Click Here for Full Article GEOPHYSICAL RESEARCH LETTERS, VOL. 36, L16804, doi:10.1029/2009gl039667, 2009 A model for the relationhip between tropical precipitation and column water vapor Caroline J. Muller, 1 Laria E. Back, 1 Paul A. O Gorman, 1 and Kerry A. Emanuel 1 Received 18 June 2009; revied 22 July 2009; accepted 28 July 2009; publihed 25 Augut 2009. [1] Several obervational tudie have hown a tight relationhip between tropical precipitation and columnintegrated water vapor. We how that the oberved relationhip in the tropic between column-integrated water vapor, precipitation, and it variance can be qualitatively reproduced by a imple and phyically motivated two-layer model. It ha previouly been argued that feature of thi relationhip could be explained by analogy with the theory of continuou phae tranition. Intead, our model explicitly aume that the onet of precipitation i governed by a tability threhold involving boundary-layer water vapor. Thi allow u to explain the precipitation-humidity relationhip over a broader range of water vapor value, and may explain the oberved temperature dependence of the relationhip. Citation: Muller, C. J., L. E. Back, P. A. O Gorman, and K. A. Emanuel (2009), A model for the relationhip between tropical precipitation and column water vapor, Geophy. Re. Lett., 36, L16804, doi:10.1029/2009gl039667. 1. Introduction [2] Rainfall and column-integrated water vapor are cloely related in the tropic [e.g., Bretherton et al., 2004; Peter and Neelin, 2006; Neelin et al., 2009; Holloway and Neelin, 2009]. The exitence of a poitive correlation between rainfall and humidity i unurpriing: high humidity can be both a caue and conequence of deep convection and rainfall. In fact, apect of thi relationhip are an integral part of theorie for explaining tropical phenomena including the MJO [e.g., Bony and Emanuel, 2005; Khouider and Majda, 2006; Raymond, 2000; Raymond and Fuch, 2009], convectively-coupled wave [e.g., Neelin and Yu, 1994; Kuang, 2008] and hurricane [e.g., Emanuel, 1995]. However, we do not at preent have a full undertanding of the mechanim underlying the oberved humidity-rainfall relationhip. In thi work, we propoe a poible framework for undertanding ome key apect of thi relationhip. [3] Our work evolved from a dicuion about the obervation preented by Neelin et al. [2009] (hereinafter referred to a NPH). They how that intantaneou precipitation, P, on 25 25 km cale increae with column water vapor, w, with a harp increae near a critical value of w and then a omewhat lower increae at higher w (Figure 1). The harp increae (or pickup) i aociated with a peak in precipitation variance near the critical w. NPH and a preceding tudy [Peter and Neelin, 2006] alo argue that precipitation at high w follow a power law with a 1 Department of Earth, Atmopheric and Planetary Science, Maachuett Intitute of Technology, Cambridge, Maachuett, USA. Copyright 2009 by the American Geophyical Union. 0094-8276/09/2009GL039667$05.00 univeral (temperature independent) power of about 0.21 0.26. [4] NPH (and earlier work by thee author) ue thee reult to argue that elf-organized criticality (SOC) i a ueful theoretical framework to explain the dependence of precipitation on water vapor. In their view, precipitation i a critical phenomenon, and the ytem elf-organize toward the critical point of the tranition to trong precipitation. While thi idea i intereting, the empirical evidence for SOC i eentially circumtantial. Until a clear phyical mechanim i provided, the evidence for SOC from the humidity-rainfall relationhip relie heavily on a power law fit to very high humidity value, at which obervation are limited. It i alo unclear how applicable a theory baed on very high humidity and precipitation value i to undertanding the relevant phyic in more typical rainy condition. [5] In light of thee iue, we wondered if a imple et of eaily phyically jutified aumption could explain the key feature of the relationhip NPH document: the harp pickup aociated with a peak in rainfall variance and a flattening of the humidity-rainfall relationhip at higher humidity value. In thi work, we introduce a imple, phyically-baed two-layer model that reproduce thee feature. Our model ha ome commonalitie with NPH interpretation, which we dicu further in our concluion. However, we believe that the phyical jutification for the aumption in our model i more traightforward. In addition, our model can explain the relationhip between humidity and rainfall over a broader range of column water vapor than the power law fit that NPH bae their interpretation on. Hence, our interpretation i le enitive to very high humidity value at which atellite retrieval may be problematic. 2. Model Aumption [6] We firtly aume independent Gauian ditribution of humidity in the boundary layer and in the free tropophere. A motivation for thi aumption, Figure 2 how the probability denity function of water vapor path averaged to 24 24 km reolution, below and above 850 mbar in a radiative convective equilibrium imulation with a Cloud Reolving Model (CRM). The CRM wa run to tatitical radiative convective equilibrium on a 1024 1024 km horizontal domain with 4 km horizontal reolution and 64 vertical level, with pecified radiative cooling conitent with a imilar maller domain 300 K urface temperature run. The wind wa relaxed (time cale of two hour) toward a background wind profile with hear of 5 m/ linearly decreaing from the urface to 16 km. Detail about the CRM are given by Khairoutdinov and L16804 1of5
Figure 1. Figure 1 from Peter and Neelin [2006], howing precipitation rate and their variance veru column water vapor w for two region of the tropical Pacific, a well a a power-law fit above the critical point (olid line). Both the precipitation and the water vapor path are recaled by empirical contant o that the curve collape. The inet how on double-logarithmic cale the precipitation rate a a function of (w w critical )/w critical, where w critical i the critical water vapor path at which precipitation pick up. Reprinted by permiion from Macmillan Publiher Ltd: [Nature Phyic] [Peter and Neelin, 2006], copyright (2006). Randall [2003]. We ee that the Gauian aumption i a reaonable approximation, although there are ome departure from it. In the CRM, the boundary layer and free tropopheric water vapor path are lightly correlated with a correlation coefficient of 0.2; in our model they are aumed to be independent, but thi i not a crucial aumption (ee ection 4). Alo the mean of the boundary layer and free tropopheric water vapor path differ, but thi depend on the choice for the preure cutoff between the two layer. For definitene, we define each of our model layer a contributing roughly half of the column-integrated water vapor. [7] Our econd aumption i that precipitation occur only when the lower layer water vapor exceed a critical value (we ue the term critical to be conitent with NPH terminology, but here critical need not imply the preence of long-range correlation, cale-free behavior, etc. In our uage, the term imply refer to a threhold). In the tropic, horizontal temperature gradient are mall due to the large Roby deformation radiu, o intability, a meaured by convective available potential energy (CAPE), depend primarily on low-level humidity. Thu, our econd model aumption correpond looely to auming that rainfall doe not occur below a critical CAPE value. [8] The third aumption in our model i that when the lower-layer water vapor exceed the critical value, precipitation i a linear function of column-integrated water vapor. We expect rainfall to be modulated by humidity at many level [e.g., Bretherton et al., 2004; Holloway and Neelin, 2009] for reaon we decribe in more detail below. The linear functional form of the column humidity-rainfall relationhip in rainy condition i choen for implicity. Thi aumption i not crucial to the idea underlying our model and could be modified beyond the cope of thi paper. [9] The aumption that when deep convection i occurring, more free tropopheric humidity lead to more rainfall can be rationalized in two way. Mot air parcel riing in deep convective updraft are trongly diluted by mixing with environmental air, o their buoyancy i affected by moiture at many level. When riing parcel entrain moi- Figure 2. (a) Illutration of the two-layer model for precipitation. H v denote the Heaviide function. (b and c) Probability denity function of water vapor (olid line) below and above 850 mbar in a cloud-reolving model with a fixed ea urface temperature (SST) of 300 K. The dahed line how Gauian denitie with the correponding mean and tandard deviation. 2of5
Figure 3. Precipitation and it variance veru column water vapor for m = w c = 0.5 and = 0.025. The exact olution from equation (2) and (3) are alo hown; they agree very well with the numerical olution. (right) Precipitation veru (w 2w c )/(2w c ). (left) For = 0.025, the mean log-log lope i 0.2. ter environmental air, they remain poitively buoyant longer, rie further and the convection i more vigorou than in drier condition with imilar temperature profile. Alternatively, the moiture-precipitation relationhip can alo be jutified uing boundary layer quai-equilibrium [Raymond, 1995; Emanuel, 1995], which potulate a balance between moitening of the boundary layer by urface evaporation and drying of the boundary layer by precipitation-driven cold pool. In moiter condition, fewer precipitation-driven downdraft occur, o more deep convection and rainfall i needed to balance a given urface forcing. [10] Our model i a promiing alternative interpretation of NPH obervation, but at thi tage we conider it a toy model. Further reearch, potentially uing a cloud-reolving model (CRM) or obervation, i needed to tet and contrain the model in more detail before we would conider it a fully quantitative model of the relationhip between rainfall and humidity. 3. Analytical Decription and Exact Solution [11] Our goal i to examine the relationhip between a normalized precipitation P and column water vapor w in a ytem following the aumption decribed above. We ue a imple model of the atmophere with two layer, whoe water vapor path are modeled by independent random variable w lower and w upper, a illutrated in Figure 2a. We aume that w lower and w upper are normally ditributed with the ame mean m and tandard deviation. In the following, w lower and w upper are normalized by the total column-integrated water vapor path o that the mean of the upper layer and lower layer water vapor path are both equal to m = 0.5. [12] In our model, the precipitation i non-zero only if the low-layer water vapor exceed a threhold value w c. Then it i given by the total column water vapor P ¼ w upper þ w lower Hv ðw lower w c Þ; ð1þ where H v denote the Heaviide function, and where P i a non-dimenional precipitation, normalized by an arbitrary time cale. [13] The expected value of precipitation for a given total column water vapor w = w lower + w upper i therefore hpiðwþ ¼ w 2 erfc w c w=2 ; where erfcðx Þ 2 Z 1 pffiffiffi e t2 dt; p X ð2þ and it variance varpðwþ ¼ w2 2 erfc w c w=2 1 1 2 erfc w c w=2 : ð3þ Interetingly, neither hpi nor the conditional variance depend on the mean m. The mean only affect the probability denity function of w, not the relationhip between w and hpi. Note that thi would not be true if the mean of the upper and lower water vapor path were different. [14] In the following ection, we compare thee exact olution with reult from Monte Carlo imulation. In the numerical imulation, we alo enforce that both w upper and w lower are poitive. But even with thi light difference, equation (2) and (3) are in very good agreement with the numerical olution. 4. Reult [15] Figure 3 (left) how the precipitation and it variance a a function of the column water vapor obtained from Monte Carlo imulation (the Monte Carlo imulation were performed with typical ample ize 10 7, and the total column water vapor i forced to remain above 0) for m = w c = 0.5 and for = 0.025. The value choen for i baed on a CRM imulation; from the probability denity function in Figure 2, an approximate value for the tandard deviation i.05m, hence m = 0.5 yield 0.025. The exact olution given in equation (2) and (3) are alo hown, they agree very well with the numerical olution. Depite it implicity, our model predict the pickup in precipitation around w =2w c, a well a the peak in variance for that value of (compare Figure 1 and Figure 3, left). The reaon why the variance peak at the critical water vapor path i traightforward: at low w it doe not rain, and at high w it almot alway rain, ince it i hard to reach 3of5
high value of w without having w lower > w c. Therefore the larget variability in precipitation i expected between thee two limit, i.e., near the critical value of w. [16] We checked the enitivity of our reult to the variou parameter of our model. The hape of precipitation veru water vapor relationhip i upriingly robut to parameter change. A mentioned earlier, the reult are independent of the mean m (ee equation (2) and (3)). In particular, the location of the pickup in precipitation, w = 2w c, only depend on the low-layer critical water vapor, not on the mean m. Changing w c alo doe not affect the hape of hpi(w), it primarily hift the location of the critical w, where precipitation pick up and it variance reache a maximum. Similarly, the key feature of the relationhip hpi(w) do not depend on the tandard deviation ; varying make the pickup in precipitation more or le localized near the critical water vapor path (the reult for variou value of are given in the auxiliary material). 1 [17] One could alo chooe a different equation for the precipitation in equation (1), uch that for intance the precipitation intenity only depend on the upper layer water vapor hpi(w) =w upper H v (w lower w c ). The hape of hpi(w) and of it variance found are almot identical in thi cae. Chooing equal mean for w upper and w lower i alo not crucial: allowing thee mean to be different only hift the location of the critical w. If we relax the aumption that w upper and w lower are independent by allowing for a correlation between them, the reult are till unchanged up to trong correlation above 0.8 or o in abolute value (the correlation in the CRM imulation i 0.2). [18] The only change that doe make a light difference i if the low-layer water vapor i enforced to remain at or below it critical value, i.e. w lower = min(w lower, w c ). The main reult are unchanged, but the pickup in precipitation i not a teep, and the variance doe not decreae all the way to zero for w above the critical value. [19] We note in paing that although there i no clear evidence for a power law dependence of precipitation at high w, the mean power i about 0.2 (ee Figure 3, right), in agreement with NPH who derive a power 0.21 0.26 from data. 5. Concluion [20] We have hown that a very imple, phyically motivated two-layer model can reproduce the oberved relationhip between column-integrated water vapor, precipitation and it variance, a hown by NPH. In our model, humidity in the boundary layer and in the free tropophere are aumed to be independent and rainfall occur only when the boundary layer humidity exceed a critical value, below which the atmophere i aumed table. The amount of rainfall then depend on column-integrated humidity. [21] In thi model, rainfall increae rapidly with columnintegrated humidity cloe to a critical humidity, and the lope of thi relationhip decreae at very high humiditie. Alo, a in obervation, rainfall variance a a function of humidity i maximum near the critical humidity. Qualitatively at leat, our model explain the humidity-rainfall 1 Auxiliary material are available in the HTML. doi:10.1029/ 2009GL039667. relationhip over a wider range of humiditie than NPH and aociated work. [22] Our model may alo explain the reult in NPH that the oberved critical water vapor doe not cale with column-integrated aturation humidity, but intead like the lower tropopheric aturation humidity (ee Figure 3 of NPH). The location of the pickup in precipitation, w =2w c, only depend on the low-layer critical water vapor w c.ifwe aume that w c correpond to a critical relative humidity r c independent of temperature, then the pickup in precipitation occur when the column-integrated water vapor i w(t) = 2w c (T) =2r c w lower,at (T), where w lower,at (T) i the lowlayer aturation water vapor and where T denote temperature. Therefore, a temperature change, the critical column-integrated water vapor in our model cale with the low-layer aturation humidity. [23] Peter and Neelin [2006] ued the oberved relationhip between humidity and rainfall (Figure 1) to argue that phyic analogou to that occurring in continuou phae tranition are important to the dynamic of the moit atmophere at the cale in quetion. Our model alo contain an inherent ingularity where lower tropopheric humidity approache the critical value (dp/dw goe to infinity at thi point). Our model i alo not inconitent with NPH interpretation that the tropical atmophere elforganize toward a tability threhold like that potulated in our model. In thi view, urface evaporation provide a low, continuou forcing bringing the atmophere toward thi threhold, while convection event rapidly diipate intability after it i generated (the quai-equilibrium potulate [e.g., Arakawa and Schubert, 1974; Emanuel et al., 1994]). [24] However, in contrat to Peter and Neelin [2006], our model aume that tranition phyic i unimportant and intead that the relevant tranition can imply be decribed uing a Heaviide function. A dicued in the introduction, we view thi a phyically correponding to a tability threhold that a column mut exceed in order for deep convection and rainfall to occur. Alo, in contrat to NPH interpretation, in our model, the high end of the water vapor-precipitation curve doe not need to be univeral. In fact, a een in equation (2), thi curve depend on the aumed probability denity function of water vapor path near the tability threhold. Our model formulation i alo agnotic about time-pace caling (unlike the critical phenomenon analogy). [25] In our view, the mere exitence of a tability threhold and an approximate power-law behavior of rainfall near that threhold doe not etablih that tropical convection i an example of SOC. Peter et al. [2002] do preent evidence that rainfall event ize follow a power law ditribution, which ugget that the theory of critical phenomena i relevant to atmopheric convection. Further evidence could come from detailed analyi of the patiotemporal character of convection [e.g., Cohen and Craig, 2006; Peter et al., 2009]. [26] Acknowledgment. Thank to Eric Downe, David Neelin, Ole Peter, Daniel Rothman and Bo Yang for ueful dicuion on thi work. Reference Arakawa, A., and W. H. Schubert (1974), Interaction of a cumulu cloud enemble with the large cale environment, part I, J. Atmo. Sci., 31, 674 701. 4of5
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