THE UNIVERSITY OF MI C HI GAN COLLEGE OF ENGINEERING Department of Meteorology and Oceanography Techn.ical Report No. 1 FORMULATION OF AN OBSERVATIONAL STUDY OF KINETIC ENERGY CONVERSIONS IN THE ATMOSPHERE A. Wiin.n-Nielsen ORA Project 06372 under contract with.o NATIONAL SCIENCE FOUNDATION GRANT NO. GP-2561 WASHINGTON, D. C. administered through~, OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR January 1966

TABLE OF CONTENTS Page ABSTRACT v 1.i INTRODUCTION 3 2. BASIC EQUATIONS FOR THE ENERGY CONVERSIONS 3 REFERENCES 16 iii

ABSTRACT The total kinetic energy in the atmosphere has been subdivided into four energy reservoirs. The partition of the kinetic energy is accomplished by dividing the total flow into the vertical mean flow (the barotropic component) and the vertical shear flow (the baroclinic component). Each of these components is subdivided into the zonal components and the eddy components. The complete energy exchange diagram is derived by dividing a given energy conversion in the contribution from the quasi-non-divergent flow and the contribution from the divergent flow. Such a subdivision of the energy conversion is advantageous because the calculations are based on geopotential data. Calculations will be carried out for five months (January, April, July, October, 1962, and January, 1963) based on five isobaric surfaces (20, 30, 50, 70, and 85 cb). The complete energy diagrams will be presented for each month together with an averaged diagram representing the annual mean. v

1. INTRODUCTION The major components of the energetics of the atmosphere originally formulated by Lorenz [2], have been investigated in great detail during recent years. A comprehensive summary of calculations based on observations has been given by Oort [31 who also has included some results of studies of the general circulation based on long-term numerical integrations of theoretical models of the atmosphere The kinetic energy of the atmosphere has been subdivided into the kinetic energy of the zonally averaged flow and the kinetic energy of the remaining flow, the eddies. Several estimates have been made of the energy conversions between the eddy kinetic energy and the zonal kinetic energy, (Starr [8], Saltzman and Fleisher [4], Wiin-Nielsen, Brown and Drake [12, 13]) A different subdivisi.on of the kinetic energy was introduced by Wiin-Nielsen [10] in close collaboration with Smagorinsky [71 who used this subdivision to describe the energetics of his basic general circulation experimento The new subdivision consists of dividing the atmospheric flow into the vertically averaged flow (the so-called barotropic component) and the deviation from the vertical mean flow, the vertical shear flow (or the baroclinic component) of the atmospheric flowo The original pilot study by Wiin-N.ielsen [10] was later extended (Wiin-Nielsen and Drake [14,15]) to cover much larger data samples and a greater vertical resclut:on in addition to an estimate of the contribution from the divergent part of the wind to the energy conversion 'between the vertical shear flow and the vertical mean flow. Smagorinsky [7] extended the study of the energy conversions between the different forms of kinetic energy to cover all possible components of the energy transformation between the four forms of energy. (1) The kinetic energy of the zonally averaged vertical mean flow; (2) the kinetic energy of the eddy component of the vertical mean flow; (3) the kinetic energy of the zonally averaged vertical shear flow; and (4) the kinetic energy of the eddy component of the vertical shear flowo We shall in the following secti.ons denote these components by the symbols' KMZ KME, KSZ, KSE. Numerical values of the many energy conversions between the four energy forms can be found in Smagorinsky~ s paper [7] for his model, The same energy conversions have not to the knowledge of the authors been calculated from atmospheric datao It is the main purpose of this paper to present the formulation of such calculati.ons based on atmospheric height data from five isobaric levels: 85, 70, 50, 30 and 20 cb, The height data will be the routine objective analysis carried out by the National Meteorological Center, UoSo Weather Bureau in connection with its short range numerical prediction program The data which have been used in several studies of the energetics 1

of the atmosphere (Wii.n-Nielsen [11]) were originally made available to us oy Dr. George P. Cressm.anr There are significant differences between an observational study of atmospher.c energy conversions and a study of the same quantities based on a numerical i.ntegration of a set of equations which simulates the thermohydrodynamic behavior of the atmosphere on a large time-scale including modeling of the atmospheric heat sources and the frictional dissipationo One of the main differences is the fact that atmospheric parameters such as the vertical velocity, the horizontal divergence, and the distribution of the atmospheric heat sources escape ordinary synoptic analysis and must be computed by more or less realistic indirect methods while they are readily available as by-products of the numerical integrations of the model equations with an accuracy as great as the electronic computer employed for the experiment will permit. Due to this fact it has been found advantageous to transform the original expressions for the kinetic energy ccnversions in such a way that we isolate easily computable quantities such as the hcrizontal wind and the vertical components of the vorticity in separate integrals while the quantities which must be computed by indirect methods (divergence etco) are isolated in other integrals. This subdivision is made possible by using well known identities between terms in the hydrodynamic equationso As shown in the earlier paper (Wiin-Nielsen [10]), we can in this way distinguish between terms which will make contributions in a quasi-geostrophic formulation of the atmospheric dynamics and the terms which will contribute only in an atmospheric model based on the primitive equations. Section 2 of this paper contains an outline of the basic framework and the formulas which will be used in the calculat:i.ons. 2

2. BASIC EQUATIONS FOR THE ENiG, GY CONVERSIONS The integrals which are used for the calc!i- Lation of the energy conversions are naturally very similar to those derived by Smlagorinsky [7], although his derivations apply to a two-Level representatil of the vertical structure of the atmosphere. We begin by defining the wini coimponents used in this study. The vertically averaged wind vM is defined by -he expression vM - P (.)1o, (2.1) in which v is the horizontal wind vectorx with o. omponents u and v, p is pressure and po = 100 cb. The deviation of the wind from the vertical mean vM, the shear vector, is defined by the relation vS = v.- vH. (2.2) Each of the wind components will be subdivided in the zonal average defined by the relation ( ) 1 di a, ( -.) where A is longitude, while the eddy componert is defined as the following deviation: (2.4) ( )E ( )z * (2 4) The amount of kinetic energy in thre zonna part of the vertical mean flow can be evaluated from the formula: KZ ' — u 2) dS (2.5) 9

2 in which dS = a cos d d dp is the area element (a is the rad'ius cf the earth, and cp is lat:itude), while S is the total area of.ntegrat:.on, The corresponding expressions for the three additional energy forms are KME ' PC / g 1 (u 2.+ N 2) dS 2 ( 2 6) Ksz PO 1 0 S 1 2 - (USZ2 2 + V 2 ) dS dp and KSE PO 1 1 5 E 2 SE dS dp. During the derivations it has been assumed that the boundary co:ndi ticins for (c are () = 0 for p = 0 and p = po. It follows then from the general cc:ntinu-ity equation in pressure coordinates that 7 ' ^ =: 0 and V" v =' ~,. (2.9) When we average V vM 0 in the zonal direction we cbta:in 1 a VMZ cos cp a cos cp ac fromrr whh. ch.1t:follows poles we obtain v -MZ that vMZcos cp is constant, but s:ince vT. cos c = 0 at the 0 everywhere, The express:.on (2.5) reduces therefore to KMZ - -P j dS S 2 S ('2,10) The complete equations for the energy transformations are cbtaned 'by deriv:,ng equations for the rate o:f change of KMZ, KME, KSz a:nd KSE- From (2ol0) and (2 06) we obtain dKM7/dt PC & S uMZ MZ dS at (2,11)

and Po 'j-ME~TV~E -~t- 7 bME dK. /t- [ UME ME E dS (2,12) ME g S with expressions analogous to (2.12) obtained from (2o7) and (2,8). It is seen from (2o12) that we must go through the following steps in order to derive an equation for dKME/dto First, we must derive the equations governing the local rate of change of uME and vME. The next step is to multiply the first of these equations by uME and the second by vME, The final step Ais to add the resulting equations and integrate over the domain So A similar procedure must be followed in order to obtain equations for dKsz/dt and dKgs/"dto The derivation of these equations is rather straightforward althou.gh labori.ous and cumbersome due to the many different subscripts and the two different averaging procedures which are being usedo However, the equation for dKMz/dt is rather easy to easy to get according to (2o11)o We shall therefore derive this equation in detall and also obtain the equation for dKME/dt while the remaL.ning two equations will be given without derivationo The first equation of motion will be used i:n the form u. u_ v au. au. 1 a +-U -A.... fv at a sa cos p s a 6 -p a cos cp a\ 2,, 13) UV *uv tan cp + F, a x where c, dp/dt is the verti'cal velocity in the pressure system., 9 =- gz the geopotential of the m.sobaric surfaces, f the Cor-:cl.s parameter and Fx the frictional force per u.n:.t mass in the zonal direct?.ion. We first perform an averaging process in the vertical direction by applying the operator (2ol) to (2.13)o Writi:rng u - uM + uS, v: vrM:- v etc, c we obtain buM uM uM vM -u S c u, v$ du s VS. -t —o 4 - a - acos - a 7- + -a ' a S 4 )t a cos c Pa a bc a cos c bA a 6cp S 6Cp 'JM 5

as M + UMVM +M (USVS)M + 2 + f v-M + tan ~ + FxM ( 2. a cos cp a a Our next step is to apply the operator (2..5) to (2.14). Writing UM = uZ + uME and analogous expressions for the other dependent variables we obtain auMZ VME aUME VSZ auSZ - + [ - - ] + [ ]a at a a Z a aq) M VSE uSE a acp MZ r uSZ + [ ]1 + sz p M [wSE auSE ]MZ [SE MZ Op (2.15) (UMEVME )Z USZVSZ M ( SESE )MZ -a -tan c + tan cp + tan + Fx MZ In order to obtain a more convenient and shorter form of (2.15) we make use of the continuity equations for the zonal and eddy parts of the vertical mean flow and the vertical shear flow, i.e., 1 a cos cp auME avME cos c [^ + ] ah aOcR =0 ( r 1 - C \, ci )J0 1 - uSZ vsz v cos cp awSZ a s +- ]+ = a cos p aac ap ( 2. 17) and 1 _ duSE a cos cp La 6 SE cos auSE + ] + bqC bp = 0 (2.18) We can then write (2.15) in the so-called momentum form and obtain 6uyz ~t+ 1 a coso Cp a(usZvSZ cos2r)M ac (2.19) 1 a cos2Cp (UMEME cos )Z dcp 1 a(USESME cos C2 a cos2Cp cp = FxMZ 6

The equation for dKM /dt is now obtained by multiplying (2cl9) by -y2 and integrating over the region S which we assume to be bounded in such a way that v cos cp = 0 at the boundary. This requirement will certainly be fulfilled if S is the region of the whole eartho We obtain dKMZ dt C(KME, KMZ + C(, K ) C(KS K C(K KMZ) - D(KMZ) (2.20) in which C(KME KMZ) 2n a po 9 (P2 r,. 2 a (u MvE z cos C - I (UEME)Z Co u-M ) dcP coscp (2.21) C(KsE, KMz) 2-: a po _. cpi CP2 1~~~~'~ coscp dcp (2,-22) C(Ksz, KM) 2n a p0 g CP2 P 'U.2 _MZ ' (usvsz) cos - ( ) dcp C1J DZI bz 5p Cvdp ccscp ePz (2.23) (2,24) and D(KM) -?a p 2 2 (2 2t a P / U - uE mzx cos c dc g cpi It will be noticed that the expressions (2o21) to (2o23) are obtai.ned after an integrati.on by parts using the lateral boundary conditiono The integral (2o21) can only be called the energy conversion from KME to KM when it has been shown that the same integral with opposite sign appears in the equation for dKME./dt, Similar remarks apply to the integrals (2~22) and (2.253) An inspection of the integrals (2o21) to (2o23) shows that they have the same form. as the well-known energy conversion between the eddy motiocn and the zonal motiono The physical interpretation of energy conversions of this klind has been discussed by several authors (Kuo [1], Starr [8], Wiin-Nielsen etoalo, [12,131]) The kinetic energy of the zonally averaged vertical mean flew will increase if we have a positive correlatcion between the proper momentum. transport and the shear of the zcnal, vertical mean current uM,. It is obvious that the three integrals represent physical processes cf the same nature but while a numerical estimate of the first two integrals ((2o21) and (2o22)) can be obtained from the nondivergent assumption:.t is evident that the third 7 1

integral depends entirely on a divergent wind component (vvz) because vSZ vanishes when we substitute the assumption of a nondivergent wind. If we therefore want to calculate the integral (2.23) we must be able to calculate the vertical velocity, the horizontal divergence and thereby the divergent wind components. The procedure which is used for this purpose will be described in a later section. We are next turning our attention to the derivation of the equation for dKME/dt. The first goal is to derive equations for the rate of change of UME and vME with respect to time. The equation for the rate of change of uM has already been given as Eq. (2.14). The corresponding equation for vM is derived in a similar manner from the equation + u v + at a coscp?\ 1 v v+ o v a acp ap - - fu - tancp + F a acp a y (2.25) and we obtain avM uM avM at a coscp a\ 1 (M 2 CP M VM 6vM us vs, V 6vs avs + [- + I ' ]I a acp a coscp a? a acp S ap M UM2 + (us2)M - ( tan cp + FyM a (2.26) In order to obtain the equation for auME/v t we must subtract Eq. (2.14) from Eq. (2o13). After some rearrangement we arrive at the equation ME UMZ uME UME UME VME auM vME aUME at a coscp a a coscp aS a a a C ( UME) + a ep z USz auSE SE auSE + Vz uSE a coscp + + p a cp a coscp ah a coscp a a a + vSE Usz + a acp vSE aSE (SE uSE) + w USE au a a ( Z S p dpSE p dSE 8uSE ap / SE ( 'SE ) ] = - pZ M 1 aME uMZvME - -- -- - f VE + tancp + a cosCP (p ME a 8

UMEVME ( UMEVME 7 _ _SSE )M ( ib.SEVSE )M tnc - tanc + tancp + a a a a USE_ _ (u'SE SZ)M. tanp( t p + _ - tan + tanc F (2 27) a a xi We obtain the equation for avME/' t by subtracting (2~26) from (2.25).o After a similar rearrangement we obtain dv ~ UM yE ME 'jMZ aVME 4 — - + — dt a ccscp dcp a.oME 6ME a coscp a\ 1__ME aVME ME VME a cosp \ ) Z a cp Z a8.rME avME U. s ( a icp 'z a cosp VSZ avSE vSE aVsz a cp a dcp aVSZ aVSE aUSE, p ', SE p )vSE USE vSE USE SE d\ ' a coscP S a coscp ah aZ vSE SE SE vSE v SE -{ ) + + a acp a ap) Z )Z+ p VSE SE1 1 - SE dp Z'M a f UME M;' —ME uMEm iME ( 'ME tanc - t a:ncp - ta::,-a tancp a a a a ( U.zSE N a ( uSE 'SZ) M tancp - tanD - a /', ) ( U 2. \ ( SE )M uSE') MZ — tancp - t- tanp - F a a Y? ME (2- 2;8) 'TME the By The following step cons.i'sts of multiplying (2o27) by UME, (2o28) by and adding the resulting equationso The resulting equation will express 1 (U.ME2 + 2 local rate of change of the eddy kinetic energy~ kME =- ( uME2- ME2 o ing use of the contnuity equatns we obtain the llowing equati makhing use of the continuity eqU.atLions 'e obtain the following equation. kME + at uMZ akME a coscp 7\ UMj aE kMJE d-ME UM MEVEME 8 ulMZ a costp a7 a 6cp a cp 9

+Iu [ i((USZuSE) 1 (VSZUSE) tan cp + UME - 2 USEVSZ a coscp a cp a S + 1 a(USEUSE) + 1 (USEVSE) 2 tancp --- -- - -- S ---- "2 u v a coscp a p a SE SE, 1 _ _(USZuSE) 1 s( USZvE) tancp + + - 2 -- USZVSE]M a coscp a a a a L a (VSEUSZ) 1 a(VSEVSZ) + VME [ (+vEUSZ _ ( a coscp a a acp tancp VSEVsz+ a tancp uSEUSZ + 1 a coscp 1 +a coscp a coscp a( SEUSE) ad 1 a(vSEvSE) tancp a — USEUSZ+ a a6c a tancp USE USE a ( vSZuSE) 1 ( vSzvsE) tancp a p a aa a a a tancp v v + SE SZ a SE SZ SE SZ M - 1 6aME 1 6aME UMEVE uaE - - VME - a coscp 6A a aCp a V MEFy, NE UMZ tancp + UMEFME + (2 29) The equation for dKM/dt is now obtained by integrating (2.29) over the region S employing the boundary conditions mentioned earlier. One could naturally be satisfied by simply listing the terms which would appear according to the form of (2.29). However, in order to distinguish between the terms depending essentially on the rotational and the irrotational part of the flow which in turn, according to our present experience, will be quantities of first and second order of magnitude, it is advantageous to rewrite Eq. (2.29) in terms of vorticity and divergence. This can be done by using a number of well known identities. Writing the terms in the new equations in the same order in which they appear in (2.29) we obtain: 10

akME at UMEVME a uMZ akME UME a coscp ah a coscp bkME vME dkME _+ +- a aW a a6( cOS(s ac 'UMZ ( -cop ME coscp 1 a cosa coscp [USZUSE + VSZVSE ( USE2 VSE 2 ) VSZ SE + USE V S SZ 1a c a coscp - VSE SE + uSEV SE 1 a USZUSE + VszsE ] -VSESZ C USZ V 'VSE M 2 a- cp 2 C + USZ SE + VSE V CSZ 1 ( SE SE" a + U S + VE 'V " + USE SE VSE V SE M SESE SE SE SE `SE SE SE M - 1 U ME I _ a coscp UME a bME aVME +; +u TF EacD 4ME x,ME + v F ME y,ME E (2W) (P.z~.5) When Eqo (2o30) is integrated over the region S we make use of the fact that the windfield vE i. s:nondivergento The resulting equation can be written in the form. dKME.-. C( K KMZ, ME). C(KSZKME) C (KSE,KE + at C(KSE>[KS ], Z] ) -D(K ), where the symbols have their usual meaning except that the term C(KSE,[KSz iKME) represent an energy conversion from KSE to KME 'in which KSZ participates al though it remains unchanged. Such an energy conversion is called catalytic by Smagorinsky [7] in close analogy with the well known chemical reactions. The integral for C(KMzKME) - - C(KMEKMz) has already been given in Eq. (221) o The remaining integrals are: 11

C(KszKME) P CP2 2= 2ata [((UMESE)ZVSZ - (VMESE)ZuSZ - o cp1 ((UMEUSE)Z+ (VMEVSE)Z) V 'VSZ] coscp d dp (2.32) ( KSE,'KME) Po CP2 2:t a g J J J ~ cpi ~ [ ( UMSE - ESE) SE - (U ME SE + VMEVSE) V "VSE] coscp dd dcp dp (2,533) C(KSE, [Ksz],KME) P P2 = a" 0 [ (u vSE-vMEuSE)Z SZ o cpi -(U E+ (V SE)ZS cos d dp ((uME VWVSE)ZUSZ + (VMEV'vSE)ZvSZl coScp dp dp (2.34) In order to complete all energy conversions it is necessary to derive the equations for d Ksz/dt and d KsE/dto This procedure is necessary because we can check the calculations leading to (2.20) and (2o31), and because we want to check that our interpretation of the catalytic energy conversion (2534 ) is correct. It suffices to reproduce the main results dKsz -d C(AzKsz) C(KSEKSZ). C(KME,KSZ) + C(KMZsKS ) - D(KSZ) dt (2,35) and dKSE d- =- CC(AE,KSE) - C(KSE,KSZ) - C(KSEKME) - C(KSEKMZ) dt - C(KSE,[KSZ],KME) - D(KSE) o (2236) The first terms on the right-hand side are the conversions from the zonal and eddy available potential energy to the zonal and eddy shear flow kinetic energy, respectively. They have been calculated by Saltzman and Fleisher [5], [6] and Wiin-Nielsen [91 The only other term which appears in (2 35) and (2,36) and which are not found in (2O20) or (2c31) is the energy conversion C(KSE, KSZ) 12

whiich turns out to be P ~2 SE ) _ r2a2 J0 [(v SEME)ZU SZ (- SE-ME)ZSZ 0 '.1 8USE avSE (ME p )Z USZ - (E ) Vsz] cos dc dp d (27) One would normally expect that an energy conversion like C(KSEKSZ,) woulld contain quantities whith subscripts SE and SZ only. In (2~37) we find, however, also the subscript ME, A similar remark can be made with respect to Eq (2.32) expressing the energy conversion C(KSZ,KME) In addition to the subscripts SZ and ME we find also the subscript SE in (2~32)> The reason for this special condition in (2.32) and (2o37) is that the zonal shear flow energy act as a catalyst for the energy conversion from KSE to KME. The energy conversion, C(KSE,KSZ) and C(KSZ,KME), given in our formulas (2537) and (2o32) are in both cases the noncatalytic part of the total energy conversion~ Hence the appearance cf the additional subscripts, Smagorinksy ['7] gives the necessary formulas to find the catalytic and noncatalytic parts of a given energy conversiono Suimmarizing the results of the derivations of the energy conversions involving the four forms of kine-tic energy we find a total of seven energy conversions given in Eqo (2'21.), (2 22), (2 23), (2o32), (233), (2o34),:. d <_5,7)0 The additional terms concerning a conversion from available potential energy or a frictional dissipation will not be considered in this paper. A schematic diagram given the seven energy conversions is shown in Figure 1, where the dashed line indicates the catalytic energy conversion, Only the integrand in the convers. on integrals is shown in the diagramo The integration is carried out over the area S, where dS - a coscp dh dcp, and -with respect to pressure, oeo, 1 dp The rectangular boxes:n Figure 1 represent the amounts of energy while ~he hexagonal boxes show the energy conversions and the integrand which has to be used for the numerical evaluation of the integral, It will be noticed from the formulas for the energy conversions and from Figure 1 that there are three different energy conversionso If any energy conversion requires any component: of the mean meridional circulatiaon, a divergence or a vertical veloc'ty, it will be classified as a divergent component of the energy conversion and denoted by a subscript. D The remaining parts of the energy conversions will be classified as nondivergent components and will have a subscript: NDo We find by inspection of the formulas that one energy conversion C(Ksz,KMz) - CD(KSzKMZ), while two conversions: C(KMEKM) = CND(KME,KM) and C(KSEKME) CN(KSEKM) contain only nondivergent componentso The remaining four kinetic energy conversions contain both nondivergent and divergent components The nondivergent components wwill be present ii....:asi-geostrophic model of the atmospheric motion, while we must use the primitive equations to incorporate the divergent components in a numer.tical integration 13

a a CosUMZ C( KS K )-/C(ME IKSZ)~ c (VSESME) USZ -(USEM(ME SE) USZ-(UMSE)VSZ ( US E a dVSEu sE USME SEMEVsE ME M-(Ep Usz- (ME ) I, --, (,., ) KME'KMZ)-' 'ME) Coso r-MZ 0 dCos H 4:: — KSE KSE No-* --- / L- L &-J i- vi. -C ( UMEVSE-USEVME)SZ -(UMEUSZ +VMEVSZ)V.' \ 00, C(KSE KME) (UMEVSE- USEVME)SE -(UMEUSE+VMEVSE)V. 'SE Jigure 1. Schematic energy diagram showing all energy conversions between the four energy forms: Ksz, KSE, KMZ and KME. The hexagonal boxes show the energy conversion together with the integrand in the energy conversion integral.

The nondivergent energy components create no special problem in a calcuilat:ocn based on atmospheric data. Only height data are available for the calculations to be performed in this study From the height data we will obtain a socalled geostrophic streamfunction ' by solving the equation 2 2 1. V\ - V f ~ 7 (2.8) f f2 The vortic.ity wi.ll be computed as = V f while the wind components wi1 be obtained from u - - -l and v The streamfunction.;s first a ac a coscp a obtained by solving (2o38) at the five data levels mentioned in the introducticono The four components, MZ,' ME, *SZ' and VSE are next computed using the definiti.ons (2,1) to (2,4). The remai.ning calculations necessary to obtain all the com.pcnents CND will be completed using a computat'ional procedure as described by Win.:n-Nielsen and Drake [14]o The energy conversions CD require a knowledge of a vertical velocity, a divergence or the components vSg of the mean meriLdional circulation. In order to obtaln a numerical estimate of the conversions CD we will follow a procedure as outlined by Wiin-Nielsen and Drake [15] by computing the vertical velocity, from the so-called (,.-equation, obtain the hor.:izontal divergence from the continuity equation and then calculate aE, V`vrSE and V\VSZ using the averaging procedures Finally, the component vSZ is obtained by integration of the conc tinultty equation for the zonally averaged flow 1 v cosc sc a coscp -cp 9z Equation (2,39) can be integrated starti ng from the North Pole where vZcosc = O iZ1 15

REFERENCES 1. Ho L. Kuo: "A Note on the Kinetic Energy Balance of the Zonal Wind Systems," Tellus, 3, pp. 205-207, 1951. 2. Eo No Lorenz: "Available Potential Energy and the Maintenance of the General Circulation," Tellus, 7 No. 2, pp. 157-167, 1955. 35 A. H, Cort: "On Estimates of the Atmospheric Energy Cycle," Monthly Weather Review, 92, No 11, pp. 483-493, 1964. 4. B. Saltzman and A. Fleisher: "Spectrum of Kinetic Energy Transfer due to Large-Scale Horizontal Eddy Stresses" Tellus, 12, ppo 110-111, 1960. 5. B, Saltzman and A. Fleisher: "The Modes of Release of Available Potential Energy in the Atmosphere," Journal of Geophysical Research, 65, pp. 1215 -1222, 1960, 6. Bo Saltzman and Ao Fleisher: "Further Statistics of the Modes of Release of Available Potential Energy," Journal of Geophysical Research, 66 pp. 2271-2273, 1961. 7 o J Srmagorinsky~ "General Circulation Experiments with the Primitive Equations, I-The Bas.ic Experiment," Monthly Weather Review, 91, pp. 99-165, 19635 8 V. P. Starr: "Note Concerning the Nature of the Large-Scale Eddies in the Atmosphere," Tellus, 5, ppo 494-498, 19535 9o A. Wiin-Nielsen: "A Study of Energy Conversion and Meridional Circulations for the Large-S ale Motion in the Atmosphere," Monthly Weather Review, 87, pp. 319-332, 1959. 10o Ao Wiin-Nielsen~ "On Transformation of Kinetic Energy between the Vertical Shear Flow and the Vertical Mean Flow in the Atmosphere,"' Monthly Weather Review, 90, pp. 311-323, 1962. 11. A, Wiin-Nielsen: "Some New Observational Studies of Energy and Energy Transformations in the Atmosphere," Proceedings from WMO-IUGG Symposium on Research and Development Aspects of Longe Range Forecasting, Technical Note No 66, 1964. 12. A,, Wiin-Nielsen, J. A, Brown and Mo Drake 1 "On Atmospheric Energy Conversions between the Zonal Flow and the Eddies," Tellus, 15, pp. 261-279, 1963 16

13 Ao Wiin-Nielsen, J. A. Brown, and M, Drake: "Further Studies of Energy Exchange between the Zonal Flow and the Eddies," Tellus 16, ppo 168-180, 1964. 14. A. Wiin-Nielsen, and Mo Drake: "On the Energy Exchange between the Baroclin.c and Barotropic Components of Atmospheric F.ow," Monthly Weather Review, 93, pp 79-92, 1965G 15. Ao Wiin-Nielsen and Mo Drake: "The Contribution of Divergent Wind Components to the Energy Exchange between the Baroclinic and Barotropic Components," accepted for publication in Monthly Weather Review, 94, 1966. 17

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