Tropical Continents Rainier Than Expected From Geometrical Constraints

Abundant rainfall over tropical land masses sustains rich ecosystems, a crucial source of biodiversity and sink of carbon. Here, we use two characteristics of the observed tropical precipitation distribution, its distinctive zonal arrangement and its partitioning between land and ocean, to understand whether land conditions the climate to receive more than its fair share of precipitation as set by the land‐sea distribution. Our analysis demonstrates that it is not possible to explain the tropics‐wide partitioning of precipitation unless one assumes that rain is favored over land. Land receives more than its fair share of precipitation by broadening and letting the tropical rainbelts move more, effectively underpinning a negative feedback between surface water storage and precipitation. In contrast, rain is disfavored over land in climate models. Our findings suggest that the abundance of rainfall that shapes the terrestrial tropical biosphere is more robust to perturbations than models have suggested.

In this study, we ask how the presence of land influences the partitioning of tropical precipitation between land and ocean. From this answer, we draw some inferences as to the existence of feedbacks between the state of the land surface and precipitation as well as the stability of the wet tropics over land. While it seems obvious that the location and size of land masses must somehow matter as the climate is not zonally symmetric, what is not obvious is whether the land with its distinctive surface characteristics influences the atmosphere in ways that unduly favors it to collect more than its fair share of precipitation as given by geometrical constraints. We answer this question by taking advantage of two features of the observed distribution of tropical precipitation: its distinctive zonal arrangement, which allows us to conceptualize the precipitation distribution as a rainbelt whose parameters can easily be varied as a function of the land-sea distribution; and the ratio between precipitation averaged over tropical land and over tropical ocean, which we denote χ(t) with t indexing the month.

Considered Data Sets and Climate Models
To investigate the extent to which the land can affect χ(t), predictions of our conceptual model are compared to observations. Given the uncertainties plaguing the retrieval of precipitation, we consider seven data sets, namely GPCP, PERSIANN-CDR, abbreviated as PERSIANN in the reminder of the text, GPCC, HOAPS, CMORPH, TRMM, and IMERG (see Table 1). As reviewed by Sun et al. (2018), the data sets differ in the instruments and retrieval methods they use, their temporal and spatial coverage as well as their spatial resolution. Even within one data set, retrieval methods may slightly differ between land and ocean. This could cause artificial precipitation contrasts between land and ocean, a further motivation for including as many data sets as possible.
For IMERG, we only consider the time period 2015-2020, the time period the new Global Precipitation Measurement (GPM) satellite constellation has been in operation. The version 6 of the IMERG data set also contains reprocessed precipitation data from the TRMM satellite era (2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014). We disregard this period as we observed a discontinuity in the precipitation amounts over ocean between these two periods. Each year after 2015 exhibits a lower mean tropical precipitation than any of the years before, in mean by about 0.2 mm day −1 , something that is not seen in other observational data sets. The release notes of IMERG also mention that the start of the GPM calibration shifts the probability distribution function of monthly precipitation rates to lower values (see Figure 7 in Huffman et al. (2020)). For the observational data set labeled GPCCHOAPS in Figure 5, we use GPCC over land and HOAPS over ocean as these two separate data sets are only available over land (GPCC) and over ocean (HOAPS). For all the data sets, the analysis is performed on their native grid with their provided land-sea mask if available. Except if noted otherwise, TRMM is always used to derive the required parameters of our conceptual model. From other available data sets, we do not include in our analysis CPC Merged Analysis of Precipitation and the rain-gauge corrected version of Global Satellite Mapping of Precipitation. In these two satellite-based data sets, the yearly averages of tropical precipitation over land are smaller than in the rain gauge-based data set of GPCC, whereas rain gauges are thought to provide a lower estimate of precipitation. We also eliminate PERSIANN_CCS which exhibits a very low mean value of tropical precipitation over ocean, around 2.4 mm day −1 against at least 3 mm day −1 in the other data sets.
The ability of state-of-the-art General Circulation Models to represent χ(t) and its underlying controls is also investigated. We consider the historical simulations from the CMIP6 intercomparison project and focus on the Note. GPCC is only available over land and HOAPS only over ocean.  Fiedler et al. (2020) for their assessment of precipitation biases across CMIP phases. We also added 21 further models, which became available later on, even though this does not affect the statistics and our conclusions. The considered models are listed in Table 2. The model outputs are regridded on a T63 grid to compute the multi model mean.

The Surprising Zonality of Tropical Precipitation
In the tropics, the precipitation falls in well-defined, zonally extended bands (Figure 1a), a fact recognized in the widespread use of the term "rainbelt" to designate tropical rainfall. But the rains fall in an even more perfect belt than a simple rendering of the long-term mean rainfall, as shown in Figure 1a Note. First 18 models as employed in Fiedler et al. (2020). this, we define the hydrological equator to be the latitude of the maximum of the monthly mean precipitation at each longitude and month. The hydrological equator is visualized by the red dots in Figures 2a and 3a for two example months. We then plot the distribution of rainfall relative to the hydrological equator ( Figure 1b). The shifting offsets the well-known seasonal migration of the rains that blurs the belt structure in a rendering of the long-term mean. The new rendering reveals the true zonality of the rainbelts, with the presence of one single and noteworthy narrow rainbelt in the long-term mean. Outside a band of ±1.9° around the hydrological equator, precipitation intensities already fall to less than half of their maximum. This indicates that one clear precipitation maximum indeed generally exists at each longitude for a given month. Moreover, the uncovered rainbelt exhibits surpris ingly little zonal variations in intensity and width, with the imprint of land masses being less easily recognizable.
This property of the precipitation climatology motivates its interpretation using a simple conceptual "rainbelt" model. In this conceptualization, the precipitation falls in one single, zonally continuous belt of half-width W(λ, t), intensity I(λ, t) and latitudinal location L(λ, t). This conceptualization is visualized in Figures 2 and 3 for particular configurations of our rainbelt model, configurations that are explained in the next sections. The intersection between the rainbelt and the underlying land-sea mask Γ(λ, ϕ) provides theoretical values of the expected . In this rendering, the case of asymmetric intensity (aI_RB) would look like U_RB but the intensity over land (orange points) and ocean (blue points) would be different. tropics-wide partitioning of precipitation χ(t). These estimates can be contrasted with observed values of χ(t) to understand the underlying controls on χ(t).

Land Receives More Than Its Fair Share of Precipitation
We start by investigating the null hypothesis that the land receives its fair share of precipitation as set by geometrical constraints. If the land did not statistically influence the distribution of precipitation, the parameters of the rainbelt model would not vary in space. This case is illustrated for a winter month in Figure 2b and for a summer month in Figure 3b. It is designated as the uniform rainbelt, abbreviated U_RB. Figure 1b suggests this is not as bad as an assumption as one might first guess given that the rainbelt characteristics do not exhibit strong discontinuities between land and ocean. In this case: (2) ( ) and ̃o ( ) denote the monthly mean tropical precipitation (mm day −1 ) obtained from the rainbelt model, conditionally averaged over tropical land and over tropical ocean, respectively. They can be derived knowing the three parameters of the rainbelt model as Equation 3 assumes that the intensity is zero outside of the rainbelt and that neither intensity, nor location, nor width varies zonally, in agreement with our null hypothesis. Equation 3 also assumes that the half-width does not vary in time. This assumption is motivated by the analysis presented in Figure 1b where the width of the rainbelt,   Table 1), configurations of our rainbelt model with uniform rainbelt (U_RB, Equation 5), asymmetric intensity (aI_RB, Equation 7), asymmetric width (aW_RB, Equation 9), asymmetric location (aL_RB, Equation 10), and for the CMIP6 multi model mean with (a) χ and (b) frequency of χ(t) being out of the bounds given by U_RB. Orange means that the land receives more than its fair share of precipitation, stealing some rain that ought to fall over ocean, blue that the land receives less than its fair share of precipitation. Vertical bars in panel (a) for the standard error except in CMIP6 where it represents one ensemble standard deviation. For the rainbelt models, the required input parameters are always derived from TRMM.
where we drop λ from the notation in Equation 5 to denote the zonal average. Note that Γ ℓ (ϕ) + Γ o (ϕ) = 1 and α is the known ratio between tropical land area and ocean area: The uniform rainbelt model (Equation 5) reveals that the values that χ(t) can take are constrained by the zonal extent of the land, Γ ℓ (ϕ). The latter is displayed in Figure 4a. The meridional profile indicates that the land fraction is relatively constant south of 15°N, with a mean of 0.23. Assuming thus first that the rainbelt just straddles the latitudes south of 15°N, a value of 1.
0.348 0.23 1 − 0.23 = 0.86 is expected for χ. Hence, under the null hypothesis that the tropics-wide partitioning of precipitation is purely set by the location and size of land masses, a ratio of 0.86, not 1.0, is expected. A ratio of 1.0 would require that the land fraction that the rainbelt sees is the same as the tropics-wide land fraction, 1 + = 0.26 . Second, we can contrast this expected value to observed values. Temporal averages of observed χ(t) from various data sets are displayed in Figure 5a and denoted χ. The values lie between 0.90% and 1.04%, 5%-21% larger than 0.86. To be able to explain by geometrical constraints the largest of the observed χ, 1.04 in PERSIANN, a mean land fraction of 0.27 would be required. As indicated by the second red vertical line in Figure 4a, this would require the rainbelt to often reside past 15°N. Systematically testing all potential combinations of rainbelt width and location in Figure 4b confirms that either a narrow uniform rainbelt located poleward of 15°N or, otherwise, Figure 6. (a) Kernel density estimate of the enhancement of precipitation required over land to match observed χ(t) (black) compared to land enhancement resulting from intensity (brown) and width (green) changes. The Kernel density estimates were computed with NCL 6.5.0 and its built-in function using a Gaussian kernel and a bin size of 0.02 following Engel et al. (1994). Using Kernel density estimates leads to smoother curves than using histograms. Panel (b) shows the mean seasonal cycle in χ(t) from observations and from our rainbelt model with uniform rainbelt (U_RB, Equation 5), asymmetric width (aW_RB, Equation 9), and asymmetric location (aL_RB, Equation 10). Required observations taken from TRMM.
a wide enough rainbelt is needed to explain the observed χ. Both allow the rainbelt to see the larger land fraction poleward of 15°N. Both are nevertheless unlikely. Figure 1b already revealed the narrowness of the rainbelt in observations. Moreover, the monthly mean zonal averages of the observed hydrological equator, indicated by the blue bar in Figure 4a and taken as the expected observed locations L(t) of the rainbelt, never lie poleward of 15°N.
As a final check, we recompute the distribution of χ(t) using now monthly mean and zonally averaged values of observed rainbelt half-width W and location L(t) as input for our rainbelt model (Equation 5). W is set to 1.9°, in agreement with Figure 1b, and L(t) is derived by zonally averaging the hydrological equator. The results are displayed in Figure 5a by the horizontal lines. Not surprisingly, a mean value of 0.86 is obtained, which is robustly lower than any of the observed values.
Using these observed values of rainbelt half-width W and location L(t) as input for the rainbelt model also indicates that the probability distribution function of χ(t) lies between the two values of 0.76 and 0.95. These extrema are geometrically constrained by the extrema of the land fraction (see Figure 4a) and the rainbelt width, as given by Equation 5. Figure 5b indicates how often observed values of χ(t) lie outside these bounds. Whereas the lower bound is roughly consistent with the data, observed values exceed the upper bound 35% (IMERG) to 78% (GPCP) of the time, that is, for 4-9 months per year.

Land Overcompensates Its Geometrical Deficit by Making the Rainbelt Wider and Letting It Move More
Our conceptual model refutes the null hypothesis and indicates that the land receives more than its fair share of precipitation. How is the geometrical deficit of the land overcompensated? Given that storms are more explosive over land (Bang & Zipser, 2015), one answer could be that the anomalous precipitation over land arises from asymmetries in intensity between land and ocean. Figure 6a quantifies how much the intensity in our rainbelt model needs to be enhanced over land to explain the observed χ(t) (black curve) and compares this enhancement to observed ratios of intensity between land and ocean (brown curve). The intensity I(λ, t) is defined for each longitude and month as the observed monthly mean precipitation rate at the location of the meridional precipitation maximum (the hydrological equator), motivated by the results of Figure 1, and conditionally averaged over land (I ℓ ) and over ocean (I o ) precipitation maxima. Using other definitions, such as mean precipitation larger than a certain threshold or precipitation averaged over a certain area, would confound effects resulting from intensity, width, and rainbelt movement. Figure 6a indicates that asymmetric intensities cannot explain the observed values of χ being larger than 0.86. I ℓ is actually smaller than I o (see also Figure 1b) with a value of 14.6 versus 17.2 mm day −1 in observations. This hypothesis of asymmetric intensities can be further tested by allowing for asymmetric intensities between land and ocean in our rainbelt model, effectively replacing I(t) by I ℓ,o (t) in Equation 3. This leads to: The result of evaluating Equation 7 is displayed in Figure 5a as the asymmetric intensity rainbelt, aI_RB, now using observed values of I ℓ (t), I o (t), L(t) and W as input. Allowing for distinct intensities over land and ocean lowers χ to 0.74, further away from observations.
In contrast, letting the rainbelt width over ocean and land be different (see green curve in Figure 6a) reveals larger (∼20%) width over land in observations. The needed half-width values W(λ, t) are here computed for every longitude and month as: where ϕ 2 and ϕ 1 correspond to the first latitudes north and south of the hydrological equator where the precipitation drops to a fraction C of its value at the hydrological equator. If this condition is not met inside the tropics, then 30° is used for the corresponding bound. In mathematical terms: P(λ, ϕ, t) > C ⋅ P(λ, L, t) ∀ ϕ ∈ [ϕ 1 , ϕ 2 ] and ‖ϕ‖ ≤ 30°. We use C = 0.5 based on the results of Figure 1b. The half-width values are then conditionally averaged over land (W ℓ ) and ocean (W o ) using L(λ, t) to distinguish between land and ocean points. This is an approximation, as in coastal areas the points enclosed by W(λ, t) would belong both to the ocean and to the land. This approximation nevertheless does not affect the overall results.
As shown in Figure 6a, the enhancement due to asymmetric width is strong enough to compensate for the land deficit. This is further demonstrated by recomputing χ(t) solely allowing for asymmetric width between land and ocean in our rainbelt model, a configuration visualized in Figures 2c and 3c for a winter and a summer month. In this case, χ(t) equals: Note that Equation 9, in contrast to the uniform rainbelt model in Equation 5, retains the temporal variations in width as those are large over land, as can be seen by comparing Figures 2c and 3c. Equation 9 is then again evaluated using observations as input and the results are summarized in Figure 5 as the asymmetric width rainbelt, aW_RB. Equation 9 leads to a mean χ of 1.03. More importantly, χ(t) being larger than 0.95, the upper bound But the rainbelt also sporadically visits locations poleward of 15°N in observations, a feature associated with the Asian monsoon (see Figure 3a). These land points are not seen in the uniform rainbelt model when using the observed zonally averaged hydrological equator as input for L(t) since only a small fraction of these locations, at most 7% for a given month, lies poleward of 15°N. We thus allow the rainbelt in our model to wander following L(λ, t), see Figures 2d and 3d. This is a more extreme case than just allowing two different values for L ℓ (t) and L o (t) over land and over ocean, as done for the intensity and width. We do so especially to allow the rainbelt model to see land points poleward of 15°N in summer. Recomputing χ(t) by replacing L(t) by L(λ, t) in Equation 4 leads to: The case is denoted as the asymmetric location rainbelt, aL_RB, and the obtained χ(t) values from Equation 10 are summarized in Figure 5. As expected, the frequency of χ(t) values larger than 0.95 increases and χ also increases in comparison to the uniform case, leading to a better agreement with observations.
The fact that the land broadens the rainbelt more and moves it more makes the continents rainier than expected from geometrical constraints. Considering the seasonal cycle of χ(t) in Figure 6b reveals a distinct seasonality in these two processes. This distinct seasonality is also well visible when comparing the spatial rendering of our rainbelt model configurations for a winter ( Figure 2) and summer (Figure 3) month. The asymmetric width is needed to explain the observed enhancement of χ(t) during boreal winter, although the rainbelt model (aW_RB) overinflates this peak. In contrast, allowing for asymmetric locations in aL_RB only affects simulated boreal summer values and strongly overestimates those as compared to observations. This overestimation illustrates a limitation of our rainbelt model: aL_RB retains mostly the maxima associated with the Asian monsoon (see Figure 3a), whereas in observations, secondary maxima appear over the Indian Ocean. Hence, our assumption of one single rainbelt partly breaks down in boreal summer, as also visible in Figure 7 when averaging the shifted precipitation distribution per season. This limitation of our rainbelt model may also illustrate that there is a limit in reality as to how strongly the geometrical deficit of the land can be compensated by attracting precipitation poleward in regions of much higher land fraction, as anomalously poleward displacements of the rainbelt over land can be compensated by the emergence of secondary bands of precipitation over ocean.

Summary and Discussion
Overall, our conceptualization of the distribution of tropical precipitation in the form of a rainbelt model indicates that the land receives more than its fair share of precipitation as set by the size and location of land masses: all the observational data sets reveal a tropical land-to-ocean precipitation ratio larger than 0.86. In the uniform rainbelt model, not distinguishing between land and ocean controls on the atmospheric properties implies that the mean evapotranspiration averaged over the land points equals the mean evaporation averaged over the ocean points. It is thus larger than the evapotranspiration averaged over a true land surface. But at the same time, the land points in the uniform rainbelt model receive in mean less precipitation than over a true land surface, given the smaller χ and assuming the same tropics-wide mean precipitation. In summary, implicitly putting land to ocean in the uniform rainbelt model leads to an increased evapotranspiration and decreased precipitation compared to a true land surface, a finding that we interpret as being indicative of the existence of a negative feedback between evapotranspiration and precipitation. A negative feedback is needed to explain the long-term tropics-wide partitioning of precipitation between land and ocean and, on a monthly basis, the negative feedback dominates. This finding seems at odds with climate models which predominantly simulate a positive feedback between evapotranspiration and precipitation in response to land surface perturbations, such as an increase in soil moisture. Are climate models thus misrepresenting the feedback, here deduced in a distinct way by quantifying the strength of the land control on the tropics-wide partitioning of precipitation? Looking at χ in the CMIP6 multi model mean in Figure 5a reveals a value of 0.85, too low compared to observations, and even indicative of a slight tendency of the land to receive less than its fair share of precipitation, that is, to lose some precipitation to the ocean. More strikingly, on a monthly basis, ocean and land steal from each other rain as frequently, whereas it should be at most one out of nine cases (see IMERG in Figure 5b) that the land receives less than its fair share of precipitation. Twenty two of the 39 CMIP6 models do indicate that the land receives more than its fair share of precipitation, albeit with a too weak magnitude, and most (18) for the wrong reason, by having I ℓ larger than I o . The 17 CMIP6 models where land receives less than its fair share of precipitation do so for two reasons. First, only four of these models capture the expected width enhancement over land and eight of the 17 actually have W ℓ smaller than W o . Second, too often the hydrological equator falls over ocean instead of land. This can be seen by using the hydrological equator derived for each individual CMIP6 model as input for L (λ, t) in the asymmetric location rainbelt aL_RB. For the 17 CMIP6 models where land receives less than its fair share of precipitation, this analysis reveals that 13 of them have a χ smaller than 0.93, the value obtained for aL_RB using observations as input. Except in one model, which strongly underestimates I ℓ as compared to I o , intensity cannot be invoked to explain the failure of CMIP6 models to capture precipitation enhancement over land. The fact that climate models cannot represent the effect of a perturbation of the surface characteristics, from ocean to land, on a basic property of the precipitation distribution, its partitioning, calls into question the ability of climate models to answer more subtle questions like future tropical regional precipitation changes. The results also suggest that the land may be more resilient to perturbations than models might have led us to believe, with the land being less dependent on its own water supply and hence more resilient to the occurrence of tipping points.
Our study demonstrates the power of geometrical constraints to explain basic features of the precipitation distribution and the potential usefulness of considering the precipitation ratio, rather than precipitation amounts per se. The use of χ(t) is best justified taking the view that the precipitation averaged over the whole tropics is set and land and ocean have to compete for this given resource. We cannot demonstrate this assumption at the outset, but we note that it is broadly consistent with the fact that monthly mean precipitation amounts over land and over ocean are negatively correlated (see Figure 8). A potential explanation could be that stronger ascent over land (more precipitation) has to be compensated by stronger subsidence over ocean (less precipitation), assuming that the export of water to the extratropics is held fixed.
The findings of our study are conditioned on the ability of our rainbelt model to capture the precipitation dynamics to a first order. Correlation between χ(t) derived from TRMM and from our simplest conceptualization, the uniform rainbelt model, is surprisingly high, 0.55, compared to 0.45 for the CMIP6 multi model mean. Also, the seasonal cycle in χ(t) is very well reproduced, albeit being shifted to lower values (see Figure 6b). The findings are also robust to the chosen definition of the tropics and to the observational data set chosen to derive the input parameters of the rainbelt model. The only noteworthy sensitivity is in the case of the asymmetric location (aL_RB). All the data sets show an increased occurrence of χ(t) being larger than 0.95. All the data sets except IMERG also lead to an increase of χ as compared to the uniform case (U_RB). With IMERG, χ does not change.
We also tested the sensitivity of the results to the definition of the rainbelt width, of relevance for the asymmetric width rainbelt (aW_RB). Using a more stringent threshold of 75% to define W(λ, t) reduces χ to 0.95 and the frequency of χ(t) values larger than 0.95%-36%, compared to a χ of 1.03 and a frequency of 58% when using a threshold of 50%. This is expected from the land-sea distribution, where the mere presence of a wider belt tends to help the land, as seen in Figure 4b.
The results of our study are only valid for the given land-sea distribution. Changing the location or size of land masses would affect the precipitation distribution (Dirmeyer, 1998;Xie & Saito, 2001) and its propensity to be represented by a rainbelt. Also, although we demonstrated that the land receives more than its fair share of precipitation by having a wider rainbelt and moving it more, and not via changes in intensity, it especially remains an open question why the width is larger over land. Recent studies based on idealized frameworks (Hohenegger & Stevens, 2018;Leutwyler & Hohenegger, 2020) and/or looking at other properties of the climate system (Byrne, 2021) have argued that the limited water availability of the land is a decisive property for making land distinct from ocean. Other suspects could be its stronger heterogeneity and roughness or a smaller effective surface heat capacity, the latter known to be important for the propagation of the monsoons (Halley, 1686). Mountains could also affect χ(t), but their expected direct effect would be via an increased precipitation intensity or by violating the rainbelt assumption, both of which we can rule out from our analysis. Studies have begun exploring theoretical frameworks to understand the processes that set the width of the zonally averaged tropical rainbelt (see Windmiller and Hohenegger (2019) and review by Byrne et al. (2018)), but these studies haven't yet attempted to explain differences observed between land and ocean.
In our interpretation of the precipitation behavior over the tropics with our conceptual model, the fact that the land supports a wider rainbelt and moves it more leads to the existence of a negative feedback between precipitation and evapotranspiration. Scheff (2014) showed that including ocean heat transport in simulations with an idealized representation of continents leads to an enhancement of precipitation over land. We nevertheless note that in his simulations, evapotranspiration increases as well, unlike in our findings. Precipitation and evaporation are connected via the water balance equation. The land, in contrast to the ocean, continuously loses water due to runoff. This implies that, over land, the ratio between precipitation and evaporation has to be larger than over ocean. This nevertheless also does not provide any constraint on the sign of precipitation changes, P ℓ − P o , versus evapotranspiration changes, E ℓ − E o . More importantly, our study indicates that the magnitude of χ(t), being larger or smaller than 0.86, can be used as an indicator of the sign of the feedback between precipitation and evapotranspiration. Given that most of the observational spread in χ(t) comes from uncertain precipitation amounts over ocean, better understanding precipitation over land and its constraints will especially require a better ability to measure precipitation amounts over ocean.