Held-Hou model (review) -Summary mperature constrained Review angular momentum and thermal wind balance. 6(0)-6(o)22a2sin4p Smaller than the 2 日o 2gH cos2o RE temp gradient 日e(o,=1=2△HP(sin0) 9. ■ Extent of Hadley Cell: 5gH△H 八1/2 pHY20.4° EQUATOR 8H POLE φH= 322a2 LATITUDE 1.6 Strength of Hadley Cell: (gH3/2△2 1.4 U a223T△V 12 2 Upper jet: in2Φ [u=2a 三UM 1.0 ZONAL WIND】 AT Z=H cos o .8 ■Surface winds. .6 Cu(0)≈ 25g2H3△ 18a323r△v [()-g()八+（品）月 3 1/2 surface easterlies H ZONAL WIND AT Z=0 2
Held-Hou model (review) -Summary 授课教师：张洋 2 ! Extent of Hadley Cell: ! Strength of Hadley Cell: H = ✓5 3 gHH ⌦2a2 ◆1/2 ⇥˜ (0) ⇥˜ () ⇥o = ⌦2a2 2gH sin4 cos2 v ⇠ (gH) 3/2 5/2 H a2⌦3⌧V ! Upper jet: [u] = ⌦a sin2 cos ⌘ UM Smaller than the RE temp gradient ! Surface winds: < ✓3 7 ◆1/2 surface easterlies H Cu(0) ⇡ 25 18 g2H33 H a3⌦3⌧V "✓ H ◆2 10 3 ✓ H ◆4 + 7 3 ✓ H ◆6 # ⇥˜ E(, z) ⇥o = 1 2 3 HP2(sin ) H ⇠ 20.4o ! Distribution of temperature constrained by the conservation of angular momentum and thermal wind balance. Review
Held-Hou model (review) -Discussion Review Upper jet:right place,but too large and discontinue. ■ Extent of Hadley Cell:only a finite extent. 0eo.习_i-号arB(6im) 2 9. Hadley cell cannot carry heat from equator to pH￥20.4° EQUATOR 94 POLE the pole,thus cannot be responsible to the LATITUDE 1.6 observed equator-pole temperature difference.So does the wind,momentum and 1.4 heat flux distribution. 12 1.0 ZONAL WIND】 Axisymmetric flow:roles of eddies are AT Z=H neglected. ■ Moisture effect is neglected. Seasonal variation and asymmetry on the equator? ZONAL WIND_ AT Z=0 13
Held-Hou model (review) -Discussion 授课教师：张洋 3 ! Axisymmetric flow: roles of eddies are neglected. ! Upper jet: right place, but too large and discontinue. ! Extent of Hadley Cell: only a finite extent. Hadley cell cannot carry heat from equator to the pole, thus cannot be responsible to the observed equator-pole temperature difference. So does the wind, momentum and heat flux distribution. ! Moisture effect is neglected. ! Seasonal variation and asymmetry on the equator? ⇥˜ E(, z) ⇥o = 1 2 3 HP2(sin ) H ⇠ 20.4o Review
&售 Hadley Cell-Theory: Asymmetry about the equator* Review Lindzen-Hou (1988) Tropopause Angular momentum conserving flow Still make the assumptions: Large zonal flow aloft the circulation is steady or quasi-steady Warm Cool ascent descent (the flow adjusts to a steady circulation on a timescale faster than that on which the solar radiation varies), the upper branch conserves angular Frictional return flow momentum;surface zonal winds are 《 weak; Ground Weak zonal flow at surface summer equator winter the circulation is in thermal wind hemisphere hemisphere balance; the only difference:the heating is centered off the equator. 授课教师：张洋 A
授课教师：张洋 4 Hadley Cell - Theory: Asymmetry about the equator* ! Lindzen-Hou (1988) Still make the assumptions: ! the circulation is steady or quasi-steady (the flow adjusts to a steady circulation on a timescale faster than that on which the solar radiation varies); ! the upper branch conserves angular momentum; surface zonal winds are weak; ! the circulation is in thermal wind balance; 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) summer equator hemisphere winter hemisphere ! the only difference: the heating is centered off the equator. Review
设雾 Asymmetry about the equator* -Angular momentum Review In an inviscid,axisymmetric flow,the angular momentum is conserve upper branch of Hadley Cell. M =(2acosΦ+u)acosΦ 23H Sa2 cos2 1 Sa(cos2 1-cos2 UM coSΦ o1-latitude of the rising motion 中m 中0 中1 中H FIG.3.Schematic illustration of the Hadley circulation. 授课教师：张洋5
1 - latitude of the rising motion 授课教师：张洋 5 Asymmetry about the equator* -Angular momentum ! In an inviscid, axisymmetric flow, the angular momentum is conserved for the upper branch of Hadley Cell. M = (⌦a cos + u)a cos = ⌦a2 cos2 1 UM = ⌦a(cos2 1 cos2 ) cos Review
设雾 Hadley Cell-Theory: Asymmetry about the e Review 1.0 —△1=1/3 △m=1/6 0.8h 中r 0中0中 中n: FIG.3.Schematic illustration of the Hadley circulation. 0.6 As the heating shifts off the equator,both cells and the center of the raising branch shift poleward,with a wider Hadley cell cross the equator and a narrower Hadley cell in the 0.2 summer hemisphere; As the diabatic heating varies stronger in the 0.05 0.0 0.15 meridional direction,both cells shift -。(radians) poleward and become wider. FIG.4.and as functions of(see text for definitions). Open circles show results from numerical integration forand when△H='%.(Note1°of latitude≈0.0175 radians.) 授课教师：张洋6
授课教师：张洋 6 Hadley Cell - Theory: Asymmetry about the equator* ! As the heating shifts off the equator, both cells and the center of the raising branch shift poleward, with a wider Hadley cell cross the equator and a narrower Hadley cell in the summer hemisphere; ! As the diabatic heating varies stronger in the meridional direction, both cells shift poleward and become wider. Review
&鲁 Hadley Cell Discussion: UNN Asymmetry about the eguator* Review Lindzen-Hou (1988) Quasi-steady assumption,however the seasonal cycle is temporally progressing; The lack of angular momentum conservation in reality,especially when the angular momentum transport by eddies are significant. The moisture effect is still neglected; 授课教师：张洋7
授课教师：张洋 7 Hadley Cell - Discussion: Asymmetry about the equator* ! Lindzen-Hou (1988) ! Quasi-steady assumption, however the seasonal cycle is temporally progressing; ! The lack of angular momentum conservation in reality, especially when the angular momentum transport by eddies are significant. ! The moisture effect is still neglected; Review
设雾 Hadley Cell Role of eddies* Qualitative considerations 易M=-,C+where品-品+ D∂ 0 acosΦa入a0b In an axisymmetric flow ([M]=M): 2l-aosiA In a 3-D(with eddies)flow,the zonally averaged angular momentum: ade品adso+是o 0m+1 O arde品(m-是mo+aasB 1 1 vjos2到-品uror1es+as9B acosΦ0Φ Neglecting vertical advection,vertical eddy fluxes and friction: m]+回am=-1} O a 8o acos(a cos) 授课教师：张洋8
授课教师：张洋 @[m] @t + [v] a @[m] @ = 1 a cos @ @([u⇤v⇤]a cos2 ) 8 Hadley Cell : Role of eddies* ! Qualitative considerations In a 3-D (with eddies) flow, the zonally averaged angular momentum: D Dt M = 1 ⇢ @p @ + a cos F In an axisymmetric flow ([M]=M): D Dt[M] = a cos [F] D Dt = @ @t + u a cos @ @ + v a @ @ + w @ @z , where Neglecting vertical advection, vertical eddy fluxes and friction: @[m] @t + 1 a cos @ @ ([v][m] cos ) + @ @z ([w][m]) = 1 a cos @ @([m⇤v⇤] cos ) @ @z ([m⇤w⇤]) + a cos [F] = 1 a cos @ @([u⇤v⇤]a cos2 ) @ @z ([u⇤w⇤]a cos ) + a cos [F]
设雾 Hadley Cell Role of eddies* Qualitative considerations D 8 Di-at acosΦaλaab In an axisymmetric flow ([M]=M): -cmi In a 3-D(with eddies)flow,the zonally averaged angular momentum: Neglecting vertical advection,vertical eddy fluxes and friction: olm]lolm] acaei2erhoms2 1 20 [w] b Eddy momentum flux always 10 acts to decrease the angular momentum of the zonal flow, and the zonal velocity is lower than UM. Observations -20 10 授课教师：张洋9
授课教师：张洋 @[m] @t + [v] a @[m] @ = 1 a cos @ @([u⇤v⇤]a cos2 ) 9 Hadley Cell : Role of eddies* ! Qualitative considerations D Dt M = 1 ⇢ @p @ + a cos F In an axisymmetric flow ([M]=M): D Dt[M] = a cos [F] In a 3-D (with eddies) flow, the zonally averaged angular momentum: D Dt = @ @t + u a cos @ @ + v a @ @ + w @ @z , where Neglecting vertical advection, vertical eddy fluxes and friction: http://www.adultpdf.com Created by Image To PDF trial version, to remove this mark, please register this software. Observations Eddy momentum flux always acts to decrease the angular momentum of the zonal flow, and the zonal velocity is lower than UM
设雾 Hadley Cell Role of eddies* Numerical solutions: Vallis,2006, axisymmetric numerical results from idealized GCM 0.0 40 um n uo 0.5 20 us 0 1.0 Narrower but stronger 0.0 Hadley cell 3-D flow.with eddies 40 1化m E Upper jet shifts with weakeruo 0.5 20 maximum leuoz Stronger surface winds 0 Us 1.0 60 -30 0 30 60 -60 -30 0 30 60 Latitude Latitude
授课教师：张洋 10 Hadley Cell : Role of eddies* ! Numerical solutions: axisymmetric 3-D flow, with eddies Vallis, 2006, numerical results from idealized GCM Narrower but stronger Hadley cell Upper jet shifts with weaker maximum Stronger surface winds
设 Hadley Cell Outline Observations Held-Hou theory (axisymmetric flow,a model that is symmetric about the equator) Lindzen-Hou theory (axisymmetric flow,a model that is asymmetry about the equator) The role of eddies Moisture effects Discussions 授课教师：张洋11
授课教师：张洋 11 Hadley Cell - Outline ! Observations ! Held-Hou theory (axisymmetric flow, a model that is symmetric about the equator) ! Lindzen-Hou theory (axisymmetric flow, a model that is asymmetry about the equator) ! The role of eddies ! Moisture effects ! Discussions