FOR A LANDFORM UNIT DERIVED FROM THE MOST PROBABLE AREA−ALTITUDE DISTRIBUTION

ATHERMODYNAMIC ANALOGY

FOR A LANDFORM UNIT DERIVED FROM THE MOST PROBABLE AREA−ALTITUDE DISTRIBUTION

Eiji ToKuNAGA＊

∠4bstract The geomorphic temperature and the partition function of a landform unit are defined by assuming the canonical distribution of the potential energies of material forming the surface of the landform unit． These two quantities provide the variables analogous to the intemal energy， the entropy， and the free energy in the thermodynamic system． The area−altitude distributions of the earth surface and the four Japanese main islands imply that landform units at the equilibrium state satisfy the canonical distribution． However， further examinations of area−altitude distributions of actual landform units are needed to ensure more evidentially the thermodynamic analogy composed of the quantities and the variables mentioned above．

Key words：Geomorphic temperature， thermodynamic analogy， area・altitude distribu−

tion， canonical distribution， equilibrium state

1．Introduction

  In the thermodynamic analogy for landscapes introduced by Leopold and Langbein

（1962），the height h of the land above some base line is regarded as to be equivalent to the absolute temperature T and the mass M to the heat quantity Q． Then the change of entropy in the landscape evolution is given by dM／h． Scheidegger（1967）derived the equations to define the heat・capacity coefficient， the pressure， and the work in the analogy． The analogy is certainly formalized． The author， however， has a question about the prerequisite postulate for the analogy that the height of the land corresponds to the temperature． The height of the land means the potential energy per unit volume of material forming it above the base line． On the other hand， the temperature in ther−

modynamics is an intensive variable and therefore does not depend on mass． Physical meanings of variables should be examined carefully to set up a thermodynamic analogy．

  Culling（1988b）introduced the temperature of particulate systems in geomorphic settings based on classical approaches in statistical mechanics． This temperature， which

＊Faculty of Economics， Chuo University

might be called the Culling temperature ， is clearly defined as an intensive variable independent from mass and available to explain behaviors of particles in many kinds of geomorphic processes． The geomorphic temperature proposed in this paper means the

temperature of a landform unit， differing from the Culling temperature ． The approa・

ching method is， however， similar to that by Culling（1988b）in the point that the canonical distribution is postulated to set up the analogy．

2．］Most Probable Area・altitude】）istribution

  Athree dimensional map is often used as a heuristic tool to explain shapes of land−

forms． In Fig．1the thick lines show the outline of such a map and the thin line shows the profile of a landform unit， where the unit of length is given arbitrarily． In this paper alandform unit means a set of landforms which is relatively independent from the surroundings． This definition is vague， because it is not easy to distinguish an indepen−

dent area from its surroundings on the land． H owever， we can start the study by imaging alandform unit like an isolated island．

  The landform unit is divided into zones with the height interva16H and each zone is ordered by increasing the ordinal number from the lowest toward the uppermost as shown in Fig．1． When the landform unit is expressed by a three dimensional map， the altitude of a zone can be represented by that of the corresponding iso−level plane． The ith zone is regarded as to have the altitude（i−1）δH above the base level in Fig．1．

  Let denote the area of the ith zone by ai and the total area of the landform unit by A．

Then／1 is given by the following equation．

｝δH

Fig．1 Alandform unit and the zones with the height interva1δH

     ．4一Σαi      （1）

         i＝l

where m is the ordinal of the highest zone． Let denote the volume of the landform unit byγ． Then the following equation is derived．

         m

      γ÷ΣaiδH（i−1）       （2）

         i＝1

The approximation sign in the above equation can be replaced by the equality sign， when δHis sufficiently small and m is sufficiently large． Namely，

         m

      γ＝ΣaiδH（i−1）      （3）

         i＝l

  Next each ordered zone is divided into sections of unit area and all the sections are enumerated according to some geomorphic measure． We can enumerate the sections by their altitudes， when we assume that each section has the altitude microscopically different from others in an ordered zone， though the altitude of the whole ordered zone is represented by that of the corresponding iso・level plane． It is also possible to enumer。

ate the sections assuming differences in relief between them

In any case， the ith zone is divided into ai enumerated sections． Then the value of ai results in being given by an integer． When the enumerated sections distribute randomly in each ordered zone， the number g of configurations of all the sections in the zones from first to mth is given as follows：

9−al！α2！……am！

Hence we obtain the equation which gives the most probable state of configurations of the sections under the condition that the area and the volume of the landform unit are respectively constant． When g has a maximum value， a small variation in the distribution will leave this value unchanged． The variation must be so that， from Eqs．（1）and（3）， A and V remain unchanged． The variation may be made， for example， by transferring one section from the second zone to the first zone and one from the second zone to the third zone． Then the number g of configurations is given as follows：

8！＝（al−｛−1）！（a2−2）！（α3→−1）！・・・…  am！

The ratio g ／g is given by

（al→−1）！（a2−2）！（a3十1）！   （a1十1）（a3十1）

al！a2！a3！ ^{a、（a、−1）}

For very large numbers of sections this expression may be replaced by

 グ9＿ala3

  −    2

9

    a2

If g is not noticeably changed by the rearrangement， then g ／g＝1 and thus a2  a3

al  a2

Asimilar relationship applies for the higher zones． The condition for a maximum value

ofgis thus

a2  a3

−＝：−＝ ■■コ    ニ

al  a2

am

am＿1

（6）

This equation states the most probable area−altitude distribution of the landform unit．

The restrictive condition given by Eqs．（1）and（3）， however， fixes the value of al for the relation that ai＝砺＝……＝ am． Then the problem is whether the geometrical progression

｛ai｝ is increasing or decreasing． The author has the intuition that the decreasing geometrical progression is suitable for the landform unit which has been formed by the normal erosion． The landform unit may take the most various shapes under the condition that the most probable area・altitude distribution is sustained．

3．Variables Analogous to Thermodynamic Functions

Let ai／ai＿1 be 1／c， then ｛ai｝ is written as follows：

al， a・−alC 1 ^{C a、＝＝ a1C 2}

C…… ^{C am−a、C m＋1}

where c＞1 for the decreasing progression． The representative altitudes of the zones corresponding to these areas are

O， 6H，2δH，……，（m−1）δH

Then it follows that

      一βδH（i−1）

ai＝ale ^（7）

whereβ＝（6H）−iln c is a positive quantity． Here let denote the density of material forming the landform unit byρand gravitational acceleration by g． Then ai is given by the following equation．

      一≠γρ9δH（i−1）

ai＝al6 ^（8）

whereβノ＝（ρgδH）−11n c andρis constant for all the i values from 1 to m． This equation states the relation area ai of the ith zone to potential energy P9δH（i−1）of unit volurne of the material forming the zone measured from the base leve1． From Eqs．（1）and（8）we

     P＋蓋叢1、，      （9）

       i＝l

where P is the probability that surface material of unit volume has the potential energy ofρgδHσ一1）． Equation（9）is comparable with the equation which describes the canoni・

cal distribution in thermodynamics． H owever， the value of 6H（i−1）is bounded， because there exists limitation for thinness of landforms． And Eq．（9）also differs from the equation to describe the moleculer system or particulate systems proposed by Culling

（1988a，1988b）in the point that the energy term in Eq．（9）is given by the potential energy

while it by the kinetic energy in the latter cases．

  Let give the unknown constantβ in Eq．（8）by

       1      β ＝

      （10）

         K．T．

Then we can compare T． to the absolute temperature T and K． to the Boltzman constantκ．

  Equation（8）shows that the relation of log ai to（SH（i−1）on ordinate is expressed by astraight line on Cartesian co・ordinates． Then the gradient of the straight line is given by−1／（ρgβ 10gの． For a large value ofβ that means the small value of T．，αゴdecreases rapidly as the altitude becomes high． From Eqs．（1），（8）， and（10）， A＝mal for Tc＝0， and then all the iso−level planes coincide with the base level． Hence T． is defined as a

geomorphic temperature ． Substitution ofβノ＝1／K6T． into Eq．（8）yields

     の＝α、〆9δH（i 1）／K・ Tc       （11）

From Eqs．（1）and（11）we obtain

     al＝ m

         Σe一ρ9｛盟（i−1）／κ・Tc          i；1

  The sum appearing in the denominator is regarded as to correspond to the partition function in thermodynamics． Accordingly the partition function ZG of the landform unit is defined as follows：

      m

     ZG＝Σe一ρ酬（i−1）／KG Tc          i＝1

 The geomorphic temperature and the partition function of the landform unit

provide variables analogous to thermodynamic functions． Three examples will be shown in this paper． The total potential energy乙IG of material forming the iso−level planes，

which is almost same with that of surface material of the landform unit， is given by

      m

      ．4Σρ9δπ（i−1）e一ρ96H（ゴー1）／K・Tc      σG＝ i＝l

       ZG

The total potential energy U． is analogous to the intemal energy in thermodynamics．

The entropy S』 of the landform unit is given by the following equation．

     SG−KG ln gmax

       m

where gmax is the maximum value of g． And ln g＝Σ（ln ai！）＝Σ（ailn ai 一 ai）． Using this

       ゴ苫1

      ご＝1

relation we obtain

     S・−K・A（ln A−・）−KbA lnZ・一舞

Then the free energy瓦 of the landform unit is

     F．＝Uc−TbSG

4．Discussion

  The geomorphic temperature and the partition function of a landform unit were derived from the most probable distribution of the potential energies of surface material of the landform unit． These two fundamental quantities readily provided the variables analogous to the internal energy， the entropy， and the free energy in the thermodynamic system． In thermodynamics the most probable distribution of energies in a system means that the system is at an equilibrium state． Therefore the landform unit of the most probable area・altitude distribution can be regarded as to have attained the equilibrium state． Then it becomes a problem that actual landform units satisfy the distribution mentioned above with what extents of deviations．

  Figure 2 shows the area・altitude distribution of the earth surface above sea level， The values of altitude and area to plot the points were obtained from the graph used by Wegener（1924）for explanation of his continental drift theory， Therefore the figures under the abscisa axis mean percentages of areas to the area of the whole earth surface including the part below sea level． The points distribute around the straight line．

Accordingly it might be possible to interpret the figure as mentioned hereafter． The super continent attained the state to have the most probable area・altitude distribution after the lapse of a long period． That distribution is still now preserved essentially in spite of the disturbances by crustal movements and erosion after the continental drift．

  Sakaguchi（1964）studied the area・altitude relation of the four Japanese main islands．

In his study these main islands are regarded as to consist a morphogenetic unit and their area−altitude relation is analysed en bloc． He showed that the relation of area y to

3000

0 0 0 2

1000

0

、

Area（％）

Fig．2 Area−altitude distribution of the earth surface above sea level

The values to plot the points were obtained from Wegener（1924）． The figures

under the abscisa axis are percentages of areas to the whole area of the earth

surface．

altitude x is expressed by three exponential equations． Namely， y＝5．408×10ioe−o・oo73x for 2，700m≦x≦3，100 m，夕＝84220θ一〇・oo27x for 700 m≦x≦2，700 m， and夕＝63330θ一〇・oo24x for 100m≦x≦700 m， where x is given by the medium value of the height of each zone with the interval of 200 m and y is given by the number of intersection points of latitude and longitude lines which are drawn on topographic maps of 1：50，000 with the intervals of one mlnute．

  Sakaguchi（1964）inferred that each equation corresponds to one stage of mor−

phogenetic movement and therefore the geomorphic history of the main islands is divided into the three stages． The landform unit proposed in this paper differs from the morphogenetic unit in definition． The above three equations imply， however， that the thermodynamic analogy derived from the most probable area−altitude distribution can be applied to the morphogenetic unit． Sakaguchi（1964）also inferred that the absolute value of coefficient of x is smaller the shorter is the period elapsed since the cessation of upwarping or the larger the upwarping rate． This is interpreted as that the geomorphic temperature falls with the time elapsed since the upwarping ceased．

  The two examples mentioned above imply that the analogy introduced in this paper will be widely applied． However， further examinations of area・altitude distributions of actual landform units are necessary to give the evidential bases to the analogy．

  It takes a long time for a landform unit to attain the equilibrium state and meanwhile morphogenetic movements will probably happen． This means that we must be compre−

hensive for deviations from the idealistic area・altitude distribution．

  Some students tried to define the equilibrium state of a landform by the most probable state of potential energy loss of running water（Leopold and Langbein，1962；Yang，1971；

Kashiwaya，1979）． There is the problem how to connect the most probable area・altitude distribution to the most probable state mentioned above．

Acknowledgment

 Iwish to dedicate this paper to Prof． Kaizuka in comemoration of his retirement from Tokyo Metropolitan University． Part of this study is financially supported through the Special Research Fund（1988）of Chuo University．

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（＊：in Japanese with English abstract）