Fluid Dynamics Is atmospheric convection organised?: information entropy analysis

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Fluid Dynamics Is atmospheric convection organised?:

information entropy analysis

Yuanlong Li, Jun-Ichi Yano, Yanluan Lin

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Yuanlong Li, Jun-Ichi Yano, Yanluan Lin. Fluid Dynamics Is atmospheric convection organised?:

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2019, �10.1080/03091929.2018.1506449�. �hal-02396030�

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Is atmospheric convection organised?: information entropy analysis

Yuanlong Li, Jun-Ichi Yano & Yanluan Lin

To cite this article: Yuanlong Li, Jun-Ichi Yano & Yanluan Lin (2019) Is atmospheric convection organised?: information entropy analysis, Geophysical & Astrophysical Fluid Dynamics, 113:5-6, 553-573, DOI: 10.1080/03091929.2018.1506449

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2019, VOL. 113, NOS. 5–6, 553–573

https://doi.org/10.1080/03091929.2018.1506449

Is atmospheric convection organised?: information entropy analysis

Yuanlong Li^a, Jun-Ichi Yano^b∗and Yanluan Lin^a

aMinistry of Education Key Laboratory for Earth System Modeling, Department of Earth System Science, and Joint Center for Global Change Studies (JCGCS), Tsinghua University, Beijing, China;^bCNRM, Météo-France and CNRS, Toulouse, France

ABSTRACT

In order to quantify the degree of organisation of atmospheric con- vection, an analysis based on the information entropy, which is widely considered a measure of organisation in information science, is performed. Here, the information entropy is defined in terms of the spectrum of the empirical orthogonal functions (EOFs). Satellite- based brightness temperature data from CLAUS (Cloud Archive User Service) is used over the domain covering the Indian Ocean and the Western Pacific with a spatial resolution of 2/3^◦from January 1985 to June 2009. The information entropy remains close to a mean value of 0.899 with a very small standard deviation of 2.7×10⁻³, suggest- ing that the atmospheric convection is always disorganised under a measure of the information entropy, which is against our common understanding. To better interpret this result, some basic theoretical analyses are performed, and the values of the information entropy for different systems (English literature texts, turbulent flows) from pre- vious studies are reviewed. The same analysis is further performed on the Ising model, which is characterised by a clustering tendency of spin distribution, akin to convective organisation morphologically, at the critical temperature. The study suggests a need for a careful use of the term “organised”. Atmospheric convection represents a ten- dency for clustering up to the planetary scale in analogous manner as the critical-point behaviour of the Ising model. However, neither is considered an “ordered” state under a measure of the information entropy.

ARTICLE HISTORY Received 16 April 2018 Accepted 27 July 2018 KEYWORDS

Organisation; atmospheric convection; information entropy; EOFs; Ising model

1. Introduction

We often consider that atmospheric convection is “organised” (cf. Yano

1998, Mon-

crieff

2010). Its most basic elements, the few-kilometer-scale convective towers, tend to

cluster into a mesoscale (the squall line, mesoscale clusters: Houze

2004); mesoscale con-

vection is in turn, further clustered into the synoptic scale, and so on, forming a hierarchy of convective systems of various horizontal scales over the tropics (Nakazawa

1988).

CONTACT Jun-Ichi Yano jiy.gfder@gmail.com

∗Current address: CNRM, Météo-France, 42 av Coriolis, Toulouse 31057 Cedex, France

© 2018 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group

This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives License (http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited, and is not altered, transformed, or built upon in any way.

One of the most prominent features in the largest scale of the hierarchy is the tropical Madden–Julian oscillation (MJO: Zhang

2005).

However, is it really appropriate to call this tendency an “organisation”? If that is the case, what do we really mean by “organised”? In fact, is atmospheric con- vection really “organised”? In order to address these rather epistemological ques- tions in a critically constructive manner, the present paper adopts, as a working hypothesis, the information entropy (Shannon

1948, Guiasu 1977) as a measure of

organisation.

The information entropy, as originally introduced by Shannon (1948), has opened up a new discipline called information science (Guiasu

1977), and is often considered an

objective measure of the organisation of a given system. The basic idea of the informa- tion entropy is to measure a complexity (or an order) of a system (or of information) by decomposing it into a number,

N_q

, of the “quanta”. Here, the quantum is a unit for measur- ing the frequency of the occurrence of the “states” of the system. A frequency of occurrence for a particular state, designated by

k, is defined by

p_k=n_k/Nq

, (1)

when the number of quanta found in this state is

nk

. A continuous formulation is obtained by taking a limit of

N_q→ ∞

as assumed in the following.

Under this formulation, the complexity of the system is measured by the number of total possible combinations for re-arranging the partitioning of those quanta, but on preserving a given frequency distribution. When all the quanta occupy a single state, it is impos- sible to re-shuffle, or re-arrange these quanta, because there is no other state to move a quantum. When the quanta occupy two states, it is now possible to re-shuffle the quanta between these two states without changing the distribution. Thus, the total number of pos- sible re-arrangements of the quanta becomes greater than one. The number of possibilities for re-arrangements increases as the quanta occupy more states, and also when they are distributed more homogeneously amongst the possible states. The information entropy is defined as a logarithmic measure of the total possible number of re-arrangements, which we define more formally in the next section.

In the present study, we apply the information entropy analysis to the infrared mea- surement from satellites, as detailed in Section

3. Here, we define the states by empirical

orthogonal functions (EOFs) obtained by a decomposition of the system (following Aubry

et al.1991). Thus, when a system is characterised only by a single EOF, the system is con-

sidered the most organised, and the information entropy is the lowest; when a system is characterised by an equal distribution over all possible EOF modes, the system is con- sidered least organised, and the information entropy is the highest. The present study uses this definition as a working hypothesis for measuring the degree of organisation of atmospheric convection in the large scale. A rather surprising result from the present anal- ysis, presented in Section

4, is that according to this measure, atmospheric convection is

not significantly organised. Our analysis shows that the information entropy is 0.899 on average, a rather large value, with a very weak variability with the standard deviation of 2.7

×

10

⁻³

.

The remainder of the paper is focussed on interpreting this rather unintuitive result:

why does the information entropy of large-scale atmospheric convection remain so large

regardless of its phase? This question is addressed from several different aspects. In Section

5, we first perform some basic theoretical analyses on the information entropy.

The first question posed is: how large is the value of 0.899? To answer this question, we translate the value of information entropy to the total number of possible rearrangements of the states. The latter depends on the total number,

N, of the modes available for a given

system (e.g. the total number of the measurement points for the present data), or the number of freedom of the system. Thus, the fractional number for re-arrangements (i.e.

the number of total possible re-arrangements under a given distribution normalised by the maximum possible number of re-arrangements obtained after considering every pos- sible distribution) is plotted as a function of the number of freedom for given values of the information entropy. In Section

5, we also investigate a “structural” reason why the

information entropy of atmospheric convection is so large. The satellite data-based analy- sis shows that the EOF spectrum closely follows a power law with a relatively small power exponent. We quantify the relationship between the power exponent and the information entropy.

In Section

6, we review typical values of the information entropy for various systems

found in the literature. The most important point of reference is the information entropy associated with the human text, because the original intention of Shannon (1948) for introducing the information entropy was to measure a degree of information content in transmitted electronic signal, which typically consists of a human text. Another key ques- tion is to compare the obtained information entropy value with those obtained for other types of fluid flows.

The power law spectrum of EOF found in the present study may be interpreted as a manifestation of the self-organised criticality based on the fact that the latter is typically associated with the former (Sornette

2000). See Wilson (1979) as an introduction for the

critical phenomena. The question of self-organised criticality motivates us to examine the information entropy (as defined by Aubry

et al.1991) of a simple system in statistical

mechanics at a criticality. As such an example, in Section

7, we choose an Ising system

(Chandler

1987, Chapter 5; Yeomans 1992, Chapter 4). The paper closes with further

discussions on the implications in Section

8.

2. Information entropy

The information entropy measures the total possible number,

W, of re-arrangements of the

quanta of a system in logarithmic scale. Here, the number of quanta occupying a system state, designated by an index,

k, isnk

, with

N

the number of possible states,

Nq

the total number of the quanta, then

W

is given by

W= Nq

!

n1

!n

2

!

· · ·nN

! . (2) In the limit of large

Nq

, we obtain

1

Nq

log

W − N k=1

pk

log

pk

(3)

by invoking Stirling’s formula

log

k!=k

log(k)

+

O(k) for every integer

k. Recall thatpk

is defined by (1).

Note that the maximum for the quantity (3) is given by log

N

when all the states are occupied homogeneously (i.e.

pk=

1/N for

k=

1,

. . .

,

N). Thus, we define the information

entropy,

H, such that its maximum becomes unity:

H= −

1 log

N

N k=1

p_k

log

p_k

. (4)

For measuring the information entropy for any physical variable, say,

u(r,t), defined in

space,

r, and time,t, we first decompose the variable,u, by use of the EOFs,ϕk(r):

u(r,t)= N k=1

αkψk(t)ϕk(r)

(5)

(following Aubry

et al. 1991). Here, αk

are the eigenvalues in descending order, and

ψk

are the time-dependent expansion coefficients (k

=

1,

. . .

,

N). Note the orthogonality

relationships

1

S

Sϕk(r)ϕl(r)

dr

=δkl

, 1

T

_T

0 ψk(t)ψl(t)

dt

=δkl

. (6a,b) Here,

δkl

is Kronecker’s delta, and the integrals are performed over the analysis domain,

S, and the analysis period, T, respectively. The integrals (6a,b) are further normalised

by the analysis domain and period, respectively. Note that the choice of this decom- position is unique, because it essentially constitutes a singular vector decomposition in linear algebra (cf. Aubry

et al.1991). In geophysical applications, the spatial patterns

(EOFs) are often defined by

αkϕk(r)

so that they directly measure degrees of spatial varaibilities.

Under this EOF decomposition, we consider two types of entropy. The first is a global entropy,

Hg

, that characterises the system for the entire analysis period by defining the frequencies of the states as

pk=αk

N k=1

αk

. (7)

The second is a spatial entropy,

Hs(t), that characterises the organisation at every instant

of the time series by defining the frequencies of the states as:

pk(t)=αk|ψk(t)|

N k=1

αk|ψk(t)|.

(8)

Here, the definition of the distribution (7) for global entropy is slightly modified from that of Aubry

et al.

(1991), in which a relation

pk∼α²_k

is assumed instead, in order to retain a symmetry between the definitions of the two entropies above.

3. Data set

The study adopts the global brightness temperature data from the Cloud Archive User Service (CLAUS, Robinson

2002). The data are based on the operational meteorological

satellites obtained form the National Aeronautics and Space Administration (NASA) Langley Atmospheric Sciences Data Center (LASDC). The data provide global thermal infrared imagery of the Earth based on the level B3 10 micron radiance.

The data cover the period from January 1985 to June 2009 with a 3-h interval. The horizontal resolution is 1/3

^◦

both in longitude and latitude. This resolution is degraded to 2/3

^◦

in the study to reduce the analysis burden. Further data issues, including those of the resolution, are discussed separately in the

Appendix.

To focus the analysis on tropical convection, the analysis domain is taken to be the rect- angle (30

^◦

S–30

^◦

N, 60

^◦

E–120

^◦

W), which covers the Indian Ocean and the Western Pacific, where convection is most active.

4. Data analysis result

The first three EOF patterns obtained are shown in figure

1

with the corresponding time series in figure

2. The first mode represents an annual cycle associated with the inter-

tropical convergence zone over the Maritime Continent. The second mode is interannual, likely related to the El Nino cycle, whereas the third is intraseasonal, likely linked to the MJO. These first three EOFs describe, respectively, 7.3%, 2.5% and 2.2% of the total vari- ance, and are hardly representative of the whole convective variability. We present these first three EOFs merely for a purpose of providing a general feeling of the data set.

The global entropy evaluated by the frequency distribution (7) for this data set is 0.926.

By definition, the entropy value is dictated by a distribution of the frequency,

pk

, which is plotted in figure

3: it decreases with the increasing indexk

only in algebraic manner, i.e. a power law of the index number,

k^−a

, where

a

is a positive constant. This is an indication of the self-organised criticality (cf. Sornette

2000): See Section7

below for further discussions.

We also find the power exponent is

a

0.55, thus

_N

k=1p_k

log

p_k

diverges in the limit of

N→ ∞, as explicitly demonstrated later in Section5.

The spatial entropy is plotted as a function of time in figure

4. Rather strikingly, the

entropy time series is almost constant about 0.899, with a very weak variability with the

standard deviation of 2.7

×

10

⁻³

. To get a concrete feeling about how the spatial entropy

measures the convective organisation, the two snapshots corresponding to the minimum

and the maximum spatial entropies in the time series are shown in figure

5. Convection

with the minimum entropy (a) is better organised, whereas more scattered convection is

seen with the maximum entropy (b). To that extent, the information entropy quantifies a

degree of the organisation of convection. However, the state with the minimum entropy is

hardly one that corresponds to the best organised state in the large scale, say, at a peak of the

MJO activity. The state with the maximum entropy, representing a clear sign of mesoscale

convective organisation, is again hardly the least organised state of tropical convection.

Figure 1.The first three EOFs,ϕ1,ϕ2andϕ3, of the CLAUS satellite-based brightness temperature.

5. Basic considerations

So how large is the information entropy of 0.9? To obtain a better feeling for this, we

examine the relative number,

nr

, of the total possible re-arrangements for a system with

an information entropy,

H, against that of the totally disorganised state withH=

1. It is

defined in terms of the total number of modes,

N, and the total number of the quota,

Figure 2.The time series of the coefficients,ψ1,ψ2andψ3, for the first three EOFs shown in figure1.

Nq

by

nr =

e

^(NqlogN)H

e

^NqlogN

.

Here, recall that the maximum possible number of re-arrangements with the

N

modes is given by e

^NqlogN =N^Nq

, whereas the possible total number of re-arrangements to a given system is e

^(NqlogN)H=N^NqH

, when the information entropy is

H. Dependence ofnr

on

Nq

can be made implicit by factorising the above form by

n^N/N_r ^q=

e

^(NlogN)H

e

^NlogN

. (9)

This factorised ratio,

n^N_r ^/Nq

, is plotted in figure

6. Note that the actual relative number,nr

,

decreases with the increasing

Nq

.

Figure 3.Plot of the frequencypk(see (1)) as a function of the index,k. A slope with−0.55 is also shown by the dashed line.

Figure 4.The time series of spatial entropy (in red). The monthly mean is also shown in green (Colour online).

As expected from the exponential dependence of

n^N/Nr ^q

on

N

in (9), the factorised rel- ative number,

n^N/Nr ^q

, for the total arrangements rapidly decreases as the number,

N, of

modes increases: above

N =

10

³

, the fractional number,

n^N/N_r ^q

, becomes extremely small (i.e.

n^Nr^/Nq <

10

⁻³⁰⁰

) even when the entropy is relatively large (e.g.

H=

0.9). Thus, unfor- tunately, we have to conclude that it is difficult to get a sense of a largeness of the entropy by this approach.

To obtain further insight, we note that our analysis suggests that the frequency distri- bution approximately follows (cf. figure

3)

pk=CNk^−a

(10)

with

CN

and

a

constants. Here, the constant,

CN

, is defined by the normalisation condition

N

k=1

p_k=

1,

Figure 5.The distribution of brightness temperature corresponding to (a) the minimum (12 December 1988 UTC 00) and (b) the maximum (25 April 1998 UTC 06) of the spatial entropies.

Figure 6.The relative number, n^N/Nr ^q, of the total possible re-arrangements factorised by N/Nq

(equation (9)) in logarithmic scale as a function of the total number of modes,N, for various entropy values,H.

and it leads to

CN = n

k=1

k^−a ₋₁

.

Figure 7.The information entropy,HN, defined by (11) with the power law distribution of the EOF spec- trum (see (10)) as a function of the total number of modes,N, for the power exponentsa=0.6, 0.8, 1, 1.2 (Colour online).

The entropy may be evaluated as a function of

N, i.e.

HN = −

1 log

N

N k=1

pk

log

pk

. (11)

Note that the sum

_N

k=1k^−a

with

a<

1 diverges as

N→ ∞. Since k^−a<

−k^−a

log

k^−a=ak^−a

log

k

when

a

log

k>

1 (or

k^a>

e), the sum

−_N

k=1k^−a

log

k^−a

with

a<

1 is also divergent. Furthermore, it suggests that

HN →

1 as

N→ ∞

when

a<

1, corresponding to a completely disorganised state in entropy measure. In order to see this tendency more quantitatively, figure

7

plots

HN

as a function of

N

for the range of

N=

10–10

⁷

for selected values of

a. Whena=

1, this divergent tendency is weak, and the entropy value,

HN

, stays close to 0.7 for the range of

N =

10

⁴

–10

⁷

. However, once

a≤

0.8, a diverging tendency is clearly seen for

N >

10

³

. When

a=

0.8,

HN

remains close to 0.9 over a whole range considered in the plot, but gradually increases with the increas- ing

N

above 10

⁴

, being consistent with the entropy values obtained from the convective data analysis in Section

3. Thus, if the tendency predicted by (10) witha

0.55 contin- ues for the observed convection field for

N→ ∞, we obtainH→

1. If that is the case, we must conclude that atmospheric large-scale convection is completely disorganised from the information entropy point of view.

6. Information entropy of other systems

To glean a better sense of the values of the information entropy obtained for large-scale atmospheric convection, this section surveys typical values of the information entropy of other systems.

The first category to investigate is, naturally, those for the human-written texts, because

the original intention of Shannon (1948) was to measure the degree of the order in

human-sent signals, that more than often consist of human texts. Shannon (1951) already

considered this question. By quoting Dewey (1923), he points out that the frequency dis-

tribution of the use of the English words follows (10) with

a=

1. As already pointed out in

the last section, this is a divergent series, and the information entropy increases gradually as the number of words considered increases.

A more recent study by Rosso

et al.

(2009) sheds more light on this issue by exam- ining a digitised full data set for Renaissance-period English literature (consisting of 185 texts). They found that the entropy falls to a range of 0.75–0.85 when the vocabulary for the analysis is limited to 400, being consistent with the original suggestion of Shannon (1951) that the information entropy for the typical English text could be fairly high. On the other hand, when the full vocabulary of the texts is considered, the entropy decreases to a range of 0.5–0.6, suggesting that the frequency of the use of the words drops more rapidly than expected by the form (10) with

a=

1 for the higher order words. The result further sug- gests that human texts are structured more heavily relying on the basic vocabularies than unusual words.

The next category to examine is the information entropy of the fluid flows. Here, the information entropy may be defined in various different manners. For consistency with the present analysis, we focus on the analysis of the entropy based on the definition by (Aubry

et al.1991) as in the present study. Aubryet al.

(1991,

1994) present the information entropy

analysis of the rotating-disk turbulent flow based on a laboratory experiment. They find that the information entropy of the system is relatively low, remains constant close to 0.4, under a low Reynolds number (Re) laminar flow regime. On the other hand, by crossing a critical point (around

Re=

1.2

×

10

⁵

), the system transits into a turbulent regime, and the entropy suddenly increases. The entropy is

H

0.8 at

Re=

1.7

×

10

⁵

, and it reaches

H=

0.97 at

Re=

2.8

×

10

⁵

. Thus, if an analogy with this laboratory-experiment flow is taken, we may conclude that the implications from a high entropy value associated with large-scale atmospheric convection is a simple manifestation that it is also fully turbulent.

7. Analysis of the self-organised Ising system

There is evidence from observations (e.g. Yano

et al.2001,2004, Peters and Neelin2006)

and modelling (Yano

et al.2012) that atmospheric convection may be at self-organised

criticality. Yano and Plant (2012) discuss this concept in reviewing the convective quasi- equilibrium (see question 1.7, Section 2.1 of Yano

et al.

(2015) for more extensive refer- ences). The power law distribution of the EOF spectrum (see (10)) provides an additional support for this possibility.

Since the self-organised criticality is a macroscopic counterpart to the criticality that is found in the microscopic systems, it might be insightful to examine the behaviour of the information entropy for a simple theoretical system that represents a classical critical behaviour. As such a model, we consider here the Ising model that explains the sponta- neous magnetisation of ferromagnet below a critical temperature (called the Curie point by honouring the discoverer Pierre Curie): above the critical temperature the magnetic spins of the ferromagnet point to random directions, whereas below the critical tempera- ture the spins tend to align to the same direction, with the tendency enhances towards the lower temperature. The Ising model is designed to explain this behaviour under a simple configuration (see Chandler

1987, Chapter 5; Yeomans1992, Chapter 4 for more details).

We consider a two-dimensional Ising model consisting of 1000

×

1000 spins. Each spin,

s_k

, with

k

designating the position over the two-dimensional space, can take the values of

either

sk=

1 or

−1. The Hamiltonian (energy) of this system (after a simplification with a

normalisation) is given by

H= −

k,l

sksl

, (12)

where

<k,l>

suggests a sum over the nearest neighbours. There are four nearest- neighbour points,

l, for a givenk

when a rectangular spin distribution is assumed as herein.

The equilibrium state for the magnetisation at a normalised temperature,

T, is determined

by

{s}

¯ se^−H/T

{s}

e

^−H/T

, (13)

where

{s}

suggests a sum over all the possible configurations for the spins,

s=(s1

,

s2

,

. . .),

and

¯s

is the mean magnetisation under a given particular configuration,

s.

However, a direct application of formula (13) is impractical due to a huge number of possible configurations to be evaluated. Moreover, we are interested with spin patterns of the equilibrium state at a given temperature. The above procedure does not directly provide this information. For these reasons, instead, we adopt a Monte–Carlo method (as described in Chapter 6 of Chandler

1987

and Chapter 7 of Yeomans

1992) for evaluating

the equilibrium state of the Ising system as a function of the temperature. The Monte–Carlo computation is always initiated with the state

sk=

1 for all the spins so that a unique sign for the equilibrium magnetisation in the low-temperature limit is ensured. The number of Monte–Carlo iterations is 10

¹⁰

for

T<

3 and 10

⁹

for

T ≥

3. More iterations are required to ensure a better convergence at lower temperatures. For statistical analyses, 1000-ensemble runs are performed at each temperature, and their averages are presented in the following.

The equilibrium magnetisation evaluated by this method is plotted in figure

8

as a func- tion of temperature. Magnetisation of the Ising system suddenly begins to increase right below the critical temperature,

T =Tc

2.27, and asymptotically approaches unity in the low-temperature limit. Unfortunately, we note a problem with the Monte–Carlo procedure adopted: although the theory predicts that the transition to the spontaneous magnetisation state is discontinuous over the critical temperature, no discontinuity is identified in the plot with the critical value also slightly shifting towards

Tc

2.33. However, we did not

Figure 8.Magnetisation (orange dashed) and the global entropy (red) of the Ising model as a function of temperature as averages over 1000 random realisations (Colour online).

attempt to obtain the exact critical temperature by further increasing the iteration num- ber. Our purpose was merely to qualitatively reproduce the change of the spin pattern by crossing the critical temperature,

T=Tc

. Rather, the key question to be addressed is how information entropy quantifies this change.

Figure

9

shows the change of the spin distribution by changing the temperature. In the high-temperature limit (a:

T=

3), the distribution of spins is completely random with no structure to be seen. However, as the temperature decreases, we notice a tendency of neg- ative spins to form clusters ( (b):

T=

2.4 ; (c):

T=

2.35). At the critical temperature ( (d):

T=

2.33

Tc

), the clusters become most extensive, being connected together for a long distance. Note that regardless of the presence of a structure, it is important to note that pos- itive and negative spins are equally distributed in numbers, leading to a zero magnetisation above the critical temperature. Below the critical temperature, the number of negative spins begins to decrease relative to the positive spins, reflecting the fact that the spontaneous magnetisation, consisting of a spin distribution with a constant value, is realised towards the lower temperature. From a structural point of view, this tendency is accomplished by a gradual decrease of the size of the negative-spin clusters as seen at

T=

2.27 (e). Already at

T=

2.2 (f), the negative spins are seen only at localised locations.

How does the information entropy measure the change of the spin-distribution pattern with the decreasing temperature? In the Ising problem, there is no notion of the temporal evolution. For the purpose of computing the information entropy in a manner described in Section

2, we treat the 1000 realisations of the spin distribution for every temperature with

Figure 9.Spatial spin patterns realised by the Ising model with (a)T=3, (b) 2.4, (c) 2.35, (c) 2.33 (close to a numerical critical point), (e) 2.27, and (f ) 2.2. The spin values with 1 and−1 are shown, respectively, by white and black.

a realisation index as a substitute for the time. The 1000 corresponding EOFs are evaluated by taking the “lag” correlations between all the realisations. The global entropy is evalu- ated on the basis of the distribution defined by (7). The result is shown by a red curve in figure

8.

The obtained entropy curve may be, to some extent, considered a mirror image of the magnetisation curve, however not perfectly. Due to a completely random distribution of spins in the high-temperature limit, the entropy value is also high in this limit, close to 0.95.

Following the argument associated with the distribution (10), we expect that the entropy approaches unity as the sample size is increased.

As the temperature decreases, the entropy also decreases. However, an onset of decrease of the entropy is not recognised until the temperature decreases well below the critical temperature. The entropy crosses the value of 0.9, at

T=

1.1, well below the numerically identified critical temperature,

T_c

2.33. Below

T=

0.9, the entropy goes down to zero due to a full realisation of the spontaneous magnetisation, and it remains so down to the low-temperature limit.

In this manner, the “order” of the system associated with the spontaneous magnetisation is well measured by the global information entropy. However, the most unintuitive aspect of this analysis is that a strong clustering tendency of the spins at the critical temperature is hardly detected by the entropy. Under the information entropy measure, the clustered

“organised” state at the criticality is equally disorganised in the high-temperature limit.

8. Discussions

8.1. Organisation and information entropy

The most general lesson to learn from the present study is the need to be precise with the terminology in any scientific discourse. Many words found in common life are not defined in any precise manner, and usually, that does not cause any practical problem.

On the other hand, in science, the meaning of the terminology must be precise so that an objective discourse becomes possible. Having said that as a matter of principle, its practice is not easy.

So-called convective organisation is just such an example. As briefly reviewed in the introduction, convection organisation can visually be recognised by examining the images of convective clouds seen by radar as well as satellites. These structures range from the mesoscale organisations such as squall lines to planetary-scale coherencies such as the MJO. For most researchers, it would be hard to argue against what they recognise clearly from visual inspections.

However, to be precise with the terminology, we must be rather epistemological: would it be really appropriate to call these structures “organised”? Or alternatively, what do we mean more precisely when we say that convection is “organised”? It is relatively easy to pick up any statistical definitions that claim to measure a degree of “ organisation”, and use them for the purpose of

objectively

quantifing the degree of “ organisation”. However, any such approach would not answer those rather epistemological questions.

In the present study, we have adopted the information entropy as such a measure,

because it is well established in information science as a measure of such. However, the

results of our analysis are rather disappointing: the information entropy has failed to

identify convective organisations seen by eye. Instead, the time series of the information entropy for the large-scale tropical convective variability is almost constant with time, and always remains a relatively high value of about 0.9. Its fluctuations are extremely small with a standard deviation of about 2.7

×

10

⁻³

. Here, more specifically, we have followed the definition of the information entropy based on the EOF decomposition proposed by Aubry

et al.

(1991). The satellite-measured CLAUS global brightness temperature is used for the analysis.

The entropy value obtained from satellite data is compared with those found in different systems. When the number of occurance of words is used for the definition of the probabil- ity, it is shown that typical human (English) texts have the information entropy around 0.5 (Rosso

et al.2009). For the fluid flows, the information entropy increases from around 0.4

at a laminar state to above 0.9 at a fully developed turbulent state (Aubry

et al.1991,1994).

Based on those references, we may conclude that atmospheric convection is not organised as much as human texts as well as laminar fluid flows, but its state is rather close to that of fully developed turbulence. However, what do all these numbers mean?

8.2. Order and clustering

In order to put our interpretation into a more solid context, we have turned to an analysis on the evolution of the spin pattern with change of the temperature under a two-dimensional Ising model. As the temperature decreases, the spin distribution gradually transforms form a state of a complete randomness to a perfectly “ordered” state with a single sign every- where. Intriguingly, at an intermediate-temperature state, the spins tend to cluster around with the same signs, representing a structure reminiscent of large-scale convective organ- isation, and this clustering tendency becomes most extensive at the point called “critical”.

However, the information entropy simply monotonically decreases with the decreasing temperature towards the final “ordered” state in the low-temperature limit without any indication of clustering tendency in transition. In this manner, though the information entropy may measure a degree of the order of a system, not necessarily a degree of organisation, as seen as clustering at the critical point.

This conclusion also appears to be applicable to the atmospheric convective organisa- tion: when we talk about it, more precisely, we are talking about the clustering tendency of convection into larger scales, rather than any order. These clusterings of convection may still be considered organisation in the sense that they permit more compact descriptions than otherwise. Various methodologies such as composites and EOFs can efficiently

extract

an organisation of convection such as mesoscale squall lines, convective clusters, or the MJO from the whole system, leading to compact descriptions of these “clustered” organi- sations. For example, the main features of the MJO can be described only by two leading EOFs (e.g. Matthews

2000, Wheeler and Hendon2004).

However, here, we have to distinguish the two closely related, but subtly different issues:

existence of an organisation (i.e. clusters) that can be described in a compact manner and

the question of describing the whole system in a compact manner. The information entropy

adopted in the present study quantifies the second aspect, which turns out to be rather large

for the large-scale atmospheric convective system, as well as at the critical point of the Ising

system. The result, thus, suggests that although the atmospheric convection system may

contain some organisations (i.e. clusters), that may be described in a compact manner, the

system as a whole does not permit a compact description, because of many “disorganised”

structures co-existing with these organisations. The clusters in the Ising system as well as in convective systems such as MJO may be considered “organised” from our visual inspec- tions. However, these organisations are associated with fluctuations in many scales. Thus, they are considered “ disorganised” from a point of view of the information entropy.

We need to clearly distinguish the two senses of the word “ organisation”, that of being well “clustered” and that of being under an “order”. The information entropy measures the degree of the “order” of the system. However, it does not measure a degree of “clustering”.

8.3. Definition of the information entropy

Before further discussing the implications of the result, it is important to establish its robustness first. The major ambiguity in defining an information entropy for any physi- cal system is the choice of a distribution,

pk

, in (4). In the present study (following Aubry

et al.1991), we have chosen a normalised EOF spectrum forpk

. However, this is just one of the possible choices found in the literature (e.g. Powell and Percival

1979, Grassberger

and Procaccia

1983, Isaacson2013). Finding alternative possible choices for the mode

decomposition rather than choosing EOF (cf. Schmid

2010, Rossoet al.2009) is merely

one particular aspect.

The mode decomposition is hardly a unique choice for constructing a distribution,

p_k

, for computing the information entropy. A more obvious alternative choice is to take a dis- tribution of a physical variable itself as

pk

. This is a standard definition in the statistical mechanics (cf. Mandl

1988). In case of the Ising model, the distribution reduces to that

of between the spin values

+1 and−1. After an appropriate normalisation, this definition

reduces to the standard thermodynamic entropy. In case of the present satellite data anal- ysis, an obvious choice is to consider the distribution of the brightness temperature. In the preliminary analysis, we also considered this choice, and confirmed that the result is essentially the same as that of based on the EOF spectrum.

Another obvious possibility, though the definition of entropy itself must also be modified, is to compute Kolmogorov’s entropy. This is attempted by Yano and Muk- ougawa (1992) for a quasi-geostrophic system. However, unfortunately, evaluating this entropy from an observational data set is rather difficult, because this definition is based on the Liapunov-exponent spectrum.

8.4. Complexity, degree of freedom and system dimension

Another important aspect elucidated by the present study is the fact that atmospheric con- vection as a whole consists of very many degrees of freedom, as the case of the critical state (cf. Wilson

1983) as well as fully developed turbulence (cf. Fritsch1995). This high dimen-

sionality of atmospheric convection leads to a high entropy state. It is important to realise that a compact description of the atmospheric convection system,

as a whole, is not pos-

sible in spite of its organisation (i.e. clustering) tendency, due to its high dimensionality.

It is a common custom to extract those convective clusters by the methodologies such as

composites and EOF. However, these standard procedures are likely to lose some essence

of atmospheric convection that is fundamentally an extremely high-dimensional process.

Naively speaking, the information entropy measures a structuredness of the informa- tion contained in a given system, and the more structured the system, the more compact its possible description, and henceforth with a smaller effective dimension. However, we should realise that a link between the entropy and a dimension of the system (effective or not) is rather tenuous. The point is made explicit with Kolmogorov’s entropy, which is defined as a sum of positive Liapunov exponents. The (attractor) dimension of the system is best measured by Kaplan–Yorke dimension, which is defined as an effective index (after an interpolation) such that the sum of the Liapunov exponents up to this index become zero. Clearly, these two measures are related, but not in quantitative manner.

Yano and Mukougawa (1992) make a rather pedagogic point about the attractor dimen- sion by taking a particular setting of a quasi-geostorphic system, in which the system rapidly settles to a quasi-steady state as a truncation wavenumber increases. We may expect that an effective dimension of such a system would be very small. However, their investi- gation reveals that the attractor dimension of this system does not indicate any sign of decrease with the increasing truncation wavenumber.

In the same manner, an organised tropical variability such as the MJO is likely a system with a very high attractor dimension, as the present study also suggests. This speculation is supported by an attempt of estimating a correlation dimension of the MJO by constructing an embedded phase space from the two MJO EOFs extracted by Matthews (2000). How- ever, no tendency for convergence is identified within a reasonable limit of the embedded dimension (Martin Döge

2000). However, thisdoes not

mean that a forecast of MJO with a low-dimensional system

is not

possible. In fact, Maharaj and Wheeler (2005) and Jiang

et al.

(2008) show that the two leading EOF indices for MJO introduced by Wheeler and Hendon (2004) are enough for constructing a reliable statistical forecast system for MJO.

8.5. Perspectives

We meteorologists and climate scientists tend to think of convective organisation in terms of clustering (similar to the Ising model clusters near criticality), but the information entropy does not measure clusterings as being organised.

MJO would probably be a good example to make the point. Indeed, this is clearly a well clustered system. However, the information content, as measured by the informa- tion entropy, associated with the MJO is most likely not at all small, as suggested by the present study, and also more explicitly demonstrated by a nonconverging attractor dimen- sion under an embedded-dimension analysis aforementioned. Although the MJO is indeed clustered in a compact manner, its structure is not at all simple, but highly complex, prob- ably associated with the fractality (cf. Yano and Takeuchi

1987). The information entropy

measures the latter aspect of a system, but

not

a degree of clustering. Another objective measure must still be sought for quantifying the degree of clustering of a system. Wavelet could be a possible answer (cf. Yano

et al.2001a,2001b; Yano and Jakubiak2016).

To be fair, the question of the information entropy associated with a single MJO event is

still to be answered. A formal procedure to answer this question would be to first extract a

single MJO from the whole convective field as measured by, for example, the satellite-based

brightness temperature. However, the answer would depend on how MJO is extracted from

the whole field. If a standard EOF method is used, MJO would be represented only by a few

EOFs, and as it follows, a low information entropy would be obtained under the definition

of Aubry

et al.

(1991). On the other hand, if the wavelet is used instead, it is likely that a high information entropy value is obtained. This is to be explicitly demonstrated by a future study.

Clustering and complexity (or structuredness) of atmospheric systems must be clearly distinguished from each other. Although the former concept suggests an organisation of a system, the latter suggests an intrinsically high dimensionality with an implication of an disorganisation. These two separate aspects are still to be carefully quantified for many atmospheric phenomena, including various convective organisations. At the same time, an intrinsically high dimensionality of the system does not automatically exclude a pos- sibility of a compact description of the given system for predictions. It has already been demonstrated that such a simple statistical forecast is possible for MJO (Maharaj and Wheeler

2005, Jianget al.2008) in spite of a high complexity of this phenomena.

The present study suggests an importance of clearly distinguishing between the closely related concepts in atmospheric research, and defining each of them carefully: organisa- tion, order, clustering, complexity, structuredness. When a certain statistical measure is introduced, we must be extremely careful in knowing what this measure actually measures.

The present study with the information entropy provides a useful lesson in this respect.

Acknowledgements

J.-I. Y. acknowledges Samson Hagos for originally suggesting to take the information entropy as a measure of the organisation of atmospheric convection. He also acknowledges the discussions with Chidong Zhang for various occasions. The authors thank Georg Gottwald for working as an editor, two anonymous reviewers for their constructive comments, and Nabeela Sadaf for her careful proofreading of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

This work was supported by the National Natural Science Foundation of China (41775098).

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Appendices. Data issues

The purpose this appendix is to investigate data issues encountered in due course of the study. These are (i) the data resolution and (ii) the surface effects on data.

Appendix 1. Resolution dependence

For practical reasons, the present analysis was performed with a half (2/3^◦) of the horizontal resolu- tion of the full data set (i.e. 1/3^◦). The purpose of this section is to infer an effect on the result using the degraded resolution.

For this purpose, we have attempted to reconstruct the full-resolution result from the obtained half-resolution result in the following manner. For the distinctions, in the following, the full- and the half-resolution variables are designated by the subscripts,FandL, respectively. The full-resolution EOF pattern,ϕk,F(r), can be inferred from the half-resolution EOF coefficient time seriesψk,L(t), invoking its orthogonality, and also assuming that it is still a good approximation under the full resolution

αk,Fϕk,F(r)= _T

0 u(r,t)ψk,L(t)dt T. (A1)

Here, the eignevalues,αk,F, under the full resolution may be diagnosed by the normalisation αk,F=

S(αk,Fϕk,F)²dr S _1/2

. (A2)

For verifying the self-consistency of this diagnosis, we have further estimated the full-resolution EOF coefficient time seriesψk,F(t)by invoking the orthogonality ofϕk,F, i.e.

ψk,F(t)=

u(x,y,t)ϕk,F(r)dr Sαk,F. (A3) The global entropy for the full-resolution data obtained in this manner is found to agree well with that of the half resolution. On the other hand, the spatial entropy for the full resolution turns out to be about 2% larger in average than the half-resolution case. The reason for this minor discrepancy is traced to the fact that the EOF time series,ψk,F(t), obtained in this manner do not satisfy the normalisation condition (6a) well fork>3000.

Appendix 2. Surface effects

A major limitation of the use of the infrared radiation data as a measure of convective activity is that the high infrared radiation (i.e. high brightness temperature) values are, to a great degree, a representation of the surface temperature. Thus, it also influences the whole variability of the data.

The analysis is repeated by re-setting all the high brightness temperature values above a threshold to the threshold temperature. The threshold temperature is set to 250 K, low enough that most of the

surface influences are removed, but high enough so that a bulk of cloud-top temperature information is retained.

The leading order EOF patterns and time series remain qualitatively the same, although modifi- cations are indeed noticeable both in patterns and time series. The most significant modification is a reversal of the order of the 2^ndand the 3^rdEOF modes after filtering out the surface effects. The time series of the spatial entropy increases both in average (0.908) and the standard deviation (4×10⁻³).

However, these minor modifications hardly affect the main conclusions in the main text.