Atmospheric Wave-Mean Flow Interactions by Declan Q u i n n B . S c , T h e National University of Ireland, 1996 M . S c , T h e National University of Ireland, 1997 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF THE REQUIREMENTS FOR T H E DEGREE OF M A S T E R OF SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES (Department of Mathematics) We accept this thesis as conforming s$o the required standard T H E UNIVERSITY OF BRITISH COLUMBIA August 2000 © Declan Quinn, 2000 In presenting degree freely at this the available copying of department publication of in partial fulfilment of the University of British Columbia, I agree for this or thesis reference thesis by this for his thesis and scholarly or for her Department DE-6 (2/88) Columbia I further purposes gain that agree may be It is representatives. financial permission. T h e U n i v e r s i t y o f British Vancouver, Canada study. requirements shall not that the Library permission granted by understood be for allowed an advanced shall for the that without head make it extensive of my copying or my written Abstract The interactions between the zonally-averagedflowand 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 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. ii CONTENTS Abstract L i s t of Figures i i vi 1 Introduction 1 2 Notation 3 3 Model 6 4 Data 10 4.1 Preprocessing of diabatic heating data 11 4.2 Calculated quantities 14 4.3 Comparison w i t h data of previous studies 15 5 N u m e r i c a l solution of meridional streamfunction equation 6 Distributions of zonal wind, potential temperature and diabatic heating 17 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 C a l c u l a t i o n of heat- and momentum-flux forcings 43 7 Streamfunction ip 49 8 M o m e n t u m and heat budgets 55 8.1 Budgets of eddy-driven circulation . 55 8.2 Budgets of diabatic heating-driven circulation 70 iii 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 w i n d 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 potent i a l temperature equations B.4 C D 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 w i n d equation . Ill Governing equations for zonally-averaged motion 112 Derivation of diagnostic streamfunction equation for mean meridional circulation 114 M o n t h l y plots 120 D.l Distributions of zonal w i n d , potential temperature and diabatic heating 121 iv D.2 E d d y 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 E d d y momentum and heat fluxes 155 D.4 Streamfunction ip 164 D.5 Zonal K i n e t i c Energy P r o d u c t i o n 171 v List of Figures 6.1 A n n u a l and seasonal-mean distributions of [it]. U n i t s are m s and the contour interval is 5 m s" 23 A n n u a l and seasonal-mean distributions of [0]. U n i t s are K and the contour interval is 10 K 24 - 1 6.2 6.3 A n n u a l and seasonal-mean distributions of are K s 6.4 1 _ 1 and the contour interval is 1 x 1 0 - 5 K . A n n u a l and seasonal-mean distributions of [Tt* v*} cos <f) (stationary eddies). Units are m s~ and the contour interval is 5 m s 28 2 2 - 2 A n n u a l and seasonal-mean distributions of [u] [v] cos <f> (transient, zonally-symmetric eddies). U n i t s are m s~ and the contour interval is 0.5 m s~ 29 2 2 2 6.6 2 2 cos (j) A n n u a l and seasonal-mean distributions of i t ' v 2 L J (transient, zonally-asymmetric eddies). U n i t s are m s~ and the contour interval is 5 m s~ A n n u a l and seasonal-mean distributions of [Tt* u*] (stationary eddies). U n i t s are 1 0 m Pa s~ and the contour intervals are 2 x 10~ m Pa s~ (light contours) and 10 x 1 0 m Pa s~ (dark contours) 2 2 6.7 2 32 2 A n n u a l and seasonal-mean distributions of [u] [u/] (transient, zonally-symmetric eddies). U n i t s are 10~ m Pa s~ and the contour intervals are 0.5 x 10~ m Pa s~ (light contours) and 5 x 10~ m Pa s~ (dark contours). . . . 33 2 2 2 6.9 30 2 2 - 2 2 2 - 2 6.8 26 2 2 6.5 Units Cp 2 2 2 A n n u a l and seasonal-mean distributions of it* o r (tran- sient, zonally-asymmetric eddies). U n i t s are 10~ m Pa s~ and the contour intervals are 5 x 10~ m Pa s~ . . . . 34 2 2 2 2 6.10 A n n u a l and seasonal-mean distributions of 6 v* COS(f> (stationary eddies). U n i t s are K s interval is 2 i f s - 1 vi 1 and the contour 36 6.11 A n n u a l and seasonal-mean distributions of [8'] [v] cosc/> (transient, zonally-symmetric eddies). U n i t s are K s and the contour interval is 0.1 K s~ _ 1 37 l 6.12 A n n u a l and seasonal-mean distributions of 8*'v*'COS0 (transient, zonally-asymmetric eddies). U n i t s are K and the contour interval is 2 K s s~ l 38 _ 1 6.13 A n n u a l and seasonal-mean distributions of 8 tionary eddies). Units are 1 0 Pa K s - 2 - 1 UJ* (sta- and the con- tour intervals are 2 x 10~ Pa K s~ (light contours) and 30 x 10~ Pa K s~ (dark contours). . 2 2 x 40 l 6.14 A n n u a l and seasonal-mean distributions of \8'\ [CJ'] (transient, zonally-symmetric eddies). U n i t s are 10~ Pa K s~ and the contour intervals are 1 x 10~ Pa K (light contours) and 10 x 10~ Pa K s" (dark contours). . . 41 2 l 2 2 1 6.15 A n n u a l and seasonal-mean distributions of 8*'u>*'(transient, zonally-asymmetric eddies). U n i t s are 10~ Pa K s and the contour intervals are 5 x 1 0 Pa K s~ . . . . 42 2 - 2 _ 1 l 6.16 A n n u a l and seasonal-mean distributions of northward momentum flux. U n i t s are m s~ and the contour interval is 10 m s~ 44 6.17 A n n u a l and seasonal-mean distributions of upward momentum flux. U n i t s are 1 0 m Pa s~ and the contour intervals are 4 x 1 0 m Pa s~ (light contours) and 20 x 10~ m Pa s~ (dark contours) 45 6.18 A n n u a l and seasonal-mean distributions of northward heat flux. Units are K s~ and the contour interval is 5 K s~ 46 6.19 A n n u a l and seasonal-mean distributions of upward heat flux. U n i t s are 10~ Pa K and the contour i n tervals are 5 x 1 0 Pa K s (light contours) and 50 x 1 0 Pa K s' (dark contours) 47 2 2 2 2 - 2 - 2 2 2 2 2 l l 2 - 2 - 2 _ 1 1 vii 7.1 A n n u a l and seasonal-mean distributions of ip due to diabatic heating (Hadley circulation, ipHadiey)- U n i t s are 10 m Pa s' and the contour intervals are 1 x 10 m Pa s~ (light contours) and 5 x 10 m Pa s (dark contours). 52 A n n u a l and seasonal-mean distributions of the eddyinduced component of ip (Ferrel circulation, ipFerrei)U n i t s are 10 m Pa s and the contour interval is 1 x 10 m Pa s~ . 53 A n n u a l and seasonal-mean distributions of ip. U n i t s are 10 m Pa s and the contour intervals are 1 x 10 m Pa s (light contours) and 5 x 10 m Pa s (dark contours) 54 3 1 3 7.2 l 3 3 3 _ 1 3 8.1 _ 1 1 3 7.3 - 1 - 1 3 _ 1 A n n u a 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~ m s~ and the contour interval is 0.5 x 10~ m s~ . For the b o t t o m panel, the units are 10~ m s~ and the contour interval is 5 x 10~ m s~ 5 5 5 5 8.2 2 2 2 3 3 59 Average winter ( D J F ) momentum budget (top three panels) and zonal kinetic energy production due to eddyflux forcing. For the top three panels, the units are 10~ m s~ and the contour interval is 0.5 x 1 0 m s~ . For the b o t t o m panel, the units are 10~ m s~ and 5 2 - 5 5 the contour interval is 5 x 10~ m 5 8.3 2 s~ 2 2 2 3 60 3 Average spring ( M A M ) momentum budget (top three panels) and zonal kinetic energy production due to eddyflux forcing. For the top three panels, the units are 10~ m s~ and the contour interval is 0.5 x 10~ m s~ . For the b o t t o m panel, the units are 10~ m s~ and 5 2 5 5 the contour interval is 5 x 10~ ra s 5 viii 2 - 3 2 2 3 61 8.4 Average summer ( J J A ) momentum budget (top three panels) and zonal kinetic energy production due to eddyflux forcing. For the top three panels, the units are 10~ m s~ and the contour interval is 0.5 x 10~ m s~ . For the b o t t o m panel, the units are 10~ m s~ and 5 2 5 5 the contour interval is 5 x 1 0 8.5 - 5 m 2 s~ 2 3 62 3 Average autumn ( S O N ) momentum budget (top three panels) and zonal kinetic energy production due to eddyflux forcing. For the top three panels, the units are 10~ m s~ and the contour interval is 0.5 x 1 0 m s" . For the b o t t o m panel, the units are 10~ m s~ and the contour interval is 5 x 10~ m s~ 5 2 - 5 5 5 8.6 2 2 2 2 3 A n n u a l - and seasonal-average zonal kinetic energy production due to eddy-flux forcing. T h e units are 1 0 m s~ and the contour interval is 5 x 10~ m s~ 64 - 5 5 8.7 63 3 2 2 3 3 A n n u a l average heat budget due to eddy-flux forcing. T h e units are 10~ K s" and the contour interval is 0.5 x l O " K s' 65 Average winter ( D J F ) heat budget due to eddy-flux forcing. T h e units are 10~ K s~ and the contour interval is 0.5 x 10~ K s~ 66 Average spring ( M A M ) heat budget due to eddy-flux forcing. T h e units are 10~ K s" and the contour interval is 0.5 x 10~ K s 67 8.10 Average summer ( J J A ) heat budget due to eddy-flux forcing. T h e units are 10~ K s~ and the contour interval is 0.5 x 1 0 " K s 68 8.11 Average autumn ( S O N ) heat budget due to eddy-flux forcing. T h e units are 10~ K s~ and the contour interval is 0.5 x 10~ K s' 69 5 5 8.8 1 1 5 5 8.9 5 5 l - 1 5 5 1 _ 1 5 5 l l 1 ix l 8.12 A n n u a l average momentum tendency (top panel) and zonal kinetic energy production due to diabatic heating . For the top panel, the units are 1 0 m s~ and the contour interval is 0.5 x 10~ m s~ . For the bottom panel, the units are 10~ m s and the contour interval is 5 x 10~ m s~ 72 8.13 Average winter ( D J F ) momentum tendency (top panel) and zonal kinetic energy production due to diabatic heating. For the top panel, the units are 10~ m s~ and the contour interval is 0.5 x 10~ m s~ . For the b o t t o m panel, the units are 10~ m s and the contour interval is 5 x 10~ m s~ 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~ m s~ and the contour interval is 0.5 x 1 0 m s~ . F o r the b o t t o m panel, the units are 1 0 m s~ and the contour interval is 5 x 10~ m s~ 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~ m s~ and the contour interval is 0.5 x 10~ m s~ . F o r the b o t t o m panel, the units are 10~ m s~ and the contour interval is 5 x 10~ m s~ 75 8.16 Average a u t u m n ( S O N ) momentum tendency (top panel) and zonal kinetic energy production due to diabatic heating. For the top panel, the units are 10~ m s~ and the contour interval is 0.5 x 1 0 m s~ . For the b o t t o m panel, the units are 10~ m s and the contour interval is 5 x 10~ m s~ 76 8.17 A n n u a l average heat budget due to diabatic heating. T h e units are 1 0 K s~ and the contour interval is O.SxlO^Xs" 77 - 5 5 5 5 2 2 2 2 - 3 3 5 5 5 5 2 2 - 3 3 5 - 5 - 5 5 2 2 2 3 5 5 2 2 2 3 5 5 - 5 2 2 3 - 5 5 2 3 5 5 2 2 2 2 2 - 3 3 l 1 x 8.18 Average winter ( D J F ) heat budget due to diabatic heating. T h e units are 10~ K s~ and the contour interval is 0.5 x 10~ K s" 78 8.19 Average spring ( M A M ) heat budget due to diabatic heating. T h e units are 10~ K s~ and the contour interval is 0.5 x 1 0 " K s~ 79 8.20 Average summer ( J J A ) heat budget due to diabatic heating. T h e units are 10~ K s~ and the contour interval is 0.5 x 1 0 " K s' 80 8.21 Average autumn ( S O N ) heat budget due to diabatic heating. T h e units are 10~ K s~ and the contour interval is 0.5 x 10~ K s~ 81 8.22 A n n u a 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 panels, the units are 10~ m s~ and the contour interval is 0.5 x 1 0 m s~ . For the b o t t o m panel, the units are 10~ m s~ and the contour interval is 5 x 1 0 m s~ . 85 8.23 Average winter ( D J F ) 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~ m s~ and the contour interval is 0.5 x 10"~ m s~ . For the b o t t o m panel, the units are 10~ m s~ and the contour interval is 5 x 10~ m s' 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 m s~ and the contour interval is 0.5 x 1 0 m s~ . For the b o t t o m panel, the units are 1 0 m s~ and the contour interval is 5 x 10~ m s~ 87 5 5 l 1 5 5 l l 5 5 l 1 5 5 l 5 - 5 5 2 l 2 2 3 - 5 5 5 5 5 2 2 2 2 3 - 5 - 5 2 3 3 - 5 5 2 2 3 xi 2 2 3 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~ m s~~ and the contour interval is 0.5 x 10~ m s~ . For the b o t t o m panel, the units are 1 0 m s~ and the contour interval is 5 x 1 0 " m s~ 88 8.26 Average autumn ( S O N ) momentum budget (top three panels) and zonal kinetic energy production due to the combined effects of the eddies and diabatic heating. F o r the top three panels, the units are 10~ m s~ and the contour interval is 0.5 x 10~ m s~ . For the b o t t o m panel, the units are 10~ m s~ and the contour interval is 5 x 1 0 " m s~ 89 8.27 A n n u a l average heat budget for the combined action of diabatic heating and eddy fluxes of heat and moment u m . T h e units are 1 0 K s and the contour interval is 0.5 x 10~ K s" 90 8.28 Average winter ( D J F ) heat budget for the combined action of diabatic heating and eddy fluxes of heat and momentum. T h e units are 1 0 K s and the contour interval is 0.5 x 10~ K s" 91 8.29 Average spring ( M A M ) heat budget for the combined action of diabatic heating and eddy fluxes of heat and momentum. T h e units are 1 0 K s and the contour interval is 0.5 x 1 0 " K s~ 92 8.30 Average summer ( J J A ) heat budget for the combined action of diabatic heating and eddy fluxes of heat and momentum. T h e units are 10~ K s and the contour interval is 0.5 x 10~ K s~ 93 8.31 Average autumn ( S O N ) heat budget for the combined action of diabatic heating and eddy fluxes of heat and momentum. T h e units are 10~ K s and the contour interval is 0.5 x 10~ K s~ 94 5 5 - 5 5 2 2 2 3 3 5 5 5 5 2 2 2 2 3 3 - 5 5 2 _ 1 1 - 5 5 _ 1 1 - 5 5 5 5 _ 1 l 5 5 _ 1 l l xii _ 1 D.1 January-June distributions of [it]. U n i t s are m s the contour interval is 5 m s 1 and 122 _ 1 D.2 July-December distributions of [it]. U n i t s are m and the contour interval is 5 m s~ s~ l 123 l D . 3 January-June distributions of [0]. U n i t s are i f and the contour interval is 10 i f 124 D.4 July-December distributions of [0]. U n i t s are i f and the contour interval is 10 i f 125 D.5 January-June distributions of [Q]. U n i t s are i f s the contour interval is 1 x 10~ i f s 5 D.6 _ 1 and 126 _ 1 July-December distributions of [Q]. U n i t s are i f and the contour interval is 1 x 10~ i f s 5 s - 1 _ 1 127 D . 7 January-June distributions of [u*v*] cos <f) for stationary eddies. U n i t s are m s~ and the contour interval is5m s129 2 2 2 D.8 2 2 July-December distributions of [u*v*] cos <fi for stationary eddies. U n i t s are m s~ and the contour interval is 5 m s' 130 2 2 2 D.9 2 2 January-June distributions of [u*u*\ for stationary eddies. U n i t s are 10~ m Pa s~ and the contour i n tervals are 2 x 1 0 m Pa s~ (light contours) and 10 x 10~ m Pa s~ (dark contours) 131 2 2 - 2 2 2 2 D.10 July-December distributions of [u*u*] for stationary eddies. U n i t s are 10~ m Pa s~ and the contour i n tervals are 2 x 10~ m Pa s (light contours) and 10 x 1 0 m Pa s~ (dark contours) 132 2 2 2 - 2 - 2 2 D . l l January-June distributions of [v*0*] cos<f> for stationary eddies. U n i t s are i f s~ and the contour interval is l 2Ks~ 133 l D.12 July-December distributions of [v*0*]cos<fi for stationary eddies. U n i t s are i f s and the contour interval is _ 1 2Ks~ 134 x xiii D.13 January-June distributions of [u*6*] for stationary eddies. U n i t s are 1 0 Pa K s~ and the contour i n tervals are 2 x 10~ Pa K s" (light contours) and 30 x 1 0 " Pa K s' (dark contours) 135 - 2 l 2 2 1 1 D.14 July-December distributions of [u*6*\ for stationary eddies. U n i t s are 1 0 Pa K s" and the contour i n tervals are 2 x 1 0 Pa K s" (light contours) and 30 x 10~ Pa K s~ (dark contours) 136 - 2 1 - 2 2 1 l D . 15 January-June distributions of [u*v*] cos <j> for transient, zonally-symmetric eddies. U n i t s are m s~ and the contour interval is 0.5 m s~~ 138 2 2 2 2 2 D . 16 July-December distributions of [u*v*] cos c/> for transient, zonally-symmetric eddies. U n i t s are m s~ and the contour interval is 0.5 m s~ 2 2 2 2 2 139 D.17 January-June distributions of [u*ui*\ for transient, zonallysymmetric eddies. U n i t s are 10~ m Pa s~ and the contour intervals are 0.5 x 1 0 m Pa s~ (light contours) and 5 x 10~ m Pa s~ (dark contours) 140 2 2 - 2 2 2 2 D.18 July-December distributions of [u*a;*] for transient, zonallysymmetric eddies. U n i t s are 10~ m Pa s~ and the contour intervals are 0.5 x 10~ m Pa s~ (light contours) and 5 x 10~ m Pa s~ (dark contours) 141 2 2 2 2 2 2 D.19 January-June distributions of [v*9*] cose/) for transient, zonally-symmetric eddies. U n i t s are K s~ and the contour interval is 0.1 K s~ 142 l l D.20 July-December distributions of [v*9*] cos 4> for transient, zonally-symmetric eddies. U n i t s are K s~ and the contour interval is 0.1 X s" 143 l 1 D.21 January-June distributions of [u*6*] for transient, zonallysymmetric eddies. U n i t s are 1 0 Pa K s and the contour intervals are 1 x 10~ Pa K s (light contours) and 10 x 10~ Pa K s~ (dark contours) 144 - 2 2 2 l xiv _ 1 _ 1 D.22 July-December distributions of [u*8*] for transient, zonallysymmetric eddies. U n i t s are 10 Pa i f s and the contour intervals are 1 x 10~ Pa i f s (light contours) and 10 x 1 0 Pa i f s' (dark contours) 145 2 1 2 - 2 _ 1 1 D.23 January-June distributions of [u*v*] cos (j> for transient, zonally-asymmetric eddies. U n i t s are m s~ and the contour interval is 5 m s~ 147 2 2 2 2 2 D.24 July-December distributions of [u*v*] cos <p for transient, zonally-asymmetric eddies. U n i t s are m s~ and the contour interval is 5 m s~ 148 2 2 2 2 2 D.25 January-June distributions of [u*u*] for transient, zonallyasymmetric eddies. U n i t s are 1 0 m Pa s~ and the contour intervals are 5 x 10~ m Pa s~ 149 - 2 2 2 2 D.26 July-December distributions of [«*_>*] for transient, zonallyasymmetric eddies. U n i t s are 1 0 m Pa s~~ and the contour intervals are 5 x 10~ m Pa s~ 150 - 2 2 2 2 D.27 January-June distributions of [v*9*] coscfi for transient, zonally-asymmetric eddies. U n i t s are i f s~ and the contour interval is 2 i f s~ 151 l l D.28 July-December distributions of [v*6*] cos <f> for transient, zonally-asymmetric eddies. U n i t s are K s and the contour interval is 2 i f s 152 - 1 _ 1 D.29 January-June distributions of [u>*9*] for transient, zonallyasymmetric eddies. U n i t s are 1 0 Pa i f s and the contour intervals are 5 x 10~ Pa i f s 153 - 2 2 _ 1 _ 1 D.30 July-December distributions of [u*0*] for transient, zonallyasymmetric eddies. U n i t s are 10~ Pa i f s' and the contour intervals are 5 x 1 0 Pa i f s" 154 2 - 2 1 1 D.31 January-June distributions of northward momentum flux. U n i t s are m s~ and the contour interval is 10 m s . 156 2 2 2 - 2 D.32 July-December distributions of northward momentum flux. Units are m s~ and the contour interval is 10 m s~ 2 2 2 2 xv 157 D.33 January-June distributions of upward momentum flux. U n i t s are 10~ m Pa s~ and the contour intervals are 4 x 10~ m Pa s~ (light contours) and 20 x 1CT m Pa s~ (dark contours) 158 2 2 2 2 2 2 D.34 July-December distributions of upward momentum flux. U n i t s are 10~ m Pa s~ and the contour intervals are 4 x 10~ m Pa s~ (light contours) and 20x 1 0 " ra Pa s~ (dark contours) 159 2 2 2 2 2 2 D.35 January-June distributions of northward heat flux. U n i t s are K s~ and the contour interval is 5 K s 160 l _ 1 D.36 July-December distributions of northward heat flux. U n i t s are K s~ and the contour interval is 5 K s~ 161 l l D.37 January-June distributions of upward heat flux. U n i t s are 10~ Pa K s and the contour intervals are 5 x 10~ Pa K s' (light contours) and 50 x 10~ Pa K s~ (dark contours) 162 D.38 July-December distributions of upward heat flux. U n i t s are 1 0 Pa K s~ and the contour intervals are 5 x 10~ Pa K s" (light contours) and 50 x 10~ Pa K s~ (dark contours) 163 2 _ 1 2 1 2 - 2 l l 2 1 2 l D.39 January-June distributions of tp due to diabatic heating (Hadley circulation). U n i t s are 10 m Pa s" and the contour intervals are 1 x 10 ra Pa s (light contours) and 5 x 10 m Pa s (dark contours) 165 3 3 3 1 - 1 _ 1 D.40 July-December distributions of ip due to diabatic heating (Hadley circulation). U n i t s are 10 ra Pa s and the contour intervals are 1 x 10 ra Pa s (light contours) and 5 x 10 ra Pa s (dark contours) 166 D.41 January-June distributions of the eddy-induced component of ip (Ferrel circulation). U n i t s are 10 ra Pa s and the contour interval is 1 x 10 ra Pa s" 167 3 3 3 _ 1 - 1 - 1 3 3 _ 1 1 D.42 July-December distributions of the eddy-induced component of ip (Ferrel circulation). U n i t s are 10 ra Pa s" and the contour interval is 1 x 10 ra Pa s~ 3 3 xvi x 1 168 D.43 January-June distributions of ib. U n i t s are 10 m Pa s" and the contour intervals are 1 x 10 m Pa s~ (light contours) and 5 x 10 m Pa s~ (dark contours) 169 3 3 3 1 x l D.44 July-December distributions of ib. U n i t s are 10 m Pa s and the contour intervals are 1 x 10 m Pa s" (light contours) and 5 x 10 ra Pa s (dark contours) 170 3 3 3 _ 1 1 - 1 D.45 January-June zonal kinetic energy production due to eddy-flux forcing. T h e units are 10~ m s~ and the contour interval is 5 x 10~ m s~ 172 D.46 July-December zonal kinetic energy production due to eddy-flux forcing. T h e units are 10~ m s~ and the contour interval is 5 x 10~ m s~ 173 5 5 2 xvii 2 3 3 5 5 2 3 2 s 1. Introduction O f perennial interest i n the meteorological community are the processes maintaining and affecting the westerly upper-air jet streams along w i t h the effect of these wind-speed m a x i m a on synoptic-scale atmospheric waves. F i r s t investigated i n depth during the Second W o r l d W a r , the subsequent rise of c i v i l aviation has provided the impetus for many studies of mid-latitude upper-air phenomena. T h e 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 A r c t i c region; also, to improve the quality of these results i n 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 synopticscale 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 primarily the availability of high-quality data that makes these aims feasible and, although much of the data are still somewhat questionable, w i t h some data being more artifacts of the reanalysis model t h a n actual measured quantities, the results should give insight into the processes maintaining the thermal-wind balance i n the atmosphere. 2 2. Notation Definition 2.1. The zonal-averaging operator is 1 f* 2 w-si A i X The departure from this average is A* = A-[A]. Definition 2.2. The time-averaging operator is — 1 f A — —-— Ad h — h Jt t2 x The departure from this average is A' = A-A. 3 U s i n g these definitions , one may write 1 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 i o n i n the case A = u) 4. a time-varying, zonally-asymmetic component (e.g. a quantity associated w i t h a storm or other transient, local event) Other useful identities, which w i l l be used implicitly throughout the present investigation, may be derived: [AB] [A] [B] + [A*B* (2.1) (2.2) This notational scheme is a reversal of that of [Pfe81], but is in keeping with the more accepted conventions of [P092] and [OR71]. 1 4 A'B' = [A]' [B]' + [A'*B'* (2.3) Throughout the b o d y 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 • meridional transport: Av] = [A] [v] + \A*v*] + [A] [v] + [A'*v'*\ • downward (2.4) transport: 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 circulat i o n i n the ((j), p)-plane. 5 3. M o d e l T h e model used for the calculations is briefly described below; a detailed derivation of the model equations is given i n A p p e n d i x B . T h e basic equations are the zonally-averaged momentum equation (refer to A p p e n d i x B.3.3) d\u] _ dt ( LJ a([«]008fl\ V a(K^] QS20) d[u] R cos <pd(j) J [ i d[u*U*\ C dp Rco&<j)d<t> dp ' L J (3-1) the zonally-averaged thermodynamic equation (refer to A p p e n d i x B.3.4) dt R dcj) M dp Rcos4>d(j) dp \p J c ' p { ' } and the zonally-averaged thermal w i n d equation (refer to A p p e n d i x B.3.5) jd[u] dp where / = (j + M ^ 2 a n < R *- ld[6] l = dP P00K R dcj) ^ _ Here, t represents time, R is the earth's radius, Rd is the gas constant, 4> represents latitude, p represents pressure, p o is a reference pressure 0 level (1000 hPa), u is eastward velocity, v is northward velocity, u (— ^ ) is the 6 vertical velocity i n pressure coordinates, 6 is potential temperature, / (= 2£) sin </>) is the Coriolis parameter, FX is the zonal forcing, Q is the diabatic heating, c is p the constant pressure heat capacity of air and K = ^ . Also, it is useful to define SL Cp the static stability parameter S = — ^ r f j ; p One may define a streamfunction ip associated w i t h the mean meridional circulation ([v], [UJ]) and derive the following diagnostic equation for the streamfunction (see A p p e n d i x C ) d ip , D d[9] 2 K N <9»7 P 2 Poo ^ K d ip , D Kd[9] ^r/dp Poo 2 P P K (K- drj \ 1 dip\ P E>^d^[0\dip i J Poo dr K 1 dr 2 9 dp (3-4) where = D [S ] P H _L(f 9 cose/) \ (M c o s < Rdr] ^) Rdl_ p R 2 pn d[9] Poo K R^' \ \ 00 f P d ((Poo\[Q]_ 1 P KR X c cos (pdp — + or] op c ( %) p J d([9*v*}coscb) _ c K Rdr} p (d([u*v*}co ct>) cose/) V d[9*u*}\ 2 S Rcosqjdrj d[u*u*} dp 7 dp J \ 1 J Xi V ' ' V dip cos <j> dp 1 (3.7) (3.8) Rdrj 77 = sin (p. A description of each variable, and its associated units, is given i n A p p e n d i x A. T h e above model equations differ from those of [Pfe81] by their presentation i n terms of potential temperature rather than absolute temperature . T h e equa2 tions originally presented i n [Kuo56] differ substantially from the more modern presentations of this model, w i t h K u o ' 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, w i t h no assumptions being made implicitly. W h i 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 Also, it appears that there is an error, or at least an implicit assumption, in the published formulae of [Pfe81], with the contribution by — ^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. 2 9 8 not necessary for the type of investigation presented herein. T h e principal m o t i vation for use of a T E M formulation of a zonally-averaged model is to combine the direct effect of eddy motions w i t h that of the eddy-induced circulation. It shall be seen, however, that the eddies still play an important role i n affecting the zonally-averaged jet core. 9 4. Data T h e data used i n this model are a l l products of the N C E P / N C A R Reanalysis project. A comprehensive overview of the project is provided by [ K K K + 9 6 ] ; a precis relevant only to the data actually used w i l l be provided here. T h e 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. T h e daily data is provided o n a 2.5° x 2.5° grid a n d 12 pressure levels are chosen. T h e 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 a n d latitudes 8 7 . 5 ° 5 - 87.5°iV are chosen for the five-year periods 1959-1963 a n d 1989-1993. M o n t h l y - m e a n values of each of the variables and fluxes are calculated using the flux-decompositions given i n equations (2.4) a n d (2.5). T h e results presented here are for 1989-1993. Calculations for 1959-1963 for the eddy-induced circulation were also carried out to check the implementation of the model through comparison w i t h the results of [Pfe81]. T h e original N C E P / N C A R Reanalysis Project daily-averaged data were downloaded from the N O A A web-site i n 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. 3 10 ( H G T ) , temperature T ( T M P ) , u w i n d ( U G R D ) , v w i n d ( V G R D ) and pressure vertical velocity UJ ( W E L ) for each of the years. GrADS control files were then 1 created for the five years of each variable and the corresponding index files were created using the gribmap utility. T h e data was then written to binary files using GrADS and imported into Matlab, where a l l further calculations were performed. 4.1. Preprocessing of diabatic heating data T h e monthly-mean model-derived diabatic heating fields 5 for 1979-1993° were downloaded from the N O A A archive i n the form of GRIB files and then processed to ultimately produce the data required for model calculations. T h e data are provided on 27 model cr-levels on a Gaussian grid w i t h a longitudinal resolution of 1.875° and a latitudinal resolution of 94 Gaussian levels (an approximate resolution of 1.9°). T h e vertical levels for a = 2- and their approximate pressure-level equivalents are: 4 5 See http://www.iges.org/grads for a description of the software. The supplied data appears to have been weighted as ^ by N C E P , although this is undocumented. T h e data for January 1982 were inexplicably absent. 6 11 a-level nearest p-level (hPa) a-level 0.9950 0.9820 0.9640 0.9420 0.9155 0.8835 0.8454 1000 0.3717 0.3122 0.2582 0.2102 925 0.1682 0.1323 0.1023 0.0778 0.0578 0.0413 0.0278 0.0174 850 0.8009 0.7499 0.6934 700 0.6323 0.5678 0.5012 0.4352 500 nearest p-level (hPa) 400 300 250 200 150 100 70 50 30 20 10 0.0093 T h e extraction and processing procedure is as follows: • T h e data for the period 1989-93 are indexed to Gr^4Z)5-compatible format by using the gribmap utility. • T h e fields, provided by N C A R , relevant to determining diabatic heating 7 are combined by Q = CNVHR + LRGHR + LWHR It should be noted that these fields are all model-derived fields rather than calculated directly from observational data. However, other repositories of diabatic heating data use a finite difference form of the thermodynamic equation to determine Q (— c T ^ ^ ) , providing data of dubious quality, which are fundamentally unsuitable for budget calculations. 7 p 12 +SHAHR + SWHR + VDFHR, (4.1) where CNVHR LRGHR LWHR SHAHR SWHR VDFHR — Deep convective heating rate — Large-scale condensation heating rate = Long-wave radiative heating rate = Shallow convective heating rate — Short-wave radiative heating rate = Vertical diffusion heating rate • T h e 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 i n the previous section using 1 — D cubic splines at each grid-point. • T h e N C E P / N C A R data-set also provides monthly-mean surface pressure; this is used to discard a l l those values of Q which are on v i r t u a l pressure levels (i.e. pressure levels which are below the E a r t h ' s surface) . 8 This procedure, of course, introduces an unquantifiable bias into the boundary-layer diabatic heating data. 8 13 • A n n u a 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. T h i s is accomplished by cubic spline extrapolation at each grid-point. • To convert the Gaussian-grid data to a latitudinal resolution of 2.5° compatible w i t h the data of the previous section a bi-cubic spline interpolation technique i n the (f) — p plane is used. T o 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. • T h e resulting Q field on p-levels w i t h a latitudinal resolution of 2.5° is the sole source for all calculations involving diabatic heating. 4.2. C a l c u l a t e d quantities T h e following are the quantities calculated from the above d a t a for use i n the numerical solution of the model equation and the budget calculations: 14 u steady, zonally-averaged zonal w i n d e Q steady, zonally-averaged potential temperature 0* V* meridional heat transport by stationary eddies steady, zonally-averaged diabatic heating meridional heat transport by zonally-symmetric transient eddies 9*'v*' [ff] w 6*'UJ*' u* V*] [«'] [«'] u*'v*' u* u* [ti'] [u/] u*'u>*' meridional heat transport by zonally-asymmetric transient eddies downward heat transport by stationary eddies downward heat transport by zonally-symmetric transient eddies downward heat transport by zonally-asymmetric transient eddies northward zonal momentum transport by stationary eddies northward zonal momentum transport by zonally-symmetric transient eddies northward zonal momentum transport by zonally-asymmetric transient eddies downward zonal momentum transport by stationary eddies downward zonal momentum transport by zonally-symmetric transient eddies downward zonal momentum transport by zonally-asymmetric transient eddies 4.3. Comparison with data of previous studies T h e use of reanalysis model data represents a significant departure from previous studies to investigate wave-mean flow interactions i n the atmosphere. T h e 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 i n 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 i n 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 i n some measurements - these are particularly evident i n [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 equation T h e streamfunction is discretised on a 71 x 12 grid (covering latitudes 8 7 . 5 ° 5 — 87.5°N 9 and levels 1000 hPa - 100 hPa) w i t h latitudinal coordinate rj (= s i n 0 ) and vertical coordinate p. Thus, the gridpoint intervals are unequal i n both directions: • T h e l a t i t u d i n a l coordinate is discretised as t]i = sin (i x 2.5° - 90°) = sin ((i - 36) x 2 . 5 ° ) . • T h e vertical coordinate is discretised as pj, w i t h 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 T h e various spatial derivatives of the streamfunction are discretised as follows : 10 To avoid complexities from dealing with the removable singularities of terms involving »?L 9 o, the poles are not included in this discretisation scheme. Where possible, central difference approximations are used; at the boundaries of the domain forward- and backward-differences are used, as appropriate. 9 =± 0 1 0 17 Mm,Pi)->P(m,p.i) f d± i o r for i = 2 , . . . , 70 tb(m^,Pi)-^(rn-uPj) 4>(Vi,P2)-i>(r)i,Pi) £ - o r ? ' P2—Pi V'fa.P.7+i)-i/'fa»P.;-i) £ ]_ = ^ o r 7— 2 11 Pi+l—Pi-1 (Vi,Pj) , for j = 12 1p(Vi,Pl2)-ll'{.'ni,Pll) Pl2~Pll J>(ri3,Pj)—4>(ri\,Pj) »('?2,Pj)-V'(')l,Pj) 13 "''I »?2—^1 b dry i = (vi+i •PJ ) 2 (m-PJ ) (it . P J ) -22^ ( n - 1 .PJ ) , for i = 2 , . . . , 70 it+i-it-i (Vi,Pj) 2 V'(l71.Pj)-'/'(l70.Pj) */* (l71 .Pj ) ~V (l69 .Pj ) 171 -170 171 -169 1 Z ?771-7)70 V (lj.P3)- /'(lt.Pl) P3-P1 , 1 i 8 7/> 2 '(li'Pj+l)-^(li'Pj) , for j = 1 ^'(li.Pj)-^'(l.'Pj-l) y (ii.pi2)-v (ii-pii) , , v (it.pii)- / (ii.pio) , P12—Pll d ip 2 dndp dr] , for j = 2 , . . . , 11 Pt+i-Pj-i 2 (Vi,Pj) or , Pi-Pi-1 2 -f j _ 71 V'(li.P2)-V (lt.Pl) P2—Pi dp , for» = 1 i , , for j = 12 is evaluated using the formulae for | ^ (Vi,Pj) . (Vi,Pj) and (m,Pj) (Vi,Pj) T h e streamfunction problem was solved by direct inversion of the 852 x 852 matrix, assuming zero boundary conditions o n b o t h the lateral, upper a n d lower 18 boundaries. Other solution methods were considered, but given that the t y p i - cal solution time for the direct inversion is ~ 30 s i n comparison w i t h the d a t a preprocessing which takes hours or days, a fast solution scheme was not deemed necessary. Additionally, the difficulty of obtaining a good initial 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. T h e 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 circulations and the circulation due to their combined effects, giving a t o t a l of 57 m a t r i x inversions . 11 Throughout this study, the forcing by surface w i n d stress is ignored (i.e. F A = 0). W h i l e estimates of the surface w i n d stress do e x i s t , these are particularly 12 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. The most comprehensive appears to be those of the E C M W F , although these are only available for a short range of dates. 1 1 1 2 19 unreliable over land areas, indicating that such archives of data are, at best, appropriate o n l y for regional, oceanic studies. T h e derived quantities and the results shown i n the body of this work w i l l be those for annual- and seasonal-mean conditions; for readability, the corresponding plots for monthly-mean conditions w i l l be deferred to A p p e n d i x D . 20 6. Distributions of zonal wind, potential temperature and diabatic heating and eddyfluxesof 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 w i n d [u] and potential temperature 6 , respectively. Evident i n these plots are the jet-stream m a x i m a i n the mid-latitude upper troposphere and the corresponding strong meridional temperature gradient implied by the thermal w i n d relationship A l s o evident are the predominant easterlies i n the trade-wind belt, extending all the way to the upper troposphere. G i v e n the high terrain of A n t a r c t i c a , as well as the sparse data available, the apparent easterlies south of 70° S are rather suspect . 13 T h e zonal w i n d speed maxima, found at 200 mb i n a l l seasons and i n the annual average, are aligned w i t h the regions of the largest values of the meridional temperature gradient, i n accordance w i t h the requirements of thermal T h e 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. iy 21 w i n d balance, w i t h the m a x i m u m values of [it] being found during the winter season i n each hemisphere. A s one might expect from the greater temperature variability i n the Northern Hemisphere (which is due to less oceanic coverage a n d 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 i n Figure 6.3 are the annual and seasonal means of zonal-mean diabatic heating. T h e dominant feature is the quasi-barotropic region of diabatic heating extending for a range of approximately 20° near the equator. T h e 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 i n the A n t a r c t i c 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. E x a m i n a t i o n 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: A n n u a l and seasonal-mean distributions of [0]. U n i t s are K and the contour interval is 10 K. 24 high altitudes. 6.2. E d d y fluxes of heat a n d m o m e n t u m Shown i n Figures 6.4, 6.5 and 6.6 are the annual and seasonal means of stationaryeddy, transient zonally-symmetric eddy and transient zonally-asymmetric eddy components of northward momentum transport. T h e northward m o m e n t u m flux is concentrated at 200 hPa at a l l times throughout the year, w i t h the transient zonally-symmetric eddies making an insignificant contribution. T h e stationaryeddy component, \u* v*\, is significant only i n the N o r t h e r n Hemisphere m i d latitudes and equatorial upper troposphere, w i t h net northward transport i n the midlatitudes peaking i n the boreal winter and a seasonally-reversing equatorward transport at the Equator. E x a m i n i n g the transient zonally-asymmetric component, | , one sees that this is the dominant contribution, w i t h a clear annual u*'v*' cycle i n the N o r t h e r n Hemisphere, peaking i n wintertime, being reinforced by the aforementioned standing eddy component. T h e net southward transport i n the Southern Hemisphere has typically larger magnitudes t h a n its boreal counterpart, w i t h the annual cycle being less evident. In contrast w i t h the results of [P092], which shows a region of strong northward momentum transport i n the A n t a r c tic troposphere by the transient eddies, Figure 6.6 shows little transport (this is 25 26 i n agreement w i t h [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. T h e 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 w i t h consideration 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 i n the N C E P / N C A R reanalysis model. For this reason a contouring scheme is chosen whereby b o t h the very large (and hence suspect) values are plotted, along w i t h values i n the range expected from previous studies. Ignoring these possibly misleading results, F i g u r e 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, w i t h no particular seasonality i n evidence. Comparison w i t h the results of [Pfe81] reveals a similar centre of upward flux i n the m i d troposphere at 4 0 ° N during winter and spring and the same pattern of E q u a t o r i a l upper-tropospheric maxi m a throughout the year. N o other significant similarities are apparent. T h e tran- 27 80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 Figure 6.4: A n n u a l and seasonal-mean distributions of [it* v*] cos <fr (stationary eddies). U n i t s are m s~ and the contour interval is 5 m s~ . 2 2 2 2 28 2 80 Figure 6.5: A n n u a l and seasonal-mean distributions of [u] [v] cos <f> (transient, zonally-symmetric eddies). Units are m s~ and the contour interval is 0.5 m s~ . 2 2 2 2 2 3 > c > a TI 4 > o Figure 6.6: A n n u a l and seasonal-mean distributions of zonally-asymmetric eddies). Units are m 2 30 s 2 u*'v*' cos (b (transient, 2 and the contour interval is 5 m 2 s 2 . sient, zonally-asymmetric eddy component depicted i n Figure 6.9 has strong net upward flux of zonal momentum i n both hemispheres, w i t h peak values reached at about 350 hPa between 20° — 40° i n 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 momenturn is the region of relatively significant values of u*'u*' at 600 hPa between 60°S — 80°S, w i t h a strong annual cycle peaking i n the austral winter. T h e origin of this region of strong downward zonal momentum flux is unclear. T h e 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. T h e stationary-eddy component of heat transport (Figure 6.10), 9* v* cos<{>, is negligible almost everywhere, except i n the N o r t h e r n Hemisphere mid-latitude upper troposphere at 200 hPa between October and M a r c h . A m a x i m u m at 850 hPa also appears about this time at the 31 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 Figure 6.7: A n n u a l and seasonal-mean distributions of [it* u*] (stationary eddies). U n i t s are 1CT m Pa s~ and the contour intervals are 2 x 10~ m Pa s~ (light contours) and 10 x 10~ m Pa s~ (dark contours). 2 2 2 2 2 32 2 80 Figure 6.8: A n n u a l and seasonal-mean distributions of [«'] [UJ'} (transient, zonallysymmetric eddies). Units are 10~ m Pa s" and the contour intervals are 0.5 x 10~ m Pa s~ (light contours) and 5 x 10~ m Pa s~ (dark contours). 2 2 2 2 2 33 2 c > a w O Latitude f) Figure 6.9: A n n u a l and seasonal-mean distributions of u*'u>*' asymmetric eddies). 5 x 10" 2 U n i t s are 1 0 - 2 m Pa s~ 2 m Pa s~ . 2 34 (transient, zonally- and the contour intervals are same latitude. T h e Southern Hemisphere upper troposphere appears to have an intruding region of southward heat transport between August and November, possibly an indicator of the more active lower stratosphere during the austral Spring (as remarked upon by [TW98]).The m a x i m a i n the lower troposphere here are, however, quite suspect for reasons previously discussed. T h e transient, zonallyasymmetric meridional heat transport (Figure 6.12), 9*'v*' coscf), exhibits the same directions of transport characteristic of the standing eddy component, but w i t h greater symmetry about the equator. Strong mid-latitude regions of pole- ward transport appear at b o t h 200 hPa and 850 hPa throughout the year, w i t h a seasonal cycle of magnitude strongly i n evidence; reinforcement by the stationary eddy component is also i n evidence^ along w i t h the springtime intrusion of poleward heat transport i n the Southern Hemisphere. T h e transient, zonally- symmetric component (Figure 6.11), \9'] [v] cos<j), again makes m i n i m a l contribution. 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, w i t h regions of weak downward heat transport to be found i n 35 c > 2> 3 S £ 400 500 600 700 H J < ^ l ^ > L CO O Figure 6.10: Annual and seasonal-mean distributions of 6 v* cosc/> (stationary eddies). Units are K s 1 and the contour interval is 2 K s . 1 36 Figure 6.11: A n n u a l and seasonal-mean distributions of [6] [v']cos(j) (transient, zonally-symmetric eddies). Units are K s and the contour interval is 0.1 K s~ . _ 1 37 l -80 -70 -50 -40 -30 -20 0 -10 10 20 30 40 50 Figure 6.12: A n n u a l and seasonal-mean distributions of 8*'v*' cost/) (transient, zonally-asymmetric eddies). Units are K s~ l 38 and the contour interval is 2 iv" 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 E q u a t o r i a l surface, w i t h the latitude of the m a x i m a possibly following the seasonal migration of the Inter-Tropical Convergence Zone; this is readily apparent only from consideration of monthly-mean plots. South of 6 0 ° 5 , the large values of upward heat transport cannot be explained by consideration of orography alone; no anecdot a l evidence appears to exist which would provide a possible explanation. transient, zonally-asymmetric component (Figure 6.15), 0*V The again appears to exhibit the greatest symmetry about the Equator, along w i t h more significant values t h a n the stationary-eddy contribution and the most clearly defined annual cycle. M o s t apparent are the regions of broad upward heat flux between 30° —80° and 1000 hPa — 250 hPa i n b o t h hemispheres, dominating the total heat flux. A region of weak downward heat transport appears i n the equatorial m i d - troposphere, along w i t h 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 M a r c h and September; as this region extends up to 700 hPa, this phenomenon may be associated w i t h the surface albedo being altered b y seasonal changes i n the extent of the ice-cover near A n t a r c t i c a . Consideration of the transient, zonally-symmetric component (Figure 6.14), [9'\ [u], reveals a negligible 39 contribution by this quantity. 6.3. C a l c u l a t i o n of heat- a n d momentum-flux forcings T h e t o t a l northward and upward momentum fluxes are shown i n Figures 6.16 and 6.17 respectively. In b o t h cases, it is the transient, zonally-asymmetric contribution, w i t h occasional reinforcement by the stationary eddies, which dominates. T h e predominant poleward and upward transport of zonal m o m e n t u m is a reflection of the requirement of northward angular momentum transport, chiefly accomplished by synoptic-scale motions, as outlined by [P092]. T h e t o t a l northward and upward heat fluxes are shown i n Figures 6.18 and 6.19 respectively. A g a i n , it is the transient, zonally-asymmetric component which is most evident, w i t h contributions by the stationary-eddy component i n the highlatitude lower troposphere. T h e strong poleward and upward heat flux i n the m i d latitude troposphere, principally by synoptic-scale systems, supports the results of [Cha47] for the global energy balance. These values, together w i t h diabatic heating data, are used to calculate the fluxes of heat and momentum, H and x respectively, associated w i t h the Hadley 40 C > CO o -80 -70 -60 -50 -40 -30 -20 -10 0 Latitude f> 10 20 30 40 50 60 70 80 Figure 6.13: Annual and seasonal-mean distributions of 9* u>* (stationary eddies). Units are 1 0 Pa K s and the contour intervals are 2 x 10~ Pa K s~ (light contours) and 30 x 10~ Pa K (dark contours). - 2 - 1 2 2 41 l Figure 6.14: A n n u a l and seasonal-mean distributions of [&'] [UJ'} (transient, zonallysymmetric eddies). U n i t s are 10~ Pa K s' and the contour intervals are 1 x 1 0 Pa K s~ (light contours) and 10 x 10~ Pa K s~ (dark contours). 2 - 2 1 l 2 42 x Figure 6.15: Annual and seasonal-mean distributions of zonally-asymmetric eddies). Units are 10~ Pa K s' are 5 x 1 ( T Pa K s~\ 2 2 43 1 6*'u>*' (transient, and the contour intervals -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 Figure 6.16: A n n u a l and seasonal-mean distributions of northward momentum flux. U n i t s are m s~ and the contour interval is 10 m s~ . 2 2 2 44 2 80 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 Figure 6.17: Annual and seasonal-mean distributions of upward momentum flux. Units are 10~ m Pa s~ and the contour intervals are 4 x 1 0 m Pa s~ (light contours) and 20 x 10~ m Pa s~ (dark contours). 2 2 2 - 2 2 45 2 80 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 Latitude f) Figure 6.18: A n n u a l and seasonal-mean distributions of northward heat flux. U n i t s are K s~ and the contour interval is 5 iv" s . l _ 1 46 80 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 Figure 6.19: A n n u a l and seasonal-mean distributions of upward heat flux. U n i t s are 10~ Pa K s and the contour intervals are 5 x 10~ Pa K s~ (light contours) and 50 x 10~ Pa K s~ (dark contours). 2 _ 1 2 2 l 47 l 80 and Ferrel cells, along w i t h their combined effect: _ -"Hadloy R D P 00 RdP P KR K ~ -- P KR XHadley = c p ^ P ) r c Rdr) p dp 0 f XFerrel I \ \ 00 K , d[9*uj* \(d{[9*v*]cos(j)) Rdr) dp x 00 — H I V P J P KR •"Ferrel [Q] (poo\ ^ f7 I — (d cose/), (\ = f X cos <j>) 2 d*u*}\ [u )p dp D J.A„ Rcos<bdr) /d([u*v*]cos <f>) cos(f> \ i \ r> ) d[u*uj* 2 R cos (bdr) dp remembering that [F\] = 0. F r o m these quantities, the heat and momentum flux forcing functions are calculated: 4- 4- T a C • t o t a l eddy flux forcing = total diabatic heating forcing c • * = • total streamfunction forcing = <9#Ferrel . ^XFerrel — drj h dp d#Hadley dr) dH d X ——h — = dn dp d V " Ferrel + #Hadley) : dr) —I . ^XFerrel dp and the streamfunction equation is solved numerically for each of the three choices of forcing, w i t h momentum and thermodynamic budgets being calculated for each 48 resulting distribution of ip. 49 7. Streamfunction ip T h e 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 i n Figures 7.1 and 7.2 respectively. T h e streamfunction corresponding 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 w i t h equations (3.7) and (3.8)). T h e effect of diabatic heating is seen to principally drive a direct circulation around the equator; this is the Hadley c i r c u l a t i o n . B o t h the latitude a n d mag14 nitude of the circulation is seen to strongly follow the angle of insolation, w i t h the regions of subtropical subsidence also being seen to vary w i t h the seasons. The effective disappearance of the Hadley cell i n the summer hemisphere is strongly evident (as w i t h the mass streamfunction computed by [P092]). Notable from Figure 7.3, i n spite of the apparent strong diabatic heating i n the lower troposphere 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 n u l l boundary condition is the appearance of large values of ipn- d\e A y near the E q u a t o r i a l surface. A historical overview of the origins of the names for the various circulations may be found i n [Lor67]. 1 4 50 T h e 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. T h e 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 i n intensity, w i t h 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. T h e assumption of [Pfe81] that the circulation shows strong E q u a t o r i a l symmetry across the seasons is thus seen to be fallacious. F r o m consideration of the individual circulations depicted i n Figures 7.1 and 7.2, one expects a broad reinforcement of the direct circulation near the equator, w i t h opposing values outside the tropics. T h e picture of the net circulation provided by the combined effect of eddy-fluxes and diabatic heating, depicted i n F i g u r e 7.3, is one of strong direct circulation near the E q u a t o r w i t h a substantially weaker indirect circulation i n the mid-latitudes, varying b o t h i n latitude and intensity w i t h the seasons. T h e polar direct circulations are barely i n evidence, being only readily apparent i n the Southern Hemisphere during the austral 51 a u t u m n and winter. -80 -70 -60 -50 -40 -30 -20 0 -10 10 20 30 40 50 60 70 Figure 7.1: A n n u a l and seasonal-mean distributions of ip due to diabatic heating (Hadley circulation, ipHadiey)Units are 10 m Pa s~ and the contour intervals are 1 x 10 m Pa s~ (light contours) and 5 x 10 m Pa s (dark contours). l 3 3 l 3 53 _ 1 80 -80 -70 -60 -50 -40 -30 -20 -10 0 Latitude f) 10 20 30 40 50 60 70 Figure 7.2: A n n u a l and seasonal-mean distributions of the eddy-induced component of ip (Ferrel circulation, ipFerrei)Units are 10 m Pa s and the contour interval is 1 x 10 m Pa s . 3 3 - 1 54 - 1 80 Figure 7.3: A n n u a l and seasonal-mean distributions of ip. U n i t s are 10 m Pa s~ and the contour intervals are 1 x 10 m Pa s" (light contours) and 5 x 10 m Pa s (dark contours). 3 3 3 - 1 55 1 l 8. Momentum and heat budgets T h e 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] i n equations (3.5) and (3.6). 8.1. B u d g e t s o f e d d y - d r i v e n c i r c u l a t i o n For the eddy-driven circulation, the momentum tendency is given b y 9[u] j , / , =[ \[JV V dt 1 J d ([u] cos c/>) \ d [u] d ([u*v*] cos <f>) d [u*u* AXA ~ M — Rcos<pd<p J dp Rcos <pd<p ' dp 2 1 D J 2 S~ r (v,u) terms Eddy terms and the potential temperature tendency is given by d[6] dt [v]d[6] R d(p d[9] — 1 \UJ\ J dp d([9*v*] cos cb) d[6*u* Rcos(pd(p (v,to) terms dp Eddy terms where the streamfunction equation (3.4) is solved w i t h forcing of and the values of [v] and [u] are determined from Vtorrci- 9 H ^ n l + gp &X Bl Zonal-momentum bud- gets for annual- and seasonal-mean conditions are shown i n Figures 8.1-8.5, w i t h 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 circulation i n the Ferrel cell is concentrated i n the upper part of the (200 hPa — 250 hPa), w i t h the m a x i m u m migrating between 35° — 45° and strengthening w i t h the annual cycle to reach a peak i n wintertime. t r i b u t i o n to ^ atmosphere T h e con- by the eddy motions is very similar to this, although opposite i n sign and w i t h magnitudes ~ 25% greater. M o m e n t u m tendencies due to the (v, UJ) terms are always opposite i n the upper and lower troposphere, w i t h the secondary centres appearing near the surface. T h e boundary-layer peaks i n m o m e n t u m tendency 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 w i t h a large sensitivity to the choice of contouring levels. There is, apparently, strong wave-mean flow interaction, w i t h values of ~ 2 x 10~ m s~ 5 l day' 1 surprisingly evident i n the wintertime upper troposphere, providing deceleration of the mean zonal flow far equatorward of the jet-stream m a x i m a . Comparison of the polarity of the momentum tendency i n regions equatorward and poleward of the jet-stream m a x i m a (from F i g u r e 6.1) shows that the momentum tendencies produced by eddy motions strongly m i m i c 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 t h adjacent peaks i n the wintertime upper tropical atmosphere. T h e regions of strong production, shown i n Figure 8.6, are at exactly the same latitudes and altitudes as the jet-stream m a x i m a of Figure 6.1, i n sharp contrast w i t h the results of [Pfe81]. T h e contribution to the zonally-averaged potential temperature tendency by the eddy-induced meridional circulation is opposite i n sign to that induced d i rectly by the eddies and is typically smaller i n magnitude. T h e 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), w i t h the largest values appearing during the winter i n 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. E x a m i n i n g the vertical variations of ^ , one m a y see that the general effect of the eddies i n the southern polar regions is to destabilise the atmosphere. T h e general structure of the potential temperature tendency i n 58 the higher latitudes is heating throughout the troposphere, w i t h the secondary circulation apparently moving the eddy-induced heating m a x i m u m slightly poleward. Comparison of the plots of potential temperature tendency w i t h Figure 6.2 shows that the distribution of ^ can account for some changes i n the meridional temperature gradient across the seasons. 59 Latitude f) Figure 8.1: A n n u a l average momentum budget (top three panels) energy production due to eddy-flux forcing. For the top three are 10~ m s~ and the contour interval is 0.5 x 10~ m s~ . panel, the units are 10~ m s~ and the contour interval is 5 x 5 2 5 5 2 3 60 2 and zonal kinetic panels, the units For the b o t t o m 10~ m s~ . 5 2 3 Latitude f) Figure 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 1 0 m s~ and the contour interval is 0.5 x 1 0 " m s~ . For the b o t t o m panel, the units are 10~ m s~ and the contour interval is 5 x 10~ m s~ . - 5 2 5 5 2 3 2 5 61 2 3 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 1 0 " m s~ and the contour interval is 0.5 x 1 0 m s~ . For the b o t t o m panel, the units are 1 0 m s~ and the contour interval is 5 x 10" m s- . 5 2 - 5 - 5 5 2 2 3 62 3 2 feflL^^g^dU>. i I L |_J 1U 100 150 200 25C 300 400 500 600 700 Re c 11 _ k.\ > i y K — i o ,. i u 0 10 Latitude f) Figure 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~ m s~ and the contour interval is 0.5 x 10~ m s For the b o t t o m panel, the units are 10~ m s~ and the contour interval is 5 x 10~ m s~ . 5 2 5 5 5 2 2 3 63 3 r Latitude f ) Figure 8.5: Average autumn ( S O N ) momentum budget (top three panels) and zonal kinetic energy production due to eddy-flux forcing. For the top three panels, the units are 10~ m s~ and the contour interval is 0.5 x 10~ m s~ . For the b o t t o m panel, the units are 10~ m s~ and the contour interval is 5 x l O " m s" . 5 2 5 5 5 2 2 3 64 3 2 Figure 8.6: A n n u a l - and seasonal-average zonal kinetic energy production due to eddy-flux forcing. T h e units are 10~ m s~ and the contour interval is 5 x 10" m s~ . 5 5 2 3 65 2 3 66 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 Latitude t) Figure 8.8: Average winter ( D J F ) heat budget due to eddy-flux forcing. T h e units are 10~ K s~ and the contour interval is 0.5 x 10~ K s~ . 5 l 5 67 l 80 68 69 70 8.2. B u d g e t s of d i a b a t i c heating-driven c i r c u l a t i o n For the circulation driven by diabatic heating, the momentum tendency is given by (v,u>) terms and the potential temperature tendency is given by Diabatic Heating where the streamfunction equation (3.4) is solved w i t h forcing of the values of [v] and [u] are determined from V'Hadicy — a n d Zonal-momentum budgets for annual- and seasonal-mean conditions are shown i n Figures 8.12-8.16, w i t h potential temperature budgets for annual- and seasonal-mean conditions shown i n Figures 8.17-8.21. T h e m o m e n t u m 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 w i n - tertime between 100 hPa — 250 hPa, w i t h decelerations of comparable magnitude i n the boundary layer, slightly equatorward of these peaks. 71 There is substan- t i a l zonal kinetic energy production i n regions poleward of the equatorial upper tropospheric m a x i m a i n zonal w i n d tendency, w i t h clear m a x i m a evident b o t h throughout the year and i n the annual average. T h e location of the m a x i m a i n momentum tendency are, as w i t h the eddy-induced tendency, far equatorward of the mid-latitude jet stream maxima, thus indicating that diabatic heating cannot alone account for the variations i n the zonally averaged jet core. E x a m i n a t i o n 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, w i t h m a x i m a obtained i n the polar upper troposphere. Considerating, i n 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 i n the equatorial mid-troposphere and relatively strong near-surface heating between 60° S — 60°N, w i t h significant cooling throughout the remainder of the atmosphere and particularly strong cooling i n the A r c t i c and A n t a r c t i c (with typical peak values of ~ 1.5 K 72 day ). -1 Figure 8.12: A n n u a l average momentum tendency (top panel) and zonal kinetic energy production due to diabatic heating . For the top panel, the units are 1 0 m s~ and the contour interval is 0.5 x 10~ m s~ . For the b o t t o m panel, the units are 10~ m s~ and the contour interval is 5 x 10~ m s~ . - 5 2 5 5 2 3 2 5 73 2 3 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 Figure 8.13: Average winter ( D J F ) momentum tendency (top panel) and zonal kinetic energy production due to diabatic heating. For the top panel, the units are 10~ m s~ and the contour interval is 0.5 x 10~ m s~ . For the b o t t o m panel, the units are 10~ m s~ and the contour interval is 5 x 10~ m s~ . 5 2 5 5 2 3 2 5 74 2 3 80 -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~ m s~ and the contour interval is 0.5 x 1 0 m s~ . For the b o t t o m panel, the units are 1 0 m s~ and the contour interval is 5 x 10~ m s~ . 5 2 - 5 - 5 2 3 2 5 75 2 3 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~ m s~ and the contour interval is 0.5 x 10~ m s~ . For the b o t t o m panel, the units are 10~ m s~ and the contour interval is 5 x 10~ m s~ . 5 2 5 5 2 3 2 5 76 2 3 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 40 30 50 60 70 Latitude f) Figure 8.16: Average autumn ( S O N ) momentum tendency (top panel) and zonal kinetic energy production due to diabatic heating. For the top panel, the units are 10~ m s~ and the contour interval is 0.5 x 10~ m s~ . For the b o t t o m panel, the units are 10~ m s~ and the contour interval is 5 x 1 0 m s~ . 5 2 5 5 2 3 2 - 5 77 2 3 80 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 Figure 8.17: A n n u a l average heat budget due to diabatic heating. T h e units are 1 0 K s~ and the contour interval is 0.5 x 10~ K s~ . - 5 l 5 78 l 80 I CD 2. 5' CO -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 Latitude f) Figure 8.18: Average winter ( D J F ) heat budget due to diabatic heating. units are 10~ K s~ and the contour interval is 0.5 x 10~ K s . 5 l 5 79 - 1 The 80 -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. units are 10~ K and the contour interval is 0.5 x 1 0 K s~ . 5 - 5 80 l The -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 Figure 8.20: Average summer ( J J A ) heat budget due to diabatic heating. T h e units are 10~ K s' and the contour interval is 0.5 x 10~ K s . 5 1 5 81 - 1 80 82 8.3. Budgets of total circulation For the t o t a l circulation, the momentum tendency is given by d[u] dt r i / , \ 5([«]cos0)\ i? cos 050 J > (V,UJ) r ,d[u] dp 1 d([u*v*}cos (f)) Rcos 4>d(p 2 1 >>> terms 2 * d[u*u* dp v— Eddy terms and the potential temperature tendency is given b y d\6) _ dt > [v]d[9] R d<f> (u,uj) d[9] d([9*v*]cos<P) d[9*u*\ / p \ ^[Q] dp Rcoscpdcp dp \ p ) c | L J ... terms Eddy terms 0 0 p *— Diabatic heating where the streamfunction equation (3.4) is solved w i t h forcing of { d 9 x ^ r r o 1 H l c I I e l ^ i H l " * ) _|_ U a a n d the values of [v] and [u] are determined from ip. Zonal-momentum budgets for annual- a n d seasonal-mean conditions are shown i n Figures 8.22-8.26, w i t h potential temperature budgets for annual- and seasonal-mean conditions shown i n Figures 8.27-8.31. U n l i k e the study of [Pfe81], where data from vastly different sources were drawn together i n a n attempt to synthesise the combined effect of eddy- a n d diabatic heating-induced motions, the current investigation relies solely o n d a t a from one source; however, even i f one ignores the inevitable distorting effect of 83 the various interpolation schemes, one is still left w i t h the drawback that there is an unquantifiable influence of the reanalysis model's various parameterisation and assimilation schemes inherent i n the results. A s s u m i n g that, at the very least, the qualitative nature of the results are correct, and m a k i n g the further assumption that comparison w i t h the most trustworthy of the synoptic fields w i l l reveal questionable results, some conclusions on the effect of the combined influences m a y be drawn. Focusing attention on the momemtum tendency, ^ , the principal regions of net positive acceleration are i n the mid-latitudes, w i t h peaks i n the boundary layer near 40° and i n the sub-tropics, w i t h m a x i m a obtained i n the upper troposphere around 30°. Comparison of these tendencies w i t h those resulting from the Hadley and Ferrel circulations alone suggest that the peak positive tendencies at 150 hPa are p r i m a r i l y due to the eddy-induced circulation and the near-surface peaks result from the diabatic heating-induced circulation. Strong cancellation b o t h i n the momentum tendency and the zonal kinetic energy production is evident between the eddy-induced and diabatic heating-induced circulations. Deceleration appears to mostly occur i n the equatorial lower troposphere and more weakly near the A r c t i c surface, w i t h weak upper tropospheric decelerations consistent w i t h the requirements of thermal w i n d balance (equation (3.3)) associated w i t h 84 the calculated potential temperature tendencies discussed presently. E x a m i n i n g 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 i n the boundary layer, occasionally extending as far as 80°; the general tendency is for a cooling elsewhere throughout the year. T h e diabatic heating provides the principal contribution to ^ , w i t h a broader region of heating i n the equatorial mid-troposphere t h a n would be provided by the Hadley circulation alone and the Ferrel circulation providing tendencies of slightly lower magnitude i n the m i d 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 m i d to upper troposphere throughout the year and to reduce it i n the lower troposphere between the A r c t i c and A n t a r c t i c circles; this is compatible w i t h the observed pattern of positive momentum tendencies i n 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 Latitude f) Figure 8.22: A n n u a 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 panels, the units are 10~ m s~ and the contour interval is 0.5 x 1 0 " m s~ . For the b o t t o m panel, the units are 1 0 m s~ and the contour interval is 5 x 10~ m s~ . 5 5 2 2 - 5 5 2 3 86 2 3 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 diabatic heating. For the top three panels, the units are 10~ m s~ and the contour interval is 0.5 x 10~ m s~ . For the bottom panel, the units are 10~ m s~ and the contour interval is 5 x 10~ m s~ . 5 5 2 2 5 5 2 3 87 2 3 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 diabatic heating. For the top three panels, the units are 10~ m s~ and the contour interval is 0.5 x 1 0 m s~ . For the b o t t o m panel, the units are 1 0 m s and the contour interval is 5 x 10~ m s~ . 5 - 5 2 2 - 5 5 2 3 88 2 - 3 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 diabatic heating. For the top three panels, the units are 1 0 m s~ and the contour interval is 0.5 x 10~ m s~ . For the b o t t o m panel, the units are 10~ m s~ and the contour interval is 5 x 10~ m s~ . - 5 5 2 2 5 5 2 3 89 2 3 Latitude f ) Figure 8.26: Average autumn ( S O N ) 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~ m s~ and the contour interval is 0.5 x 1 0 " m s~ . For the b o t t o m panel, the units are 10~ m s~ and the contour interval is 5 x 10~ m s~ . 5 5 2 2 5 5 2 3 90 2 3 Latitude (•) Figure 8.27: A n n u a l average heat budget for the combined action of diabatic heating and eddy fluxes of heat and momentum. T h e units are 10~ K s~ and the contour interval is 0.5 x 10~ K s~ . 5 5 l 91 l Latitude 0 Figure 8.28: Average winter ( D J F ) heat budget for the combined action of diabatic heating and eddy fluxes of heat and momentum. T h e units are 1 0 K s~ and the contour interval is 0.5 x 10~ K s . -5 5 - 1 92 l Figure 8.29: Average spring ( M A M ) heat budget for the combined action of diabatic heating and eddy fluxes of heat and momentum. T h e units are 10~ K s and the contour interval is 0.5 x 10~ K s~ . 5 5 l 93 - 1 Figure 8.30: Average summer ( J J A ) heat budget for the combined action of diabatic heating and eddy fluxes of heat and momentum. T h e units are 10~ K s~ and the contour interval is 0.5 x 10~ K s~ . 5 l 5 94 l Latitude f) Figure 8.31: Average autumn ( S O N ) heat budget for the combined action of diabatic heating and eddy fluxes of heat and momentum. T h e units are 10~ K s~ and the contour interval is 0.5 x 10~ K s~ . 5 l 5 95 l 9. Conclusions T h e results of this thesis suggest indicate that there is considerable wave-mean flow interaction i n the Earth's atmosphere, w i t h the circulation induced by eddy motions being primarily 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 w i l l lead to westerly winds ~ 45 m s ~ 60 ms" 1 - 1 i n the upper troposphere and near the surface over the course of a month. T h e 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 eddyinduced 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 w i n d stress F A ) and ~ 1 m s - 1 i n the tropical upper troposphere. T h e streamfunction results determined by the model presented herein m a y 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 i n the E q u a t o r i a l upper troposphere to its regions of destruction i n the polar lower-stratosphere takes place. Additionally, the results presented i n this study provide a baseline for further investigations of the effect of the E N S O phenomenon on the global circulation of the atmosphere (as 1989-1993 were n o n - E l N i n o years). T h e 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 i n the derivation of the model equations. 97 References [AHL87] D a v i d 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 . M c l n t y r e . Planetary waves i n horizontal a n d vertical shear: T h e generalized eliassen-palm relation a n d the mean zonal acceleration. Journal of the Atmospheric Sciences, 33(11):20312048, November 1976. [BBC+94] R . Barrett, M . Berry, T . F . C h a n , 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. T h e 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 . Kistler, W . Collins, L . G a n d i n , M . Iredell, S. Saha, D . Deaven, G . W h i t e , J . Woollen, Y . Z h u , A . Leetmaa, R . Reynolds, M . Chelliah, W . E b i s u z a k i , W . Higgins, 98 J . Janowiak, K . C . M o , C . Ropelewski, J . Wang, R o y Jenne, a n d D e n nis Joseph. T h e 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 i n the atmosphere. Journal of Meteorology, 13:561-568, December 1956. [Lor67] E d w a r d N . Lorenz. The Nature and Theory of the General Circulation of the Atmosphere. W o r l d Meteorolgical Organization, 1967. [NVDF69] R . E . Newell, D . G . Vincent, T . G . Dopplick, and D . Ferruzza. T h e energy balance of the global atmosphere. I n G . A . Corby, editor, The Global Circulation of the Atmosphere, pages 42-90. R o y a l Meteorological Society, August 1969. [OR71] A b r a h a m H . Oort and Eugene M . Rasmusson. Atmospheric circulation statistics. Technical report, N a t i o n a l Oceanographic and Atmospheric A d m i n i s t r a t i o n , 1971. [Pfe81] R i c h a r d L . Pfeffer. Wave-mean flow interactions i n the atmosphere. Journal of the Atmospheric Sciences, 38:1340-1359, J u l y 1981. 99 [P092] Jose P. Peixoto and A b r a h a m H . Oort. Physics of Climate. A m e r i c a n 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 i n the atmosphere. Pure and Applied Geophysics, 80:346-357, 1970. [TW98] D a v i d W . J . T h o m p s o n and J o h n M . Wallace. Structure of the arctic and antarctic oscillations. Journal of Climate (submitted), October 1998. 100 A. Table of symbols Symbol Cp f 9 P Poo t u V Units j kq K 1 m Description atmospheric specific heat at constant pressure ( = 1004 J kg~ l Coriolis parameter ( = 2f2 sin (fi) acceleration due to gravity (— 9.81 m s~ at sea level) pressure reference pressure (1000 hPa) time zonal velocity meridional velocity geometric height above the E a r t h ' s surface 2 x* Pa Pa s m .s m .s z m Symbol FX Fcfi Units Description m zonal forcing meridional forcing H Q R Rd Sp T Symbol «2 m «2 kq s k m j kq K K Pa heat flux forcing diabatic heating radius of the E a r t h gas constant for dry air (= 287 J kg' 1 temperature K Units Description (none) (= sin (fi) potential temperature K K (none) A rad longitude kg m. density latitude momentum flux forcing streamfunction X 3 rad m *3 m Pa ,s Pa s n 1 static stability (= 9 P 4> K" ) rad fi Cp pressure vertical velocity rotation rate of the E a r t h (= 7.292 x 1 0 101 - 5 rad s' ) 1 K' ) 1 B. Derivation of model equations B . l . G o v e r n i n g equations In the (A, c/>,p, t) system, the appropriately scaled equations of horizontal m o t i o n are du -TT dt = tanc/> gdz —z-uv + fv-— T^T+F , R R cos (pd\ civ * . . (B.l) X tanc/> 2 = -^r 9dz f - u f u ~ m + , m F * > ( R 2 ' } ) where It ~ dt + Rcos<bd\ U Rd<b +V dp T h e equation of continuity is R cos (bd\ { (B.4) UJ du dp' +U djvcoscp) duj R cos (bdcb dp 102 The hydrostatic equation is dp d - -pg- z (-) B 6 The ideal-gas law and the expression for potential temperature are P = pRdT, 0 where p 00 = (B.7) T ^ - J , (B.8) is a reference level (~ 1000 hPa) and K = ^ ~ 0.286. The first law of thermodynamics is "it' Q' where c ~ 1004 J kg' 1 p K" 1 (a9) 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): 0a! tan 6 —ir" , 9 103 - gdz } u - m - Differentiating w.r.t. p gives 2tan</> \ du f + ~~R~ ) dp U _ d ~ ~dp (gdz Rd(j) \Rd4 9 dz 2 Rdpdtfi Changing the order of differentiation and using the hydrostatic equation (B.6) together w i t h the idea gas law (B.7) gives 2tan0 f + R \ du ) U _ dp 1 (p) 9 R d(f> 1 (-£r) d Rp d<f> 2 v RR g(a dcj) 2 dP ( _ P RR 2 p d - 2 ) ^ dT RR p T 2 r ' d4> v dP 2 d(f)' which gives / J + ^ R ^ = ^ . J dp pRdtp 104 (B.10) ' v U s i n g the definition of potential temperature (B.8) gives R 1 d 2tanc/> \ du R d J dp Poo) \ p R d<f> RaP^^dO P Poo P R p ~ 1 d6 Rd( K l d Poo R d(f> which finally results i n the alternative thermal w i n d equation . / 2tan0\0u + - r B s R ~ K 1 d9 l dP i r w = ( B - n ) B.3. Derivation of zonally-averaged equations of motion B.3.1. Derivation of zonally-averaged mass-continuity equation T h e mass continuity equation (B.5) may be zonally-averaged to give 0 ( M c o B 0 R cos (bd(p + g M = a ( B - 1 2 ) dp M u l t i p l y i n g the original mass continuity equation (B.5) by some quantity A 105 and zonally-averaging gives ^ du ^d(vcos(f)) R cos <j)d(j) dp J R cos 4>d\ [A] du d (v cos <fi) dui R cos (pdX R cos <fid<fi du* + J A* R cos dp , d (v* cos <f>) + (f)d\ ^du + R cos <pd<p A* du* dp which, upon using equation (B.5), becomes A* du* + Rcos 4>dX ,d(v* cos0) .du* + (B.13) dp R cos 4>d(j) W i t h the choice A = u this gives the relationship u du* R cos (fidX + . d (v* cos cp) u R cos (f)d<j) + .du* u dp = 0, which yields the useful identity u . d (v* cos (f>) R cos <pd(j)+ 106 u .du* dp = 0. (B.14) 0 0, B . 3 . 2 . D e r i v a t i o n of the zonally-averaged ideal gas law a n d p o t e n t i a l t e m p e r a t u r e equations T h e ideal gas law equation (B.7) may be decomposed into p = R [p] [T]+R [p*T*]. d d Alternatively, one may write P RT = d which may be zonally-averaged to (B.15) Rd[T]=p T h e expression for potential temperature (B.8) may be zonally-averaged to give M = PI and Q* _ rp* (POO \ 107 P (B.16) 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 E u l e r i a n form du du du dt R cos (pdX Rdq> du tanc6 dp R gdz + F R cos <pdX X Taking the zonal average of this equation, one arrives at d\u) u dt du du du '~Rd4> Rcos <f)dX\ du du UJ Rd<b UJ taxi 6 dp gdz R cos (bdX dp uv] + f[v] + + [F ]. x U s i n g identity (2.1) one may write d [u] [v] d [u] dt R 1 <du* du UJ d<f> R tanc/> , r y , tanc/> , * _ 1 L / r i . m dp which may be rewritten as d [u = v f + = H / ^ l u R " L J ld[u 1 R d<p R cos c/>^ — sin tf) [u] a; 108 dp du 1 (90 Rcoscb c5u to— dp + ^ K » " ] + [FA] = M l / - 5 ([it] cos 0) cos 090 du* 50 5M t a; 5p ! ^ [ „ V ] + + | ] ft Now, 5it 5 [it dp cu 5p 0[u] = - OJ - OJ 5p 0Ju] 5p 5u* dp djvTu\ dp 5 [it*a;*; dp + 5cu* dp u*d (v* cos0) cos 050 it from equation (B.14). Substituting this into the expression for the evolution of zonal velocity yields 5 [it = 5 ([it] cos 0) R cos 050 [v] / tan /2 V*] [F ] + ( R 5it* 50 J 5([w]cos0)\ li*5 (v* COS0) 1 5j/ n* cos0i?cos0 50 sin 0 i? COS 050 i?cos0 5 ([li] COS 0) - M = M R COS 050 5 [u] dp if- 5 [u] 5p 5 [u*u;* it*5 (v* cos 0)' i? COS 050 5p A M ( 7 - R cos 050 / =M { / - a; / i?COS0 D — cu 5 [it] 5 [it*a>* 5p 5p , 5u* it*5 (i;* cos 0) cos 0-^-r H —^-7 50 50 5 [u*cu*] dp 5 ([it] cos 0) COS 050 / 5 (li*U* COS 0) i?COS0 109 50 — sin 0u* v* . , „. sm 0it v M = dp + [Fx] 1 d ([it] cos c/>) [v] / r -—- l dp R cos 05c/» y ,d[u] R cos 0 2 d [u*u , <9p r cos 0 3 ( « * w * cos 0 ) , sm 0 cos <pu v dq> ^ . dp d([u]cos0)\ <9([u"V]cos 0) <9[u*u;* 2 cos 0 3 0 / R cos 0r90 2 — UJ dp dp + [Fx B.3.4. Derivation of the zonally-averaged thermodynamic equation Using the expression 6dt p ^ for the first law of thermodynamics (B.9), one may write dt \ p J c p which, in the Eulerian framework, may be rewritten as dt RcoscbdX Rdcf) 110 dp \ P J c p Taking the zonal average of this equation gives d[6] Ot = 86 1 u Rcos (pOXj R m -u u Poo\ [Q] K U Q j)_ , ( Op + P 06 R cos cj)dX . 06' 06* R cos <pOX 06 u* R cos (pOXj 1_ 06_ R '0<p R 0& UJ Op , OJ— 04> } L dP + PooY [Q] P + pwYiQ] p Now, 06* u R cos (pOX 06* u R cos (pOX 0& UJ o\ P M 06* 0[6] Op Op 0[8] 0 [UJ*6* Op Ou*' 6* Op Op 0 (v* cos (p) 6* R cos (pO(p R cos (pOX u + Op [OJ] u R cos (pOX Ou* - Op [OJ] 06* R cos (pOX Op 0[6] _ 0 [UJ*6* u R cos (pOX 0 [u*8*] 9_mUJ.06* 0 [6] [OJ] Op 0 [UJ*6*] , 0 (v* cos (p) Op ^0(v*cos(p) 6 R cos (pO(p R cos <pO(p 0 [u*6* Op using equation (B.13). R e t u r n i n g to the zonally-averaged thermodynamic equation and substituting 111 ^d(v* cos <f>) d [u*8* Poo [Q] + R cos (bd(b dp p „ 9 (v* cos 4>) d [u*e* Poo d[6) _ _1 - CO + p dp dp R d(b R ' d<b\ d(v* cos 1 V* C O S (p • MP W <9p R cos 0 R d<p d(f> R cos <f)d(j) d[9] M dp R [ d(b\ dt K V R cos (pd<p <p) de e* d [u*6*] + dp Poo V p Finally, one arrives at d[6] _ dt [v}d[e] R d(p d[w*e*} d([v*8*}cos(f)) R cos (pd<p 9[g] dp dp (PooY [Q] \ p ) c | p B.3.5. D e r i v a t i o n of t h e zonally-averaged t h e r m a l w i n d e q u a t i o n T a k i n g the zonal average of the thermal w i n d equation (B.10) gives ,d[u] dp „d\u\ dp 2tanc6 r R + R 2tanc/> ,d\u] 2 [u] t a n / 2 tan R dp R (p\ d [u] ) ~dp~ + u dp du* dp <p t a n d(u*f ^T 112 (pu du dp RddT pR d<p RddT pR d(p RddT pR dcp' K [Q] If the variance of u is neglected, one arrives at + 2 [u] t a n <p\ d [u] _ R dT d R J dp pRd(f> In terms of potential temperature, this relationship is 2Mtan0\5M_^- l5[0] 1 J + R J dp p^ R d<P • ^ ' B.4. G o v e r n i n g equations for z o n a l l y - a v e r a g e d m o t i o n T o summarise the equations derived above: • T h e zonally-averaged zonal momentum equation is djv] = c% U ( d([u] cos 0 ) \ V i?cos<pd(j) J o>] 1 1 dp dk^*] g(K^]cos20) dp Rcos ^ 2 " 1 J (B.18) • T h e zonally-averaged thermodynamic equation is d[0] dt [v]d[6] - R d(fi d[0] \U>\ -r— dp d([6*v*]cos<i>) d[9*u*\ — R cos <j)d(j) ~ dp (p \ m h \ p ) [Q] \K c p (B.19) 113 ; T h e zonally-averaged mass continuity equation is a_M^ R cos (pdtp + m . dp (B.20) = 0 • T h e thermal-wind equation is or jd [u] dp where / = (/ + = R^P K 00 M^±y 114 Id [9] 1 R d(f> C. Derivation of diagnostic streamfunction equation for mean meridional circulation F i r s t , it is convenient to make the following change of coordinates r] — sin cp, (C-l) so d dn d cos <pd(p T h e zonally-averaged mass continuity equation (B.20) may now be w r i t t e n as °A R ) , 9 dr\ M = 0 dp T h i s allows the definition of a streamfunction ip: cos (p dp Rdrj R e w r i t i n g equations (B.18) and (B.19) i n terms of derivates w.r.t. n and p 115 gives d[u] ( d([u] cos<p)\ dt V Rdr J 9 [u*u*] d([u*v*}cos (f>) 3H 1 , 2 ' dp dp , Rcos<bdn and d [fl] dt = 0 [fl] [v] cos09[fl] R drj d([9*v*]cos<j>) dp M Rdr] [Q] d[6*u*] (p \ K 00 \ p J dp c ' p 1 ' } Changing meridional coordinate a n d differentiating equation (B.21) w.r.t. t allows one t o write a(( / + ^ [ „ ] ) M ) _ R d P .-i l c o s ( f ) d Poo & dt 2 [0] dtdv C o m b i n i n g 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) dp _R d P ^l^d{f) pfjo R 116 dn into which expressions (C.3) and (C.4) may be substituted: ^)-Mf-ag - Sg^ 3 9 ( / ( M (/ - £( 1 + Ifil) ) dp Hcos^afe] ^.coz4d( r, _ RdP 1 Poo C O b V > C ' \ k d\e] ayr**] d(\9*v*] cos <p) U d r , M d p R [Q]\ S p ^ ^ P ) fl0»j °P ) v d Using the streamfunction definition (C.2) gives I J I u ^ (f cos</> dp \J 1 rlfff dCMcos^A Rdr, . J ^ l dip d[u) Rdn d\u'u'\ dp dp , A\ d([u*v*}cos->4>) Rcoscpdn ' i *1 ) I C O S c65p Rp l U d Poo \ Rdp dr, ^ Rdr, dp Rdr, R dp \ P J c P J drj M o v i n g terms involving ib to the left-hand side and the remaining terms to the right-hand side gives d(f(-^ ^ ( f - ^ r ^ ) + i f ¥ ) ) y yoscp dp y Rdr, J fl dn dp J J cos <pdp = ^9(- f l e ^ g a - 9 a R K di J— u I cos0 I d 2 u dp Rcos<t>dn ti-dP p K R (?)' f) a + t _ a(V*']«» ») (- l *"*] 1 0Q dr] POO PrA-i t X\ , F] ' i 1 dp 117 M J J ^f-^fl + ^f1) \ R°P "V R dr) dp dn J which may be simplified t o d ( f ( - ^ f ( f - ^ ^ ) + ^ ¥ ¥ ) ) V •> V c o s <f> dp \ RdP K Rp X °{~ Ran J cos R9n ~ ttdP fidp p + { y ) 0 0 \ ^"P *• dn R K R an dp J c) K p dn cos<p \ 7* 7f* d{[ + d ( ^ - J ^ l d ( ^ ^ - ^ f ^ ) i R dn dp J I dp K 0 0 J op ] } ~*~ ~ [FX])) L Rcoscpon / / dp Defining H R -w K = d([e*v*]cas<i>) d[e*u*\ dP POQR Rdv \ /p y[g] 00 | dp \ p J c p f (d([u*v*]cos*<j>) d[u*u*} + —dp — i ^ J cos0 \ R cos cfidn allows one t o write V c o s e i \ J Rdn cos <pdp J °P J \ p J 71 R cos <pdp T h i s may be rewritten, taking C = ^ 118 ^dP t y dn Poo R dp dn 2 (/ - ^ dp ^ ) dn J Oil OX dn , [S ] = - ^ f j a n d P dp a (eg) a(/fg) , i ^ - a ( f g - f t Q R cos (pdp PQ dn { cos (pdp PJ V , ° \ d dp 0 dpdn) J | Dp R cos <pdp a{c%) cos (pdp y dn dp J PQ ° \p^ dp dn) dn Poo SH 3X dn dp + dH = dn ,.a(f§*) D j R cos (pdp ^ dn 0 a ( / f » ) _l_ _ dn a(\s ]%) r _j_ d\ | a H = = dn dp dn d x _l_ — _ dp T a k i n g the left-hand side of this equation, expanding the derivative w.r.t. p i n the second t e r m and using the zonally-averaged thermal w i n d relationship (B.17) gives B { %) c + cos (pdp ,D P K S{^) PK dn 00 0 0 ( t) cos (pdp d dp p K R dn 9(IS P ] Q0 K dn + ( \ M Rcos(p dp dpdn %) 0 C / a ay dn s 2 K + 2 I i? p o drydp cos </><9p P + d a» ft.i a(^'f) <9p dn \ dn R cos 0 dp dpoVy dn d[S ] dn D p {%w)d[0]\ dp d + 119 dip dn J dn + f d[u] Rcos(p dp dpdn PK dn 00 d { %) , / C cos (bdp d[S ] , , dn { t) , C { { %) P*- d[6\di> , l D , P* d[6] /K-ldr/f ' ^ P Q O ^ dn \ p D \ P 9 K D cos (bdp D PK 00 P K p K 00 e r dn d { t) d V C ] cos (bdp W 9 W (llzl?t dn / \ p d[u] d j; DpK {^%) 2 pn d[0] d *b p K PK D | K P 2 ^L\ dn 2 d *P D [ s ] dn p 2 d iP dr; 2 | [ D ] s ] d^ 00 + dn 2 | dn 00 d | D [ s ] p { ^ % ) d p K d ip\ DpK (^%) dn ' dpdn J ' p /s 2 { dn 00 dn dpdn 00 d iP d | 2 D 2 Rcos(b dp dpdn Rcos(b dp dpdn | d ib i P d[6] d ip D 2 K dpdn J p K 00 dn dndp en, + dp 2 dn dn dn 00 C dn | PK d[9\dj> X) p p K ( t) cos (bdp dn ) dn dn dn 00 dip^j_d[u] 00 l l)D cos (bdp i 2 pK C d dn p cos 4>dp 9 p K dn OQ . / - ^ [ ^ p \ P K 9 dn p dn JJ P K d[8] U - l d ^ \ p K, dn \ p dn J | D Kd[e\ d ib | D Kd [9}dib p K dn dndp dn 2 2 P 00 00 2 9 | P p K 00 2 D dp [ S 2 ] p ^ dn 2 T h i s finally gives _ . d tb 2 dn 2 Dp d [8] d ip K p rz 00 2 dn dndp Dpn d[9) (K-\ p K Q0 dn \ 120 p dip\ dn J Dpn d [9] dib 2 p K 0Q dn 2 dp 9 ip%) cos <pdp dH dn d dp X D. Monthly plots 121 D.1. D i s t r i b u t i o n s of z o n a l w i n d , p o t e n t i a l t e m p e r a t u r e a n d d i a b a t i c heating 122 123 124 L a W u d e f) LaWude O Figure D . 3 : January-June distributions of [0]. interval is 10 K. 125 U n i t s are K and the contour - 8 0 -70 - 6 0 - 5 0 -40 - 3 0 -20 -10 0 10 20 30 40 50 60 70 80 -80 -70 - 6 0 - 5 0 - 4 0 Latitude (') -30 - 2 0 - 1 0 0 10 20 30 40 50 Latitude t ) Figure D.4: July-December distributions of [9]. Units are K and the contour interval is 10 K. 126 60 70 127 128 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. U n i t s are 10~ m Pa s~ and the contour intervals are 2 x 10~ m Pa s~ (light contours) and 10 x 10~ m Pa s~ (dark contours). 2 2 2 2 2 132 2 contours) and 10 x 10 2 m Pa s 2 (dark contours). 133 W t Jl I: ; (Tf 1 <c i \V. 1 I f\fl.(, ) M -80 -70 -60 -50 -40 -30 -20 -10 0 ^ l(?rm#fl -80 -70 -60 -50 -40 -30 -20 -10 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 Latitude ( ) Figure D . l l : January-June distributions of [v* *] cos <j) for stationary eddies. U n i t s are K s~ and the contour interval is 2 K s~ l l 134 F ii 3 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 1 V 75 40 50 60 70 Latitude ( ) Figure D.12: July-December distributions Units are K s" and the contour interval is 1 135 f [v*6*] cos <ft for stationary eddies. K s~\ 80 100 150 200 250 •S a. f 1 3 °- 3 0 0 400 500 600 700 925 1000 > TJ TJ -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.13: January-June distributions of [u*8*]forstationary eddies. U n i t s are 10~ Pa K s' and the contour intervals are 2 x 10~ Pa K s~ (light contours) and 30 x 10~ Pa K s" (dark contours). 2 1 2 2 1 136 x m Tl -70 -60 -50 -40 -30 -20 -10 0 10 20 Latitude () 30 40 50 60 70 -80 -70 -60 -50 -40 -30 -20 -10 0 10 Latitude () 20 30 40 50 60 Figure D.14: July-December distributions of [uj*9*] for stationary eddies. Units are 10~ Pa K s and the contour intervals are 2 x 10~ Pa K s~ (light contours) and 30 x 10~ Pa K s~ (dark contours). 2 - 1 2 2 l 137 l 70 D.2.2. Transient, z o n a l l y - s y m m e t r i c eddies 138 > lOu I If > TJ > -< J TO. -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Latitude f ) •70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 Latitude 0 Figure D.15: January-June distributions of [u*v*] cos 0 for transient, zonallysymmetric eddies. U n i t s are m s~ and the contour interval is 0.5 m s~ . 2 2 2 2 139 2 140 (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, zonallysymmetric eddies. U n i t s are 10~ m Pa s~ and the contour intervals are 0.5 x 10~ m Pa s~ (light contours) and 5 x 10~ m Pa s~ (dark contours). 2 2 2 2 2 142 2 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 -80 -70 -60 -50 -40 -30 -20 -10 80 0 10 20 30 40 50 60 70 Figure D.19: January-June distributions of [v*6*] cos (ft for transient, zonallysymmetric eddies. Units are K s~ and the contour interval is 0.1 K s~ . l l 143 80 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 -80 -70 -60 -50 -40 Latitude f) -30 -20 -10 0 10 20 30 40 50 60 Latitude (') Figure D.20: July-December distributions of [v*9*]cos(p for transient, zonallysymmetric eddies. Units are K s~ and the contour interval is 0.1 K s . l -1 144 70 80 -80 -70 -60 -50 -40 0 -30 -20 -10 10 20 30 40 50 60 70 80 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 Figure D.21: January-June distributions of [ui*9*\ for transient, zonally-symmetric eddies. Units are 10" Pa K s" and the contour intervals are 1 x 10~ Pa K s~ (light contours) and 10 x 1 0 Pa K s" (dark contours). 2 1 - 2 2 1 145 x 70 80 C/3 m !S 500 -70 - 6 0 - 5 0 - 4 0 - 3 0 - 2 0 - 1 0 0 10 20 30 40 50 60 -80 - 7 0 - 6 0 - 5 0 - 4 0 - 3 0 - 2 0 - 1 0 70 Latttude ( ) 0 10 20 30 40 50 Latitude ( ) Figure D.22: July-December distributions of [UJ*6*] for transient, zonallysymmetric eddies. Units are 10~ Pa K s and the contour intervals are 1 x 1 0 " Pa K s~ (light contours) and 10 x 1 0 " Pa K s~ (dark contours). 2 2 _ 1 l 2 146 l 60 70 D . 2 . 3 . Transient, z o n a l l y - a s y m m e t r i c 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 Latitude f ) 0 10 20 30 40 50 60 Latitude (") Figure D.25: January-June distributions of [u*u>*] for transient, zonallyasymmetric eddies. Units are 10~ m Pa s~ and the contour intervals are 5 x 1CT m Pa s~ . 2 2 2 2 150 70 80 151 -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 Latitude P) 0 10 20 30 40 50 60 70 Latitude () Figure D.27: January-June distributions of [v*9*] cos <f> for transient, asymmetric eddies. Units are K s' and the contour interval is 2 K s . 1 _ 1 152 zonally- 80 -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 10 0 20 30 40 50 60 Latitude t) Figure D.28: July-December distributions of [ v*6*]cos<fr for transient, zonallyasymmetric eddies. U n i t s are K s~ and the co:ntour interval is 2 K s . l _ 1 153 70 80 154 u5 -a- , , , i V ^ B Figure D.30: July-December distributions of [u;*0*] for transient, zonallyasymmetric eddies. U n i t s are 10~ P a X s and the contour intervals are 5 x 1 0 " Pa K s~ . 2 2 - 1 l 155 . A C D.3. E d d y m o m e n t u m and heat fluxes 156 Wim i i i |i 8T .J. m oo A > ill c © -70 -60 -50 -40 -30 -20 -10 0 10 20 Latitude () 30 40 50 60 70 -70 -60 -50 -40 -30 -20 -10 0 10 Latitude (') 80 20 30 2 40 50 Figure D.31: January-June distributions of northward momentum flux. Units are m s~ and the contour interval is 10 m s~ . 2 2 2 157 2 60 70 Figure D.32: July-December distributions of northward momentum flux. U n i t s are m s~ and the contour interval is 10 m s~ . 2 2 2 158 2 -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 L a M u d e (') 0 10 20 30 40 50 60 70 80 Latitude (•) Figure D.33: January-June distributions of upward momentum flux. U n i t s are 1 0 " m Pa s' and the contour intervals are 4 x 1 0 " m Pa s~ (light contours) and 20 x 1 0 " m Pa s~ (dark contours). 2 2 2 2 2 159 2 -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. U n i t s are 1 0 m Pa s~ and the contour intervals are 4 x 10~ m Pa s~ (light contours) and 20 x 10~ m Pa s~ (dark contours). - 2 2 2 2 2 160 2 161 162 Figure D.37: January-June distributions of upward heat flux. U n i t s are 10~ Pa K s~ and the contour intervals are 5 x 10~ Pa K s~ (light contours) and 50 x 10~ Pa K s~~ (dark contours). 2 l 2 2 l 163 l -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. U n i t s are 10~ Pa K s~ and the contour intervals are 5 x 10~ Pa K (light contours) and 50 x 10~ Pa K s~ (dark contours). 2 l 2 2 l 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 Latitude f) 0 10 20 30 40 50 Latitude () Figure D.39: January-June distributions of ip due to diabatic heating (Hadley circulation). Units are 10 m Pa s" and the contour intervals are 1 x 10 m Pa s (light contours) and 5 x 10 m Pa s~ (dark contours). 3 1 3 3 l 166 - 1 60 70 80 k lj : . " — •80 -70 -60 -60 -40 -30 -20 -10 0 * £ v , .r » ••» 10 20 30 40 50 60 70 80 »• » r i "gra^j -80 -70 -60 -50 -40 -30 -20 -10 0 i. JJ Figure D.40: July-December distributions of ip due to diabatic heating (Hadley circulation). Units are 10 m Pa s and the contour intervals are 1 x 10 m Pa s (light contours) and 5 x 10 ra Pa s~ (dark contours). 3 3 - 1 - 1 3 167 l u , 10 20 30 40 50 60 70 80 > -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.41: January-June distributions of the eddy-induced component of -0 (Ferrel circulation). Units are 10 m Pa s and the contour interval is 1 x 10 m Pa s~ . 3 3 - 1 l 168 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Latitude C) -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Latitude () Figure D.42: July-December distributions of the eddy-induced component of ip (Ferrel circulation). U n i t s are 10 m Pa and the contour interval is 1 x 10 m Pa s . 3 3 _ 1 169 - 8 0 -70 -60 -50 - 4 0 - 3 0 -20 -10 0 10 20 30 40 50 60 70 80 -80 -70 - 6 0 - 5 0 - 4 0 - 3 0 - 2 0 - 1 0 0 10 20 30 40 50 Figure D.43: January-June distributions of ip. U n i t s are 1 0 m Pa s and the contour intervals are 1 x 1 0 m Pa s (light contours) and 5 x 1 0 m Pa s" (dark contours). 3 3 _ 1 170 - 1 3 1 60 70 80 Figure D.44: July-December distributions of ib. U n i t s are 10 m Pa s~ and the contour intervals are 1 x 10 m Pa s (light contours) and 5 x 10 m Pa s~ (dark contours). 3 3 - 1 171 l 3 l D.5. Z o n a l K i n e t i c E n e r g y P r o d u c t i o n 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 Latitude () 0 10 20 30 40 50 Latitude () Figure D.45: January-June zonal kinetic energy production due to eddy-flux forcing. T h e units are 10~ ra s~ and the contour interval is 5 x 10~ m s~ . 5 2 3 5 173 2 3 60 70 80 174
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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 |
File Format | 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 |
Graduation Date | 2000-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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