Atmospheric Wave-Mean Flow Interactions by Declan Quinn B . S c , The National University of Ireland, 1996 M . S c , The National University of Ireland, 1997 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S (Department of Mathematics) We accept this thesis as conforming s$o the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A August 2000 © Declan Quinn, 2000 In p resen t i ng this thesis in partial fu l f i lment of the requ i rements for an a d v a n c e d d e g r e e at the Un ivers i ty of Brit ish C o l u m b i a , I agree that the Library shall m a k e it f ree ly avai lable fo r re fe rence and s tudy. I fur ther agree that pe rm iss i on for ex tens ive c o p y i n g of th is thes is fo r scho lar ly p u r p o s e s may b e gran ted by the h e a d o f m y d e p a r t m e n t o r by his o r her representat ives. It is u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n of this thesis for f inancia l gain shal l n o t b e a l l o w e d w i t h o u t m y wr i t t en p e r m i s s i o n . D e p a r t m e n t T h e Univers i ty o f Bri t ish C o l u m b i a V a n c o u v e r , C a n a d a D E - 6 (2/88) Abstract The interactions between the zonally-averaged flow and the various eddy circulations in the atmosphere are studied using N C E P / N C A R Reanalysis data. The contributions to zonal- and potential temperature-tendency and zonal kinetic energy production due to both the Ferrel and Hadley circu-lations are determined. Inferences are then made regarding the processes which affect the strong cores of the jet stream and the forms of zonally-averaged models of the atmosphere that are appropriate for studies of this type. It will be shown that there is considerable wave-mean flow interaction in the atmosphere, with the circulation induced by eddy motions being primarily responsible for the maintainance and seasonal variations of the zonally-averaged jet core. This study also provides a repository of derived atmospheric quantities for the years 1989-1993. ii CONTENTS Abstract i i L i s t of Figures v i 1 Introduction 1 2 Notat ion 3 3 Mode l 6 4 Da ta 10 4.1 Preprocessing of diabatic heating data 11 4.2 Calculated quantities 14 4.3 Comparison wi th data of previous studies 15 5 Numerical solution of meridional streamfunction equation 17 6 Distributions of zonal wind, potential temperature and diabatic heating and eddy fluxes of heat and momentum 21 6.1 Zonal wind, potential temperature and diabatic heating . . . . . . 21 6.2 E d d y fluxes of heat and momentum 25 6.3 Calculat ion of heat- and momentum-flux forcings 43 7 Streamfunction ip 49 8 Momentum and heat budgets 55 8.1 Budgets of eddy-driven circulation . 55 8.2 Budgets of diabatic heating-driven circulation 70 i i i 8.3 Budgets of total circulation 82 9 Conclusions 95 A Table of symbols 100 B Derivation of model equations 101 B . l Governing equations 101 B.2 Derivation of thermal wind equation 102 B .3 Derivation of zonally-averaged equations of motion 104 B.3.1 Derivation of zonally-averaged mass-continuity equation . . 104 B.3.2 Derivation of the zonally-averaged ideal gas law and poten-t ia l temperature equations 106 B.3.3 Derivation of the zonally-averaged zonal momentum equationl07 B.3.4 Derivation of the zonally-averaged thermodynamic equation 109 B.3.5 Derivation of the zonally-averaged thermal wind equation . I l l B .4 Governing equations for zonally-averaged motion 112 C Derivation of diagnostic streamfunction equation for mean meridional circulation 114 D Month ly plots 120 D . l Distributions of zonal wind, potential temperature and diabatic heating 121 iv D.2 Eddy fluxes of heat and momentum 128 D.2.1 Stationary eddies 128 D.2.2 Transient, zonaUy-symmetric eddies 137 D.2.3 Transient, zonally-asymmetric eddies 146 D.3 Eddy momentum and heat fluxes 155 D.4 Streamfunction ip 164 D.5 Zonal Kinet ic Energy Product ion 171 v List of Figures 6.1 Annua l and seasonal-mean distributions of [it]. Uni ts are m s - 1 and the contour interval is 5 m s"1 23 6.2 Annua l and seasonal-mean distributions of [0]. Uni ts are K and the contour interval is 10 K 24 6.3 Annua l and seasonal-mean distributions of Uni ts Cp are K s _ 1 and the contour interval is 1 x 1 0 - 5 K . 26 6.4 Annua l and seasonal-mean distributions of [Tt* v*} cos 2 <f) (stationary eddies). Units are m2 s~2 and the contour interval is 5 m 2 s - 2 28 6.5 Annua l and seasonal-mean distributions of [u] [v] cos 2 <f> (transient, zonally-symmetric eddies). Units are m2 s~2 and the contour interval is 0.5 m2 s~2 29 i t ' v cos2 (j) 6.6 Annua l and seasonal-mean distributions of L J (transient, zonally-asymmetric eddies). Uni ts are m2 s~2 and the contour interval is 5 m2 s~2 30 6.7 Annua l and seasonal-mean distributions of [Tt* u*] (sta-tionary eddies). Units are 1 0 - 2 m Pa s~2 and the con-tour intervals are 2 x 10~ 2 m Pa s~2 (light contours) and 10 x 1 0 - 2 m Pa s~2 (dark contours) 32 6.8 Annua l and seasonal-mean distributions of [u] [u/] (tran-sient, zonally-symmetric eddies). Uni ts are 10~ 2 m Pa s~2 and the contour intervals are 0.5 x 10~ 2 m Pa s~2 (light contours) and 5 x 10~ 2 m Pa s~2 (dark contours). . . . 33 it* or (tran-6.9 Annua l and seasonal-mean distributions of sient, zonally-asymmetric eddies). Units are 10~ 2 m Pa s~2 and the contour intervals are 5 x 10~ 2 m Pa s~2. . . . 34 6.10 Annua l and seasonal-mean distributions of 6 v* COS(f> (stationary eddies). Units are K s 1 and the contour interval is 2 i f s - 1 36 v i 6.11 Annua l and seasonal-mean distributions of [8'] [v] cosc/> (transient, zonally-symmetric eddies). Uni ts are K s _ 1 and the contour interval is 0.1 K s~l 37 8*'v*' COS0 6.12 Annua l and seasonal-mean distributions of (transient, zonally-asymmetric eddies). Units are K s~l and the contour interval is 2 K s _ 1 38 8 UJ* (sta-6.13 Annua l and seasonal-mean distributions of tionary eddies). Units are 1 0 - 2 Pa K s - 1 and the con-tour intervals are 2 x 10~ 2 Pa K s~x (light contours) and 30 x 10~ 2 Pa K s~l (dark contours). . 40 6.14 Annua l and seasonal-mean distributions of \8'\ [CJ'] (tran-sient, zonally-symmetric eddies). Units are 10~ 2 Pa K s~l and the contour intervals are 1 x 10~ 2 Pa K (light contours) and 10 x 10~ 2 Pa K s"1 (dark contours). . . 41 8*'u>*' (tran-6.15 Annua l and seasonal-mean distributions of sient, zonally-asymmetric eddies). Units are 10~ 2 Pa K s _ 1 and the contour intervals are 5 x 1 0 - 2 Pa K s~l. . . . 42 6.16 Annua l and seasonal-mean distributions of northward momentum flux. Units are m2 s~2 and the contour interval is 10 m2 s~2 44 6.17 Annua l and seasonal-mean distributions of upward mo-mentum flux. Units are 1 0 - 2 m Pa s~2 and the contour intervals are 4 x 1 0 - 2 m Pa s~2 (light contours) and 20 x 10~ 2 m Pa s~2 (dark contours) 45 6.18 Annua l and seasonal-mean distributions of northward heat flux. Units are K s~l and the contour interval is 5 K s~l 46 6.19 Annua l and seasonal-mean distributions of upward heat flux. Uni ts are 10~ 2 Pa K and the contour in-tervals are 5 x 1 0 - 2 Pa K s _ 1 (light contours) and 50 x 1 0 - 2 Pa K s'1 (dark contours) 47 v i i 7.1 Annua l and seasonal-mean distributions of ip due to diabatic heating (Hadley circulation, ipHadiey)- Uni ts are 10 3 m Pa s'1 and the contour intervals are 1 x 10 3 m Pa s~l (light contours) and 5 x 10 3 m Pa s - 1 (dark contours). 52 7.2 Annua l and seasonal-mean distributions of the eddy-induced component of ip (Ferrel circulation, ipFerrei)-Units are 10 3 m Pa s _ 1 and the contour interval is 1 x 10 3 m Pa s~1. 53 7.3 Annua l and seasonal-mean distributions of ip. Uni ts are 10 3 m Pa s _ 1 and the contour intervals are 1 x 10 3 m Pa s - 1 (light contours) and 5 x 10 3 m Pa s _ 1 (dark contours) 54 8.1 Annua l average momentum budget (top three panels) and zonal kinetic energy production due to eddy-flux forcing. For the top three panels, the units are 10~ 5 m s~2 and the contour interval is 0.5 x 10~ 5 m s~2. For the bottom panel, the units are 10~ 5 m2 s~3 and the con-tour interval is 5 x 10~ 5 m 2 s~ 3 59 8.2 Average winter ( D J F ) momentum budget (top three panels) and zonal kinetic energy production due to eddy-flux forcing. For the top three panels, the units are 10~ 5 m s~2 and the contour interval is 0.5 x 1 0 - 5 m s~2. For the bot tom panel, the units are 10~ 5 m2 s~3 and the contour interval is 5 x 10~ 5 m2 s~3 60 8.3 Average spring ( M A M ) momentum budget (top three panels) and zonal kinetic energy production due to eddy-flux forcing. For the top three panels, the units are 10~ 5 m s~2 and the contour interval is 0.5 x 10~ 5 m s~2. For the bot tom panel, the units are 10~ 5 m2 s~ 3 and the contour interval is 5 x 10~ 5 ra2 s - 3 61 v i i i 8.4 Average summer ( J J A ) momentum budget (top three panels) and zonal kinetic energy production due to eddy-flux forcing. For the top three panels, the units are 10~ 5 m s~2 and the contour interval is 0.5 x 10~ 5 m s~2. For the bot tom panel, the units are 10~ 5 m2 s~3 and the contour interval is 5 x 1 0 - 5 m2 s~ 3 62 8.5 Average autumn (SON) momentum budget (top three panels) and zonal kinetic energy production due to eddy-flux forcing. For the top three panels, the units are 10~ 5 m s~2 and the contour interval is 0.5 x 1 0 - 5 m s"2. For the bot tom panel, the units are 10~ 5 m 2 s~3 and the contour interval is 5 x 10~ 5 m2 s~ 3 63 8.6 Annual - and seasonal-average zonal kinetic energy pro-duction due to eddy-flux forcing. The units are 1 0 - 5 m2 s~3 and the contour interval is 5 x 10~ 5 m 2 s~3 64 8.7 Annua l average heat budget due to eddy-flux forcing. The units are 10~ 5 K s"1 and the contour interval is 0.5 x l O " 5 K s'1 65 8.8 Average winter (DJF) heat budget due to eddy-flux forcing. The units are 10~ 5 K s~l and the contour interval is 0.5 x 10~ 5 K s~l 66 8.9 Average spring ( M A M ) heat budget due to eddy-flux forcing. The units are 10~ 5 K s"1 and the contour interval is 0.5 x 10~ 5 K s _ 1 67 8.10 Average summer ( J J A ) heat budget due to eddy-flux forcing. The units are 10~ 5 K s~l and the contour interval is 0.5 x 10" 5 K s - 1 68 8.11 Average autumn (SON) heat budget due to eddy-flux forcing. The units are 10~ 5 K s~l and the contour interval is 0.5 x 10~ 5 K s'1 69 ix 8.12 Annua l average momentum tendency (top panel) and zonal kinetic energy production due to diabatic heat-ing . For the top panel, the units are 1 0 - 5 m s~2 and the contour interval is 0.5 x 10~ 5 m s~2. For the bot-tom panel, the units are 10~ 5 m2 s - 3 and the contour interval is 5 x 10~ 5 m2 s~3 72 8.13 Average winter (DJF) momentum tendency (top panel) and zonal kinetic energy production due to diabatic heating. For the top panel, the units are 10~ 5 m s~2 and the contour interval is 0.5 x 10~ 5 m s~2. For the bot tom panel, the units are 10~ 5 m2 s - 3 and the con-tour interval is 5 x 10~ 5 m2 s~3 73 8.14 Average spring ( M A M ) momentum tendency (top panel) and zonal kinetic energy production due to diabatic heating. For the top panel, the units are 10~ 5 m s~2 and the contour interval is 0.5 x 1 0 - 5 m s~2. For the bot tom panel, the units are 1 0 - 5 m2 s~3 and the con-tour interval is 5 x 10~ 5 m2 s~3 74 8.15 Average summer ( J J A ) momentum tendency (top panel) and zonal kinetic energy production due to diabatic heating. For the top panel, the units are 10~ 5 m s~2 and the contour interval is 0.5 x 10~ 5 m s~2. For the bot tom panel, the units are 10~ 5 m2 s~3 and the con-tour interval is 5 x 10~ 5 m2 s~3 75 8.16 Average autumn (SON) momentum tendency (top panel) and zonal kinetic energy production due to diabatic heating. For the top panel, the units are 10~ 5 m s~2 and the contour interval is 0.5 x 1 0 - 5 m s~2. For the bot tom panel, the units are 10~ 5 m2 s - 3 and the con-tour interval is 5 x 10~ 5 m2 s~3 76 8.17 Annua l average heat budget due to diabatic heating. The units are 1 0 - 5 K s~l and the contour interval is O . S x l O ^ X s " 1 77 x 8.18 Average winter (DJF) heat budget due to diabatic heat-ing. The units are 10~ 5 K s~l and the contour interval is 0.5 x 10~ 5 K s"1 78 8.19 Average spring ( M A M ) heat budget due to diabatic heating. The units are 10~ 5 K s~l and the contour interval is 0.5 x 10" 5 K s~l 79 8.20 Average summer ( J JA) heat budget due to diabatic heating. The units are 10~ 5 K s~l and the contour interval is 0.5 x 10" 5 K s'1 80 8.21 Average autumn (SON) heat budget due to diabatic heating. The units are 10~ 5 K s~l and the contour interval is 0.5 x 10~ 5 K s~l 81 8.22 Annua l momentum budget (top three panels) and zonal kinetic energy production due to the combined effects of the eddies and diabatic heating. For the top three pan-els, the units are 10~ 5 m s~2 and the contour interval is 0.5 x 1 0 - 5 m s~2. For the bottom panel, the units are 10~ 5 m2 s~3 and the contour interval is 5 x 1 0 - 5 m2 s~3. 85 8.23 Average winter (DJF) momentum budget (top three panels) and zonal kinetic energy production due to the combined effects of the eddies and diabatic heating. For the top three panels, the units are 10~ 5 m s~2 and the contour interval is 0.5 x 10"~5 m s~2. For the bot tom panel, the units are 10~ 5 m2 s~3 and the contour inter-val is 5 x 10~ 5 m2 s'3 86 8.24 Average spring ( M A M ) momentum budget (top three panels) and zonal kinetic energy production due to the combined effects of the eddies and diabatic heating. For the top three panels, the units are 1 0 - 5 m s~2 and the contour interval is 0.5 x 1 0 - 5 m s~2. For the bot tom panel, the units are 1 0 - 5 m2 s~3 and the contour inter-val is 5 x 10~ 5 m2 s~3 87 x i 8.25 Average summer ( J J A ) momentum budget (top three panels) and zonal kinetic energy production due to the combined effects of the eddies and diabatic heating. For the top three panels, the units are 10~ 5 m s~~2 and the contour interval is 0.5 x 10~ 5 m s~2. For the bot tom panel, the units are 1 0 - 5 m2 s~3 and the contour inter-val is 5 x 10" 5 m2 s~3 88 8.26 Average autumn (SON) momentum budget (top three panels) and zonal kinetic energy production due to the combined effects of the eddies and diabatic heating. For the top three panels, the units are 10~ 5 m s~2 and the contour interval is 0.5 x 10~ 5 m s~2. For the bot tom panel, the units are 10~ 5 m2 s~3 and the contour inter-val is 5 x 10" 5 m2 s~3 89 8.27 Annua l average heat budget for the combined action of diabatic heating and eddy fluxes of heat and momen-tum. The units are 1 0 - 5 K s _ 1 and the contour interval is 0.5 x 10~ 5 K s" 1 90 8.28 Average winter (DJF) heat budget for the combined action of diabatic heating and eddy fluxes of heat and momentum. The units are 1 0 - 5 K s _ 1 and the contour interval is 0.5 x 10~ 5 K s" 1 91 8.29 Average spring ( M A M ) heat budget for the combined action of diabatic heating and eddy fluxes of heat and momentum. The units are 1 0 - 5 K s _ 1 and the contour interval is 0.5 x 10" 5 K s~l 92 8.30 Average summer ( J J A ) heat budget for the combined action of diabatic heating and eddy fluxes of heat and momentum. The units are 10~ 5 K s _ 1 and the contour interval is 0.5 x 10~ 5 K s~l 93 8.31 Average autumn (SON) heat budget for the combined action of diabatic heating and eddy fluxes of heat and momentum. The units are 10~ 5 K s _ 1 and the contour interval is 0.5 x 10~ 5 K s~l 94 x i i D.1 January-June distributions of [it]. Units are m s 1 and the contour interval is 5 m s _ 1 122 D.2 July-December distributions of [it]. Units are m s~l and the contour interval is 5 m s~l 123 D.3 January-June distributions of [0]. Units are i f and the contour interval is 10 i f 124 D.4 July-December distributions of [0]. Units are i f and the contour interval is 10 i f 125 D.5 January-June distributions of [Q]. Units are i f s _ 1 and the contour interval is 1 x 10~ 5 i f s _ 1 126 D.6 July-December distributions of [Q]. Units are i f s - 1 and the contour interval is 1 x 10~ 5 i f s _ 1 127 D.7 January-June distributions of [u*v*] cos 2 <f) for station-ary eddies. Units are m 2 s~2 and the contour interval i s 5 m 2 s - 2 129 D.8 July-December distributions of [u*v*] cos 2 <fi for station-ary eddies. Units are m2 s~2 and the contour interval is 5 m 2 s'2 130 D.9 January-June distributions of [u*u*\ for stationary ed-dies. Units are 10~ 2 m Pa s~2 and the contour in-tervals are 2 x 1 0 - 2 m Pa s~2 (light contours) and 10 x 10~ 2 m Pa s~2 (dark contours) 131 D.10 July-December distributions of [u*u*] for stationary ed-dies. Units are 10~ 2 m Pa s~2 and the contour in-tervals are 2 x 10~ 2 m Pa s - 2 (light contours) and 10 x 1 0 - 2 m Pa s~2 (dark contours) 132 D . l l January-June distributions of [v*0*] cos<f> for stationary eddies. Units are i f s~l and the contour interval is 2Ks~l 133 D.12 July-December distributions of [v*0*]cos<fi for station-ary eddies. Units are i f s _ 1 and the contour interval is 2Ks~x 134 x i i i D.13 January-June distributions of [u*6*] for stationary ed-dies. Units are 1 0 - 2 Pa K s~l and the contour in -tervals are 2 x 10~ 2 Pa K s"1 (light contours) and 30 x 10" 2 Pa K s'1 (dark contours) 135 D.14 July-December distributions of [u*6*\ for stationary ed-dies. Units are 1 0 - 2 Pa K s"1 and the contour in-tervals are 2 x 1 0 - 2 Pa K s"1 (light contours) and 30 x 10~ 2 Pa K s~l (dark contours) 136 D . 15 January-June distributions of [u*v*] cos 2 <j> for tran-sient, zonally-symmetric eddies. Units are m2 s~2 and the contour interval is 0.5 m2 s~~2 138 D . 16 July-December distributions of [u*v*] cos2c/> for tran-sient, zonally-symmetric eddies. Units are m2 s~2 and the contour interval is 0.5 m2 s~2 139 D.17 January-June distributions of [u*ui*\ for transient, zonally-symmetric eddies. Units are 10~ 2 m Pa s~2 and the contour intervals are 0.5 x 1 0 - 2 m Pa s~2 (light con-tours) and 5 x 10~ 2 m Pa s~2 (dark contours) 140 D.18 July-December distributions of [u*a;*] for transient, zonally-symmetric eddies. Units are 10~ 2 m Pa s~2 and the contour intervals are 0.5 x 10~ 2 m Pa s~2 (light con-tours) and 5 x 10~ 2 m Pa s~2 (dark contours) 141 D.19 January-June distributions of [v*9*] cose/) for transient, zonally-symmetric eddies. Units are K s~l and the contour interval is 0.1 K s~l 142 D.20 July-December distributions of [v*9*] cos 4> for transient, zonally-symmetric eddies. Units are K s~l and the contour interval is 0.1 X s" 1 143 D.21 January-June distributions of [u*6*] for transient, zonally-symmetric eddies. Units are 1 0 - 2 Pa K s _ 1 and the contour intervals are 1 x 10~ 2 Pa K s _ 1 (light contours) and 10 x 10~ 2 Pa K s~l (dark contours) 144 xiv D.22 July-December distributions of [u*8*] for transient, zonally-symmetric eddies. Units are 10 2 Pa i f s 1 and the contour intervals are 1 x 10~ 2 Pa i f s _ 1 (light contours) and 10 x 1 0 - 2 Pa i f s'1 (dark contours) 145 D.23 January-June distributions of [u*v*] cos 2 (j> for tran-sient, zonally-asymmetric eddies. Uni ts are m2 s~2 and the contour interval is 5 m2 s~2 147 D.24 July-December distributions of [u*v*] cos 2 <p for tran-sient, zonally-asymmetric eddies. Units are m2 s~2 and the contour interval is 5 m2 s~2 148 D.25 January-June distributions of [u*u*] for transient, zonally-asymmetric eddies. Units are 1 0 - 2 m Pa s~2 and the contour intervals are 5 x 10~ 2 m Pa s~2 149 D.26 July-December distributions of [«*_>*] for transient, zonally-asymmetric eddies. Uni ts are 1 0 - 2 m Pa s~~2 and the contour intervals are 5 x 10~ 2 m Pa s~2 150 D.27 January-June distributions of [v*9*] coscfi for transient, zonally-asymmetric eddies. Units are i f s~l and the contour interval is 2 i f s~l 151 D.28 July-December distributions of [v*6*] cos <f> for transient, zonally-asymmetric eddies. Units are K s - 1 and the contour interval is 2 i f s _ 1 152 D.29 January-June distributions of [u>*9*] for transient, zonally-asymmetric eddies. Units are 1 0 - 2 Pa i f s _ 1 and the contour intervals are 5 x 10~ 2 Pa i f s _ 1 153 D.30 July-December distributions of [u*0*] for transient, zonally-asymmetric eddies. Units are 10~ 2 Pa i f s'1 and the contour intervals are 5 x 1 0 - 2 Pa i f s"1 154 D.31 January-June distributions of northward momentum flux. Uni ts are m2 s~2 and the contour interval is 10 m2 s - 2 . 156 D.32 July-December distributions of northward momentum flux. Units are m2 s~2 and the contour interval is 10 m 2 s~2 157 xv D.33 January-June distributions of upward momentum flux. Units are 10~ 2 m Pa s~2 and the contour intervals are 4 x 10~ 2 m Pa s~2 (light contours) and 20 x 1CT 2 m Pa s~2 (dark contours) 158 D.34 July-December distributions of upward momentum flux. Units are 10~ 2 m Pa s~2 and the contour intervals are 4 x 10~ 2 m Pa s~2 (light contours) and 20x 10" 2 ra Pa s~2 (dark contours) 159 D.35 January-June distributions of northward heat flux. Uni ts are K s~l and the contour interval is 5 K s _ 1 160 D.36 July-December distributions of northward heat flux. Uni ts are K s~l and the contour interval is 5 K s~l 161 D.37 January-June distributions of upward heat flux. Uni ts are 10~ 2 Pa K s _ 1 and the contour intervals are 5 x 10~ 2 Pa K s'1 (light contours) and 50 x 10~ 2 Pa K s~l (dark contours) 162 D.38 July-December distributions of upward heat flux. Uni ts are 1 0 - 2 Pa K s~l and the contour intervals are 5 x 10~ 2 Pa K s" 1 (light contours) and 50 x 10~ 2 Pa K s~l (dark contours) 163 D.39 January-June distributions of tp due to diabatic heating (Hadley circulation). Units are 10 3 m Pa s"1 and the contour intervals are 1 x 10 3 ra Pa s - 1 (light contours) and 5 x 10 3 m Pa s _ 1 (dark contours) 165 D.40 July-December distributions of ip due to diabatic heat-ing (Hadley circulation). Units are 10 3 ra Pa s _ 1 and the contour intervals are 1 x 10 3 ra Pa s - 1 (light con-tours) and 5 x 10 3 ra Pa s - 1 (dark contours) 166 D.41 January-June distributions of the eddy-induced compo-nent of ip (Ferrel circulation). Units are 10 3 ra Pa s _ 1 and the contour interval is 1 x 10 3 ra Pa s"1 167 D.42 July-December distributions of the eddy-induced com-ponent of ip (Ferrel circulation). Uni ts are 10 3 ra Pa s"1 and the contour interval is 1 x 10 3 ra Pa s~x 168 x v i D.43 January-June distributions of ib. Units are 10 3 m Pa s"1 and the contour intervals are 1 x 10 3 m Pa s~x (light contours) and 5 x 10 3 m Pa s~l (dark contours) 169 D.44 July-December distributions of ib. Uni ts are 10 3 m Pa s _ 1 and the contour intervals are 1 x 10 3 m Pa s"1 (light contours) and 5 x 10 3 ra Pa s - 1 (dark contours) 170 D.45 January-June zonal kinetic energy production due to eddy-flux forcing. The units are 10~ 5 m2 s~3 and the contour interval is 5 x 10~ 5 m 2 s~3 172 D.46 July-December zonal kinetic energy production due to eddy-flux forcing. The units are 10~ 5 m2 s~s and the contour interval is 5 x 10~ 5 m2 s~3 173 x v i i 1. Introduction O f perennial interest in the meteorological community are the processes main-taining and affecting the westerly upper-air jet streams along wi th the effect of these wind-speed maxima on synoptic-scale atmospheric waves. Firs t investigated i n depth during the Second Wor ld War, the subsequent rise of c iv i l aviation has provided the impetus for many studies of mid-latitude upper-air phenomena. The aims of this project are: • to study the processes which affect the strong cores of the jet-stream • to extend the results of [Pfe81] to the Southern Hemisphere and to the Arc t i c region; also, to improve the quality of these results in the equatorial regions • to determine the relative importance of transient eddy and stationary eddy upward fluxes of heat and momentum and thus investigate the conjecture of [SPS70] that the vertical transient eddy flux of momentum by the synoptic-scale eddies is much greater than the vertical stationary eddy flux • to extend the eddy-transport statistics computed by [P092] to seasonal- and monthly-mean conditions • to provide a baseline for possible futher investigations of the effect of the 1 E N S O phenomenon on the global circulation of the atmosphere • to provide a mathematical framework for investigating the transport of ozone from its region of manufacture at the Equator to the region of destruction near the poles It is primari ly the availability of high-quality data that makes these aims feasible and, although much of the data are sti l l somewhat questionable, wi th some data being more artifacts of the reanalysis model than actual measured quantities, the results should give insight into the processes maintaining the thermal-wind balance in the atmosphere. 2 2. Notation Definition 2.1. The zonal-averaging operator is 1 f2* w - s i A i X The departure from this average is A* = A-[A]. Definition 2.2. The time-averaging operator is — 1 ft2 A — —-— Ad h — h Jtx The departure from this average is A' = A-A. 3 Using these definitions 1, one may write A = [A\ + A* + [A]' + A These may be interpreted as decomposing a variable into: 1. a steady, zonally-symmetric component 2. a steady, zonally-asymmetric (i.e. stationary eddy) component 3. a time-varying, zonally-symmetric component (e.g. an index cycle fluctua-t ion in the case A = u) 4. a time-varying, zonally-asymmetic component (e.g. a quantity associated wi th a storm or other transient, local event) Other useful identities, which wi l l be used implici t ly throughout the present investigation, may be derived: 1This notational scheme is a reversal of that of [Pfe81], but is in keeping with the more accepted conventions of [P092] and [OR71]. [AB] [A] [B] + [A*B* (2.1) (2.2) 4 A'B' = [A]' [B]' + [A'*B'* (2.3) Throughout the body of this work, reference is made to the decomposition of fluxes into various stationary and transient, zonally symmetric and asymmetric components; the decompositions used are • m e r i d i o n a l t r a n s p o r t : Av] = [A] [v] + \A*v*] + [A] [v] + [A'*v'*\ (2.4) • d o w n w a r d t r a n s p o r t : Aw] = p ] [w] + \A*u*] + [A] [u] + [A'*u'*\ (2.5) where \A\ [v] and \A\ [UJ] represent the transport of A by the steady circula-t ion i n the ((j), p)-plane. 5 3. Mode l The model used for the calculations is briefly described below; a detailed derivation of the model equations is given in Appendix B . The basic equations are the zonally-averaged momentum equation (refer to Appendix B.3.3) d\u] _ ( a ( [ « ] 0 0 8 f l \ d[u] a ( K ^ ] C Q S 2 0 ) d[u*U*\ dt L JV R cos <pd(j) J [ i dp Rco&<j)d<t> dp L J' (3-1) the zonally-averaged thermodynamic equation (refer to Appendix B.3.4) dt R dcj) M dp Rcos4>d(j) dp \ p J cp ' { ' } and the zonally-averaged thermal wind equation (refer to Appendix B.3.5) jd[u] = RdP*-lld[6] dp P00K R dcj) where / = (j + 2 M ^ a n < ^ _ Here, t represents time, R is the earth's radius, Rd is the gas constant, 4> represents latitude, p represents pressure, p 0o is a reference pressure level (1000 hPa), u is eastward velocity, v is northward velocity, u (— ^ ) is the 6 vertical velocity in pressure coordinates, 6 is potential temperature, / (= 2£) sin </>) is the Coriolis parameter, FX is the zonal forcing, Q is the diabatic heating, cp is the constant pressure heat capacity of air and K = ^SL. Also , it is useful to define Cp the static stability parameter Sp = — ^ r f j ; -One may define a streamfunction ip associated wi th the mean meridional circu-lation ([v], [UJ]) and derive the following diagnostic equation for the streamfunction (see Appendix C) d2ip , NDPKd[9] d2ip , DPKd[9] (K- 1 dip\ E>^d^[0\dip 9 (c%) P <9»72 PooK ^ ^r/dp PooK drj \ P dri J PooK dr12 dp cos (pdp — + (3-4) or] op where c = _L(f 9 ( M c o s < ^ ) D [SP] H cose/) \ Rdr] Rdl_ p R2 pn d[9] PooK dP R ^ ' 1 ((Poo\[Q]_ d([9*v*}coscb) _ d[9*u*}\ P00KR \ \ p J K cp Rdr} dp J f (d([u*v*}coS2ct>) d[u*u*} \ X cose/) V Rcosqjdrj dp 1 XiJ V ' ' 7 V 1 dip cos <j> dp (3.7) Rdrj (3.8) 77 = sin (p. A description of each variable, and its associated units, is given in Appendix The above model equations differ from those of [Pfe81] by their presentation i n terms of potential temperature rather than absolute temperature 2 . The equa-tions originally presented in [Kuo56] differ substantially from the more modern presentations of this model, wi th Kuo ' s decomposition across time-scales leading to a bewildering array of model equations. It is felt that the current approach is more appropriate to the type of model calculations undertaken herein, wi th no assumptions being made implicitly. Whi l e the Transformed Eulerian-Mean formulation of such a zonally-averaged model of the atmosphere (as introduced by [AM76] and further discussed by [AHL87]) is possible, it shall be seen that such a cogitated set of equations is 2 Also, it appears that there is an error, or at least an implicit assumption, in the published formulae of [Pfe81], with the contribution by — 9^Qp 1 having been transformed incorrectly in the formulation of H. The effect of this term is, according to the derivations of [P092] and [Kuo56], expected to be secondary under the quasi-geostrophic assumption. A . 8 not necessary for the type of investigation presented herein. The principal moti-vation for use of a T E M formulation of a zonally-averaged model is to combine the direct effect of eddy motions wi th that of the eddy-induced circulation. It shall be seen, however, that the eddies st i l l play an important role in affecting the zonally-averaged jet core. 9 4. Data The data used in this model are al l products of the N C E P / N C A R Reanalysis project. A comprehensive overview of the project is provided by [KKK+96] ; a precis relevant only to the data actually used wi l l be provided here. The bulk of the data used is taken from gridded N C E P / N C A R reanalyses; a description of the treatment of the diabatic heating data is given below. The daily data is provided on a 2.5° x 2.5° grid and 12 pressure levels are chosen. The selected levels are 1000 hPa, 925 hPa, 850 hPa, 700 hPa, 600 hPa, 500 hPa, 400 hPa, 300 hPa, 250 hPa, 200 hPa, 150 hPa, 100 hPa and latitudes 87 .5°5 - 87.5°iV are chosen for the five-year periods 1959-1963 and 1989-1993. Monthly-mean values of each of the variables and fluxes are calculated using the flux-decompositions given in equations (2.4) and (2.5). The results presented here are for 1989-1993. Calculations for 1959-1963 for the eddy-induced circu-lation were also carried out to check the implementation of the model through comparison wi th the results of [Pfe81]. The original N C E P / N C A R Reanalysis Project daily-averaged data were down-loaded from the N O A A web-site 3 in the form of GRIB files for geopotential height 3 T h e daily data are stored at http://sgi62.wwb.noaa.gov:8080/reanlm//test.daily.prs/, with an overview of the N C E P / N C A R Reanalysis Project to be found at http://wesley. wwb. noaa. gov/reanalysis.html. 10 ( H G T ) , temperature T ( T M P ) , u wind ( U G R D ) , v wind ( V G R D ) and pressure vertical velocity UJ ( W E L ) for each of the years. GrADS1 control files were then created for the five years of each variable and the corresponding index files were created using the gribmap utility. The data was then writ ten to binary files using GrADS and imported into Matlab, where a l l further calculations were performed. 4.1. Preprocessing of diabatic heating data The monthly-mean model-derived diabatic heating fields5 for 1979-1993° were downloaded from the N O A A archive in the form of GRIB files and then processed to ultimately produce the data required for model calculations. The data are provided on 27 model cr-levels on a Gaussian grid wi th a longitudinal resolution of 1.875° and a lat i tudinal resolution of 94 Gaussian levels (an approximate reso-lution of 1.9°). The vertical levels for a = 2- and their approximate pressure-level equivalents are: 4 See http://www.iges.org/grads for a description of the software. 5 The supplied data appears to have been weighted as ^ by N C E P , although this is undocumented. 6 T h e data for January 1982 were inexplicably absent. 11 a-level nearest p-level (hPa) a-level nearest p-level (hPa) 0.9950 1000 0.3717 400 0.9820 0.3122 300 0.9640 0.2582 250 0.9420 0.2102 200 0.9155 925 0.1682 150 0.8835 0.1323 0.8454 850 0.1023 100 0.8009 0.0778 70 0.7499 0.0578 50 0.6934 700 0.0413 0.6323 0.0278 30 0.5678 0.0174 20 0.5012 500 0.0093 10 0.4352 The extraction and processing procedure is as follows: • The data for the period 1989-93 are indexed to Gr^4Z)5-compatible format by using the gribmap utility. • The fields, provided by N C A R , relevant to determining diabatic heating 7 are combined by Q = CNVHR + LRGHR + LWHR 7 It should be noted that these fields are all model-derived fields rather than calculated di-rectly from observational data. However, other repositories of diabatic heating data use a finite difference form of the thermodynamic equation to determine Q (— c p T ^ ^ ) , providing data of dubious quality, which are fundamentally unsuitable for budget calculations. 12 +SHAHR + SWHR + VDFHR, (4.1) where CNVHR — Deep convective heating rate LRGHR — Large-scale condensation heating rate LWHR = Long-wave radiative heating rate SHAHR = Shallow convective heating rate SWHR — Short-wave radiative heating rate VDFHR = Vertical diffusion heating rate • The Q field (on the er-level Gaussian grid) is calculated from the GRIB data using the above expression and the resulting values are then interpolated and extrapolated on to the same pressure levels as the atmospheric data in the previous section using 1 — D cubic splines at each grid-point. • The N C E P / N C A R data-set also provides monthly-mean surface pressure; this is used to discard al l those values of Q which are on vi r tua l pressure levels (i.e. pressure levels which are below the Earth 's surface) 8. 8 This procedure, of course, introduces an unquantifiable bias into the boundary-layer diabatic heating data. 13 • Annua l , seasonal and monthly mean values of Q are now calculated and zonally averaged. • B y experiment, it was determined that it is now reasonable to extrapolate downwards for one pressure level from the Q values retained i n the last step. This is accomplished by cubic spline extrapolation at each grid-point. • To convert the Gaussian-grid data to a lat i tudinal resolution of 2.5° com-patible wi th the data of the previous section a bi-cubic spline interpolation technique in the (f) — p plane is used. To calculate usable values at the polar 1000 hPa latitudes, symmetry extending for 2 grid-points is assumed about the poles and the interpolation is performed on this extended grid. • The resulting Q field on p-levels wi th a lati tudinal resolution of 2.5° is the sole source for al l calculations involving diabatic heating. 4.2. Ca lcu la ted quantities The following are the quantities calculated from the above data for use i n the numerical solution of the model equation and the budget calculations: 14 u steady, zonally-averaged zonal wind e steady, zonally-averaged potential temperature Q steady, zonally-averaged diabatic heating 0* V* meridional heat transport by stationary eddies meridional heat transport by zonally-symmetric transient eddies 9*'v*' meridional heat transport by zonally-asymmetric transient eddies downward heat transport by stationary eddies [ff] w downward heat transport by zonally-symmetric transient eddies 6*'UJ*' downward heat transport by zonally-asymmetric transient eddies u* V*] northward zonal momentum transport by stationary eddies [«'] [«'] northward zonal momentum transport by zonally-symmetric transient eddies u*'v*' northward zonal momentum transport by zonally-asymmetric transient eddies u* u* downward zonal momentum transport by stationary eddies [ti'] [u/] downward zonal momentum transport by zonally-symmetric transient eddies u*'u>*' downward zonal momentum transport by zonally-asymmetric transient eddies 4.3. Comparison with data of previous studies The use of reanalysis model data represents a significant departure from previous studies to investigate wave-mean flow interactions in the atmosphere. The data used by [Pfe81] are principally taken from two sources: 15 • the tabulated data of [OR71] (stemming from a large data compilation project coordinated by the Massachusetts Institute of Technology) • the energy-balance investigations of [NVDF69] (which is a compilation of results from many different studies) These sources suffer from a significant number of disadvantages: • poor upper-air coverage - the number of radiosonde measurements providing information in the upper troposphere and lower stratosphere is extremely small • poor spatial coverage - over oceanic areas, even to this day, there are vast data voids which may only partially be filled by satellite data • large inaccuracy in some fields - a particular instance of this is the vertical velocity field of [OR71], whose authors suggest that one should only consider the qualitative nature of the statistics (i.e. whether fluxes are upwards or downwards) • gross simplifications in some measurements - these are particularly evident in [NVDF69] where, as an example, boundary layer heating is taken as being between 1000 hPa and 700 hPa 16 5. Numerical solution of meridional streamfunction equa-tion The streamfunction is discretised on a 71 x 12 grid (covering latitudes 87 .5°5 — 87.5°N 9 and levels 1000 hPa - 100 hPa) wi th lati tudinal coordinate rj (= s in0) and vertical coordinate p. Thus, the gridpoint intervals are unequal in both directions: • The lat i tudinal coordinate is discretised as t]i = sin (i x 2.5° - 90°) = sin ((i - 36) x 2 .5° ) . • The vertical coordinate is discretised as pj, wi th j 1 2 3 4 5 6 7 8 9 10 11 12 Pj (in hPa) 1000 925 850 700 600 500 400 300 250 200 150 100 The various spatial derivatives of the streamfunction are discretised as fol-lows 1 0 : 9 To avoid complexities from dealing with the removable singularities of terms involving »?L = ± 9 0 o, the poles are not included in this discretisation scheme. 1 0 Where possible, central difference approximations are used; at the boundaries of the domain forward- and backward-differences are used, as appropriate. 17 d± Mm,Pi)->P(m,p.i) f o r i = i tb(m^,Pi)-^(rn-uPj) for i = 2 , . . . , 70 (Vi,Pj) 4>(Vi,P2)-i>(r)i,Pi) £ o r ?- = ]_ P2—Pi ' ^ V ' f a . P . 7 + i ) - i / ' f a » P . ; - i ) £ o r 7 — 2 11 Pi+l—Pi-1 1p(Vi,Pl2)-ll'{.'ni,Pll) Pl2~Pll , for j = 12 dry2 (Vi,Pj) J>(ri3,Pj)—4>(ri\,Pj) » ( '?2 ,Pj ) -V ' ( ' ) l ,Pj) 13 " ' ' I -22^ , for» = 1 »?2—^1 b (vi+i •PJ ) (m-PJ ) (it . P J ) (n - 1 . P J ) , for i = 2 , . . . , 70 i t + i - i t - i 2 V ' ( l71.Pj)-' / '( l70.Pj) */* (l71 .Pj ) ~V1 (l69 .Pj ) 171 -170 Z 171 -169 -f o r j _ 71 ?771-7)70 827/> dp2 V , (lj .P3)- 1 / '( lt .Pl) V'(li.P2)-V ,(lt.Pl) P3-P1 P2—Pi i ' ( l i ' P j + l ) - ^ ( l i ' P j ) ^ ' ( l i . P j ) -^ ' ( l . ' P j - l ) , for j = 1 (Vi,Pj) Pi-Pi-1 , for j = 2 , . . . , 11 P t + i - P j - i 2 y ,(ii.pi2)-v ,(ii-pii) v ,(it.pii)- i/ ,(ii.pio) , for j = 12 P12—Pll d2ip dndp (Vi,Pj) . is evaluated using the formulae for | ^ dr] (Vi,Pj) (m,Pj) and (Vi,Pj) The streamfunction problem was solved by direct inversion of the 852 x 852 matrix, assuming zero boundary conditions on both the lateral, upper and lower 18 boundaries. Other solution methods were considered, but given that the typi -cal solution time for the direct inversion is ~ 30 s i n comparison wi th the data preprocessing which takes hours or days, a fast solution scheme was not deemed necessary. Addit ionally, the difficulty of obtaining a good ini t ia l estimate of the solution renders many iterative schemes unworkable. A range of schemes from [ B B C + 9 4 ] was attempted, but convergence difficulties, accompanied by imple-mentational complications, rendered these unsuitable for the required once-off solutions of the discretised streamfunction equation. The streamfunction equation is solved: • for each month and season of the year and for the annual average, • to separately determine the eddy-induced and diabatic heating-induced cir-culations and the circulation due to their combined effects, giving a total of 57 matrix inversions 1 1 . Throughout this study, the forcing by surface wind stress is ignored (i.e. F A = 0). Whi l e estimates of the surface wind stress do exis t 1 2 , these are particularly 1 1 A relaxation scheme was chosen by [Pfe81], possibly reflecting the available computational resources at the time of that study; however, no details of the numerical solution scheme are given. 1 2 The most comprehensive appears to be those of the ECMWF, although these are only available for a short range of dates. 19 unreliable over land areas, indicating that such archives of data are, at best, appropriate only for regional, oceanic studies. The derived quantities and the results shown in the body of this work wi l l be those for annual- and seasonal-mean conditions; for readability, the corresponding plots for monthly-mean conditions wi l l be deferred to Appendix D . 20 6. Distributions of zonal wind, potential temperature and diabatic heating and eddy fluxes of heat and momentum 6.1. Zonal wind, potential temperature and diabatic heating Shown i n Figures 6.1 and 6.2 are the annual and seasonal means of zonal-mean jet-stream maxima in the mid-latitude upper troposphere and the corresponding strong meridional temperature gradient implied by the thermal wind relationship Also evident are the predominant easterlies in the trade-wind belt, extending al l the way to the upper troposphere. Given the high terrain of Antarc t ica , as well as the sparse data available, the apparent easterlies south of 70° S are rather suspect 1 3 . The zonal wind speed maxima, found at 200 mb in al l seasons and in the annual average, are aligned wi th the regions of the largest values of the meridional temperature gradient, in accordance wi th the requirements of thermal i y The data compiled by [P092] shows a strong weakening of westerly win as one approaches the Antarctic; the authors do note, however, that such statistics may stem either from a selective loss of weather balloons or the sparse synoptic network in south of 60° S. wind [u] and potential temperature 6 , respectively. Evident i n these plots are the 21 wind balance, wi th the maximum values of [it] being found during the winter season i n each hemisphere. A s one might expect from the greater temperature variabili ty in the Northern Hemisphere (which is due to less oceanic coverage and hence the reduced thermal inertia reflected by the lower-tropospheric values of [#]), the annual cycle is most pronounced i n the boreal mid-latitudes. The largest values of [it], however, are found i n the Southern Hemisphere, reflecting the largely zonal nature of the global circulation there. Shown in Figure 6.3 are the annual and seasonal means of zonal-mean diabatic heating. The dominant feature is the quasi-barotropic region of diabatic heating extending for a range of approximately 20° near the equator. The migration of the region of strong heating is seen to very clearly follow the annual cycle of the angle of insolation. It can also be seen that there is some diabatic heating i n the boundary layer extending from the equatorial region to the midlatitudes. It is not apparent whether the large values of diabatic cooling in the Antarct ic lower troposphere are an effect of the large surface albedo there (due to the ice-albedo feedback mechanism outlined by [P092]) or simply the high-terrain or poor data coverage of that region. Examinat ion of the separate contributions to diabatic heating (from equation (4.1)) shows that the release of latent heat dominates near the Equator and it is radiative cooling that dominates near the poles and at 22 23 Latitude f) Figure 6.2: Annua l and seasonal-mean distributions of [0]. Units are K and the contour interval is 10 K. 24 high altitudes. 6.2. E d d y fluxes of heat and momentum Shown i n Figures 6.4, 6.5 and 6.6 are the annual and seasonal means of stationary-eddy, transient zonally-symmetric eddy and transient zonally-asymmetric eddy components of northward momentum transport. The northward momentum flux is concentrated at 200 hPa at a l l times throughout the year, w i th the transient zonally-symmetric eddies making an insignificant contribution. The stationary-eddy component, \u* v*\, is significant only i n the Northern Hemisphere mid-latitudes and equatorial upper troposphere, wi th net northward transport i n the midlatitudes peaking in the boreal winter and a seasonally-reversing equatorward transport at the Equator. Examining the transient zonally-asymmetric compo-| , one sees that this is the dominant contribution, wi th a clear annual u*'v*' nent, cycle in the Northern Hemisphere, peaking i n wintertime, being reinforced by the aforementioned standing eddy component. The net southward transport i n the Southern Hemisphere has typically larger magnitudes than its boreal counterpart, w i th the annual cycle being less evident. In contrast wi th the results of [P092], which shows a region of strong northward momentum transport i n the Antarc-tic troposphere by the transient eddies, Figure 6.6 shows litt le transport (this is 25 26 i n agreement wi th [Pfe81]); the sparse synoptic network for the data-sources of [P092] may again be responsible for this discrepancy. Shown i n Figures 6.7, 6.8 and 6.9 are the annual and seasonal means of stationary-eddy, transient zonally-symmetric eddy and transient zonally-asymmetric eddy components of downward momentum transport. The predominant values appear to be the lower-tropospheric stationary eddy contribution, [u* uJ*], south of 65°S'. Closer consideration of the topography of this region, along wi th con-sideration of the data quality-control mechanism used, reveals that these results stem from an analysis that, at best, is heavily skewed and, at worst, could result from some poor physical parameterisation in the N C E P / N C A R reanalysis model. For this reason a contouring scheme is chosen whereby both the very large (and hence suspect) values are plotted, along wi th values in the range expected from previous studies. Ignoring these possibly misleading results, Figure 6.7 suggests that the stationary eddy component makes a significant contribution to the net downward momentum flux i n the equatorial upper troposphere around 250 hPa, wi th no particular seasonality i n evidence. Comparison wi th the results of [Pfe81] reveals a similar centre of upward flux i n the mid troposphere at 40°N during winter and spring and the same pattern of Equatorial upper-tropospheric max-ima throughout the year. No other significant similarities are apparent. The tran-27 80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Figure 6.4: Annua l and seasonal-mean distributions of [it* v*] cos 2 <fr (stationary eddies). Units are m2 s~2 and the contour interval is 5 m2 s~2. 28 Figure 6.5: Annua l and seasonal-mean distributions of [u] [v] cos 2 <f> (tran-sient, zonally-symmetric eddies). Units are m2 s~2 and the contour interval is 0.5 m2 s~2. 3 > c > a TI 4 > o Figure 6.6: Annua l and seasonal-mean distributions of u*'v*' cos 2 (b (transient, zonally-asymmetric eddies). Units are m2 s 2 and the contour interval is 5 m2 s 2 . 30 sient, zonally-asymmetric eddy component depicted i n Figure 6.9 has strong net upward flux of zonal momentum in both hemispheres, wi th peak values reached at about 350 hPa between 20° — 40° in each of the corresponding winters. E x a m -ination of the magnitude of the contributions confirms the conjecture of [SPS70] that the vertical flux of momentum by synoptic-scale eddies dwarfs that of the standing eddies. A previously unnoted feature of the downward flux of momen-at 600 hPa between u*'u*' turn is the region of relatively significant values of 60°S — 80°S, w i th a strong annual cycle peaking i n the austral winter. The origin of this region of strong downward zonal momentum flux is unclear. The transient, zonally-symmetric fluxes of downward momentum transport, shown i n Figure 6.8, are negligible through the year. Shown i n Figures 6.10, 6.11 and 6.12 are the annual and seasonal means of stationary-eddy, transient zonally-symmetric eddy and transient zonally-asymmetric eddy components of northward heat transport, respectively. Throughout the year, the standing and synoptic eddy-fluxes of heat are directed equatorward between 20°S — 20°N and poleward elsewhere. The stationary-eddy component of heat transport (Figure 6.10), 9* v* cos<{>, is negligible almost everywhere, except i n the Northern Hemisphere mid-latitude upper troposphere at 200 hPa between October and March . A maximum at 850 hPa also appears about this t ime at the 31 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Figure 6.7: Annua l and seasonal-mean distributions of [it* u*] (stationary eddies). Units are 1CT 2 m Pa s~2 and the contour intervals are 2 x 10~ 2 m Pa s~2 (light contours) and 10 x 10~ 2 m Pa s~2 (dark contours). 32 Figure 6.8: Annua l and seasonal-mean distributions of [«'] [UJ'} (transient, zonally-symmetric eddies). Units are 10~ 2 m Pa s"2 and the contour intervals are 0.5 x 10~ 2 m Pa s~2 (light contours) and 5 x 10~ 2 m Pa s~2 (dark contours). 33 c > a w O Latitude f) Figure 6.9: Annua l and seasonal-mean distributions of u*'u>*' (transient, zonally-asymmetric eddies). Units are 1 0 - 2 m Pa s~2 and the contour intervals are 5 x 10" 2 m Pa s~2. 34 same latitude. The Southern Hemisphere upper troposphere appears to have an intruding region of southward heat transport between August and November, pos-sibly an indicator of the more active lower stratosphere during the austral Spring (as remarked upon by [TW98]).The maxima in the lower troposphere here are, however, quite suspect for reasons previously discussed. The transient, zonally-asymmetric meridional heat transport (Figure 6.12), 9*'v*' coscf), exhibits the same directions of transport characteristic of the standing eddy component, but wi th greater symmetry about the equator. Strong mid-latitude regions of pole-ward transport appear at both 200 hPa and 850 hPa throughout the year, w i th a seasonal cycle of magnitude strongly in evidence; reinforcement by the station-ary eddy component is also in evidence^ along wi th the springtime intrusion of poleward heat transport in the Southern Hemisphere. The transient, zonally-symmetric component (Figure 6.11), \9'] [v] cos<j), again makes minimal contribu-tion. Shown i n Figures 6.13, 6.14 and 6.15 are the annual and seasonal means of stationary-eddy, transient zonally-symmetric eddy and transient zonally-asymmetric eddy components of downward heat transport, respectively. Shown i n Figure 6.13 is the stationary-eddy component, 9 uT , which is upward throughout most of the atmosphere, wi th regions of weak downward heat transport to be found i n 35 2> 400 3 500 S 600 £ 700 c > H > J < ^ l ^ L CO O Figure 6.10: Annual and seasonal-mean distributions of 6 v* eddies). Units are K s 1 and the contour interval is 2 K s 1. cosc/> (stationary 36 Figure 6.11: Annua l and seasonal-mean distributions of [6] [v']cos(j) (transient, zonally-symmetric eddies). Units are K s _ 1 and the contour interval is 0.1 K s~l. 37 -80 -70 -50 -40 -30 -20 -10 0 10 20 30 40 50 Figure 6.12: Annua l and seasonal-mean distributions of 8*'v*' cost/) (transient, zonally-asymmetric eddies). Units are K s~l and the contour interval is 2 iv" 38 the lower stratosphere and i n the Southern Hemisphere between 60° S — 40° S. Large values of upward heat transport are to be found near the Equatorial sur-face, w i th the latitude of the maxima possibly following the seasonal migration of the Inter-Tropical Convergence Zone; this is readily apparent only from consid-eration of monthly-mean plots. South of 60°5 , the large values of upward heat transport cannot be explained by consideration of orography alone; no anecdo-ta l evidence appears to exist which would provide a possible explanation. The transient, zonally-asymmetric component (Figure 6.15), 0*V again appears to exhibit the greatest symmetry about the Equator, along wi th more signifi-cant values than the stationary-eddy contribution and the most clearly defined annual cycle. Most apparent are the regions of broad upward heat flux between 30° —80° and 1000 hPa — 250 hPa i n both hemispheres, dominating the total heat flux. A region of weak downward heat transport appears i n the equatorial mid-troposphere, along wi th weak downward heat transport i n the lower stratosphere. A small region of significant upward heat transport appears at the surface at about 70°S between March and September; as this region extends up to 700 hPa, this phenomenon may be associated wi th the surface albedo being altered by seasonal changes i n the extent of the ice-cover near Antarct ica. Consideration of the tran-sient, zonally-symmetric component (Figure 6.14), [9'\ [u], reveals a negligible 39 contribution by this quantity. 6.3. Ca lcu la t ion of heat- and momentum-flux forcings The total northward and upward momentum fluxes are shown i n Figures 6.16 and 6.17 respectively. In both cases, it is the transient, zonally-asymmetric contri-bution, wi th occasional reinforcement by the stationary eddies, which dominates. The predominant poleward and upward transport of zonal momentum is a re-flection of the requirement of northward angular momentum transport, chiefly accomplished by synoptic-scale motions, as outlined by [P092]. The total northward and upward heat fluxes are shown i n Figures 6.18 and 6.19 respectively. Again , it is the transient, zonally-asymmetric component which is most evident, wi th contributions by the stationary-eddy component i n the high-latitude lower troposphere. The strong poleward and upward heat flux in the mid-latitude troposphere, principally by synoptic-scale systems, supports the results of [Cha47] for the global energy balance. These values, together wi th diabatic heating data, are used to calculate the fluxes of heat and momentum, H and x respectively, associated w i th the Hadley 40 C > CO o -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Latitude f> Figure 6.13: Annual and seasonal-mean distributions of 9* u>* (stationary ed-dies). Units are 1 0 - 2 Pa K s - 1 and the contour intervals are 2 x 10~2 Pa K s~l (light contours) and 30 x 10~2 Pa K (dark contours). 41 Figure 6.14: Annua l and seasonal-mean distributions of [&'] [UJ'} (transient, zonally-symmetric eddies). Units are 10~ 2 Pa K s'1 and the contour intervals are 1 x 1 0 - 2 Pa K s~l (light contours) and 10 x 10~ 2 Pa K s~x (dark contours). 42 Figure 6.15: Annual and seasonal-mean distributions of 6*'u>*' (transient, zonally-asymmetric eddies). Units are 10~2 Pa K s'1 and the contour intervals are 5 x 1(T 2 Pa K s~\ 43 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Figure 6.16: Annua l and seasonal-mean distributions of northward momentum flux. Units are m2 s~2 and the contour interval is 10 m2 s~2. 44 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Figure 6.17: Annual and seasonal-mean distributions of upward momentum flux. Units are 10~2 m Pa s~2 and the contour intervals are 4 x 1 0 - 2 m Pa s~2 (light contours) and 20 x 10~2 m Pa s~2 (dark contours). 45 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Latitude f) Figure 6.18: Annua l and seasonal-mean distributions of northward heat flux. Units are K s~l and the contour interval is 5 iv" s _ 1 . 46 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Figure 6.19: Annua l and seasonal-mean distributions of upward heat flux. Units are 10~ 2 Pa K s _ 1 and the contour intervals are 5 x 10~ 2 Pa K s~l (light contours) and 50 x 10~ 2 Pa K s~l (dark contours). 47 and Ferrel cells, along wi th their combined effect: _ R D P ^ (poo\ [Q] -"Hadloy — f7 I I K P00KR V P J c p RdPK-x (d{[9*v*]cos(j)) , d[9*uj* •"Ferrel — ~ I ^ r H --XHadley = 0 P00KR \ Rdr) dp P00KR \ \ P ) cp Rdr) dp f (d cos 2 <j>) d [u XFerrel = , ( D J.A„ cose/) \ Rcos<bdr) dp f /d([u*v*]cos2 <f>) d[u*uj* X i \ r> *u*}\ ) ) cos(f> \ R cos (bdr) dp remembering that [F\] = 0. From these quantities, the heat and momentum flux forcing functions are calculated: 4- 4- T a C • <9#Ferrel . ^XFerrel total eddy flux forcing = — h total diabatic heating forcing = drj dp d#Hadley dr) c • * • dH dX d V " Ferrel + #Hadley) . ^XFerrel total streamfunction forcing = — — h — = : —I dn dp dr) dp and the streamfunction equation is solved numerically for each of the three choices of forcing, wi th momentum and thermodynamic budgets being calculated for each 48 resulting distribution of ip. 49 7. Streamfunction ip The streamfunctions depicting the diabatic heating- and eddy- induced mean meridional circulations i n the annual mean and for all the seasons, ipn-Miiey and •0Fcrrei, are shown in Figures 7.1 and 7.2 respectively. The streamfunction corre-sponding to the combined effects, ip, is shown i n Figure 7.3. In each case, positive values represent represent clockwise motion and negative values an anticlockwise circulation (consistent wi th equations (3.7) and (3.8)). The effect of diabatic heating is seen to principally drive a direct circulation around the equator; this is the Hadley c i rcula t ion 1 4 . B o t h the latitude and mag-nitude of the circulation is seen to strongly follow the angle of insolation, wi th the regions of subtropical subsidence also being seen to vary wi th the seasons. The effective disappearance of the Hadley cell i n the summer hemisphere is strongly evident (as wi th the mass streamfunction computed by [P092]). Notable from Figure 7.3, i n spite of the apparent strong diabatic heating i n the lower tro-posphere south of 7 0 ° 5 , is the absence of any thermally-forced circulation there. A n indication that is not simply a consequence of the imposed nul l boundary condition is the appearance of large values of ipn-Ad\ey near the Equatorial surface. 1 4 A historical overview of the origins of the names for the various circulations may be found in [Lor67]. 50 The eddy fluxes of heat and momentum drive, i n each hemisphere, a relatively strong direct circulation near the Equator, a strong indirect circulation i n the midlatitudes and a weak direct circulation i n the polar latitudes. The tropical cells exhibit the same pattern of seasonal variability as those generated by diabatic heating. However, the extratropical cells do not appear to shift i n position during the course of the year, although the circulations' peak valus show strong seasonal variation in intensity, wi th small seasonal shift i n latitude. It should be noted that the peak values of the streamfunction i n the Hadley circulation (Figure 7.1) w i l l substantially dominate those of the Ferrel circulation (Figure 7.2), except i n the mid-latitudes. The assumption of [Pfe81] that the circulation shows strong Equatorial symmetry across the seasons is thus seen to be fallacious. From consideration of the individual circulations depicted in Figures 7.1 and 7.2, one expects a broad reinforcement of the direct circulation near the equa-tor, w i th opposing values outside the tropics. The picture of the net circulation provided by the combined effect of eddy-fluxes and diabatic heating, depicted i n Figure 7.3, is one of strong direct circulation near the Equator wi th a substan-t ia l ly weaker indirect circulation i n the mid-latitudes, varying both i n latitude and intensity wi th the seasons. The polar direct circulations are barely i n evi-dence, being only readily apparent i n the Southern Hemisphere during the austral 51 autumn and winter. -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Figure 7.1: Annua l and seasonal-mean distributions of ip due to diabatic heating (Hadley circulation, ipHadiey)- Units are 10 3 m Pa s~l and the contour intervals are 1 x 10 3 m Pa s~l (light contours) and 5 x 10 3 m Pa s _ 1 (dark contours). 53 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Latitude f) Figure 7.2: Annua l and seasonal-mean distributions of the eddy-induced compo-nent of ip (Ferrel circulation, ipFerrei)- Units are 10 3 m Pa s - 1 and the contour interval is 1 x 10 3 m Pa s - 1 . 54 Figure 7.3: Annua l and seasonal-mean distributions of ip. Units are 103 m Pa s~l and the contour intervals are 1 x 103 m Pa s"1 (light contours) and 5 x 103 m Pa s - 1 (dark contours). 55 8. Momentum and heat budgets The values of ip determined by numerical solution of the streamfunction equation (3.4) for the cases of eddy-driven, diabatic heating-driven and total circulation are used to evaluate (through equations (3.7) and (3.8)) the terms involving [v] and [UJ] in equations (3.5) and (3.6). 8.1. Budgets of eddy-dr iven c i rcu lat ion For the eddy-driven circulation, the momentum tendency is given by 9[u] j , / , d ([u] cos c/>) \ d [u] d ([u*v*] cos 2 <f>) d [u*u* =[V\[J- D AXA ~ M — dt 1 J V Rcos<pd<p J 1 J dp Rcos2<pd<p ' dp r S~ (v,u) terms Eddy terms and the potential temperature tendency is given by d[6] [v]d[6] d[9] d([9*v*] cos cb) d[6*u* — \UJ\ dt R d(p 1 J dp Rcos(pd(p dp (v,to) terms Eddy terms where the streamfunction equation (3.4) is solved wi th forcing of 9 H ^ n l + &XgpBl and the values of [v] and [u] are determined from Vtorrci- Zonal-momentum bud-gets for annual- and seasonal-mean conditions are shown i n Figures 8.1-8.5, wi th 56 potential temperature budgets for annual- and seasonal-mean conditions shown i n Figures 8.7-8.11. Throughout the year, the contribution to the momentum tendency by the cir-culation i n the Ferrel cell is concentrated i n the upper part of the atmosphere (200 hPa — 250 hPa), wi th the maximum migrating between 35° — 45° and strengthening wi th the annual cycle to reach a peak i n wintertime. The con-tr ibut ion to ^ by the eddy motions is very similar to this, although opposite i n sign and wi th magnitudes ~ 25% greater. Momentum tendencies due to the (v, UJ) terms are always opposite in the upper and lower troposphere, wi th the secondary centres appearing near the surface. The boundary-layer peaks i n momentum ten-dency found by [Pfe81] are not so much i n evidence i n the current investigation, possibly due to the model-based nature of the N C E P / N C A R data set and the consequent interpolation from <7-levels, along wi th a large sensitivity to the choice of contouring levels. There is, apparently, strong wave-mean flow interaction, wi th values of ~ 2 x 10~ 5 m s~l day'1 surprisingly evident i n the wintertime upper troposphere, providing deceleration of the mean zonal flow far equatorward of the jet-stream maxima. Comparison of the polarity of the momentum tendency in regions equatorward and poleward of the jet-stream maxima (from Figure 6.1) shows that the momentum tendencies produced by eddy motions strongly mimic 57 the migration of the jet stream. Consideration of the production of zonal kinetic energy, [u] ^ , by the eddy motions and the associated eddy-induced mean meridional circulation shows a net production i n the mid-latitudes and a decrease near the equator and the higher latitudes; an annual cycle is strongly evident, w i th adjacent peaks i n the wintertime upper tropical atmosphere. The regions of strong production, shown i n Figure 8.6, are at exactly the same latitudes and altitudes as the jet-stream maxima of Figure 6.1, in sharp contrast wi th the results of [Pfe81]. The contribution to the zonally-averaged potential temperature tendency by the eddy-induced meridional circulation is opposite in sign to that induced d i -rectly by the eddies and is typically smaller in magnitude. The eddy-induced tendency is for heating throughout the atmosphere poleward of 45° (although this region extends to 30° at the altitude of the jet-stream maxima), wi th the largest values appearing during the winter in each hemisphere. There is also a strong eddy-induced upper-tropospheric cooling at 25° during the boreal winter. In contrast, the eddies typically have a cooling influence throughout the tropical and subtropical atmosphere. Examining the vertical variations of ^ , one may see that the general effect of the eddies in the southern polar regions is to destabilise the atmosphere. The general structure of the potential temperature tendency i n 58 the higher latitudes is heating throughout the troposphere, wi th the secondary circulation apparently moving the eddy-induced heating maximum slightly pole-ward. Comparison of the plots of potential temperature tendency wi th Figure 6.2 shows that the distribution of ^ can account for some changes in the meridional temperature gradient across the seasons. 59 Latitude f) Figure 8 .1: Annua l average momentum budget (top three panels) and zonal kinetic energy production due to eddy-flux forcing. For the top three panels, the units are 10~ 5 m s~2 and the contour interval is 0.5 x 10~ 5 m s~2. For the bot tom panel, the units are 10~ 5 m2 s~3 and the contour interval is 5 x 10~ 5 m2 s~3. 60 Latitude f) Figure 8.2: Average winter (DJF) momentum budget (top three panels) and zonal kinetic energy production due to eddy-flux forcing. For the top three panels, the units are 1 0 - 5 m s~2 and the contour interval is 0.5 x 10" 5 m s~2. For the bot tom panel, the units are 10~ 5 m2 s~3 and the contour interval is 5 x 10~ 5 m2 s~3. 61 Figure 8.3: Average spring ( M A M ) momentum budget (top three panels) and zonal kinetic energy production due to eddy-flux forcing. For the top three panels, the units are 10" 5 m s~2 and the contour interval is 0.5 x 1 0 - 5 m s~2. For the bot tom panel, the units are 1 0 - 5 m2 s~3 and the contour interval is 5 x 10" 5 m2 s - 3 . 62 100 150 200 25C 300 400 500 600 700 feflL^^g^dU>. i I L | _J 1U 1 1 _ > k. \ K — i i y u 0 10 Latitude f) Re c . o , i r Figure 8.4: Average summer ( J JA) momentum budget (top three panels) and zonal kinetic energy production due to eddy-flux forcing. For the top three panels, the units are 10~ 5 m s~2 and the contour interval is 0.5 x 10~ 5 m s For the bot tom panel, the units are 10~ 5 m2 s~3 and the contour interval is 5 x 10~ 5 m2 s~3. 63 Lat i tude f ) Figure 8.5: Average autumn (SON) momentum budget (top three panels) and zonal kinetic energy production due to eddy-flux forcing. For the top three panels, the units are 10~ 5 m s~2 and the contour interval is 0.5 x 10~ 5 m s~2. For the bot tom panel, the units are 10~ 5 m2 s~3 and the contour interval is 5 x l O " 5 m2 s" 3 . 64 Figure 8.6: Annual - and seasonal-average zonal kinetic energy production due to eddy-flux forcing. The units are 10~ 5 m 2 s~3 and the contour interval is 5 x 10" 5 m2 s~3. 65 66 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Latitude t) Figure 8.8: Average winter (DJF) heat budget due to eddy-flux forcing. The units are 10~ 5 K s~l and the contour interval is 0.5 x 10~ 5 K s~l. 67 68 69 70 8.2. Budgets of d iabat ic heat ing-dr iven c i rculat ion For the circulation driven by diabatic heating, the momentum tendency is given where the streamfunction equation (3.4) is solved wi th forcing of — a n d the values of [v] and [u] are determined from V'Hadicy Zonal-momentum budgets for annual- and seasonal-mean conditions are shown i n Figures 8.12-8.16, wi th potential temperature budgets for annual- and seasonal-mean conditions shown i n Figures 8.17-8.21. The momentum tendency induced by the Hadley circulation (precisely the (v,u) terms outlined above) show significant values only between 30°S — 30°N; significant accelerations on the order of 2 — 4 m s _ 1 day"1 are evident i n the win-tertime between 100 hPa — 250 hPa, wi th decelerations of comparable magnitude i n the boundary layer, slightly equatorward of these peaks. There is substan-by (v,u>) terms and the potential temperature tendency is given by Diabatic Heating 71 t i a l zonal kinetic energy production in regions poleward of the equatorial upper tropospheric maxima in zonal wind tendency, wi th clear maxima evident both throughout the year and in the annual average. The location of the maxima i n momentum tendency are, as wi th the eddy-induced tendency, far equatorward of the mid-latitude jet stream maxima, thus indicating that diabatic heating cannot alone account for the variations in the zonally averaged jet core. Examinat ion of the diabatic heating, (^fj reveals a structure of strong heating i n the equatorial troposphere between 15° S—15° N, weaker heating i n the equatorial lower stratosphere between 30° S — 30°N and relatively strong heating i n the planetary boundary layer, up to about 850 hPa and between 60°S — 60°N; the remainder of the troposphere shows strong cooling, wi th maxima obtained i n the polar upper troposphere. Considerating, in addition, the effect of the induced Hadley circulation, the structure of the potential temperature tendency, ^ , w i l l give a relatively weak net heating in the equatorial mid-troposphere and rela-tively strong near-surface heating between 60° S — 60°N, w i th significant cooling throughout the remainder of the atmosphere and particularly strong cooling i n the Arc t i c and Antarct ic (with typical peak values of ~ 1.5 K day-1). 72 Figure 8.12: Annua l average momentum tendency (top panel) and zonal kinetic energy production due to diabatic heating . For the top panel, the units are 1 0 - 5 m s~2 and the contour interval is 0.5 x 10~ 5 m s~2. For the bot tom panel, the units are 10~ 5 m2 s~3 and the contour interval is 5 x 10~ 5 m2 s~3. 73 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Figure 8.13: Average winter (DJF) momentum tendency (top panel) and zonal kinetic energy production due to diabatic heating. For the top panel, the units are 10~ 5 m s~2 and the contour interval is 0.5 x 10~ 5 m s~2. For the bot tom panel, the units are 10~ 5 m2 s~3 and the contour interval is 5 x 10~ 5 m2 s~3. 74 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Figure 8.14: Average spring ( M A M ) momentum tendency (top panel) and zonal kinetic energy production due to diabatic heating. For the top panel, the units are 10~ 5 m s~2 and the contour interval is 0.5 x 1 0 - 5 m s~2. For the bot tom panel, the units are 1 0 - 5 m2 s~ 3 and the contour interval is 5 x 10~ 5 m2 s~3. 75 Figure 8.15: Average summer ( J J A ) momentum tendency (top panel) and zonal kinetic energy production due to diabatic heating. For the top panel, the units are 10~ 5 m s~2 and the contour interval is 0.5 x 10~ 5 m s~2. For the bot tom panel, the units are 10~ 5 m2 s~3 and the contour interval is 5 x 10~ 5 m2 s~3. 76 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Latitude f) Figure 8.16: Average autumn (SON) momentum tendency (top panel) and zonal kinetic energy production due to diabatic heating. For the top panel, the units are 10~ 5 m s~2 and the contour interval is 0.5 x 10~ 5 m s~2. For the bot tom panel, the units are 10~ 5 m2 s~3 and the contour interval is 5 x 1 0 - 5 m2 s~3. 77 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Figure 8.17: Annua l average heat budget due to diabatic heating. The units are 1 0 - 5 K s~l and the contour interval is 0.5 x 10~ 5 K s~l. 78 I CD 2. 5' CO -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Latitude f) Figure 8.18: Average winter ( D J F ) heat budget due to diabatic heating. The units are 10~ 5 K s~l and the contour interval is 0.5 x 10~ 5 K s - 1 . 79 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 Latitude 0 Figure 8.19: Average spring ( M A M ) heat budget due to diabatic heating. The units are 10~ 5 K and the contour interval is 0.5 x 1 0 - 5 K s~l. 80 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Figure 8.20: Average summer ( J J A ) heat budget due to diabatic heating. The units are 10~ 5 K s'1 and the contour interval is 0.5 x 10~ 5 K s - 1 . 81 82 8.3. Budgets of total circulation For the total circulation, the momentum tendency is given by d[u] r i / , 5([«]cos0)\ r ,d[u] d([u*v*}cos2(f)) d[u*u* dt \ i? cos 050 J 1 1 dp Rcos24>d(p dp > >>> * v— (V,UJ) terms Eddy terms and the potential temperature tendency is given by d\6) _ [v]d[9] d[9] d([9*v*]cos<P) d[9*u*\ | / p 0 0 \ [^Q] dt R d<f> L J dp Rcoscpdcp dp \ p ) c p > ... *— (u ,uj) terms Eddy terms Diabatic heating where the streamfunction equation (3.4) is solved wi th forcing of d { H l c I I e l ^ i H l " U a * ) _|_ 9 x ^ r r o 1 and the values of [v] and [u] are determined from ip. Zonal-momentum budgets for annual- and seasonal-mean conditions are shown i n Figures 8.22-8.26, wi th potential temperature budgets for annual- and seasonal-mean conditions shown i n Figures 8.27-8.31. Unl ike the study of [Pfe81], where data from vastly different sources were drawn together in an attempt to synthesise the combined effect of eddy- and diabatic heating-induced motions, the current investigation relies solely on data from one source; however, even if one ignores the inevitable distorting effect of 83 the various interpolation schemes, one is st i l l left wi th the drawback that there is an unquantifiable influence of the reanalysis model's various parameterisation and assimilation schemes inherent i n the results. Assuming that, at the very least, the qualitative nature of the results are correct, and making the further assumption that comparison wi th the most trustworthy of the synoptic fields wi l l reveal questionable results, some conclusions on the effect of the combined influences may be drawn. Focusing attention on the momemtum tendency, ^ , the principal regions of net positive acceleration are i n the mid-latitudes, wi th peaks i n the boundary layer near 40° and i n the sub-tropics, wi th maxima obtained i n the upper troposphere around 30°. Comparison of these tendencies wi th those resulting from the Hadley and Ferrel circulations alone suggest that the peak positive tendencies at 150 hPa are primari ly due to the eddy-induced circulation and the near-surface peaks re-sult from the diabatic heating-induced circulation. Strong cancellation both i n the momentum tendency and the zonal kinetic energy production is evident be-tween the eddy-induced and diabatic heating-induced circulations. Deceleration appears to mostly occur in the equatorial lower troposphere and more weakly near the Arc t i c surface, wi th weak upper tropospheric decelerations consistent w i th the requirements of thermal wind balance (equation (3.3)) associated w i t h 84 the calculated potential temperature tendencies discussed presently. Examin ing the potential temperature tendency of the combined motions, one sees that the general pattern throughout the year is of a weak heating i n the equatorial mid-troposphere and heating of a somewhat greater magnitude in the boundary layer, occasionally extending as far as 80°; the general tendency is for a cooling elsewhere throughout the year. The diabatic heating provides the principal contribution to ^ , wi th a broader region of heating in the equatorial mid-troposphere than would be provided by the Hadley circulation alone and the Ferrel circulation providing tendencies of slightly lower magnitude in the mid-latitude troposphere. In summary, one sees that the combined effects of the eddy- and diabatic heating-induced circulations is to increase the equator-to-pole temperature gradient i n the mid to upper troposphere throughout the year and to reduce it i n the lower troposphere between the Arc t i c and Antarct ic circles; this is compatible wi th the observed pattern of positive momentum tendencies in the upper atmosphere. There is also tendency to destabilise the lower troposphere between 1000 hPa - 700 hPa and the upper troposphere between 30° S - 30°N. 85 Lati tude f) Figure 8.22: Annua l momentum budget (top three panels) and zonal kinetic en-ergy production due to the combined effects of the eddies and diabatic heating. For the top three panels, the units are 10~ 5 m s~2 and the contour interval is 0.5 x 10" 5 m s~2. For the bottom panel, the units are 1 0 - 5 m 2 s~3 and the contour interval is 5 x 10~ 5 m 2 s~3. 86 Latitude f) Figure 8.23: Average winter (DJF) momentum budget (top three panels) and zonal kinetic energy production due to the combined effects of the eddies and dia-batic heating. For the top three panels, the units are 10~5 m s~2 and the contour interval is 0.5 x 10~5 m s~2. For the bottom panel, the units are 10~5 m2 s~3 and the contour interval is 5 x 10~5 m2 s~3. 87 Figure 8.24: Average spring ( M A M ) momentum budget (top three panels) and zonal kinetic energy production due to the combined effects of the eddies and dia-batic heating. For the top three panels, the units are 10~ 5 m s~2 and the contour interval is 0.5 x 1 0 - 5 m s~2. For the bottom panel, the units are 1 0 - 5 m2 s - 3 and the contour interval is 5 x 10~ 5 m2 s~3. 88 Latitude f) Figure 8.25: Average summer ( J J A ) momentum budget (top three panels) and zonal kinetic energy production due to the combined effects of the eddies and dia-batic heating. For the top three panels, the units are 1 0 - 5 m s~2 and the contour interval is 0.5 x 10~ 5 m s~2. For the bottom panel, the units are 10~ 5 m2 s~ 3 and the contour interval is 5 x 10~ 5 m2 s~3. 89 Lati tude f ) Figure 8.26: Average autumn (SON) momentum budget (top three panels) and zonal kinetic energy production due to the combined effects of the eddies and dia-batic heating. For the top three panels, the units are 10~ 5 m s~2 and the contour interval is 0.5 x 10" 5 m s~2. For the bottom panel, the units are 10~ 5 m2 s~3 and the contour interval is 5 x 10~ 5 m2 s~3. 90 Lati tude (•) Figure 8.27: Annua l average heat budget for the combined action of diabatic heating and eddy fluxes of heat and momentum. The units are 10~ 5 K s~l and the contour interval is 0.5 x 10~ 5 K s~l. 91 Latitude 0 Figure 8.28: Average winter (DJF) heat budget for the combined action of diabatic heating and eddy fluxes of heat and momentum. The units are 10 - 5 K s~l and the contour interval is 0.5 x 10~5 K s - 1 . 92 Figure 8.29: Average spring ( M A M ) heat budget for the combined action of dia-batic heating and eddy fluxes of heat and momentum. The units are 10~ 5 K s - 1 and the contour interval is 0.5 x 10~ 5 K s~l. 93 Figure 8.30: Average summer ( J JA) heat budget for the combined action of dia-batic heating and eddy fluxes of heat and momentum. The units are 10~ 5 K s~l and the contour interval is 0.5 x 10~ 5 K s~l. 94 Latitude f) Figure 8.31: Average autumn (SON) heat budget for the combined action of dia-batic heating and eddy fluxes of heat and momentum. The units are 10~ 5 K s~l and the contour interval is 0.5 x 10~ 5 K s~l. 95 9. Conclusions The results of this thesis suggest indicate that there is considerable wave-mean flow interaction i n the Earth's atmosphere, wi th the circulation induced by eddy motions being primari ly responsible for the maintenance and seasonal variations of the zonally-averaged jet core. In the mid-latitudes, the peak values of momentum tendency due to the Ferrel circulation wi l l lead to westerly winds ~ 45 m s - 1 in the upper troposphere and ~ 60 ms"1 near the surface over the course of a month. The Hadley circulation, i n contrast, leads to centres of significantly stronger momentum tendencies i n the equatorial and subtropical upper- and lower-troposphere. These, however, are strongly counterbalanced by momentum tendencies resulting from the eddy-induced circulation, leaving peak easterly residual tendencies of ~ 3 ms'1 i n the tropical boundary layer (which one would expect to be substantially decreased by the previously-neglected surface wind stress FA) and ~ 1 m s - 1 i n the tropical upper troposphere. The streamfunction results determined by the model presented herein may be used i n investigations of material transport i n the atmosphere (as outlined by [AHL87]) and so may give an indication of the processes by which the zonally-96 symmetric transport of ozone i n its region of production in the Equator ia l upper troposphere to its regions of destruction i n the polar lower-stratosphere takes place. Addit ional ly, the results presented in this study provide a baseline for fur-ther investigations of the effect of the E N S O phenomenon on the global circulation of the atmosphere (as 1989-1993 were non-El Nino years). The validity of these conclusions rests, of course, on the veracity of the data supplied through the N C E P / N C A R Reanalysis project and the unimportance of those terms neglected in the derivation of the model equations. 97 References [AHL87] Dav id G . Andrews, James R . Holton, and Conway B . Leovy. Middle Atmosphere Dynamics, volume 40 of International Geophysics Series. Academic Press, 1987. [AM76] D . G . Andrews and M . E . Mclntyre . Planetary waves i n horizontal and vertical shear: The generalized eliassen-palm relation and the mean zonal acceleration. Journal of the Atmospheric Sciences, 33(11):2031-2048, November 1976. [BBC+94] R . Barrett, M . Berry, T . F . Chan, J . Demmel, J . Donato, J . Dongarra, V . Eijkhout, R . Pozo, C . Romine, and H . V a n der Vorst. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd Edition. S I A M , Philadelphia, P A , 1994. [Cha47] J . G . Charney. The dynamics of long waves i n a baroclinic westerly current. Journal of Meteorology, 4(5):135-162, October 1947. [KKK+96] E . Kalnay, M Kanamitsu , R . Kist ler , W . Collins, D . Deaven, L . Gandin, M . Iredell, S. Saha, G . Whi te , J . Woollen, Y . Zhu, A . Leetmaa, R. Reynolds, M . Chell iah, W . Ebisuzaki , W . Higgins, 98 J . Janowiak, K . C . M o , C . Ropelewski, J . Wang, Roy Jenne, and Den-nis Joseph. The ncep/ncar 40-year reanalysis project. Bulletin of the American Meteorological Society, 77(3):437-471, M a r c h 1996. [Kuo56] H . - L . K u o . Forced and free meridional circulations in the atmosphere. Journal of Meteorology, 13:561-568, December 1956. [Lor67] Edward N . Lorenz. The Nature and Theory of the General Circulation of the Atmosphere. Wor ld Meteorolgical Organization, 1967. [NVDF69] R . E . Newell, D . G . Vincent, T . G . Dopplick, and D . Ferruzza. The energy balance of the global atmosphere. In G . A . Corby, editor, The Global Circulation of the Atmosphere, pages 42-90. Royal Meteorolog-ical Society, August 1969. [OR71] Abraham H . Oort and Eugene M . Rasmusson. Atmospheric circulation statistics. Technical report, Nat ional Oceanographic and Atmospheric Administrat ion, 1971. [Pfe81] Richard L . Pfeffer. Wave-mean flow interactions in the atmosphere. Journal of the Atmospheric Sciences, 38:1340-1359, Ju ly 1981. 99 [P092] Jose P. Peixoto and Abraham H . Oort. Physics of Climate. Amer ican Institute of Physics, 1992. [SPS70] V . P. Starr, J . P. Peixoto, and J . E . Sims. A method for the study of the zonal kinetic energy balance in the atmosphere. Pure and Applied Geophysics, 80:346-357, 1970. [TW98] Dav id W . J . Thompson and John M . Wallace. Structure of the arctic and antarctic oscillations. Journal of Climate (submitted), October 1998. 100 A. Table of symbols Symbol Units Description Cp j kq K atmospheric specific heat at constant pressure (= 1004 J kg~l K'1) f 1 Coriolis parameter (= 2f2 sin (fi) 9 m x* acceleration due to gravity (— 9.81 m s~2 at sea level) P Pa pressure Poo Pa reference pressure (1000 hPa) t s t ime u m .s zonal velocity V m .s meridional velocity z m geometric height above the Earth 's surface Symbol Units Description FX m «2 zonal forcing Fcfi m «2 meridional forcing H kq s heat flux forcing Q k diabatic heating R m radius of the Ea r th Rd j kq K gas constant for dry air (= 287 J kg'1 K"1) Sp K Pa static stability (= T K temperature Symbol Units Description (none) (= sin (fi) 9 K potential temperature K (none) Cp A rad longitude P kg m.3 density 4> rad latitude X m *3 momentum flux forcing m Pa ,s streamfunction Pa s pressure vertical velocity n rad fi rotation rate of the Ea r th (= 7.292 x 1 0- 5 rad s'1) 101 B. Derivation of model equations B . l . Govern ing equations In the (A, c/>,p, t) system, the appropriately scaled equations of horizontal motion are where du tanc/> gdz . . -TT = —z-uv + fv-— T^T+FX, (B. l) dt R R cos (pd\ civ tanc/> 2 f 9dz m , * = -^ru - f u ~ m + F * > ( R 2 ) It ~ dt + URcos<bd\ +VRd<b +Udp' { ' } UJ dp The equation of continuity is R cos (bd\ R cos (bdcb dp (B.4) du djvcoscp) duj 102 The hydrostatic equation is dp d z - -pg- (B-6) The ideal-gas law and the expression for potential temperature are P = pRdT, (B.7) 0 = T ^ - J , (B.8) where p00 is a reference level (~ 1000 hPa) and K = ^ ~ 0.286. The first law of thermodynamics is "it' Q' (a9) where cp ~ 1004 J kg'1 K"1 and Q is the net heating rate per unit mass. B.2. Derivation of thermal wind equation The thermal wind equation for u may be found by scale analysis of equation (B.2): tan 6 9 , gdz 0 a ! —ir" - } u - m -103 Differentiating w.r.t. p gives 2tan</> \ du _ d (gdz f + ~~R~U) dp ~ ~dp \Rd4 Rd(j) 9 d2z Rdpdtfi Changing the order of differentiation and using the hydrostatic equation (B.6) together wi th the idea gas law (B.7) gives which gives 2 t a n 0 \ du _ 1 9 ( p ) f + R U) dp R d(f> 1 d(-£r) Rp2 d<f> v g(a RRdP2 dcj) P ( _ r - 2 ) ^ RRdP2 v ' d4> p dT RRdp2T2 d(f)' / + ^ ^ = ^ . (B.10) J R J dp pRdtp v ' 104 Using the definition of potential temperature (B.8) gives 2tanc/> \ du Rd 1 d \ Poo) R J dp p R d<f> RaP^^dO P PooRd(P RdpK~l 1 d6 Poo R d(f> which finally results i n the alternative thermal wind equation . 2 t a n 0 \ 0 u RdPK~l 1 d9 / + - r B s = i r w ( B - n ) B.3. Derivation of zonally-averaged equations of motion B.3.1. Derivation of zonally-averaged mass-continuity equation The mass continuity equation (B.5) may be zonally-averaged to give 0 ( M c o B 0 + g M = a ( B - 1 2 ) R cos (bd(p dp Mul t ip ly ing the original mass continuity equation (B.5) by some quantity A 105 and zonally-averaging gives [A] du d (v cos <fi) dui R cos (pdX R cos <fid<fi dp J + A* du* ^ du ^d(vcos(f)) ^du R cos 4>d\ R cos <j)d(j) dp J du* R cos (f)d\ + , d (v* cos <f>) R cos <pd<p + A* dp 0 0, which, upon using equation (B.5), becomes A* du* Rcos 4>dX + ,d(v* cos0) R cos 4>d(j) + .du* dp (B.13) W i t h the choice A = u this gives the relationship du* u R cos (fidX + u . d (v* cos cp) R cos (f)d<j) + u .du* dp = 0, which yields the useful identity u . d (v* cos (f>) R cos <pd(j) + u .du* dp = 0. (B.14) 106 B.3.2. Der iva t ion of the zonal ly-averaged ideal gas law and potent ia l temperature equations The ideal gas law equation (B.7) may be decomposed into p = Rd [p] [T]+Rd[p*T*]. Alternatively, one may write RdT = P which may be zonally-averaged to Rd[T]=p (B.15) The expression for potential temperature (B.8) may be zonally-averaged to give M = PI (B.16) and Q* _ rp* (POO \ P 107 B.3.3. Derivation of the zonally-averaged zonal momentum equation Starting from equation ( B . l ) one may express the evolution of zonal velocity i n the Euler ian form du du du du tanc6 gdz dt R cos (pdX Rdq> dp R R cos <pdX + FX Taking the zonal average of this equation, one arrives at d\u) dt du u Rcos <f)dX\ du Rd<b UJ du dp du '~Rd4> taxi 6 UJ du dp gdz R cos (bdX + uv] + f[v] + [Fx]. Using identity (2.1) one may write d [u] [v] d [u] 1 dt R d<f> R <du* UJ du dp tanc/> , r y , tanc/> , * 1 _ L / r i . m which may be rewritten as d [u = v f + ^ l u ld[u = H / R L"J R d<p cos c/>^ — sin tf) [u] Rcoscb 1 R 1 (90 a; c5u dp du to— dp + ^ K » " ] + [FA] 108 = M l / -5 ([it] cos 0) cos 090 tdu* 50 a; 5M 5p + ! ^ [ „ V ] + | f t ] Now, cu 5it 5p = - OJ - OJ 5 [it dp 0[u] 5p 0Ju] 5p 5u* dp djvTu\ dp 5 [it*a;*; dp + 5cu* it dp u*d (v* cos0) cos 050 from equation (B.14). Substituting this into the expression for the evolution of zonal velocity yields 5 [it = [v] / 5 ([it] cos 0) R cos 050 tanV*] + [F A ] R (5it* 50 J a; 5 [u] 5 [u*u;* 5p 5p it*5 (v* cos 0)' i? COS 050 /2 M ( 7 -5([w]cos0)\ 1 R cos 050 / i?cos0 n* cos0-5j/ 50 — cu 5 [it] 5 [it*a>* 5p 5p li*5 (v* COS0) i? COS 050 sin 0 i?cos0 = M { / -- M 5 ([li] COS 0) R COS 050 / i?COS0 5 [u] 5 [u*cu*] , 5u* it*5 (i;* cos 0) . , „ . D cos 0-^-r H —^-7 sm 0it v = M if-dp dp 5 ([it] cos 0) COS 050 / i?COS0 50 5 (li*U* COS 0) 50 50 — sin 0u* v* 109 M l-—- + [Fx] = [v] / dp dp d ([it] cos c/>) 1 R cos 05c/» y R cos2 0 3 ( « * w * cos 0 ) , cos 0 sm 0 cos <pu v dq> r ,d[u] d [u*u , r ^ . <9p dp d([u]cos0)\ <9([u"V]cos20) <9[u*u;* cos 0 3 0 / R cos2 0r90 — UJ dp dp + [Fx B.3.4. Derivation of the zonally-averaged thermodynamic equation Using the expression p6dt ^ for the first law of thermodynamics (B.9), one may write dt \ p J cp which, in the Eulerian framework, may be rewritten as dt RcoscbdX Rdcf) dp \ P J cp 110 Taking the zonal average of this equation gives d[6] Ot 86 u = -u u Rcos (pOXj m R cos cj)dX . 06* R cos <pOX 1 R u* 06' ,Q(j)_ 06 U Op + R cos (pOXj R 1_ R 06_ '0<p UJ 0& Op Poo\K [Q] P }04> + 06 , OJ— L dP pwYiQ] p + PooY [Q] P Now, 06* u R cos (pOX UJ 0& oP\ 06* u u u R cos (pOX 06* R cos (pOX 06* R cos (pOX Ou* R cos (pOX 0 [u*8*] M [OJ] [OJ] 6* 9_m Op UJ .06* Op 0[6] _ 0 [UJ*6* Op 0[8] Op 0 [UJ*6* + R cos (pOX [OJ] Op Op 0 (v* cos (p) R cos (pO(p 0 [6] 0 [UJ*6*] 6* Ou*' Op Op Op , 0 (v* cos (p) R cos <pO(p - u 0[6] Op 6 ^0(v*cos(p) R cos (pO(p 0 [u*6* Op using equation (B.13). Returning to the zonally-averaged thermodynamic equation and substituting 111 dt R [Vd(b\ R d(b R d<p d [u*6*] - M _1 R • M d[9] dp ' d<b\ ^d(v* cos <f>) R cos (bd(b d [u*8* - CO + Poo dp V p P W 1 <9p R cos 0 d[6) _ dp V* C O S (p dp „ 9 (v* cos 4>) + Poo p K[Q] R cos (pd<p de d [u*e* d(f> e* dp d(v* cos <p) R cos <f)d(j) + Poo p K [Q] Final ly, one arrives at d[6] _ [v}d[e] 9[g] d([v*8*}cos(f)) d[w*e*} | (PooY [Q] dt R d(p dp R cos (pd<p dp \ p ) cp B.3.5. Der iva t ion of the zonal ly-averaged thermal w ind equat ion Taking the zonal average of the thermal wind equation (B.10) gives ,d[u] 2 tan (p dp R du u „d\u\ 2tanc/> r ,d\u] 2tanc6 dp R dp R dp du* u 2 [u] tan (p\ d [u] tan <p / + R ) ~dp~ + ^T dp d(u*f dp RddT pR d<p RddT pR d(p RddT pR dcp' 112 If the variance of u is neglected, one arrives at + 2 [u] tan <p\ d [u] _ Rd dT R J dp pRd(f> In terms of potential temperature, this relationship is 2 M t a n 0 \ 5 M _ ^ - 1 l 5 [ 0 ] J + R J dp p^ R d<P • ^ ' ; B.4. Govern ing equations for zonal ly-averaged mot ion To summarise the equations derived above: • The zonally-averaged zonal momentum equation is djv] = ( d([u] cos 0 ) \ o> ] d k ^ * ] g ( K ^ ] c o s 2 0 ) c% U V i?cos<pd(j) J 1 1 dp dp R c o s 2 ^ 1 J" (B.18) • The zonally-averaged thermodynamic equation is d[0] [v]d[6] d[0] d([6*v*]cos<i>) d[9*u*\ ( p m \ [Q] - \U>\ -r— — ~ h \ K dt R d(fi dp R cos <j)d(j) dp \ p ) cp (B.19) 113 The zonally-averaged mass continuity equation is a_M^ + m = 0 . (B.20) R cos (pdtp dp • The thermal-wind equation is or jd [u] = R ^ - 1 Id [9] dp P00K R d(f> where / = ( / + M^±y 114 C. Derivation of diagnostic streamfunction equation for mean meridional circulation Firs t , it is convenient to make the following change of coordinates r] — sin cp, ( C - l ) so d d dn cos <pd(p The zonally-averaged mass continuity equation (B.20) may now be wri t ten as °A R ) , 9 M = 0 dr\ dp This allows the definition of a streamfunction ip: cos (p dp Rdrj Rewri t ing equations (B.18) and (B.19) in terms of derivates w.r.t. n and p 115 gives d[u] ( d([u] cos<p)\ 3 H 9 [u*u*] d([u*v*}cos2(f>) , , dt V Rdr J 1 ' dp dp Rcos<bdn and d [fl] = [v] cos09[fl] 0 [fl] d([9*v*]cos<j>) d[6*u*] (p00\K [Q] dt R drj M dp Rdr] dp \ p J cp ' 1 ' } Changing meridional coordinate and differentiating equation (B .21) w.r.t. t allows one to write a ( ( / + ^ [ „ ] ) M ) _ R d P . - i l c o s ( f ) d 2 [0] dt Poo & dtdv Combining the derivative w.r.t p, making a change i n the order of differentiation and expanding the derivative w.r.t. t gives K/g) _ R d P ^ l ^ d { f ) dp pfjo R dn 116 into which expressions (C.3) and (C.4) may be substituted: 9 ( / ( M (/ - ^) - M f -ag 3 - £ ( Sg^ 1 + I f i l ) ) dp r, ^.coz4d( Hcos^ afe] Ud\e] d(\9*v*] cos <p) ayr**] [Q]\ _ RdP 1 C O b V > C ' \ k d r , M d p fl0»j S p ^ ^ P ) °P ) Poo R dv Using the streamfunction definition (C.2) gives rlfff 1 ^ (f dCMcos^A . l dip d[u) d\u'u'\ d([u*v*}cos->4>) , A \ u I J I cos</> dp \J Rdr, J ^ Rdn dp dp Rcoscpdn ' i *1 ) I C O S c65p Rdp lU \ Rdp dr, ^ Rdr, dp Rdr, dp \ P J cP J Poo R drj Moving terms involving ib to the left-hand side and the remaining terms to the right-hand side gives d(f(-^ ^ ( f - ^ r ^ ) + i f ¥ ) ) PrA-i 1 ^ f-^fl + ^ f1) y yoscp dp y Rdr, J fl dn dp J J ti-dP \ R°P "V R dr) dp J cos <pdp p0QK R dn = ^ 9 ( - f l e ^ g a - a 9 a + (?)'tf) RPOOK dr] di J— (-dlu*"*] _ a(V*']«» 2») , tF]X\ u I c o s 0 I dp Rcos<t>dn ' i1 M J J dp 117 which may be simplified to d ( f ( - ^ f ( f - ^ ^ ) + ^ ¥ ¥ ) ) i d ( ^ ^ - ^ f ^ ) V •> V c o s <f> dp \ J Ran J R dn dp J I ttdP *• \ ^"P R an dp J cos fidp p 0 0 K R dn RdPK-X °{~ R9n ~ dp + { y ) K cp) R p 0 0 K dn d ( ^ - J ^ + d{[7*]7f*} ~ [FX])) l c o s < p \ op Rcoscpon L / / ~*~ dp Defining H = RdPK-w d([e*v*]cas<i>) d[e*u*\ | /p00y[g] POQR \ Rdv dp \ p J cp f (d([u*v*]cos*<j>) d[u*u*} + — — i ^ J cos0 \ R cos cfidn dp allows one to write V c o s e i \ J Rdn J °P J \ p 71J ^dP t y dn dp dp dn J Oil OX cos <pdp R cos <pdp Poo R2 dn dn dp This may be rewritten, taking C = ^ ( / - ^ ^ ) , [SP] = - ^ f j and 118 a (eg) { a(/fg) , i ^ - a ( f g - f t Q _ SH + 3X cos (pdp R cos (pdp PQ0 dn dn dp V dPJ , ° \ J dpdn) | Dp y dn dp J ^ ° \p^ dp dn) = dH | d\ dp R cos <pdp PQ0 dn dn dn dp a{c%) a ( / f » ) D j , . a ( f § * ) a(\sr]%) a H d x _ l _ _j_ = = _ l _ — _ cos (pdp R cos (pdp Poo dn dn dn dp Taking the left-hand side of this equation, expanding the derivative w.r.t. p i n the second term and using the zonally-averaged thermal wind relationship (B.17) gives B { c %) + a ( ^ ' f ) ft.i a» + / aMay cos (pdp dp pQ0K R2 dn Rcos(p dp dpdn , D P K S { ^ ) + d 9 ( I S P ] % ) P00K dn s dn cos </><9p I i? 2 p 0 o K drydp <9p dn \ dn R cos 0 dp dpoVy P 0 0 K dn dn d(Ct) + ( Dd[Sp] + d{%w)d[0]\ dip + f d[u] cos (pdp \ dn dp dn J dn Rcos(p dp dpdn 119 P00K dn dn d {C%) , / d[Sp] , , . / - ^ [ ^ dip^j_d[u] d2ib cos (bdp \ dn p00K dn ) dn Rcos(b dp dpdn POQK dn p dn2 dn dn 9{Ct) { , l)DP*-ld[6\di> | / d[u] d2j; | DpKd{^%) { D [ s ] d 2 i P cos 4>dp p00K dn dn Rcos(b dp dpdn P00K dn p dn2 9{C%) l , X)D PK d[9\dj> | Dpn d[0] d2*b | D P K d { ^ % ) | D [ s ] d 2 * P cos (bdp p p00K dn dn p00K dn dpdn p00K dn p dn2 d(Ct) , DP* d[6] /K-ldr/f | d2ip\ | DpKd(^%) [ D ] s ] d 2 i P cos (bdp ' ^ P Q O ^ dn \ p dn ' dpdn J ' p 0 0 /s d^ dr; 9 \ D P K 9 W (llzl?t + ^L\ i DPKd[6] d2ip cos (bdp p00K dn \ p dn dpdn J p00K dn dndp i D P K e r V W + en, P00K dn2 dp p dn d {Ct) ] JJ PK d[8] U - l d ^ \ | 2DPKd[e\ d2ib | DPKd2[9}dib | D [ S ] 9 2 ^ cos (bdp p00K, dn \ p dn J p00K dn dndp p00K dn2 dp p dn2 This finally gives _ . d2tb DpK d [8] d2ip Dpn d[9) (K-\ dip\ Dpn d2 [9] dib 9 ip%) dH dX dn2 p00rz dn dndp pQ0K dn \ p dn J p0QK dn2 dp cos <pdp dn dp 120 D. Monthly plots 121 D .1. D is t r ibu t ions of zonal w ind , potent ia l temperature and d iabat ic heat ing 122 123 124 LaWude f) L a W u d e O Figure D.3: January-June distributions of [0]. Units are K and the contour interval is 10 K. 125 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 5 0 60 70 Lati tude (') Lati tude t ) Figure D.4: July-December distributions of [9]. Units are K and the contour interval is 10 K. 126 127 128 D.2. Eddy fluxes of heat and momentum D.2.1. Stationary eddies 129 130 131 Figure D.9: January-June distributions of [U*OJ*] for stationary eddies. Units are 10~ 2 m Pa s~2 and the contour intervals are 2 x 10~ 2 m Pa s~2 (light contours) and 10 x 10~ 2 m Pa s~2 (dark contours). 132 contours) and 10 x 10 2 m Pa s 2 (dark contours). 133 t W I: I i 1 1 <c \ V . -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Figure D . l l : January-June distributions of [v* are K s~l and the contour interval is 2 K s~l Jl ; (Tf f \ f l . ( , ) M ^ l(?rm#fl -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Lat i tude ( ) *] cos <j) for stationary eddies. Units 134 Figure D.12: July-December distributions Units are K s"1 and the contour interval is F ii 75 V 1 3 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Latitude ( ) f [v*6*] cos <ft for stationary eddies. K s~\ 135 100 150 200 250 •S 3 0 0 a. f 400 1 500 3 °- 600 700 925 1000 > TJ TJ -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Latitude () -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 Latitude {) Figure D.13: January-June distributions of [u*8*] for stationary eddies. Uni ts are 10~ 2 Pa K s'1 and the contour intervals are 2 x 10~ 2 Pa K s~x (light contours) and 30 x 10~ 2 Pa K s"1 (dark contours). 136 m T l -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 Latitude () -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 Latitude () Figure D.14: July-December distributions of [uj*9*] for stationary eddies. Units are 10~ 2 Pa K s - 1 and the contour intervals are 2 x 10~ 2 Pa K s~l (light contours) and 30 x 10~ 2 Pa K s~l (dark contours). 137 D.2.2. Transient, zonal ly-symmetr ic eddies 138 l O u I If T O . J > > TJ > -< •70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 Latitude 0 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Latitude f) Figure D.15: January-June distributions of [u*v*] cos 2 0 for transient, zonally-symmetric eddies. Units are m2 s~2 and the contour interval is 0.5 m2 s~2. 139 140 (light contours) and 5 x 10 2 m Pa s 2 (dark contours). 141 Figure D.18: July-December distributions of [u*u*] for transient, zonally-symmetric eddies. Units are 10~ 2 m Pa s~2 and the contour intervals are 0.5 x 10~ 2 m Pa s~2 (light contours) and 5 x 10~ 2 m Pa s~2 (dark contours). 142 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Figure D.19: January-June distributions of [v*6*] cos (ft for transient, zonally-symmetric eddies. Units are K s~l and the contour interval is 0.1 K s~l. 143 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Latitude f) Latitude (') Figure D.20: July-December distributions of [v*9*]cos(p for transient, zonally-symmetric eddies. Units are K s~l and the contour interval is 0.1 K s - 1 . 144 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Figure D.21: January-June distributions of [ui*9*\ for transient, zonally-symmetric eddies. Units are 10" 2 Pa K s"1 and the contour intervals are 1 x 10~2 Pa K s~x (light contours) and 10 x 1 0 - 2 Pa K s"1 (dark contours). 145 !S 5 0 0 C/3 m -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 Latttude ( ) -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 Lati tude ( ) Figure D.22: July-December distributions of [UJ*6*] for transient, zonally-symmetric eddies. Units are 10~ 2 Pa K s _ 1 and the contour intervals are 1 x 10" 2 Pa K s~l (light contours) and 10 x 10" 2 Pa K s~l (dark contours). 146 D.2.3 . Transient, zonal ly-asymmetric eddies 147 148 149 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Lati tude f ) Lat i tude (") Figure D.25: January-June distributions of [u*u>*] for transient, zonally-asymmetric eddies. Units are 10~2 m Pa s~2 and the contour intervals are 5 x 1CT2 m Pa s~2. 150 151 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 -80 -70 -60 -50 -40 -30 -20 -10 0 10 2 0 30 40 50 60 70 80 Latitude P) Latitude () Figure D.27: January-June distributions of [v*9*] cos <f> for transient, zonally-asymmetric eddies. Units are K s'1 and the contour interval is 2 K s _ 1 . 152 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Figure D.28: July-December distributions of [ asymmetric eddies. Units are K s~l and the co: -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Latitude t) v*6*]cos<fr for transient, zonally-ntour interval is 2 K s _ 1 . 153 154 u 5 -a- , , , i V B ^ . A C Figure D.30: July-December distributions of [u;*0*] for transient, zonally-asymmetric eddies. Units are 10~ 2 P a X s - 1 and the contour intervals are 5 x 10" 2 Pa K s~l. 155 D.3. E d d y momentum and heat fluxes 156 Wi m i i i A | i i l l © .J. 8T m oo > c 2 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Latitude () -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 Latitude (') Figure D.31: January-June distributions of northward momentum flux. Units are m2 s~2 and the contour interval is 10 m2 s~2. 157 Figure D.32: July-December distributions of northward momentum flux. Units are m2 s~2 and the contour interval is 10 m2 s~2. 158 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 L a M u d e (') Latitude (•) Figure D.33: January-June distributions of upward momentum flux. Uni ts are 10" 2 m Pa s'2 and the contour intervals are 4 x 10" 2 m Pa s~2 (light contours) and 20 x 10" 2 m Pa s~2 (dark contours). 159 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Figure D.34: July-December distributions of upward momentum flux. Units are 1 0 - 2 m Pa s~2 and the contour intervals are 4 x 10~ 2 m Pa s~2 (light contours) and 20 x 10~ 2 m Pa s~2 (dark contours). 160 161 162 Figure D.37: January-June distributions of upward heat flux. Units are 10~ 2 Pa K s~l and the contour intervals are 5 x 10~ 2 Pa K s~l (light contours) and 50 x 10~ 2 Pa K s~~l (dark contours). 163 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Figure D.38: July-December distributions of upward heat flux. Units are 10~ 2 Pa K s~l and the contour intervals are 5 x 10~ 2 Pa K (light contours) and 50 x 10~ 2 Pa K s~l (dark contours). 164 D.4. Streamfunction tp 165 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Latitude f) Latitude () Figure D.39: January-June distributions of ip due to diabatic heating (Hadley cir-culation). Units are 103 m Pa s"1 and the contour intervals are 1 x 103 m Pa s - 1 (light contours) and 5 x 103 m Pa s~l (dark contours). 166 k l j :. " — * £ v , . r » • • » » • » r i "gra^j i. J J u , •80 -70 -60 -60 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Figure D.40: July-December distributions of ip due to diabatic heating (Hadley circulation). Units are 103 m Pa s - 1 and the contour intervals are 1 x 103 m Pa s - 1 (light contours) and 5 x 103 ra Pa s~l (dark contours). 167 > •70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 Latitude () -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Latitude () Figure D.41: January-June distributions of the eddy-induced component of -0 (Ferrel circulation). Units are 10 3 m Pa s - 1 and the contour interval is 1 x 10 3 m Pa s~l. 168 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Latitude C) Latitude () Figure D.42: July-December distributions of the eddy-induced component of ip (Ferrel circulation). Units are 10 3 m Pa and the contour interval is 1 x 10 3 m Pa s _ 1 . 169 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Figure D.43: January-June distributions of ip. Units are 10 3 m Pa s - 1 and the contour intervals are 1 x 10 3 m Pa s _ 1 (light contours) and 5 x 10 3 m Pa s"1 (dark contours). 170 Figure D.44: July-December distributions of ib. Units are 103 m Pa s~l and the contour intervals are 1 x 103 m Pa s - 1 (light contours) and 5 x 103 m Pa s~l (dark contours). 171 D .5. Zonal K i n e t i c Energy P roduc t ion 172 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Latitude () Latitude () Figure D.45: January-June zonal kinetic energy production due to eddy-flux forc-ing. The units are 10~ 5 ra2 s~3 and the contour interval is 5 x 10~ 5 m2 s~3. 173 174
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Atmospheric wave-mean flow interactions Quinn, Declan 2000
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Title | Atmospheric wave-mean flow interactions |
Creator |
Quinn, Declan |
Date Issued | 2000 |
Description | The interactions between the zonally-averaged flow and the various eddy circulations in the atmosphere are studied using NCEP/NCAR Reanalysis data. The contributions to zonal- and potential temperature-tendency and zonal kinetic energy production due to both the Ferrel and Hadley circulations are determined. Inferences are then made regarding the processes which affect the strong cores of the jet stream and the forms of zonallyaveraged models of the atmosphere that are appropriate for studies of this type. It will be shown that there is considerable wave-mean flow interaction in the atmosphere, with the circulation induced by eddy motions being primarily responsible for the maintainance and seasonal variations of the zonally-averaged jet core. This study also provides a repository of derived atmospheric quantities for the years 1989-1993. |
Extent | 42158300 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-07-13 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080037 |
URI | http://hdl.handle.net/2429/10770 |
Degree |
Master of Science - MSc |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2000-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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