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Negro and Danube are mirror rivers

R. Gonçalves ^a & A. Pinto ^b

a Faculdade de Engenharia do Porto , R. Dr. Roberto Frias s/n, 4200-465, Porto, Portugal

b Universidade do Minho , Campus de Gualtar, 4710-057, Braga, Portugal

Published online: 19 Nov 2010.

To cite this article: R. Gonçalves & A. Pinto (2010) Negro and Danube are mirror rivers, Journal of Difference Equations and Applications, 16:12, 1491-1499, DOI: 10.1080/10236190902845662 To link to this article: http://dx.doi.org/10.1080/10236190902845662

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Negro and Danube are mirror rivers

R. Gonc¸alves^a* and A. Pinto^b

aFaculdade de Engenharia do Porto, R. Dr. Roberto Frias s/n, 4200-465 Porto, Portugal;

bUniversidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal (Received 4 February 2009; final version received 19 February 2009)

Dedicated to Maurı´cio Peixoto and David Rand

We study the European river Danube and the South American river Negro daily water levels. We present a fit for the Negro daily water level period and standard deviation.

Unexpectedly, we discover that the river Negro and Danube are mirror rivers in the sense that the daily water levels fluctuations histograms are close to the universal non- parametric BHP and reversed BHP, respectively. Hence, the probability of a certain positive fluctuation range in the river Negro is, approximately, equal to the probability of the corresponding symmetric negative fluctuation range in the river Danube.

Keywords:dynamical systems; universality; hydrological statistics; data analysis AMS Subject Classification: 37M10; 37N10; 62P12; 86A05; 82B27

1. Introduction

Ja´nosi and Gallas [10] analyzed statistics of the Alpine river Danube daily water level collected, over the period 1901 – 1997, at Nagymaros, Hungary. The authors found, in the daily logarithmic rate of change of the river water level, similar characteristics to those of company growth (see Stanley et al. [13]) which shows that the properties seen in company data are present in a wider class of complex systems. Bramwell et al. [3] defined a daily water level mean and variance and computed the daily river water level fluctuations. They have shown a data collapse of the Danube daily water level fluctuations histogram to the reversed Bramwell – Holdsworth – Pinton (BHP) probability density function (pdf). As we point out in the next section, the BHP pdf is universal, non-parametric with heavy tails and non-zero skewness. Later, Dahlstedt and Jensen [5] studied the statistical properties of several river systems. They did a careful study of the size of the basin areas influence in the data collapse of the rivers water level and runoff fluctuations. They have shown that not all rivers water level and runoff fluctuations have the same statistical behaviour. In particular, they compared and discussed the lack of similarity between the histogram of the South American river Negro water level fluctuations, at Manaus (104 years), the gaussian pdf and the reversed BHP pdf. In this paper, we show the data collapse of the histogram of the Negro water level fluctuations to the BHP pdf. Putting together the result of this paper with the result of Bramwell et al. [3], we conclude that the fluctuations of the river Danube and Negro follow mirror pdfs, i.e. the probability of a certain positive fluctuation range in the river Negro is, approximately, equal to the probability of the corresponding symmetric negative fluctuation range in the river Danube. Furthermore, we compute and present a

ISSN 1023-6198 print/ISSN 1563-5120 online q2010 Taylor & Francis

DOI: 10.1080/10236190902845662 http://www.informaworld.com

*Corresponding author. Email: [email protected] Vol. 16, No. 12, December 2010, 1491–1499

Downloaded by [b-on: Biblioteca do conhecimento online UP] at 16:12 20 October 2013

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cyclic fit for the Negro daily water level period and for the Negro daily water level standard deviation. Using these fits and the BHP pdf, we give, for any given day of the year, an estimation for the probability of any range of Negro daily water level.

2. Universality of the BHP distribution

The universal non-parametric BHP pdf was discovered by Bramwell et al. [1]. The universal non-parametric BHP pdf is the pdf of the fluctuations of the total magnetization, in the strong coupling (low temperature) regime for a two-dimensional spin model (2dXY), using the spin wave approximation. The magnetization distribution, that they found, is named, after them, the BHP distribution. The BHP pdf is given by

pðmÞ ¼ ð1

21

dx 2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

2N² X

N21

k¼1

1 l²_k vu

ut e

ixm ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2N2

P^N21

k¼1 1 l2 k

q

2P^N21

k¼1 ix 2N1

lk2₂ⁱarctan _Nl^x

k

h i

£e

2P^N21

k¼1 1 4ln 1þ^x²

N2l2 k

; ð1Þ

where the {lk}^L_k¼1are the eigenvalues, as determined in Ref. [2], of the adjacency matrix.

It follows, from the formula of the BHP pdf, that the asymptotic values for large deviations, below and above the mean, are exponential and double exponential, respectively (in this article, we use the approximation of the BHP pdf obtained by takingL¼10 andN¼L²in equation (1)). As one can see, the BHP distribution does not have any parameter (except the mean that is normalized to 0 and the standard deviation that is normalized to 1).

Furthermore, the BHP distribution is universal in the sense that appears in several physical phenomena. For instance, the universal non-parametric BHP distribution is a good model to explain the fluctuations of order parameters in theoretical examples such as the Sneppen model (see Refs. [1,4]), auto-ignition fire models (see Ref. [12]), self-organized models, and percolation models (see Ref. [1]). The universal non-parametric BHP distribution is, also, an explanatory model for fluctuations of several phenomenon such as width power in steady state systems (see Bramwell et al. [1]), river heights and flow (see Bramwell et al. [2], Gonçalves et al. [8] and Dahlstedt and Jensen [5]), Plasma density and electrostatic turbulent fluxes measured at the scrape-off layer of the Alcator C-Mod tokamak (see van Milligen [11]), Wolf’s sunspot numbers (see Gonçalves et al. [7]), the Standard & Poor’s S&P100 re-scaled stock index and the re-scaled daily returns of its constituent stocks (see Gonçalves and Pinto [6]), and the re-scaled Dow Jones’s index and the re-scaled daily returns of its constituent stocks (see Gonçalves and Pinto [6]).

3. BHP and the Danube and Negro data

As Bramwell et al. [3], we define the Danube daily water level period ^l_mðsÞ, for s[{1;. . .;365}¹, by

^l_mðsÞ ¼ 1 Tl

X^T21

j¼0

Xðsþ365jÞ; ð2Þ

whereT_l¼87 is the number of observed years andX(t) is the Danube daily water level time series witht[{1;. . .;365 £ 87}. In Gonc¸alves et al. [9], it is given a fit to the Danube daily water level period^l_mðtÞ(see Figure 1).

R. Gonc¸alves and A. Pinto 1492

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We define the Negro daily water level periodn^_mðtÞfor,s[{1;. . .;365} [2], by

^

n_mðsÞ ¼ 1 T_n

X^T21

j¼0

Yðsþ365jÞ; ð3Þ

whereTn¼104 is the number of observed years andY(t) is the Negro daily water level time series with t[{1;. . .;365 £ 104}. We use the first four sub-harmonics of the Fourier series to give a fit

~

n_mðsÞ ¼a0

2 þX⁴

n¼1

a_ncos 2snp 365

þb_nsin 2snp 365

ð4Þ

of the Negro daily water level periodn^_mðsÞ(see Figure 2). The Fourier coefficientsa_nand b_n of the fit are given in Table 1. The percentage of variance explained by the fit is R²¼99:9%.

As in Bramwell et al. [3] (see Figure 3), we define the Danube daily water level standard deviation^l_sðsÞfor,s[{1;. . .;365}, by

^l_sðsÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PT21

j¼0Xðsþ365jÞ² Tl

2^l_mðsÞ² s

: ð5Þ

We define the Negro daily water level standard deviationn^_sðsÞfor,s[{1;. . .;365}, by

^ n_sðsÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PT21

j¼0Yðsþ365jÞ²

T_n 2n^_mðsÞ² s

: ð6Þ

The standard deviation n^_sðsÞ attains its minimum point in the range ½160;180 that coincides with the range where the water level periodn^_mðtÞattains its maximum point (see Figures 2 and 4). We observe that, both, the standard deviationn^_sðsÞand the coefficient of variationn^_sðsÞ=^n_mðsÞare higher for values ofsclose to 284 (peak of the dry season).

0 50 100 150 200 250 300 350

140 160 180 200 220 240 260 280 300 320

Days

Height in m

Danube mean.

fit

Figure 1. Chronograph of the Danube daily water level period^l_mðtÞ, in a semi-log plot.

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We use the first 10 sub-harmonics of the Fourier series to give a fit

~

n_sðsÞ ¼a0

2 þX¹⁰

n¼1

ancos 2snp 365

þbnsin 2snp 365

ð7Þ

of the Negro daily water level standard deviation. The Fourier coefficientsa_nandb_nof the fit are given in Table 2. The percentage of variance explained by the fit isR²¼88:6%.

Following Bramwell et al. [3], we define the Danube daily water level fluctuations lfðtÞby

lfðtÞ ¼lðtÞ2^l_mðtmod 365Þ

^l_sðtmod 365Þ ; ð8Þ

where t[{1;. . .;365 £ 87}. Bramwell et al. [3] have shown the data collapse of the histogram of the Danube daily water level fluctuations to the reversed BHP pdf (see Figure 5). Let the non-normalized BHP(m,s) be the BHP pdf with mean and standard

0 50 100 150 200 250 300 350

1800 2000 2200 2400 2600 2800

Days

Water level in centimeters

Negro mean fit

Figure 2. Chronogram of the Negro daily water level periodn^_mðtÞand fitn~_mðtÞ.

Table 1. Fourier coefficients for the Negro daily water level periodn^mðtÞ.

n an bn

0 4671.2 –

1 2386.92 195.28

2 73.672 74.577

3 29.0336 212.473

4 29.9726 213.724

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deviation re-scaled such that the mean ismand the standard deviation iss. Fixing any given days[{1;. . .;365} of the year, the probability pdf of the Danube daily water level is approximated by the non-normalized reversed BHP(~l_mðsÞ;^l_sðsÞ) pdf. For any given day

s[{1;. . .;365}, this gives an estimation for the probability of any range of the Danube

0 50 100 150 200 250 300 350

60 70 80 90 100 110 120

Days

Height in m

Danube St.Dev.

Figure 3. Chronogram of the Danube daily water level standard deviation^lsðtÞ.

0 50 100 150 200 250 300 350

120 140 160 180 200 220 240

Days

Water level in centimeters

Negro St. Dev.

fit

Figure 4. Chronogram of the Negro daily water level standard deviationn^_sðtÞand fitn~_sðtÞ.

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water level. For s[{1;. . .;365} and k[{0;. . .;103}, the Danube daily water level Xðsþ365kÞis given by

Xðsþ365kÞ ¼^l_sðsÞl_fðsþ365kÞ þ^l_mðsÞ: ð9Þ

Hence, fixing the day s[{1;. . .;365} of the year, we observe that the pairs ðXðsþ365kÞ;lfðsþ365kÞÞ, fork[{0;. . .;108}, vary along a lineL(s), in the plane, with slope 1=^l_sðsÞand passing trough the pointð0;2^l_mðsÞ=^l_sðsÞÞ(see Figure 6). We callL(s) the Danube water level fluctuation lines. The observed slopes of the Danube water level fluctuations lines vary between 0.0096 and 0.0148. The highest fluctuation occurs for the highest water levelX(t).

Table 2. Fourier coefficients for the Negro daily water level standard deviationn^_sðtÞ.

n an bn

0 304.1 –

1 28.505 33.983

2 228.086 1.1481

3 22.6941 28.1938

4 6.5597 21.9674

5 23.6529 1.6374

6 2.1779 1.2313

7 20.2847 21.2559

8 20.5713 20.6129

9 1.7276 0.54441

10 21.0358 0.64293

–3 –2 –1 0 1 2 3 4

–6 –5 –4 –3 –2 –1 0

l_f(t) ln(p(lf(t)))

l_f(t) histogram BHP pdf

Figure 5. Data collapse of the histogram of the Danube water level fluctuations to the BHP pdf, in the semi-log scale.

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We define the Negro daily water level fluctuationsnfðtÞby

n_fðtÞ ¼YðtÞ2n^_mðtmod 365Þ

n^_sðtmod 365Þ ; ð10Þ

wheret[{1;. . .;365 £ 104}. In Figure 7, we show the data collapse of the histogram of Negro daily water level fluctuations to the BHP pdf. Hence, we conclude that the probability of a certain positive fluctuation range in the river Danube is, approximately,

–100 0 100 200 300 400 500 600 700 800

–3 –2 –1 0 1 2 3 4 5 6

X(t) water level in m lf(t)

Figure 6. Danube water level fluctuation lines.

–6 –5 –4 –3 –2 –1 0 1 2 3

–6 –5 –4 –3 –2 –1 0

l_f(t) ln(p(nf(t)))

n_f(t) histogram BHP pdf

Figure 7. Histogram of the Negro daily water level fluctuations for the full year, in the semi-log scale, with the BHP on top.

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equal to the probability of the corresponding symmetric negative fluctuation range in the river Negro. Fixing any given days[{1;. . .;365} of the year, the probability pdf of the Negro daily water level is approximated by the non-normalized BHP(~n_mðsÞ;n~_sðsÞ). For any given days[{1;. . .;365}, this gives an estimation for the probability of any range of Negro water level.

Fors[{1;. . .;365} andk[{0;. . .;103}, the Negro daily water levelyðsþ365kÞ

is given by

Yðsþ365kÞ ¼n^_sðsÞn_fðsþ365kÞ þn^_mðsÞ: ð11Þ

Hence, fixing the day s[{1;. . .;365} of the year, we observe that the pairs ðYðsþ365kÞ;n_fðsþ365kÞÞ, fork[{0;. . .;108}, vary along a lineN(s), in the plane, with slope 1=^n_sðsÞand passing trough the pointð0;2n^_mðsÞ=n^_sðsÞÞ(see Figure 8). We call N(s) the Negro water level fluctuation lines. The observed slopes of the Negro water level fluctuations lines vary between 0.0043 and 0.0086. We observe that the Negro water level valuesY(t) smaller than the level 1800 cm correspond to the negative fluctuations and the water level values higher than the level 2800 cm correspond to positive fluctuations. The highest fluctuation occurs for the water levelY(t) close to the level 2200 cm that is not close to the highest water level values.

4. Conclusions

We computed and presented a cyclic fit for the Negro daily water level period and for the Negro daily water level standard deviation using the first 4 and 10 sub-harmonics, respectively. The histogram of the Negro daily water level fluctuations is close to the BHP pdf. Using these fits and the BHP pdf, our result gives, for any given day of the year, an estimation for the probability of any range of Negro water level. We observed that the fluctuations of the river Danube and Negro follow mirror pdfs, i.e. the probability of Figure 8. Negro water level fluctuation lines.

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a certain positive fluctuation range in the river Danube is, approximately, equal to the probability of the corresponding symmetric negative fluctuation range in the river Negro.

This puzzeling result needs further investigation to understand the physical reasons why these rivers have mirror probability for fluctuations.

Acknowledgements

We thank Peter Holdsworth, Henrik Jensen and Nico Stollenwerk for showing us the relevance of the Bramwell – Holdsworth – Pinton distribution. We, also thank Imre Ja´nosi for providing the river Danube data and Mrs. Andrelina Santos of the Ageˆncia Nacional de A´ guas of Brazil for providing the river Negro data.

Note

1. All the observations of the days 29th of February were eliminated.

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