# Self-Organized Criticality of Rainfall in Central China

Authors: Zhiliang Wang; Chunyan Huang
Journal: Advances in Meteorology
 (2012)

---

## Abstract

Rainfall is a complexity dynamics process. In this paper, our objective is to find the evidence of self-organized criticality (SOC) for rain datasets in China by employing the theory and method of SOC. For this reason, we analyzed the long-term rain records of five meteorological stations in Henan, a central province of China. Three concepts, that is, rain duration, drought duration, accumulated rain amount, are proposed to characterize these rain events processes. We investigate their dynamics property by using scale invariant and found that the long-term rain processes in central China indeed exhibit the feature of self-organized criticality. The proposed theory and method may be suitable to analyze other datasets from different climate zones in China.


---

## Body

## 1. Introduction

China is not only a big country for its population but also a big agriculture one. Rain is the main source of irrigation water, and it plays a key role in the crop growing period. No rain will cause drought while storm may cause flood. To keep sufficient agriculture production sustainable, it is necessary to identify the role of the rain clearly and to understand the characteristics of the rain deeply. In particular, analyzing the rain in central China is more important because this region is the main crop source and the population density is very high.Rainis liquidprecipitation, as opposed to nonliquid kinds of precipitation such assnow andhail and so on. Rainfall is the result of the atmosphere movement, which is influenced by sun radiation, sea water evaporation, and earth rotation. In the fact, the long-term rain record is a time series which can be regarded as a random process. The rainfall process is actually a complexity system because there are too many influencing factors.In previous studies, many mathematical methods have been applied to find the rainfall pattern, such as periodic, trend, change point, and fractal. Based on the last 1033 years historic data set, Jiang analyzed the temporal and spatial climate variability by using a “Mexican hat” wavelet transform [1]. Bordi used Standardized Precipitation Index (SPI) to assess the climatic condition of this region and applied principal component to capture the pattern of co-variability of the index at different gauge stations [2]. The results suggest that the northern part of east-central China is experiencing dry conditions more frequently from the 1970s onwards indicated by a negative trend in the SPI time series. Applying the binary cubic interpolation and optimal fitting method, Wang et al. set up a statistical model [3] and Yu used the application of gray and fuzzy methods [4], to make the rain forecast. Appling chaos dynamics theory on rainfall, Rodriguez-Iturbe et al. [5] and Wang et al. [6, 7] found that both the characteristics of the correlation integral and the Lyapunov exponents of the historical data give preliminary support to the presence of chaotic dynamics with a strange attractor. Using the correlation dimension method, the inverse approach of the nonlinear prediction method, and the method of surrogate data, Siva Kumar found that the rainfall data exhibit nonlinear behavior and possibly low-dimensional chaos, which imply that short-term prediction based on nonlinear dynamics might be possible [8, 9]. Ramirez used a feedforward neural network and resilient propagation learning algorithm to analyze the relation between the rain data and potential temperature, vertical component of the wind, specific humidity, air temperature, precipitable water, relative vorticity, and moisture divergence flux [10].All the previous works are based on an assumption that the long-term rain process is stochastic or chaotic one. Nevertheless, the rain process is located at the brink between the chaos and determination. It is necessary to develop the new theory and methodology to address this context.In recent years, a new perspective, which is called self-organized criticality (SOC), attracted applied mathematicians, meteorologists, climatologists, and environmentalists. Self-organized criticality is proposed by Bak et al. [11–15]. The term self-organized criticality refers to the tendency of many systems driven by an energy input at a slow and constant rate to enter states characterized by scale-free behavior. The statistics of the systems then resemble those of equilibrium systems near the critical point of a phase transition. Self-organized criticality is one of a number of important discoveries made in statistical physics and related fields over the latter half of the 20th century, discoveries which relate particularly to the study of complexity in nature. The most classical instances of SOC include the common natural phenomena, such as earthquakes and avalanches [16–19]. A rainfall event can be considered as an earthquake-like or an avalanche-like event [20, 21]. Further more, a long-term rain event series can be also seen as a similar event which is a complexity dynamics process and exhibits the feature of self-organized criticality.Andrade analyzed long-term daily rain records of weather stations around the world with a special emphasis on the semiarid regions and found that there existed some evidences of SOC with these data [22]. Peters et al. investigated the European rain and fund it exhibits the feature of SOC [23–25]. However, up to now, we have not seen any report which related the China rain to SOC in the literature. In this work, we chose five meteorological stations in Henan, a central province of China, to try to find out the SOC evidence.First of all we assume that the rainfall events that occurred in this region follow the power law distribution. And then based on the theory and method, we look for the SOC evidence through our calculating and analyzing. It is our aim for us to confirm the existence of SOC.Henan is a leading province in grain, wheat, and oil seed output, and it is also an important producer of beef, cotton, pork, animal oil, and corn. With a population of approximately 93.6 million, Henan is the second most populous Chinese province after Guangdong. In this sense Henan is the big agriculture and population province in China. Precipitation, especially rain, has a dramatic effect on agriculture. All plants need at least some water to survive; therefore rain is important to agriculture. A regular rain pattern is usually vital to healthy plants; too much or too little rainfall can be harmful, even devastating to crops. Drought can kill crops and increase erosion while overly wet weather can cause harmful fungus growth. So studying the characteristics of the rainfall event is important to understand the dry and wet spell. It is consequently helpful to local flood and drought management.


## 2. Data Sets

We downloaded the data sets from China Meteorological Data Sharing Service System. These stations’ geographic positions are displayed in Figure1 in which Xinyang and Zhumadian are in the northern of Henan, Anyang and Zhengzhou in the northern part, Lushi in the western region. From the point of view of meteorological classification, the former two stations belong to humid subtropical climate zone while the later three stations are in the temperate climate zone.Figure 1
Distribution of stations, named Anyang, Zhengzhou, Lushi, Zhumadian, and Xinyang.The site name, site number, operation period and location are listed in Table1. Station number is the general international code which is the WMO number. Obliviously the operation period is not equal in different stations. The format of degree, minute, and second is used to represent the location of the gauge station.Table 1
Observation sites with station number, corresponding time periods estimated, and location.
Site name
Station number
Operation period
Location
Zhengzhou
57083
195101–200712
34° 43′N
113° 39′E
Anyang
53898
195102–200712
36° 07′N
114° 22′E
Zhumadian
57290
195801–200712
33° 00′N
114° 01′E
Xinyang
57297
195101–200712
32° 08′N
114° 03′E
Lushi
57067
195207–200712
34° 03′N
111° 02′E
Longitude and latitude of the location is represented as degree and minute.The original data sets should have contained the daily rain, but in some cases missing data may occur. Therefore these need to be pretreated by the method of interpolation before further analysis. After data pretreatment, we plot the five series in the Figure2 whose horizontal axis represents the time of the rainfall day and vertical axis displays the daily rain amount (0.1 mm).Figure 2
Plot of five rain series in Henan province.


## 3. Method of Analysis

### 3.1. Scale Invariant

One of the great successes of physics in the last decades has been in the understanding of phenomena with fluctuations over many scales. In high-energy physics, critical phenomena and hydrodynamics it is often possible to establish the existence of a scaling or scale invariant regime in which the fluctuation (Δx) in the field of interest (x) at small scaleΔt and at large scaleλΔt(λ>1)is amplified by the factorλ-τ, whereτis the scaling parameter. This may be written more concisely as
(1)Δx(λΔt)=λ-τΔx(Δt),
whereΔt=t1-t0,Δx(Δt)=x(t1)-x(t0), t2=t0+λ(t1-t0), Δx(λΔt)=x(t2)-x(t0), and equality is understood in the sense of probability distributions, that is,F(X)=F(Y)if   Pr(X>c)=Pr(Y>c)for all c, Pr means Probability.In Section1, we said that our work objective is to find the evidence of the SOC of rain process in central china. To look it out, we need construct mathematical models to mine the relations between the SOC and the long-term rain records. In the literatures on the SOC, power law distribution (scaling or scale invariance) is often used as a tool to prove a complexity dynamic process with the feature of SOC. To reveal the existing of SOC in the rain process, in the beginning we define a rain event as a sequence of consecutive nonzero measurements of the rain rate. Consequently, we use 3 variables, which contain rain duration, drought durations and accumulated rain amount to describe the feature of rain event in the region.The probability density function of a physical variableX (such as rain duration,drought duration and accumulated rain amount in a rainfall event) is denoted by n(x), which accumulated probability distribution (PDF) isN~(x). PDF can be defined as
(2)N~(x)=∫xxMn(x)dx,
where xM is the maximum value in the data set. By using the integrated description instead of histograms we avoid data fluctuations in the low (high) value regime induced by the choice of logarithmic. Ifn(x)∝x-τ, xM→∞and τ→1, thenN~(x)∝x-τ+1. Our rainfall data are generally confined to ranges 1<I<200daysand0.1mm<x<400mm, while higher values are observed only in extreme situations. Therefore, we cannot replaceMby∞in (2) and obtain
(3)N(x)=N~(x)x∝1xτ[1-(xxM)τ-1].
Thus, the log-log plot of N(x) versusxdefinitely departs from a straight line asx→xM.


### 3.2. Algorithm

Based on the principle of scale invariant, we develop 2 algorithms corresponding to rain duration, drought duration and accumulated rain amount, in which first algorithm is about the former two items and the second is about the accumulated rain amount.


#### 3.2.1. Procedure for Rain Duration and Drought Duration

Rain duration is the life time of a successive rainfall, and drought duration is the waiting time between two rain events. Let RD and DD stand for rain duration and drought, respectively. Their distribution densities are discrete which are listed as shown in Table2.Table 2
Distribution model of rain duration and drought duration.
RD/DD
x
1
x
2
x
M
⋯
n
(
x
)
n
(
x
1
)
n
(
x
2
)
⋯
n
(
x
M
)
N
~
(
x
)
∑
i
=
1
M
n
(
x
i
)
∑
i
=
2
M
n
(
x
i
)
⋯
∑
i
=
M
M
n
(
x
i
)
N
(
x
)
(
∑
i
=
1
M
n
(
x
i
)
)
/
x
1
(
∑
i
=
2
M
n
(
x
i
)
)
/
x
2
⋯
∑
i
=
M
M
n
(
x
i
)
/
x
M
x
i stands for the length of every rain or drought event. M is the longest time period.To test the scaling of rain duration and drought duration in central China, analysis can be conducted by the following procedure.Algorithm 1 (Accumulated probability distribution computation and plot of rain duration and drought duration).
We have the following:Step 1.
Wash and treat the original data.Step 2.
Plot the daily rain size on the rain date (see Figure2).Step 3.
Count the numbers of every kind of rain events and assignn(x)the number listed in the second row of Table 2.Step 4.
Calculate the accumulated numberN~(x)=∑i=jMn(xi), j=1,2,…,Mand list it in the third row.Step 5.
Calculate the number densityN(x)=(∑i=jMn(xj))/xj of rain event corresponding to the fourth row and list it in the fourth row.Step 6.
According to formula (3), we take the logarithm of the first and fourth row. By the linear regression technology, we will obtainτ.Step 7.
Make the double-log distribution plot.


#### 3.2.2. Procedure for Accumulated Rain Amount

In the present paper, the accumulated rain amount, which is symbolized as ARA, represents the total rain amount in a single rain event. Then, based on Step1 and Step 2 in Algorithm 1, the procedure of ARA analysis can be followed by Algorithm 2  below.Algorithm 2 (Accumulated probability distribution computation and plot of ARA).
We have the following:Step 1.
By calculating the ARA we get a new ARA seriesy1,y2,…,yN, where N stands for the number of the rainfall event in the long-term record.Step 2.
Letymin=min(y1,y2,…,yN), ymax=max(y1,y2,…yN), and partition the closed interval [ymin,ymax] into subinterval[y,y+Δy]. Here,Δy can be computed by using the following equation:
(4)y+Δyy=101/5.Step 3.
IfSis the number of all the subintervals, we calculate the times of the occurrences of y1,y2,…yNin every interval and note themn(s), s=1,2,…,S.Step 4.
Calculate the accumulated numberN~(s)=∑s=jSn(s), j=1,2,…S.Step 5.
Calculate the number densityN(s)=(∑s=jSn(s))/Δsj.Step 6.
Take the logarithm ofN(s) and left end point of[y,y+Δy]. By the linear regression technology, we will obtain τin terms of Formula (3).Step 7.
Make the double-log distribution plot.


## 3.1. Scale Invariant

One of the great successes of physics in the last decades has been in the understanding of phenomena with fluctuations over many scales. In high-energy physics, critical phenomena and hydrodynamics it is often possible to establish the existence of a scaling or scale invariant regime in which the fluctuation (Δx) in the field of interest (x) at small scaleΔt and at large scaleλΔt(λ>1)is amplified by the factorλ-τ, whereτis the scaling parameter. This may be written more concisely as
(1)Δx(λΔt)=λ-τΔx(Δt),
whereΔt=t1-t0,Δx(Δt)=x(t1)-x(t0), t2=t0+λ(t1-t0), Δx(λΔt)=x(t2)-x(t0), and equality is understood in the sense of probability distributions, that is,F(X)=F(Y)if   Pr(X>c)=Pr(Y>c)for all c, Pr means Probability.In Section1, we said that our work objective is to find the evidence of the SOC of rain process in central china. To look it out, we need construct mathematical models to mine the relations between the SOC and the long-term rain records. In the literatures on the SOC, power law distribution (scaling or scale invariance) is often used as a tool to prove a complexity dynamic process with the feature of SOC. To reveal the existing of SOC in the rain process, in the beginning we define a rain event as a sequence of consecutive nonzero measurements of the rain rate. Consequently, we use 3 variables, which contain rain duration, drought durations and accumulated rain amount to describe the feature of rain event in the region.The probability density function of a physical variableX (such as rain duration,drought duration and accumulated rain amount in a rainfall event) is denoted by n(x), which accumulated probability distribution (PDF) isN~(x). PDF can be defined as
(2)N~(x)=∫xxMn(x)dx,
where xM is the maximum value in the data set. By using the integrated description instead of histograms we avoid data fluctuations in the low (high) value regime induced by the choice of logarithmic. Ifn(x)∝x-τ, xM→∞and τ→1, thenN~(x)∝x-τ+1. Our rainfall data are generally confined to ranges 1<I<200daysand0.1mm<x<400mm, while higher values are observed only in extreme situations. Therefore, we cannot replaceMby∞in (2) and obtain
(3)N(x)=N~(x)x∝1xτ[1-(xxM)τ-1].
Thus, the log-log plot of N(x) versusxdefinitely departs from a straight line asx→xM.


## 3.2. Algorithm

Based on the principle of scale invariant, we develop 2 algorithms corresponding to rain duration, drought duration and accumulated rain amount, in which first algorithm is about the former two items and the second is about the accumulated rain amount.


### 3.2.1. Procedure for Rain Duration and Drought Duration

Rain duration is the life time of a successive rainfall, and drought duration is the waiting time between two rain events. Let RD and DD stand for rain duration and drought, respectively. Their distribution densities are discrete which are listed as shown in Table2.Table 2
Distribution model of rain duration and drought duration.
RD/DD
x
1
x
2
x
M
⋯
n
(
x
)
n
(
x
1
)
n
(
x
2
)
⋯
n
(
x
M
)
N
~
(
x
)
∑
i
=
1
M
n
(
x
i
)
∑
i
=
2
M
n
(
x
i
)
⋯
∑
i
=
M
M
n
(
x
i
)
N
(
x
)
(
∑
i
=
1
M
n
(
x
i
)
)
/
x
1
(
∑
i
=
2
M
n
(
x
i
)
)
/
x
2
⋯
∑
i
=
M
M
n
(
x
i
)
/
x
M
x
i stands for the length of every rain or drought event. M is the longest time period.To test the scaling of rain duration and drought duration in central China, analysis can be conducted by the following procedure.Algorithm 1 (Accumulated probability distribution computation and plot of rain duration and drought duration).
We have the following:Step 1.
Wash and treat the original data.Step 2.
Plot the daily rain size on the rain date (see Figure2).Step 3.
Count the numbers of every kind of rain events and assignn(x)the number listed in the second row of Table 2.Step 4.
Calculate the accumulated numberN~(x)=∑i=jMn(xi), j=1,2,…,Mand list it in the third row.Step 5.
Calculate the number densityN(x)=(∑i=jMn(xj))/xj of rain event corresponding to the fourth row and list it in the fourth row.Step 6.
According to formula (3), we take the logarithm of the first and fourth row. By the linear regression technology, we will obtainτ.Step 7.
Make the double-log distribution plot.


### 3.2.2. Procedure for Accumulated Rain Amount

In the present paper, the accumulated rain amount, which is symbolized as ARA, represents the total rain amount in a single rain event. Then, based on Step1 and Step 2 in Algorithm 1, the procedure of ARA analysis can be followed by Algorithm 2  below.Algorithm 2 (Accumulated probability distribution computation and plot of ARA).
We have the following:Step 1.
By calculating the ARA we get a new ARA seriesy1,y2,…,yN, where N stands for the number of the rainfall event in the long-term record.Step 2.
Letymin=min(y1,y2,…,yN), ymax=max(y1,y2,…yN), and partition the closed interval [ymin,ymax] into subinterval[y,y+Δy]. Here,Δy can be computed by using the following equation:
(4)y+Δyy=101/5.Step 3.
IfSis the number of all the subintervals, we calculate the times of the occurrences of y1,y2,…yNin every interval and note themn(s), s=1,2,…,S.Step 4.
Calculate the accumulated numberN~(s)=∑s=jSn(s), j=1,2,…S.Step 5.
Calculate the number densityN(s)=(∑s=jSn(s))/Δsj.Step 6.
Take the logarithm ofN(s) and left end point of[y,y+Δy]. By the linear regression technology, we will obtain τin terms of Formula (3).Step 7.
Make the double-log distribution plot.


## 3.2.1. Procedure for Rain Duration and Drought Duration

Rain duration is the life time of a successive rainfall, and drought duration is the waiting time between two rain events. Let RD and DD stand for rain duration and drought, respectively. Their distribution densities are discrete which are listed as shown in Table2.Table 2
Distribution model of rain duration and drought duration.
RD/DD
x
1
x
2
x
M
⋯
n
(
x
)
n
(
x
1
)
n
(
x
2
)
⋯
n
(
x
M
)
N
~
(
x
)
∑
i
=
1
M
n
(
x
i
)
∑
i
=
2
M
n
(
x
i
)
⋯
∑
i
=
M
M
n
(
x
i
)
N
(
x
)
(
∑
i
=
1
M
n
(
x
i
)
)
/
x
1
(
∑
i
=
2
M
n
(
x
i
)
)
/
x
2
⋯
∑
i
=
M
M
n
(
x
i
)
/
x
M
x
i stands for the length of every rain or drought event. M is the longest time period.To test the scaling of rain duration and drought duration in central China, analysis can be conducted by the following procedure.Algorithm 1 (Accumulated probability distribution computation and plot of rain duration and drought duration).
We have the following:Step 1.
Wash and treat the original data.Step 2.
Plot the daily rain size on the rain date (see Figure2).Step 3.
Count the numbers of every kind of rain events and assignn(x)the number listed in the second row of Table 2.Step 4.
Calculate the accumulated numberN~(x)=∑i=jMn(xi), j=1,2,…,Mand list it in the third row.Step 5.
Calculate the number densityN(x)=(∑i=jMn(xj))/xj of rain event corresponding to the fourth row and list it in the fourth row.Step 6.
According to formula (3), we take the logarithm of the first and fourth row. By the linear regression technology, we will obtainτ.Step 7.
Make the double-log distribution plot.


## 3.2.2. Procedure for Accumulated Rain Amount

In the present paper, the accumulated rain amount, which is symbolized as ARA, represents the total rain amount in a single rain event. Then, based on Step1 and Step 2 in Algorithm 1, the procedure of ARA analysis can be followed by Algorithm 2  below.Algorithm 2 (Accumulated probability distribution computation and plot of ARA).
We have the following:Step 1.
By calculating the ARA we get a new ARA seriesy1,y2,…,yN, where N stands for the number of the rainfall event in the long-term record.Step 2.
Letymin=min(y1,y2,…,yN), ymax=max(y1,y2,…yN), and partition the closed interval [ymin,ymax] into subinterval[y,y+Δy]. Here,Δy can be computed by using the following equation:
(4)y+Δyy=101/5.Step 3.
IfSis the number of all the subintervals, we calculate the times of the occurrences of y1,y2,…yNin every interval and note themn(s), s=1,2,…,S.Step 4.
Calculate the accumulated numberN~(s)=∑s=jSn(s), j=1,2,…S.Step 5.
Calculate the number densityN(s)=(∑s=jSn(s))/Δsj.Step 6.
Take the logarithm ofN(s) and left end point of[y,y+Δy]. By the linear regression technology, we will obtain τin terms of Formula (3).Step 7.
Make the double-log distribution plot.


## 4. Result and Discussion

According to the steps of Algorithm1, the long-term rains are shown in Figure 2. Xinyang station and Zhumadian station, which are located in the southern part of Henan Province, are plotted above. Contrast Anyang station, Zhenzhou station, and Lushi station, which are located in the northern part of this province.


### 4.1. Rain Duration and Drought Duration

Corresponding to Table2, the calculation results of rain duration are listed in Table 3. For the dataset is too large, here we only list out Zhenzhou station’s data.Table 3
Days of RD and DD; ARA (in 0.1 mm) per rain event.
RD
N (RD)
DD
N (DD)
DD
N (DD)
S.N
ARA
S.N
ARA
S.N
ARA
1
4309
1
14586
37
44
1
48
37
92
⋯
⋯
2
2427
2
11564
38
35
2
92
38
42
2501
15
3
1418
3
9553
39
27
3
7
39
18
2502
60
4
812
4
7615
40
21
4
61
40
55
2503
1
5
463
5
6205
41
17
5
39
41
14
2504
212
6
257
6
5084
42
13
6
8
42
20
2505
26
7
141
7
4179
43
11
7
6
43
88
2506
80
8
78
8
3448
44
9
8
3
44
1
2507
9
9
42
9
2868
45
8
9
42
45
320
2508
66
10
26
10
2425
46
7
10
35
46
95
2509
297
11
17
11
2072
47
6
11
75
47
56
2510
60
12
10
12
1783
48
5
12
58
48
296
2511
8
13
6
13
1542
49
4
13
780
49
68
2512
10
14
4
14
1333
50
3
14
85
50
38
2513
18
15
2
15
1161
51
2
15
7
51
93
2514
53
16
1
16
1105
52
1
16
278
52
16
2515
51
17
0
17
879
53
0
17
44
53
25
2516
223
18
0
18
762
54
0
18
468
54
121
2517
299
19
0
19
660
55
0
19
18
55
1
2518
66
20
0
20
566
56
0
20
555
56
653
2519
437
21
0
21
486
57
0
21
50
57
43
2520
213
22
0
22
421
58
0
22
59
58
30
2521
104
23
0
23
365
59
0
23
34
59
29
2522
11
24
0
24
317
60
0
24
21
60
79
2523
605
25
0
25
278
61
0
25
213
61
52
2524
381
26
0
26
241
62
0
26
225
62
237
2525
1561
27
0
27
211
63
0
27
13
63
1
2526
237
28
0
28
187
64
0
28
96
64
21
2527
355
29
0
29
164
65
0
29
11
65
8
2528
125
30
0
30
143
66
0
30
2
66
384
2529
267
31
0
31
125
67
0
31
812
67
47
2530
398
32
0
32
109
68
0
32
198
68
7
2531
422
33
0
33
95
69
0
33
7
69
15
2532
25
34
0
34
81
70
0
34
146
70
111
2533
0
35
0
35
68
71
0
35
189
71
71
2534
0
36
0
36
56
72
0
36
8
72
76
2535
0The computation results about rain duration and drought duration show that the rainfall times and the rain duration, the drought days and the drought duration exhibit the power law relation. Taking the logarithm value of number density as the vertical axis and the logarithm value of the rain duration as horizontal axis, we find that the distribution of the rain duration number in Xinyang station, Zhumadian station, Zhengzhou station, and Lushi station is similar with their function profile (see Figure3) while Anyang’s number density of rain duration exhibits the different feature. The scale-free region ranges from 1 to 10 in Anyang and from 1 to 11 in other places. Here, τanyang=-1.35, other τs fluctuate around −1.68. Influenced by the monsoon climate, the rain season begins in the early June and ends in the lately September. In fact, the Anyang is this region where the rain duration is the shortest in Henan province.Figure 3
Log-log plot of the probability number density for rain duration.Looking at Figure3 and Figure 4, it seems that the number density of drought duration displays the same distribution characteristics with the rain durations. Actually, the number density of rain duration is different in some aspects, such as the maximum time period and scaleless region. Furthermore, attention should be paid to the number density of drought duration in Zhengzhou and Anyang (see Figure 5). The scale region of Anyang station ranges from 1 day to 100 days, and Zhengzhou’s scaling left end point closes to 100 but Xingyang’s and Zhumadian’s are about 50.Figure 4
Log-log plot of the probability number density for drought duration.Figure 5
Log-log plot of the probability number density for accumulated rain amount.Here,τxinyang,τxinyang≈-1.71,and otherτis about −1.5.


### 4.2. Accumulated Rain Amount

Comparing with the analysis on rain duration and drought duration, it is more difficult to understand the behavior characteristics of long-term ARA dynamics process. To test if the ARA is scale invariant, we calculated the ARA number density in the partitioned intervals in terms of Algorithm2.The results show that all five series follow the power law relation in the scale range from 0.1 mm to 60 mm. Hereτxingyang=-1.61,  τzhumadian=-1.63,τanyang=-1.72,τzhengzhou=-1.52,τlushi=-1.55. The ARA number density distributions of Xinyang and Zhumadian displays the similar profile since they are in the same climate classification region, the subtropical zone. Like Section 4.1, the feature of ARA in Anyang station is also different from the other stations. Although Lushi station is located in the western mountain region, however, the ARA behavior characteristic in Zhengzhou station is similar to it.


### 4.3. Discussion

In Sections4.1 and 4.2 we have investigated the long-term series of precipitation records and succeeded in finding the evidence for power law distribution even though the five stations are located in different position in central China. This implies that our guess (i.e., the rainfall in central China takes on the feature of SOC) is right. This also supports the view that atmospheric dynamics is governed, at least in part, by SOC.As described in the introduction, the concept of SOC refers to the state of nonequilibrium systems driven by slow constant energy input to organize themselves into critical systems. The intermediately stored energy is eventually released in sudden bursts with no typical scale. From this point of view, rainfall events which occur in the central China are not very different from those ones in the other place. Like earthquakes, a rainfall event is driven by a slow and constant energy input from the sun and water is evaporated from the Western Pacific Ocean. The energy is stored in the form of water vapor in the atmosphere. It is then suddenly released in bursts when the vapors condense to water drops. The power-law distribution of the number density of rain events is equivalent to the Gutenberg-Richter law for earthquakes.Although there are different geographic coordinates at the five stations, theseτs show little change because the five stations are neighborhoods in one province after all. Except for the topography and landform the driving factors of rainfall, such as the sun radiation and the pattern of atmospheric circulation are nearly the same.


## 4.1. Rain Duration and Drought Duration

Corresponding to Table2, the calculation results of rain duration are listed in Table 3. For the dataset is too large, here we only list out Zhenzhou station’s data.Table 3
Days of RD and DD; ARA (in 0.1 mm) per rain event.
RD
N (RD)
DD
N (DD)
DD
N (DD)
S.N
ARA
S.N
ARA
S.N
ARA
1
4309
1
14586
37
44
1
48
37
92
⋯
⋯
2
2427
2
11564
38
35
2
92
38
42
2501
15
3
1418
3
9553
39
27
3
7
39
18
2502
60
4
812
4
7615
40
21
4
61
40
55
2503
1
5
463
5
6205
41
17
5
39
41
14
2504
212
6
257
6
5084
42
13
6
8
42
20
2505
26
7
141
7
4179
43
11
7
6
43
88
2506
80
8
78
8
3448
44
9
8
3
44
1
2507
9
9
42
9
2868
45
8
9
42
45
320
2508
66
10
26
10
2425
46
7
10
35
46
95
2509
297
11
17
11
2072
47
6
11
75
47
56
2510
60
12
10
12
1783
48
5
12
58
48
296
2511
8
13
6
13
1542
49
4
13
780
49
68
2512
10
14
4
14
1333
50
3
14
85
50
38
2513
18
15
2
15
1161
51
2
15
7
51
93
2514
53
16
1
16
1105
52
1
16
278
52
16
2515
51
17
0
17
879
53
0
17
44
53
25
2516
223
18
0
18
762
54
0
18
468
54
121
2517
299
19
0
19
660
55
0
19
18
55
1
2518
66
20
0
20
566
56
0
20
555
56
653
2519
437
21
0
21
486
57
0
21
50
57
43
2520
213
22
0
22
421
58
0
22
59
58
30
2521
104
23
0
23
365
59
0
23
34
59
29
2522
11
24
0
24
317
60
0
24
21
60
79
2523
605
25
0
25
278
61
0
25
213
61
52
2524
381
26
0
26
241
62
0
26
225
62
237
2525
1561
27
0
27
211
63
0
27
13
63
1
2526
237
28
0
28
187
64
0
28
96
64
21
2527
355
29
0
29
164
65
0
29
11
65
8
2528
125
30
0
30
143
66
0
30
2
66
384
2529
267
31
0
31
125
67
0
31
812
67
47
2530
398
32
0
32
109
68
0
32
198
68
7
2531
422
33
0
33
95
69
0
33
7
69
15
2532
25
34
0
34
81
70
0
34
146
70
111
2533
0
35
0
35
68
71
0
35
189
71
71
2534
0
36
0
36
56
72
0
36
8
72
76
2535
0The computation results about rain duration and drought duration show that the rainfall times and the rain duration, the drought days and the drought duration exhibit the power law relation. Taking the logarithm value of number density as the vertical axis and the logarithm value of the rain duration as horizontal axis, we find that the distribution of the rain duration number in Xinyang station, Zhumadian station, Zhengzhou station, and Lushi station is similar with their function profile (see Figure3) while Anyang’s number density of rain duration exhibits the different feature. The scale-free region ranges from 1 to 10 in Anyang and from 1 to 11 in other places. Here, τanyang=-1.35, other τs fluctuate around −1.68. Influenced by the monsoon climate, the rain season begins in the early June and ends in the lately September. In fact, the Anyang is this region where the rain duration is the shortest in Henan province.Figure 3
Log-log plot of the probability number density for rain duration.Looking at Figure3 and Figure 4, it seems that the number density of drought duration displays the same distribution characteristics with the rain durations. Actually, the number density of rain duration is different in some aspects, such as the maximum time period and scaleless region. Furthermore, attention should be paid to the number density of drought duration in Zhengzhou and Anyang (see Figure 5). The scale region of Anyang station ranges from 1 day to 100 days, and Zhengzhou’s scaling left end point closes to 100 but Xingyang’s and Zhumadian’s are about 50.Figure 4
Log-log plot of the probability number density for drought duration.Figure 5
Log-log plot of the probability number density for accumulated rain amount.Here,τxinyang,τxinyang≈-1.71,and otherτis about −1.5.


## 4.2. Accumulated Rain Amount

Comparing with the analysis on rain duration and drought duration, it is more difficult to understand the behavior characteristics of long-term ARA dynamics process. To test if the ARA is scale invariant, we calculated the ARA number density in the partitioned intervals in terms of Algorithm2.The results show that all five series follow the power law relation in the scale range from 0.1 mm to 60 mm. Hereτxingyang=-1.61,  τzhumadian=-1.63,τanyang=-1.72,τzhengzhou=-1.52,τlushi=-1.55. The ARA number density distributions of Xinyang and Zhumadian displays the similar profile since they are in the same climate classification region, the subtropical zone. Like Section 4.1, the feature of ARA in Anyang station is also different from the other stations. Although Lushi station is located in the western mountain region, however, the ARA behavior characteristic in Zhengzhou station is similar to it.


## 4.3. Discussion

In Sections4.1 and 4.2 we have investigated the long-term series of precipitation records and succeeded in finding the evidence for power law distribution even though the five stations are located in different position in central China. This implies that our guess (i.e., the rainfall in central China takes on the feature of SOC) is right. This also supports the view that atmospheric dynamics is governed, at least in part, by SOC.As described in the introduction, the concept of SOC refers to the state of nonequilibrium systems driven by slow constant energy input to organize themselves into critical systems. The intermediately stored energy is eventually released in sudden bursts with no typical scale. From this point of view, rainfall events which occur in the central China are not very different from those ones in the other place. Like earthquakes, a rainfall event is driven by a slow and constant energy input from the sun and water is evaporated from the Western Pacific Ocean. The energy is stored in the form of water vapor in the atmosphere. It is then suddenly released in bursts when the vapors condense to water drops. The power-law distribution of the number density of rain events is equivalent to the Gutenberg-Richter law for earthquakes.Although there are different geographic coordinates at the five stations, theseτs show little change because the five stations are neighborhoods in one province after all. Except for the topography and landform the driving factors of rainfall, such as the sun radiation and the pattern of atmospheric circulation are nearly the same.


## 5. Conclusion

Testing the hypothesis, the existence of self-organized criticality in the long rain process in central China is our objective in this work. By calculating the number density of the rain duration, the drought duration, and the accumulated rain amount, we found that the relationship between the number density functions and rain duration, drought duration and accumulated amount exhibits the feature of power law. In other words, we have looked out the evidence of self-organized criticality in the long-term rain processes in central region of China. It is turned out that the long dry and wet process is indeed the complexity dynamics process.Henan province is in the climate transition zone where the climate changes from the southern humid subtropical monsoon region to the northern semiarid temperature monsoon. It is also the topography transition zone. The western region is mountain and the eastern is the great plain. Because of these reasons, the weather and the climate change drastically in a year and interyears. This point has also been turned out from our results and plots. The feature of wet and dry spell is significantly different between Anyang station and Xinyang station.In this work we only studied five meteorological stations due to the limitation of datasets. And the minimum time interval is a day which appears to be not small enough for the analysis of rainfall event. We will therefore collect more data sets in central China and other regions in the future work. The self-organized criticality needs to be tested in a small temporal interval and in a larger spatial scale, such as in the level of hour or second and in the level of the whole country. Furthermore, Algorithm2 proposed should be improved on the number density computation of accumulated rain amount. The concept of number density of duration and intension should be redefined over and over again in the future work.



---

*Source: 203682-2012-10-31.xml*