The Vertical Structure of the Eddy Diffusivity and the Equilibration of the Extratropical Atmosphere

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1 The Vertical Structure o the Eddy Diuivity and the Equilibration o the Extratropical Atmophere The MIT Faculty ha made thi article openly available. Pleae hare how thi acce beneit you. Your tory matter. Citation A Publihed Publiher Janen, Malte, and Raaele Ferrari. The Vertical Structure o the Eddy Diuivity and the Equilibration o the Extratropical Atmophere. Journal o the Atmopheric Science 70, no. 5 (May 2013): American Meteorological Society American Meteorological Society Verion Final publihed verion Acceed Tue Nov 06 07:21:54 EST 2018 Citable Link Term o Ue Detailed Term Article i made available in accordance with the publiher' policy and may be ubject to US copyright law. Pleae reer to the publiher' ite or term o ue.

2 1456 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 70 The Vertical Structure o the Eddy Diuivity and the Equilibration o the Extratropical Atmophere MALTE JANSEN AND RAFFAELE FERRARI Maachuett Intitute o Technology, Cambridge, Maachuett (Manucript received 9 March 2012, in inal orm 11 November 2012) ABSTRACT Obervation ugget that the time- and zonal-mean tate o the ratropical atmophere adjut itel uch that the o-called criticality parameter (which relate the vertical tratiication to the horizontal temperature gradient) i cloe to one. T. Schneider ha argued that the criticality parameter i kept near one by a contraint on the zonal momentum budget in primitive equation. The contraint relie on a diuive cloure or the eddy lux o potential vorticity (PV) with an eddy diuivity that i approximately contant in the vertical. The diuive cloure or the eddy PV lux, however, depend crucially on the deinition o average along ientrope that interect the urace. It i argued that the deinition avored by Schneider reult in eddy PV luxe whoe phyical interpretation i unclear and that do not atiy the propoed cloure in numerical imulation. An alternative deinition, irtpropoedbyt.-y.kohandr.a.plumb,ipreerred. A diuive cloure or the eddy PV lux under thi deinition i upported by analyi o the PV variance budget and can be ued to cloe the near-urace zonal momentum budget in idealized numerical imulation. Following thi approach, it i hown that O(1) criticalitie are obtained i the eddy diuivity decay rom it urace value to about zero over the depth o the tropophere, which i likely to be the cae in Earth atmophere. Large criticality parameter, however, are poible i the eddy diuivity decay only weakly in the vertical, conitent with reult rom quaigeotrophic model. Thi provide theoretical upport or recent numerical tudie that have ound upercritical mean tate in primitive equation model. 1. Introduction Obervation ugget that the time- and zonal-mean tate o the ratropical atmophere adjut itel uch that ientrope leaving the urace in the ubtropic reach the tropopaue near the pole, and thu j [ (a/h) ; O(1), where a i the planetary radiu, H the height o the tropopaue, and the lope o the ientrope. The parameter j (in thi or related ormulation) i commonly reerred to a the criticality parameter, ince the condition that j ; O(1) bear reemblance to the marginal criticality condition in the two-layer quaigeotrophic (QG) model, where mall criticality parameter denote a tate that i table to baroclinic intability, while large criticality parameter denote a trongly untable tate (Phillip 1954; Stone 1978). Numerical tudie ugget that the Correponding author addre: Malte Janen, Maachuett Intitute o Technology, 77 Maachuett Avenue, , Cambridge, MA mjanen@mit.edu oberved adjutment o the ratropical atmophere to O(1) criticalitie hold over a wide range o orcing and parameter (e.g., Schneider and Walker 2006). Other tudie, however, have hown that more upercritical tate can alo be obtained i ernal parameter are uiciently changed (e.g., Zurita-Gotor 2008; Zurita- Gotor and Valli 2010; Janen and Ferrari 2012). Variou theorie have been put orward to explain the oberved equilibration o the midlatitude atmophere to a tate o marginal criticality. The original argument brought orward by Stone (1978) i baed on the analogy to the two-layer QG model, where j 5 1 denote the tranition rom baroclinically table to untable tate. The implication i that at a criticality o order 1, eddy luxe tranition rom being ineicient to being highly eicient in reducing the ientropic lope. Thi argument, however, i not very compelling becaue the neutrality condition, a deined above, applie only to the two-layer model, and the atmophere i not generally adjuted to baroclinic neutrality (e.g., Barry et al. 2000). DOI: /JAS-D Ó 2013 American Meteorological Society

3 MAY 2013 J A N S E N A N D F E R R A R I 1457 Another approach i to argue in term o contraint on the zonal momentum budget. The vertically integrated QG zonal momentum budget can be ued to derive a balance condition between the eddy luxe o potential vorticity (PV) in the interior and the eddy luxe o potential temperature at the urace. Uing a diuive cloure or the eddy luxe o PV and urace potential temperature, thi budget give a relation between the criticality parameter and the eddy diuivity. Mot importantly, the relation how that the criticality parameter can be inite only i the eddy diuivity varie in the vertical (Green 1970). Held (1978, 1982) ued thi reult (and ome aumption or the vertical tructure o the PV gradient) to argue that atmopheric mean tate with j 1 are obtained i the eddy diuivity decay in the vertical over the depth cale o the tropophere. Schneider (2004, 2005) call into quetion the relevance o the QG reult o Green (1970) and Held (1978, 1982), arguing that imilar reult are not recovered in primitive equation. The breakdown o the QG reult in primitive equation i attributed to the inadequate repreentation o ientropic outcrop at the urace. In primitive equation, a relation between the turbulent eddy luxe and the atmopheric mean tate i mot naturally derived rom the zonal momentum budget in ientropic coordinate (Schneider 2005; Koh and Plumb 2004). A in QG, the latter can be ued to derive a relation between the vertically integrated ientropic eddy lux o PV and the eddy lux o potential temperature at the urace. Quetion, however, arie about the appropriate deinition o ientropic average and eddy luxe on ientrope that interect with the urace. Schneider (2005) argue that there are two poible way to eparate the eddy PV lux into a mean and eddy lux component on outcropping ientrope. One approach, which wa originally propoed by Koh and Plumb (2004), reult in a ormulation or the ientropic zonal momentum budget, which more cloely reemble the zonal momentum budget in QG but i argued to be le uitable or eddy lux cloure becaue the reulting PV gradient oten change ign on mall cale. Schneider (2005) propoe an alternative approach, under which the PV luxe and gradient on outcropping ientrope are generally larger and o uniorm ign. Thi alternative approach i ued in Schneider (2004) to derive a contraint or the criticality parameter in primitive equation. The reult ugget that, unlike in QG, the zonal momentum budget in primitive equation can be cloed, auming that the eddy diuivity i contant in the vertical throughout the whole depth o the tropophere. Moreover, the choice o a vertically contant eddy diuivity implie that j 5 O(1). In thi paper we reviit the argument preented in Schneider (2005, 2004). We ind that the deinition o the ientropic mean and eddy PV luxe propoed by Koh and Plumb (2004) reult rom a well-deined average over the exiting part o the ientrope, while the deinition ued in Schneider (2004) include contribution rom ientrope below the urace, which are diicult to jutiy mathematically. Furthermore, a diuive cloure or the eddy PV lux can be jutiied phyically by analyi o the PV variance budget with the approach o Koh and Plumb (2004), while the phyical jutiication or a diuive cloure with the approach ued in Schneider (2004) remain unclear to the author o thi paper. In addition, we how that idealized numerical imulation upport the ue o a diuive cloure ollowing the deinition o Koh and Plumb (2004), while a diuive cloure generate inconitent reult i the approach o Schneider (2004) i applied to analyze our imulation. We thereore repeat Schneider (2004) derivation or the relation between the criticality parameter and the vertical tructure o the eddy diuivity but uing the averaging approach advocated by Koh and Plumb (2004). Uing thi approach, we can end the QG reult or the relation between the criticality parameter and the eddy diuivity to inite ientropic lope. In particular, we how that the eddy diuivity decreae in the vertical or any inite criticality, conitent with reult obtained uing QG caling and in contrat to the reult in Schneider (2004). The revied caling relation urther how that criticality parameter much larger than one are poible i the eddy diuivity decay only weakly in the vertical. Thi provide theoretical upport or recent numerical tudie that have ound upercritical mean tate in primitive equation model (e.g., Zurita-Gotor 2008; Zurita-Gotor and Valli 2010; Janen and Ferrari 2012). Thee reult are alo in qualitative agreement with the vertical decay o paive tracer eddy diuivitie ound in atmopheric GCM (Plumb and Mahlman 1987) and reanalyi data (Hayne and Shuckburgh 2000). Thi paper i organized a ollow. In ection 2 we illutrate the contraint o the momentum budget on the vertical tructure o the eddy PV diuivity uing a imple QG model. In ection 3, we dicu the two dierent approache that have been propoed to cloe the ientropic zonal momentum budget in primitive equation. In ection 4, we tet the diuive cloure againt idealized numerical imulation. In ection 5, we derive a revied condition or the criticality parameter rom the vertically integrated ientropic zonal momentum budget. The concluion are ummarized in ection 6.

4 1458 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME Eddy diuivity and criticality in QG Thi tudy dicue the relation between the criticality parameter and the vertical tructure o the eddy diuivity in the tropophere. The problem i well undertood in model uing the QG approximation. Schneider (2004), however, pointed out that the relationhip may be quite dierent in primitive equation ytem that allow or teep ientrope that interect with the urace. In the n ew ection we will reviit thee argument and how that the reult o QG theory i recovered i care i taken in computing average along ientropic urace. It thereore eem appropriate to tart with a review o the QG reult to et the tage or the ret o the paper. We will illutrate the QG argument uing what i arguably the implet coniguration: a zonally reentrant channel compoed o two eparate layer with dierent denity. The QG argument goe back to Green (1970), who dicued the contraint on the vertical tructure o the eddy diuivity in the continuouly tratiied cae. A dicuion o the two-layer cae wa, to our knowledge, irt given by Marhall (1981). The qualitative reult rom the two-layer model, dicued here, i imilar to the continuou QG cae. Ignoring rictional orce, the teady tate zonal-mean zonal momentum budget in the two-layer QG model can be expreed in term o a balance between the Corioli acceleration, acting on the mean low, and the Reynold tre: 0 y i 5 y u 0 i y0 i, (1) where i 5 1, 2 i the model layer, (u i, y i ) are the zonal and meridional velocitie in each layer, 0 i the Corioli requency, the overbar denote a time and zonal mean, and primed quantitie denote deviation thereo. Introducing a reidual mean low a y i * 5 y i 1 Hi 21 y 0 i h0 i, with h i and H i denoting the in itu and mean layer depth, repectively, the momentum budget can be rewritten a a balance between the Corioli acceleration, acting on the reidual mean low, and the eddy lux o PV (e.g., Valli 2006): 0 y i * 52y 0 i q0 i. (2) Here q i 5 by 1 z i 2 0 h i /H i i the QG potential vorticity, where b i the planetary vorticity gradient and z i 5 x y i 2 y u i i the relative vorticity. Uing a diuive cloure or the eddy luxe o PV and ignoring the contribution o relative vorticity to the mean PV gradient, we can approximate the reidual low in each layer a y i * y 0 i q0 i [ 21 0 D i y q i D i b 0 7 H i, (3) where [ y h 1 52 y h 2 denote the lope o the interace. For eae o dicuion, we here want to aume that 0. 0, a in Earth ratropical atmophere. The negative ign in the lat term on the rh o Eq. (3) hold or the lower layer (layer 1) where the planetary vorticity and thickne contribution to the PV gradient have oppoite ign, while the poitive ign hold or the upper layer (layer 2) where the planetary vorticity and thickne contribution to the PV gradient have the ame ign. Ma conervation provide the additional contraint that H 1 y 1 * 1 H 2 y 2 * 5 0. (4) Combining Eq. (3) and (4) yield H1 b H2 b D 1 2 2D 2 1. (5) 0 0 Thi yield a relation between the vertical tructure o the eddy diuivity and the criticality parameter a j [ 0 bh ; [D] DD, (6) where H 5 H 1 1 H 2 denote the total depth, 0 /b i the planetary cale, [D] [ (H 1 D 1 1 H 2 D 2 )/H i the vertical mean eddy PV diuivity, and DD [ D 1 2 D 2 i the dierence in the diuivitie. Equation (6) tate that the criticality parameter i inverely proportional to the relative vertical variation o the eddy diuivity. Criticality parameter O(1) are aociated with vertical variation in D on the ame order o D itel, while trongly upercritical tate are aociated with weak vertical variation in D. The criticality parameter j i alo related to the condition or baroclinic intability. In the two-layer model, the condition or intability i that the PV gradient in the lower layer i negative that i, (b 2 0 /H 1 ) # 0, or, uing the deinition o the criticality parameter, j $ H 1 /H. Hence, weak criticalitie are aociated with weakly turbulent tate and vertically variable diuivitie, while large criticalitie characterize trongly turbulent tate with vertically uniorm eddy diuivitie. A imilar relation between criticality and vertical tructure o the eddy diuivity i obtained with the continuouly tratiied QG model a hown by Green (1970). In ection 5, we how that a reult analogou to Eq. (6) can alo be derived in primitive equation i the tropophere i divided into two vertical layer. The

5 MAY 2013 J A N S E N A N D F E R R A R I 1459 primitive equation problem alo illutrate what part o the atmophere ought to be interpreted a the lower layer, which i otherwie arbitrarily deined in the QG problem. 3. The ientropic zonal momentum budget in primitive equation The eminal paper o Schneider (2004, 2005) and Koh and Plumb (2004) how that the vertically integrated ientropic zonal momentum budget in primitive equation can be ued to relate the meridional ma tranport in the tropophere to the eddy luxe o PV and urace potential temperature. Schneider (2004) point out that the integral o the ma tranport approximately vanihe i the integral i ended all the way to the top o the tropopaue. One i thereore let with a relationhip between the eddy luxe o PV and urace potential temperature, which can be converted into a relationhip between the interior PV gradient and the urace potential temperate gradient i one aume that the luxe are down their mean gradient. Schneider urther how that thi relationhip can alo be cat a a condition or the criticality o the ratropical atmophere. Beore dicuing the implication or the criticality parameter, we need an appropriate ormulation o the ientropic zonal-mean zonal momentum budget on ientrope that interect with the urace. Thi i the crucial tep that lead Schneider (2004) to obtain a relationhip undamentally dierent rom the QG equation (6). For implicity, we will aume that both large-cale mean and eddy low have a mall Roby number (Ro), which i generally a good approximation in the tropophere. Departing rom the QG approximation, however, we do not make an aumption o mall ientropic lope (e.g., Valli 2006). The latter i inadequate or tudie o the large-cale atmopheric circulation and crucial to undertand the dierence between the QG reult and thoe preented in Schneider (2004). For mall Ro, the ientropic zonal momentum budget reduce to the geotrophic wind relation y x M, (7) where i the planetary vorticity, y i the meridional velocity, and M 5 c p T 1 gz i the Montgomery treamunction (c p denote the peciic heat capacity o air at contant preure, T i temperature, g i the gravitational acceleration, and z i height). The partial zonal derivative x i taken at contant potential temperature u. In addition to term that are higher order in Ro, we neglected the eect o diabatic and rictional orcing in the momentum equation. The ull budget i dicued in Schneider (2005) and Koh and Plumb (2004), but the neglected term are not relevant or the key reult preented here. To derive an equation or the ientropic ma tranport, we take zonal and temporal average o the momentum budget along ientrope. Thi raie the iue o how to deine thee average when ientrope interect the urace. We can introduce a normalized zonal and temporal integral along any ientrope (that may or may not outcrop at ome longitude or time) a in Koh and Plumb (2004): (u, x, y, t) ) [ 1 TL ðð u.u (x,y,t) (u, x, y, t) dx dt, (8) where L i the width o the (zonally reentrant) domain and the temporal integral i taken over ome time period T. For implicity we ue Carteian coordinate, but the generalization to pherical coordinate i traightorward. I (u, x, y, t) i continually ended to ientrope below the urace, the normalized integral may or convenience be written a (u, x, y, t) ) 5 1 ðð H(u 2 u TL ) (u, x, y, t) dx dt [ H(u 2 u ) (u, x, y, t), (9) where the overbar denote the ientropic zonal and temporal mean, and H(u 2 u ) i the Heaviide unction, which guarantee that there are no contribution to the integral rom the uburace part o an ientrope. Uing the deinition in Eq. (9), the normalized integral o the momentum budget in Eq. (7) can be written a H(u 2 u ) y H(u 2 u ) x M. (10) Equation (10) i the tarting point to illutrate the two dierent approache to rewrite the zonal-mean zonal momentum budget in term o mean and eddy luxe o PV, a dicued in Schneider (2005). a. Two ormulation or mean and eddy luxe o PV To derive an equation or the ientropic ma tranport rom the momentum budget in Eq. (10), we expre the Corioli term on the lh o Eq. (10) in term o thickne-weighted mean and eddy luxe o PV. Upon diviion by the mean PV, thi yield an equation or the thickne-weighted velocity, which can be integrated in the vertical to yield the total ientropic ma tranport. Two dierent approache have been recommended to eparate the term H(u 2 u )y into mean and eddy lux

6 1460 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 70 term. The approach propoed by Koh and Plumb (2004) can be written a H(u 2 u )y H(u 2 u )y 5 5 H(u 2 u )yp 5 y 5 (y* 1 ^y ^ ), (11) where we introduced the ientropic thickne, [ 2g 21 u p, and the planetary PV, P [ /. We urther introduced a generalized thickne a 5 H(u 2 u ) and a generalized thickne-weighted zonal average, () * 5 ( ) ) / ) 5 ()/ (ee alo Andrew 1983; Jucke et al. 1994). Note that () * i imply the thickneweighted zonal average, taken over the above-urace part o the ientrope. Deviation rom thi average are denoted by (b) [ ()2 () *. Schneider (2004, 2005) intead recommend to deine ru H(u 2 u H(u 2 u )y 5 )y 5 H(u 2 u )yp 5 H(u 2 u )yp * 5 (y *P * 1 ^y ^P * ), (12) and introduce an ended planetary PV, P [ / (notice the generalized thickne,, in the denominator), and an ended velocity, y [ H(u 2 u )y, deined to exit on ientrope both above and below the urace. The irt tep in Eq. (12) i mathematically ill deined ince it implie multiplying and dividing by zero whenever the ientrope i below the urace and vanihe. Furthermore, the ended PV, P [ /,i ininite on ientrope below the urace, and thu both the generalized thickne-weighted average o P and ^y ^P are ill deined. Schneider (2004) evaluate P * a ru 5 [ ; (13) that i, the ratio o in the numerator and in the denominator o Eq. (13) i aumed equal to one everywhere, including the part o ientrope below the urace where 5 0. Schneider (2005) dicue the relative merit o the two ormulation. He argue that the two approache can be undertood a ariing rom dierent evaluation o the ill-deined ratio in Eq. (13). The ormulation o Koh and Plumb (2004) i recovered i the term ( / )i aumed to vanih when ientrope are below the urace, while Schneider (2004) approach aume that it i equal to. Notice, however, that the approach o Koh and Plumb can intead be derived without making any aumption or quantitie on the uburace part o ientrope, a per Eq. (11). The problem i that, by introducing the generalized quantitie P and y, the ended PV approach require arbitrary aumption down the line to obtain well-deined thickne-weighted average. Taking a dierent point o view, which we will adopt in the ollowing, the eparation into mean and eddy term in Eq. (12), together with the deinition or the thickneweighted average o the ended PV in Eq. (13), may be regarded a a deinition or the ended eddy PV lux ^y ^P *. One could then regard Eq. (12) a a dierent choice or eparating mean and eddy luxe. The quetion i whether one can rationalize what the mean and eddy lux term phyically repreent in thi cae, a they are no longer deined a the mean and eddy contribution rom a Reynold decompoition. The phyical meaning o the mean and eddy lux term become particularly important i we want to relate the eddy PV lux to the mean PV gradient through a diuive cloure o a to cloe the momentum budget in term o mean tate variable. Tung (1986) oer upport or a diuive cloure or the thickne-weighted ientropic eddy PV lux uing mall amplitude theory. He doe, however, not conider ientropic outcrop, which are crucial or the dicuion here. In appendix A, we preent a general jutiication or a diuive cloure or Koh and Plumb (2004) interior eddy PV lux, ^y ^, baed on the PV variance budget, that account or ientropic outcrop. We were not able to derive a imilar relation or the ended PV variance ince ^P 2 * i not well deined on outcropping ientrope. Schneider (2004, 2005) jutiie hi cloure by howing that the ended eddy lux ^y ^P * i down the gradient o P * in hi numerical imulation and proceed to ue a diuive cloure ^y ^P * 52D y P *. However, we will argue below that uch a relationhip i not jutiied in general. It may be argued that the approach o Koh and Plumb (2004) become problematic when average are taken along the coldet ientrope, which exit only in a ew place. In particular, one could quetion whether a downgradient cloure can be deended when average are taken on mall region. Thee ientrope, however, contribute little to the net PV lux and aociated ma tranport. In ection 4 we will how that the cloure or Koh and Plumb interior PV i well upported by idealized numerical experiment.

7 MAY 2013 J A N S E N A N D F E R R A R I 1461 b. An equation or the meridional ma tranport In thi ection we will dicu how the ientropic zonal momentum budget can be tranormed into an equation or the meridional ientropic ma tranport, ollowing the ame approach outlined in Schneider (2005) and Koh and Plumb (2004). Focuing on the role o ientropic outcrop, we will here analyze the ma tranport integrated over the urace layer (SL), deined a the layer including all ientrope that interect the urace at ome longitude or time. Schneider (2005) how the ma tranport over the SL can be etimated by dividing the zonal momentum equation (10) by and integrating vertically over the SL. Subtituting Eq. (11) into Eq. (10), dividing by, and integrating rom the urace to the top o the SL, one obtain an etimate or the tranport in the SL: r y* du 2 u^y ^ du 2 (u llllllll{zllllllll} ) y0 u0. (14) lllllll{zlllllll} (1) (2) Here i the minimum potential temperature over the domain, and u i i the potential temperature at the top o the SL, the ubcript denote urace quantitie, and () denote an average along the urace. The urace potential temperature lux term on the rh o Eq. (14) repreent the bottom orm tre, which arie a the boundary contribution rom the Montgomery treamunction term in Eq. (10) (ee Koh and Plumb 2004; Schneider 2005). The derivation o thi term i unambiguou and thereore not repeated in detail here. Equation (14) tate that the total meridional ma tranport between the urace and the ientropic urace u i, i driven by ientropic eddy PV luxe and the eddy lux o urace potential temperature. The ame equation can be derived uing the ended PV approach advocated by Schneider (2005), who ind that * r y* du 2 u^y ^P du 2 llllllllllllll{zllllllllllllll} (u ) y0 u0. llllllllll{zllllllllll} (1) (2) (15) The total ma tranport the term on the lh o Eq. (14) and (15) i independent o the approach, but the eparation between the contribution aociated with the eddy lux o PV, term 1, and the contribution aociated with the eddy lux o urace potential temperature, term 2, dier in the two approache. Thi can be een by noting that P * (u ) 6¼ (u ). The preactor multiplying the eddy lux o urace potential temperature, term 2, thu dier in the two approache. Thi dierence i aborbed by a correponding dierence in the PV lux contribution, term 1. Uing diuive cloure or the eddy luxe o interior PV and urace potential temperature, Eq. (14) become y* du D P* y du 1 (u ) D y u ; (16) that i, the SL ma tranport i given by a contribution aociated with the interior PV gradient and a contribution aociated with the urace potential temperature gradient (decribing the orm drag on outcropping ientrope). Uing intead a diuive cloure or the eddy lux o ended PV, along with a cloure or the urace potential temperature lux, Eq. (15) become y* du D y du 1 (u ) D y u. (17) One would now like to aume that the eddy diuivity i a undamental property o the low, decribing the rate o mixing by the turbulent eddie independently o the tracer being tirred. Since the SL i centered around the median urace temperature, it eem appropriate to aume that D D D near the urace. Numerical imulation dicued in ection 4 how that the budget in Eq. (16) can be cloed, auming D D, while Eq. (17) trongly overetimate the near-urace ma tranport i we aume that D D. 4. Numerical imulation We proceed to tet the argument preented in the previou ection, and in particular the applicability o diuive cloure or the eddy PV luxe in the two approache, in an idealized numerical imulation. The model i et up to limit boundary eect that arie rom explicit parameterization o boundary layer phyic. While the model etup might be a le realitic decription o the real atmophere, it help to ocu excluively on the dynamic that are the topic o the preent paper and keep any additional phyic a imple a poible.

8 1462 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 70 FIG. 1. Equilibrium potential temperature (K) or thermal retoring, contour interval (CI): 5 K. FIG. 2. Time- and zonal-mean ield o potential temperature (K) (color hading, CI: 5 K), eddy kinetic energy (thin black line, CI: 2 m 2 22 ), zonal wind (gray line, CI: 1 m 21 ), and the tropopaue height, deined a the height at which du/dz Km 21 (thick black line). a. Model etup We ue a hydrotatic, Bouineq, 1 Carteian coordinate coniguration o the Maachuett Intitute o Technology general circulation model (Marhall et al. 1997). The etup i imilar to the imulation dicued in Janen and Ferrari (2012): a zonally reentrant b-plane channel km long, bounded meridionally by idewall at y km and vertically by rigid lid at z 5 H km and z 5 0. Diering rom the imulation in Janen and Ferrari (2012), we ue ree-lip boundary condition on all boundarie to limit rictional eect. Kinetic energy i removed by a linear Rayleigh drag, with a contant drag coeicient r 5 (40 day) 21, acting over the whole domain. Following Janen and Ferrari (2012), we ue a linear equation o tate with a thermal expanion coeicient o a K 21 ; that i, b 5 ga(u 2 u 0 ) in which b i buoyancy, u 0 a reerence potential temperature, and g the acceleration o gravity. The thermal expanion coeicient i maller than that o air to obtain a better eparation between the eddy and domain cale and to allow the model to et it criticality (ee Janen and Ferrari 2012). The imulation are orced through relaxation to the equilibrium temperature proile hown in Fig. 1, which i the radiative convective equilibrium olution o the proile ued in Janen and Ferrari (2012). It i characterized by a baroclinic zone 7000 km wide, acro which the temperature drop by about 55 K. The equilibrium tratiication i tatically neutral over the lower part o the domain, while a tatically table equilibrium 1 Notice that the theoretical dicuion above are or the more general cae o a compreible luid, in direct analogy to the derivation in Schneider (2004) and Koh and Plumb (2004). However, a dicued in Janen and Ferrari (2012), the ame reult are obtained in the Bouineq limit. tratiication i precribed at higher altitude to mimic the table radiative-equilibrium proile o the tratophere. The relaxation time cale i contant over the domain at t 5 50 day. The imulation i pun up until a quai-teady tate i reached. Ientropic diagnotic are calculated rom 200 naphot aved every 10 day ater the olution i equilibrated. b. Reult The equilibrated time- and zonal-mean tate o the imulation i hown in Fig. 2. Signiicant baroclinicity i ound over mot o the domain, with a jet centered around y km. The jet i aociated with a maximum in baroclinicity, a well a maxima in the barotropic weterly wind and eddy kinetic energy. Figure 3a how the three term that appear in the zonal momentum budget a per Eq. (14). The integral i taken rom the urace to the top o the urace layer, deined to include all ientrope up to the 95% quantile o potential temperature value ound at the urace, along a latitude circle. The budget i hown only or the SL, becaue that i where we expect dierence in the averaging approache (all approache are equivalent alot where there are no ientrope that interect the urace). The continuou gray line how the total SL ientropic ma lux: F tot 5 y* du. (18) The dotted black line how the um o the ma luxe aociated with the eddy PV lux and the urace potential temperature lux. Uing the averaging approach o Koh and Plumb (2004), thee are

9 MAY 2013 J A N S E N A N D F E R R A R I 1463 FIG. 3. Surace layer ma luxe over the baroclinically orced zone. Shown i the total ma tranport integrated vertically over the SL, it contribution due to eddy PV and urace potential temperature lux, a well a etimate o the repective luxe reulting rom a diuive cloure uing the urace eddy diuivity (ee legend). Reult (a) uing the interior PV approach or the eparation o SL ma luxe into a contribution rom the eddy PV lux and urace potential temperature luxe and (b) uing the ended PV approach. The SL i here deined a including all ientrope up to the 95% quantile o the urace potential temperature. F^y ^P* [2 ^y ^ du and F y 0 [2 u0 (u ) y0 u0 The budget derived in Eq. (14) predict that. (19) F tot F^y ^P* 1 F y 0 u0. (20) Figure 3a how that thi relation i atiied by the numerical imulation. We ind that mot o the SL ma tranport i aociated with the urace contribution F y 0 with a maller (but igniicant) contribution rom u, 0 the interior eddy PV lux component F^y ^P*. Alternatively we can eparate the eddy ma lux component according to the ended PV approach o Schneider (2004), given in Eq. (15): F tot F ^y ^P * 1 F y 0, (21) u0 where * F r ^y ^P * [2 u^y ^P du and F y 0 u0 [2 (u ) y0 u0. (22) * The ended eddy PV lux ^y ^P i deined a the reidual between the total PV lux y and the mean advection o the generalized PV y * P *, a dicued in ection 3a. The ma lux budget in Eq. (21) i imilarly conirmed by the model imulation (Fig. 3b). Compared to the budget in Fig. 3b, the total ma tranport ha a larger contribution rom the ended eddy PV lux term and a correpondingly maller contribution rom the eddy lux o urace potential temperature. In act, it can be hown eaily that F ð1/2þf y 0 u0 y 0 u 0 by comparing Eq. (19) and (22), but realizing that (u ) 5 H(u 2 u ) (u ) 1 2 (u ) (u ). (23) Here we ued that the mean urace potential temperature i well approximated by it median value and, hence, H(u 2 u ) 1/2. Since the total ma tranport i independent o the averaging approach, a imilar but oppoite dierence exit between the eddy PV lux contribution F^y ^P * and F ^y ^P *. We now tet the diuive cloure or the eddy luxe o PV uing the two approache. We calculate the eddy diuivity rom the urace potential temperature lux/ gradient relationhip a 2 2 Notice that the urace eddy diuivity i here calculated uing the actual urace potential temperature lux. Following the ull derivation hown in Schneider (2005), or Koh and Plumb (2004), the urace term in Eq. (14) and (15) hould generally be decribed by the geotrophic eddy lux o urace potential temperature. In the preented imulation, where the Roby number i mall and no enhanced urace drag i ued, the dierence between the ull eddy lux and the geotrophic eddy lux i negligible.

10 1464 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 70 D [2y 0 u 0 / y u. (24) Uing thi deinition, the diuive cloure or the ma tranport aociated with the urace eddy lux o potential temperature, F D y [ u (u ) D y u and F [ D y u (u ) D y u, (25) reproduce, by contruction, the correponding lux term F y 0 u 0 and F. y 0 u 0 Schneider (2004) argue that a diuive cloure hold alo or the eddy PV lux with the ame eddy diuivity D, a one ought to expect i eddie mix all tracer at the ame rate and i thi mixing rate varie little over the depth o the SL. Thi i teted by computing D F D y [ P* y du and F D y D y [ du. (26) For the interior PV approach, we ind that the PV lux and thereore the ull SL ma tranport i decribed reaonably well by the diuive cloure (Fig. 3a), uggeting that the PV lux in the SL can be decribed a a diuive lux with an eddy diuivity imilar to that or urace potential temperature. However, with the ended PV approach, we ind that the eddy PV lux, and thereore the total SL ma lux, i coniderably overetimated by the diuive cloure. Interetingly, the diuive approximation or the ended PV lux contribution, F D y P *, alone i a better approximation o the total SL ma tranport F tot (Fig. 3b). Thi reult can be explained in term o the dierent approache taken to treat outcropping ientrope, a dicued in the ollowing. c. Explaining the reult The dierence between the SL ma balance in the two approache can be undertood by rewriting the ended PV lux and gradient in term o the contribution aociated with luxe and gradient o interior PV and urace potential temperature. Uing again that F (1/2)F y 0 u0 y 0, a hown above in Eq. (23), we ind u 0 that (ee appendix B or a more detailed derivation) F ^y ^ F^y ^ F y 0. (27) u0 In the ended PV ormulation o the ma budget [Eq. (21)], about hal o the urace potential temperature lux contribution to the ientropic ma tranport i thu aborbed into the ended PV lux term. However, the diuive cloure or the lux aociated with the ended PV gradient contain the contribution aociated with the interior PV gradient, plu the ull contribution aociated with the urace potential temperature gradient (ee appendix B or a derivation): F D y F D y 1 F D y u. (28) I the interior eddy PV lux near the urace can be cloed with a diuivity imilar to that or urace potential temperature (a conirmed in our imulation), a imilar cloure or the ended eddy PV lux thu automatically overetimate the lux. Conitent with the numerical reult hown in Fig. 3, a comparion o Eq. (28) and (20) ugget that F D y P * i by itel an approximation or the ull SL ma tranport, F tot. Beore returning to the implication o our inding or the criticality parameter, we remark that the numerical reult dicued here have been conirmed in a whole uite o imulation uing a imilar idealized etup but with diering parameter and retoring proile the reult hold a long a the ytem doe not become convective, in which cae the tranormation into ientropic coordinate ail. Reult rom thee imulation will be reported in an upcoming paper, where we dicu their implication or oundertanding o the equilibration o the ratropical tropophere. Schneider (2005) argue that in hi imulation a diuive cloure or the ended eddy PV lux i better jutiied than or the eddy lux o interior PV ince the gradient o the latter are generally weak and vary on mall cale in the urace layer. While thi make it impractical to etimate an eddy diuivity rom a luxgradient relationhip or the interior PV, it doe not mean that the integrated SL momentum budget cannot be cloed with a diuive cloure or the eddy lux o interior PV. I the eddy lux i imilarly weak, the integrated momentum budget can till be cloed (and in act become rather independent o the exact choice or the interior PV diuivity). We ran ome example imulation that are in thi limit and were able to cloe all budget. In cae where both the eddy lux and gradient o interior PV are very weak, the ended eddy PV lux may appear to be well deined and downgradient, a both the lux and the gradient are dominated by the urace potential temperature contribution [ee Eq. (27) and (28)]. Such a cenario wa conidered in Schneider (2005)

11 MAY 2013 J A N S E N A N D F E R R A R I 1465 and ued to conclude that the ended PV approach i more uitable or a cloure. However, a dicued above, the ma tranport aociated with the ended eddy PV lux in thi cae i overetimated by a actor o 2 i it i cloed with an eddy diuivity imilar to that o urace potential temperature. Thi become crucial in the derivation o a relation between the criticality parameter and the vertical tructure o the eddy diuivity, a will be dicued in ection 5. The ended eddy PV lux cloure can correctly repreent the SL ma budget only i a very weak interior PV gradient were aociated with a trong eddy lux o interior PV. Thi may be true in ome o the imulation analyzed by Schneider (2005). However, we urmie that uch ituation may a well be explained by nonconervative boundary layer eect, which can caue both interior PV and urace potential temperature to behave dierently rom conervative tracer. The eect o nonconervative boundary layer term, however, cannot be alvaged by redeining the zonal average. The problem in uch ituation i that a imple diuive cloure i no longer appropriate. We plan to careully analyze more realitic atmopheric imulation and obervational data to tet whether the cloure advocated here hold up in a more realitic etting. 5. The integrated momentum budget and the criticality Schneider (2004) derive a condition or the tate o the ratropical atmophere by integrating the ientropic ma lux balance (15) to the top o the tropophere u t. He urther make a diuive cloure or both the ended eddy PV lux ^y ^P * and the urace potential temperature lux y 0 u0, with the ame vertically contant eddy diuivity. We howed above that thi i generally not expected to hold. Intead, we argued that a diuive cloure or the eddy lux o interior PV, a deined in Koh and Plumb (2004), i better upported by our numerical imulation and can be jutiied by phyical argument. We will thereore repeat the derivation o Schneider (2004), but uing diuive cloure or theinterioreddypvluxe^y ^ and urace potential temperature lux y 0 u0. We ind that, except in the limit cae o ininite upercriticality, the momentum balance cannot be atiied with an eddy diuivity that i vertically contant throughout the entire tropophere, conitent with theorie baed on the QG approximation (Green 1970). We then dicu how the revied momentum balance can be ued to derive a criticality condition, or more generally a relation between the criticality parameter and the vertical tructure o the eddy diuivity. a. The vertically integrated ientropic momentum budget I the integration in the ma balance equation (14) i ended all the way up to the tropopaue and it i aumed that the ma tranport above the tropophere i negligible, one get ^y ^ du 2 (u ) ~y0 u0. (29) ð ut Auming a diuive cloure or the eddy lux o PV and urace potential temperature then yield ð ut D 2 P* y du (u ) D y u. (30) While we ound in the previou ection that D may be aumed imilar to D within the SL, we now ue Eq. (30) to how that D cannot be contant throughout the depth o the entire tropophere, but generally decreae toward the tropopaue. The thickne-weighted mean PV can be approximated a P / ; thu y P* b 1 P* P y P2P* y. (31) Equation (30) then become ð ut ð ru ut 2 D b 2 y du 2 D P y P du D (u ) P(u ) y u (32) The econd term on the lh o Eq. (32) can be approximated a ð ut 2 D r ð ut u P yp du 52 D P y H(u 2 u ) du which leave u with 5 D ru (u ) P(u ) y u r D u (u ) P(u ) y u, (33) ð ut ru 2 D b 2 y du 0: (34) Equation (34) tate that the diuive lux aociated with the planetary vorticity gradient and the generalized thickne gradient ha to integrate to zero over the depth o the tropophere..

12 1466 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 70 Comparion o Eq. (32) to Eq. (11) in Schneider (2004) how that the irt integral on the lh i imilar in the two approache, but Schneider doe not include the econd integral on the lh, which appear becaue o the actor P in the deinition o the thickne-weighted mean PV. Thi term balance the urace contribution on the rh o Eq. (32). Schneider (2004) thereore remain with an additional urace potential temperature gradient contribution on the rh o Eq. (34). Thi additional term allow him to balance the momentum budget with a vertically contant eddy diuivity. Equation (34) how that D cannot be contant in general. Thi can be een by noting that the econd term in the integral in Eq. (34) approximately integrate to zero i D i aumed vertically contant between and u t : ð ut D y du 52Dg 21 [ y j u p(u t ) 2 y p ] 0. (35) A jutiied in Schneider (2004), we here aumed that both the ientropic lope at the tropopaue, y j u p(u t ), and the urace preure gradient, y p, are negligible. The irt term in the integral in Eq. (34), however, i poitive deinite. A in the QG model, we thu ind that the eddy diuivity D in general cannot be vertically contant throughout the entire depth o the tropophere. b. A caling relation or the criticality parameter Equation (34) can be ued to obtain a caling relation between the criticality parameter and the vertical tructure o the eddy diuivity. A argued above, the momentum budget cannot generally be atiied i the eddy diuivity i aumed vertically contant. Intead, we will how that the eddy diuivity i generally expected to be larger near the urace and decay toward the tropopaue, a in the QG cae dicued in ection 2. To derive a caling argument or the vertical tructure o the eddy diuivity, we eparate the integral o the ma tranport in Eq. (34) into two part above and below ome level u 1 (y), which yield D 1 ð u1 ru ð ut b 2 y du 52D 2 u 1 ru b 2 y du, (36) where we deined bulk diuivitie D 1 and D 2 or the two layer, which ormally can be decribed a weighted vertical average o the eddy diuivity over the repective layer. We urther aume that the lower layer i choen uch that it include the entire urace layer, which, at any given latitude, comprie all ientrope that interect with the urace at ome time or longitude. We thu require that H(u 1 2 u ) 5 1 at all time and longitude. Ignoring again contribution due to the ientropic lope at the tropopaue, y j u p(u t ), and the urace preure gradient, y p, we then ind that [p(u1 ) 2 p D ]b 1 2 y j u p(u 1 ) [p(ut ) 2 p(u 2D 1 )]b 2 1 y j u p(u 1 ). (37) Equation (37) i the generalization o the two-layer QG relation in Eq. (5), where layer thicknee have been replaced by the correponding preure range, and the ientropic lope become y j u p z p y j u z. A in the QG problem, the ma luxe in the two layer can be written in term o the eddy diuivity multiplied by the um o a contribution aociated with the layer-integrated planetary vorticity gradient b and a contribution aociated with the thickne gradient in each layer, which in turn i given by the ientropic lope at the interace. Analog to the QG cae, Eq. (37) can be ued to derive a caling relation or the criticality parameter a j ; bh ; [D] DD, (38) where [D] 5 [p 2 p(u 1)]D 1 1 [p(u 1 ) 2 p(u t )]D 2 g/[p 2 p(u t )] denote the vertical mean o the eddy diuivity, DD [ D 1 2 D 2, [ y j u z(u 1 ), and H [2 p z(u 1 )[p 2 p(u t )]. Notice that we required that the lower layer include the entire urace layer. Apart rom that requirement, Eq. (38) can technically be derived or any choice o the height o the layer interace. However, the bulk diuivitie D 1 and D 2 repreent poorly deined layer average i the weighting actor, [( /)b2 y ], take on large poitive and negative value within a ingle layer. The ign o thi actor i given by the ign o the ended PV gradient and i typically negative throughout the SL, owing to the dominant contribution aociated with the ientropic outcrop. An adequate choice o the layer interace i thu given by the irt level above the SL where the PV gradient become poitive. Schneider (2005) ind that thi i typically jut above the SL; in the imulation dicued here, the ign change occur omewhat higher than the top o the SL. Relation (38) how that vertical variation in the eddy diuivity over the tropophere may be mall (compared to it mean value) only i j 1, which i in agreement with the act that eddie tend to be more barotropic in the limit o large upercriticalitie. I, on the other hand, the criticality i O(1), vertical variation

13 MAY 2013 J A N S E N A N D F E R R A R I Concluion FIG. 4. Surace (olid) and upper-tropopheric (dahed) eddy diuivity. The urace eddy diuivity i calculated rom a luxgradient relationhip o urace potential temperature a in Eq. (24). The upper-tropopheric eddy diuivity i calculated a an eective bulk eddy diuivity integrated rom the irt level above the SL where the PV gradient become poitive, u 1, to the top o the tropophere a D UT 5 Ð ut (1 u^y ^ / )/ Ð ut (1 u y / ) du. Notice that both quantitie are independent o the deinition o PV luxe and gradient on near-urace ientrope. in the eddy diuivity are expected to be on the ame order a the eddy diuivity itel. Thi i in agreement with the numerical imulation dicued in ection 4. A rough etimate or the criticality in the imulation can be obtained rom Fig. 2. With a 5 0 /b km, we have j 5 (a/h) 2, which i in qualitative agreement with the oberved upper-tropopheric eddy diuivity being approximately 50% maller than the eddy diuivity near the urace (ee Fig. 4). An enive erie o numerical imulation upporting the caling relation (38) will be dicued in a ollow-up paper. The argument or the relationhip between the criticality and the vertical tructure o D i cloely related to the analyi by Held (1978, 1982), who argue that the criticality parameter need to be cloe to one, i the depth o the tropopaue i et by the depth cale o the baroclinic eddie, and thereore equal the vertical cale over which the eddy diuivity decay. Thi i likely to be the cae in Earth preent atmophere and or a coniderable range o parameter around the preent tate, which would explain why the criticality o the atmophere i O(1), and i reported to be rather inenitive to change in the ernal orcing in numerical imulation (Schneider 2004; Schneider and Walker 2006). However, Zurita-Gotor (2008) and Janen and Ferrari (2012) ind trongly upercritical mean tate when they orce the ytem to be very dierent rom today atmophere. According to relation (38), uch tate are poible i the eddy diuivity decay only weakly over the depth o the tropophere. We ued the ientropic zonal momentum budget o the tropophere to derive a caling relation, which link the criticality parameter to the vertical tructure o the eddy diuivity [Eq. (38)]. We ound that the criticality parameter i inverely proportional to the relative change o eddy diuivity between the urace and the tropopaue. Marginally critical tate are expected i the cale over which the eddy diuivity decay i imilar to the depth o the tropophere that i, i the relative vertical variation o the eddy diuivity i O(1). Supercritical tate are predicted i the eddy diuivity varie only weakly in the vertical. Subcritical tate would require the eddy diuivity to decreae rapidly over a cale much maller than the depth o the tropophere. Thi reult i in contrat to that o Schneider (2004), who argued that, in primitive equation, marginally critical tate are obtained with an eddy diuivity that i vertically contant throughout the whole depth o the tropophere, while trongly upercritical tate are impoible (Schneider and Walker 2006). The dierence tem rom the dierent approache taken to compute average along ientrope in the urace layer. We ue a cloure or the interior eddy PV lux a deined in Koh and Plumb (2004), which can be jutiied by analyi o the PV variance budget and i upported by idealized numerical imulation. Schneider (2004) intead ue a cloure or an ended eddy PV lux, which include contribution rom ientrope below the urace. We ind that thi cloure overrepreent the urace layer ma tranport aociated with the orm drag on outcropping ientrope. With our approach, the relation between the criticality and the vertical tructure o the eddy diuivity i a direct enion o Green (1970) reult baed on QG theory. Dierence rom the QG reult are only quantitative in nature, and arie becaue o the inite ientropic lope and interection o ientrope with the ground. While thee reult are illutrated through one imulation in thi paper, we ran many more example that conirm our reult (Janen 2012). It would alo be intructive to tet our caling relation in le idealized imulation. The revied caling relationhip in Eq. (38) ha important implication or the macroturbulent equilibration o an atmophere. In particular, the criticality parameter can become much larger than one. Both marginally critical and trongly upercritical tate are poible i the magnitude and vertical tructure o the eddy diuivity can change. Notice, however, that the caling relation i not a predictive theory or the criticality. Such a theory would require an independent prediction or the magnitude and vertical tructure o the eddy diuivity.