Course.8, General Circulation of the Earth's Atmosphere Prof. Peter Stone Section 8: Lorenz Enery Cycle Enery Forms: As we saw in our discussion of the heat budet, the enery content of the atmosphere per unit mass can be expressed as the sum of four terms: E = I + Φ + LH + K I = C V T = internal enery Φ = z = potential enery K = (u + v + w ) (u + v ) = kinetic enery LH L V q = latent heat. We assume C V and L V are constants. If we assume hydrostatic equilibrium, then for the total internal enery in the column of the atmosphere we have: ρidz = ρc V Tdz = p s (z=) C V T dp; for the potential enery we have ps (z = ) ps (z=) ρφdz = ρzdz = zdp = d(zp) pdz ps RT = pdz = ρrtdz = dp ; (we choose z = to be a level below the lowest surface, z s min, and for convenience, in the presence of toporaphy, we take p = p s for z < z s. We will see later why. Note that the zero point for potential enery is arbitrary.) p s (z=) + R p s (z=) Thus ρ(i + Φ)dz = ( C V ) dp = C P T dp. Therefore in a column of the atmosphere the sum of I.E. and P.E. equals the enthalpy, and all three are equal within a constant factor. This is also sometimes called the total potential enery = TPE. This can be re-written in terms of the potential temperature, θ = T p p p Tdp = θ p R where = : 7 C P dp = θ ( + ) p dp+ ; θdp + = d ( θp + ) p + dθ;
interatin by parts we find for the total potential enery: p s (z=) C Tdp = C P P (+ ) p θp + p=ps,z= p s (z=) p= p + dθ What are the limits on θ? We now adopt the convention that below the surface, z < z s, θ < θ S = surface θ, and p = p s = constant, and θ at z =. (We implicitly assume that dθ >, i.e. the system is always stably stratified.) dz θ p + p s CP = and + ( ) p p + dθ = TPE However, we note that in a stably stratified atmosphere like ours much of the TPE can never be released or converted to KE. In particular, consider a typical arranement of the isentropic surfaces as shown in the first diaram below. Fiure by MIT OCW. θ 6 > θ 5... > θ. As we see there are horizontal radients and because of this PE can be released by the motions in spite of the stable stratification. Consider for example a displacement like that shown below.
Fiure by MIT OCW. For this displacement hih θ air is bein introduced into lower θ surroundins, so θ' >, but w ' > also, i.e., w 'θ' >, warm air is risin, and PE is bein converted into KE This is of course the mechanism of baroclinic instability. Now suppose we re-arrane the isentropes adiabatically (so no new enery is added) and while conservin mass, in such a way as to eliminate any horizontal radients. Since in H.E. p is just the amount of mass above any iven level, conservin mass means the interated or averae p above any θ surface is conserved. The averae p is p! ( θ) = p x,y,θ)da and usin this definition, one can now construct the state with no A ( horizontal radients, as shown below. 3
! + >, ( p ) Fiure by MIT OCW. In this new state, since θ >, there are no displacements which can release PE, z because now for any displacement, w 'θ' <. Thus the PE of this state is unavailable. This led Lorenz (955) to define the available potential enery as the difference between the TPE in the actual atmosphere and that in the adjusted state just described. If we call this P, then for a iven system or reion, C P = P dxdy ( + ) p A Since p! is already interated over the area, we can rewrite the interal as + + A dθ ( p! p" ). This is the so-called exact formula. Note that, since dθ ( p + p! + ) > ( p" ) +, i.e. P is accordin to our definition always positive. Lorenz derived an approximation for P that is commonly used. Let p = p! + p', and similarly for other variables: 4
p + = p! + + p' ( + = p! + +) p' ( +) p' + + + +... p! p! p! " ( +) p # ' p + = p! + + + +... p! C P A ( +) P + ( ) p p! + p # ' p! p A dθ = C P dxdy p! p' dθ As a test of the accuracy of this approximation, we can take from Oort (97) the followin numbers for θ = 3 K in the annual mean: p ( equator) = 98mb, p ( 7N) = 435mb, p! ~ 78mb, p ' ~ 73mb, and the ratio of the st nelected term in the series to the first retained term is ( ) p'.4 73 = =.5 3 p! 3 78 Carryin out an analysis in θ coordinates is not very convenient, because we would have to interpolate from pressure coordinates, which would introduce error. Lorenz introduced another approximation. This is based on the approximation that both isentropes and pressure surfaces are quasi-horizontal with very small slopes: θ 4 z ο y = ~ 4 km ~ 3 and y θ 5 ο θ z km z p y ρƒu 4 = ~ ~ ~ 4 y p. p ρ z Consider for example an isentrope θ on which p has an averae p! ; now consider a point A on θ = constant. At this point, there will be a pressure deviation p'. 5
Fiure by MIT OCW. Now consider the θ variations on p! at the same latitude, φ. Since the θ variations are primarily vertical, or equivalently, vary primarily with pressure, we can write: p' p! θ'=. The minus sin is necessary since <, (Note that as drawn, p', θ' < ).! p θ θ Substitutin this into our interand for P, and chanin coordinates from θ to p! : dθ = d!p!p θ ( ), before interatin over x,y we have that the interand C P p! = p! θ' dp;! Note that θ' is now the variation of θ alon a constant p surface, p θ p since θ = T. p θ' T' θ θ = ; Also + = (Γ d Γ ) (use H.E.) where Γ d = θ T P Γ d p p! = θ Γ = d p! p, and θ = T ; θ p! ( Γ Γ) θ p d Substitutin this, the interand simplifies to C P p!p =!p p Γ d Γ d Γ!p Γ d Γ!p θ θ T T' d!p p!p T' T d!p C P. 6
T' = T (Γ Γ ) dp;! Now we substitute into the interation and absorb the minus sin by d reversin the limits of interation: P = A dxdy P S T' dp! T (Γ Γ ). d Finally Lorenz assumed that the coefficient of T' in the interand varied slowly compared to T', and replaced T and Γ by their area weihted means T! and Γ!. (i.e., 3 rd order quantities are aain bein nelected.) This is not as ood an approximation as the earlier ones. (From Oort s data I estimate ~% error, primarily from T.) Once this assumption is made we can write (notin that now p! is a dummy variable) p S P = dp T! (Γ d Γ ) A dxdyt'. Note that dxdyt' = dxdy T T! ) A A ( = dxdy ( T T T! + T! ) A dxdy T T! ; P = p S dp dxdy T T! T! (Γ Γ! ) ( ) = A ( ) The first term can be expanded in the space domain: T = T + T *, + T T = T * we can decompose P! into mean and eddy components; the mean is p S P M = T! (Γ dp d Γ! ) dxdy T T!. p S (You may verify that P M = dxdy dp T' A T! ( Γ ).) d Γ! d We have included the T! term in the mean available potential enery because it has no zonal variations. The eddy component is the remainin part, P E = p S dp T! (Γ d Γ! ) T * dydx. Now let us compare P to TPE: 7
p S dp dxdyt' T! (Γ d Γ! ) T' ~ p S C!! P TT(Γ d Γ) C P dp dxdyt T ~ T;! = Γd ; T' ~ ΔT; C P P (ΔT) ~ ; ΔT ~ 3 ο, T ~ 5 ο ; Γ! ~ Γ TPE d T Γ! Γ d P 3 ~ ; Hence only ~ to % of the TPE is available for conversion to TPE 5 ~ 7 kinetic enery. Balance Equations for Kinetic Enery: We have already derived this for the eddy kinetic enery, interated over the whole atmosphere, (remember, K E = u + v ), i.e.!!!! K t E dydp = ω ω Z Z dydp + p p ( v H F v H F )dydp + u uv + uω dydp + v v + vω dydp. y p y p Let us expand these terms in the space domain, u = u + u*, etc. Consider the first term: expandin, and also substitutin for H.E., Z α = = ; we have ( ω ωα α )dydp = ω * α * dydp. p ρ The term C ( P E,K E ) ω * α * dydp represents a conversion of eddy potential enery to eddy kinetic enery, i.e. in pressure coordinates in zonal planes, if warm air is risin (α* >,ω* < ) and cold air is sinkin α* <,ω* > ), then C P,K ) >, etc. The factor is necessary to make C have ( ( E E units of enery per unit time. Also, recall that this is total potential enery, i.e. internal enery plus conventional potential enery, not just the latter. 8
Similarly, the second term when expanded becomes simply u * F *+v * F * dydp = D K x y ( E ),! where D ( K ) represents the dissipation of eddy kinetic enery due to friction. Since v E! and F are neatively correlated, the dissipation, D(K E ) is positive. Now consider the third term, and expand it: u uv + uω dydp y p = u ( u v + u * v * ) + ( u ω + u * ω * ) dydp. y p The first and third terms can be combined and rewritten: expandin the derivatives, we have: v ω u + u u v + + u u u ω y y p p = v u + ω u = v! u, y p because of continuity. Now we have a pure diverence, and can interate by Gauss! theorem, and applyin the B.C. v nˆ =, we et no contribution from these terms. Only the eddy terms survive. The same thin happens with the fourth term in the equation, and these terms are combined to define another conversion, this time a conversion from eddy kinetic enery to mean kinetic enery: C ( K,K ) = u u * v * + u u * ω * + v v * + v v * ω * dydp E M y p y p Note that in the Q-G approximation, the vertical flux terms would disappear and the v term would be small, of order Ro. Thus when there is converence of eddy momentum flux positively correlated with the momentum itself, either zonal or meridional, then C K,K ) >. (Peixoto and Oort ( E M (99) ive the conversions in the more complicated spherical coordinates.) Thus we can now write for our eddy kinetic enery equation: t K E dydp = C ( P E,K E ) C ( K E,K M ) D ( K E ) Also, we now need merely to substitute this into our earlier result for the sum of K E and u + K M to et an equation for K M = v : 9
K M dydp = ω Z dydp + v! F! dydp H K E dydp; t p t Rewritin the ω term as before, and substitutin for C ( P E, K E ) and D ( K ) we et E!!!! K v H F v t ωα dydp + ω * α * dydp + dydp M dydp = H * F * dydp + C K E,K M!! = F ω α dydp + v H dydp + C ( K E,K M ). ( ) Define C P M,K M ( ) = ω α dydp; i.e. now warm air risin and cold air sinkin in meridional planes converts zonal mean total potential enery to zonal mean kinetic enery; and define D K ) =!! ( v H F dydp M = dissipation of the zonal mean flow, and we have t K M dydp = C ( K E,K M ) + C ( P M,K M ) D ( K M ). Balance equation for mean available potential enery: We need an equation for P M = P S dxdy dp T' ; T' = T T;! θ' = θ θ T! (Γ Γ! )!, d T! = T! ( ), etc. We start by writin the heat balance equation in p coordinates with θ as the dependent variable: dt dp = H;! dp p R ω = ; θ = T CP ; = dt ρ dt dt p C P T ω p θ p ω = ω RT p ρc + p θ P p p p pc P p T Substitute and divide throuh by C = C P P θ θ θ θ θ p + ω H! + u + v = = θ + v " θ. t x y p p C P t p
Substitute θ = θ' + θ! ( p); also H! = H! ' + H " ( ) " + "! (θ'+ θ! ) + v H (θ'+ θ! ) + (ωθ') + (ωθ! ) = p ( H ' H ). t p p p C P Because of continuity, v H θ! + (ωθ! ) = ω θ!. Substitutin this into the equation and p p! nˆ =, we have take the x,y averae, ( ~ ) ; usin the B.C., v θ + + " θ ω! p H = ; ω! = because of continuity. t p p p C P Note that this is precisely the equation on which lobal mean radiative-convective models are based (cf. Hantel, 976). Also, this equation in effect tells us that in equilibrium, the mean diabatic heatin is balanced by the mean eddy vertical heat flux, ωθ'!. Now subtract this equation from the θ'+ θ! equation:! ( ωθ' ) +! K #! θ' + v! H θ'+ " θ # p H $ ' ( ωθ' ωθ' ) + ω = t p p p C P Take the zonal mean of this equation, and multiply by G θ' where p G = p T! (Γ d Γ! ) ; G t θ' + G θ' y vθ' + G θ " G p θ' = ωθ' ωθ' + G θ' ω p C P p I II III IV V VI (! ) p H # ' θ' Now consider what will happens to each of these terms when interated over y,p. (The interation over x is implicit.) In I, in order to obtain the desired term, P M t, we have to take G inside the time derivative, i.e., we have to nelect the time variations in T! and Γ! in this term. Since these are comparable in manitude to the already nelected y variations, it is no worse an assumption. If we are doin a lobal analysis, it is a much better approximation because the seasonal chanes in the two hemispheres counteract each other. (See Oort & Peixoto, 983, Fi. 47). Substitutin for G and θ' the I term becomes: t p p!t Γ d! Γ ( ) p p T' dydp = P M. t Also note that this approximation does not affect the calculation of the mean balance from this equation.
Terms II and III can be combined and simplified by interatin the perfect derivatives, as follows: dydpg θ' vθ' + ωθ' y p ' (G θ' ) y p y p = dydp vθ' G θ' + G θ' θ'ω vθ' (G θ' ) ωθ = dydp ( v θ' + v * θ'* ) The first and third terms can be rewritten as ( G θ' y ) + ω θ' + ω * θ'* ( G θ' p ) v G θ' + ω G θ' = v! G θ', y p because of continuity, and this interates to zero. Notin that θ* = θ'*, we now have II + III = dydp v * θ * (G θ' ) + ω * θ * (G θ' ) and substitutin for θ', θ * y p and G, this becomes T' ω * T * p T' dydp v * T * y T! (Γ Γ! d ) + p p T! (Γ Γ = C P M,P E d ) ( ). This represents the conversion from mean to eddy available potential enery. Thus if there are neative correlations between the eddy transports and the radients of T', (actually in the ω * T * case it is not the radient of T', but the coefficient of T' does not vary stronly in p), i.e. if the transports are down radient, then C P,P ) > ( M E and P M is converted to P E (see diaram below). In effect, lonitudinal T radients are bein created from the meridional and vertical radients. Note that in the quasi eostrophic approximation, the vertical eddy flux term would not appear.
Fiure by MIT OCW. ( ) ωθ'! Next consider the term IV: dydpg θ' p =, because ωθ! ' and G are independent of y, θ! ' = by definition, and θ ' interates to zero over y. Now consider the term V:! =. We find p p Γ d p V = dydpg θ' ω θ! ; substitute for G, θ' and θ! (Γ Γ d ) θ! V = dydp ω α' = +C ( P M,K M ) Note that α' and α are interchaneable in the interand because the interal of ω is. The factor of comes in because we are interatin in pressure coordinates. Note that α = = RT ; α' = RT' ; if warm air rises and cold air sinks, ω and α' ρ p p in meridional planes are neatively correlated, and P M is converted to K M. Also note that we can replace α' by α because ω α! interates to zero. Finally, the VI term, upon substitutin for G and θ', becomes: VI = G K p H! ' θ' dydp C P p 3
H! ' = T' dydp G P M CP T" ( Γ Γ " d ) ( ) mean available potential enery is enerated if the zonal mean T and H! fluctuation are correlated, i.e. if we have heatin where T is hih and coolin where T is low. Note that # T' can aain be replaced by T because H! ' T" =. Now we put it all toether to et the P M equation: P M = G ( P M ) C ( P M, P E ) C ( P M, K M ). t Finally, a P E equation can be derived in similar fashion by derivin an equation for T * from the same thermodynamic equation. The already defined conversions for C ( P E,K E ) and C( P M,P E ) will appear aain, and the only new term is a eneration term ( M exactly analoous to G P ), i.e.! H * T * ( ) G P = dydp E " CP T) (Γ Γ ) d Thus our equation is P E = G ( P E ) + C ( P M, P E ) C( P E,K E ). t Observational Analyses: We shall follow Oort & Peixoto (983). Oort & Rasmussen (99) cite Oort and Peixoto (983) as their source, but there are minor discrepancies. We will only look at the Northern Hemisphere analyses, because of the better data base. Note however that if we do, then in eneral there are boundary fluxes across the equator which must be added to the equations above. Oort & Peixoto (974) calculated these and found that they were enerally not important except for the flux of P M in the seasonal ( M extremes, B P ), associated with the stron seasonal winter Hadley Cells and the term v G θ'. Thus we add B P ) to the P M equation, P M ( = B ( P M ) +... M The analyses are based on the years in the MIT eneral circulation library, and they calculated the four eneries, C ( P M,K M ), C( P M,P E ), and C( K,K ). Note that in E M evaluatin the first two conversions they assumed Ro <<, and only retained the leadin terms. G ( P M ) and G ( P E ) cannot be calculated because of lack of data on diabatic t 4
( E E heatin, C P,K ) cannot be calculated because of the lack of data on ω * α *, and D( K ) and D K ) because of lack of data on friction. However if any one of the three M ( E G ( P ), C P,K ) and D K ) were known, all the other unknowns can be calculated as E ( E E ( E residuals, assumin a stationary state (see Fi., top). Oort & Peixoto (983) assumed values for G ( P E ) and calculated the others for the annual mean and for the December- January-February and June-July-Auust seasons. They did not explain the basis for their assumed G ( P E ) values in any of their papers. We can reard them as educated uesses. They assumed positive values, which could be explained as bein due to a positive correlation between H! * and T! * (i.e. in zonal planes) due to baroclinic eddies releasin latent heat (which is contained in H! in the enery equations), i.e. warm air movin poleward rises, leadin to condensational heatin. 5
Fiure 6
Fiure 7
Fi. (bottom) shows the annual mean Northern Hemisphere enery cycle, and Fi. shows the Northern Hemisphere seasonal cycles. (Note apparent small errors in the residual calculations in the annual mean.) Lookin first at the annual mean, we note:. The main input is due to G P ), caused by heatin in warmer low latitudes and ( M coolin in cooler hih latitudes. The radiative fluxes cause this both directly and indirectly by drivin low latitude moist convection and condensation.. From P M enery flows mainly to P E, because of baroclinic eddies i.e., their downradient heat transport. 3. P E enery flows mainly to K E, i.e. K E is enerated by upward eddy heat fluxes, i.e. ω * T * >. 4. This flow is mainly balanced by dissipation, i.e., mainly by friction near the surface. 5. There is however some flow of enery from K E to K M, i.e., there is an upradient transport of momentum by the eddies, on balance. In fact this is the main source of K M. 6. There is some flow of enery from K M to P M, i.e., on balance ω T <, i.e. on balance this conversion is dominated by the indirect Ferrel Cell. 7. D K ( ) are relatively small. ( E ) and G P E Fi. shows the seasonal chanes. Note:. In winter the dominant enery source is B ( P M ), i.e. P M is mainly bein enerated in the Southern Hemisphere, but a lot of it is bein transported across the equator to the Northern Hemisphere.. G ( P M ) is reater in June-July-Auust, presumably because tropical moist convection and the associated LH release is reater then. 3. The eddy converences are all much weaker in summer and thus the flow of enery is weaker in summer, and all the enery forms are less. 4. The Ferrel Cell does not dominate C ( P, K ) in December-January- M M February, because it has to contend with the very stron winter Hadley cell. Because of the difficulty in calculatin the enery cycle (calculatin G repeatedly for example) there have been very few attempts to calculate it from models. The only one I know of was with an early version of the GISS GCM (Somerville et al, 974). It used 4 lat (lon ( ) x5 ) resolution with 9 vertical levels, and a sinle January was simulated. No boundary fluxes were calculated; the Northern Hemisphere simulated enery cycle is shown in Fi. a in Somerville et al (974). Comparin with the Peixoto and Oort winter results, we note. The enery cycles are qualitatively similar.. Most of the model s eneries are too hih (except for K E ) 8
3. Most of the conversions, eneration and dissipation terms are too weak, except for C P M,P E ( ) ( ) is.6, close to Oort and Peixoto s assumed value. 4. Gratifyinly, G P E Other features for the GISS simulation indicate that the differences are not due to the resolution or short time base. The lare K M may be in part because the tropospheric jet is not closed. Note that the results for the Enery Cycle tell us that the EP theorem is not at all satisfied in the trosposhere, because there is a net exchane of enery between the eddies and the mean flow. C P,P ) is about four times C K,K ( M E ( E M ) 9