The seventh International Conference on Urban Climate,
29 June - 3 July 2009, Yokohama, Japan
FRACTAL GEOMETRY OF THE GROUND SURFACE
AND URBAN HEAT ISLAND
Satoshi Sakai*, Isao Iizawa**, Masanori Onishi†, Miki Nakamura*, Kei Kobayashi*,
Makoto Mitsunaga*, Kimie Furuya*
*Kyoto University, Kyoto, Japan; ** Kyoto Municipal Horikawa Senior High School, Kyoto, Japan
† National Museum of Emerging Science and Innovation (Mirai-kan)
Abstract
The surfaces of the urban areas are covered by large flat surfaces while those of the rural areas are covered by
many plants which have fractal structures. This difference in the surface geometry has great impact on the energy
balance at the ground surface. A small leaf has large thermal conductivity to the surrounding air flow comparing
with a anthropogenic large flat surface in an urban area. This difference in geometrical structure of the surface
results in very high surface temperature at the urban area comparing with that of the rural area.
Key words: Heat Island, Fractal Geometry, Heat transfer
1. INTRODUCTION
In 1980s, the satellite observations revealed that the daytime surface temperatures of the urban areas were very
high compared with the rural area and the heat island was distinct in the daytime (Goward 1981, Carlson et al.
1981). The air temperature, however, shows distinct heat island in the night time and the heat island of the air
temperature in the daytime is not significant. This has been a mystery of the urban climate and it has been
attributed to the complicated urban structures (Roth et al. 1989, Arnfield 2003, Voogt and Oke 2003). This paper
proposes a very simple mechanism which explains this old mystery.
Figure 1 : Real car and miniture model cars under sun shine.
Correnponding authour: Satoshi Sakai , Graduate school of human and environmental studies, Kyoto University,
Kyoto 606-8501, JAPAN
e-mail: [email protected]
The seventh International Conference on Urban Climate,
29 June - 3 July 2009, Yokohama, Japan
2. SCALE EFFECT
It is well known that the body surface of the cars becomes very hot under sunshine, but it is not true for miniature
cars. Figure 1 shows the surface temperature of the full scale car under sunshine and those of miniature cars
placed in front of the full-scale car. It is clear that the temperature of the full-scale car is very high while the
temperatures of the miniature cars are lower by about 20ºC. This can be explained by the scale effect as follows.
Any object under the sunshine becomes hot due to the strong radiation. The temperature difference Δ T
compared to the air temperature increases inversely proportional to the heat transfer coefficient h. This coefficient
for many shapes of objects under constant wind has been intensively measured by wind tunnel experiments in
terms of Nusselt number Nu ≡ h L / λ , where L and λ denote the size of the object and the thermal conductivity
m
of the air. The results show that the Nusselt number depends on Reynolds number as in a power law Nu = c Re ,
where c is a constant, m is a multiple factor ranging from 0.5 to 0.8 depending on the shape of the object (Jakob
1949, Whitaker 1972). The Reynolds number is defined by Re ≡ U L / ν, where U andνdenote the wind speed
and the kinematic viscosity of the air. Substituting Nu and Re, the power law above yields h ∝ Lm -1 (m - 1 < 0),
indicating that larger objects have smaller heat transfer coefficients and higher temperature for given radiation
intensity.
3. STRUCTURE OF THE NATURAL TREE
The leaves of trees are kept relatively cool even under the strong sunshine. In general it is accepted that this is
due to transpiration effect of the tree, but this cooling strategy is not always applicable because the sunshine is so
strong and much larger amount of water than the precipitation is required to compensate the energy of the
sunshine. Clearly, the tree should have alternative strategy to cool down their leaves to save the enzyme for the
photosynthesis.
A possible strategy is to make the leaves small, because small objects have large heat transfer coefficient
comparing with large objects. The trees, however, need large number of small leaves to absorb the sun light for
the photosynthesis and the leaves have to be distributed to minimize the thermal interference. How the leaves of
the trees are distributed in 3-D space?
We have conducted 3-dimensional laser measurements and analyzed their fractal dimensions (Figure 2). The
analysis revealed that the trees have a fractal dimension of 2 although they are spread in a 3-dimensional space.
Mathematically speaking, it means that the trees have 2-dimensional areas but do not have volume, in the limit of
infinite self similarity. This geometry is favorable to absorb the sunlight using 2-dimensional area with minimum
interruption of the 3-dimensional air flow. Therefore, this structure serves to keep surface temperature of the
leaves low.
Over view
Cross section
Canphora
Zelkova
Figure 2 Fractal dimension (Box dimension) of natural trees.
Box dimension
The seventh International Conference on Urban Climate,
29 June - 3 July 2009, Yokohama, Japan
Figure 3: Sierpinski Tetrahedron
4. SIRPINSKI’S FOREST
Being motivated by the results above, we have constructed a fractal sun roof consists of Sierpinski’s tetrahedrons
(Figure 3) which have fractal dimension of 2. The roof was placed at a shopping center at the center of Kyoto city
(Shin-pu kan) in summer 2008 (Figure 4). The roof consisted of about 150 units of Sierpinski’s tetrahedrons and
2
covered about 100 m . It shaded sunshine completely at 13:00 in summer. When the sunlight slightly declined, it
was leaked down through the roof and created pleasant environment as in a forest.
Figure 5 shows fish-eye views under the fractal roof and a parasol in different rays. It was brighter in visible ray
but darker in near infra-red ray under the fractal roof than the parasol. The mean radiant temperature under the
fractal roof was about 8ºC lower than the parasol. It is because the surface temperature of the fractal roof is
relatively low due to high efficiency of heat transfer to the air.
Similar test will be carried out at National Museum of Emerging Science and Innovation (Mirai-kan) in Tokyo from
23th June 2009.
5. CONCLUSION
It was shown that the fractal object consists of “small leaves” can suppress its surface temperature under the
strong sunshine. It can be one of cause of the heat island in daytime shown by satellites observations. It also
suggests new approach to mitigate the heat island.
References
Goward, S.N. 1981. Thermal behavior of urban landscapes and the urban heat island. Physical Geography 2, 1933.
Carlson, T.N., Dodd, J.K., Benjamin, S.G., Cooper, J.N., 1981. Satellite Estimation of the surface energy
balance, moisture availability and thermal inertia. J. Applied Meteol., 20, 67-87.
Roth, M., Oke, T.R., Emery, W.J., 1989. Satellite-derived urban heat islands from three coastal cities and the
utilization of such data inurban climatology. International J. Remote Sensing, 10, 1699–1720
Arnfield, A. J., 2003. Two decades of urban climate research: A review of turbulence, exchanges of energy and
water, and the urban heat island. International J. Climatology, 23, 1-26
Voogt, J. A. & Oke, T. R., 2003. Thermal remote sensing of urban climates. Remote Sensing of Environment, 86,
370−384
Jakob, M. 1949, Heat Transfer. John Wilery & Sons, Inc.
Whitaker, S. 1972. Forced convection Heat Transfer Correlations for Flow in Pipes, Past Flat Plates, Single
Cylinders, Single Spheres, and for Flow I Packed Beds and Tube Bundles. AIChE J., 18, 361-371.
The seventh International Conference on Urban Climate,
29 June - 3 July 2009, Yokohama, Japan
Figure 4 Sierpinski’s fractal roof at Shin-pu kan in Kyoto.
Over View
Visible
Near IR
Thermal IR
Fractal
Roof
Parasol
Figure 5 Fish-eye views under the fractal roof and a parasol in visible, near infra-red, thermal infra-red rays.