Full Text
Formation of Cellular Structures in Layers of Liquid or Gas
N. S. Shishkin
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
I. Bénard Cells under Laboratory Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462
II. Bénard Cells in Nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
III. Causes of the Formation of Cellular Structures under Natural Conditions . . . . . . . . . . . . . . . . 473
IV. Quantitative Theory of the Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
Introduction
The article considers the emergence of convective motions in a layer of liquid or gas in the presence of an unstable state.
Such a state arises, for example, in a liquid uniformly heated from below and cooled from above by evaporation (with a normal dependence of density on temperature). For water, equilibrium will also be unstable in the case when the lower level of the layer has a temperature of \(0^\circ\text{C}\), while the upper level is maintained at a positive temperature up to \(+4^\circ\text{C}\). In both cases the density of the liquid increases with height.
Under random disturbances of equilibrium, the lighter liquid below tends to rise upward in the form of separate jets, while the heavier liquid tends to descend from the upper levels downward. If the anomalous distribution of density is maintained for a sufficiently long time, an ordered convective motion is gradually established in the liquid. The liquid acquires a cellular structure. In each of the cells there is a closed circulation of liquid.
Such cells are called Bénard cells¹, after the French scientist who studied them in detail in 1900 under laboratory conditions. Among later laboratory investigations of the cellular structure of liquids, one should note the experiments of Mel² with solidifying spermaceti containing suspended aluminum powder, and the experiments of Schmidt and Saunders³ with water.
Under natural conditions an unstable state of a liquid, favoring the formation of cells, arises in the Arctic and Subarctic—
in springtime thawing of soils containing a large amount of moisture, and in the presence of permafrost, which maintains a temperature of about \(0^\circ\mathrm{C}\) at the lower level of the liquid layer; and in the summer–autumn period, if the temperature of the upper level is positive and close to \(4^\circ\mathrm{C}\) (see the papers by Low \(^{4}\) and Gripp \(^{5}\)).
In zones where there is no winter freezing of the soil, Bénard cells may arise during the drying of liquefied soil owing to the fact that the upper layer of the liquid is intensively cooled through evaporation (a description of surface cells on solonchak soils in Iran is given, for example, in the paper by Shtekh \(^{6}\)).
Under atmospheric conditions the formation of convective cells is possible if the potential temperature of the layer under consideration decreases with height (see the paper by Brunt \(^{7}\)). For example, with a superadiabatic temperature gradient and unsaturated air, the temperature of a packet of overheated air that has begun to rise decreases according to the adiabatic law, and it will remain warmer than the surrounding air. Consequently, the ascent will continue until it enters a layer with a smaller temperature gradient or an inversion layer.
The ascent of air in some places will cause its compensating descent in other places, and, in the presence of a sufficient temperature difference between the lower and upper bases in a layer of considerable horizontal extent (as compared with the thickness of the layer), a regular convective circulation will arise. This is, in particular, the cause of the formation of clouds of cellular structure.
Under laboratory conditions, the occurrence of cells was observed in a layer of smoke introduced into a Walker chamber, heated from below and cooled from above (see the papers by Chandra \(^{8}\) and Brunt \(^{7}\)).
Theoretically, the question of the emergence of a cellular structure was first investigated by Rayleigh \(^{9}\) in 1916 for the simplest case of cells having the form of a rectangular four-sided prism. He derived a criterion for the conditions under which convective motion arises. Further development of the theory was given by Jeffreys \(^{10}\) in a series of papers in 1926–1928 and especially by Pellew and Southwell \(^{11}\) (in 1940), who investigated the question for the general case of cells of regular symmetrical form.
The application to the atmosphere, taking into account the phenomenon of turbulent exchange, was considered by Leyle \(^{12}\) in 1941.
I. BÉNARD CELLS UNDER LABORATORY CONDITIONS
Bénard \(^{1}\) placed a thin layer of liquid up to \(1\ \mathrm{mm}\) thick on a metal plate heated from below and maintained at a definite temperature. From above the liquid was cooled owing to evaporation. In this way a layer of liquid was produced with density increasing from bottom to top. In the initial stage numerous
disordered jets directed from below upward, and compensating descending streams. Then the number of jets decreased and cells of irregular shape appeared, so that the surface was divided into polygons with 4–7 sides.
Finally, after a time determined for the given liquid and the given temperature of the underlying surface, a stationary state was established in which all the cells had a regular hexagonal shape (Fig. 1)*.
The time required for the establishment of the stationary state at a temperature of the metal plate of \(100^\circ\mathrm{C}\) was, for alcohol and benzine, 1–2 sec.; for paraffin, 10 sec.; for viscous oils—several minutes.
In the center of each cell there was an ascending stream of liquid; at the periphery of the cell, a descending motion (Fig. 2).
Fig. 1. View of Bénard cells in spermaceti with graphite.
Fig. 2. Vertical section of a Bénard cell with the circulation scheme.
The width of each cell was approximately equal to three times the depth of the liquid layer.
Mál\(^2\), in 1931, obtained a completely analogous pattern for spermaceti mixed with aluminum powder. He further established that, when the temperature difference between the lower and upper boundaries of the liquid is increased, the horizontal dimensions of the cells increase. If the bottom of the vessel is inclined, then, as the depth of the layer decreases, the cell dimensions decrease up to a certain limit (determined by the temperature regime), beyond which cells are absent. When the depth of the layer is increased, the cells grow, but, beginning with some definite depth, the growth ceases and the cells retain their size (apparently, in this case the circulation embraces not the whole liquid layer, but only a bounded part of it).
Interesting phenomena were observed by Mál during the solidification of spermaceti and its subsequent melting owing to heating from below. When only a thin crust of solid substance remained on the surface, regular cells of hexagonal shape arose, and
* The figures are borrowed from the cited works.
in the middle of the cells a bright circle with a dark “star” in the center was formed (Fig. 3).
The “star” was formed because the ascending stream of heated spermaceti caused the crust to melt in the center.
At the periphery the crust broke owing to the descending stream of liquid; the circle thus represented a line of equal velocities of the descending stream. Theoretically this was explained by Pellew and Southwell (see IV).
After the entire upper solid film had melted, the ordinary pattern of cells was restored.
Schmidt and Saunders3 investigated by an optical method the formation of cellular structures in a layer of water, making use of the dependence of the refractive index on temperature. The deviation of rays in the vertical direction, dependent on the vertical temperature gradient, proved to be stronger for the lower part of the layer than for the upper. In the middle part of the layer the deviation of rays was insignificant, i.e. the vertical temperature gradient was small. Convection, as in the experiments of Bénard and Mêl, likewise had ascending streams in the centers of the cells.
Fig. 3. Cells in spermaceti, heated from below, in the presence of a thin solid film on the surface. Taken at the moment when destruction of the film began.
Analogous experiments were carried out by them for air. The number of cells in air, for the same dimensions of the layer under investigation, proved equal to the number of cells in water.
Chandra6,7 performed a series of experiments with smoke admitted into Walker’s chamber. The chamber had a metal bottom, heated from below by means of an electric furnace, and a glass cover, cooled by water or liquid air.
Fig. 4. View of cells on the surface of smoke in Walker’s chamber.
The side walls had felt thermal insulation.
Circulation at a chamber height of 10 mm arose when the temperature difference between the bottom and the cover reached 11.4°. On the surface the smoke formed polygons (Fig. 4).
In those places where there were translational motions of the mass of smoke, the cells had an elongated form.
If the upper lid was moved in the horizontal direction by means of an electric motor (the lid was made 3 feet long), then at first the cells were stretched out along the direction of motion of the lid, as though crawling over one another, and then merged, forming smoke “ridges.”
With a small temperature difference the convection had a descending flow in the center of the cell, while when it was increased to a certain critical value, owing to heating from below, a transition occurred to circulation with an ascending flow in the center.
It is interesting to note that, with a chamber height of 6 mm, it proved impossible to obtain motion of the first type when the lid was cooled with water, however much the bottom of the chamber was heated, whereas it was obtained if the lid was cooled with liquid air.
Fig. 5. Convection in a thin layer of smoke covering the bottom of Walker’s chamber.
In a chamber 12 mm high, in a thin layer of smoke lying on the bottom of the chamber, it was possible to obtain a pattern of fairly regularly arranged round clearings, associated with descending air currents in the centers of the cells (Fig. 5).
Thus, a number of authors were able to establish experimentally that a layer of liquid or gas heated from below and cooled from above breaks up into cells. Within each of the cells a convective motion arises, continuing as long as the necessary temperature difference between the upper and lower bases of the layer is maintained.
II. BÉNARD CELLS IN NATURE
As has already been indicated in the introduction, cellular structures occur in nature in two principal forms:
a) the microstructure of clouds,
b) the microrelief of the earth’s surface under strong moistening of the surface layer.
§ 1. Description of cellular structures in the atmosphere
A regular microstructure of clouds is observed mainly in upper-level clouds, where it is possible for homogeneous conditions to be created in a layer of considerable horizontal extent.
The main types of cellular structure of clouds are:
1) Systems of small cloudlets of round or irregular shape (Fig. 6) may form both through the breaking up of a continuous stratiform cloud and in a cloudless sky. In the latter case, as they gradually spread out, they may lead to the formation of a continuous veil.
2) Systems of rectangular clouds (usually of considerable size), separated by narrow gaps forming mutually perpendicular lines. As in the preceding case, a rectangular
Fig. 6. Cellular structure of a cloud with an ascending current at the center of the cell.
structure may form both from continuous stratiform cloudiness and in a cloudless sky. The ratio between the horizontal dimensions of the cloud and the thickness of the layer must be 4:1 for a square cloud shape (according to Rayleigh’s theory).
3) A “lace” structure with regularly arranged round or oval gaps (Fig. 7). The shape of the cloud indicates that there is descending motion at the center of the cells.
4) Systems of ridges (strips), separated by more or less parallel gaps (Fig. 8). The distance between neighboring ridges, according to the data of Sprung and Schöring (see Malk’s article[^2]), averages 250 m when the thickness of the cloud layer is 70 m, i.e., the ratio of the width of the cell to its height is 3.5.
The ridges have a great extent in length; their outlines are blurred and their form is often curved.
The stretching of cells into ridges occurs in the presence of variation of wind with height. The direction of the ridges coincides with the direction of the relative
… of a strong wind at the upper and lower boundaries of that layer, in which an unstable state was created.
Ridges should not be confused with Helmholtz waves, which have an entirely different nature. Waves form at the boundary of two layers under stable stratification, owing to a disturbance of the equilibrium state (similar to waves on the sea surface). The external appearance of Helmholtz waves is characterized by smoother edges and greater regularity of structure. Their orientation in space also differs from the orientation of ridges. Helmholtz waves are arranged perpendicular to the relative wind characterizing both air layers. They form both in a cloudless sky, in the form of rather long bands, and from continuous stratiform clouds—in the form of a small number of short bands, usually perpendicular to the general elongation of the cloud system.
Fig. 7. Cellular structure of a cloud with a descending current in the center of the cell.
Fig. 8. Cloud ridges (strips).
In addition to the cases considered above, when the division into cells embraces the entire thickness of the cloud, the formation of cells on the upper or lower surface of the cloud is also observed.
In the first case the upper level of the cloud resembles a hilly surface with a more or less regular structure (Fig. 9). However, the hilliness of the upper surface of a stratiform cloud, seen from an airplane, by no means always indicates an established circulation. This observation applies especially to the lower cloud tier, where the influence of the surface layer is strong and individual jets may penetrate into the inversion layer, creating irregularities of the upper surface.
A nonequilibrium state at the lower surface of clouds leads to the formation of mammato cumulus (Fig. 10). These are udder-shaped hemispherical clouds hanging downward. When there is a relative velocity of motion in the underlying layer of air, these hemispheres may be drawn out into bands (like ridges), sharply visible against the background of the cloud.
Let us also point out that clouds not infrequently exhibit a cellular structure of several orders, when smaller cells arise within large cells. For example, in the case of rectangular clouds, several rectangular checkers as it were combine into one
Fig. 9. View of the surface of a layered cloud from an airplane.
group, separated from neighboring ones by wider openings. The same phenomenon can be observed in the case of ridges.
As for cumulus clouds, they arise under conditions in which the influence of the earth’s surface is still quite strong. Therefore the homogeneity of conditions at the upper and lower boundaries of a layer of great horizontal extent, necessary for the establishment of a regular structure, is in fact never attained. Apparently, the phenomenon is limited to the phase of separate disordered jets.
It is also interesting to note that the scheme of air currents proposed by Palmen¹³ to explain funnels and mountains of the tropopause very much resembles the circulation in cells corresponding to the antisymmetric solution of the problem (see Fig. 22).
However, it should be borne in mind that the theory of steady convective circulation is not applicable to cyclones and anticyclones, since the occurrence of homogeneous boundary conditions required for its establishment over considerable areas of the earth’s surface, with linear dimensions of thousands of kilometers, is impossible.
It may rather be said that, as in the case of cumulus clouds, the process develops only in its first phase—the phase of separate jets in those places where the disturbance has arisen. Above a cyclone there is an ascending current in the troposphere. The development of vortices around a vertical axis causes the formation of funnels of the tropopause, with its level being lowered in individual cases by 4–5 km.
In the stratosphere, accordingly, a descending current arises. At the periphery of the cyclone compensating movements of air occur. Above an anticyclone the direction of the air currents is the reverse.
Fig. 10. Mammato cumulus.
It can hardly be admitted that these currents close into a regular circulation, all the more so since in that case the ratio between the horizontal and vertical dimensions of the cells would be excessively large, of the order of 100:1 (the distance between the cyclone and anticyclone being of the order of 1–2 thousand km, and the height of the tropopause, i.e. half the height of the cell, of the order of 10 km).
§ 2 Description of Cellular Structures of the Microrelief of the Earth’s Surface
Let us proceed to a brief description of some cellular forms of microrelief encountered in areas with considerable moistening of the upper soil layer, characteristic mainly of the Arctic and subarctic zones.
Three principal forms of microrelief of cellular structure may be indicated:
1) Hummocky microrelief6,14 arises on level, waterlogged terrain. The hummocks are distributed rather uniformly over the whole area (Fig. 11).
The area between hummocks often consists of boggy “quagmires.” The height of the hummocks usually does not exceed 0.4–0.5 m, but hummocks up to several meters high (“bultunyakhs”) also occur. Cases have been noted^15 when the hummocks do not reach the bottom of the bog, but float in a fluid, quaking mass.
Fig. 11. Finely hummocky microrelief in Iceland.
It is characteristic that the height of the hummocks is usually the greater, the deeper the layer of permafrost lies.
2) Ridges^6 usually form on slopes with a steepness of more than 2° and extend downward along the slope, resembling garden beds (Fig. 12).
3) Fissured polygons^14,16,17 are characteristic of areas with homogeneous fine-grained soils. Cracks of regular shape form on the surface either hexagons or quadrilaterals. When their dimensions are small, the areas of the hexagons are usually convex in the middle (Fig. 13).
Fig. 12. Ridges extending along a slope, in Reykjavík.
But hexagonal polygons of larger dimensions (up to 40 m), the so-called Taimyr polygons, also occur (Fig. 14).
A fine photograph of the Taimyr polygons, taken from the airship Graf Zeppelin during its flight over the Arctic in 1932, is reproduced in the article by A. I. Gusev^16.
Orthogonal systems of cracks (Fig. 15) often have raised edges, which is explained by their being pushed upward when the water in the cracks freezes. The fields of quadrilaterals are usually level, swampy, and sometimes have a small lake in the middle.
With heterogeneity of the material covering the surface, forms of microrelief of cellular structure also arise.
Thus, Low^4 describes mud spots forming on the surface of a flat moraine during the spring thawing of the ground (Fig. 16). The mud spots have a shape close to circular and are surrounded by clean stones. The distance between the centers of the spots is 3–4 m, which agrees well with Bénard’s laboratory data for the ratio between the horizontal and vertical dimensions of cells, if one assumes that the ground thaws to a depth of up to 1 m.
Fig. 13. Hexagonal polygons on Spitsbergen.
Apparently, a more developed form is represented by the so-called “stone rings” and “stone polygons”^18,19, which arise when there is a significant quantity of fine earth in the composition of the soil.
Fig. 14. “Taimyr polygons” on Spitsbergen.
They consist of fine-grained material in the center, forming a convex circle or polygon, and of a stone border around it. The borders are either isolated stone rings or a stone network.
Accumulations of fine earth have the form of a cone with its apex turned downward (see the article by A. A. Grigor'ev^18). The surface of the fine-earth
Fig. 15. Rectangular polygons on the second terrace of the Beder-anara River (a tributary of the Lena River).
core is convex; its width reaches 1–1.5 m; the top of the core rises 20–25 cm above the rim. The height of the stony border above the same level is about 30 cm, and the width of the border is 30–50 cm. Differentiation of the materials is observed to a depth of 50–60 cm.
Fig. 16. Mud spots formed during the thawing of soil in Spitsbergen.
Similar polygons, but of smaller size (diameter up to 50 cm), are encountered on glaciers covered with fine earth.^19
In the regions of the boreal zone, the tussock bog constitutes a formation analogous to the fine-hummocky microrelief of the subarctic.
It is quite possible that cellular structures also form in the liquid metallic core of the terrestrial globe, where the temperature increases toward the center of the earth, while the density practically does not change with pressure. Ya. I. Frenkel'^20 proceeds from the existence of such structures in his theory of terrestrial magnetism.
III. CAUSES OF THE ORIGIN OF CELLULAR STRUCTURES UNDER NATURAL CONDITIONS
(A Qualitative Theory)
After having given a brief survey of cellular structures arising in layers of gas or liquid under natural conditions, we shall proceed to elucidate the causes determining the concrete forms of these formations.
§ 1. Cellular structures in the atmosphere
As was already indicated in the introduction, the condition of instability for a layer of air consists in the fact that in this layer the potential temperature decreases with height, i.e. the temperature decreases with height more rapidly than according to the adiabatic law. When this condition is fulfilled, convective motions may arise in the layer, and consequently cellular structures as well, which are revealed especially clearly in the structure of clouds.
For the vertical temperature gradient in stratiform clouds, divided into small cloudlets or into ridges, Mell² gives the following data, obtained during airplane ascents:
Fig. 17. Stone polygons on Spitsbergen.
| Date | Maximum temperature gradient (in degrees per 100 m of height) | Thickness of the layer in which this temperature gradient is observed (in m) |
|---|---|---|
| 1/IX 1928 | 1.40 | 65 |
| 13/IX 1928 | 1.10 | 88 |
| 14/IX 1928 | 1.23 | 171 |
| 15/IX 1928 | 1.11 | 144 |
| 24/X 1928 | 1.02 | 98 |
| 5/XI 1928 | 1.10 | 75 |
| 7/IX 1929 | 2.30 | 55 |
| 16/IX 1929 | 1.00 | 103 |
The mean of the maximum temperature gradients according to the data of 8 observations is equal to 1.28. These data are not entirely reliable, since, first, the airplane ascended along an inclined curve and could enter different temperature conditions not coinciding with the vertical
distribution of temperature, and, secondly, the temperature measurements themselves by aircraft instruments could give considerable errors.
It is, however, indisputable that in the case of clouds of cellular structure the temperature gradients reach appreciably larger values than in ordinary layered clouds.
Favorable conditions for the formation of cellular structures in cloud layers may arise in two ways (see Brent’s article[^7]):
1) Through the absorption and emission of long-wave radiation by layered clouds. The vertical temperature gradient may reach large values if the lower part of the cloud is heated by absorption of radiation coming from below, while the upper part is cooled by emission of long-wave radiation upward. Short-wave radiation, as Brent points out, is more likely reflected by clouds of the St and As types than absorbed, and cannot substantially affect their temperature regime.
Fig. 18. Scheme of circulation in cloud layers in the presence of an unstable state.
2) Through the raising or lowering of the entire mass of air containing both cloud layers and layers of dry air. In this case the temperature of the dry air changes by \(1^\circ\mathrm{C}\) when the height changes by \(100\ \mathrm{m}\), while the temperature of moist air changes by \(0.5^\circ\mathrm{C}\). As a result, during general lifting the cloud layer proves to be warmer than the adjacent layers of dry air, and an unstable state arises with respect to the overlying layer. When moist air descends it becomes colder than dry air, and instability of the cloud mass arises with respect to the underlying layer. General lifting occurs, for example, during motion along a frontal surface, while the sinking of air masses is observed in the rear of thunderstorms.
To explain the possible forms of clouds of cellular structure, we propose the following scheme (Fig. 18).
a) Unstable state at the upper boundary of the cloud. (Cells of the upper row.) The cloud has a higher temperature than the overlying dry air, for example, owing to general
rising of the air mass. In case \(A\), when a jet of moist air rises into a layer with a lower temperature, additional condensation of water vapor occurs, accompanied by the release of heat. This, in turn, will promote further ascent. The ascending jet is active; it is, so to speak, the leading jet. At the same time, a compensating, slower motion occurs in the space surrounding the jet. A circulation develops with an upward current at the center of the cell. The surface of the cloud will thereby acquire a hilly appearance, as in Fig. 9. Reverse circulation (case \(A'\)) cannot develop, since a jet of cold dry air directed downward, when mixed with the cloud mass, will produce a decrease in humidity, and some of the droplets will evaporate, causing cooling of the cloud. The temperature difference between the cloud and the overlying layer of dry air will begin to decrease, and the process dies out.
b) An unstable state within the cloud layer. (Cells of the middle row.) It may arise through cooling of its upper layers by radiation directed upward, and heating of the lower layers due to absorption of radiation coming from below. With a temperature gradient in the cloud greater than the moist-adiabatic one, a rising parcel of moist air remains at all times warmer than the surrounding cloud, while a descending parcel remains at all times colder than it. Compensating flows arise outside the jet. As a result, convective motion is produced, encompassing the layers of air adjacent to the cloud as well. The direction of the circulation must depend on additional causes, since both directions of motion are active. Such causes may be: 1) the temperature distribution, 2) the humidity distribution.
Since a convective cell may be regarded as a thermodynamic engine, for which the underlying layer is the heater and the overlying layer the refrigerator, the temperature of the air beneath the cloud must in any case be no lower than the temperature of the lower base of the cloud, and the temperature of the air above the cloud must be no higher than the temperature of the upper base of the cloud.
The magnitude of the temperature gradient in the upper and lower parts of the nonequilibrium layer may prove important. A high temperature gradient in the lower part of the layer should promote the formation of ascending jets, increasing the upward transfer of heat. In exactly the same way, a high temperature gradient in the upper part of the layer should cause the appearance of descending jets. This conclusion agrees with the results of the laboratory experiments described above by Schmidt and Saunders for water and air, and of Chandrasekhar’s experiments for smoke.
The influence of the humidity of the air beneath the cloud may also be significant in establishing one or another type of circulation. If the humidity is high, then the rising jets produce convection at the level of the cloud base and, consequently, additional release
heat, which activates the ascending current. The descending current is not active. With dry air beneath the cloud, on the contrary, the downward transport of cloud mass causes evaporation of droplets and, consequently, cooling of the surrounding air, which activates the descending current. The ascending current in this case is inhibited.
Thus, in the presence of a nonequilibrium state within the cloud, one may expect the stratiform cloud to break up into separate small cloudlets (Fig. 6) in the case of convection with an ascending current at the center of the cell (case \(B\)), and to break up into a lace-like structure with regularly arranged round openings (Fig. 7) under convection with a descending current at the center of the cell (case \(B'\)).
c) A nonequilibrium state at the lower base of the cloud. (Cells of the lower row.) It arises either when the entire air mass, containing both moist and dry air, descends, or as a result of warming of the layer of air beneath the cloud owing to turbulent inflow of air from below, from the earth’s surface. In the first case the relative humidity of the air beneath the cloud is, as a rule, low. An ascending current of warm dry air penetrating into the cloud causes evaporation of some quantity of droplets (and therefore cooling of the surrounding air mass) and is inhibited. Conversely, a descending current of moist air into a layer of dry air is active. If the cloud is sufficiently thick, ever new cloud masses will be carried downward, the humidity of the air will increase, and evaporation will weaken. A cloud of the mammato cumulus type arises (see Fig. 10). With considerable cooling of the entire mass of the underlying layer of dry air, the cause of the instability disappears and the mammatos dissipate. This process proceeds especially rapidly if rain begins to fall from the cloud.
If the air beneath a stratiform cloud is moist, then the ascending current develops in the same way as in a cumulus cloud and, as indicated above, regular circulation cannot become established. A cloud of the Castellatus type arises. Descending currents, accompanied by slow evaporation, lead to gradual cooling of the underlying layer and, consequently, contain within themselves the cause of their own extinction.
The cellular structure of clouds considered above occurs when, in the cloud layer and at its boundaries, there is no change of the horizontal wind speed with height. In the presence of relative wind there is observed, as in the picture of smoke in Walker’s chamber, elongation of the cells and their transformation, under the corresponding conditions, into cloud rows (see Fig. 8). It is precisely a slight relative wind that causes deformation of the streamlines and, consequently, asymmetry of the circulation. If the drift by the relative wind during one cycle is of the same order of magnitude as the horizontal size of an ordinary cell, then the visible boundaries between neighboring cloud cells in the direction of the relative wind are erased, and the group of cells merges into a row.
§ 2. Cellular structures of the microrelief of the earth’s surface
The role of convective circulation in the formation of cellular structures of the microrelief of the Arctic and Subarctic has already been pointed out by Low4, Gripp9, and a number of other authors. Their ideas, however, have met with numerous objections. The most important of these are the following (see, for example, Gladtsin’s article19):
1) The temperature of the ground surface undergoes considerable fluctuations, and a state in which a definite temperature difference is maintained even for several hours in some layer (about \(4^\circ\) C above and \(0^\circ\) C at depth) is seldom realized. At the same time, convective motions at high viscosity are slow.
2) The moisture content of soils in many cellular structures of the microrelief is insufficient to satisfy the Rayleigh criterion; according to Steche’s calculations6, the necessary moisture content is of the order of 60%, i.e., it lies at the limit of soil saturation.
3) The mechanical forces developed during convective circulation are insufficient to move stones and, consequently, with its aid it is impossible to explain the formation of stone rings, stone polygons, and stone stripes.
These objections are based chiefly on Steche’s calculation, which rests on a misunderstanding. First, Steche took for the coefficient of thermal diffusivity \(\chi\) of liquefied soils the value 0.2, whereas in reality the mean value is of the order of \(0.005\ \text{cm}^2/\text{sec}\)23; second, for the case of liquefied soils situated on permafrost or dense ground, one must apply the criterion \(K \geqslant 1100.65\) (see IV, § 3), and not the Rayleigh criterion (23).
Introducing these corrections leads to the conclusions that: a) circulation can arise at a much lower soil moisture content than that obtained by Steche; b) fluctuations of the temperature of the upper surface by several degrees will not violate the instability criterion when the layer is sufficiently thick.
To clarify the possible role of convective motions, let us turn to concrete forms of microrelief.
- Hummocky microrelief. Let us consider, by way of example, the description of hummocks in the region of Lake Shchuchye on the Kola Peninsula, given by M. I. Sumgin15. The hummocks formed in a bog consisting of a liquid quicksand-like mass up to 2 m deep, covered above by a floating layer of peat. The average height of the hummocks is about 1 m. Excavation of pits in the hummocks showed that they consist of an upper peat layer and a frozen mineral core containing sandy loam and a silty-clayey mass. Upon thawing, this mass turns into quicksand. The frozen core often does not reach the bottom,
and floats in the liquid mass of the bog. In the intervals between the mounds the peat layer thaws completely.
Having visited the same area four years later, M. I. Sumgin^21 found that the mounds he had investigated had settled onto the firm ground forming the bottom of the bog, and had become lower.
Summer measurements of the temperature of the floating mass of the bog showed that at its bottom the temperature is \(2^\circ\mathrm{C}\), and immediately beneath the peat layer \(3.6^\circ\mathrm{C}\).
Thus, the conditions for the occurrence of convective motion in the liquid layer are present. There is the necessary inversion in the distribution of density with height. The low thermal conductivity of the floating peat layer ensures considerable stability of temperature at the upper level of the floating mass. Ascending circulation currents bring upward soil particles, which are retained in the peat layer. Conditions are provided for a certain raising of the peat cover above the centers of the cells. At the beginning of winter, beneath the mounds, in their initial stage of development, lenses of frozen floating mass arise both because of the better conditions for freezing of the hummocky relief and because of the inflow from below of liquid having a lower temperature. Expansion of the soil during freezing promotes the further bulging of the mound. In turn, the frozen lens favors the development of descending currents of the heavier liquid mass, precisely in the intervals between the mounds, since the temperature of the floating mass near the lens is lower, and consequently its specific gravity is less. In subsequent years the circulation develops in the same places, and the process of growth continues.
The more uniform the conditions over the entire area, the more regular the system of mounds will be. The fact that the size of the mounds (and the distances between them) is the greater, the thicker the layer of floating mass, is in full agreement with experimental data under laboratory conditions. The ratio of the distance between mounds to the thickness of the layer of liquid soil should be of the order of \(3:1\).
The participation of convection in the formation of mud spots (see Fig. 16) on the level moraine fields of Spitsbergen, when sufficiently moist soils are present during spring thawing, appears quite natural. Stones serve, as it were, as a rigid framework that does not disturb the basic conditions of circulation.
As for stone rings and polygons, convection during the thawing of the upper soil layer may lead to a redistribution of the fine earth, increasing its share in places of ascending currents and washing it downward where descending currents occur. Subsequently other forces also act, connected with colloidal swelling of the soil (see the article by Steche^6), the action of frost, etc.
The question of the origin of convection under these conditions still requires investigation. It is possible that the most favorable conditions for circulation are created in the case when there is a значи-
...relative proportion of fine-grained material, insoluble in water, that permits the circulation of water in the spaces between particles.
The removal of the finest particles by water during periods when there is no circulation may lead to the formation of regularly arranged islands of gravel among large stones.
The migration of stones is apparently conditioned by the processes of their being washed out of the fine earth during alternating freezing and thawing of the ground (Auffrieren in Steche \(^{6}\)) and the subsequent sliding downward of stones raised onto the convex surface of the fine-grained core.
We touch upon this question only briefly, wishing merely to draw attention to it, without attempting to give an exhaustive explanation, since there is not enough material for a definitive conclusion.
In the formation of bog hummock fields, spring thawing of the soil must play an essential role, when an unstable state exists in the shallow thawed layer. Mineral particles brought to the roots of the vegetation cover by the ascending currents of the circulation favor the growth of vegetation (various species of sedges and other plants). Descending flows, on the contrary, expose the root system and contribute to the death of the vegetation cover (if it existed), and create depressions in the relief, in which water usually accumulates. Annual deposits of organic mass lead to the further growth of the hummock field.
-
Ridges. The formation of ridges on slopes in the presence of quick soils sliding downward over a solid frozen underlying surface is analogous to the formation of cloud streets. The braking of the lower layer by friction against the solid surface causes the velocity to change with height and, consequently, the cells to be stretched along the slope. Such elongated cells, as it were overlapping one another, are in fact observed on gentle slopes. Under appropriate conditions (steepness of the slope, viscosity of the liquid soil, etc.) ridges arise.
-
Fissure polygons. The formation of fissure polygons is usually explained by the cracking of soil either as it dries, or during severe frosts (see the articles by Steche \(^{6}\), S. V. Obruchev \(^{17}\), A. I. Gusev \(^{16}\)). However, the regularity of the arrangement of fissures, often forming a strictly orthogonal system or a system of hexagons, has not found sufficient explanation in these theories. At the same time, the exceptional similarity of the form of fissure hexagons to the pattern of the surface of a liquid in the laboratory experiments of Bénard, Mesl, and other authors, as well as the similarity of orthogonal fissures to the gaps of quadrangular clouds, is striking.
One may imagine the occurrence of convection in clayey liquid soils with a gradual decrease in moisture under conditions close to the cessation of fluidity. Further drying of the soil may lead to the fixing of the established form of the surface—
...to the appearance of cracks in places where the microrelief is depressed. In exactly the same way, frost cracking will be facilitated in those same places.
Thus, convection in soil that has not yet solidified can prepare the conditions for subsequent cracking under the action of other forces. Then the natural and convex form of the hexagonal cups between the cracks becomes clear.
Fig. 19. Polygons of the 1st and 2nd orders on Spitsbergen.
The appearance of cells of the 2nd order (Fig. 19) may be explained by the fact that, with subsequent moistening of the upper soil layer, the moisture no longer penetrates to a great depth because of the cracks, and an unstable state is created in a thinner layer, which leads to the formation of cells of smaller size; moreover, cells of the 2nd order are formed chiefly in the depressed parts of the polygons, where moistening is greatest.
Hexagonal figures on the soil surface are also encountered in southern regions (for example in Iran, see Fig. 20, borrowed from the article by Shtekhe[^6]). There the unstable state in the layer of liquid soil arises owing to intense evaporation from the surface, accompanied by cooling.
This may also explain the flat or even concave form of the cups, since drying of the surface proceeds very intensely and a strong contraction of the upper layer of the crust arises.
Fig. 20. Polygons on desiccated solonchak soil in Iran.
The rectangular form of polygons, just like the hexagonal one, is fully explained by the theory (see below), since the appearance of an orthogonal system of cracks is entirely possible under the corresponding conditions. What these conditions are, determining the form of the polygons, is still unclear. It is possible that conditions at the lateral boundaries play an essential role.
layer. It has been noted that the direction of the cracks in the orthogonal system is always parallel and perpendicular to the edge of the terrace or the shore of the reservoir. With curvilinear boundaries of the area on which polygons arise, they have a hexagonal or irregular shape.
IV. QUANTITATIVE THEORY OF THE PHENOMENON
In the present chapter we give a brief exposition of the article by Pellew and Southwell[^11], in which a theory is developed for convective motions in a layer of liquid with an anomalous density distribution, for any symmetric form of the cells.
Let us consider a layer of liquid of height \(h\), the liquid being assumed incompressible. At the sides the layer is bounded by heat-insulating walls. Under laboratory conditions this was achieved by covering the side walls of the chamber with felt or other materials.
Under natural conditions heat transfer proceeds chiefly in the vertical direction, and heat losses to the sides may practically be neglected (in the absence of general transfer of the whole mass of liquid).
The lower and upper surfaces of the layer are assumed to be horizontal and heat-conducting. If the latter condition were not fulfilled even for one of the surfaces, convective motion could not become established, since the temperature throughout the whole layer would gradually be equalized by heat conduction. The impossibility of convective motion in this case also follows directly from the second law of thermodynamics.
The density of the liquid in the layer increases with height, for example, owing to uniform heating from below and cooling from above.
§ 1. General equations
In order to investigate the motion of the liquid, we shall write the system of equations:
a) the equation of motion in Eulerian form
\[ \rho \frac{d}{dt}(u, v, w)=\rho(0,0,-g)-\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right)p+\nu\rho\Delta(u,v,w), \tag{1} \]
where \(u, v, w\) are the velocity components of a liquid particle passing through the given point \((x,y,z)\), \(\rho\) is the density of the liquid, \(p\) is the pressure, \(\nu\) is the coefficient of kinematic viscosity, \(\Delta\) is the Laplace operator.
The first term on the right gives the force of gravity (the \(Z\)-axis is directed upward), the second is the pressure gradient, and the third is the frictional force.
b) the equation of continuity
\[ \frac{d\rho}{dt}+\rho\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}\right)=0. \tag{2} \]
c) The equation for the dependence of density on temperature
\[ \rho=\rho_0(1-a\vartheta')=\rho_0(1-a\beta z), \tag{3} \]
where \(a\) is the coefficient of expansion, \(\vartheta'\) is the temperature of the liquid, and \(\beta\) is the vertical temperature gradient (assumed constant). At the lower level \((z=0)\) it is assumed that \(\vartheta'=0\).
d) The heat-conduction equation
\[ \frac{d\vartheta'}{dt}=k\Delta\vartheta', \tag{4} \]
where \(k\) is the coefficient of thermal conductivity.
To solve this system of equations, the following simplifying assumption is made: the velocities \(u, v, w\) in the convective motion and the deviations of temperature, density, and pressure from their equilibrium values (caused by convection) are assumed to be small, so that their second and higher powers may be neglected.
Then, instead of (2), one may write
\[ \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0, \tag{2'} \]
since \(\frac{1}{\rho}\frac{d\rho}{dt}\) will be a quantity of the second order of smallness if (3) is substituted and it is taken into account that \(a\) is small.
Eliminating \(u, v, p\) from the three equations (1) and equation \((2')\), we obtain, instead of (1):
\[ \left(\frac{\partial}{\partial t}-\nu\Delta\right)\Delta w=ga\Delta_1\vartheta, \tag{5} \]
where \(\vartheta\) denotes the deviation of the temperature from its equilibrium value and
\[ \Delta_1=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}. \]
Instead of (4), after a series of transformations, we obtain:
\[ \left(\frac{\partial}{\partial t}-k\Delta\right)\vartheta=-\beta w. \tag{6} \]
From the two equations (5) and (6) one may eliminate \(\vartheta\), and we shall have one sixth-order equation for determining \(w\):
\[ \left[\left(\frac{\partial}{\partial t}-\nu\Delta\right)\left(\frac{\partial}{\partial t}-k\Delta\right)\Delta+a\beta g\Delta_1\right]w=0 \tag{7} \]
or, eliminating \(w\), an analogous equation for \(\vartheta\).
Consequently, it is sufficient to find the form of one of these functions, and the problem will be solved.
As for the other components of the velocity, \(u\) and \(v\), they are connected with \(w\) by a simple dependence, which will be given below.
§ 2. Boundary Conditions
We shall now clarify the boundary conditions that the solution must satisfy.
1. Conditions on the lateral surface of a cell:
a) In a layer of liquid unlimited in area, the lateral surfaces of the cells are surfaces of symmetry; for them the following conditions must be satisfied:
\[ \frac{\partial w}{\partial n}=0,\qquad \frac{\partial \vartheta}{\partial n}=0, \tag{8} \]
where \(n\) is the outward normal to the surface of the cell.
b) If the boundary is a heat-insulating solid surface along which there is no slip, then the conditions on it will be:
\[ w=0,\qquad \frac{\partial \vartheta}{\partial n}=0. \tag{9} \]
2. Conditions on the upper and lower boundaries:
a) For a free surface we have the boundary conditions:
\[ w=0,\qquad \vartheta=0, \tag{10} \]
i.e. the vertical component of velocity is absent for all \(x,y,t\), and the temperature is maintained constant (the deviation from the equilibrium temperature is equal to zero).
In addition, for the horizontal components of velocity the conditions
\[ \frac{\partial u}{\partial z}=\frac{\partial v}{\partial z}=0 \]
must be satisfied.
Differentiating \((2')\) with respect to \(z\) and using (5), one can show that
\[ \frac{\partial^{2}w}{\partial z^{2}}=\frac{\partial^{4}w}{\partial z^{4}}=\ldots=0, \tag{11} \]
i.e. all even derivatives of \(w\) with respect to \(z\) are equal to zero.
b) For a solid surface along which there is no slip and which is maintained at a constant temperature, we have:
\[ u=v=w=0,\qquad \vartheta=0. \tag{12} \]
Moreover,
\[ \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}=0, \]
and therefore, by \((2')\),
\[ \frac{\partial w}{\partial z}=0. \tag{13} \]
Instead of the condition for \(\vartheta\), one may, with the aid of (5), write the third condition for \(w\):
\[ \left(\frac{\partial}{\partial t}-\nu \Delta\right)\Delta w=0. \tag{14} \]
§ 3. Solution of Equation (7)
In order to obtain the solution of equation (7) corresponding to the cellular structure of the fluid, we must seek it in the form*):
\[ w(x,y,z,t)=f(x,y)\cdot F(z)\cdot \Phi(t), \tag{15} \]
where \(f(x,y)\) is a periodic function, so that for \(\Delta_1 f\) one may write
\[ \Delta_1 f=-\frac{a^2}{h^2}f, \tag{16} \]
where \(a\) is a constant characterizing the horizontal dimensions of the cell.
The dependence on time is assumed exponential,
\[ \Phi(t)=e^{\sigma t}. \tag{17} \]
In the general case \(\sigma\) may be taken as a complex quantity,
\[ \sigma=p+iq. \]
Pellew and Southwell show, analyzing equations (5) and (6) and using Green’s formula, that for \(\alpha\beta>0\) the real part of \(\sigma\) is \(p<0\), i.e. the oscillatory motion can only be damped.
If \(\alpha\) and \(\beta\) have different signs, i.e. in the case of an anomalous distribution of density with height that is of interest to us, then \(q=0\), i.e. \(\sigma\) is real. A stationary convective motion is established in the case \(p=0\). For this case, therefore, we must put
\[ \frac{\partial}{\partial t}\equiv 0. \]
The equation of stationary convective motion takes, with (16) taken into account, the form
\[ k\nu\,\Delta^3 w-\alpha\beta g\,\frac{a^2}{h^2}w=0 \tag{18} \]
or, introducing the new variable \(\zeta=\dfrac{z}{h}\) and denoting
\[ D=\frac{\partial}{\partial \zeta},\quad \lambda^3=-\frac{\alpha\beta g}{k\nu}\frac{h^4}{a^4}\quad(\lambda>0), \]
we obtain:
\[ (D^2-a^2)^3 w+\lambda^3 a^6 w=0. \tag{18'} \]
The general solution of this equation is:
\[ w=\sum_{i=1}^{3}(A_i\,\operatorname{ch}2\psi_i\zeta+B_i\,\operatorname{sh}2\psi_i\zeta), \tag{19} \]
*) It can be shown\(^{11}\) that the variables do not separate if even one of the horizontal boundaries is heat-insulating.
where \(A_i, B_i\) are arbitrary constants, and \(2\mu_i\) are the roots of the characteristic equation for \(D\):
\[ (D^2-a^2)^3=-\lambda^3 a^6. \tag{18''} \]
One of these roots is imaginary and two are complex.
Since there are 6 arbitrary constants, 3 conditions must be satisfied at each of the horizontal boundaries.
If the origin of coordinates is placed at the center of the cell, then at the boundaries \(\zeta=\pm \frac12\), and in (19), the even and the odd solutions can be separated.
Let us consider the various possible forms of the upper and lower bases of the cell:
a) Two free surfaces (Rayleigh’s case). For both surfaces the boundary conditions (10) and (11) must be satisfied.
Let us first consider the even solution (symmetry with respect to the plane \(\zeta=0\), see Fig. 21; the direction of circulation is chosen arbitrarily):
\[ w_{\text{even}}=\sum_{i=1}^{3} A_i \operatorname{ch} 2\mu_i \zeta . \]
Fig. 21. Convection scheme for a symmetric solution.
This solution corresponds to the cellular structure of clouds.
Forming \(\dfrac{\partial^2 w}{\partial \zeta^2}\) and \(\dfrac{\partial^4 w}{\partial \zeta^4}\), and taking into account that \(\mu_1 \ne \mu_2 \ne \mu_3\), we obtain, for example, for \(A_1 \ne 0\) \((A_2=A_3=0)\),
\[ \mu_1=i n\frac{\pi}{2} \qquad (n\text{ odd}), \tag{20} \]
and the solution satisfying the boundary conditions will be
\[ w_{\text{even}}=A\cos n\pi\zeta \tag{21} \]
(\(n\) is the number of cells vertically).
Substituting (20) into the characteristic equation (18''), we find the dependence of \(\lambda\) on \(a\):
\[ \lambda^3=\left(1+\frac{n^2\pi^2}{a^2}\right)^3. \tag{22} \]
Let us introduce the quantity \(K=\lambda^3 a^4\), called the characteristic number.
The smallest value of \(K\) at which convective motion is established is found for \(a^2=n^2\dfrac{\pi^2}{2}\) (for the smallest value of \(n\), equal to 1).
Consequently, the condition
\[ K=-\frac{\alpha\beta gh^4}{kv}\geq \frac{27\pi^4}{4}=657.5 \]
is the condition for the establishment of stationary convective motion.
Passing to Rayleigh’s notation:
\[ \beta=\frac{\vartheta'_1-\vartheta'_0}{h},\qquad \alpha=\frac{\rho_1-\rho_0}{\rho_0(\vartheta'_0-\vartheta'_1)}, \]
where the subscript “0” refers to the lower boundary, and the subscript “1” to the upper boundary, we arrive at Rayleigh’s criterion:
\[ \frac{\rho_1-\rho_0}{\rho_0}\geq \frac{27\pi^4 kv}{4gh^3}. \tag{23} \]
Fig. 22. Scheme of convection for an antisymmetric solution.
Similarly, the odd solution is found (antisymmetry with respect to the plane \(\zeta=0\), Fig. 22. The directions of circulation are chosen arbitrarily):
\[ w_{\text{odd}}=B\sin 2m\pi\zeta \tag{21} \]
\[ (m\text{—an integer, not zero}), \]
and the dependence between \(\lambda\) and \(a\) is expressed by the formula
\[ \lambda^3=\left(1+\frac{4m^2\pi^2}{a^2}\right)^3. \tag{22'} \]
The smallest value \(m=1\) corresponds to \(n=2\) in (22), i.e. two tiers of cells arise in the layer.
b) Two rigid boundaries. The boundary conditions (12), (13), and (14) for \(w\) must be satisfied.
For the even solution we obtain:
\[ \begin{vmatrix} \operatorname{ch}\mu_1 & \operatorname{ch}\mu_2 & \operatorname{ch}\mu_3\\ \mu_1\operatorname{sh}\mu_1 & \mu_2\operatorname{sh}\mu_2 & \mu_3\operatorname{sh}\mu_3\\ (4\mu_1^2-a^2)^2\operatorname{ch}\mu_1 & (4\mu_2^2-a^2)^2\operatorname{ch}\mu_2 & (4\mu_3^2-a^2)^2\operatorname{ch}\mu_3 \end{vmatrix}=0. \]
The first root of this equation is:
\[ 2\mu_1=ia\sqrt{\lambda-1}. \]
We put the remaining two complex conjugate roots in the form:
\[ 2\mu_2=(\eta-i\varkappa)a,\qquad 2\mu_3=(\eta+i\varkappa)a. \]
For the dependence of \(\lambda\) on \(a\) we find:
\[ -\sqrt{\lambda-1}\cdot \operatorname{tg}\left(\frac{a}{2}\sqrt{\lambda-1}\right) = \frac{(\eta+\sqrt{3}\varkappa)\operatorname{sh}a\eta+(\sqrt{3}\eta-\varkappa)\sin a\varkappa} {\operatorname{ch}a\eta+\cos a\varkappa}. \]
The solution of this equation is found graphically. The condition for stability of the convective motion takes the final form
\[ K \geq 1707.8 \quad \text{for} \quad a \simeq 3.13. \]
The odd solution is studied similarly.
c) One free surface and one rigid boundary. It turns out that the odd solution for case (b) satisfies the boundary conditions (10) and (11) for \(\zeta = \pm \frac12\) and the conditions (12), (13), and (14) for \(\zeta = 0\).
Therefore we may extend it to case (c), assuming that the thickness of the layer is half as large as in case (b). The stability condition in this case has the form
\[ K \geq 1100.65 \quad \text{for} \quad a \simeq 5.36. \]
§ 4. Dependence of the vertical component of velocity \(w\) on \(x\) and \(y\)
To find the dependence \(w(x,y)\), we must satisfy the condition on the lateral boundaries of the cell. If this is a plane of symmetry, then the condition
\[ \frac{\partial w}{\partial n}=0. \tag{8} \]
must be fulfilled.
The symmetry conditions require[^11] that the number of faces of the prism be expressed by the formula
\[ N = 2 + \frac{4}{l-2}, \]
where \(l\) is an integer.
There are three possibilities:
1) at the base of the prism there is an equilateral triangle with side \(L\),
2) » » a rectangle with sides \(L_1\) and \(L_2\),
3) » » a regular hexagon with side \(L\).
Case (1) will not be considered, since the solution for case (3) also extends to (1).
Case (2) was considered by Rayleigh[^9]. The boundary condition (8) will be satisfied if one sets
\[ w = w_0(z)\cos \frac{m\pi x}{L_1}\cos \frac{n\pi y}{L_2}. \tag{24} \]
The origin of coordinates is taken at the center of the quadrilateral prism, and \(m\) and \(n\) are even numbers.
The relation between the quantities entering into (24) and \(a\) is found from condition (16):
\[ \frac{m^2}{L_1^2}+\frac{n^2}{L_2^2}=\frac{a^2}{h^2}. \tag{25} \]
Case (3). The solution is found by Christopherson’s method. A solution is chosen which is symmetric with respect to the polar axis \(\varphi=0\) and does not change when the angle \(\varphi\) is increased by \(\dfrac{\pi}{3}\). Placing the origin of coordinates at the center of the hexagon, we may put:
\[ \begin{aligned} w&=\frac{1}{3}w_0\left[\cos\frac{2\pi}{3L}(\sqrt{3}\,x+y)+\cos\frac{2\pi}{3L}(\sqrt{3}\,x-y)+\cos\frac{4\pi y}{3L}\right] \\ &=\frac{1}{3}w_0\left[\cos\frac{2\pi r}{3L}(\sqrt{3}\cos\varphi+\sin\varphi)+\right.\\ &\qquad\left.+\cos\frac{2\pi r}{3L}(\sqrt{3}\cos\varphi-\sin\varphi)+\cos\frac{4\pi r}{3L}(\sin\varphi)\right], \end{aligned} \tag{26} \]
where \(r\) is the radius in polar coordinates, and \(w_0\) is the value of \(w\) at the origin of coordinates.
Fig. 23. Lines of equal vertical components of the circulation velocity for a horizontal section passing through the center of a hexagonal cell.
It is easy to verify that the function (26) satisfies the boundary condition (8), if \(n\) is an integer.
The relation between \(a\) and the quantities entering into (26) is found from (16):
\[ \frac{a}{h}=\frac{4\pi}{3L}. \tag{27} \]
The form of the function (26) is given in Fig. 23. The numbers express the ratio \(\dfrac{w}{w_0}\) for the plane \(z=0\).
At the center of the cell there is an ascending current, and at the periphery a descending one.
The radius of the circle with zero vertical velocity component is equal to \(r_0=0.58L\). Consequently, the area through which the ascending flow passes is \(S_+=1.07L^2\), and the cross-section of the descending flow is \(S_-=1.53L^2\).
After the vertical velocity component \(w\) has been found, the horizontal components \(u\) and \(v\) can be calculated from the formulas
\[ u=\frac{h^2}{a^2}\frac{\partial^2 w}{\partial z\,\partial x},\qquad v=\frac{h^2}{a^2}\frac{\partial^2 w}{\partial z\,\partial y}, \tag{28} \]
which are obtained from the first two equations (1) and equations (2′) and (16).
§ 5. Application of the theory to the atmosphere
For the application of the theory to the atmosphere it is necessary to take into account that here turbulent viscosity plays a greater role than molecular viscosity, and the transfer of heat by turbulent exchange is more significant than by molecular thermal conductivity.
Therefore Leile \(^{12}\) proposed replacing the 3rd term in equation (1) by the exchange term
\[ \nu\rho\,\Delta(u,v,w)\longrightarrow A\Delta(u,v,w) \]
and, instead of equation (4), writing the equation for heat transfer by exchange
\[ \frac{d\theta}{dt}=\frac{A}{\rho}\Delta\theta, \tag{29} \]
where \(A\) is the coefficient of exchange, and \(\theta\) is the potential temperature. Then the basic equation for \(w\) takes the form
\[ \left(\frac{\partial}{\partial t}-\frac{A}{\rho}\Delta\right)^2\Delta w +ga\beta\,\Delta_1 w=0 \tag{30} \]
and is solved analogously to (7).
In the Rayleigh criterion it is only necessary to replace the product
\[ k\nu\longrightarrow \left(\frac{A}{\rho}\right)^2. \]
For example, with a cloud layer of thickness \(100\ \mathrm{m}\), convective motion will develop if the condition
\[ \frac{\rho_1-\rho_0}{\rho_0}> \frac{27\pi^4 A^2}{4gh^3\rho^2}=0.004. \]
is satisfied.
For the calculation we took \(A=10^2\ \dfrac{\mathrm{g}}{\mathrm{cm}\,\mathrm{sec}}\), \(\rho=1.3\cdot10^{-3}\ \dfrac{\mathrm{g}}{\mathrm{cm}^3}\).
Since the coefficient of expansion of gases is \(a=\dfrac{1}{273}\ \mathrm{deg}^{-1}\), then, for the formation of a cellular structure from a continuous stratified cloud, it is sufficient that the temperature difference between its lower and upper boundaries reach \(1^\circ\). This value, in order of magnitude, agrees well with experimental data.
Conclusion
Cellular structures in layers of liquid and gas in the presence of an unstable state have, as we have seen above, a fairly wide occurrence in nature.
Their study, apart from its scientific interest, is also of practical significance.
Thus, for example, knowing the regularities of the formation of cells in clouds, one can judge, from their external appearance, the state of the corresponding layers of the atmosphere.
The study of cellular structures of soils is of substantial importance for the economy of the Subarctic.
For the further development of the theory, it is important to clarify such data as the magnitude of the circulation velocity, the conditions for the formation of one or another form of cell, the temperature distribution in the unstable layer, and the propagation of heat in the presence of convective motion. For soils in which cellular structures arise, it is necessary to accumulate experimental data on moisture, viscosity, and thermal conductivity, and on the temperature distribution during the thawing of soil.
In conclusion I cannot fail to express my deep gratitude to Ya. I. Frenkel and E. S. Selezneva for a number of valuable suggestions.
Cited Literature
- Bénard H. Ann. d. chim. et phys., 23, 62 (1901).
- Mal. S. Beitr. z. Phys. d. fr. Atm., 17, 40 (1931).
- Schmidt R. a. Saunders O. Proc. Roy. Soc., A 165, 216 (1938).
- Low A. Nature, 115, 299 (1925).
- Gripp K. Abh. d. Naturwiss. Ver. Hamburg, 21, H. 3 (1927).
- Steche H. Ber. Verh. d. sächs. Akad. d. Wiss., Leipzig, 85, H. IV (1933).
- Brunt D. Q. J. Roy. Met. Soc., 63, 277 (1937).
- Chandra K. Proc. Roy. Soc., A 164, 231 (1938).
- Rayleigh. Phil. Mag., 32, 529 (1916).
- Jeffreys H. Phil. Mag., 2, 833 (1926); Proc. Roy. Soc., A 118, 195 (1928).
- Pellew A. a. Southwell R. Proc. Roy. Soc., A 176, 312 (1940).
- Löh1e F. Met. Zschr., 58, 278 (1941).
- Bergeron T. Three-Dimensionally Related Synoptic Analysis (1934).
- Grigor’ev A. A. The Subarctic. Publishing House of the Academy of Sciences of the USSR (1946).
- Sumgin M. I. Transactions of the Commission on Permafrost, vol. III (1934).
- Gusev A. I. Izv. State Geographical Society, 70, issue 3 (1938).
- Obruсhev S. V. Izv. State Geographical Society, 70, issue 6 (1938).
- Grigor’ev A. A. Problems of Physical Geography, vol. VI (1939).
- Gladtsin I. N. Izv. State Geographical Society, 68, issue 6 (1936).
- Frenkel Ya. I. DAN, 49, 98 (1945).
- Sumgin M. I. Transactions of the Commission on Permafrost, vol. VI (1938).
- Andrianov P. I. Transactions of the Commission on Permafrost, vol. VII (1939).