# Scaling and Kinetic Exchange Like Behavior of Hirsch Index and Total
Citation Distributions: Scopus-CiteScore Data Analysis 111Dedicated to the
loving memory of Prof. Amit Dutta, IIT Kanpur, India.

Asim Ghosh<EMAIL_ADDRESS>Department of Physics, Raghunathpur
College, Raghunathpur, Purulia 723133, India.  Bikas K. Chakrabarti
<EMAIL_ADDRESS>Saha Institute of Nuclear Physics, Kolkata
700064, India.  Economic Research Unit, Indian Statistical Institute, Kolkata
700108, India.

###### Abstract

We analyze the data distributions $f(h)$, $f(N_{c}$) and $f(N_{p})$ of the
Hirsch index $(h)$, total citations ($N_{c}$) and total number of papers
($N_{p}$) of the top scoring 120,000 authors (scientists) from the Stanford
cite-score (or c-score) 2022 list and their corresponding $h$ ($3\leq h\leq
284$), $N_{c}(1009\leq N_{c}\leq 428620$) and $N_{p}$ ($3\leq N_{p}\leq 3791$)
statistics from the Scopus data. For reasons explained in the text, we divided
the data of these top scorers (c-scores in the range 5.6125 to 3.3461) into
six successive equal-sized Groups of 20,000 authors or scientists. We tried to
fit, in each Group, $f(h)$, $f(Nc)$ and $f(Np)$ with Gamma distributions,
viewing them as the “wealth distributions” in the fixed saving-propensity
kinetic exchange models and found $f(h)\sim
h^{\gamma_{h}}\mathrm{exp}(-h/T_{h})$ with fitting noise level or temperature
level ($T_{h}$) and average value of $h$, and the power $\gamma_{h}$
determined by the “citation saving propensity” in each Group. We further
showed that using some earlier proposed power law scaling like
$h=D_{c}N_{c}^{\alpha_{c}}$ (or $h=D_{p}N_{p}^{\alpha_{p}}$) with
$\alpha_{c}=1/2=\alpha_{p}$, we can derive the observed $f(h)$ from the
observed $f(N_{c})$ or $f(N_{p})$, with $D_{c}=0.5$, but $D_{p}$ depending on
the Group considered. This observation suggests that the average citations per
paper ($N_{c}/N_{p}$) in each group ($=(D_{p}/D_{c})^{2}=4D_{p}^{2}$) vary
(from 58 to 29) with the c-score range of the six Groups considered here,
implying different effective Dunbar-like coordination numbers of the
scientists belonging to different groups or networks.

##  I Introduction

A popular measure of the success of individual scientist or author (called
scientist here generally) has been the Hirsch Index 1Hirsch2005  or h-index,
which can be viewed as the fixed point 2Ghosh2022  of the non-linear function
relating the monotonically decreasing number of publications ($n_{p}$) with
increasing number of citations ($n_{c}$): $n_{p}=h=n_{c}$ of the scientist.
Mapping the citation function to a combinatorial Fermi one, Yong proposed
3Yong2014  the relationship

$h=D_{c}N_{c}^{\alpha_{c}},$ (1)

with $D_{c}\simeq 0.54$ and $\alpha_{c}=1/2$ for any scientist with Hirsch
index value $h$ and total citations $N_{c}=\sum_{p}n_{c}$ from all his or her
publications (denoted by $p$), in the limit $N_{c}\rightarrow\infty$. Several
attempts to check the validity of such a relationship between $h$ and $N_{c}$
have been made, see e.g., Redner 4Redner2010  (supporting the relation (1)
with the exponent $\alpha_{c}$ value equal to 0.5, from the data analysis for
255 scientists) and Radicchi and Castellano 5Radicchi2013  ( analysing a much
larger set of data for 83,897 scientists) who found the best fit value of the
exponent $\alpha_{c}\simeq 0.42$. Ghosh et al. 2Ghosh2022  studied the Widom-
Stauffer like scaling behavior of the Hirsch index for the fiber bundle as
well as percolation models away from the “critical” breaking point or stress
and percolation point respectively and proposed

$h\sim\sqrt{N}_{c}/[\mathrm{log}N_{c}],$ (2)

for the citations of individual scientists, giving reasonable agreement with
the google scholar data for 1000 scientists (with $h$-indices in the range
$17\leq h\leq 221$ and total number of citations $N_{c}$ in the range $996\leq
Nc\leq 348680$).

Figure 1: Frequency distribution (normalized) for the Hirsch index ($h$) of
the authors in Groups I to VI, shown in Figs. 1a to 1f. The fitting Gamma
functions are also shown. Figure 2: Frequency distribution (normalized) of the
total number of citations ($N_{c}$) of the authors in Groups I to VI, shown in
Figs. 2a to 2f. The fitting Gamma functions are also shown. Figure 3:
Frequency distribution (normalized) of the total number of papers ($N_{p}$) of
the authors in Groups I to VI, shown in Figs. 3a to 3f. The fitting Gamma
functions are also shown. Figure 4: Frequency distribution for Hirsch index
($h$), obtained directly from the data source (denoted by symbol o) and that
obtained from the number of total citations ($N_{c}$) using relation (1) with
$\alpha_{c}=0.5$ (Yong 3Yong2014 ) and $D_{c}=0.5$ (denoted by symbol
$\times$). The observed overlaps confirm the relation (1) with
$\alpha_{c}=0.5$. The insets show the same with $\alpha_{c}=0.42$
5Radicchi2013  (left) and that with $\alpha_{c}=0.5$ with an inverse
$\mathrm{log}N_{c}$ correction 2Ghosh2022  (right); there seem to be
considerable mismatches. Figure 5: Frequency distribution of Hirsch index
($h$) obtained directly from the data source and those obtained from the
number ($N_{p}$) of total publication by the authors of the Group I to VI
using relation (5) $\alpha_{p}=0.5$ Figure 6: Extrapolated (statistical) value
of the prefactor $D_{p}$ in eqn. (5) for top ranking ($r=1$) scientist. We
plot the values of $D_{p}$ for different Groups from table 2 against the
middle rank ($r$) of the corresponding rank range (the same plot for the
prefactor $D_{c}$ in Eqn. (1) remains a horizontal line) .

We find here, in each of the six equal-size Groups of twenty thousand top
ranking scientists from the Elsevier Stanford c-score list Ioannidis2019 ;
6scopus ; 7elsevier  (total one hundred and twenty thousand top cited
scientists), the distributions (frequencies) $f(h)$, $f(Nc)$ and $f(Np)$ of
their Hirsch index ($h$), total citations ($N_{c}=\sum_{p}n_{c}$) and total
number of papers ($N_{p}=\sum_{p}n_{p}$) all fit very well with Gamma function
form:

$f(h)\sim h^{\gamma_{h}}[\mathrm{exp}(-h/T_{h})],$ (3a) $f(N_{c})\sim
N_{c}^{\gamma_{c}}[\mathrm{exp}(-N_{c}/T_{c})],$ (3b) $f(N_{p})\sim
h^{\gamma_{p}}[\mathrm{exp}(-N_{p}/T_{p})],$ (3c)

with the exponent values $\gamma_{h}\simeq 11.0$, $\gamma_{c}\simeq 3.0$,
$\gamma_{p}\simeq 2.2$, and the noise levels $T_{h},T_{c},T_{P}$, dependent on
the c-score range generally decreases with decreasing c-score (see Figs. 1, 2
and 3 in the next section on data analysis). We tried to obtain $f(h)$ from
$f(N_{c})$ using the power law relation (1) and got excellent fit (for
$D_{c}=0.5$ and $\alpha_{c}=1/2$; see Figs. 4). We also tried obtain $f(h)$
from $f(N_{p})$, using the relation $h=D_{p}N_{p}^{\alpha_{p}}$ 2Ghosh2022 ,
with $\alpha_{p}=1/2$ but $D_{p}$ value depending on the Group. Again we got
good fit (see Figs 5).

As mentioned above already, the values of $h$ calculated from the
corresponding values of $N_{c}$, using relation (1) proposed by Yong 3Yong2014
, but with $D_{c}=0.5$ (not with $D_{c}\simeq 0.54$ as suggested by Yong),
gives extremely good fit to the distributions $f(h)$ as shown in Figs. 1.
Other scaling relations $h\sim N_{c}^{0.42}$ suggested by Radicchi and
Castellano 5Radicchi2013  in 2013, or $h\sim N_{c}^{0.50}/\mathrm{log}N_{c}$
suggested by Ghosh et al. 2Ghosh2022  in 2022 do not give comparable good
fits.

In addition, we find the $T_{h}$ values for each Group (the six c-score ranges
I-VI), calculated using the relation

$T_{h}=h_{av}/(\gamma_{h}+1),$ (4)

where $h_{av}$ denotes the average of $f(h)$ in each Group, compares very well
with the observed values. This relation suggests a strong correlation of the
Chakraborti-Chakrabarti model 8Chakraborti  of “wealth” distribution where a
fixed saving fraction of the wealth (which determines the exponent value of
$\gamma$ in the resulting Gamma distribution of wealth) is retained in each
kinetic exchange or interaction (see 9Sen2014 ; 10Patriarca2004 ). If we
consider a similar stochastic dynamics of paper citations, where the fixed
fraction of (confident or core Group) “citations” (like wealth) in each paper-
writing (interaction) determines the exponent $\gamma_{h}$ value and the
corresponding noise level $T_{h}$ in the resulting Gamma distribution $f(h)$
of the (wealth) h-index. The equivalent wealth conservation may be assumed to
come (see e.g., Pareschi2014 ) from the overall constancy (node coordination
number) of the citation network, discussed later. In other words, as in the
model 8Chakraborti  of “wealth” distribution, discussed above, where a fixed
saving fraction of the wealth (which determines the value of the exponent
$\gamma_{c}$ in the Gamma distribution) is retained in each kinetic exchange
or interaction, suggests here a similar stochastic dynamics of paper
citations, where the fixed fraction of (confident or core Group) “citations”
(or wealth) in each paper-writing (interaction) determines the exponent
$\gamma_{c}$ value and the corresponding noise level $T_{h}$ in the Hirsch
index $f(h)$ distributions in each Group.

We also found an interesting feature of the citation network. As we mentioned
in connection with relation (3c), using the scaling relation

$h=D_{p}N_{p}^{\alpha_{p}},$ (5)

with $\alpha_{p}=0.5$, we get $D_{p}\simeq$ 3.8, 3.4, 3.2, 3.0, 2.8 and 2.7
respectively for the successively decreasing six c-score Groups I to VI.
Comparison of the relations (3a) and (3c) with $\alpha_{c}=0.5=\alpha_{p}$,
and as discussed above the best fit value of $D_{c}=0.5$, suggests the value
of the average citation per paper $N_{c}/N_{p}$ for any of these scientists
depends on average on the Group in which the scientist belongs, and is given
by $(D_{p}/D_{c})^{2}=4D_{p}^{2}$, ranging from 58 (for Group I) to 29 (for
Group VI). This may be identified as the effective social coordination number
(or Dunbar number Ghosh2021 ; Dunbar1992 ) giving the group size of the
scientists, as in the primates.

##  II SCOPUS DATA ANALYSIS

As mentioned already, we analyzed here the Elsevier Scopus 6scopus  data for
the Hirsch index $h$ and the corresponding number $N_{c}$ of total citations
for 120,000 scientists who came at the top of Stanford c-score list 7elsevier
last year (2022). We divided the set into six equal Groups of 20,000
scientists having c-sore rank ranges I [1-20000], II [20001-40000], III
[40001-60000], IV [60001-80000], V [80001-100000], and VI [100001-120000]. We
observed that for the scientists in each of these ranges, both the $h$-index
values and the total citations numbers $N_{c}$ have similar Gamma-like
distributions (see Fig. 1 (1a to 1f) for distributions of $h$ and of $N_{c}$
for the six ranges of c-score ranks mentioned above). We observe (see Figs 2;
2a to 2f) that the $h$-index distribution $f(h)$ in each of these six score
ranges fit very well to the Maxwell-Boltzmann like Gamma function form 3(a)
with $\gamma_{h}\simeq 11$ and the effective noise (temperature) decreases
with increasing range (from I through VI). In Table 1, we give for each of the
six ranges (I-VI) the estimated values of the most probable value of Hirsch
index $h^{mp}$, its average $h_{av}$ and the respective noise level or
temperature $T_{h}$.

Table 1: Hirsch index ($h$) data fitting parameters obtained from relation (3a).  Group | c-score rank | $\gamma_{h}$ | $h^{mp}$ | $h_{av}$ | $T_{h}$ |  $h_{av}/(\gamma_{h}+1$)
---|---|---|---|---|---|---
I | 1-20K | 11.0 | 61.0 | 68.8 | 5.53 | 5.73
II | 20k-40K | 11.0 | 45.5 | 49.8 | 4.14 | 4.15
III | 40K-60K | 11.0 | 39.0 | 43.4 | 3.57 | 3.61
IV | 60K-80K | 11.0 | 35.0 | 39.5 | 3.21 | 3.29
V | 80K-100K | 11.0 | 32.5 | 36.6 | 2.95 | 3.05
VI | 100K-120K | 11.0 | 29.9 | 34.5 | 2.76 | 2.88
Table 2: Fitting parameters for the total number of citations ($N_{c}$) and the total number of papers ($N_{p}$) obtained using the relations (1) and (5). Group | c-score rank | $\alpha_{c}$ | $\alpha_{p}$ |  $D_{c}$ (from relation (1)) |  $D_{p}$ (from relation (5)) | $(D_{p}/D_{c})^{2}$
---|---|---|---|---|---|---
I | 1-20K | 0.5 | 0.5 | 0.5 | 3.8 | 58
II | 20k-40K | 0.5 | 0.5 | 0.5 | 3.4 | 46
III | 40K-60K | 0.5 | 0.5 | 0.5 | 3.2 | 41
IV | 60K-80K | 0.5 | 0.5 | 0.5 | 3.0 | 36
V | 80K-100K | 0.5 | 0.5 | 0.5 | 2.8 | 31
VI | 100K-120K | 0.5 | 0.5 | 0.5 | 2.7 | 29

In Figs. 4 (4a to 4f) we compare the above-mentioned observed distribution of
$f(h)$ with those obtained by using the Yong’s scaling relation (1) with
$\alpha=0.5$ and the best fit value of $D_{c}=0.5$ for the six different
ranges I to VI of c-score ranks. The overlap seems to be very good and
encouraging. In contrast, insets of Figs. 4 we compared the same $h$-index
distributions $f(h)$ in the six different ranges of c-score ranks with the $h$
values obtained the $N_{c}$ values using relation (1) with $\alpha=0.42$ (as
observed in 5Radicchi2013 , mentioned above) and the best fit value of the
prefactor. The level of misfit is obvious. The same is true when one uses the
relation (2) between $h$ and $N_{C}$ with the best fit value ($4.5$) of the
prefactor (as suggested in 2Ghosh2022 ). Again the distributions of $h$ and
those obtained using relation (2) do not match (see the insets of Fig. 4).

Our analysis (see Figs. 2) for the Hirsch indices and the corresponding values
of the total citations (from Scopus data) for the top ranking c-score authors
therefore confirms the relation (1) with the exponent $\alpha_{c}=1/2$, as
obtained by Yong 3Yong2014 . This is because of the lack of matches (inset of
the Figs. 4) with $\alpha_{c}=0.42$ 5Radicchi2013  or $\alpha_{c}=1/2$) with a
log correction in relation (1) 2Ghosh2022 ).

An important observation (see Figs. 1) has been the Gamma function for the
distribution of the $h$ indices for all these (arbitrarily) divided six ranges
of top scorers. This indicates a Chakraborti-Chakrabarti type kinetic exchange
model 8Chakraborti ; 9Sen2014  of citation dynamics for each new paper, with a
random citation sharing fraction over a fixed (saved) faction of citations of
the close-circle papers. This “saving” fraction determines (see e.g.,
10Patriarca2004 ) the exponent $\gamma_{h}$ in the distribution (3a) and the
conservation of the total citations in such “social dynamics” of citations is
practically determined by the total publications within the “aging” period
(see e.g., 11Basu2005  and the references therein). Indeed, for such a Gamma
distributed statistics (3) in the Chakrabort-Chakrabarti kinetic exchange
model (with fixed fraction close circle citation propensity), the analysis of
Patriarca et al. 10Patriarca2004  suggests the relation (4). Such a relation
fits extremely well with the values of the noise level (temperature) $T$
obtained by fitting the Hirsch index distribution data to the relation (4) and
the value of $h_{av}$ obtained from distribution (3a) of $h$ together with its
$\gamma_{h}$ value. As mentioned already, this indicates an effective kinetic
exchange like stochastic dynamics for citations where each author has a fixed
share of core-group citations and allows the rest from the literature. The
dynamics give the total citations per paper constant on an average (constant
value weakly dependent on the c-score rank or the Group).

In fact, the relation (5) fits very well with the data set for each Group with
$\alpha_{p}=0.5$ (see Figs. 5). Combining relations (1) and (5) with
$\alpha_{c}=0.5=\alpha_{p}$ and $D_{c}=0.5$, one gets the average citations
per paper $N_{c}/N_{p}$ or the average coordination number of the citations
network equal to $4D_{p}^{2}$, which ranges from 58 to 29 (see Table 2). This
was observed and reported earlier Ghosh2021  and can be viewed as an effective
Dunbar number Dunbar1992  for the citations network.

Unlike the fitting value (0.50; see Table 2 ) of the prefactor $D_{c}$ in eqn
(1). The fitting values of the prefactor $D_{p}(r)$ in eqn. (5) increase with
the rank $r$ (see table 2). Fig. 6 gives the extrapolated value of $D_{p}$ for
the top rank ($r=1$) to be about 4.31, which gives the limiting value of the
citation network coordination number (network average of citations per paper)
to be $4[D_{p}(r=1)]^{2}\simeq 75$.

##  III Summary and Conclusion

We analyze the distributions $f(h)$, $f(N_{c})$ and $f(N_{p})$ of the Hirsch
index ($h$), total citations ($N_{c}$) and total number of papers ($N_{p}$) of
the top 120,000 scorers (scientists with c-score in the range 5.6125 to
3.3461) from the Stanford cite-score 2022 list, dividing them into six
successive and equal sized Groups, and their corresponding $h$ ($3\leq h\leq
284$), $N_{c}$ ($1009\leq N_{c}\leq 428620)$ and $N_{p}$ ($3\leq N_{p}\leq
3791$) from the Scopus data. It may be mentioned that all these authors fall
within (indeed the toppers of) the top 2% scientists in the Stanford cite-
score (2022) selection list 6scopus ; 7elsevier . As may be seen from Table 2,
while fitting the $h$ index data to those for $N_{c}$ or $N_{p}$, using the
power law relations (1) and (5) respectively, we found that while the powers
$\alpha_{c}$ and $\alpha_{p}$ both assumes the same value (1/2), and the value
of $D_{c}$ remains also 0.5 across the Groups, the value of $D_{p}$ varies
very slowly with the c-score value of the scientist. In order to get
sufficient statistics, yet to capture the changes in the values of $D_{p}$, we
divided (arbitrarily) the scientists into six equal Groups (I, II, III IV, V
and VI), each having 20,000 scientists according to their successive c-score
ranks. We find in each Group $f(h)$, $f(N_{c})$ and $f(N_{p})$ fit well with
Gamma function form (3a), (3b) and (3c) (see Figs 1, 2, and 3), e.g.,
$f(h)\sim h^{\gamma_{h}}[\mathrm{exp}(-h/T_{h})]$, with the exponent
$\gamma_{h}\simeq 11.0$, $\gamma_{c}\simeq 3.0$ and $\gamma_{p}\simeq 2.2$ and
the noise levels $T_{h}$, $T_{c}$ and $T_{p}$ dependent on the c-score range
considered. We compared the data (directly obtained from Scopus) for $f(h)$ in
all the six Groups (see Figs. 1) with those obtained from the data for
$f(N_{c})$, using the relation (1) with $\alpha_{c}$ = 1/2 and $D_{c}=0.5$,
and got excellent overlap (see Figs. 4). We did also same, obtained from the
$f(N_{p})$ data, using the relation (5), again with $\alpha_{p}=1/2$, but
$D_{p}$ dependent on the Group (see Table 2), showing again very good fit (see
Figs. 5). Other suggestions like $\alpha_{c}\simeq 0.42$ 5Radicchi2013  or
$\alpha_{c}=0.5$ but with an inverse $\mathrm{log}N_{c}$ correction term
2Ghosh2022  do not give good fits (see the insets of Fig. 4). In fact, a very
recent extensive analysis biro2023  of the statistical relation (1) between
$h$ and $N_{c}$ from the Google Scholar data gave $\alpha_{c}=0.5$ and $D_{c}$
= 0.5 (as obtained here), confirming a much earlier study Glanzel2006 . In our
study, apart from similar relationship between $h$ and $N_{p}$ and the
relationships (3a-3c), we find here the $T_{h}$ values (in relation (3a)) for
each of the six c-score ranges fit very well with the relation
$T_{h}=h_{av}/(\gamma_{c}+1)$ where $h_{av}$ is the average of $f(h)$ in each
Group. This compares very well with the Chakraborti-Chakrabarti model
8Chakraborti ; 10Patriarca2004 ; Pareschi2014  of “wealth” distribution where
a fixed saving fraction of the wealth (which determines the value of the
exponent $\gamma_{c}$ in the Gamma distribution) is retained in each kinetic
exchange or interaction, suggesting a similar stochastic dynamics of paper
citations, where the fixed fraction of (confident or core Group) “citations”
(wealth) in each paper-writing (interaction) determines the exponent
$\gamma_{c}$ value and the corresponding noise level $T_{h}$ in $f(h)$. We
also observe an interesting feature of the citation network. The observation
(relations (1) and (5)) $h=D_{c}N_{c}^{\alpha_{c}}=D_{p}N_{p}^{\alpha_{p}}$,
where $\alpha_{c}=\alpha_{p}$ = 0.5, $D_{c}=0.5$ and $2.7\leq D_{p}\leq 3.8$
depending on the Group, suggesting the value
($N_{c}/N_{p}=(D_{p}/D_{c})^{2}=4D_{p}^{2}$) of the average citation per paper
shown in Table 2 depends on the Group the scientist belongs to and ranges from
58 (for Group I) to 29 (for Group IV). As discussed at the end of the last
section (see Figs. 6), the limiting value of this citation-network
coordination number (network average of citations per paper) gets extrapolated
to about 75. This then may be identified as the effective social coordination
number (or Dunbar number Dunbar1992 ) giving the group size of the top-rated
scientists today, as in the primates.

## Acknowledgement

We are thankful to Soumyajyoti Biswas and Parongama Sen for several useful
comments on the manuscript. BKC is grateful to the Indian National Science
Academy for their Senior Scientist Research Grant.

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one may expect from increased shocks contributing to high overall heating
rates in the simulation. However, the inclusion of magnetic fields changes
this trend. Instead we see the highest temperature peak in our 8 km/s MHD
model, as well as more gas in the higher temperature range ($\geq 100\rm{K}$)
than the other models.

Figure 18: Density and temperature PDFs for all of our models.

##  Appendix B Line mass stability analysis

We introduce in this paper a simple and rudimentary approach to analyzing the
line mass of filaments in our simulations, and here describe the method
followed. We define the line mass as the thermal line mass, that is where the
critical line mass accounts for sound speed but not the velocity dispersion if
the gas. We then have a critical line mass of the form

$M_{crit}=\frac{2c_{s}^{2}}{G}$ (6)

The actual line mass we calculate as total mass of a cell divided by its
length, a physical line mass. The ratio of this value to the critical line
mass gives an indication of criticality, as discussed in Sec. 2. This cell by
cell construction allows us to visualize the line mass in multiple ways. In
Figure 19, we show the line mass ratios for a slice through our disp8 and
disp8MHD models. As all the cells have an associated line mass ratio, one
could also create a 3D map of the line masses in order to show a map of the
criticality throughout the filaments.

In Sec. 4.6, we show the projected line masses of both of our 8 km/s models.
In that case we have taken all the cells stacked on top of each other in the
direction of the projection, and integrated the line mass field over the path
length. Due to this integration, we then correct for the extra term of total
path length by dividing the value in each cell by the length of the box. We
note here that this is relatively easy to do given that our box is
symmetrical, but non-symmetric boxes would of course require the definition of
as many fields as projection directions one wishes to take. From this
correction, we thus have both 3D and 2D planar information about the critical
line masses of filaments, and we note the similarities in criticality between
the slices above and the projections in the body of this paper.

Figure 19: Examples of slices through the cube of our disp8 and disp8MHD
models showing the line mass in evaluated in the slice cell by cell.

##  Appendix C Turbulent Energy Spectra

We show here our turbulent energy spectra for our disp8MHD model at times of
1.90 Myr and 3.79 Myr. These show that our turbulent energy is not fully
saturated to the Burger’s relation until sink formation begins, therefore no
appreciable decay happens in the timescales we study.

Figure 20: Turbulent kinetic energy spectra for our disp8MHD model at a time
of 1.90 Myr (top) and 3.79 Myr (bottom). The dotted line represents the
Kolmogorov spectrum $E(k)\propto k^{-5/3}$. The dash dotted line shows the
Burgers spectrum $E(k)\propto k^{-2}$. The maximum wavenumber (k) corresponds
to a cell of 0.48 pc width (our highest resolution), but we note that some
resolution effects still arise in large k values. These spikes in energy are
likely due to small numbers of cells of the specific corresponding resolution.