Annual cycle and longitudinal structure of tropical eddy and mean momentum fluxes
AAnnual cycle and longitudinal structure of tropical eddy andmean momentum fluxes
Abu Bakar Siddiqui Thakur ∗ and Jai Sukhatme Centre for Atmospheric and Oceanic Sciences, Indian Institute of Science, Bangalore560012, India Divecha Centre for Climate Change, Indian Institute of Science, Bangalore 560012,India
Abstract
The longitudinal structure and annual cycle of mean meridional and eddy momentumfluxes in the tropical upper troposphere are studied. In zonal mean, these two terms opposeeach other and peak during the Indian summer monsoon. This zonal mean character arisesfrom a rich longitudinal structure that is revealed by splitting the globe into three zones,namely, the Asia-West Pacific (AWP), central Pacific-West Atlantic (CP-WA) and Africansectors. The mean convergence term is cohesive across all three regions, has a single peak inthe boreal summer and always acts to decelerate the zonal flow. A Helmholtz decompositionshows that the advection of absolute vorticity by the divergent meridional wind in localizedcross-equatorial cells is responsible for the coherent nature of the mean convergence acrossall sectors.On the other hand, the eddy convergence goes from being small and seasonally invariant inthe African region to one with large seasonal maxima (minima) in AWP (CP-WA) sectorthat accelerate (decelerate) the zonal mean flow. The disparate nature the eddy flux in theAWP and CP-WA regions is in the winter season and is due to the tropical and extratropi-cal origin of waves, respectively. In summer, the AWP region accounts for almost all of theeddy flux convergence. In fact, the leading role of the rotational zonal - divergent meridionalcomponent in the zonal mean eddy flux does not hold in individual sectors. Finally, throughthe year, the CP-WA region is where the local overturning cell is strongly influenced byeddy activity.
Key Words: Momentum flux, tropics, eddy and mean flow, local Hadley cells
The Hadley Cell (HC) is the dominant feature of the zonally averaged tropical troposphericcirculation. Towards the later half of the twentieth century, theoretical efforts aimed at un-derstanding the HC emphasized axisymmetric dynamics with rising motion at the equatorforced by latent heating of cumulus convection and subsidence in the subtropics (Schneider andLindzen, 1977; Schneider, 1977). In this context, Held and Hou (1980) arrived at simple analyt-ical relations between parameters predicting the latitudinal extent and strength of the tropicaloverturning motion. Extensions to a moist setting (Satoh, 1994) as well as the time dependent ∗ Corresponding author: A B S Thakur, [email protected] a r X i v : . [ phy s i c s . a o - ph ] A ugug
The Hadley Cell (HC) is the dominant feature of the zonally averaged tropical troposphericcirculation. Towards the later half of the twentieth century, theoretical efforts aimed at un-derstanding the HC emphasized axisymmetric dynamics with rising motion at the equatorforced by latent heating of cumulus convection and subsidence in the subtropics (Schneider andLindzen, 1977; Schneider, 1977). In this context, Held and Hou (1980) arrived at simple analyt-ical relations between parameters predicting the latitudinal extent and strength of the tropicaloverturning motion. Extensions to a moist setting (Satoh, 1994) as well as the time dependent ∗ Corresponding author: A B S Thakur, [email protected] a r X i v : . [ phy s i c s . a o - ph ] A ugug ature of the flow (Fang and Tung, 1999) were also explored in this axis-symmetric scenario.The cross-equatorial nature of the cell and an abrupt threshold behaviour of the angular mo-mentum conserving zonal mean circulation in response to off-equatorial heating was elucidatedby Lindzen and Hou (1988) and Plumb and Hou (1992), respectively. The nature of this axis-symmetric cell with subtropical heating has also been studied in a moist framework with an eyetowards explaining the rather sudden onset of the south-Asian monsoon flow (Emanuel, 1995;Boos and Emanuel, 2008).Though the possible importance of non-axis-symmetric dynamics on the tropical circulationhas been known for a long time (Kuo, 1956; Schneider, 1984), the role of midlatitude eddies ininfluencing the HC has been brought to the fore in recent years (Bordoni and Schneider, 2010;Singh et al., 2017). Indeed, three dimensional primitive equation based simulations and generalcirculation model experiments in a host of climate regimes have seen to yield results that differfrom axis-symmetric predictions (Becker et al., 1997; Kim and Lee, 2001; Walker and Schneider,2006; Frierson et al., 2007; Korty and Schneider, 2008). In particular, Walker and Schneider(2006) showed that the HC falls in regimes intermediate to angular momentum conserving(flow responds directly to changes in thermal driving) and eddy-driven limits. This has alsobeen found to be applicable in the seasonal cycle of the overturning motions (Schneider andBordoni, 2008; Bordoni and Schneider, 2008). In fact, recent results highlight the role of eddiesin allowing the HC extend down to the surface from the upper troposphere (Davis and Birner,2019). Further, studies using reanalysis data have provided evidence of extratropical eddyinfluences on the tropics on interannual timescales (Caballero, 2007; Caballero and Anderson,2009).In all, given their prominent role, it is important to understand the nature and transport ofmomentum by the eddies in the deep tropics. In this context, an examination of the uppertropospheric annually averaged zonal momentum budget showed that eddies induce a westerlyacceleration in a zonal mean sense (Lee, 1999). This is overwhelmed by an easterly accelera-tion due to the mean flow advection of zonal momentum by the seasonally reversing HC andcauses the equatorial upper tropospheric flow to be easterly in nature. Further, the westerlyacceleration in the equatorial upper troposphere was shown to be due to the convergence ofeddy momentum flux by the thermally forced climatological Rossby gyres present throughoutthe year (Dima et al., 2005). More detailed analysis revealed that the eddy momentum fluxconvergence in the deep tropics is dominated by correlations between the rotational zonal anddivergent meridional wind (Zurita-Gotor, 2019). These tropical stationary eddies, forced by lon-gitudinal thermal contrasts (Wang and Ting, 1999), have an inherent zonal dependence whichis lost in averaging. But, even in the context of the zonal mean picture, there are a numberof basic questions that arise here. For example, does the eddy momentum flux have a similarcharacter through the year? Does the flux convergence always accelerate the zonal mean flow?Given that the extratropical waves can reach the tropics (Hoskins and Ambrizzi, 1993), is therea competition between eddies of differing origin? In other words, are there periods in the yearwhere the eddies of a midlatitude origin “win” and the eddy fluxes actually act to deceleratethe mean zonal flow that is part and parcel of the HC?Along with the recognition of the role of eddies, it is also important to note that the tropicaloverturning circulation is longitudinally localized. These localized circulations result from thelongitudinal heterogeneity in the distribution of thermal forcing associated with the world’smonsoons. Although, the pioneering study of Gill (1980) formulated the steady linear responseof the atmosphere to a localized heat source, interest in the zonal structure of the responsehas re-awakened recently and most studies, in this regard, have focused on the interannualvariability of the resulting longitudinally limited overturning motions (Zhang and Wang, 2013;Nguyen et al., 2017; Sun et al., 2018). Hoskins et al. (2020) presented a detailed analysis of the2orthern Hemisphere (NH) summer HC and showed that the Asian monsoon circulation is thedominant contributor to the zonally averaged circulation during the boreal summer. Further,the rapid transition of southern cell during the Asian monsoon from an eddy-driven regimeto a thermally-driven one (Bordoni and Schneider, 2008) was seen to be not so abrupt whenthe forcing itself was zonally asymmetric (Zhai and Boos, 2015). Indeed, regime transitions ofthe HC have been examined in the presence of inhomogeneous boundary forcing (Shaw, 2014;Geen et al., 2018). More insight into the localised nature of the HC has been possible througha coupled Eulerian-Lagrangian analysis that suggests a ”conveyor belt” type circulation withlarge-scale ascent in the Indo-Pacific region and subsequent descent over the Americas (Raiteret al., 2020).Thus, bringing the two threads together, the issue at hand is to understand the temporalvariation and longitudinal nature of eddy fluxes in the deep tropics and subsequently put themin perspective with the local overturning circulations. In essence, the questions we posed aboveneed to be taken a step further and our aim in this study is to first of all study the annual cycleof momentum fluxes and then probe these longitudinal variations in eddy and mean fluxes inthe tropical upper troposphere. The outline of the paper is as follows: In Section 2 we describethe data used and the methods employed to examine the regional tropical HC and momentumfluxes. Section 3 deals with the annual cycle of the momentum fluxes. Section 4 presents thesefluxes and the regional overturning cells during the NH winter, summer and spring equinoctialperiod. Section 5 contains a summary of results and their discussion. The data used in this study comprises of four-times daily horizontal wind at a resolution of 2 . ◦ across 17 pressure levels from ERA-Interim (Dee et al., 2011) for a 40-year period from 1979-2018. We also make use of monthly mean GPCP Precipitation data (Adler et al., 2003) withthe same resolution and over the same period provided by NOAA ( https://psl.noaa.gov/ ).The zonally averaged zonal momentum equation reads (Dima et al., 2005; Kraucunas and Hart-mann, 2005), ∂ [ u ] ∂t = [ v ] (cid:18) f − φ ∂ [ u ] cos φ∂y (cid:19) − φ ∂ [ u ∗ v ∗ ] cos φ∂y − [ ω ] ∂ [ u ] ∂p − ∂ [ u ∗ ω ∗ ] ∂p + [ X ] (1)The notation above is standard. Specifically, square braces denote a zonal mean and asterisksdenote a deviation from this mean. The first term on the right is the mean meridional mo-mentum flux convergence, the second term is the meridional eddy momentum flux convergence.The third and fourth terms are the mean vertical advection and vertical eddy flux convergence.Consistent with previous studies, we find these to be small in comparison to the first two termson the RHS. The last term is a residual and accounts for all sub grid-scale processes. The firstpart of this study focuses on the Day of Year variation of the first two terms on the RHS ofEquation 1. In particular, daily estimates of each variable are calculated and then these areaveraged over the respective days through the 40 years on record.Utilising the flux decomposition of Lee (1999), the eddy and mean flow convergence terms ofEquation 1 can broken down into five components. Specifically, − φ ∂ [ u ∗ v ∗ ] cos φ∂y = − φ ∂ [ ¯ u ∗ ¯ v ∗ ] cos φ∂y − φ ∂ [ u ∗ v ∗ ] cos φ∂y (2)3 v ] (cid:18) f − φ ∂ [ u ] cos φ∂y (cid:19) = f [ v ] − [ v ] 1cos φ ∂ [ u ] cos φ∂y − [ v ] 1cos φ ∂ [ u ] cos φ∂y (3)The two terms on the RHS of Equation 2 are momentum convergences associated with thestationary eddies (SE) and transient eddies (TE). The three terms on the RHS of Equation 3 aredue to the Coriolis force (CF), momentum convergences linked to mean meridional circulation(MMC) and transient meridional circulation (TMC). Apart from the notation presented before,overbars indicate a temporal mean and primes indicate a deviation from this mean.For part of our analysis, we make use of a rotational-divergent partition. Specifically, treatingthe horizontal wind field on each pressure level as a two-dimensional vector field, we split it intorotational and divergent components via a Helmholtz decomposition. The horizontal velocityfield is expressed as, ~v = ∇ χ − ~k × ∇ ψ. (4)The first term on the RHS describes the irrotational component while the second term on theRHS describes the non-divergent component of the velocity field. The rotational and divergentcomponents will be denoted by the subscripts r and d respectively.Seasonal averages of these terms are computed by time averaging the respective quantities overthe duration of a season for every year on record and then presented as a 40-year mean. Themajor seasons considered in this study are NH winter, spring and summer. To obtain resultsthat are consistent with and comparable against previous studies, the seasonal estimates arecomputed by temporal averaging over the dates mentioned in Table 1 of Dima et al. (2005).Seasonal estimates of winter, summer and spring are presented. Figure 1 describes the annual cycle of the zonal mean eddy momentum flux convergence and themean meridional momentum flux convergence, i.e., the two major terms in Equation 1 for thezonally averaged momentum budget of the tropical upper troposphere. In an annual mean, themean meridional term exceeds the eddy convergence and leads to the tropical upper troposphericeasterlies (Lee, 1999). Compared to the rest of the year, both the mean and eddy momentumconvergence terms are particularly enhanced during the Asian summer monsoon season (Junethrough September) and are close to zero during the equinoctial periods. There is a strongsense of anti-correlation (correlation coefficient of -0.93, Table 1) between the mean meridionaland eddy momentum flux convergence throughout the year. This has been noted previously onseasonal timescales (Dima et al., 2005; Kelly and Mapes, 2011). In fact, while emphasising onthe roles of planetary and sub-planetary scale eddies in the winter-summer transition of the NHcirculation, a similar balance between mean meridional convergence and planetary scale eddyflux convergence was noted (Shaw, 2014).The dynamical phenomena behind this compensation in the upper tropospheric zonally averagedzonal mean momentum budget have also been explored previously. Dima et al. (2005) observedthat there is a coincidence between the zonally averaged eddy flux and mean meridional windsin the upper troposphere along with a negative correlation between the climatological [ v ] and − ∂ y [ u ]. They argue that this tendency for opposition between the two convergence termsoccurs because of the preference of the zonal mean tropical rain belts and eddy forcing to occurin the same latitude band. The zonally averaged heating generates the meridional overturningcirculation and a resultant upper tropospheric mass flux divergence, while the eddy forcingresults in a momentum flux convergence into the source region. In fact, stationary Rossby wave4nergy can propagate meridionally through easterlies in the presence of a meridional backgroundflow (Watterson and Schneider, 1987; Li et al., 2015). This line of reasoning is supported byidealised modelling efforts (Kraucunas and Hartmann, 2007). Further, the imbalance createdby any eddy flux convergence is nullified by a mean flow adjustment to maintain thermal windbalance (Kelly and Mapes, 2011). Day of Year A cc e l e r a t i o n ( m s d a y ) Eddy Mean
Figure 1: Climatologial Day of Year variation of the zonally averaged mean meridional momen-tum flux convergence (orange) and eddy momentum flux convergence (blue), as Equation 1,averaged over 150-300 hPa, about ± ◦ of the equator. A 20-day low-pass filter has been appliedprior to presentation. ms Figure 2: Spatial map of the annual mean difference between upper (150hPa) and lower level(925 hPa) divergent meridional winds (colors), along with time mean precipitation (contours).The contours of precipitation are at 5., 7., 9. and 11. mm/day.Recent work highlights that the seasonally varying zonally averaged mean meridional circu-lation is actually composed of longitudinally limited overturning circulations (Hoskins et al.,2020). Following the argument that it is the divergent motions which contribute to the north-south meridional overturning circulations (Zhang and Wang, 2013; Schwendike et al., 2014), weconstruct a spatial map of the difference between the upper and lower tropospheric divergentmeridional wind as an annual mean overlaid with contours of annual mean precipitation. Such adifference strengthens the upper level divergent motion forced by the monsoonal heating. Figure2 shows that, the tropical overturning activity can be partitioned into three sectors. They areAsia-West Pacific (40E - 150W; abbreviated as AWP), Central Pacific - West Atlantic (150W5 30W; CP-WA) and African (30W - 40E) regions. These three sectors have been motivatedby the regional monsoonal circulations that exist within them. Of course, the exact choice oflongitudinal extents for each regional sector is subjective, the results are roughly insensitive tosuch alternatives.
Day of Year A cc e l e r a t i o n ( m s d a y ) Figure 3: Same as Figure 1 except that the zonal mean and eddy computations are over respec-tive longitudinal bands mentioned in reference to Figure 2. The thick solid lines represent themomentum flux convergence computed and averaged over AWP sector, the thinner dotted linesare for those over CP-WA and the broken dash-dot lines are for the African regions respectively.The solid lines in Figure 3 show mean and eddy momentum flux convergences as in Figure1, but when computed only over the AWP sector. This region, covering the Asian summermonsoon and Australian monsoon sectors, is larger and associated with stronger overturningmotions than the other two longitudinal zones mentioned with reference to Figure 2 (Hoskinset al., 2020). On comparison with the quantities computed over the global domain in Figure1, the two mean convergence terms agree quite well with each other (thick orange curves inFigures 1 and 3; correlation coefficient of 0.88 as in Table 1), but the eddy convergences arequite different during a large part of the year (thick blue lines; 0.59). This disagreement betweenthe two eddy momentum convergence estimates begins around the decay of the Indian monsoon(approximately Day 220) and persists into spring season of the following year. Like the globaldomain, the mean and the eddy momentum flux convergences computed over this region alsoexhibit a weaker anti-correlation of -0.56. The eddy momentum flux convergence in this regionis almost always positive and shows two peaks, one in the winter and one in the summer season.The mean meridional momentum flux tries to retard the zonal mean flow and peaks in thesummer season.The set of dotted curves in Figure 3 show the mean and eddy momentum flux convergence overCP-WA sector. This region is associated with the American monsoons and exhibits a divergenceof eddy momentum flux over most of the year except during spring. The mean meridional fluxis also mostly negative through the year, and in summer, is greater in magnitude than thatcomputed over the global domain. Annual mean values of the eddy and mean meridional fluxconvergences over this region are − . − . ms − day − , respectively. In contrast tothe roles associated with the eddy and mean fluxes over global domain (Lee, 1999), both theflux convergence terms in the CP-WA region try to decelerate the flow over most of the year.From Table 1, the anti-correlation between the two fluxes here is weaker than in the AWPsector. Further, Table 1 also shows that when compared against the global domain, the linear6egion EMFC vs MMFC Global vs regionalEMFC MMFCGlobal -0.93 - -AWP -0.56 0.59 0.88CP-WA -0.27 0.30 0.93Africa -0.77 0.84 0.96Table 1: Pearson’s correlation coefficients computed (first column) between eddy momentumflux convergence (EMFC) and mean meridional momentum flux convergence (MMFC) overdifferent regions (Figures 1 and 3), between global and regional (second column) EMFC and(third column) MMFC.association between the two eddy (mean) convergences is weak (strong).Fluxes computed over the African sector are shown in dash-dot lines in Figure 3. This regionis associated with the African monsoons, and is smaller in longitudinal extent (70 ◦ ) comparedto AWP (170 ◦ ) and CP-WA (120 ◦ ) but has overturning circulation of strength comparable tothat of CP-WA region (illustrated by the strength of V d in Figure 2; consistent with Figure 5 ofHoskins et al. (2020)). The eddy momentum flux convergence in the African region is small ( ≤ . ms − day − ) over the course of the annual cycle, although the mean meridional momentumflux convergence is very similar in form to that of the global domain (linear association of 0.96,Table 1) and has the largest peak of all the three regions.In all, we see that the mean meridional flux is mostly negative in all regions and this is reflectedin the global mean in Figure 1. Interestingly, despite representing different longitudinal zones,the magnitude of this flux convergence in each zone peaks in the boreal summer. Further,the nature of the mean flux convergence is similar in all three regions throughout the year.Eddy fluxes on the other hand are much more diverse in character with minimal strength inthe African sector through much of the year and a decelerating (accelerating) influence on thezonal flow in the CP-WA (AWP) region during winter. In addition, in AWP region, the eddyflux convergence peaks in both the boreal summer and winter. Whereas, in CP-WA, there is asingle peak in the winter season. For a more fundamental understanding of these convergenceterms, we now examine how the rotational and divergent components of the horizontal windscontribute to the total momentum fluxes. After which, to probe changing character of the eddyfluxes we consider the winter, summer and spring time balances separately. Figure 4 shows panels of the eddy and mean flux convergence computed using the rotationalzonal wind and divergent meridional wind (solid lines) along with those computed using the totalvelocity field (dashed lines) over the individual sectors considered in this study. Convergencecomputed using other combinations of the rotational-divergent components of winds are notshown as the mean terms obtained from them are small compared to the ones computed using u r and v d . For simplicity, we will use uv and u r v d to denote flux computed from the full velocityfield and the u r - v d components respectively.The most striking feature of Figure 4a is that the u r v d contribution and total mean convergence(solid and dashed orange curves) are identical. The reason behind this identity can be elucidatedsimply. On splitting Equation 4 into its zonal and meridional wind components, it can be seenthat that v r and u d are zonal gradients of a stream function and velocity potential, respectively,and go to zero when integrated over all longitudes. Further, as seen in Figure 4b,c and d, the7 A cc e l e r a t i o n ( m s d a y ) a b0 100 200 300Day of Year10 A cc e l e r a t i o n ( m s d a y ) c 0 100 200 300Day of Yeard Figure 4: Same as Figures 1 and 3 except that solid lines represent the u r v d eddy and meanmomentum flux convergences (Equation 4) computed over a) global longitudes, b) AWP, c)CP-WA and d) African sectors. The dashed lines in each panel represent the full eddy andmean flux convergences over each of these regions.dominance of the u r v d contribution to the mean flux also holds individually in every region.Figure 5 shows the annual cycle of v d and − ∂ y u r as a zonal mean over the different regions.The annual cycle of zonally averaged v d is a near perfect sinusoid with an amplitude of 2 ms − .The zonally averaged − ∂ y u r is simply the zonally averaged relative vorticity and it’s annualcycle is orthogonal to that of [ v d ], though it’s magnitude is diminished in the winter comparedto summer. This arrangement of these two quantities hints at advection of vorticity from thesummer hemisphere into the winter hemisphere by the divergent mean meridional wind (Hoskinset al., 2020) and the resultant year-round southward flux of absolute vorticity (Zurita-Gotor,2019). In fact, this theme of cross-equatorial transport of vorticity by the thermally directcirculation is consistent across all sectors and explains why there exists a strong correlationbetween the mean convergence computed over individual regions and the zonally averaged meanconvergence (Table 1). The larger magnitude of zonal mean vorticity in the summer leads tothe single seasonal peak in the mean flux convergence.In Figure 4, the zonal mean eddy momentum flux convergence is dominated by the u r v d termbecause of the prominence of the thermally forced stationary Rossby waves in the tropics (Zurita-Gotor, 2019). However, the picture is quite different in the localised sectors. In the AWP andCP-WA regions, the two eddy convergence terms (solid and dashed blue lines) align only aroundthe spring season. This implies that the other components of the total eddy momentum fluxdie down during the spring and become important during other times of the year. Amongstall the sectors, the largest contribution from the u r v d eddy convergence is in the AWP sector.The CP-WA sector is, in fact, dominated by the u r v r component (not shown), which impliesa role of extratropical eddies (Zurita-Gotor, 2019). However, these two convergence terms lineup quite nicely during the NH monsoon season in the African sector. This fact suggests thatthe prominent role of the u r v d flux in the zonal mean eddy convergence results from internalcompensations between other components of the eddy momentum flux.8
50 100 150 200 250 300 350Day of Year32101234 V d ( m s ) ; × y U r ( d a y ) Figure 5: Climatological Day of Year variation of v d (blue) and − × ∂ y u r (orange) zonallyaveraged over all longitudes (solid), AWP (dashed), CP-WA (dash-dotted) and Africa (dotted).As Figures 1, all quantities are averaged over 150-300 hPa, about ± ◦ and a 20-day low-passfilter has been applied prior to presentation. As we have seen, the nature of momentum fluxes varies with season and region. Given thatthese are likely to influence different longitudinal sectors in the tropics, here, we explore thespatial character of the eddy fluxes and the associated overturning cells in the tropics duringthe winter, summer and spring equinox seasons. ms ms day Figure 6: Spatial map of the upper tropospheric (150-300 hPa) eddy momentum flux con-vergence overlain with quivers of eddy wind vectors, seasonally averaged over winter. Thequiver-key is shown in the bottom right of the figure. The green boxes are centred over theequator at 105E and 120W respectively and are 10 ◦ wide in both cases.Figure 6 shows the upper tropospheric spatial distribution of the eddy momentum flux conver-gence term seasonally averaged for winter. The aforementioned winter-time mismatch between9he global and local domain estimates (solid blue lines in Figures 1 and 3) is evident here.When the eddy convergence is calculated globally, there is a cancellation between the positivecontribution around 120E and the negative contribution from approximately 120W. The fea-tures in Figure 6 responsible for this mutual cancellation are highlighted by the green boxes.In fact, the eddy momentum flux divergence due to the 120W box can be seen in the dottedcurve (CP-WA region) of Figure 3. The positive eddy convergence in the 120E box is linked tothe two off-equatorial anti-cyclonic Rossby gyres straddling the equator around the MaritimeContinent — see also the top panel of Figure 9 in Dima et al. (2005) — of which the SouthernHemispheric (SH) gyre is tied to the Australian Monsoon. The eddy momentum convergencefeatures associated with the 120W box are related to the Pacific Ocean upper tropospherictroughs (Kelly and Mapes, 2016, and the references therein). These eddies can also be seen inthe stationary wave patterns during the winter season (Held et al., 2002).These regions of eddy momentum convergence/divergence can be explained using the eddy windvectors ( u ∗ , v ∗ ). Considering Figure 6, in AWP the combination of negative u ∗ with negative(positive) v ∗ southwards (northwards) of the Equator with zero v ∗ at the Equator results ina northward (southward) eddy momentum flux to the south (north) of the Equator with zeroflux at the Equator. This arrangement of eddy momentum fluxes leads to an accumulation ofmomentum at the Equator. For the region in the East Pacific, positive u ∗ along with negative v ∗ over most of the area gives a southward momentum flux. Moving away from the equator,this southward flux increases as y (distance from equator, positive northwards) decreases andresults in positive or almost zero convergence close to southern edge of the box.The NH subtropical and extratropical features of Figure 6 compare nicely with wave activityconvergence features (Caballero and Anderson, 2009). This is testimony to the fact that azone of eddy momentum flux convergence (divergence) is a zone of wave activity divergence(convergence). This is synonymous with the theory that Rossby wave breaking (convergenceof wave activity) causes mean-flow deceleration. From their results, it is worthwhile to notethat the divergence of eddy momentum flux in the 120W box of Figure 6 is due to a weakconvergence of wave activity emanating from SH (see also Figure S1). These waves, seeminglygenerated in SH Pacific ocean, propagate across the equator into NH through the ”westerlywindow” in the 120W box (Hoskins and Ambrizzi, 1993; Li et al., 2015). These waves travelnorthwards to join the waves coming in from Asia to form a region of confluence off the NorthAmerican coast in the subtropics of eastern Pacific (Figure S1).Despite there being a pocket ofwesterlies in the equatorial Atlantic, the Rossby waves from the North Atlantic are absorbedin North Africa before they can reach the equator and there is no cross-equatorial propagationhere (Caballero and Anderson, 2009).The cross-equatorial propagation observed during winteris not present in the other seasons (Figure S1).The top row of Figure 7 shows the vertical structure of zonally averaged zonal wind as a seasonalmean for winter. Similar to the global domain, the vertical structure of the zonal mean zonalwind over the AWP sector has mid-latitude westerlies and easterlies over most of the tropicalatmosphere, except the weak surface westerlies near the equator in SH. Both domains showa maxima in the mid-latitude westerly winds in NH. Unlike these two domains, the CP-WAsector shows westerlies over the entire the upper troposphere. As noted before, this is also thewesterly duct associated with cross-equatorial wave propagation. These westerlies are linkedto the two eddies flanking the 120W box in Figure 6. Coupled with the weaker westerlies overtropical Atlantic, this is manifested in the form of weak equatorial superrotation in the zonalmean picture (Zurita-Gotor, 2019).In the second row, Figure 7 shows the zonally averaged meridional overturning stream function(black contours) over the respective sectors. These are computed using Equation 6 of Zhang andWang (2013). Consistent with Figure 2, the AWP sector has stronger overturning circulations10 [ u ] All Longitudes 40E-150W 150W-30W7010010020030050070010001000 [ m ] .
00 0 . . . . . . . . . . . . . . . . . . . . . .
00 0 . . . . . . . . . . . . . . . . . . . . .
00 0 . . . . . . . . . . . . . . . . . . . . . S E
30 S 0 30 N7010010020030050070010001000 C F
30 S 0 30 N 30 S 0 30 N6 4 2 0 2 4 6 ms day Figure 7: Winter time distribution of (top row) zonal wind, (second row) angular momentum perunit mass, (third row) stationary eddies and (fourth row) coriolis force computed over the globaland local domains (Figure 2) along with meridional overturning stream function (black contours,maximum stream function is marked for each domain by red cross). All quantities are presentedas a zonal mean over the respective domains. In all panels, solid (dashed) contours indicatepositive (negative) values and the zero contour is dotted (not provided for the overturningstream function). For [ u ], the contour interval is 4 ms − . For [ m ], heavier grey contoursare used which are multiples of Ω a (Ω is the angular velocity of Earth about it’s own axisand a is its radius). For the overturning stream functions, a two point moving average hasbeen applied along latitudes prior to presentation; positive (negative) values indicate clockwise(counter-clockwise) circulations and the contour interval is 5 × kgs − .than the CP-WA region. The broad similarities between the zonally averaged quantities ofglobal and 40E - 150W regions is evident here too, though the latter has stronger cells withgreater meridional extent in both hemispheres and extends farther polewards and into theupper troposphere. The red marker in these panels denotes the position of the stream functionmaximum. It is clear that conventional HC is simply a area-weighted average of these localisedoverturning motions.This discussion of individual cells colluding together to give the conventional HC is appropriatedmore importance once the zonally averaged angular momentum (second row of Figure 7) isconsidered. Angular momentum per unit mass is expressed as m = (Ω a cos φ + u ) a cos φ , where φ is latitude. From Figure 7, the solitary upper tropospheric angular momentum maximaabove the equator in the global domain comes from the CP-WA sector. It is noteworthy thatthis maximum in angular momentum is tied to the weak superrotation identified in the previousparagraph and that it is a result of the zonal gradients in geopotential that exist in this regionin the annual mean (Figure 4 of Dima et al. (2005)). It should also be noted that the flow11treamlines cross [ m ] contours, immediately suggesting a strong influence of the eddy flux inthis region (Walker and Schneider, 2006). In the AWP region, the rising branch of the HC inthe SH and the upper tropospheric cross-equatorial flow does not cross [ m ] contours suggestingthe conservation of angular momentum and a lack of eddy influence in this region. In bothregions, as well as over the entire globe, the deviation of [ m ] contours from the vertical at thepoleward extremity of the HC in the NH indicates a regime with vorticity based Rossby numberbetween 0 and 1 (Walker and Schneider, 2006).It is well known that the winter HC is influenced by eddies (Caballero, 2007; Zurita-Gotor and´Alvarez-Zapatero, 2018), but here we have shown that the most significant affect of the eddiesis in the 150W-30W (CP-WA) region. In fact, the AWP sector has the strongest overturningcirculation and a large eddy momentum flux convergence but is not a region of strong eddyinfluence. In particular, the AWP winter cell is shielded from the SH eddies due to the band ofupper level tropical easterlies as seen in Figure 6 (Walker and Schneider, 2006). As seen, thisallows its rising branch to be nearly angular momentum conserving and to extend farther intothe winter hemisphere (Schneider and Bordoni, 2008). However, the NH midlatitude eddies doexert an influence on AWP winter cell. This is via a deceleration of the subtropical jet due tothe breaking of these waves on the poleward flank of this cell (Becker and Schmitz, 2001; Heldet al., 2002).The dominant terms in Equations 2 and 3, during winter are the stationary eddy flux in Equation2 and the Coriolis term in Equation 3; these are presented in the third and fourth rows of Figure7. The tropical eddy momentum convergence in this season is largely due to the stationarywave contribution from AWP. As discussed previously, this is because of the presence of thestationary Rossby waves straddling the Maritime Continent in the West Pacific. In fact, thislarge eddy momentum flux convergence during winter in this sector is seen in Figure 3. Incomparison the stationary eddy features in the CP-WA are much weaker and maximize ata lower level (200 hPa) than the AWP features. The convergence of eddy momentum flux(divergence of wave activity) from the Pacific Ocean upper tropospheric troughs in SH andthe associated tropical and subtropical divergences (convergences) in NH, discussed previously,are evident here. Also, the eddy momentum convergences (divergences) that occur because ofthe generation (absorption) of extratropical Rossby waves in the midlatitudes are visible herenorthwards (southwards) of 30 ◦ N in both the longitudinally limited sectors (Held et al., 2002;Caballero and Anderson, 2009). In contrast, the Coriolis force has a major role to play acrossall longitudinal regions. In fact, in the deep tropics, it is the Coriolis force alone that balancesthe positive acceleration due to the stationary eddies in the AWP sector. Though, in the NHsubtropics, the balance is largely between Coriolis, MMC and SE terms (Shaw, 2014).From the discussion so far, it is clear that the contrasting nature of eddy momentum convergencein the AWP and CP-WA sectors, observed in Figure 3, can be attributed to their respectiveorigins. Reiterating, the positive acceleration in AWP sector is due to the convergence of west-erly momentum by the tropical Rossby waves there. The deceleration in the CP-WA sector isdue to the absorption/breaking of the waves coming towards the equator from the midlatitudes.The rather weak response (˜ − . ms − day − ) of the mean momentum convergences during thisseason is due to the feeble and symmetric nature of the Coriolis force close to the equator andbecause the non-linear momentum advection by the MMC term takes effect somewhat north-ward of the equator during this season. The CP-WA region, with a weak meridional gradientof zonal winds, also has a small MMC contribution.12 ms ms day Figure 8: Same as Figure 6 except for summer. The green box is centered over the equator at60E.
Moving to the boreal summer, Figure 8 shows the spatial distribution of the upper troposphericeddy momentum flux convergence during this season. The dominant feature, highlighted bythe box in Figure 8, responsible for the eddy flux convergence in the deep tropics during thisseason in Figure 1 (see also Figure 3) is the upper tropospheric return flow of the Asian summermonsoon (Dima et al., 2005; Hoskins et al., 2020). As discussed previously for winter, similararguments can be made here regarding the eddy momentum flux coupled with the nature of themeridional divergence operator to explain the eddy flux convergence over the tropical IndianOcean. In comparison to winter, the Pacific Ocean gyres have a much weaker tilt over theEast Pacific sector during winter. This results in the weak contribution to the zonally averagedmomentum budget from this region, consistent with the dotted blue lines in Figure 3.Figure 9 shows the vertical structure of the zonally averaged diagnostics for summer. The toprow shows that the equatorial upper troposphere is dominated by easterlies which extend upto the subtropics in NH. As in Figure 7 for winter, the AWP sector has similar but strongerfeatures in comparison to the global domain. Both these panels show the near-surface westerliesin tropical NH that are characteristic of the lower-level monsoonal flow. In contrast to winter,the CP-WA sector also shows equatorial upper tropospheric easterlies during summer. However,there does appear to be a hint of weak equatorial westerly winds over AWP (CP-WA) sector at70 (150) hPa level.The regional stream function and angular momentum contours during summer, shown in thesecond row of Figure 9, are quite similar to their winter season counterparts, though the globallyaveraged HC has a different character. Specifically, in the deep tropics the upper branches ofwinter hemisphere cross-equatorial cells in the global and AWP domains do not cross angu-lar momentum contours and conserve angular momentum (Hoskins et al., 2020; Bordoni andSchneider, 2008). Now, a band of tropical easterlies shield the rising branch of the local cell fromthe influence of NH midlatitude eddies propagating towards the tropics (Bordoni and Schneider,2008; Schneider and Bordoni, 2008). Once again, there is an isolated angular momentum maxi-mum in the CP-WA sector, and another one in the stratosphere of the AWP sector. In both thelongitudinally limited sectors, these isolated maxima appear to be tied to the weak signatureof the equatorial westerly winds. The cross-equatorial CP-WA cell does cross the isolated [ m ]contour at 200 hPa in the SH deep tropics indicating an eddy influence in this sector; however,this affect doesn’t appear to carry through to the globally averaged HC. Thus, the upper branchof globally averaged cross-equatorial HC does not cross angular momentum contours and is freeof eddy influence (Hoskins et al., 2020; Walker and Schneider, 2006), which is in contrast to the13 [ u ] All Longitudes 40E-150W 150W-30W [ m ] .
00 0 . . . . . . . . . . . . . . . . . . . .
00 0 . . . . . . . . . . . . . . . . . . . .
00 0 . . . . . . . . . . . . . . . . . . . . S E C F
30 S 0 30 N7010010020030050070010001000 MM C
30 S 0 30 N 30 S 0 30 N6 4 2 0 2 4 6 ms day Figure 9: Same as Figure 7 except for summer. The blue dots denote the mimima of theoverturning stream function.global cell in the winter season.As discussed with regard to Figure 8 and evident in the third row of Figure 9, the dominantcontribution to the eddy momentum flux convergence in the equatorial upper troposphere is bythe monsoonal return flow of the AWP sector. The amplitudes of SE during this season are muchreduced than that observed in winter. Though diminished with respect to the winter season,the stationary waves generated in the NH are still visible and the extratropical stationary eddysignature is dominated by the CP-WA sector. The NH subtropical region in AWP that wasobserved to be a zone of strong eddy momentum flux divergence during winter, is one of weakconvergence during summer. The lower strength of these waves is probably accounted for bythe weakened NH subtropical jet and that the NH meridional temperature gradient is reducedduring this season (Held et al., 2002; Lee and Kim, 2003). The transient eddy contribution isdominantly via waves from the midlatitudes of the SH (Ambrizzi et al., 1995). The featuresof eddy momentum convergence/divergence observed in Figure 8 around 30 ◦ S are signatures ofthese transient eddies. They are small in the tropics with respect to SE and hence, left out ofconsideration here.Similar to the winter calculations, here too the Coriolis force plays a major role. Consistentwith the observations of Shaw (2014), the dominant balance in the deep tropics is now betweenthree terms; CF, MMC and SE. Towards the SH subtropics, the balance is between Coriolis,14MC and TE (not shown) with a relatively small contribution from SE. Bordoni and Schneider(2008) note the importance of high frequency transient eddies in this balance with a negligiblerole for the stationary eddies, before and after the monsoon onset over the monsoon sector. Thenear-surface balance in the tropics is between Coriolis and friction (Shaw, 2014), dominated bythe AWP sector. This explains the NH tropical near-surface westerlies. As during winter, theMMC has one dominant feature in the subtropics of the winter hemisphere and it is controlledby the AWP sector. Though, near the equator, the MMC flux convergence in the CP-WA regionis actually greater in magnitude than that in the AWP region (Figure 3).Putting the tendencies of summer season (Figure 3) in perspective, the mean meridional mo-mentum flux divergence from AWP is much smaller than the other two sectors. As the AWPsector has weaker − ∂ y [ u ] in the deep tropics despite having stronger overturning circulation([ v ]), the MMC contribution from this region gains traction farther away from the equator.This coupled with the non-existent Coriolis term close to the equator explains the weak meanmeridional flux divergence linked to this sector during summer. However, the CP-WA sectorhas a larger zonal wind gradient close to the equator which accounts for its stronger MMCresponse. With regard to the eddy fluxes, they are almost entirely accounted for by the AWPregion and their tropical origin leads to an accelerating tendency on the zonal mean flow. ms ms day Figure 10: Same as Figure 6 except for spring.The features of the equinoctial spring season can be seen as a transition from NH winter toNH summer. Figure 10 shows the spatial features of the equatorial upper troposphere as aseasonal average for spring. On comparison with Figure 6 and Figure 8, it is apparent thatthere is a weakening of the eddy momentum flux convergence over the Maritime Continentand gradual strengthening over the tropical Indian Ocean. The strong eddy momentum fluxconvergence/ divergence features observed near 120W in the Pacific (Figure 6) has decreased toan intermediate value. This is due to the decreasing tilt of the Pacific Ocean subtropical gyresas the seasons progress from winter to summer, reaffirmed by the near zero eddy momentumconvergence in CP-WA (blue dotted lines in Figure 3). Apart from these features in the deeptropics, there are considerable differences in the subtropics. The decreasing strength of the NHstationary eddies in the Pacific and Atlantic Ocean and increasing strength of the SH barocliniczones is clear.Figure 11 shows the vertical structure of the zonal mean diagnostics seasonally averaged overspring. In comparison to the winter and summer, the zonal mean zonal winds are more sym-metric about the equator. As in the winter season, the CP-WA sector shows upper tropospheric15 [ u ] All Longitudes 40E-150W 150W-30W7010010020030050070010001000 [ m ] .
00 0 . . . . . . . . . . . . . . . . . . . . .
00 0 . . . . . . . . . . . . . . . . . . . . .
00 0 . . . . . . . . . . . . . . . . . . . . T E C F
30 S 0 30 N7010010020030050070010001000 MM C
30 S 0 30 N 30 S 0 30 N6 4 2 0 2 4 6 ms day Figure 11: Same as Figure 7 except for spring. The blue dots (red crosses) denote the mimima(maxima) of the meridional overturning stream function.westerly winds contributed by the Pacific ocean upper tropospheric troughs, though they aremuch weaker. The AWP sector has easterlies throughout the depth of the equatorial tropo-sphere. The globally averaged zonal winds exhibit a weak superrotation limited to 150 hPalevel and above.The meridional overturning cells show equatorial symmetry across all regions, although theregion of ascent is displaced slightly into NH (Dima et al., 2005). The NH cells extend slightlydeeper into the troposphere. There is an isolated maximum of angular momentum away fromthe surface in the global domain as well as the CP-WA estimate. The deviation of the angularmomentum contours from the vertical in all sectors implies a regime intermediate betweenangular momentum conserving and non-conserving schemes (Walker and Schneider, 2006). Thisimplies a greater role for the eddies across all sectors in comparison to the solstitial seasons.From the discussions related to Figure 1, we know that both the mean meridional and eddymomentum flux convergences in the deep tropics are near-zero during the equinoctial seasons.This is evident from the subsequent panels in Figure 11. The dominant balance during thisseason is between the subtropical features of TE, CF and MMC. Compared to the two solstitialseasons, the SE term has a inconsequential role here and it’s contribution in the deep tropicsis similar to TE. It should also be noted that the mild westerly acceleration by the TE is alsoobserved during the other seasons but it gets overshadowed by the strong SE contribution of16he climatological Rossby gyres.In all, the spring season showcases a high degree of equatorial symmetry marked by a pairof nearly symmetric overturning cells (Figure 11). This is due to a lack of eddy forcing ofequatorial Rossby waves and the primary thermal forcing during this season is the incomingsolar radiation (Norton, 2006). This is evident here due the lack of Rossby gyres (Figure 10)seen during the two solstitial seasons (Figure 6 and 8). This lack of eddy forcing is evidentin Figure 1 with eddy momentum flux convergence ≤ . ms − day − during this season fromDay 100-150. Interestingly, the mean meridional flux convergence provides a weak accelerationduring this period. This happens because of the non-zero Coriolis effect on the southern cell asit’s rising edge is positioned over the equator (Figure 10). As discussed in the Introduction, the zonal mean tropical eddy and mean momentum fluxesand the behaviour of the conventional Hadley cell is fairly well documented in literature. Here,we have focused on the temporal structure of these zonal mean entities and also studied thelongitudinal structure underlying them. At the outset, the zonal mean eddy and mean fluxesare shown to have a pronounced annual cycle that peaks in the boreal summer. Further, thedeep tropical fluxes almost vanish during times of the equinox. The nature of these fluxes isprobed further by splitting the globe into three regions based on the divergent motions forced bythe longitudinally heterogeneous monsoon heating. Specifically, these are the Asia-West Pacific(AWP), central Pacific-West Atlantic (CP-WA) and African sectors.The AWP sector provides the bulk of the eddy momentum flux convergence during both thesolstitial seasons via thermally forced stationary eddies in the tropics, while the CP-WA sectorreceives momentum flux during winter from extratropical eddies propagating into the tropicsfrom the southern hemisphere via the East Pacific. The disparate origin of the eddies in thesetwo regions causes the AWP and CP-WA regions to experience acceleration and deceleration ofthe zonal flow, respectively. The African sector shows relatively low levels of eddy flux throughthe year. Thus, the nature of the eddy flux in individual sectors is very different from the zonalmean perspective. In fact, the prominent contribution of rotational zonal - divergent meridionalcomponent to the zonal mean eddy convergence (Zurita-Gotor, 2019), is also not observed tohold in these isolated sectors.Although the AWP sector dominates the eddy convergence in the deep tropics, it comes upshort with respect to the contribution towards the mean convergence. This is due to the factthat although the meridional divergent winds are comparable across all sectors, the meridionalgradient of the zonal wind is lower in the deep tropics for AWP; however it picks up farther intothe winter hemisphere as the AWP local overturning cell outflow merges with the subtropicaljet. In fact, it is the African and CP-WA regions that contribute most significantly to themean convergence in the deep tropics. But, the qualitative nature of the mean convergenceis remarkably cohesive across all three regions and this is traced to the advection of absolutevorticity by the divergent meridional wind in localized cross-equatorial cells. Indeed, the fluxhas a single peak in the boreal summer due to the larger zonal mean vorticity in this season.Further, the high degree of compensation observed between the mean meridional and eddymomentum flux convergence in the zonal mean (Dima et al., 2005; Kraucunas and Hartmann,2007) does not hold in the individual sectors.The zonal mean overturning circulation is dominated by AWP cell (Hoskins et al., 2020). It isseen that the AWP cell is thermally-direct during both the solstitial seasons while the CP-WA17ell is eddy-driven to varying degrees. In fact, the CP-WA region shows an isolated angularmomentum extremum in the upper tropopshere through the year indicating significant eddyinfluence. This is also reflected in the zonal mean HC. For the most part, extratropical eddies areable to reach the deep tropics at upper levels only during winter and remain confined to the outerreaches of the HC during summer. In all, the summer and winter balance of fluxes is betweenthe Coriolis, stationary eddy and to a lesser extent the mean meridional contribution. Whereas,in the spring equinox, the deep tropics show very low levels of momentum flux convergence.Thus, the zonal mean HC is an aggregation of the localised responses to thermal and eddydriving of the overturning circulation.These results show the extent of internal cancellations which occur when momentum flux con-vergence is integrated over all longitudes. Indeed, the compensation between the eddy and meanmeridional flux does not actually hold in longitudinal zones. Moreover, a Helmholtz decompo-sition suggests that the mean flux is indeed controlled by the rotational zonal and divergentmeridional component, but this is not true for the eddy flux when examined in individual sec-tors. Further, the influence of midlatitudes on the overturning tropical circulation is seen to bemost prominent in the CP-WA region. The spatial structure put forth in this work not onlyhelps in uncovering the heterogeneous nature of the deep tropical momentum flux budget andregions of influence of extratropical eddies, it also shows the delicate longitudinal cancellationsin the present day climate that give rise to the familiar zonal mean picture of the tropics.
Acknowledgements
The authors would like to thank the Divecha Centre at IISc for financial assistance.
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Abu Bakar Siddiqui Thakur ∗ and Jai Sukhatme Centre for Atmospheric and Oceanic Sciences, Indian Institute of Science, Bangalore560012, India Divecha Centre for Climate Change, Indian Institute of Science, Bangalore 560012,India m s W i n t e r m s Sp r i n g m s S u mm e r ms day Figure 1: Wave Activity Flux (arrows) along with eddy momentum flux convergence (colors)for the individual seasons. The wave activity is computed as in Caballero and Anderson (2009). ∗ Corresponding author: A B S Thakur, [email protected] a r X i v : . [ phy s i c s . a o - ph ] A ugug