇡ H ⇥oV ⇥E ⇥ ⌧ = 5g2 HH2 18a2⌧⌦2V v ⇠ (gH) 3/2 5/2 H a2⌦3⌧V ⌧d = H w Characteristic overturning time scale can be estimated. w @⇥ @z ⇡ ⇥E ⇥ ⌧ D⇥ Dt = ⇥E ⇥ ⌧ n Using mass continuity: