MICHAEL J. WOLDENBERG
lORTON'S LAWS JUSTIFIED IN TERMS OF
LLLOMETRIC GROWTH AND STEADY STATE IN OPEN SYSTEMS
Abstract: Rivers are open systems and achieve a
Eady state or grow allomctrically, according to the
encral equation v = axK This equation yields a
raight line on double logarithmic paper and re
jects a lognormal or Yule stochastic process. Hori)n's laws are shown to be statements of these proc-
esses; and successive stream orders arc seen to benew logarithmic cycles to the base of the bifurcation
ratio ior a river system. A new absolute order is sug
gested, derived by raising the bifurcation ratio to
successive integer powers equivalent to stream order
minus
one.
/\
In recent years gcomorphologists have con
ceived of a river as an open system (Strahler,
p. 676; Leopold and Langbein, 1962:
}horley, 1962; Howard, 1965) which is defined
:a set of interrelated parts through which flow
multiplying by dt and integrating,
lergy and matter (Hall and Fagcn, 1956). A
laracteristic of open systems is that they tend
achieve a time-independent steady state if
/crage inflows and outflows are balanced and
main relatively constant. In the case of chan:1 flow, the geometry of a graded stream sys
tem remains relatively constant through time
[Strahler, 1964, p. 4-41).
It is also recognized that open systems may
lange form or grow in size, depending on imilances of flows through the system, or changrelations of the parts within the system,
lis type of growth has been termed allometric
wth by biologists and was first formulated as
jcneral principle by Huxley (1924; Reeve and
ixley, 1945; von Bertalanffy, 1960). Recently
>rdbeck (1965)1 has applied the term allometgrowth to the changing geometry of river
Stems,
'he law of allometric growth applied, for exle, to an animal states that the relative rate
[growth of an organ is a constant fraction of
relative rate of growth of the whole orism.
tThus if y is a measure of the size of the organ
Id .r is a comparable measure of the size of the
fW-
log v = log ii -f- b 1< >g x .
where a and b.QXt constants and </ is always posi
tive.2
Taking antilogs, v = ax*
This equation indicates that paired observa
tions for y and x, when plotted on double log
arithmic paper, yield a straight line or are best
fitted by a straight line. Asa corollary, we might
infer thatifeach oj two parts of a system is re
lated allomctrically to the whole svstem, then
they are related to each other by a power func-
>)\>[>Jr io^
1000 r-
0, cfs
jtire organism,
dy \_
v
dt
10
dx
x
Au*>
100
1000
Area, square miles
l_
dt'
y Nordbcck, 1965, The law of allometric growth
fichigan Inter-University Community of MathcmatGeographers Discussion Paper, number 7) is disibuted by Dcpt. Geography, University of Michigan
: Ann Arbor
Figure 1. Discharge versus area on double loga
rithmic paper (modified after Hack. 1957). Both
scales arc read in arithmetic values.
2 For a discussion of b sec von Bertalanffy. I960, p.
198-208; Nordbeck, 1965. For a discussion of a see
White and Gould, 1965
Jcological Society of America Bulletin, v. 77, 431-434. 4 figs.. April 1966
431
432
1
M.,. WOLDENBERG-ALLOMETRIC GROWTH-STEADY STATE IN OPEN SYSTEMS
tion. Hence, regression of any one of the geo
metrical characteristics of a river system in di
mensions ot length, area, volume', or weight
upon another geometrical characteristic of die
system will also yield a straight line on double
Jpganthmic paper.
Herbert Simon (1955) has shown that log-lo*
regressions result from stochastic processes and
thus reflect situations of maximum probability.
mat distribution reflects a system which, al
though not growing in numbers of individual
or size, is being maintained in steady state (Fij
Leopold and Langbein (1962, p. 2, 6, V
state that a condition of maximum probabilit
within the limitations of flow of the systc'
tends to be achieved and the entropy of t
system tends to approach the maximum in
1000
luk
rf\AiJ<>
Log order, u
Figure 2. The logarithm to the base 10 of discharge
versus the logarithm to the base of the bifurcation
ratio which is 3 in this case. Logs* - U — 1; where
■v is a variable of stream geometry or flow, and u is
stream order as defined by Horton (1945) and
Strahler (1952). Data are hypothetical. Both scales
are read in logarithmic values.
Although Simon here considered only the Yule
process, it has been suggested by Berry and
Garrison (1958, p. 91) that this type of graph
alternatively may be equivalent to a truncated
ognonnal distribution. (See also Aitchison and
Brown, 1957.) Simon and Bonini (1958, p. 611)
demonstrate that the Yule and lognormal dis
tributions both follow Gibrat's law of propor
tionate effect; that is that the distribution of
percentage changes in size over a period of time
in a given size class is the same for all size classes.
I he Yule distribution is generated by a con
stant rate of addition to the smallest size class.
I he lognormal distribution is generated bv as
suming a random walk of variates alreadv in the
system at the beginning of the time interval
under consideration with zero mean change in
size. Berry (1964, p. 149) states that a Io|nor-
points throughout that system. Furthermore,\
they point out that in the case of a river, the'
free available energy tends to be distributed
evenly throughout the system.
Hence double logarithmic graphs of variables |
of river geometry which yield a straight line are
actually expressions either of allometric growth]
or of steady state. It has only recently been sug
gested by Nordbeck (1965, p. 2) that relating I
the method of stream ordering proposed by
Horton (1945) and modified by Strahler (1952,,
p. 1120) to river basin geometry will yield al-'
lomctric growth curves.3 Although Nordbeck
cites double logarithmic graphs of variables of
river geometry, he does not offer graphs of!
3 The present writer reached the same conclusion dur
ing his own investigations.
SHORT NOTES
Table 1. Absolute Order of Streams
to when to, when
Rb = 2 Rb = 2.5
4
8
16
1
2.5
6.25
15.625
39.0625
if when
Rb = 3
1
3
9
27
81
«' when
/» = 3.7
1
3.7
13.69
50.653
187.4161
w when
Rb = 4
1
4
16
64
256
433
vcr.scly if the log scales are the same, the b will
diflcr. This observation implies that stream
order is itsell a logarithm.
A simple explanation of the meaning of
stream order should suffice to show the relation
ship just inferred. Assume a bifurcation ratio of
2 to 1. A second order stream has on the average
about two times as great a discharge as a first
order stream. A third order stream has four
times the discharge of a first order stream, etc.4
[stream order versus the log of a geometrical or
jrnass
variable
dealing
withlaws
riverof
basin
gcomctrv
■or
stream
flow.
Horton's
stream
num
bers, lengths, areas, and gradients consistently
[yield straight lines on semi-log plots. (See Fig.
Thus the problem is posed: How do the semi>g graphs having order scaled arithmetically on
-the abscissa equate to a double logarithmic
|giaph which reflects allometric growth or steady
State?
One possible answer is that stream order is
Idimensionlcss, hence the x scale may be con
tracted or expanded to give any slope, b, to the
lline of the graph. Thus two variables, the logs
lof which are related linearly to stream order arc
"in fact related to each other allometrically. If b
Is kept constant, the log scales may differ; con-
2
3
4
5
6
7
8
£
L o g o r d e r, u
Figure 4. Discharge versus stream order. Rb = 3,
Log3.v = // — 1; where x and // are as in Figure 2.
Data are hypothetical and identical to Figures 2
and 3. The ordinate is read in arithmetic'values
and the abscissa is read in logarithmic values.
lOOOr
27 81 243 729 2189 6567
Absolute order, w
urc 3. Discharge versus absolute order, w. w =
?£"•», where Rb = 3. The number w is the actual
agnitude of the coefficient of any given variable
diich is initially determined for streams of first
Order) at successively higher stream orders. Data
ire hypothetical and identical with Figures 2 and
4. The ordinate is read in arithmetic values and
the abscissa is read in arithmetic values for the
Icoefficient u:
Thus we may say that the Iog._..v = // — 1.
Stream order // minus 1 represents the log
arithm of some variable x to the base 2. How
ever, if the bifurcation ratio is 2.5 to 1 or 3 to 1,
then stream order minus 1 represents in the
general case the logarithm of some variable, in
this case .v. to the base II where B equals the
bifurcation ratio, or some other appropriate ra-tio of increase. Furthermore, it is seen that each
stream order begins a new cycle on the horizon
tal logarithmic scale.
It will be useful now to distinguish between
stream order, //, as defined by Horton (1945),
which is seen to be a logarithmic order and a
system of absolute order represented by the
symbol u:h The symbol w is, of course, the
* The author thanks Professor Sirahler for this insight.
5 The author is indebted to Professor Sirahler for 'this
suggestion.
434
M. J. WOLDENBERG-ALLOMETRIC GROWTH-STEADY STATE IN OPEN SYSTEMS
magnitude of the coefficient of the particular
variable being studied. Table 1 compares /, and
w for varying bifurcation ratios.
Figures 2, 3, and 4 are equivalent graphs
where Rb = 3. It should be clear now that
Figure 1, as well as Figures 2, 3 and 4, are all
double logarithmic graphs and thus reflect
either steady stale or allometric growth for the
river system.
Summary
The regularities of stream geometry and flow
versus stream order or versus other dimensional
quantities, which were observed bv Horton
(1945), Strahler (1964), and others, are in fact
double logarithmic relationships because i
tcger orders represent a logarithmic increase
absolute magnitude. These graphs give straig
regression lines, indicating a steady state or a
lometric growth; cither condition is cquivalel
to or approximates equilibrium and thus
close to the maximum entropy possible for
system.
. idmowledgments
The author is indebted to Prof. Art!
Strahler of the Geology and Geography I
partments of Columbia University and to Pi
Brian J. L. Berry of the Geography Der.
ment of the University of Chicago.
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Dept. Geock.u-iiv, Columbia University, New York, New York 100^7
Manuscript Received by the Society October 18, 1965