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Global surface area receiving daily precipitation,
wet-day frequency and probability of extreme
rainfall: Water Security and Climate Change
Rasmus E. Benestad ( rasmus.benestad@met.no )
Norwegian Meteorological Institute
Cristian Lussana
Norwegian Meteorological Institute
Andreas Dobler
Norwegian Meteorological Institute
Short Report
Keywords: Global hydrological cycle, precipitation surface area, mean precipitation intensity, extreme
precipitation
Posted Date: August 22nd, 2023
DOI: https://doi.org/10.21203/rs.3.rs-3198800/v1
License: This work is licensed under a Creative Commons Attribution 4.0 International License.
Read Full License
Additional Declarations: No competing interests reported.
Global surface area receiving daily precipitation,
wet-day frequency and probability of extreme
rainfall: Water Security and Climate Change
Rasmus E. Benestad1*, Cristian Lussana1and Andreas Dobler1†
1Research & development, The Norwegian Meteorological Institute,
Henrik Mohns plass 1, Oslo, 0313, Norway.
*Corresponding author(s). E-mail(s): rasmus.benestad@met.no;
Contributing authors: cristian.lussana@met.no;andreas.dobler@met.no;
†REB conceived and carried out the analysis while CL and AD
contributed to the interpretation and the writing.
Abstract
Both the total amount of precipitation falling on Earth’s surface and the fraction
of the surface area on which it falls represent two key global climate indicators
for Earth’s global hydrological cycle. We show that the fraction of Earth’s sur-
face area receiving daily precipitation is closely connected to the global statistics
of local wet-day frequency as well as mean precipitation intensity. Our analysis
was based on the ERA5 reanalysis which also revealed a close link between the
global mean of the mean precipitation intensity for each grid box and the total
daily precipitation falling on Earth’s surface divided by the global surface area
fraction on which it falls. The wet-day frequency and the mean precipitation
intensity are two important statistical indicators for inferring effects of climate
change on society and nature as they represent key parameters that can be used
to approximately infer the probability of heavy rainfall on local scales. We also
found a close match between the global mean temperature and both the total
planetary amount of precipitation and surface area in the ERA5 data, hinting at
a dependency between the greenhouse effect and the global hydrological cycle.
Hence, the total planetary precipitation and the daily precipitation area repre-
sent links between the global warming and extreme precipitation amounts that
traditionally have not been included in sets of essential climate indicators.
Keywords: Global hydrological cycle, precipitation surface area, mean precipitation
intensity, extreme precipitation
1
1 Introduction
Global warming caused by the strengthening of the greenhouse effect is expected to
lead to more extreme weather events [1,2], and one explanation for more extreme rain-
fall is that higher temperatures near Earth’s surface and in the air favour higher rates
of evaporation and an increase in the moisture holding capacity of the air according to
the Clausius-Clapeyron equation [3,4]. We can refer to the connection between water
vapour and temperature as the ’thermodynamic effect’ of climate on the hydrological
cycle.
Increased concentrations of greenhouse gases inhibit the transmission of longwave
radiation from the planetary surface, a branch of the energy flow from Earth’s sur-
face to the top of the atmosphere. An increased constraint in the vertical radiative
energy flow may result in an acceleration of the rate of the atmospheric overturning to
compensate for the reduced radiative energy flow with an increased latent heat flow
through deep convection [5]. This notion is based on different planetary equilibrium
states that develop over time both before and after a change in the greenhouse effect,
where the energy flux received by Earth equals its heat loss to space. Earth’s heat loss
mainly takes place near the top of the atmosphere because the atmosphere is semi-
opaque when it comes to infrared radiation [5]. Furthermore, the long-term energy flow
must be continuous within a planetary system that is in equilibrium, as a divergence
in the flow will lead to increases or decreases in the internal energy (temperature) and
changes to its spatial distribution according to the first law of thermodynamics.
The latent heat transport connects the energy flow and the circulation of H2O
(water, vapour, ice), and in a long-term steady-state global hydrological cycle, the
total mass of water evaporated (E) in a closed system, such as Earth’s climate system,
equals the total global precipitation (P) when integrated over the Earth’s surface area
(A≈4πr2
e, where re= 6371km is Earth’s radius). We use the upper-case notation
Pfor global spatial statistics and lower-case pfor local data (i.e. grid point values),
moreover the notation ⟨·⟩ stands for the spatial average, while ·is for the temporal
average. Then, we can write E=RAe dA and P=RAp dA, such that E=−Pover
a sufficiently long time period when Earth’s global hydrological cycle is in a steady
state. However, neither evaporation nor precipitation are uniform in time or space, and
evaporation takes place over a different planetary surface area (Ae) than the surface
area on which precipitation falls on a daily basis (Ap). For instance, evaporation takes
place continuously over wet surfaces such as the world oceans (cover about 70% of
Earth’s surface area), albeit with a dependence on temperature and wind. In contrast,
it doesn’t rain everywhere or all the time, and hence we expect that the intensity of
the precipitation is influenced by the fraction of Earth’s surface area receiving daily
precipitation A′
p=Ap/A.
We can express the total precipitation falling on Earth’s surface in terms of the
average precipitation and Earth’s surface area according to the expression RApdA =
⟨P⟩A, where ⟨P⟩is the spatial average of the 24-hr precipitation falling on Earth.
Since there are regions with zero 24-hr precipitation on a daily basis, it is also possible
to relate the amount of 24-hr precipitation falling on Earth to the area on which it
falls Apand its mean intensity over this area RAp dA =Ap⟨µ⟩, where ⟨µ⟩is the
2
average daily precipitation amount aggregated over the surface area Ap, or the mean
precipitation intensity for wet grid boxes in the ERA5 reanalysis estimated over space.
We also have local statistics aggregated over time, such as the local mean precip-
itation intensity µ(also referred to as the wet-day mean precipitation), the wet-day
frequency fw, and pwhich is the average of local precipitation pover time. The
subscript ptis used for a random variable at time tso that the mean precipitation
and the mean precipitation intensity are related according to p= (1/n)Pn
t=1 pt=
(nw/n)(Pn
t=1 pt)/nw=fwµ, where nwis the number of wet days, nis the total number
of days with observations, µ=Pn
t=1 pt/nw, and fw=nw/n. Hence the mean pre-
cipitation amount is the product between the wet-day frequency fwand the wet-day
mean precipitation µ.
It has been assumed that a decrease in the precipitation area Apcan explain more
extreme daily rainfall amounts and reduced frequency of wet days (fw), which result in
both more flood and droughts [6,7], but this has to the best of our knowledge not yet
been demonstrated through the analysis of observational data. The connection between
Apand rainfall amounts is expected to involve the mean precipitation intensity ⟨µ⟩as
stated above, however, it also needs to be linked to the local precipitation intensity
µ. Both precipitation intensity µand wet-day frequency fwturn out to be two key
parameters when it comes to estimating the probability of local precipitation amount
above a given threshold level x[8]:
P r(X > x) = fwexp −x
µ.(1)
If we can find a direct connection between global climate indicators A′
pand Pon the
one hand and µand fwon the other, then we can make rule-of-the-thumb generalised
statements about the combined effect of the fractional surface area with daily pre-
cipitation and the total global daily precipitation on the statistics for local extreme
rainfall. Our objective is therefore to test the assumption that a reduced fractional
area with daily rainfall A′
presults in reduced local fwand increased µin terms of
their global averages. We also briefly examine the connection between the global mean
temperature Tand A′
p.
2 Results
Figure 1shows that there is a perfect match between the global mean of the annual
wet-day frequency and the annual mean of the daily surface area fraction with precip-
itation, hence ⟨fw⟩=A′
p. This result is not unexpected since it doesn’t matter if the
aggregation is first done for the temporal and then in the spatial dimension or vice
versa. In addition, Figure 2presents a scatter plot between ⟨µ⟩and 1/A′
paggregated
on an annual basis. The distribution of data points reveals a tendency of having two
clusters, which may involve a spurious change over time due to changing input used
in data assimilation, however, they all are from the same reanalysis and are expected
to be internally consistent regardless the input data. Furthermore, both clusters seem
to indicate a connection between these two quantities. A regression analysis between
the two indicates that they are connected on a statistically significant level (p-value
3
<2×10−16). The Pearson correlation between the two was 0.86, with a 95% confi-
dence interval spanning 0.79–0.91. In other words, the correlation implies that mean
annual precipitation intensity is higher when the annual mean of the daily surface area
fraction with precipitation is smaller.
Since we also expect that ⟨µ⟩=P/Ap, we plotted ⟨µ⟩against P/Apin Figure 3. The
results indicate a strong connection between the two (correlation: 0.98), albeit with a
systematic bias with higher estimates of annual P/Apin the ERA5 data. An ordinary
linear regression analysis gave a best fit with ⟨µ⟩= 0.53+0.78P/Ap, which is not quite
consistent with the expression ⟨µ⟩=P/Ap. In this case, Phad been estimated for
both wet and dry days whereas days with less than 1 mm/day had been excluded in
the estimation of ⟨µ⟩. Hence the two quantities in this case were not exactly the same.
Figure 4shows a scatter plot with the global mean temperature Talong the x-axis
and the surface area fraction with precipitation (threshold 1 mm/day) along the y-
axis. The red dashed line shows a best-fit according to an ordinary linear regression.
The data points showed a high correlation (-0.78) and a statistically significant depen-
dency according to the regression analysis, implying that one degree global warming
is associated with reduction in the surface area with daily precipitation by -9.7 million
square km (-4.5 %/◦C).
Figure 5shows a similar analysis as Figure 4, but for the annual mean of total daily
mass of H2Ofalling on Earth’s surface. These results too indicated a strong relation-
ship between the total precipitation mass and T, and the ordinary linear regression
gave 62.3 gigatons (4 %/◦C) increase in the typical global mass of daily precipitation
for each degree global warming.
A change in global mean temperature is expected to have an effect on the precipi-
tation intensity ⟨µ⟩=P/Ap, as both Pand Apdepend on Taccording to the analysis
presented in Figures 4–5. The change in ⟨µ⟩linked to a one-degree global warming was
estimated to be 0.46 mm/day (8%/◦C) and was based on an expression for a partial
differential equation involving both Pand Ap. We estimated that 48.5% of the change
in ⟨µ⟩could be attributed to a change in Pand 51.5% to a change in Ap. In other
words, the effect of a global warming on the precipitation intensity can be explained
in terms of both changing thermodynamics as well as dynamical changes involving a
reduction in A′
pthat are about equally important.
These results suggest that both µand fwrespond to changes in the global mean
temperature through changes in Apand P. They may be representative for real-world
precipitation if the precipitation simulated by ERA5 provide a good representation
of measurements from rain gauges. A comparison between annually aggregated pre-
cipitation statistics derived from Norwegian rain gauges and corresponding quantities
derived from ERA5 data interpolated to same locations gave a correlation of 0.87 for
µand 0.80 for fw(supporting material). Furthermore, a comparison based on mul-
tivariate regression [9] also indicated a close match. A recent evaluation of ERA5
over the Alps, the Carpathians and Fennoscandia, using high-quality regional datasets
derived from dense rain-gauge data from these European sub-regions as reference, also
has shown that ERA5 agrees qualitatively well with the reference datasets and that
major mesoscale patterns in the climatology (mean, wet-day frequency, 95% quantile)
are reproduced [10]. More over, Lavers et al. evaluated the ERA5 precipitation on a
4
global scale and concluded that ERA5 precipitation is a better proxy for observed
precipitation in extra-tropical areas than in the Tropics, though visual inspection of
precipitation patterns from ERA5 and the observations broadly agrees for the extreme
events they have studied [11]. Hence, the ERA5 data appear to give a representative
picture of the annual rainfall statistics over Europe at least, and the results derived
here can be interpreted as having some validity in terms of Earth’s global hydrological
cycle.
5
3 Figures
Fig. 1 The global annual mean wet-day frequencies estimated over the days of each year for all
ERA5 gridboxes ⟨fw⟩show a perfect match with the annual mean fractional surface area with daily
precipitation, here shown as A′
p.
6
Fig. 2 The mean precipitation intensity shows a dependency on the fractional surface area with
daily precipitation, here shown as 1/A′
p. Dashed red line shows a best-fit based on ordinary linear
regression with y=−1.8+ 3.3x, with a p-value <2×10e−16 and a correlation of 0.86 (95% confidence
interval: 0.79–0.91).
Fig. 3 There is a close relationship between the global mean precipitation intensity estimated
through temporal aggregation and P /Ap, albeit with a constant bias. Dashed red line shows the
expected one-to-one relationship and an ordinary linear regression with ⟨µ⟩= 0.53 + 0.78P /Ap, with
a p-value <2×10e−16 for the slope estimate and a correlation of 0.98 (95% confidence interval:
0.97–0.99).
7
Fig. 4 There is a close connection between the fractional surface area with daily precipitation and
the global mean temperature, where the fractional area decreases with higher temperature. Dashed
red line shows a best-fit based on ordinary linear regression with Ap= 0.70 −0.02T, with a p-value
= 2 ×10e−15 and a correlation of -0.78 (95% confidence interval: -0.85 – -0.66).
Fig. 5 The global total precipitation mass is dependent on the global mean temperature, as expected
through accelerated evaporation (Clausius-Clapeyron). Dashed red line shows a best-fit based on
ordinary linear regression with P= 590.7 + 62.3T, with a p-value <2.0×10e−16 and a correlation
of 0.82 (95% confidence interval: 0.73 – 0.89).
8
4 Methods
The threshold for a ’wet day’ was in our case set to 1 mm/day, and all parameters pre-
sented here were estimated for each grid box of the ERA5 reanalysis [12,13] and used
as a basis for analysing the spatial distributions of fwand µrespectively, aggregated
on an annual basis. The data period was 1940–2022. The analysis was carried out in
the R-environment [14] and the data and method are provided as an R-markdown
script in the supplementary material [15], in addition to a PDF-file with the output
of the analysis.
We used a regression analysis to estimate best-fit coefficients for the equations
P=P0+αT +ηand Ap=A0+βT +ζwhere ηand ζare noise terms. Because
the mean precipitation intensity was expected to depend on both the precipitation
area and total precipitation according to ⟨µ⟩=⟨µ0⟩+λP/Ap, we used the analyses
presented in Figure 3(taking the bias into account) and Figure 4, as both Pand Ap
were found to be sensitive to a 1°C global warming. Hence, we explored the effect of
δT on ⟨µ⟩based on the expression:
d⟨µ⟩
dT =λd(P/Ap)
dT =λ1
Ap
dP
dT −P
A2
p
dAp
dT =λ
Apα−P β
Ap,(2)
where λ= 0.78, α=dPglob
dT = 0.122 and β=dAp
dT =−0.019 are the slope estimates
from ordinary linear repression analyses. The fractional contribution to an increase in
⟨µ⟩was estimated to be 48.5% from increased mass of H2Ofalling on the surface and
51.5% from a reduced fractional surface area on which it falls.
5 Discussion
The ERA5 reanalysis data reveals a link between the global surface area receiving daily
precipitation Apand the wet-day frequency ⟨fw⟩as well as the mean precipitation
intensity ⟨µ⟩. Additional analysis of the data (supporting material) also suggest that
the fraction of surface area with heavy precipitation (exceeding 20 mm/day and 50
mm/day) also has increased between 1940 and 2020 in a consistent way with the
results obtained for ⟨fw⟩and ⟨µ⟩.
With the established relationships between the global hydro-climatological indica-
tors Pand Apand local precipitation statistics as well as the global mean temperature
T, we can provide a crude rule-of-the thumb estimate for the effect of 1°C global
warming δT on µ,fwand the changes in the probability of heavy rainfall P r(X > x)
in terms of a change in the fractional surface area receiving daily precipitation or
a change in the total mass of H2Ofalling on Earth’s surface. For example, a back-
of-the-envelope calculation of the probability of receiving more than 30 mm/day for
an ‘average’ situation, based on Eq. 1, estimates that a one-degree global warming
changes P r(X > 30mm/day) from 0.25 to 0.32%. These results also reveal a connec-
tion between space and time between A′
pand ⟨fw⟩, and our results suggest that P/Ap
corresponds well with the global mean of local ⟨µ⟩aggregated over time. Hence time
and space appear to be linked when it comes to daily precipitation statistics.
Data from the Tropical Rain Measurement mission (TRMM) suggest that the
typical daily precipitation area Apbetween 50°S and 50°N decreased over the period
9
1998–2016 [6], and a similar analysis on a global basis based on ERA5 reanalysis
data also found a downward trend in the surface area receiving daily precipitation [7].
Furthermore, the analysis of the ERA5 data suggested that the reduction in surface
area receiving 24-hr precipitation mainly took place within the 50°S and 50°N latitude
band [7]. There is a caveat with satellite and reanalysis data that potentially may
lead to misleading long-term trends due to the introduction of new observational
inputs over time, e.g. connected with the launch of new satellite missions. Dobler et
al. (in progress) analysed the daily precipitation from coupled global climate models
(GCMs) from CMIP5 and CMIP6 [16,17] and found that they too simulate downward
trends in Ap, albeit with weaker amplitude than seen in the observations and ERA5
reanalysis. This finding may suggest that there is a physical basis for the reduction
in the precipitation area that the GCMs are able to reproduce. According to the said
study, there were few exceptions when it came to a decrease in the rainfall area, both
for the global area and the area between 50°S and 50°N, except for one particular
outlier (FGOALS G3). Thus, there are multiple lines of evidence indicating that the
surface area with daily precipitation has been shrinking over time while the total
precipitation amount has increased [7].
Both Ptand Apturn out to be important global climate indicators, but are not
traditionally included in the global set of indicators proposed by either Copernicus
Climate Change Service or the World Meteorological Organisation (WMO) [7]. Both
are straight-forward to compute when we have reanalyses such as ERA5, and the
demonstration provided here suggest they are linked to extreme rainfall amounts that
may produce floods. Furthermore, both fwand Apinfluence the degree of dryness and
aridness, and may potentially be linked to droughts although this aspect is beyond the
scope of present analysis. There are also some indications that µand fwmay be linked
to sub-daily amounts, e.g. through intensity-duration-frequency (IDF) curves [18].
The Clausius-Clapeyron equation predicts a 7% increase in water vapour for a
+1◦C increase in temperature, however, the regression analysis herein gave a slope
estimate of 4% for the increase in total precipitation amount due to a +1◦C global
warming. According to the sixth assessment report of the IPCC [2], increases in global
mean precipitation are a robust response to global surface temperature that very
likely is within the range of 2–3%/°C. The discrepancy between the estimates from
our regression analysis and those published previously may indicate a presence of
inhomogeneities in ERA5, such as those reported by Lavers et al. [11], that also may
explain the two clusters of data points seen in Figure 2. Hence, there is a caveat
with these results that may be related to the potentially spurious step changes in Ap
between 1980 and 1990 and after 2015 (Dobler et al., in prep). This caveat may affect
the spread in Appresented here and its dependency on the global mean temperature.
An accelerated atmospheric overturning, which we can brand as a ’dynamic effect’
of a global warming on the hydrological cycle, implies changes in the cloud struc-
ture and rainfall patterns as well as changes in Ap. This notion is supported by the
observations of higher cloud tops [19]. Higher cloud tops with a deeper vertical cloud
structure also favour more intense rainfall since the raindrops fall over a longer vertical
distance where they can collect moisture and smaller cloud drops [20]. There have also
been suggested other explanations for changes in extreme rainfall amounts, as Ombadi
10
et al. proposed that the increase in rainfall extremes in high-elevation regions of the
Northern Hemisphere has increased by 15% per degree Celsius of warming, the double
the rate expected from increases in atmospheric water vapour, which they attributed
to is due to a warming-induced shift from snow to rain [21]. Another factor may be a
slow-down in storm movement, which means that the same spot receives more of the
precipitation that otherwise would have been spread over a larger area [22]. None of
these mechanisms exclude each other and all may be valid in different settings and
locations.
6 Conclusion
We demonstrate that both the daily total global amount of precipitation and the
fractional of Earth’s surface area on which it falls contribute to the mean precipitation
intensity µand frequency fw, based on the ERA5 reanalysis. These precipitation
statistics are connected to the probability of heavy daily rainfall amounts, and our
results highlight the importance of including both the daily total global amount of
precipitation and the fractional of Earth’s surface area in the list of essential global
climate indicators.
Supplementary information. The methods are described in detail through the
provided R-markdown script, the PDF with its output and extract of data (in R-
binary) necessary to reproduce the results.
Acknowledgments. ERA5. Frost.
•Funding: The Norwegian Meteorological Institute
•Conflict of interest/Competing interests (check journal-specific guidelines for which
heading to use) None.
•Ethics approval. NA
•Consent to participate. NA
•Consent for publication.
•Availability of data and materials. See supporting material and FigShare (https:
//doi.org/10.6084/m9.figshare.23735619.v1)
•Code availability. See supporting material and FigShare [15]
•Authors’ contributions. REB carried out the analysis. CL and AD helped writing
and reviewing the manuscript.
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