# High sensitivity of tropical precipitation to local sea-surface temperature

Peter Good, Robin Chadwick, Christopher Holloway, John Kennedy, Jason

Lowe, Romain Roehrig, Stephanie Rushley

► To cite this version:

Peter Good, Robin Chadwick, Christopher Holloway, John Kennedy, Jason Lowe, et al.. High sensitivity of tropical precipitation to local sea-surface temperature. Nature, 2020, 589 (7842), pp.408-414.
10.1038/s41586-020-2887-3 . hal-04915550

HAL Id: hal-04915550

https://hal.science/hal-04915550v1

Submitted on 27 Jan 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
1 High sensitivity of tropical precipitation to local sea-surface

2 temperature

3

4 Peter Good¹*, Robin Chadwick¹,², Christopher E. Holloway³, John Kennedy¹, Jason A.

5 Lowe¹,⁴, Romain Roehrig⁵, Stephanie S. Rushley⁶

6

7 ¹MetOffice Hadley Centre, Exeter, United Kingdom.

8 ²Global Systems Institute, University of Exeter.

9 ³Department of Meteorology, University of Reading, Reading, United Kingdom.

10 ⁴Priestley International Centre for Climate, University of Leeds, United Kingdom.

11 ⁵CNRM, Université de Toulouse, Météo-France, CNRS, Toulouse, France

12 ⁶Department of Atmospheric Sciences, University of Washington, Seattle, Washington

13 98195, USA.

14

15

16 Abstract

17

18 Precipitation and atmospheric circulation are the coupled processes through which tropical

19 ocean surface temperatures drive global weather and climate¹⁻⁵. Influences of local ocean

20 temperatures on precipitation are hard to disentangle from remote effects of conditions
elsewhere; and the full influence of the recently-discovered shallow circulations$^{6-8}$ is unclear.
Uncertainty in precipitation observations$^{9,10}$, and limited observations of circulation$^{11}$, further
obstruct understanding. Despite decades of research, persistent biases remain in many
numerical model simulations$^{12-18}$, including excessively-wide tropical rainbands$^{14,18}$, the
'double-intertropical convergence zone (ITCZ) problem' $^{12,16,17}$ and too-weak responses to the
El Niño Southern Oscillation (ENSO)$^{15}$. These demonstrate stubborn gaps in our
understanding, and reduce confidence in forecasts and projections. Here we show that the
real world has a high sensitivity of seasonal tropical precipitation to local sea-surface
temperatures (higher than in all but 4 of the 47 models studied), associated with strong
shallow circulations. Our results apply to both temporal and spatial variation, over regions
where climatological precipitation is around 1 mm/day or greater. Novel analysis of multiple
independent observations, combined with physical constraints and model data, underpin these
findings. A large spread in model behaviour is further linked to differences in shallow
convection, providing a focus for accelerated research, to improve seasonal forecasts through
multidecadal climate projections.

36

37 Main paper

38

39 We first define a measure ($k_{qsat}$) of the sensitivity of seasonal mean precipitation to variation
40 in local sea surface temperature (SST). We will show that $k_{qsat}$ is a key property of the
41 atmosphere, using it to link diverse gaps in understanding to a limited subset of physical
42 mechanisms. Precipitation increases non-linearly with SST$^{19}$. Since tropical precipitation
43 increases roughly exponentially with column atmospheric water vapour$^{20,21}$; and over
44 seasonal or longer timescales, SST variation forces variation in column water vapour$^{22}$, via
differences in saturation specific humidity of the ocean surface ($q_{sat}$, Methods), we define $k_{qsat}$ as follows:

$$ \log_e(P_1/P_0) \approx k_{qsat} * (q_{sat,1} - q_{sat,0}) + \langle \text{other processes} \rangle \qquad \text{Equation 1.} $$

This describes the variation in precipitation (from $P_0$ to $P_1$) driven by local variation in $q_{sat}$ (from $q_{sat,0}$ to $q_{sat,1}$). This approximation is validated within the calculation of $k_{qsat}$ (Methods). Moist static energy arguments¹⁹ also predict a roughly exponential relationship between $q_{sat}$ and $P_0$. $k_{qsat}$ quantifies the combined effect of the physical processes by which local SST anomalies affect precipitation at the same location. The 'other processes' term includes the effects of internal atmospheric variability independent of SST, and of remote forcing from land or SST elsewhere, which can be large at individual locations or times. In order to estimate $k_{qsat}$, we filter out these other processes, by combining information from multiple locations and times (see Methods). We evaluate $k_{qsat}$ from interannual variability, with $P_0$ and $q_{sat,0}$ taken as seasonal climatological means at each location for each season. However, our estimates of $k_{qsat}$ are shown to be also informative about spatial variations in precipitation.

$k_{qsat}$ relates most directly to the strength of percentage variations in precipitation. Writing Equation 1 in exponential form, percentage precipitation differences are a function of $k_{qsat}$ and $q_{sat}$:

$$ (P_1 - P_0)/P_0 * 100 \approx 100 [\exp(k_{qsat} * (q_{sat} - q_{sat,0})) - 1]. \qquad \text{Equation 2} $$

Absolute differences depend also on the reference precipitation $P_0$:

$$ P_1 - P_0 \approx P_0[(\exp(k_{qsat} * (q_{sat} - q_{sat,0}))) - 1], \qquad \text{Equation 3} $$
67 (in absolute terms, precipitation variations are largest in regions of large mean precipitation¹⁹.

68 However, we will show that spatial variation in P₀ itself also depends partly on kqsat.

69

70 **Validating satellite observations**

71 Given uncertainty in precipitation observations⁹,¹⁰, we perform a high-precision evaluation of
72 log(precipitation) (as Equation 1) from two satellite datasets: TRMM²³,²⁴ (3B43, v7) and
73 GPCP²⁵ (v2.3), both for 1998-2015. We do this (Methods) using in-situ raingauge data from
74 89 buoys of the Global Tropical Moored Buoy Array (GTMBA)²⁶⁻²⁸. Satellite-GTMBA
75 validation is challenging: on top of satellite error⁹,¹⁰, the GTMBA point observations include
76 noise from small-scale variability unresolved by satellite data, missing data, error in
77 individual raingauges and wind undercatch²⁹. Our method reduces this noise considerably,
78 giving a tight relationship between GTMBA and TRMM data (Figure 1a). Critically, the best
79 fit gradient ≈1, so TRMM accurately retrieves differences in log precipitation. On the other
80 hand, GPCP underestimates differences in log precipitation (Figure 1b, gradient > 1;
81 differences between TRMM and GPCP emerge primarily at low precipitation⁹), although
82 GPCP is more suitable over larger spatial scales (Extended data Figure 1). Since TRMM
83 captures differences in log(precipitation) more accurately than GPCP, TRMM is used below.

84

85 **Model precipitation simulations**

86 We first highlight precipitation biases in 28 atmospheric models from the fifth Coupled
87 Model Intercomparison Project (Methods), each forced by observed SST (CMIP5 AMIP
88 experiment; Figure 2a-f). We quantify temporal, seasonal and spatial variation in
89 precipitation: the 1997-98 El Niño divided by the mean of the 1998-2000 La Niñas; Aug-Oct
divided by Feb-Mar seasons, and precipitation scaled by its latitudinal maximum. Our metrics coincide with significant spatial or temporal differences in SST (Methods). Spatial variation across the west Pacific is excluded, for example, because spatial gradients in SST are weak there, so model differences in $k_{qsat}$ will be less important for spatial variation there.
Given the form of Equation 1, precipitation is shown on log scales as ratios. Although some models are close to the observations, in others, biases exceed a factor of five in the El Niño/La Niña ratio, seasonal cycles over the Atlantic and West Pacific, and in the Atlantic spatial pattern for the Aug-Oct season (Figure 2a-d). Biases over a factor of two occur in the spatial patterns of the East Pacific annual mean (the long-standing ‘double-ITCZ’ problem<sup>12,17</sup>) and the Indian Ocean for November-April (Figure 2e-f). Such biases are known, but their causes are not well understood.

These biases (Figure 2a-f) all correspond to excessively weak spatial/temporal variations in precipitation (precipitation ratios too close to 1; including excessively-wide inter-tropical convergence zones<sup>14,18</sup>). This suggests a hypothesis ($H_0$), that the sensitivity of seasonal precipitation to local SST ($k_{qsat}$) may be too weak in many models.

To test $H_0$ objectively, we use a method independent of Figure 2 (Figure 2 was used to propose $H_0$). This involves estimating $k_{qsat}$ for each model using different data.

## Evaluating $k_{qsat}$ in models

We evaluate $k_{qsat}$ using interannual variability in seasonal mean precipitation and SST (the AMIP SST dataset<sup>30</sup> used to drive the model experiments; using years 1980-2005). $k_{qsat}$ is
calculated using gridpoint values of seasonal precipitation and $q_{sat}$, from each location in the study region, and for each year. With these data, Equation 1 becomes a model of the effect of local interannual SST variability on precipitation:

$$ \log_e(P(x,t)/P_0(x)) \approx k_{qsat} * (q_{sat}(x,t) - q_{sat,0}(x)) + <\text{other processes}> $$

where $(x,t)$ indicates values for each gridpoint and year; and here, $P_0$ and $q_{sat,0}$ are the corresponding climatological means for each gridpoint. We estimate $k_{qsat}$ from these data using a modified regression approach (detailed in Methods), minimising the influence of other processes in Equation 1.

To minimise observational error, we exclude the 30% of the tropical oceans with the lowest climatological mean SST (Figure 3b-e, area outside white contour).

Taking logarithms means that all areas of our study region contribute relatively equally to our $k_{qsat}$ estimate (Methods). Consequently, $k_{qsat}$ is relevant over most of the tropical oceans (Figure 3a, correlations are high except for the left bar). $k_{qsat}$ is inapplicable over the coolest, driest ocean regions (Figure 3a, left bar; area masked in Figure 3b-e). The applicable region corresponds to climatological precipitation > ~1 mm/day (Figure 3b-e, orange contour).

$k_{qsat}$ is intended to be independent of large-scale SST spatial patterns. To avoid bias from the large, recurrent ENSO pattern, our 'sortav' regression method first processes the data so all years contribute equally. Linear regression is then applied to obtain $k_{qsat}$. Rankings of CMIP5 models by $k_{qsat}$ are robust: insensitive to season, to using fewer years of data, or to
excluding ENSO years - Extended data Figure 2e-g. Calculated this way, $k_{qsat}$ is less sensitive to the time period used than with simple least squares regression (Extended Data Figures 2h, 7).

138

We find that the sensitivity of precipitation to local SST variability is much stronger in some models than in others: $k_{qsat}$ varies across CMIP5 models by a factor of 2.5 (0.26-0.66 kg/g; median = 0.46). We group the models into ‘high-$k_{qsat}$’ (the 6 models with the largest $k_{qsat}$ values), ‘low-$k_{qsat}$’ (the lowest 6 $k_{qsat}$ values) and ‘mid-range’ subsets.

143

## High sensitivity of precipitation to SST

We hypothesised above ($H_0$) that $k_{qsat}$ may be too low in most models. To begin testing this, the results in Figure 2a-f are replotted, but with the 'high-$k_{qsat}$' subset of models highlighted in magenta (Figure 2g-l). The 'high-$k_{qsat}$' subset shows much better agreement with TRMM than the full ensemble, in all six panels. Conversely, the 'low-$k_{qsat}$' subset performs much worse (Extended data Figure 3). Next, we calculate $k_{qsat}^{(spatial)}$ (Methods): as $k_{qsat}$, but using spatial patterns in climate means, rather than internal variability (Figure 4b). Again, models closest to the observations (Figure 4b, horizontal line) tend to be those with high $k_{qsat}$. These results all imply that $k_{qsat}$ should be high in the real world ($H_0$).

153

These results also show that $k_{qsat}$ is relevant to both spatial and temporal variations in precipitation. We emphasise this by quantifying the overall sensitivity of precipitation to local SST ($k_{qsat}^{spattemp}$), including both spatial and temporal variations (including spatial
157 variation in P₀, Methods). *k*<sub>qsat</sub><sup>spattemp</sup> is well correlated with *k*<sub>qsat</sub> (Extended data Figure 2i).

158 This confirms that *k*<sub>qsat</sub> is a useful measure of the underlying sensitivity of precipitation to
159 local SST, relevant to spatial and temporal variations. *k*<sub>qsat</sub> does not give information about
160 tropical mean precipitation, which is governed by different processes<sup>31</sup>. *k*<sub>qsat</sub> remains our
161 primary measure of precipitation sensitivity to local SST, because it is insensitive to details of
162 SST patterns. In contrast, *k*<sub>qsat</sub><sup>spattemp</sup> and *k*<sub>qsat</sub>(spatial) may be sensitive to the specific spatial
163 patterns in climatological SST (Methods), explaining some of the noise in Figure 4b and
164 Extended data Figure 2i.

165

166 We estimate a lower bound for kqsat, using observed interannual variability (independent of
167 Figure 2; Methods). Three values of kqsat are estimated, exactly as for the models, but using
168 TRMM precipitation, and qsat from each of three different SST datasets (HadISST³² version
169 1.1, ERSST³³ version 4 and COBE³⁴ version 2). Uncertainties are estimated, from SST error
170 (including regression dilution bias) and internal variability (the TRMM observational period
171 only partly overlaps the model simulation period). A lower observational bound (95%
172 confidence) of 0.51 kg/g for kqsat is obtained.

173

174 For a central observational estimate of kqsat (details in Methods), we return to Figure 2. We
175 ask: if all CMIP5 models had the same value of kqsat, with what value would they best
176 reproduce the observations in Figure 2? We first find where, geographically, the models are
177 most sensitive to kqsat. This reveals seven intervals (shaded in Figure 2g-l). For each interval,
178 model errors relative to TRMM are regressed against modelled kqsat (Extended data Figure 6).
179 For each interval, kqsat is estimated as where the regression line intercepts the x-axis (the
180 value for a theoretical model with zero precipitation error). These seven estimates of kqsat
range from 0.56 to 0.68 kg/g (Figure 4a, white dashed lines), all larger than our lower bound
estimate. The spread of estimates comes from uncertainty in processes not quantified by
kqsat. The range of conditions used, covering spatial, seasonal and temporal variability across
different locations, helps to quantify and mitigate this uncertainty. Robustness is tested by
plotting results from the sixth model intercomparison project (CMIP6, not used to select the
seven intervals) on Extended data Figure 6. The mean of the seven kqsat values (0.6 kg/g) is
our central estimate (for 1980-2005; other periods would give slightly different values, from
internal SST variability - Extended data Figure 2h).

189

These results, from two independent methods, suggest that most models underestimate kqsat.

Our central estimate (0.6 kg/g, Figure 4a, horizontal black line; Figure 4b, vertical line) is greater than 43 of the 47 model values from CMIP5 and CMIP6. This implies that models underestimating the sensitivity of precipitation to local SST underlies a range of model biases over tropical oceans. CMIP5 and CMIP6 have similar ranges of kqsat values (Figure 4a), highlighting the need for accelerated model development.

196

Other studies³⁵ have found biases in a different aspect of the SST-precipitation relationship: model precipitation often tracks SST maxima more closely than in observations. We quantify this in each CMIP5 model as the correlation coefficient between climatological spatial patterns of precipitation and SST, for each season, then average the four seasonal values.

This 'spatial-correlation index' is uncorrelated with kqsat (r = 0.01; i.e. models with a high spatial-correlation index can have high, low or intermediate kqsat), so it involves different processes.
204

205 **Processes behind uncertainty in k<sub>qsat</sub>**

206 To guide model improvements, we explore what causes model differences in k<sub>qsat</sub>, revealing
207 links to shallow atmospheric circulations. We first note that k<sub>qsat</sub> involves processes
208 unrelated to tropical mean precipitation: the correlation across CMIP5 models between the
209 two measures is 0.03. Energy budgets constrain tropical mean precipitation<sup>31</sup>, while the value
210 of k<sub>qsat</sub> affects precipitation variation in both time and space.

211

212 Beginning with interannual variability, we define kqsat<sup>wap</sup>(p): the sensitivity of the vertical
213 pressure velocity (wap) to local SST change, at each pressure level (p). This is evaluated like
214 kqsat, using data from all seasons across the tropical oceans, but using wap(p) instead of
215 log(precipitation). A deep mode dominates tropical variability<sup>36</sup>, so the CMIP5 mean profile
216 of kqsat<sup>wap</sup> peaks around 450 hPa (Figure 5a). In contrast, model spread in kqsat is linked to
217 shallow circulations: correlations between kqsat<sup>wap</sup>(p) and kqsat (Figure 5b) peak near 700hPa (r
218 = -0.9; correlations are small at 1000hPa, as wap approaches zero near the surface). That is,
219 in models with high kqsat (as in the real world), shallow circulations respond strongly to SST
220 anomalies. Although deep circulations are important<sup>36</sup> in all models, and shallow circulations
221 have been linked to mean precipitation, especially over the Eastern Pacific<sup>7</sup>, our novel finding
222 is that shallow circulations are central to model uncertainty in SST-driven precipitation
223 variability, across the tropics.

224

225 Shallow circulations are further linked to model differences in climate means. We study
226 zonal, ocean-only means over 180W-10E (most of the Pacific, entire Atlantic), in Aug-Oct,

226
227 when meridional SST gradients are strong. CMIP5 results are used to define key regions
228 (black and orange lines in Figure 5c-e), and CMIP6 used to test the conclusions. As
229 expected<sup>4,7</sup>, CMIP5 mean vertical velocity profiles are bottom heavy, but extend throughout
230 the troposphere (Figure 5c, colours). Meridional wind (white contours) peaks near the
231 tropopause, with a weaker shallow flow between 500-700hPa<sup>37</sup>. Again, however, inter-model
232 spread in k<sub>qsat</sub> is associated with shallow circulations: models with high k<sub>qsat</sub> have stronger
233 shallow descent south of the ITCZ between 600-850hPa (Figure 5d, yellow; Extended Data
234 Figure 8a, magenta line; Extended Data Figure 8d). They also tend to have stronger shallow
235 ascent in the ITCZ (Figure 5d, blue region; Extended Data Figure 8c, magenta line), stronger
236 trade winds and stronger return flow between 500-700hPa (Figure 5d, white contours;
237 Extended Data Figure 8b, magenta line). This shallow circulation is weak in the low-k<sub>qsat</sub>
238 mean (Extended Data Figure 8a-c, blue lines).

239

240 The weak link between model differences in kqsat and deep circulations may arise partly from
241 physical constraints. In descending air, differences in vertical velocity are largest below
242 about 600 hPa (Extended Data Figure 8a,i). This is partly explained by the vertically-
243 integrated dry static energy (DSE) budget. This budget constrains vertical velocities,
244 requiring balance between radiative, sensible and latent heating, and advection of DSE
245 (Methods). In descending air above 600 hPa, there are fewer uncertain processes affecting
246 this budget, with negligible energy input from cloud and precipitation. Here, therefore,
247 downward advection of DSE is balanced mostly by dry clear-sky radiative cooling (Extended
248 Data Table 1). Further, vertical temperature profiles are similar across CMIP5 models. This
249 constrains both radiative cooling and vertical gradients of DSE, limiting model differences in
250 vertical velocity. Model temperatures are constrained, near the surface by SST, and near the
251 tropopause as modelling groups aim to reproduce observed outgoing longwave radiation.
252 Below 600hPa, there are additional sources of uncertainty, from cloud and precipitation³⁸,
253 leading to larger model differences in shallow descent, and so stronger links to differences in
254 kqsat.

255

256 The depth of the meridional return flow (500-700hPa, Figure 5e, white contours) suggests a
257 circulation driven by precipitating shallow convection³⁷,³⁹. The alternative, sea-breeze
258 mechanism reaches lower levels³⁷. Both circulation types may exist over the Galapagos
259 (Extended Data Figure 8g): here, model meridional winds between 600-700hPa (shallow-
260 convection-type) are uncorrelated with those between 700-850hPa (sea-breeze-type),
261 indicating different physical processes at these two levels. The column dry static energy
262 budget (Methods) also implicates precipitating shallow convection. Model differences in this
263 budget are predominantly a balance between vertical advection integrated over 600-1000 hPa,
264 and precipitation latent heating (Extended Data Table 1, final column): in descending air
265 below 600 hPa, high-kqsat models have stronger shallow advective warming from stronger
266 shallow descent, balancing weaker warming from weaker precipitation. As expected, the
267 ensemble mean in descending air is mostly a balance between radiative cooling and advective
268 warming.

269

270 Causality is hard to fully establish, but the most likely explanation of our results is that model
271 differences in kqsat mostly originate from model differences in the behaviour of shallow
272 precipitating convection. Such differences would affect the sensitivity of precipitation to
273 SST directly. The consequent differences in shallow latent heating also appear to lead to
274 differences in shallow circulations. This couples shallow vertical velocities in descending
and ascending air (Extended Data Figure 8e), further modifying the sensitivity of precipitation to SST.

277

Significant model differences in the physical representation of convection$^{40,41}$, including its coupling to circulation$^{5,42}$, are well established. Improvements in $k_{qsat}$ in the CNRM model from CMIP5 (0.43 kg/g) to CMIP6 (0.54 kg/g) are largely associated with convection scheme changes. Running CNRM-CM6 with the CNRM-CM5 convection scheme gives $k_{qsat}=0.36$ kg/g, even smaller than that of CNRM-CM5. That is, the effect of changing the convection scheme is partly offset by changes in other schemes. Changes from CM5 to CM6 include the shallow convection scheme and the transition from shallow to deep convection$^{43}$. These, and other physics schemes, including boundary layer, deep or mid-level convection, or microphysics, could all affect how shallow precipitating convection responds to SST.

287

Other processes may have smaller contributions. Dry shallow circulations$^{44}$ are more important over hot, dry land. A limited role for differences in radiation (Extended Data Table 1, final column), suggests cloud parameterizations are not dominant. Differences in dynamical schemes are thought to be less likely to be important for tropical precipitation biases, although coupling of dynamics to physics is important$^{45-47}$.

293

Models with stronger shallow circulations can also import more moist static energy in ascending air, driving enhanced deep convection$^{48}$. Our results support this (Extended Data Figure 8h), showing that model differences in shallow ascent are strongly positively
297

correlated with differences in deep ascent. This is an indirect link to $k_{qsat}$, which is more

298

weakly associated with deep ascent rates (Figure 5d).

299

300

## Strong real-world shallow circulations

301

Shallow circulations are challenging to observe¹¹, but our results suggest they are stronger in

302

the real world than in most models. Models with high $k_{qsat}$ (as in the real world) tend to have

303

strong shallow circulations (in both climate means and internal variability). We test this

304

further with two independent observations, in Aug-Oct. In models, there is a strong ($r=0.86$)

305

relationship between shallow descent, and northward trade winds (Figure 4c). If the shallow

306

descent is strong in the real world, we would expect strong northward trades – and this is

307

confirmed by QuikSCAT satellite observations⁴⁹ (SeaWinds scatterometer, Level 3 product,

308

years 1999-2009 - horizontal line in Figure 4c). There is, similarly, a strong ($r=0.81$)

309

relationship (Figure 4d) between 500-700hPa meridional winds over our study region, and

310

the mean 600-700hPa meridional wind over Galapagos and Christmas Islands (915 MHZ

311

wind profiler⁸,⁵⁰ observation sites, years 1994-2005 at Galapagos, 1990-2002 at Christmas

312

Island; few observations reach above 600hPa). If the large-scale 500-700hPa wind is strong

313

in the real world, we would expect strong southward winds in the observations, and this is

314

what the wind profilers show (horizontal line in Figure 4d). These results, using multiple

315

observations, confirm a previous suggestion based on a single reanalysis product⁵.

316

317

## Conclusions

318

Our results show that $k_{qsat}$ is linked to a range of model biases in precipitation and

319

atmospheric circulation. Improving $k_{qsat}$ should reduce those biases, giving greater
320 confidence in seasonal through multi-decadal model projections. kqsat affects the strength of
321 precipitation variation in both time and space. Other biases, in large-scale energy budgets,
322 and teleconnections, also affect precipitation. We show that kqsat can be constrained by
323 observations, and give evidence that improving the representation of shallow tropical
324 precipitating convection, and its coupling to SST and circulation, could improve kqsat. This
325 identifies specific model development goals and gives new ways of linking these to
326 observable physical processes.

327
328

**Methods**

329

330

**Data domain, SST, precipitation and $q_{sat}$**

331

332

333

334

335

336

337

Results are based on seasonal means of precipitation and SST, over the tropical oceans (20S-20N). $q_{sat}$ is calculated as the saturation specific humidity at the seasonal mean sea-surface temperature and 1000 hPa air pressure. Use of seasonal mean SST here means that $q_{sat}$ will be lower than the seasonal mean of saturation specific humidity calculated from daily SST (due to sub-seasonal SST variability, and nonlinearity in the humidity calculation). We use seasonal mean SST to minimise observational error: sub-seasonal SST variability is hard to observe accurately.

338

**Observations**

339

340

341

342

Satellite precipitation data are seasonal averages of monthly means from V7 of the 3B43 Tropical Rainfall Measuring Mission (TRMM)23,24 dataset, covering 1998-2015. Data from V2.3 of the Global Precipitation Climatology Project (GPCP25) retrieval are included in Figure 1 only.

343

344

345

346

347

348

349

In-situ raingauge data from 89 buoys of the Global Tropical Moored Buoy Array (all buoys with more than 1 year of precipitation data), from the Tropical Ocean-Global Atmosphere (TOGA26) observing system, the Prediction and Research Moored Array in the Atlantic (PIRATA28) and the Research Moored Array for African-Asian-Australian Monsoon Analysis and Prediction (RAMA27), were retrieved as daily means. Days with lower quality data (quality codes not equal to 1 or 2) were rejected. Monthly means were then calculated only for months with 20 or more days with code 1 or 2 data (other months are marked as
missing). Seasonal means were taken only where three consecutive months had non-missing
data.

Monthly mean SSTs are taken from four different datasets. The CMIP5 AMIP dataset³⁰ (the
dataset used to drive the AMIP SST-forced model runs) is available only for 1980-2005.

This was used to calculate kqsat for each model. For our observational estimate of a lower
bound on kqsat (using TRMM precipitation), three SST datasets were used: HadISST³² version
1.1, ERSST³³ version 4 and COBE³⁴ version 2 (the AMIP dataset used by the model
simulations was not used, due to its limited temporal overlap with the TRMM operational
period).

Surface meridional wind observations are from SeaWinds on QuikSCAT Level 3, for the
period Aug 1999-Oct 2009. Wind profiler observations at San Cristóbal, Galápagos (0.9°S,
89.7°W, 1994-2005) and Christmas Island (2.8°N, 157.5°W, 1990-2002) used 915 MHz in
low mode, as used in other studies of shallow circulation⁸.

## Model data

All model results are from atmosphere-only AMIP runs (one run per model version) forced
by observed SST, corresponding to the period 1980-2005. This includes 28 models from
CMIP5 (ACCESS1-0, ACCESS1-3, BNU-ESM, CCSM4, CESM1-CAM5, CNRM-CM5,
CSIRO-Mk3-6-0, CanAM4, GISS-E2-R, HadGEM2-A, IPSL-CM5A-LR, IPSL-CM5B-LR,
MIROC-ESM, MIROC5, MRI-AGCM3-2H, MRI-AGCM3-2S, MRI-CGCM3, NorESM1-M,
inmcm4, bcc-csm1-1-m, bcc-csm1-1, EC-EARTH, MPI-ESM-LR, MPI-ESM-MR,
FGOALS-s2, FGOALS-g2, GFDL-CM3, GFDL-HIRAM-C180) and 19 from CMIP6 (BCC-
CSM2-MR, BCC-ESM1, CAMS-CSM1-0, CanESM5, CNRM-CM6-1, CNRM-ESM2-1,
EC-Earth3-Veg, INM-CM4-8, INM-CM5-0, IPSL-CM6A-LR, MIROC6, HadGEM3-GC31-
373

LL, UKESM1-0-LL, MRI-ESM2-0, GISS-E2-1-G, GFDL-AM4, GFDL-CM4, NESM3,

374

SAM0-UNICON).

375

## Evaluation of satellite precipitation using GTMBA raingauge data

376

First, seasonal mean satellite precipitation data (for all seasons) were interpolated linearly to the GTMBA locations. Logarithms of seasonal mean precipitation were then taken, and all datasets masked at times and locations where any data (GTMBA or satellite) were missing.

377

378

379

This resulted in 1723 observations of seasonal mean precipitation from each dataset, covering the period 1998-2015 (the overlap between TRMM and GTMBA operational periods). For the remainder of the analysis, the observation location and time are ignored.

380

381

382

To reduce noise effectively, while retaining the signal of interest, we use the fact that precipitation tends to increase with SST, but the noise (as defined here) is largely independent of SST. For each dataset, the 1723 observations were grouped into 120 bins (14 observations per bin). This was done by ranking the observations by seasonal mean SST (the 14 observations corresponding to the 14 lowest SST values were placed in the first bin, and so on). The mean across each bin was then taken, giving 120 bin means of log(seasonal precipitation): giving 120 symbols in Figure 1. 120 bins were chosen, as a mean over 14 observations is sufficient to reduce noise significantly, while retaining a large number of symbols in Figure 1 to assess the method visually. Doubling the number of bins has negligible effect on the gradient in Figure 1. Results are insensitive to which SST dataset is used to bin the data (compare Extended data Figure 4, bottom two rows, with Figure 1).

383

384

385

386

387

388

389

390

391

392

393

We demonstrate that regression dilution bias is likely to be small in Figure 1a (method justification in Methods subsection ‘Estimating regression dilution bias’ below). We do this by repeating the analysis in Figure 1a, but regressing first TRMM against GPCP (Extended

394

395
396 data Figure 4, top left), then GPCP against TRMM (Extended data Figure 4, top right). The
397 product of the two regression gradients is 0.98 (close to 1), suggesting that this bias is small.

398 We also tested sensitivity of the validation to potential undercatch by the GTMBA raingauges
399 in windy conditions²⁹. This issue could only bias the gradients in Figures 1a,b if the
400 percentage undercatch varied systematically from low to high precipitation (because Figure 1
401 shows log precipitation). To test this, we recalculated the gradients in Figures 1a,b, but after
402 masking the data according to the seasonal mean wind speed (also observed by GTMBA
403 buoys). Gradients calculated for low wind (0-4 m/s; 22% of all data) and high wind (6-10
404 m/s; 27% of all data) show no significant differences from Figure 1a,b (for TRMM,
405 confidence intervals consistently spanning 1 and best estimate within 5% of 1; for GPCP,
406 confidence intervals consistently excluding 1). This suggests that the satellite validation is
407 insensitive to wind undercatch.

408 **Figure 2 data preparation (regions, seasons, time periods)**

409 For Figure 2, all data is first regridded by area-averaging to a common grid (resolution: 1.25°
410 latitude by 1.875° longitude).

411 The El Niño/La Niña ratio is based on large ENSO episodes in years both simulated by the
412 models and observed by TRMM: the 1997-1998 El Niño divided by the mean of 1998-1999
413 and 1999-2000 La Niñas, for the El Niño peak season (November-January), averaged over
414 10S-10N.

415 Other data are zonal means over the following longitude bands (with land masked out):
416 (Atlantic) 70W-25E; (East Pacific) 150-100W; (Indian Ocean) 50-100E.

417 Seasonal cycles over Atlantic and Pacific are calculated as the zonal mean for August-
418 October divided by the zonal mean for February-April). These seasons were chosen because
419

they show large differences in SST, but small differences in solar zenith angle (the latter can
affect precipitation by altering land temperature).

420

421

Spatial patterns are calculated, for each model/observational dataset, as the zonal mean at
each latitude, divided by the maximum, zonal mean for the same model/observational dataset.

422

423

This was calculated for August-October (ASO) for the Atlantic, due to the large meridional
SST gradient for this season. November-April was used for the Indian Ocean, as this basin
has a significant meridional SST gradient for this period.

424

425

426

**Estimating k<sub>qsat</sub>, part 1: data preparation**

427

kqsat is used here specifically to rank the models and compare with observations. Therefore,
the method of calculation needs to be consistent across models and observations, and to
minimise the potential for observational error.

428

429

430

For each year, for a given season, the logarithm of seasonal mean precipitation is calculated.
The spatial pattern of climatological mean precipitation (Extended data Figure 2j) is
dominated by a small area of large precipitation (occupying around 10% of the area). Hence,
if we evaluated kqsat without taking the logarithm, our result would be dominated by this
small area of the tropics. The spatial distribution of log(precipitation) is much more uniform
(Extended data Figure 2k), except for the driest 10% of the tropics (which is eliminated from
our analysis as we mask the region of coolest SST). Hence, using log(precipitation) to
calculate kqsat ensures that the result is influenced fairly equally by all parts of our analysis
region (confirmed in Figure 3a). Our results use seasonal mean precipitation. Use of other
timescales would alter kqsat, due to the (nonlinear) logarithm in Equation 1.

431

432

433

434

435

436

437

438

439

440

For models, kqsat is calculated using data on each model's native grid. For observations, qsat
is regridded linearly to the high resolution TRMM horizontal grid.

441
Before estimating k<sub>qsat</sub>, to minimise observational error, we exclude the 30% of the tropical oceans with the lowest climatological mean SST. An advantage of using a logarithm in equation 1 is that k<sub>qsat</sub> estimates are not dominated by the narrow ITCZ region. However, it could mean that error in observing the very lowest rainfall rates could cause large error in our real-world estimate of k<sub>qsat</sub>. Therefore we mask the regions with coolest SST on average.
This masking is only done in calculating k<sub>qsat</sub>. It is not done in Figure 2, as the climatological means reduce observational error there.

At each location, anomalies relative to climatological means are calculated for each year, for both qsat and log(precipitation). Locations that have missing data in any year are excluded.
These data are used in the sortav method, described below.

452 **Estimating k<sub>qsat</sub>, part 2: sortav regression method**

Once the data is prepared as above, our regression method for estimating kqsat (denoted
'sortav') is applied. For python code for this method, and an illustrative example, see Code
Availability Statement. The sortav method is designed to prevent dominance from the SST
spatial pattern associated with ENSO (an issue because ENSO features large SST anomalies
in a consistent pattern). In estimating kqsat from inter-annual variability, our aim is to reduce
the 'other-processes' term in Equation 1, by averaging over different SST patterns, with
different patterns of large-scale circulation anomalies. If a single SST pattern (ENSO) was
allowed to dominate, this would not be effective. If standard linear regression was used, the
ENSO pattern would dominate, because ENSO features large SST anomalies. In addition,
tropical means of precipitation and SST can vary over time, involving different processes
than those represented by kqsat. Our method avoids these issues.

The first step sorts each year of data. Say the data has n locations and y years. For each year,
the n anomalies in log(precipitation) are sorted in order of increasing qsat anomaly. This
466 produces, for each year, a vector of length n, with the first element corresponding to the location with the most negative $q_{sat}$ anomaly, and the last element being that with the most positive $q_{sat}$ anomaly. If Equation 1 was exactly true, with zero noise, each vector would be sorted in order of increasing anomaly in log(precipitation). Because of the noise (from large-scale processes), this is not, in general, true. This gives y sorted vectors, each of length n.

471 We then average over the y years, to produce one mean vector of length n (e.g. the 1st element of this vector is the mean of the y precipitation anomalies found over the most negative $q_{sat}$ anomaly from each year). This averaging removes much of the noise, because the noise is largely independent of $q_{sat}$. In this mean, years with large SST anomalies have the same weighting as other years, avoiding dominance by ENSO.

476 The same process is repeated for $q_{sat}$. This gives two mean vectors, each of length n: for anomalies in log(precipitation) and in $q_{sat}$.

478 The relationship between the averaged anomalies in log(precipitation) and $q_{sat}$ is relatively linear in both models and observations (e.g. Extended data Figure 2a-d), suggesting that Equation 1 is a useful approximation in this context. $k_{qsat}$ is then estimated from these vectors, by ordinary least squares linear regression (e.g. the gradients of the best fit lines in Extended data Figure 2a-d).

483 To compare with our sortav method, alternative estimates of $k_{qsat}$ (marked OLS in Extended Data Figures 2h,7) use standard linear regression between seasonal anomalies in log(precipitation) and $q_{sat}$ (without sort-averaging).

487 **Calculating $k_{qsat}^{(spatial)}$ and $k_{qsat}^{spattemp}$**
kqsat(spatial) is calculated using the same method as kqsat, except that it quantifies seasonal and
spatial variation in time-mean climate (in contrast with kqsat, which quantifies interannual
variability). First, time means are taken for each dataset and season (giving 4 season means
per grid point per dataset). For each dataset and each season separately, precipitation is
divided by the tropical mean, before taking logarithms. This is done to scale out model
variation in the tropical mean (which is controlled by the large-scale energy budget). For qsat,
for each dataset and season separately, anomalies are taken with respect to the tropical mean.
Masking, to exclude regions with low qsat, is based on annual mean qsat.

k<sub>qsat</sub><sup>spattemp</sup> is defined as follows:

$$
497 \quad P/\bar{P} \approx \exp (k_{qsat}^{spattemp} \cdot q_{sat}')
$$

This approximates variation in precipitation relative to the tropical mean, driven by variation
in qsat relative to its tropical mean. The overbar represents the tropical mean for the current
season of the current year, and qsat' is specifically the qsat anomaly with respect to the
tropical mean:

501

$$
502 \quad q'_{sat} = q_{sat} - \overline{q_{sat}}
$$

Anomalies expressed this way capture temporal and spatial variability associated with
variation in local SSTs, but exclude temporal variability in tropical mean precipitation (which
is constrained by the large-scale atmospheric energy budget).

503
504
505

k<sub>qsat</sub><sup>spattemp</sup> is estimated using the same approach as kqsat, but anomalies of log(P) and qsat
are taken relative to their tropical means for the corresponding season and year (kqsat is
evaluated using anomalies with respect to local climatological means for each location).

506
507
508
A disadvantage of both $k_{qsat}^{(spatial)}$ and $k_{qsat}^{spatialtemp}$ is that they have some sensitivity to other processes (teleconnections) associated with the specific spatial patterns in climatological mean SST (these spatial patterns are filtered out less effectively as there are only 4 seasons, compared to the 25 years of internal variability used to estimate kqsat).

513

514 A lower bound for $k_{qsat}$, using observed interannual variability

This method (estimated lower bound on $k_{qsat}$) accounts for three forms of observational error, combined using Monte Carlo sampling.

First, systematic error in the observed magnitudes of seasonal mean SST anomalies could bias $k_{qsat}$. To explore this, we first estimated $k_{qsat}$ (with the sortav method) using TRMM precipitation, and each of the three SST datasets (HadISST, ERSST, COBE) that cover the whole TRMM operational period, giving three direct, unscaled estimates of $k_{qsat}$ (Extended data Table 2a).

Second, error in the SST spatial pattern will cause a low bias in $k_{qsat}$ (regression dilution bias). Typical magnitudes of this bias are estimated (see more detail in Methods section ‘Estimating regression dilution bias’). This is done by regressing pairs of SST datasets against each other, with regression coefficients calculated using the sortav method (as used for $k_{qsat}$). Extended data Table 2b shows the results for each pair of SST datasets. The final column of Extended data Table 2b gives the 6 different estimates of regression dilution bias.

Third, the TRMM operational period does not fully overlap the AMIP SST-forced model simulation period. Thus, $k_{qsat}$ estimated for the TRMM period will be different from that obtained if TRMM were operational throughout the AMIP period (due to a different set of SST patterns during the TRMM and AMIP period). Estimates of typical magnitudes of this
532

error were obtained using samples from coupled ocean-atmosphere simulations (Extended data Figure 7), giving 85 samples of percentage error.

533

534

10000 estimates of $k_{qsat}$ were then generated using Monte Carlo sampling. For each estimate, one of the three direct estimates (Extended data Table 2a) was selected at random, ‘corrected’ by a random selection from the 85 samples of percentage error (Extended data Figure 7), and further corrected using a random selection from the 6 estimates of regression dilution bias (Extended data Table 2b).

535

536

537

538

539

The lower bound of the 95% confidence interval of these 10000 estimates is then taken. We only use the lower bound for the following reason: it seems unlikely that the true regression dilution bias is weaker than the minimum value estimated here (around 10%, Extended data Table 2b). This is because we do not expect the SST datasets to be significantly closer to the real SST than they are to each other. However, it is plausible that the regression dilution bias could be larger than estimated, so we do not quote an upper bound for $k_{qsat}$.

540

541

542

543

544

545

The appropriate lower bound for an atmosphere-only AMIP model may be even higher than our quoted result, as the observed $k_{qsat}$ value may be reduced by the effect of atmospheric internal variability on SST¹⁹. $k_{qsat}$ values from AMIP models are similar to or higher than from the equivalent coupled models. In contrast, this coupling issue should be small for our central estimate of $k_{qsat}$ (next section), as that is based on metrics where ocean dynamics (for ENSO) or forcing associated with the mean state/seasonal cycle dominate the SST differences/gradients.

546

547

548

549

550

551

552

## Central estimate of $k_{qsat}$ in the real world

553

We start by finding where, geographically, the models are most sensitive to $k_{qsat}$. This is done using correlation coefficients ($r$) between $k_{qsat}$ and the log of precipitation ratios

554
(Extended data Figure 5) for each latitude (or longitude for ENSO) in each panel in Figure 2. This reveals six discrete intervals (shaded in Figure 2) where $|r| > 0.6$ (about 50% variance explained), and a seventh, showing weaker correlations, over the Indian Ocean. For each model, we average the log of precipitation ratios over each interval, and take the difference from the equivalent value for TRMM. This gives seven error indices for each model (y-axes in Extended data Figure 6). For each interval, the 28 model error indices are regressed against $k_{qsat}$ (x-axes in Extended data Figure 6), and $k_{qsat}$ is estimated from where the line of best fit crosses zero error. This gives seven estimates of $k_{qsat}$; their mean is our central estimate.

We did not use the coupled ocean-atmosphere models to estimate $k_{qsat}$, due to the evident residual biases from SST error in these models, and because it isn't possible to use the ENSO response for coupled models, due to model differences in simulations of ENSO SST responses.

## Estimating regression dilution bias

Regression dilution bias⁵¹ arises when there is random error in the independent variable (e.g. $q_{sat}$ in estimating $k_{qsat}$). This causes the regression gradient to be biased low. This bias reduces the gradient by a factor ($\beta$) that depends only on the characteristics of the independent variable. We estimate typical magnitudes of this bias using different observations of the independent variable (e.g. different SST datasets), as follows.

Say the vector of true values of the independent variable is **x**, and we have two different observational estimates, **x**1 and **x**2. We first regress **x**1 against **x**2. This regression uses the same methodology as when regressing the dependent variable against **x**: i.e. for estimating dilution bias in observations of $k_{qsat}$, the sortav method is applied; and for the satellite precipitation validation, the SST binning is used. The regression gradient ($f_{12}$) obtained from
579 regressing **x1** against **x2**, will be biased low by a factor $\beta_2$ (from error in **x2**). We then
580 regress **x2** against **x1**. This regression gradient ($f_{21}$) will be biased low by a factor $\beta_1$ (from
581 error in **x1**).

582 We then estimate the dilution bias as:

583 $\beta \approx \sqrt{\beta_2 \cdot \beta_1} = \sqrt{f_{12} \cdot f_{21}}$

584 (with no dilution bias, $f_{12} \cdot f_{21} = 1$, by definition).

585 This method assumes that the errors in **x1** and **x2** are independent.

586 Column dry static energy budget (DSE)

587 Dry static energy (DSE) is given by $s = c_p T + gz$, where $c_p$ is the specific heat at constant
588 pressure, T is the temperature, g is the gravitational acceleration, and z is altitude). DSE is
589 affected by advection, precipitation, radiation and sensible heat:

590 $-\langle\omega \frac{\partial s}{\partial p}\rangle - \langle v \cdot \nabla s \rangle + LP + R + Q_{\text{turb}} = 0,$

591 where $\omega$ is the vertical pressure velocity, $p$ is pressure, $v$ is horizontal wind. Angle brackets
592 represent the mass-weighted vertical integral from 1000-100hPa. The first term represents
593 import of DSE via column-integrated vertical advection; the second is horizontal advection.
594 $P$ is the total surface precipitation and $L$ the latent heat of condensation; $R$ is net radiation
595 into the atmospheric column; and $Q_{\text{turb}}$ is the surface sensible heat flux.

596

597 Methods References

598

599 51. Hutcheon, J. A., Chiolero, A. & Hanley, J. A. Random measurement error and regression
dilution bias. *Br. Med. J.* **340**, (2010).
600
602 **Data availability**

603 Datafiles with estimates of kqsat for models and observations, along with sample plotting
604 code, are available from http://doi.org/10.5281/zenodo.3878691. Data from the integration of
605 CNRM-CM6 with the CM5 convection scheme (denoted CNRM-CM6-conv5) are available
606 from https://doi.org/10.5281/zenodo.3875005. Model and observational data is available at the
607 following websites. CMIP5: https://cmip.llnl.gov/cmip5/; CMIP6: https://esgf-
608 node.llnl.gov/projects/cmip6/; GTMBA: https://www.pmel.noaa.gov/gtmba/; TRMM:
609 https://pmm.nasa.gov/data-access/downloads/trmm; GPCP and COBE:
610 https://www.esrl.noaa.gov/psd/; HadISST: https://www.metoffice.gov.uk/hadobs/hadisst; ERSST:
611 http://www1.ncdc.noaa.gov/pub/data/cmb/ersst/v4/netcdf/.

612 **Code Availability**

613 Python code for calculating kqsat, including the sortav regression routine, is available from
614 http://doi.org/10.5281/zenodo.3878691.
615 References

616 1. Philander, S. G. H. El-Nino Southern Oscillation Phenomena. *Nature* **302**, 295–301 (1983).

617 2. Pierrehumbert, R. T. Thermostats, Radiator Fins, and the Local Runaway Greenhouse. *J. Atmos. Sci.* **52**, 1784–1806 (1995).

619 3. Tian, B. J. Spread of model climate sensitivity linked to double-Intertropical Convergence Zone bias. *Geophys. Res. Lett.* **42**, 4133–4141 (2015).

621 4. Back, L. E. & Bretherton, C. S. Geographic variability in the export of moist static energy and vertical motion profiles in the tropical Pacific. *Geophys. Res. Lett.* **33**, (2006).

623 5. Sherwood, S. C., Bony, S. & Dufresne, J. L. Spread in model climate sensitivity traced to atmospheric convective mixing. *Nature* **505**, 37–+ (2014).

625 6. Trenberth, K. E., Stepaniak, D. P. & Caron, J. M. The global monsoon as seen through the divergent atmospheric circulation. *J. Clim.* **13**, 3969–3993 (2000).

627 7. Back, L. E. & Bretherton, C. S. A Simple Model of Climatological Rainfall and Vertical Motion Patterns over the Tropical Oceans. *J. Clim.* **22**, 6477–6497 (2009).

629 8. Zhang, C. D., Nolan, D. S., Thorncroft, C. D. & Nguyen, H. Shallow meridional circulations in the tropical atmosphere. *J. Clim.* **21**, 3453–3470 (2008).

631 9. Li, R. & Fu, Y. F. Tropical precipitation estimated by GPCP and TRMM PR observations. *Adv. Atmos. Sci.* **22**, 852–864 (2005).

633 10. Prakash, S., Mahesh, C. & Gairola, R. M. Comparison of TRMM Multi-satellite Precipitation Analysis (TMPA)-3B43 version 6 and 7 products with rain gauge data from ocean buoys. *Remote Sens. Lett.* **4**, 677–685 (2013).

636 11. Bellon, G., Reitebuch, O. & Naumann, A. K. Shallow Circulations: Relevance and Strategies for Satellite Observation. *Surv. Geophys.* **38**, 1509–1528 (2017).

638 12. Li, G. & Xie, S. P. Tropical Biases in CMIP5 Multimodel Ensemble: The Excessive Equatorial Pacific Cold Tongue and Double ITCZ Problems. *J. Clim.* **27**, 1765–1780 (2014).

640 13. Xie, S. P. et al. Towards predictive understanding of regional climate change. *Nat. Clim. Chang.* **5**, 921–930 (2015).

642 14. Byrne, M. P. & Schneider, T. Energetic Constraints on the Width of the Intertropical Convergence Zone. *J. Clim.* **29**, 4709–4721 (2016).

644 15. Bellenger, H., Guilyardi, E., Leloup, J., Lengaigne, M. & Vialard, J. ENSO representation in climate models: from CMIP3 to CMIP5. *Clim. Dyn.* **42**, 1999–2018 (2014).

646 16. Zhang, X. X., Liu, H. L. & Zhang, M. H. Double ITCZ in Coupled Ocean-Atmosphere Models: From CMIP3 to CMIP5. *Geophys. Res. Lett.* **42**, 8651–8659 (2015).

648 17. Hirota, N. & Takayabu, Y. N. Reproducibility of precipitation distribution over the tropical oceans in CMIP5 multi-climate models compared to CMIP3. *Clim. Dyn.* **41**, 2909–2920 (2013).

650 18. Dixit, V., Geoffroy, O. & Sherwood, S. C. Control of ITCZ Width by Low-Level Radiative Heating From Upper-Level Clouds in Aquaplanet Simulations. *Geophys. Res. Lett.* **45**, 5788–5797. https://doi.org/10.1029/2018GL078292 (2018).
653 19. He, J. et al. Precipitation Sensitivity to Local Variations in Tropical Sea Surface Temperature. *J. Clim.* **31**, 9225–9238 (2018).

655 20. Rushley, S. S., Kim, D., Bretherton, C. S. & Ahn, M. S. Reexamining the Nonlinear Moisture-Precipitation Relationship Over the Tropical Oceans. *Geophys. Res. Lett.* **45**, 1133–1140 (2018).

658 21. Bretherton, C. S., Peters, M. E. & Back, L. E. Relationships between water vapor path and precipitation over the tropical oceans. *J. Clim.* **17**, 1517–1528 (2004).

660 22. Kanemaru, K. & Masunaga, H. A Satellite Study of the Relationship between Sea Surface Temperature and Column Water Vapor over Tropical and Subtropical Oceans. *J. Clim.* **26**, 4204–4218 (2013).

663 23. TRMM. Tropical Rainfall Measuring Mission, TRMM (TMPA/3B43) Rainfall Estimate L3 1 month 0.25 degree x 0.25 degree V7, Greenbelt, MD, Goddard Earth Sciences Data and Information Services Center (GES DISC), Accessed 2017. 10.5067/TRMM/TMPA/MONTH/7 (2011).

667 24. Huffman, G. J. et al. The TRMM multisatellite precipitation analysis (TMPA): Quasi-global, multiyear, combined-sensor precipitation estimates at fine scales. *J. Hydrometeorol.* **8**, 38–55 (2007).

670 25. Adler, R. F. et al. The Global Precipitation Climatology Project (GPCP) Monthly Analysis (New Version 2.3) and a Review of 2017 Global Precipitation. *Atmosphere (Basel)*. **9**, (2018).

672 26. McPhaden, M. J. et al. The tropical ocean global atmosphere observing system: A decade of progress. *J. Geophys. Res.* **103**, 14169–14240 (1998).

674 27. McPhaden, M. J. et al. RAMA The Research Moored Array for African-Asian-Australian Monsoon Analysis and Prediction. *Bull. Am. Meteorol. Soc.* **90**, 459–+ (2009).

676 28. Bourles, B. et al. The PIRATA program: History, accomplishments, and future directions. *Bull. Am. Meteorol. Soc.* **89**, 1111–+ (2008).

678 29. Serra, Y. L., A’Hearn, P., Freitag, H. P. & McPhaden, M. J. ATLAS self-siphoning rain gauge error estimates. *J. Atmos. Ocean. Technol.* **18**, 1989–2002 (2001).

680 30. Taylor, K. E., Williamson, D. & Zwiers, F. W. The sea surface temperature and sea-ice concentration boundary conditions for AMIP II simulations. *PCMDI Rep. No. 60, Progr. Clim. Model Diagnosis Intercomparison, Lawrence Livermore Natl. Lab. Livermore, California*, 25 pp (2000).

684 31. Allen, M. R. & Ingram, W. J. Constraints on future changes in climate and the hydrologic cycle. *Nature* **419**, 224–+ (2002).

686 32. Rayner, N. A. et al. Global analyses of sea surface temperature, sea ice, and night marine air temperature since the late nineteenth century. *J. Geophys. Res.* **108**, (2003).

688 33. Huang, B. Y. et al. Extended Reconstructed Sea Surface Temperature Version 4 (ERSST.v4). Part I: Upgrades and Intercomparisons. *J. Clim.* **28**, 911–930 (2015).

690 34. Hirahara, S., Ishii, M. & Fukuda, Y. Centennial-Scale Sea Surface Temperature Analysis and Its Uncertainty. *J. Clim.* **27**, 57–75 (2014).

692 35. Biasutti, M., Sobel, A. H. & Kushnir, Y. AGCM precipitation biases in the tropical Atlantic. *J.*
693 *Clim.* **19**, 935–958 (2006).

694 36. Holloway, C. E. & Neelin, J. D. The convective cold top and quasi equilibrium. *J. Atmos. Sci.*
695 (2007) doi:10.1175/JAS3907.1.

696 37. Nolan, D. S., Powell, S. W., Zhang, C. D. & Mapes, B. E. Idealized Simulations of the
697 Intertropical Convergence Zone and Its Multilevel Flows. *J. Atmos. Sci.* **67**, 4028–4053 (2010).

698 38. Knutson, T. R. & Manabe, S. Time-mean response over the tropical Pacific to increased CO2 in
699 a coupled ocean-atmosphere model. *J. Clim.* (1995) doi:10.1175/1520-
700 0442(1995)008<2181:TMROTT>2.0.CO;2.

701 39. Wu, Z. H. A shallow CISK, deep equilibrium mechanism for the interaction between large-scale convection and large-scale circulations in the tropics. *J. Atmos. Sci.* **60**, 377–392 (2003).

703 40. Lin, Y. *et al*. TWP-ICE global atmospheric model intercomparison: Convection responsiveness
704 and resolution impact. *J. Geophys. Res. Atmos.* (2012) doi:10.1029/2011JD017018.

705 41. Randall, D. A. Beyond deadlock. *Geophys. Res. Lett.* (2013) doi:10.1002/2013GL057998.

706 42. Bony, S. & Dufresne, J. L. Marine boundary layer clouds at the heart of tropical cloud
707 feedback uncertainties in climate models. *Geophys. Res. Lett.* (2005)
708 doi:10.1029/2005GL023851.

709 43. Voldoire, A. *et al*. Evaluation of CMIP6 DECK Experiments With CNRM-CM6-1. *J. Adv. Model. Earth Syst.* (2019) doi:10.1029/2019MS001683.

711 44. Nie, J., Boos, W. R. & Kuang, Z. Observational evaluation of a convective quasi-equilibrium
712 view of monsoons. *J. Clim.* (2010) doi:10.1175/2010JCLI3505.1.

713 45. Stevens, B. & Bony, S. What are climate models missing? *Science* (2013)
714 doi:10.1126/science.1237554.

715 46. Oueslati, B. & Bellon, G. Convective Entrainment and Large-Scale Organization of Tropical
716 Precipitation: Sensitivity of the CNRM-CM5 Hierarchy of Models. *J. Clim.* **26**, 2931–2946
717 (2013).

718 47. Vial, J., Bony, S., Stevens, B. & Vogel, R. Mechanisms and Model Diversity of Trade-Wind
719 Shallow Cumulus Cloud Feedbacks: A Review. *Surveys in Geophysics* (2017)
720 doi:10.1007/s10712-017-9418-2.

721 48. Fläschner, D., Mauritsen, T., Stevens, B. & Bony, S. The signature of shallow circulations, not
722 cloud radiative effects, in the spatial distribution of tropical precipitation. *J. Clim.* (2018)
723 doi:10.1175/JCLI-D-18-0230.1.

724 49. NASA. *SeaWinds on QuikSCAT Level 3 Surface Northward Wind for Climate Model Comparison. Ver. 1.* PO.DAAC, CA, USA. Dataset accessed [2019-09-06] at
725 https://doi.org/10.5067/QS SNW-CMIP1. (2012).

727 50. Gage, K. S., Williams, C. R. & Ecklund, W. L. UHF wind profilers: a new tool for diagnosing
728 tropical convective cloud systems. *Bull. - Am. Meteorol. Soc.* (1994) doi:10.1175/1520-
729 0477(1994)075<2289:UWPANT>2.0.CO;2.

730

731
732

733

734

735

736 **End Notes**

737

738 **Acknowledgements** This work was supported jointly by the Met Office Hadley Centre Climate

739 Programme funded by BEIS and Defra, and by the Newton Fund through the Met Office Climate

740 Science for Service Partnership Brazil (CSSP Brazil). SSR was supported by the National Aeronautics

741 and Space Administration Grant 80NSSC17K0227 and the Korean Meteorological Administration

742 Research and Development Program under grant KMI2018-03110. We acknowledge the GTMBA

743 Project Office of NOAA/PMEL for making the GTMBA data available. The QuikSCAT data were

744 obtained from the NASA EOSDIS Physical Oceanography Distributed Active Archive Center

745 (PO.DAAC) at the Jet Propulsion Laboratory, Pasadena, CA (http://dx.doi.org/10.5067/GHGMR-

746 4FJ01). We acknowledge NOAA/ESRL PSD for the wind profiler data. We acknowledge the World

747 Climate Research Programme's Working Group on Coupled Modelling, which is responsible for CMIP,

748 and we thank the climate modelling groups (listed in Methods) for producing and making available

749 their model output. For CMIP the U.S. Department of Energy's Program for Climate Model Diagnosis

750 and Intercomparison provides coordinating support and led development of software infrastructure

751 in partnership with the Global Organization for Earth System Science Portals. GPCP data provided by

752 the NOAA/OAR/ESRL PSD, Boulder, Colorado, USA, from their Web site at

753 https://www.esrl.noaa.gov/psd/. COBE-SST2 data provided by the NOAA/OAR/ESRL PSD, Boulder,

754 Colorado, USA, from their Web site at https://www.esrl.noaa.gov/psd/

755 **Author Contributions** P.G. conceived and designed the study and performed the analysis. All

756 authors contributed to scientific interpretation and wrote the manuscript. R.R. performed the
757 CNRM model simulations. P.G., R.C., C.E.H. and R.R. contributed understanding on physical

758 processes. J.K. provided knowledge of SST observational uncertainty and datasets.

759 **Author Information** Reprints and permissions information is available at www.nature.com/reprints.

760 The authors have no competing financial interests. Correspondence and requests for materials

761 should be addressed to peter.good@metoffice.gov.uk.

762

763

764

765
766

767

768

769

<figure><img src="image_2.png" /><figcaption>Figure 1. Validating observations of log precipitation from satellites. GTMBA in-situ raingauge observations versus satellite observations from a TRMM and b GPCP. Each symbol represents the mean of all seasonal mean data within a given SST bin (Methods). Green line: best fit line (gradient and its 95% confidence interval quoted in each figure); dotted line: 1:1 line.</figcaption></figure>

775

776
<figure><img src="image_3.png" /><figcaption>Figure 2. Model precipitation biases. (black) TRMM observations. Horizontal dashed line marks precipitation ratio=1. **a-f** all CMIP5 models are shown in grey lines; **g-i** magenta: ‘high-k<sub>qsat</sub>’ subset; grey: other models. Spatial patterns (bottom 3 rows) given by scaling zonal mean precipitation by its latitudinal maximum. Green shading marks the intervals used for the 7 estimates of k<sub>qsat</sub>. These examples were chosen as they feature large</figcaption></figure>
783 differences/gradients in SST. Precipitation ratios are plotted because of the form of Equation

784 1.

785
<figure><img src="image_4.png" /></figure>

787

788

789 Figure 3. The region of applicability of kqsat. (a) each bar represents a climatological zone
790 covering 20% of the tropical oceans, defined by the seasonal climatological SSTs (e.g. the
791 left bar is the zone with the coolest 20% of SSTs – white masked ocean in the maps below).
792 Climatological zones are defined separately for each season. Bar height: the correlation
793 coefficient, across CMIP5 models, between the standard calculation of kqsat, and that
794 calculated only over the selected climatological zone. Mean SST (°C) for each zone is also
795 shown. (b-e) Colours: mean TRMM precipitation; orange line highlights 1 mm/day contour.
796 Data is masked over the 20% of the oceans where kqsat is inapplicable (left-hand bar in panel
797 a shows low correlation). White contour shows the 30th percentile of SST: the standard
798 calculation of kqsat uses data inside this contour.
<figure><img src="image_5.png" /></figure>

Figure 4. High sensitivity of precipitation to SST, and strong shallow circulations, in the real world. **a** horizontal lines mark (white dashes) the 7 estimates of $k_{qsat}$ (Extended data Figure 6), and the central estimate (black solid); shading marks $k_{qsat}$ values above our lower-bound estimate; symbols mark sorted model $k_{qsat}$ values for (crosses) CMIP5 (blue and magenta denote low-$k_{qsat}$ and high-$k_{qsat}$ model subsets) and (circles) CMIP6. **b** $k_{qsat}$<sup>(spatial)</sup> versus $k_{qsat}$, for each (black) CMIP5 and (red) CMIP6 model; horizontal line: $k_{qsat}$<sup>(spatial)</sup> from TRMM observations; vertical line: best estimate of $k_{qsat}$. **c,d** each symbol represents one CMIP5 (black) or CMIP6 (red) model; title gives Pearson correlation coefficient. **c** surface meridional wind averaged over the mid-region (180W-10E, 1-7N) versus shallow descent index (defined in Figure 5); horizontal line marks QuikSCAT observation. **d** meridional wind averaged over Galapagos & Christmas island, 600-700hPa (few observations above 600hPa) versus meridional wind averaged over the mid region (180W-10E, 1-7N), 500-700hPa; horizontal line marks wind profiler observation.
<figure><img src="image_6.png" /></figure>

Figure 5. Linking $k_{qsat}$ to shallow circulations. **a,b** quantifies internal variability, and **c-e** climate means. **a** CMIP5 ensemble mean of $k_{qsat}^{wap}$ (Pa kg/g), at each pressure level. **b** inter-model correlations (Pearson r) between $k_{qsat}$, and $k_{qsat}^{wap}$, at each pressure level. Correlations are negative because of the definition of wap. **c-e** Aug-Oct, 180W-10E zonal means. **c** CMIP5 ensemble mean of (colours) vertical velocity (Pa/s) and (white contours) meridional wind. **d** inter-model correlations between $k_{qsat}$ and mean vertical velocity (colours) and
823

between $k_{qsat}$ and mean meridional wind (white contours). **e** as **d**, but for correlations with

824

the shallow descent index instead of $k_{qsat}$ (shallow descent index = vertical velocity averaged

825

over left-hand orange-dashed box: 5S-1N, 850-600hPa).

826

827

828
829

830

831

<figure><img src="image_7.png" /></figure>

832

833

834

835

836

837

838

839

840

841

842

843

844

845

Extended data Figure 1. Effect of low spatial resolution in GPCP satellite observations of log(seasonal precipitation). y-axis: regression gradient in validation against GTMBA raingauge data (i.e. gradients in Figure 1 for light blue and red symbols). x-axis: horizontal grid dimension relative to TRMM (e.g. the TRMM resolution is 0.25°, ten times smaller than the GPCP resolution of 2.5°, so the red symbol is placed at x=10). Dark blue symbols: results when TRMM data is regridded (by area averaging) to coarser grids. The coarser grids are chosen so the grid box edges overlap edges of the native TRMM grid. To give the errors context, the dash-dot line marks the ratio between the largest and smallest model values of kqsat (2.5). Solid black line is a quadratic least-squares best fit line through the TRMM-based data. The intercept of the TRMM best-fit curve at x=0 (i.e. infinitely fine grid) is very close to the value estimated on the TRMM native grid (light blue symbol), indicating that the TRMM grid is sufficiently fine for comparison with the raingauge data on seasonal timescales.

846
847

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

848

<figure><img src="image_10.png" /></figure>

849

Extended data Figure 2. Testing the method of estimating kqsat. **a-d:** example results of the sortav method for TRMM precipitation and HadISST SST, for different seasons: mean vectors of anomalies in (y-axis) log(precipitation) and (x-axis) qsat; kqsat is given by the gradient of the blue best-fit regression line. **e,** y-axis: kqsat calculated after excluding the 9

850

851

852

853
years with the largest absolute value of the nino3.4 index; x-axis: default kqsat (one symbol per model); kqsat is on average 6% lower when ENSO years excluded, due to a small sensitivity to the ENSO characteristic spatial pattern; but the model ranking is largely unchanged (r = 0.99). f, kqsat calculated for individual seasons versus the annual mean value; g kqsat using only years 1995-1999 versus the full 25-year estimate; h, estimating variability (due to internal variability in SST patterns) in kqsat estimated from 25 years of data: for each coupled ocean-atmosphere model, kqsat is estimated both for the full historical run, and for all 25-year chunks. Panel shows the cumulative distribution function of absolute percentage differences between the 25-year estimates and the full estimates (95% of samples are within 8% of the long-term value from the full historical run). This panel shows results for two methods of estimating kqsat: our ‘sortav’ method (as used throughout the manuscript), and standard OLS regression between seasonal anomalies in log(precipitation) and qsat. i comparing kspattemp with kqsat; each cross represents one CMIP5 model. j,k Cumulative distribution functions of j climatological mean precipitation and k log(precipitation). From HadGEM2-A, May-July season (same picture seen in other seasons).

869

870

871
<figure><img src="image_11.png" /></figure>

873 Extended data Figure 3. Model biases for the high, mid-range and low-k<sub>qsat</sub> models separately. As

874 Figure 2, for a-f high-k<sub>qsat</sub> models; g-l mid-k<sub>qsat</sub> models; m-r low-k<sub>qsat</sub> models.
<figure><img src="image_12.png" /></figure>

Extended data Figure 4. Testing potential errors in the satellite validation against GTMBA.

a,b testing for regression dilution bias from error in TRMM observations: as Figure 1, but for

a TRMM versus GPCP (both interpolated to GTMBA sites and masked as in Figure 1) and b

GPCP versus TRMM. c-f testing for effects of SST uncertainty on the binning: as Figure 1,

but using c,d ERSST and e,f COBE SST datasets to bin the data.
<figure><img src="image_13.png" /></figure>

Extended data Figure 5. Regions where models are most sensitive to kqsat. For each latitude of each region: y-axis shows Pearson correlation coefficients (r) between the 28 different CMIP5 model values kqsat, and the 28 CMIP5 model values of the logarithm of the precipitation ratio for that latitude and region (i.e. the logarithm of the grey lines in Figure 2a-f). Green bands mark the latitude intervals chosen to estimate the observational constraints on kqsat (a-e: intervals chosen where |r|> 0.6; f, a band of most negative r is chosen). Coefficients close to zero near 8N in the Atlantic and East Pacific spatial patterns correspond to the latitude of the precipitation peak in most models (the model spread in the precipitation
895 peak is scaled out; coefficients are not exactly zero as there is a small model spread in the

896 latitude of the precipitation peak).

897
<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /></figure>

Extended data Figure 6. Scatter plots underpinning the central observational estimate of kqsat.

a-g Precipitation error index versus kqsat for each of the 7 latitude intervals highlighted in Figure 2. Y-axes: logarithm of precipitation ratio, averaged over each latitude band, minus the equivalent value for TRMM observations, for (black) CMIP5 and (red) CMIP6 models.

Dotted lines: linear least-squares fits (using CMIP5 data only). Vertical black line: kqsat estimate for each latitude interval, from the intercept of the green line with zero error index
906 (dotted line). **h** Mean precipitation error index versus kqsat: mean error index is averaged over

907 the 7 indices in the other panels (after the signs of the 5 indices with negative best-fit slopes

908 were changed, to ensure a positive correlation with kqsat).
<figure><img src="image_22.png" /></figure>

909

910 Extended data Figure 7. Supporting results for observational estimate of the kqsat lower bound. Estimating error, from internal variability, due to the fact that the TRMM operational period only partly overlaps the time period simulated by the AMIP SST-forced models. Error magnitudes are estimated from the coupled ocean-atmosphere simulations, using differences between kqsat estimated from all possible overlapping 17-year (TRMM-like) and 25-year (AMIP-like) periods (with the same overlap as TRMM and the 25-year SST-forced model simulations). Results are given for two methods of estimating kqsat: our ‘sortav’ method (as used throughout the manuscript), and standard OLS regression between seasonal anomalies in log(precipitation) and qsat.

919

920
<figure><img src="image_23.png" /></figure>

Extended Data Figure 8. Atmospheric circulation measures in CMIP5 and CMIP6 models.

**a-c** thick lines are CMIP5 composite means, for (magenta) high $k_{qsat}$ subset; (blue) low $k_{qsat}$ subset and (gold) intermediate $k_{qsat}$. Thin grey lines are individual models (CMIP5 and CMIP6). Descent (5S-1N), mid (1-7N) and ascent (7-13N) regions are marked by vertical dotted lines in Figure 5c-e. **d-h**: each symbol represents one CMIP5 (black) or CMIP6 (red) model. Title gives Pearson correlation coefficient. **d** shallow descent versus $k_{qsat}$; vertical line marks our best estimate of $k_{qsat}$. **e** shallow ascent versus shallow descent. **f** shallow meridional return flow versus shallow descent. **g** shallow versus very-shallow meridional wind, over Galapagos: the negligible correlation indicates different physical processes at these
933 two levels. **h** deep versus shallow ascent. **i** standard deviation, across models, of the pressure

934 velocity (wap) at each pressure level.

935
<table><thead><tr><th>Descent region, column integrals</th><th>High k<sub>qsat</sub> mean</th><th>Mid k<sub>qsat</sub> mean</th><th>Low k<sub>qsat</sub> mean</th><th>High k<sub>qsat</sub> - Low k<sub>qsat</sub></th></tr></thead><tbody><tr><td>Net radiation + sensible heat flux</td><td>-129.4</td><td>-125.9</td><td>-127.5</td><td>-1.8</td></tr><tr><td>Latent heating by precipitation</td><td>14.4</td><td>22.9</td><td>35.5</td><td>-21.1</td></tr><tr><td>Vertical advection by mean vertical velocity, integrated over 100-1000 hPa</td><td>130.1</td><td>110.0</td><td>95.5</td><td>34.6</td></tr><tr><td>Vertical advection by mean vertical velocity, integrated over 600-1000 hPa</td><td>74.5</td><td>57.4</td><td>48.1</td><td>26.4</td></tr><tr><td>Residual advection</td><td>-15.1</td><td>-7.0</td><td>-3.5</td><td>-11.6</td></tr></tbody></table>

Extended Data Table 1. Column-integrated dry static energy budget for the descent region, Aug-Oct, averaged over the high-, mid- and low-k<sub>qsat</sub> groups of CMIP5 models. Vertical advection by mean vertical velocity is calculated using seasonal mean vertical velocity. The final row, calculated as a residual from the first three columns, includes horizontal advection, and vertical advection by transient eddies.
<table><thead><tr><th>a SST dataset</th><th>Unscaled k<sub>qsat</sub> estimate (no bias correction)</th></tr></thead><tbody><tr><td>HadISST</td><td>0.58</td></tr><tr><td>ERSST</td><td>0.49</td></tr><tr><td>COBE</td><td>0.51</td></tr></tbody></table>

<table><thead><tr><th>b SST1</th><th>SST2</th><th>K1 (X = SST1)</th><th>K2 (X = SST2)</th><th>SQRT(K1*K2)</th></tr></thead><tbody><tr><td>AMIP</td><td>HadISST</td><td>0.77</td><td>1.01</td><td>0.89</td></tr><tr><td>AMIP</td><td>COBE</td><td>0.85</td><td>1.00</td><td>0.92</td></tr><tr><td>AMIP</td><td>ERSST</td><td>0.76</td><td>0.97</td><td>0.86</td></tr><tr><td>HadISST</td><td>COBE</td><td>0.76</td><td>0.79</td><td>0.78</td></tr><tr><td>HadISST</td><td>ERSST</td><td>0.83</td><td>0.55</td><td>0.68</td></tr><tr><td>COBE</td><td>ERSST</td><td>0.92</td><td>0.54</td><td>0.70</td></tr></tbody></table>

Extended data Table 2. Supporting results for observational estimate of the k<sub>qsat</sub> lower bound (see Methods for details). **a** Unscaled estimates for k<sub>qsat</sub> directly estimated using TRMM precipitation and three different SST datasets (the AMIP SST dataset was not used due to limited temporal overlap with the TRMM operational period). These values are contaminated by regression dilution bias so do not represent central estimates. **b** Estimating typical values of regression dilution bias from each pair of SST datasets. K1 is the gradient from linear regression when regressing SST1 against SST2 (using the sortav regression method). K2 is the value obtained when regressing SST2 against SST1.