QuarterlyJournalof the RoyalMeteorologicalSociety Q. J. R. Meteorol. Soc. 140: 2484 2490, October 2014 B DOI:10.1002/qj.2312 Isentroic rimitive equations for the moist trooshere Shao-Yi Lee a * and Tieh-Yong Koh b a School of Physical and Mathematical Sciences, Nanyang Technological University, Singaore b Earth Observatory of Singaore, Singaore *Corresondence to: S.-Y. Lee, Uer Air Observatory, 36 Kim Chuan Road, S537054, Singaore. E-mail: lee shao yi@nea.gov.sg Desite the knowledge that the otential temerature of an air arcel has a deendence on its water vaour content, otential temerature is often still calculated as if the arcel were dry, assuming that this moisture deendence is negligible. We show that such a dry otential temerature aroximation is not suitable for troical regions. Moisture gradient terms are seen in the isentroic rimitive equations when Exner and Montgomery functions are generalised with moist secific heat caacities, forming a contribution to the horizontal momentum tendency comarable to that by the Montgomery function. This reflects how local horizontal gradients in otential temerature created by inhomogeneous water vaour distribution are relatively significant comared to gradients created by inhomogeneous temerature, in a large-scale background of weak horizontal temerature gradient. In such an environment, water lays an active role in troical atmosheric dynamics without the utake or release of latent heat during hase changes. Hence, we suggest that the troical trooshere is a lace where the atmoshere can behave dynamically as a binary-comonent fluid at local and regional scales. Key Words: isentroic; rimitive equations; troical; trooshere Received 5 June 2013; Revised 21 October 2013; Acceted 2 December 2013; Published online in Wiley Online Library 24 March 2014 1. Introduction Isentroic treatment of the atmoshere has historically been more widely alied in the stratoshere and midlatitudes. However, global and troosheric treatment in isentroic coordinates has also always existed e.g. Gallimore and Johnson, 1981; Townsend and Johnson, 1985) and is useful for the study of global circulation Hoskins, 1991), with recent work including wave roagation Tomas and Webster, 1994), water vaour budget analysis Schneider et al., 2006), zonal mean circulation in equivalent otential temerature coordinates Pauluis et al., 2010) and and subtroical storm-driving of zonal mean circulation Egger and Hoinka, 2010). The usefulness of isentroic treatment lies in the fact that in these coordinates, Ertel otential vorticity arises naturally as a diagnostic from Kelvin s circulation theorem Andrews, 1983; Tung, 1986), and since it is conserved on isentroic sheets under inviscid adiabatic conditions, it can be treated like the mixing ratio of a tracer substance Haynes and McIntyre, 1987, 1990). The diabatic circulation also aears exlicitly and simly as the vertical velocity in isentroic coordinates. In the trooshere, water vaour as a gas constituent alters the air arcel s secific gas constant and secific heat caacity at constant ressure from their dry values R d and c d to moist values R m and c m. Desite the knowledge that the otential temerature of an air arcel has a deendence on its water vaour content, otential temerature is often still calculated ignoring the mixing ratio in the exonent, with some excetions where water vaour deendence is included e.g. Emanuel, 1994). Although the effects on otential temerature from using a dry ressure exonent is small and not exected to change any revious results ertaining to the global circulation, it is less certain whether these effects are still negligible locally when the background horizontal otential temerature gradient itself is extremely weak. The dynamical effects of having two comonents in a fluid have been studied in the context of salty water Straub, 1999; Kalashnik, 2012), different isomeric forms of hydrogen in gas lanets Gierasch et al., 2004), and cloudy atmoshere Bannon et al., 2003). Regardless of the actual fluid in question, such effects arise because both comonents contribute energetically to the mixture Bannon, 2003; Bannon et al., 2003). On the other hand, if the contribution of the dilute comonent is so small as to be comletely negligible, it may well be treated as a assive tracer. If this were the case for water vaour in the atmoshere, the water vaour deendence in otential temerature could be safely ignored. In this article, we examine how aroriate the dry assumtion is in the comutation of otential temerature in the troical regions section 2), and derive the form that the isentroic rimitive equations take when the deendence of otential temerature on water vaour content is included sections 3.1 3.4). We will further show that the terms arising from this often-ignored moisture effect in the isentroic rimitive equations are non-negligible comared to the more conventional Montgomery function terms wherever the humidity gradient is high and the temerature gradient gentle section 3.5). c 2013 Royal Meteorological Society
Isentroic Primitive Equations 2485 Figure 1. For the monthly climatology July 1981 2010: a) otential temerature on the 700 hpa surface with black contours at 0.5 K intervals; b) otential temerature error due to the dry assumtion on the 700 hpa surface with black contours at 0.01 K intervals d ); c) and d) same as a) and b) but for the 500 hpa surface. Note the different colour scales between the left and right columns. We discuss when otential temerature s deendence on water vaour may be needed and when it may be ignored section 4). A summary of the key results is given in the Conclusions section 5). 2. Aroximation to dry otential temerature Starting from the first law of thermodynamics under quasi-static reversible conditions, the net heat inut to the air arcel is dq = du + dα = 0, where U and α are the secific internal energy and secific volume of the air arcel resectively. Assuming that moist air behaves like an ideal gas so that α = R m T/ and du = c vm dt, where T is the temerature of the air arcel, is the ressure, and c vm = c m R m is the moist secific heat caacity at constant volume, the first law for an adiabatic change becomes 0 = c vm dt + R m dt R m T d dt T = κ d m, 1) where κ m R m /c m. is defined as the temerature of an air arcel when it is transorted adiabatically and reversibly from ressure to ref 1000hPa without any condensation or evaoration. Thus, κ m does not change following the air arcel, because the water vaour mixing ratio and hence secific heat caacity c m and secific gas constant R m for the moist air arcel are constant. Integrating Eq. 1) along the thermodynamic ath of the moist air arcel, in the absence of condensation or evaoration, T 1 T dt = κ m = T ref ref 1 d ) κm. 2) If the air arcel is dry, the otential temerature is ) κd ref d = T, 3) where κ d R d /c d. Henceforth, we shall refer to d as the dry otential temerature. We calculated the error in otential temerature due to the dry assumtion and found it to be small as exected, not exceeding about 0.1 K. To illustrate this effect, see Figure 1b) and d) for the difference between otential temerature and dry otential temerature of the July climatology cf. aendix A for details of the calculation). The distribution for dry otential temerature looks effectively the same as that for otential temerature and is not shown. On the other hand, the horizontal variations of otential temerature in the troical lower to middle trooshere are weak. Comarison of the horizontal gradient of the eddy otential temerature i.e. otential temerature deviation from the zonal mean, see aendix B) and the changes in gradient due to the dry assumtion confirms the susicion that the error arising from the dry assumtion is not always negligible. Comaring the contour intervals of 0.2 K/10 3 km in Figure 2a) with those of 0.01 K/10 3 km in Figure 2b), the size of the fractional error is considerable in certain regions in the July, e.g. 15% near 50.0 W, 7.5 S), 8% near 20.0 E, 5 S) Figure 2c) and d)). Neglecting this error will alter the otential temerature landscae in the Troics sufficiently to warrant attention in quantitatively accurate work. 3. Moist isentroic rimitive equations In section 2, we saw that the errors in aroximating the exonent κ m by its dry value κ d are not always negligible in troical regions where high humidity gradients coexist with weak horizontal otential temerature gradients. For quantitatively accurate work in the troical trooshere, as well as to gain insight into the role of water vaour in the dynamics in this region, it is desirable to reformulate the isentroic rimitive equations using the actual moist) definition of. To clarify the exressions of the moist secific gas constant and secific heat caacities, the constants below may be defined: a R R v 1 0.608, R d 4) a c c v 1 0.859, c d 5) where R v and c v are the secific gas constant and secific heat caacity at constant ressure of water vaour resectively. We ignore the slight variation of c d and c v with temerature and ressure.) The following quantities deend exlicitly on secific humidity q thus: R m = 1 + a R q)r d 6) c m = 1 + a c q)c d 7) ) 1 + ar q κ m = κ. 8) 1 + a c q
2486 S.-Y. Lee and T.-Y. Koh Figure 2. For the monthly climatology July 1981 2010: a) magnitude of the gradient of eddy otential temerature cf. aendix B) on the 500 hpa surface, with black contours at 0.2 K/10 3 km intervals; b) magnitude of the gradient of otential temerature error d ) due to the dry assumtion, on the 500 hpa surface, with black contours at 0.01 K/10 3 km intervals; c) and d) same as a) and b) but zoomed in on the urle box in a). In the following derivations, this exression will aear reeatedly and so we evaluate it in advance: for s = x, y,, s log c m + log T ) s log κ m ={1 log T )} s log c m + log T ) s log R m ={1 log T )} a c 1 + a c q s + logt ) a R 1 + a R q s = s, 9) where c.f. Aendix C) { 1 a c + a } R a c 1 + a c q 1 + a R q logt ) = c d {a c + a R a c ) R d log T )}. 10) c m R m 3.1. Moist Exner and Montgomery functions Using Eq. 9) to relace the under-braced terms in Eq. 13), multilying by m c m T, and with the hel of Eq. 11) and the ideal gas law, = ρr m T, m = 1 s ρ s + m When s =, Eq 14) becomes σ = ρ g s. 14) ) m, 15) where isentroic density σ 1 g by definition, with g as the gravitational acceleration. The above equation shows clearly that isentroic density is merely a rescaled mass density with a scaling factor that, from Eqs 11) and 12), ultimately deends on the distribution of temerature and water vaour in isentroic coordinates. 3.3. Hydrostatic balance Exner s function and the related Montgomery function can be generalized in a moist atmoshere as: ) κm T m c m = c m ref, 11) M m c m T + = m +, 12) where is given by Eq. 2) and is the geootential. When the atmoshere is ) dry, m and M m reduce to the familiar forms of κ c d and M ref cd T + resectively. 3.2. Isentroic density Taking the artial derivative of the logarithm of Eq. 11) with resect to s = x, y,, ) ) s log m = κ m s log κm + log + ref ref s s log c m 1 m = κ ) m T m s s + log s log κ m + s log c m. }{{} s 13) From Eqs 12) and 14), M m s = 1 ρ s + m s + s m + s. 16) In the vertical direction under hydrostatic balance, writing d ρ = d, Mm = 1 + 3.4. Momentum balance ) m. 17) Using Eq. 16) in the horizontal direction, i.e. s = x, y, M m = 1 s ρ s + m s + s. 18) Using hydrostatic balance, note that Eq. 18) may be rearranged as: ) ) Mm + m s s = 1 ) ) ρ s s
Isentroic Primitive Equations 2487 = 1 ) 1 ρ s ρ = 1 ) ρ s ) x Thus, the zonal momentum equation is ) s. 19) z Du Dt = f k u + F M m + m q, 20) where u is the horizontal wind, f is the Coriolis arameter and F is the horizontal drag. 3.5. Moist vs. dry equations Collecting the theoretical results of the last section and aending the mass continuity and moisture tendency equations, the comlete set of rimitive equations using otential temerature that is defined in Eq. 2) as the vertical coordinate is M m = ) 1/κm 1 m ρ = ref 21) R m c m σ = ρ ) g m 22) ) 1 + m 23) Du Dt = fk u + F M m + m q 24) σ = t x σ u) + σ v) + σ Q) 25) y Dq Dt = E C 26) )} = c d c m { a c + a R a c ) R d R m log m c m z, 27) where Q D/Dt is the diabatic heating rate of the moist) otential temerature; E and C are the rates of evaoration and condensation er unit mass of moist) air resectively; m and M m are defined in Eqs 11) and 12); R m, c m, andκ m are functions of q secified in Eqs 6) 8); a c and a R are constants given in Eqs 4) and 5). This set of isentroic equations is closed under rescribed or arametrized forcings F, Q, E and C. It takes recise account of the deendence of otential temerature on water vaour content. In the absence of water vaour, the terms containing are zero and Eqs 21) 27) reduce to the dry isentroic rimitive equations e.g. see Andrews et al., 1987). In extending the dry equations to the moist equations, it is not enough to generalize Exner and Montgomery functions only. New moisture-gradient terms involving the factor are needed. The origin of these terms lie in the variation of the secific heat caacities c vm and c m with the moisture content as R m c m c vm ). Their significance to the overall moist dynamics is governed by the following ratios: In the exression for the isentroic density, Eq. 22), m m : = : log m In the exression for hydrostatic balance, Eq. 23), :1 From Eq. 24) and using Eq. 23) in the second line below, we obtain the ratio for comaring the contributions to the horizontal momentum tendency in Eq. 24), as: m q : M m ) } 1 = m {1 + m = q : M m 1 + Mm ) 1 q : Mm log ). 28) The above ratio comares the isentroic horizontal moisture gradient with the sloe of M m -surfaces in isentroic coordinates. ThevaluesoftheseratiosaresummarisedinTable1and illustrated on the 315 K isentroe in Figures 3 and 4 see Aendix D for details of how the reanalysis was interolated onto otential temerature vertical coordinates). Firstly, we observe that the vertical gradient of moisture reduces the isentroic density by u to 10% Figure 3e) and f)). Even more significant is its imact on the vertical gradient of the moist Montgomery function, reducing the latter by u to 50% Figure 3a) and b)). Finally, the contribution to the horizontal momentum tendency from the isentroic moisture gradient term is comarable to the contribution from the moist Montgomery gradient force Figure 4). All these results oint to the imortance of water vaour gradients in troical dynamics, over and above the simle generalization of Exner and Montgomery functions. 4. Discussion To understand how small variations in moisture can create significant horizontal ressure gradients in the Troics, we consider an atmoshere in a steady climate with zero horizontal temerature gradient, i.e. = z ) 1 z = 0, with a function of only. Then Eq. 18) may be written as M m = m κ m log + q). Since κ m and are of the same order of magnitude, the horizontal variations of log/ ref )andofq may be comared to determine the relative magnitudes of ressure and water vaour contributions to M m. In the Troics, surface ressure at two locations tyically differ by about 1 from 1000 hpa so thatδ log/ ref ) = 10 3. Thus, secific humidity only needs to differ by 10 3 or 1 g kg 1 to result in a ressure difference of comarable magnitude. Such a change in secific humidity is commonly seen in the troical trooshere. Since variations in otential temerature caused by the inhomogeneity in water vaour are only locally comarable, but not larger than variations in dry otential temerature on an isobaric surface in the Troics, troical otential temerature gradients remain weak in comarison with midlatitude otential temerature gradients even when water vaour effects are included. The temerature, and hence the otential temerature in the resence or absence of water vaour), on an isobaric surface is sometimes regarded as effectively constant in the Troics e.g. Sobel et al., 2001), and this weak temerature gradient aroximation for troical geostrohic motion remains valid. Under this aroximation, the total diabatic heating Q T D/Dt is not only deendent diagnostically on the dry otential temerature gradient as in Sobel 2002) but on the water vaour content as well, i.e. it can be shown that Q T w ln [ z dlnd = w + a R a c )κ d { q ln dz dz ref )}]. Practically, the net correction to ln d) on the right-hand z side is small and does not exceed a few er cent in the mean
2488 S.-Y. Lee and T.-Y. Koh Table 1. The ratios that determine the significance of the moisture gradient terms in the moist rimitive equations. Equation Reresenting -terms [A] Reresenting moist terms Comarison [A]:[B] Illustrating figure generalized from dry dynamics [B] Isentroic density Hydrostatic balance Horizontal momentum tendency log m 10% Figure 3a) and b) 1 50% Figure 3e) and f) 1 + q Mm log ) About 1 Figure 4 Figure 3. From the 1981 to 2010 monthly climatology, comarison of terms aearing in Eqs 22) and 23) on the 315 K otential temerature surface: during log m a) January and b) July; c) and d) are the same as a) and b) but showing ; e) and f) are the same as a) and b) but for log m :. climatology not shown). Thus, Q T w d remains a reasonable z diagnostic aroximation. In view of the last result, earlier zonal mean studies e.g. Gallimore and Johnson, 1981; Schneider et al., 2006; Pauluis et al., 2010) remain valid. Although dry otential temerature surfaces are now not truly conserved surfaces under the adiabatic assumtion, i.e. the Lagrangian tendency of dry otential temerature d is no longer urely due to total diabatic heating Q T, the correction to D d /Dt from water vaour is negligible outside the Troics. In other words, there is little effect on revious results about the general circulation or the transorts between the Troics and midlatitudes. Even in the Troics, the thermodynamic effect of water vaour gradients without state change is small comared to the effect of state changes in the water content of an air arcel Laliberte et al., 2012). On the other hand, situations where such effects of water vaour content would matter in analysis would be local or regional atmosheric henomena embedded in the weak temerature gradient background, or analysis in the zonal direction due to the absence of the background meridional temerature gradient in the climatology. Such effects would also be more ronounced on a day-to-day or hour-to-hour basis than in the climatology illustrated in Figure 2 because on such time-scales the boundary between moist and dry air masses can be very shar, such as in the winter monsoon cold surge from the Asian continent into the South China Sea Chang et al., 1979) or in the summer monsoon rainfall front that swees from the Gulf of Guinea into the Sahel region in West Africa Sultan and Janicot, 2003). More imortantly, we wish to highlight to the reader the concet that even without involving state changes of water and their associated utake or release of latent heat, water vaour is still caable of laying an active role in troical isentroic dynamics indeendent of the temerature distribution. The dynamical forcing roduced by water vaour in isentroic coordinates is comarable to that roduced by the temerature distribution, and takes the form of moisture gradients [ x ), y ), log ) ]scaled by m. These terms are not total derivatives and hence are retained in the otential vorticity equation, resulting in the nonconservation of otential vorticity on dry otential temerature surfaces. While hydrometeors in a cloudy atmoshere have been reviously treated as a multicomonent fluid Bannon et al., 2003), our results indicate that the water vaour and air gaseous mixture can also behave as a binary-comonent fluid under certain conditions, i.e. locally or regionally in the troical lower trooshere. Such results will not extend to the other gaseous comonents of the atmoshere since, relative to water vaour, they are well mixed and form a much smaller fraction of total air In this context, it may be worth examining the results of existing numerical models when they are run in troical regions only. Referring back to Eq. 1), adiabatic reversible change for an air arcel may be exressed as TdS = c m dt RT v dln where T v 1 + a R q)t is the virtual temerature and ds c m dln ) is the secific entroy. Water vaour s effect on the entroy of an air arcel arises from the effects on the enthaly first term) and on the density second term). Since the use of virtual temerature and otential temerature v = deft v ref )κ m are routine in the dynamical cores of numerical weather rediction NWP) models, the density effect of water vaour is included in NWP cores. However, virtual otential temerature by itself is not a measure
Isentroic Primitive Equations 2489 Figure 4. From the 1981 to 2010 monthly climatology, comarison of terms aearing in Eq. 24) on the 315 K otential temerature surface: q 10 3 1+ km 1 ) during a) January and b) July; c) and d) are the same as a) and b) but showing Mm log ) 10 3 km 1 ). Note that the ratio between the two terms was not taken as either term can be exceedingly small leading to large numerical error in the ratio. of entroy since c m dln v ) = c m dlnt v R m dln ds + a R c m dq. Deending on the imlementation in the articular NWP model, entroy or total energy as reresented by otential temerature or temerature can be the rognostic quantity, with the other as a diagnostic quantity. In the interconversion between the two, moisture-deendent ressure exonents do not seem to be routinely used. For examle, the Weather Research and Forecast WRF) model advects otential temerature and mas it back to ressure through = ref R d v / ref α) γ, as arahrased with our notation from Skamarock and Klem 2007). Since γ = c d /c vd, the ratios of secific heat caacity at constant ressure and volume for dry air, this imlicitly uses a dry ressure exonent in the interconversion between virtual otential temerature and virtual temerature, i.e. v = T v ref /) 1 γ 1 = T v ref /) κ d. Nevertheless, this is not to say such NWP models will roduce wrong results, since even without a moisture-deendent ressure exonent, the resultant numerical errors may often be small enough that the model results are still valid, esecially when other sources of uncertainties e.g. numerical diffusion) are resent. Of greater interest for future work may be the analysis of numerical model outut in the Troics as an atmoshere driven dynamically not only by moist Montgomery otential, but also by water vaour gradients. 5. Conclusions While the deendence of otential temerature on water vaour content is extremely weak, the neglect of this deendence results in relative errors in the horizontal gradient of eddy otential temerature of u to 10% locally in the climatology of the troical trooshere Figures 1 and 2). The driving of horizontal momentum by the isentroic moisture gradient is comarable to that by the moist) Montgomery gradient force in the troical trooshere Figure 4). If not for the existence of the water vaour gradient terms, the moist and dry rimitive equations would have been equivalent under the exchange of the corresonding moist and dry quantities: R m R d, c m c d, κ m κ, m, andm m M. Such an equivalence means that the core dynamical equations of motion in the Troics would have admitted the same form of solutions, leaving secific humidity q in the absence of hase changes more like a assive tracer with only imlicit influence on ressure and temerature for given solutions of m. Instead, even without the utake or release of latent heat associated with hase changes, water vaour is caable of laying an active role in troical isentroic dynamics. While forcing by horizontal water vaour gradients remains negligible when studying the general circulation, it does not seem advisable to ignore the water vaour deendence of otential temerature when studying local or regional troical dynamics. In these situations, water vaour and air can behave as a binary-comonent fluid, although further study is recommended to determine if associated henomena can be observed. Acknowledgements We would like to thank the three anonymous reviewers, as well as individuals who have seen earlier versions of this work, for their many helful comments. The NCEP Reanalysis data used in this work was rovided by the NOAA/OAR/ESRL PSD, Boulder, Colorado, USA, from their Web site at htt://www.esrl. noaa.gov/sd/. Aendix A: Calculation of otential temerature In our calculation of otential temerature on the reanalysis dataset only, we used a Taylor exansion on κ m = qr v + 1 q)r d qc v + 1 q)c d {1 + R v c v )q}κ R d c d 1 0.25q)κ, A1) Oq 2 ) and higher-order terms are neglected in Eq. A1) since q is at most O10 2 ) even in troical regions. This makes the Oq 2 ) terms at most of order 10 4, and comaratively unimortant. Thus, otential temerature is calculated as = T ref ) 1 0.25q)κ. Calculations were erformed on the 1981 2010 monthly climatology of the NCEP/NCAR Reanalysis 1 Kalnay et al., 1996). Aendix B: Calculation of otential temerature gradient The horizontal gradient of the otential temerature was not shown, because it is dominated by the background Equator Pole otential temerature gradient in the subtroics. Due to zonal
2490 S.-Y. Lee and T.-Y. Koh asymmetries on the globe it is unavoidable to include subtroical air masses in the lot, hence the gradient of the eddy otential temerature,, where is the zonal mean, was lotted instead in Figure 2, so that the subtroical air masses can be resented with the troical air masses without saturating the colour scale. Nonetheless, excursions of the band of background gradient can still be seen at the north/south edges of the Troics. Aendix C: Calculation of scaling factor In our numerical comutations on the reanalysis dataset only, the scaling factor for the moisture gradient terms are aroximated by taking R m R d and c m c d in Eq. 10) to get a c + a R a c )log ) T. The numerical error involved in this aroximation is O10 2 ) since q is O10 g kg 1 ) in the Troics in Eqs 6) and 7). Aendix D: Interolation into otential temerature vertical coordinate To avoid the question of how mixed layers are treated, regions of vertical atmosheric columns that are not monotonically increasing in otential temerature near the surface were not included in constructing the monthly climatology. Subdomains in the reanalysis dataset where the ressure is greater than surface ressure or the otential temerature is lower than surface otential temerature reresent extraolation below the surface and so were not used. Finally, the analysis is limited to isentroic surfaces away from the ground i.e. 700 hpa and above), excet for intersecting some regions of high toograhy. Monotone iecewise cubic interolation of log/ ref ) against log/ ref ) was erformed vertically from the ressure-level dataset Fritsch and Carlson, 1980). In a comarison between different linear interolation methods by Ziv and Alert 1994), was found to be better reresented by interolating in log, log) coordinates than in, ) or κ, ). The choice of cubic interolation avoids the iecewise constant vertical gradient assumtion in the linear interolation which causes discontinuity in isentroic density in the vertical direction. This is because isentroic density is the accuracy of the stability factor comuted as σ 1 g in -coordinates. We linearly interolated zonal and meridional wind, geootential and humidity from the -levels in the reanalysis dataset onto the -values corresonding to the desired -levels in log coordinates. Moist Exner and Montgomery functions are comuted last on the -levels using the interolated values of ressure, humidity and geootential. In the interolation of moisture, relative humidity H is the actual variable that is interolated. The secific humidity q is then calculated from H, T and. The temerature Tused here is estimated from otential temerature and ressure only, assuming that d.) Directly interolating a variable like q that has a highly convex vertical rofile tends to roduce severe distortions and the interolated rofile may not resemble the actual rofile reresented by the original data. On the other hand, H is bounded between 0 and 1 with a much less convex or sometimes non-convex rofile. While interolating H does not strictly conserve water in the atmosheric column, this shortcoming is not a bottleneck in the diagnostics because reanalysis oututs do not strictly conserve water between analysis times in the first lace. The choice of H for interolation also avoids creating artificial suersaturation due to indeendent interolation of moisture and otential) temerature, which would necessitate ad hoc redistribution of moisture. References Andrews DG. 1983. A finite-amlitude Eliassen Palm theorem in isentroic coordinates. J. Atmos. 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