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Relationship between area and perimeter

Kathryn Sandals

1Editorial office, Statistics and Mathematics, India

Corresponding Author:
Kathryn Sandals
Editorial office, Statistics and Mathematics, India.


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Following Mandelbrot's fractal theory, the area-perimeter relation is utilized to investigate the geometry of cloud and rain areas measured by satellite and radar between me and 1.2 x 1 square kilometer. The results corresponded to a formula in which the perimeter is roughly equal to the square root of the area raised to the power D (P VKD), where D is the perimeter's fractal dimension. The perimeters of rain and clouds are discovered to be fractals with no obvious horizontal length scale between 1 and 1000 kilometers.


Clouds, as is well known, come in a variety of sizes and shapes. There are no two clouds alike, and some are only a few meters in diameter while others are continents in size. These structures have a "cloudlike" look even in the horizontal projections of clouds recognized by meteorologists in black-and-white satellite images. Similarly, radar rain zones have different geometries that can assist meteorologists in separating rainy patches from ground clutter (noise). These simple geometric truths about clouds (and the attendant rain areas) could not be directly included into any theory until recently due to a lack of an acceptable geometric framework. Since Mandelbrot's fractal work was released, the situation has changed, and there are now simple ways for evaluating them. This paper presents the results of one such technique based on the area-perimeter connection, which demonstrates extraordinary constancy of the horizontal shapes of tropical clouds and rain areas over a six-order-of-magnitude area range. These findings support the idea proposed more than 50 years ago that the outer limit of atmospheric turbulence cannot be "attained within the atmosphere." This indicates that there is no one or preferred horizontal length scale for atmospheric processes, save from scales on the order of the earth's size.

Fractals are shapes with structure at all scales but no fixed length. Within the size range explored, cloud and rain areas are fractals (1 km2 to 1.2 x 106 kmi2). Mandelbrot proposed using area perimeter relations to investigate the structure of planar structures. On a roughly rectangular grid, the shapes investigated in this research are made up of digital clouds and rain images. These photos show the projection of cloud and rain regions onto the earth's surface. Individual clouds (or rain areas) are characterized as a collection of neighboring cloud (or rain) picture parts, with the images divided into cloud-free and rain-free regions. Perimeter P and area A can be estimated by counting the number of picture components for the area and measuring the length of the cloud (or rain) boundary for the perimeter once the individual clouds (or rain areas) have been identified. Other studies, such as aircraft measurements of wind spectra (12), found no evidence of length scales less than 10 m, signaling that greater resolution radar or satellite data is needed to extend in the direction of smaller A. Scale less behavior can extend to centimeter-scale distances, where viscosity becomes relevant, as Richardson's (13) famous atmospheric diffusion experiment shown. A fractal model of the atmosphere, assuming Richardson is true, could be useful over a large percentage of the range of meteorologically significant distances. Rain and cloud fields are physically related in the tropics because they both occur in convective updraft zones, which allow moist warm surface air to rise, cool via adiabatic expansion, and form clouds and rain in the condensation processes that follow. Even around -20° or -30°C, much of the cloud is supercoiled water rather than ice due to the short time scale of these processes. As a result, the -10°C cloud border includes parts of cumulus and cirrus clouds (made primarily of super cooled droplets and ice particles, respectively). GOES photos with a resolution of 4.8 of the Indian Ocean region.

To avoid the effects of shifting image element size, which is largely due to the earth's curvature, only data from the relatively undistorted region between 20°N and 20°S and within + 300 of longitude of the sub satellite point were used. Because the effects of the earth's curvature are the same for both area and perimeter, the A/P ratio is only changed to second order. The most noticeable aspect is the lack of any visible bend or curve over a range of more than six orders of magnitude in area. This is crucial since previous research, which was based mostly on Fourier spectra of wind changes, was confusing concerning the existence of length scales, particularly in the 1 to 1000 km range. When the atmosphere seems thin (scale height of the atmosphere is - 10 km), the gradual transition from a three-dimensional turbulent regime at small scales to a two-dimensional regime at a scale of hundreds of kilometers might possibly lead to such a length scale.

Finally, it's entertaining to speculate about the empirical value of D acquired. There may be a simple association because the figure 1.35 is so close to 4/3 (the value of turbulent isobars). As a result, understanding the physical sources of this value may be essential.