Fog Dissipation by Artificial Heating

Fog Dissipation by artificial heating

Table of contents

6 Summary……………………………………………………………………………………

Introduction………………………………………………………………………………...

PART I

GENERAL ASPECTS OF FOG DISSIPATION

1. Introduction to fog dissipation……………………………………………………….. 8

9 1.1. Historical review of weather modification…………………………………………….. 10 1.2. Historical review and methods of fog dissipation……………………………………... 11 1.2.1. Dissipation of supercooled fog……………………………………………………..

1.2.2. Dissipation of warm fog.....………...………………………………...………….....

2. Fog as a meteorological phenomenon……………………………………………….. 13

13 2.1. The fog liquid water content………………...……………………………………….... 14 2.2. Water-phase state and relative humidity of fogs.……....…………………………...… 14 2.3. Number of droplets and their distribution with respect to size………………………... 14 2.4. Vertical temperature profiles in fogs…………………...……………………………… 15 2.5. Role of the wind speed………………………………………………………………… 15 2.6. The upper boundary of fogs………………………………………………………….... 15 2.7. Impact of snow cover on fog formation………………………………………………. 16 2.8. Diurnal and annual fog variations……………………………………………………..

2.9. Formation and evolution of a radiation fog……………………………………………

18 3. Annotations of Austrian airport weather services to the profitability of fog

dissipation……………………………………………………………………………... 18

18 3.1. Innsbruck Airport……………………………………………………………………... 19 3.2. Klagenfurt Airport…………………………………………………………………….. 19 3.3. Linz Airport…………………………………………………………………………… 20 3.4. Salzburg Airport………………………………………………………………………. 20 3.5. Graz Airport……………………………………………………………………………

3.6. Vienna Airport…………………………………………………………………………

4. Fog dissipation by fuel combustion………………………………………………….. 21

22 4.1. What do we expect from artificial heating?…………………………………………....

4.2. What is the minimum temperature to dissipate fog?…………………………………..

Table of contents 2

4.3. How does the minimum temperature increment necessary for fog dissipation depend on fog temperature?…………………………………………………………………… 23 4.4. What is the critical temperature for fog dissipation by means of real fuel?…………... 24 4.5. Heat diffusion in an isothermal atmosphere…………………………………………... 25 4.5.1. Analytical solution of the heat diffusion equation………………………………… 26 4.5.2. Results of the analytical solution………………………………………………….. 28

PART II

CASE STUDIES

1. Case studies for a dry atmosphere…………………………………………………... 31

1.1. Does the numerical model work well?………………………………………………... 31 1.2. Which factors determine heat diffusion?……………………………………………… 32 1.2.1. Variable diffusion coefficient……………………………………………………... 33 1.2.2. Adiabatic mixing…………………………………………………………………... 33 1.2.3. Surface heating…………………………………………………………………….. 35 1.2.4. Initial stratification………………………………………………………………… 37

2. Case studies for a moist atmosphere………………………………………………… 39

2.1. How temperature, total water content, LWC and visibility profiles are found……….. 39 2.2. Total water content diffusion………………………………………………………….. 40 2.3. Ideal fuel heating and its consequences for a fog layer……………………………….. 42 2.4. Application of real fuel in a warm fog………………………………………………... 43 2.5. Approaching real atmospheric conditions…………………………………………….. 44 2.5.1. Artificial dissipation of a young radiation fog…………………………………….. 45 2.5.2. Artificial dissipation of a mature radiation fog……………………………………. 47 2.5.3. Implications of fog temperature and fuel type on fog dissipation………………… 49

APPENDICES

Annex A: Model equations……………………………………………………………….. 54

1. The heat diffusion equation……………………………………………………………. 54 2. The total water content equation………………………………………………………. 56 3. The diffusion equation for total water content………………………………………… 56 4. Equation for the diffusion coefficient…………………………………………………. 57 5. Equation for the saturated water vapour pressure……………………………………... 57 6. Equation of state for the water vapour density………………………………………… 58 7. Visibility equation……………………………………………………………………... 58

Annex B: Initial and boundary conditions………………………………………………. 59

1. Initial conditions……………………………………………………………………….. 59 2. Lower boundary conditions……………………………………………………………. 59

Table of contents 3

3. Upper boundary conditions……………………………………………………………. 61

Annex C: Numerical method – finite differences……………………………………….. 62

1. Equation for the diffusion coefficient………………………………………………….. 62 2. Heat and TWC diffusion equation……………………………………………………... 63 3. Lower boundary conditions of the heat and TWC diffusion equations………………... 64

Annex D: Heat and moisture production ability of various fuel………………………. 65

Acknowledgements………………………………………………………………………... 66 References…………………………………………………………………………………. 67

Summary 4

Summary

The present work essentially reports and summarizes work conducted at the Hydrometeorological University of St.Petersburg, Russia, based on ideas of the author’s supervisor Dr. V.I. Berkrjaev. The main focus of the diploma thesis is the development and application of a one-dimensional model to simulate fog dissipation by means of ground-operated artificial heating. In our case heating is realized by the combustion of fuel triggering an upward heat transport, which causes the fog layer to warm up and droplets to evaporate. As principal mechanism for the heat transport we consider turbulent diffusion, which in our simulation is adapted to the particular problem of artificial heating. Case studies have been performed in a dry and moist environment. The experiments in the dry atmosphere generally exhibit the relative importance of the diffusion coefficient, mixing processes, surface heating temperature and initial stratification for our results. We note that the atmosphere heats up more quickly, if we apply a non-constant diffusion coefficient, which depends on height and the temperature gradient. In this way we allow for a more realistic simulation of the large diffusion values near surface. Also the tendency of the atmosphere to become adiabatically stratified as a result of mixing becomes evident. The fuel type and the rate at which it is burned govern the surface heating temperature. We give quantitative results of their influence on the heating process, assuming that the fuel releases no water vapour (ideal fuel). When the general aspects of turbulent diffusion and artificial heating in a dry environment have been clarified the model is applied to a fog layer. In this context we visualize the process of fog dissipation and the improvement of visibility. For this purpose not only heat diffusion, but also diffusion of water vapour and liquid water content has to be considered, since their vertical distributions affect the dissipation process. Experiments with ideal fuel show that due to evaporation cooling fog air warms up less quickly than dry air. Given the initial vertical profile of temperature, liquid water content and relative humidity we can use the model to simulate the

Summary 5 dissipation of a young and a mature radiation fog as they may really exist. The model shows not only how fog dissipates in lower layers due to artificial heating, but also changes in the upper fog boundary due to mixing processes. In our examples the fog upper boundary drops when it mixes with drier air aloft. If the fuel releases not only heat, but also water vapour (real fuel), the dissipation results are sensitive to the initial fog temperature and to the ratio between the fuel’s moisture and heat production ability. Whereas for initial fog temperatures close to 0°C results are comparable to ideal fuel heating, considerably less success in fog dissipation can be reported for colder fog. Furthermore the capacity of different fuel types to dissipate cold or warm fog is demonstrated. In cold fog conditions methane and propane, for instance, are similarly effective, whereas for warm fog methane is the better choice. Peat on the other hand could be used in warm fog, but, as experiments show, would be quite expensive for fog temperatures below -5°C. It could be estimated that for an area of 105 m2 fuel of the order of several hundred kilogram have to be provided, if the airport management aims to dissipate a radiation fog within one hour up to a height of 60m.

Introduction

Fog is a widespread meteorological phenomenon that always has influenced and inspired mankind. It is not only a cloud touching the ground, but through its beauty and peaceful silence, motivation for innumerable poems, stories and images of rural and urban landscapes. But unfortunately fog is not always appreciated so much, mainly due the fact that it reduces visibility. It may be an expensive freak of nature, at the latest as civilisation began to drive cars, sail on ships or fly planes. Economical and partly maybe even scientific interests encouraged people to search for possibilities to dissipate inconvenient fog. Most efforts have been made on attempts to increase the runway visibility range on airports, since airline companies face millions of dollars loss every year due to fog appearance on the runway. In recent decades the development of radio navigation systems allowed aircraft to land nearly blindly, but until today the pilot has the last say. In the 20th century several methods have been proposed to dissipate fog. One of them – treated in the present work – is to burn fuel along the runway, heat the fog layer and evaporate droplets. It has been used in Great Britain during World War II to allow British bombers returning from Germany to land safely in fog conditions. In the remembrances of Terry Waddington (http://www.wartime-memories.fsnet.co.uk /northeast3.html) we read:

“Carnaby had a runway which was five times the width of normal runways and 9,000 feet long. It was one of the few airfields equipped with FIDO (Fog Investigation and Dispersal Operation). This was a perforated pipeline running down each side of the runway with additional pipes set up to form a lead in at the end of the runway. When it was foggy and aircraft couldn't see to land, petrol was pumped through the lines and ignited. The heat would raise the fog so pilots could dive under it and land in warm but clear visibility. How warm was it? You could feel the warm air from the town! Was it expensive?; yes! it consumed 1.7 million gallons in the month of December 1944 alone to save 22 aircraft and their crews. One of the most spectacular departures from Carnaby was over 70 B-24 Liberators from an American bomb group which had diverted there a couple of days earlier when fog had closed their base in East Anglia.”

Certainly, the FIDO system in Great Britain has been used to rescue crews and not to save money. But despite of its large operation costs it turned out to be an effective method for dissipation of warm fog. Without claiming that it would make sense to use the method for any reasons in the future, it is nevertheless of interest to show quantitatively what actually happens with a fog layer when it is heated from below taking into account turbulence and mixing processes. FIDO has been operated without giving scientific explanations on the efficacy of the method, but had purely practical reasons. The present work should contribute to the scientific basis of such a system. Its task is the development of a one-dimensional model that describes turbulent heat diffusion in the atmosphere due to ground operated artificial heating by fuel combustion and its consequences for a fog layer. The work is divided into two parts. The first part treats general aspects and the second part case studies of fog dissipation. The first section of the first part gives a historical review of weather modification and fog dissipation in particular, including the most popular methods of fog dissipation. Section 2 illustrates the physical background of fog structure and the evolution of radiation fog. The profitability of fog dissipation on Austrian airports is discussed in section 3 through statements of the airport weather services. In the final chapter of the first part the heat diffusion equation is solved analytically for a simplified atmospheric model. The first section of the second part compares the analytical and numerical solution and presents model results for a dry atmosphere. The relative importance of fuel type, diffusion coefficient, turbulent mixing and stratification for the efficacy of fog dissipation becomes evident. The second section is devoted to heat and water vapour diffusion in a moist atmosphere. Several case studies illustrate the dissipation of fog with time for different initial conditions and fuel types. Thereby particular attention is paid to the change of temperature, liquid water content and visibility during the heating process. Model equations and the application of finite differences can be found in annex A, B and C. Annex D shows a table of the heat and water vapour production abilities of different fuel types.

Introduction 8

Part I

General aspects of fog dissipation

______

1. Introduction to fog dissipation

1.1. Historical review of weather modification

With the rise of science, possibilities of human efforts to modify the natural weather processes were proposed. Early in the history of meteorology, proposals were made for producing rain artificially by building huge bonfires to induce convection under appropriate conditions or

9 shooting into clouds (Russia, U.S.A., New Zealand). In the first decades of the 20th century, attempts were made to stimulate precipitation by K. Wegener in Germany (operation on clouds by liquid air), V.I. Vitkevitch in the U.S.S.R. (experiments with charged particles of sand) (Mason, 1961) and Veraart in the Netherlands (experiments with solid carbon dioxide (dry ice), without giving a physical explanation) (Veraart, 1931). For systematic studies of the problem, in 1932 the Artificial Rain Institute, later renamed the Leningrad Institute for Experimental Meteorology, was established in the U.S.S.R. in under V.N. Obolensky. Increased attention to the problem in other countries followed the presentation by Tor Bergeron in 1933 of his conclusion that the simultaneous presence of ice crystals and supercooled water drops was essential for the formation of rain (Bergeron, 1935). Findeisen in Germany, in particular, was active in the search for means of artificially introducing the ice phase and thereby initiating precipitation in supercooled clouds (Findeisen, 1938). It was not until Vincent Schäfer discovered in 1946 that solid carbon dioxide (dry ice) produced ice crystals in great numbers in clouds of supercooled water drops (Schäfer,

1946). Later in 1946 Vonnegut found that silver iodide (AgJ) and lead iodide (PbJ2) could also do so (Vonnegut, 1947), that artificial stimulation of precipitation was considered generally to be within reach. From then on the number of person and agencies involved in weather

Section I.1. Introduction to fog dissipation 10 experimentation increased rapidly. Also, the notion “meteorological war” originates from this time. Much work has also be done in the field of hail suppression, but the efficacy of projects intended to mitigate the severity of hailstorms remains indeterminate, although some successful reduction of crop hail damage has been reported (for example Sulakvelidze, 1968; Bader, 1992; Simeonov, 1996). The scientific establishment of cause and effect, however, is incomplete. The same holds true for attempts to modify tropical disturbances. Hurricane modification experiments of the 1950s and 1960s were inconclusive. Although strong interest continued into the 1970s, no organized research effort was undertaken, and few studies have been devoted to this subject for the past 20 years. In the last decades of the 20th century research work and operational attempts made much new data available, the fundamental concepts regarding physical processes in clouds and fog, however, have not been affected. But recent advances allowed better understanding of the complicated interactions within both natural and seeded clouds with the improvement of two- and three- dimensional models (Levin et al., 1997; Tzivion et al., 1994). It is important to note that until present day the exact nature of the circumstances leading to one or the other effect of weather modification in most instances is not enough clarified, so that increased research is strongly needed. In the meantime, the decision to undertake operational attempts to produce specific results must be made with the awareness of the risk, that effects opposite to those desired may occur, e.g. decrease in precipitation when increases are wanted. In this context, regarding the advisability of undertaking a weather modification program, the probabilities of benefit must be weighed against the costs and risks involved. A worldwide leader in present weather modification is the Weather Modification Inc. (http://www.weathermod.com) in the U.S.A.. They operate programs in fields of rainfall enhancement, hail suppression and fog dispersion in Alberta, North Dakota, Oklahoma, Argentina, Greece and Mexico.

1.2. Historical review and methods of fog dissipation

While the largest emphasis in weather modification experimentation has been made on the attempt to increase precipitation, much work has also been done in the areas of fog dissipation. With regard to the dissipation method and probability, one has to distinguish supercooled fogs and warm fogs.

Section I.1. Introduction to fog dissipation 11

1.2.1. Dissipation of supercooled fog

The dissipation of supercooled (cold) fog, which was demonstrated in one of Schäfer’s very first experiments, has become an operational practice at various airports situated in high latitudes (including 13 airports in the north and west of the central U.S.A. and 15 airports in the U.S.S.R.) (Dennis, 1983). Supercooled fog is microphysically unstable and can be modified more readily than warm fog. Water droplets in these fogs can be frozen by the introduction of heterogeneous seeding agents such as silver iodide (for fog temperatures T<-4°C) or homogeneous agents such as dry ice (solid carbon dioxide) or liquefied propane (for temperatures between –4°C and 0°C) (Dessens, 1969). Due to the artificially enhanced freezing process the vapour pressure decreases and droplets evaporate. In the U.S.S.R. first laboratory experiments on dissipation of supercooled fog were performed in 1947 by the Central Institute of Weather Forecast, followed by experiments on real clouds with dry ice and silver iodide by the Main Geophysical Observatory (Bashkirov and Krasikov, 1957). The Central Aerological Observatory was engaged in theoretical and experimental studies on fog dissipation at airports (Borovikov, 1953). Cold fog dissipation utilizing dry ice and silver iodide introduced both by aircraft and from the ground (by silver iodide generators and rockets) has been a practice at several airports in the U.S.S.R. since the late 1950s (Mason, 1961). Similar efforts have been reported in the U.S.A. (Beckwith, 1964) and France (Dessens, 1969). At the airport of Clement Ferrand for example one generator of silver iodide was sufficient to dissipate fog in favourable meteorological conditions (T<-10°C, calm). The operation cost for the generator was about 5 Franc per hour. As for air operated seeding the aircraft releases dry-ice or silver iodide into the fog at a suitable distance upwind of the runway and awaits the development of a hole in the fog as the ice crystals grow and fall out. Unless the wind drift carries new fog onto the runway, making additional seeding necessary, the plane lands on the cleared runway, which then is available for take-off and landing of other planes (Dennis, 1983). At the airports of Lyon and Paris Orly, seeding was accomplished by releasing liquid propane through expansion orifices (Cot, 1964; Serpolay, 1965). The cooling by expansion as the propane evaporates produces ice crystals. The propane method is expensive, but has found to be worthwhile at busy airports. It has also been tested successfully in the U.S.A. (Hicks, 1967). More recently experiments in China revealed that also liquid nitrogen can be used as a cold fog refrigeration catalyst (Xuecheng and Weimin, 1996). It has been estimated that the economic benefits of supercooled fog dispersal at U.S. airports are five times its cost (Beckwith, 1966), and that about the same ratio applies in the U.S.S.R. (Gaivoronskij, Krasnovskaja and Solovjev, 1968).

Section I.1. Introduction to fog dissipation 12

1.2.2. Dissipation of warm fog

From the point of view of airport operations, the dissipation of warm fog is more important than that of cold fog, for at most of the busy airports of the world a larger portion of the hours of low visibility occurs at temperatures above freezing. It has been shown in laboratory tests that improvements in visibility can be achieved by seeding with giant hygroscopic nuclei such as dry salt. Also several field tests of the procedures have been conducted (Jiusto, Pilic and Kochmond, 1968; Silvermann and Kunkel, 1970; Reisin et al., 1996). As has been mentioned in the introduction in the 1940s attempts were made to eliminate fog around runways with large quantities of heat produced by burning fuel (Dessens, 1969). This operation was known as FIDO (Fog Investigation and Dispersal Operation). In the frame of FIDO also aircraft turbines that served their time have been used to heat the surface air. They blow hot dry air with temperatures of about 800K into the fog. The method has been successfully applied in the 1970s for warm fog at the Orly airport in Paris (Katchurin, 1990; Dennis, 1983). In Fig. 1 the effect of the method in Paris is illustrated.

Fig. 1: The upper photo shows the runway shrouded in fog before the activation of the aircraft turbines and the lower photo the cleared runway afterwards. Visibility improved from 150m to 800m (Katchurin, 1990).

Section I.1. Introduction to fog dissipation 13

In cold fog conditions, the application of the method may lead to fog thickening instead of clearing. The latter is a result of cooling and condensation of water vapour included in the hot air flow while it mixes with the surrounding cold air at a certain distance of the turbines. An analogous process leads to the well-known condensation trails of aircrafts flying in high altitudes. The combination of the large expenses of installation and operation and relatively frequent inability to cope with the advection of new fog has led to abandonment of this type of approach. New interest was reported in the U.S.S.R. in the 1960s and 1970s, but there was insufficient financing, also with regard to promising plans of automatic landing systems that would make fog dissipation unnecessary. Another method of artificial heating is to install a system of compressors (ventilators) sucking in fog air from the target area (for example the runway) (Katchurin, 1990). Upper level fog air sinks down and replaces the outflowing air. While descending it warms up and droplets evaporate. The technique may be effectively applied only, if the fog liquid water content decreases with height. Otherwise, applications of this method may have the opposite effect, namely, fog intensification near the surface. A method for which some degree of success is reported has been carried out in the United States, where helicopters flying slowly across the top surface of the fog mix warm dry air into the fog (Plank, 1969, Plank and Spatola, 1969). The downwash action of the rotors forces air from above into the fog, where it mixes, producing lower humidity and causing the fog droplets to evaporate. Tests were carried out in Florida and Virginia, and in both places cleared areas were produced in the helicopter wakes. Another method involves seeding with polyelectrolytes, which are expected to cause electric charges to develop on the drops, thereby promoting their coalescence and fallout (Osmun, 1969). Other techniques that have been tried include the use of high-frequency (ultrasonic) vibrations, heating with laser rays and seeding with carbon black to alter the radiative properties (Katchurin, 1990).

2. Fog as a meteorological phenomenon

2.1. The fog liquid water content

Generally speaking, fog is the entirety of near surface water droplets and ice crystals, suspended in the air, reducing the visibility to less than 1 km. Basically, fog appearance is a function of air humidity and air temperature. For saturated air (fog air), the liquid water (droplets) content (LWC) increases with the increase of absolute humidity, as a result of evaporation from a wet surface or horizontal and vertical mixing. Alternatively, LWC increases, if the temperature decreases due to adiabatic expansion of air as a consequence of vertical motion, due to radiation cooling or due to turbulent heat exchange with the environment. Locally, advection and vertical movements may also influence the air water content. Typical values of LWC in fogs are 0,02[g/m3] to 0,5[g/m3], for radiation fogs 0.1[g/m3] to 0.3[g/m3] (maximal values 0.7[g/m3]). The vertical LWC profile is closely connected with the temperature profile. Aircraft measurements in the Ukraine from October 1960 to June 1962 have shown that in case of advection fogs LWC typically increases with height. For young radiation fogs the data revealed maximal values of LWC near the surface, whereas for mature radiation fogs the maximal LWC can be found between the surface and the upper levels. In case of old radiation fogs the LWC profile resembles that of advection fogs.

2.2. Water-phase state and relative humidity of fogs

With regard to air temperature and the degree of air purity we distinguish water fogs, consisting entirely of water droplets, ice-crystal (ice) fogs and mixed fogs, consisting of ice crystals and liquid water droplets. Ice and mixed fogs can exist only at T<0°C, whereas water fogs are observed at both positive and negative temperatures (from –3°C to 28°C). The smaller the amount of admixtures in the air, the lower the temperature, where a water fog may exist. The relative humidity of fog air is about 100%. According to measurements in Podmoskovskoe (Russia) it varies between 96% and 100%. However at very low temperatures (from –30°C to – 40°C), the relative humidity can decrease to values of 80% to 70%.

Section I.2. Fog as a meteorological phenomenon 15

2.3.Number of droplets and their distribution with respect to size

As measurements have shown, fogs consist of various size particles. They are said to be polydispersed. The number of droplets in 1cm3 air varies from 0.5 to 100 for advection fogs, from 50 to 860 for radiation fogs, from 70 to 500 for evaporation fogs. These figures were obtained for fogs of moderate intensity. The size of fog elements can be as small as 10-7m and as large as 10-4m. The latter is true for crystal fogs. However, the majority of droplets has sizes from 2 to 18 µm (2*10-6m to 18*10-6m).

2.4. Vertical temperature profiles in fogs

At first sight it may seem that an inversion-type temperature distribution is the most appropriate situation in fog conditions. However, researches have shown that in the majority of cases the fog temperature decreases with height in the surface layer. Usually the fog appears within a thin layer of air tight against the surface. In case of advection fogs the Earth surface temperature remains low until the warm air mass passes over it. In case of radiation fogs the Earth surface temperature cools due to outgoing infrared radiation. As long as the fog depth is small the underlying surface continues to cool down due to radiation, so that in a first time period of fog formation temperature inversion persists. However, as soon as the fog becomes sufficiently deep (100 – 200m), the radiating surface is shifted to the fog upper boundary. Within the fog the thermal regime is then regulated by processes of a turbulent atmosphere. As a result, the inversion may disappear, if the turbulence is strong enough. Near the surface the air temperature may even increase due to heat influx from the soil.

2.5.Role of the wind speed

Radiation fogs form at light wind near the Earth surface, whereas for advection fogs moderate wind speed (1-6 m/s) is favourable. When a fog has formed, calm may be observed at very low levels of the atmosphere (from 2m to 10m height). In upper levels wind is always present. Generally speaking the greater the wind speed the more the vertical extent of the fog.

Section I.2. Fog as a meteorological phenomenon 16

2.6. The upper boundary of fogs

The upper boundary of fogs is believed to coincide with the upper boundary of a ground or uplifted inversion. In case the temperature of an uplifted inversion sharply increases with height, as it may be true for evaporation fogs, the fog upper boundary practically coincides with the bottom of the inversion. According to aircraft measurements in the Ukraine the average height of radiation, advection and frontal fogs is 155m, 320m and 400m, respectively. Generally the fog upper boundary is most often found at heights between 200 and 300m.

2.7. Impact of snow cover on fog formation

In middle and high latitudes the underlying surface is often covered with snow in wintertime. Since the saturation pressure is smaller over ice than over water, the conditions are not favourable for water fog formation over snow surface. When temperature is decreasing, the air reaches its saturation state with respect to the snow surface earlier than with respect to water. For instance, at T = -10°C the saturation pressure with respect to ice is 2.6hPa, and with respect to water 2.87hPa. Thus, as soon as the relative humidity reaches 91%, sublimation on the snow surface begins. The sublimation impedes condensation of the water vapour and fog droplet appearance. Moreover, if a fog was already formed in an air mass before it arrived on the snow surface, it may disperse over the snow cover. The dispersion can be blocked in case the temperature of the advected air is rapidly decreasing. In this case the fog can even intensify. The most favourable conditions for fog formation over snow cover are observed at temperatures close to zero. In this case the saturation pressure over ice and over water is almost the same, while the air cooling from the snow surface serves as strong fog forming factor. Consequently, fogs over snow cover occur at temperatures close to 0°C (from –5°C to +5°C). The most intensive dispersion of fogs over snow occur at temperatures of –8°C to –16°C.

2.8. Diurnal and annual fog variations

Since the basic reason for fog formation is air cooling, the maximum fog occurrence is observed at dawn and the minimum in the afternoon. Radiation fogs typically start to appear just after midnight and reach their maximal intensity one or two hours before sunrise. These fogs usually disappear from 1.5 to 2.5 hours after sunrise. The higher the air temperature, the more distinct diurnal fog variations may be expected. This follows from the fact that at high temperatures the saturation vapour pressure increases rapidly with temperature and the fog droplets evaporate more

Section I.2. Fog as a meteorological phenomenon 17 readily. At cold weather conditions temperatures have to increase more to obtain the same change of the saturated water vapour pressure. By this reason, summer fogs are quick to disperse after sunrise and they are never observed at daytime, while in winter the dissipation process is slower. Fog then often persists for a longer time (sometimes even for a few days). The variation of annual fog occurrence is known to be of great variety. It depends on geographical and local features. However, two basic types of annual fog variation can be suggested. For the first one, the maximum occurrence is observed in the cold part of the year (autumn, winter) and the minimum in summer. This type prevails for example in Russia. As for the second type a weak maximum is observed in summer or the fog occurrence is equally frequent all over the year. Such annual variations are typical for arctic regions.

2.9. Formation and evolution of a radiation fog

One of the preconditions for the successful application of fog dissipation by artificial heating is calm weather, as it is typical for radiation fogs. Therefore a more detailed description of the occurrence and development of radiation fogs should be given. Radiation fogs arise as a result of the Earth’s surface and lowest air layer cooling under the impact of terrestrial radiation and eddy mixing. They may be very intense. The rate of surface cooling by radiation is about 1°C per hour. There are two types of radiation fogs: on the one hand a mist that is most dense near the ground but becomes diffuse with height and on the other hand a well-mixed fog with a sharp top. The first type is a young radiation fog that had not yet the time to develop and mix. If the sun rises early enough (in summer), the short-wave radiation can penetrate the fog, heat the surface and evaporate droplets. Fogs of the second type are usually older. The liquid water content had sufficient time to distribute over the fog layer, so that the fog becomes more uniform in the vertical. Well-mixed fog can persist well into the morning, because much of the solar radiation is reflected from the top, and radiation cooling of the top continues during the day. The combined effects of absorption of solar radiation in the interior of the fog and longwave cooling at cloud top can cause the boundary layer to warm and the fog base to lift from the ground. This fog is then reclassified as a stratus cloud.

Favourable conditions for the formation of radiation fogs:

?? light surface wind, clear sky ?? high pressure fields, small horizontal pressure gradients on the synoptic scale ?? clear sky after persistent rain ?? stable stratification of the boundary layer (obviating convective vertical exchange)

Section I.2. Fog as a meteorological phenomenon 18

?? relative humidity of at least 80% at sunset (in case of moderate relative humidity considerable amounts of aerosols should exist, haze) ?? humid soil ?? humid valleys, river valleys (In the night cold air sinks down the mountain slopes and the humid valley air gets supersaturated.) ?? mountain ranges (Cold air slopes down the mountain and forces valley air to move upwards which results in condensation and low stratus clouds.) ?? turbulent water vapour and heat flux (supports or obviates fog formation depending on the vertical temperature and water vapour profile)

Life cycle of radiation fogs:

1) Surface layer cooling due to outgoing infrared radiation 2) Absolute humidity decreases in lower layers owing to vapour flux into the soil; depending on the efficacy of radiation cooling the relative humidity increases or decreases. 3) Turbulent exchange decreases due to more stability within the lower layers. 4) Surface inversion grows; radiation cooling lessens; turbulent mixing intensifies. 5) Vapour flux into the soil decreases; development of a fog layer (young radiation fog). 6) Vertical growth of the fog and shifting of the radiating surface to the fog upper boundary; the temperature of lower fog layers increases. 7) Turbulent diffusion causes isothermal or pseudoadiabatic stratification. 8) At sunrise absorption of shortwave solar radiation at the surface. 9) Evaporation of soil moisture; dew droplets let the relative humidity increase in lower layers (For a short time fog intensity increases or low fog eventually develops after fog free nights.). 10) Vertical fog extent increases due to increased turbulent mixing. 11) Soil dries up; the relative humidity of the lower layers decreases; development of low stratus clouds. 12) Full fog dissipation during forenoon.

3. Annotations of Austrian airport weather services to the profitability of fog dissipation

3.1. Innsbruck Airport

Fog dissipation by artificial heating has been tested comprehensively in the 1970s at the airport of Paris Orly. Discarded aircraft engines have been installed alongside the runway that should not only thermally but also mechanically foster air mixing. Attempts, however, have been soon abandoned. For the failure of the method in Paris it can be suggested that fog formation in calm weather conditions is relatively seldom. In most cases slow advection processes leading to new fog formation play a role. Typically, airports have been constructed in humid alluvial areas, where land is cheap and where advection of humid air and weak valley winds are common phenomena. Local heat sources, as artificial heating, causes the air to move upwards and be replaced by cold humid air from surrounding areas. In times of ecological concerns, where energy expenses at airports are well observed, such a method appears at least questionable. For the International Civil Aviation Organization (ICAO) the development of instrument landing systems that allow safe landing even in conditions of poor visibility clearly take priority over such considerations. Delayed depart and landing at big airports cause heavy costs. In the time being interesting projects on the co-operation between airlines, airports and air traffic management are under way.

3.2. Klagenfurt Airport

At the Klagenfurt airport relatively frequent fog may be observed between October and January. In many cases the morning fog transforms to low stratus clouds during forenoon. Sometimes, however, it stays on the surface, in particular if weak local winds advect fog air from other areas of more persistent fog. As a consequence – as much as operational flights are concerned – an estimated number of 10 airplanes (mainly Tyrolean Airlines flights from Vienna or Frankfurt to Klagenfurt) per year cannot land. They are forced to fly back or land at an substitute airport. This may happen even if the fog forecast was correct and the flight has been still carried out. In Klagenfurt the airport management therefore decided to improve the instrument landing system (from first to third category). Costs are estimated 7 Mio. Euro (100 Mio. ATS). After the installation of the new system fog should no longer constitute a substantial problem for Airlines. Indeed, artificial ways of fog dissipation would be advantageous, but not really necessary

20 since the airport density in Austria is quite high, i.e. there exist other possibilities to reach Carinthia (1.5h by car from Graz or Ljubljana to Klagenfurt). For the following reasons it may be suggested that fog dissipation by artificial heating would not be accepted in countries of high technological standards as Austria: In the public opinion the method sounds old- fashioned in view of an instrument landing system. Airports have to consider their image and cannot risk negative propaganda (in particular from the side of travel agencies). They have to offer modernity and not unconventionality, even if this would be the cheaper way. Moreover, the Klagenfurt airport is located near residential districts, so that the airport would have to cope with complaints about bad smell, waste gas and noise pollution. Even if the application of the method in Austria seems to be not worthwhile, it may be meaningful in countries where the installation of expensive equipment is financially questionable.

3.3. Linz Airport

On 70-80 days of the year fog may be observed at Linz Airport, most frequently from mid September to mid March. In September, October and March fog generally forms during the night and dissipates until midday. From November to February, it may be persistent and only occasionally transform to stratus clouds. The rest of the year fog cases are reported in the morning or forenoon, if rain showers are followed by clear sky in the night. Cancellation of flights is rare, because the airport is equipped with an instrument landing system of category IIIb. A one-dimensional model for the forecast of fog dissipation would be of interest for the airport, if diabatic heating as the dominant process is considered.

3.4. Salzburg Airport

Fog may be a serious problem for aviation and may cause cancellation of flights. Technological developments could minimize this problem at big international airports. In Salzburg fog is not very frequent due to valley winds. The outflowing cold air from the Alps causes southeasterly winds at the airport during the night and the morning hours. These conditions are not favourable for fog formation. Most fog cases occur in autumn and winter. Statistically, in November, December and January there are nine days of fog, where in one or two cases the fog may become intense enough to limit aviation. The

Section I.3. Annotations of Austrian airport weather services 21

Salzburg airport is equipped with an instrumental landing system of category III, which allows landing for RVR-visibility of less than 200m (i.e. 80-100m for human observation). Therefore only a few or no flights at all have to be cancelled during the winter season. Foggy conditions in Innsbruck sometimes even cause planes to land in Salzburg. Nearly all international airports have the technical standard for autolanding, i.e. in theory planes can land with zero visibility, but in practice the pilot has the final say.

3.5. Graz Airport

The effects of fog on aviation depend on the category of the instrument landing system (CAT I, II, or III), on the aircraft equipment and on the pilot’s experience and training. For modern airplanes fog generally does not constitute a problem anymore. Smaller private and commercial planes, however, equipped with less technical standards are more sensitive to fog conditions. Thus, for big airlines artificial fog dissipation does not pay, whereas private pilots would benefit, but cannot pay for it.

3.6. Vienna Airport

At the Vienna International Airport fog occurs most frequently in winter (from November to March). A mixed form of radiation and advection fog prevails (humid air flowing in from pasture areas of the Danube river and the Neusiedlersee). Fog on the runway constitutes problems only for exotic airlines or if visibility is strongly reduced (i.e. runway visual range of 100m to 300m or ceilings of 100ft to 200ft). However, as a consequence of fog occurrence larger time intervals between flights cause troubles for the airport capacity. Principally fog dissipation at the airport would be convenient, but the benefit has to match the costs.

4. Artificial heating by fuel combustion

4.1.What do we expect from artificial heating?

In section 1.2.2 a short introduction in warm fog dissipation gave first implications on research efforts in artificial heating in the 20th century. One of the methods for artificial heating is fuel combustion, on which we concentrate in the present work. The main object is to analyse heat diffusion under quasi-real atmospheric conditions and its effects on temperature, water vapour, liquid water content and visibility in the lower few hundred meters of the atmosphere, in particular in the presence of liquid water (fog). This seems not only interesting from the point of view of clarifying some aspects of efficiency of artificial heating to dissipate fog, but also can give general ideas about heat diffusion in the atmospheric boundary layer in case of a heated surface even in a fog free atmosphere. The dissipation of an inconvenient fog layer is of particular interest at airports. One way to dissipate fog is to install some sort of energy source under the fog layer that heats the layer by diffusion processes (radiation processes are not considered here (see annex A)). In this context, the airport management has to choose the energy source such as to minimize costs and maximize the effect. In the frame of the present work fossil fuel is combusted to produce the heat necessary for fog dissipation. In view of such a method we have to consider the fact that generally not only heat will be released but also water vapour. This characteristic of fuel combustion may be taken into account introducing a moisture production coefficient, typical for the type of fuel used (Katchurin, 1990). Fuel types can be for example kerosene, methane, propane, petrol, coal, peat or wood. For our studies, to have a specific example, we chose for the majority of cases kerosene, the most common heat source that has been used at airports, although for the sake of comparison, the effect of other fuel types will be shown. The primary goal, as already stated, is the way the energy (in form of heat) is transported into the atmosphere, paying particular attention to the influence of heating on the presence of a fog layer. “Favourable” conditions for fog dissipation by artificial heating are warm fog, i.e. fog in a layer of positive temperature and calm weather. The fog should ideally have emerged from cooling. We figured out that artificial heating is a very effective method to dissipate warm fog, since other more effective and practically applied methods work only for fog temperatures below 0°C. Calm is favoured, because wind may “blow away” the heat emitted from the heat source. The initial

23 atmospheric stratification in the fog layer also plays some role, but its impact on the process of fog dissipation is limited, in particular, if the temperature gradients produced due to heating at the surface allow for extraordinary large values of turbulent diffusion, which are much bigger than in usual atmospheric conditions. The value of the diffusion coefficient during heating therefore is mainly governed by the new stratification, which results from intense heating on the surface. In the present work our case studies will mainly concentrate on stable stratification, which is typical for radiation fogs. Older radiation fog may often exist under pseudoadiabatic or even unstable atmospheric conditions, but the time it will disappear by natural processes may be expected to be short enough, so that „waiting for better weather“ probably will be cheaper than artificial heating.

4.2. What is the minimum temperature to dissipate fog?

The minimal heat energy Qmin necessary to dissipate a unit volume of fog is the sum of heating used to increase the temperature to a minimal necessary value (where all the water content can be held as saturated water vapour) and the heat supplied for evaporation of droplets (Bekrjaev, 1977):

Qmin ? ? a cp ? Tmin ? Lv q0 . (4.1)

In equation 4.1 cp is the specific heat capacity of air at constant pressure, ? a the air density, Lv the -3 heat of vaporization and q0 is the fog’s liquid water content. The units are for Qmin [J m ], for -1 -1 3 3 ? Tmin [K], for cp [J kgair K ], for Lv [J/kgwater], for ? a [kgair/m ]and q0 [kgwater/m ]. We now assume that fuel, for example kerosene, is uniformly burned within a given volume of the atmosphere. During this process the fuel heats up the atmosphere and at the same time releases water vapour. Which will be the minimum temperature increase ? Tmin necessary to evaporate all liquid water content in the air? If we neglect any heat and moisture exchange with the environmental air, the value of ? Tmin may be derived when solving the equation system of the heat balance equation and total water content equation of the fog, where a certain amount of fuel is burned (Bekrjaev, 1977). The two equations yield

? a cp dT ? Lv dq ? ? dm

dq ? ? a ds ? ? v dm , (4.2)

Section I.4. Artificial heating by fuel combustion 24

where in the first equation (heat balance) dq < 0. ? and ? v are the heat and moisture production ability, respectively, s the mass of water vapour per kg air and dm the mass of fuel burned per m3. 3 The units are for ? [J/kgfuel], for ? v [kgwater vapour/kgfuel], for s [kgwater vapour/kgair] and dm [kgfuel/m ].

We eliminate dm in the equation system (4.2) and integrate from the initial parameters T0, q0 and s0 to T1, q1 = 0 and s1. Furthermore expressing s by the partial pressure of saturation water vapour E and using the Clausius-Clapeyron equation, we find

? ? v ? q0 ?1 ? Lv ? ? ? ? ? Tmin ? (4.3) ? Ra Lv E ? v? ? a ? cp ? 2 2 ? ? ??R v? p T ?,

where Ra is the gas constant for dry air, Rv the gas constant for water vapour, p the atmospheric pressure, T the fog temperature and E the corresponding saturation water vapour pressure. E is a -1 -1 function of T only. The units are for Ra and Rv [J kg K ], for p and E [Pa]. Expressing Ra and Rv by their values and bearing in mind that usually Lv ? v/? << 1, eq. (4.3) transforms to

q0 ? Tmin ? 2 ?E ? 58 ? ? v? (4.4) ? a ? ? ? ? cp ? ?p ? T ? ? ? ,

2 -1 -1 where the unit of the term (58/T) is [kgwater vapour kgair K ]. For positive temperatures ? Tmin is of the order of tenths of a degree C. Fuel that does not release water vapour we call “ideal” fuel, whereas water vapour releasing fuel is denoted as “real” fuel. For ideal fuel, for example an electro heater, ? v is 0 [kgwater vapour/kgfuel], for kerosene (real fuel) ? v is 1.4 [kgwater vapour/kgfuel]. The values of ? and ? v for different types of real fuel can be found in Annex D.

4.3. How does the minimum temperature increment necessary for fog dissipation depend on fog temperature?

In order to answer this question, we calculate ? Tmin for ideal and for different types of real fuel in a temperature range of 250K to 290K. Thereby we assume a fog liquid water content (LWC) of 0.2[g m-3] for all initial fog temperatures. The results for a unit volume of fog air can be seen in Fig. 2 (Bekrjaev, 1977).

Section I.4. Artificial heating by fuel combustion 25

Fig. 2: Minimum temperature increment for different initial fog temperatures using various types of fuel 4 5 9 8 7 1 2 3 6 7 8 6 5 4

T(min)[K] 3 ? 2 1 0 230 240 250 260 270 280 290 300 310 T[K] 1 2 3 4 ideal fuel coal (anthracite) propane kerosene 5 6 7 8 alcohol methane peat hydrogen

The colder the fog the larger is the minimum temperature increment necessary to dissipate a unit volume of fog air. This is due to the fact that in order to evaporate a certain quantity of LWC we have to heat up the air in warm fogs less than in case of cold fogs, since for a given temperature increment the saturated water vapour pressure increases less for low than for high temperatures. The additional amount of water vapour released by real fuel is responsible for the strong increase of the corresponding graphs. Apart from these findings it is interesting to see that in case of warm fog the necessary temperature increment does not depend so much on fuel type (ideal or real). It could be followed from the above that using large amounts of heat energy we could dissipate fogs of any temperature. However, making use of real fuel, this is not true for very cold fogs, as we will see in the next section.

4.4. What is the critical temperature for fog dissipation by means of real fuel?

Using real fuel implies that some amount of water vapour is released during the combustion process. At very low temperatures this water vapour immediately condenses and causes the fog to intensify instead of clearing. The critical temperature, where the method of fuel combustion does not work anymore can be found setting the denominator of equation (4.4) to zero and solving the resulting equation graphically. Introducing the parameter B as

2 ? v ? T ? B ? cp ? ? p (4.5) ? ? 58 ?

Section I.4. Artificial heating by fuel combustion 26 we can draw a graph of B as function of T. The unit of B is [Pa]. The values of B(T) show at which water vapour pressures ? Tmin gets infinite. If we superpose the graph B(T) and the graph of the saturated water vapour pressure E(T), the crossing point of the graphs gives us the critical temperature we are interested in.

Fig. 3: Critical temperature for fog dissipation by combustion of kerosene: The crossing point of the saturated water vapour pressure E(T) and parameter B(T)

6 5 4 3 2 E[hPa]; B[hPa] 1 0 225 230 235 240 245 250 255 260 265 270 275 T[K]

E(T) B(T)

In Fig. 3 it can be easily seen that the critical temperature is approximately –27°C for kerosene. This means that for a cold fog of a temperature less than -27°C the method of fuel combustion using kerosene does not work anymore.

From the above calculations we can conclude so far that the method of artificial heating by fuel burning works only for relatively warm fogs (ideally positive temperatures) and is limited by a critical temperature. Up to now we have discussed the effect of heating on a unit volume of fog air. The results would be true, if we could heat the fog uniformly all over its extent. How the air will actually heat up from below is discussed in the following chapter for an ideally simplified atmospheric layer.

4.5. Heat diffusion in an isothermal atmosphere

In reality it is not very plausible that fog heats up uniformly all over its volume. In practice therefore the heating source is located on the ground and the fog layer heated from below. In that case heat and moisture are transported upward by turbulent diffusion. Intuitively it is clear that such an application would raise the necessary heating energy, since the lower layer will be

Section I.4. Artificial heating by fuel combustion 27 superheated and it will take time until the upper layer of the fog (or part of the fog) that we would like to disappear reaches the minimal temperature increment ? Tmin necessary for its dissipation. Suppose that heating is realised by a system of thermal sources that are uniformly arranged over a certain area (for example along an airport runway). Taking the heated surface temperature as the average of the whole area (including the heat sources and the spaces in between) and neglecting advection, we may describe heat diffusion in a one-dimensional model, where variables like temperature and water vapour density change with height only. Considering the above concept, the basic equation describing heat diffusion in the atmosphere is the heat diffusion equation (Katchurin, 1990)

? ? ? ? ? T ? ?D T ? (4.6) ?? ?z? ?z ? .

Here T is the temperature in [K] and D [m2 s-1] the diffusion coefficient.

4.5.1. Analytical solution of the heat diffusion equation

The major motivation to find an analytical solution of the heat diffusion equation is to check the quality of the numerical solution developed in annex C. Apart from that the analytical solution permits first conclusions about the vertical temperature change in case of ground-operated artificial heating. For convenience we apply the following simplifications to our model atmosphere:

1) The atmospheric layer for which calculations are performed is isothermally stratified and dry. 2) The vertical diffusion coefficient is constant (in real conditions it depends on the vertical temperature gradient, height and wind). 3) The temperatures at the surface and the top of the atmospheric layer are constant with time. 4) Ideal fuel is used as heating source, i.e. no water vapour is released during the combustion process. 5) No heating energy is lost by diffusion into the ground.

The above simplifications lead us to the following initial and boundary conditions: The initial condition for the isothermal atmosphere yields

T?? ? 0?z? ? Ta ? const , (4.7)

Section I.4. Artificial heating by fuel combustion 28

where ??is the time, z the height and Ta the temperature of the entire atmospheric layer at an initial moment of time. The boundary conditions yield

T???z ? 0? ? Ts ? const

T???z ? ztop? ? Tt ? const , (4.8)

where Ts indicates the average surface temperature on the heating area and Tt the temperature at the top of the atmosphere (z = ztop). “Top of the atmosphere” implies the height where no temperature change occurs. Thus, Tt is the same as Ta. In case of heating at the surface Ts is bigger than Ta and the resulting heat flux P is equal to

?? ? P ? ?D? T ? ?z ? ?z? 0 .

(The notion P for flux is common in Russian literature.). The coefficient of turbulent diffusion D is taken to be constant with height and time, so that eq. (4.6) may be rewritten as

? ?2 T ? D T (4.9) 2 ?? ?z .

Eq. (4.9) is a one-dimensional partial differential equation. A solution of this equation can be found in the form of

? T???z? ? T ? Ta ? ?Ts ? Ta? ?1 ? erf?? ?? (4.10)

(Katchurin, 1990), where erf(? ) is the error function

? z 2 ? 2 erf?? ? ? erf? ? ? ? e? ? d? ? ? ? ? 2 D ? ? ? 0 .

If the above simplifications are satisfied the temperature change with height and time within a given atmospheric layer may be calculated by equation (4.10) using the initial and boundary

Section I.4. Artificial heating by fuel combustion 29 conditions (4.7 and 4.8). The results depend on heating time, height of the atmospheric layer, D,

Ts and Ta.

4.5.2. Results of the analytical solution of the diffusion equation

In this section we solve the analytical solution of the diffusion equation (4.10) for a dry, isothermal atmosphere. All calculations are done for an atmospheric layer of 300m that we heat for 5400s (90min). We choose 90min, thinking mainly of applications of the method on airports. If the fog dissipation would take too long, the question arises, if the efforts would be economically feasible. For the initial temperature of the fog we choose Ta = 273[K]. It is clear from eq. (4.10) that the outcome of our calculations will depend on the surface heating temperature and the diffusion coefficient of the air. In Fig. 4.1-3 the results are shown after 900s (15min), 2700s (45min) and 5400s (90min).

Fig. 4.1: Analytical solution of the diffusion equation for Ts=280[K] and D=0.1[m2/s]

320 280 240 200 160 120

Height [m] 80 40 0 272 274 276 278 280 282 284 286 288 290 292 Temperature [K]

after 900s after 2700s after 5400s initial T profile

Section I.4. Artificial heating by fuel combustion 30

Fig. 4.2: Analytical solution of the diffusion equation for Ts=290[K] and D=0.1[m²/s]

320 280 240 200 160 120

Height [m] 80 40 0 272 274 276 278 280 282 284 286 288 290 292 Temperature [K]

after 900s after 2700s after 5400s initial T profile

Fig. 4.1 and Fig. 4.2 give first implications on how the atmospheric temperature profile changes, if we vary the heating temperature. In Fig. 4.1 we heat the surface constantly at a temperature of 280K, whereas in Fig. 4.2 the surface heating temperature is 290K. The vertically uniform diffusion coefficient is chosen D=0.1[m2 s-1] for both cases. If we define the “heat transport height” as the maximum height affected by heating after a given period of time, we can see that it does not depend on the surface heating temperature. This is due to the fact that D was chosen the same for both cases. The latter implies that supplementary heat energy gives no gain for the height of dissipation. It will therefore be interesting for later analysis (see part II) to take into account a variable diffusion coefficient being dependent on the vertical temperature gradient and height. In this way a more real description of turbulent diffusion should be achieved, especially in case of artificial heating, where big temperature gradients near surface may develop. Anyway in Fig. 4.2 air of higher temperature has been transported upwards, so the temperature of the lower layers increased significantly more in this case. We can see for example that in 60m height and after

5400s of heating the temperature increased by approximately 0.5K for Ts=280K, whereas a temperature increase of about 1K can be observed for equal conditions in case of Ts=290K. In Fig 4.3 we can see the influence of a bigger diffusion coefficient. For D=0.5[m2 s-1] the temperature profile after 900s is similar to that after 2700s in Fig. 4.2, where D had been chosen 0.1[m2 s-1]. One may argue that the lower layers should be less heated, since the heat spreads upwards and is thus used to warm up higher levels. This is correct, if the available energy would be the same for both cases. In fact, however, the energy input at the surface is higher in Fig. 4.3 than in Fig. 4.2 to guarantee that despite of different values of the diffusion coefficient the surface temperature may be held constant at 290K.

Section I.4. Artificial heating by fuel combustion 31

Fig. 4.3: Analytical solution of the diffusion equation for Ts=290[K] and D=0.5[m2/s]

320 280 240 200 160 120

Height [m] 80 40 0 272 274 276 278 280 282 284 286 288 290 292 Temperature [K]

after 900s after 2700s after 5400s initial T profile

Part II

Case studies

______

1. Case studies for a dry atmosphere

The ultimate goal of the present model (see Annex A, B and C) is to simulate fog dissipation by ground-operated artificial heating. To begin with, however, we would like to show how heat actually diffuses into the atmosphere. Diffusion processes in a dry environment are expected to be generally different from those in a moist or foggy environment. In the presence of water vapour condensation and vaporization may occur that introduce or extract some amount of (latent) energy to/from the system. Moreover different concentrations of water vapour at different heights would tend to equalise. Such changes affect the amount of heating on different levels, i.e. the process of heat diffusion itself. This effect can be well seen, if we suppose as a first step a dry atmosphere and fuel that does not release water vapour (ideal fuel) and compare it to the results of our case studies for a moist atmosphere (chapter II.2). First of all, however, we would like to examine, if the model actually is able to simulate heat diffusion. The next section is devoted to this problem.

1.1. Does the numerical model work well?

An answer to this question can be given, solving the diffusion equation numerically for the same initial and boundary conditions as for the analytical solution in chapter I.4. A comparison between

the numerical and the analytical solution will tell us about the quality of the finite difference method (see Annex C). In this context it is interesting to see, how the model behaves for different values of heating temperature and diffusion coefficient. Moreover the accuracy of the numerical solution depends on the time step and grid size. Since discrete grid points for height and time are introduced, difference operators replace the differential operators of the diffusion equation and a truncation error occurs. According to the convergence criteria the resulting error of the finite difference solution should be the smaller the smaller the discretisation by height (? z) and time (???. Experiments for different sets of ? z and ?? show that the numerical solution is virtually identical to the analytical solution. Also for a bigger diffusion coefficient the analytical solution is very well represented by the numerical one. As a conclusion from the above experiments further application of the finite difference scheme is seen to be justified. Which are the tasks treated by our model will be made clear in the next section.

1.2. Which factors determine heat diffusion?

For convenience the analytical solution in chapter I.4 has been found for a simplified atmosphere. We supposed isothermal stratification, a constant surface heating temperature and a constant diffusion coefficient. In reality, however, these conditions generally are not satisfied. In case of radiation fogs, for instance, we often deal with an inversion, where the temperature increases with height, owing to radiation cooling at the surface. Moreover, if we start burning fuel near the surface we actually would expect the surface temperature to depend on several parameters, such as the quantity and type of fuel or the diffusion coefficient, rather than to remain at a constant value. The diffusion coefficient itself usually is a function of the vertical temperature gradient and height. Thus, especially in case of artificial heating, where relatively large temperature gradients are expected in lower layers, the assumption of a constant diffusion coefficient would not be appropriate. Finally, similarly to the development of a vertically well mixed layer on hot summer days, mixing processes due to artificial heating should be expected and taken into account. As a consequence of mixing the atmosphere tends to become adiabatically (in case of LWC pseudoadiabatically) stratified. Therefore we will call the latter process adiabatic (pseudoadiabatic) mixing.

Evidently, the nature of heat diffusion depends on the diffusion coefficient, adiabatic mixing, surface heating temperature and the initial stratification. It will be interesting to examine, how they contribute to the combined effect of artificial heating. The following four chapters will be devoted to this problem: To begin with, we introduce in section 1.2.1 a notion for the variable

Section II.1. Case studies for a dry atmosphere 34

diffusion coefficient, which depends on the temperature gradient and height. All case studies will then be performed for a variable diffusion coefficient that is more representative for real conditions than a constant one. In the second section 1.2.2 we discuss turbulent adiabatic mixing. The third section 1.2.3 will show how the vertical temperature structure will change with time due to the evolution of the surface heating temperature. The influence of different initial stratifications on these temperature changes will be discussed in section 1.2.4. As for the analytical solution the graphs in our results show the evolution of the temperature profile after 900s (15min), 2700s (45min) and 5400s (90min).

1.2.1. Variable diffusion coefficient

The approximate dependence of the diffusion coefficient D on the temperature gradient and height adapted to our problem of artificial heating is described by equation (A.8) in annex A. It has been mentioned there that we choose the factor A such that D is about 100[m2 s-1] for a temperature -1 2 -1 gradient of -0.5[K m ]. Assuming D0=0.1[m s ] in eq. (A.8), this is true for A=14. Such a choice of A guarantees that the warm heated air on the surface quickly moves upwards, as it may be expected in real conditions owing to the instantly developing big vertical temperature gradient in lower layers. One may argue that the results could be sensitive to the choice of A. Experiments, however, show that the sensitivity for A not too small is very low. For A=7, for instance, the average surface temperature after 90min of heating is about 0.6K higher and for A=21 only 0.3K lower than it is the case for A=14. These rather small surface temperature differences, due to different assumptions for A (at least for A>10) only slightly affect the vertical temperature profiles, even after a relatively long time of heating. For small A, however, A=2 for example, significant differences in the temperature profiles can be observed. Therefore our results will be sufficiently correct, as long as A is not too small. As a consequence, the choice of A=14 is justified.

1.2.2. Adiabatic mixing

On hot summer days the atmospheric boundary layer usually is well mixed and nearly adiabatically stratified. The same effect we might expect as a result of artificial heating on a large area at the surface. This mixing process finally leads to the establishment of a neutral stratification, where potential temperature is constant with height. In the present model we simulate this conversion of a non-adiabatic stratification to an adiabatic one. The lower boundary condition of the heat diffusion equation has been defined such that due to adiabatic mixing (not

Section II.1. Case studies for a dry atmosphere 35

counting for latent heat release and fuel combustion) for every time step the first layer above surface is adiabatically stratified (see second term on the right side of eq. B.3 in annex B). After a certain period of mixing the atmosphere tends to become entirely neutrally stratified. How this simulation looks like can be seen in Fig. 1.1-3, where we suppose some external forcing (for example diabatic heating) that initiates the atmosphere to mix up. The experiments have been performed for a neutral diffusion coefficient D0=0.1m2/s. In Fig. 1.1 results are shown for an initially positive vertical temperature gradient (inversion), in Fig. 1.2 for an isothermally and in Fig. 1.3 for an adiabatically stratified atmosphere.

Fig 1.1: Adiabatic mixing in case of inversion-type stratification

320 280 240 200 160 120 Height[m] 80 40 0 269 270 271 272 273 274 275 276 277 278 279 280 281 T[K]

after 900s after 2700s after 5400s initial T profile

Fig 1.2: Adiabatic mixing in case of isothermal stratification

320 280 240 200 160 120 Height[m] 80 40 0 269 270 271 272 273 274 275 276 277 278 279 280 281 T[K]

after 900s after 2700s after 5400s initial T profile

Section II.1. Case studies for a dry atmosphere 36

Fig 1.3: Adiabatic mixing in case of neutral stratification

320 280 240 200 160 120 Height[m] 80 40 0 269 270 271 272 273 274 275 276 277 278 279 280 281 T[K]

after 900s after 2700s after 5400s initial T profile

In the two stable cases we observe that the air in upper layers moves down and let the temperature of the lower layers increase. In Fig. 1.3 no change occurs, since in this case mixing does not have any influence on the temperature profile. Thus, for an adiabatically stratified atmosphere the surface temperature does not change with time due to mixing. In the stable case the surface temperature increases and in the unstable case it decreases.

1.2.3. Surface heating

For the analytical solution we supposed a constant heating temperature at the surface. In reality, however, the surface temperature is not constant with time, but governed by the quantity of fuel burned per second and square meter, the diffusion coefficient and the fuel’s heat production ability ? (see eq. B.3 in annex B). To see the influence of each of these three parameters, Fig. 1.4 shows for reference a solution for dm/d?=2*10-6 [kg m-2 s-1] (mass of fuel burned per second),

7 -1 2 -1 ? =4*10 [J kg ] and D0=0.1[m s ]. Hereby the initial atmospheric stratification is isothermal, D dependent on the temperature gradient and height and adiabatic mixing applies. Comparing 1.4 with the results found for the analytical solution (see Fig. 3.1 and Fig. 3.2 in part I) we observe that during heating the heat is transported upwards more quickly owing to the big diffusion coefficient at the surface, where the heating source is installed. The heat supply effectively has been delivered to higher levels. The higher degree of warming in upper layers for our case is additionally supported by the fact that the diffusion coefficient increases with height.

Section II.1. Case studies for a dry atmosphere 37

Fig. 1.4: Combustion of kerosene at a rate of dm/d?=2E-6[kg/(m²*s)]

320 280 240 200 160 120 Height[m] 80 40 0 269 270 271 272 273 274 275 276 277 278 279 280 281 T[K]

after 900s after 2700s after 5400s initial T profile

Reducing the quantity of fuel available for combustion we expect the surface temperature to increase less. This may be confirmed by Fig 1.5, where dm/d? is reduced to 1*10-6 [kg m-2 s-1], i.e. twice less than in the former experiment.

Fig. 1.5: Combustion of kerosene at a rate of dm/d?=1E-6[kg/(m²*s)]

320 280 240 200 160 120 Height[m] 80 40 0 269 270 271 272 273 274 275 276 277 278 279 280 281 T[K]

after 900s after 2700s after 5400s initial T profile

Whereas in Fig. 1.4 the surface temperature increased by about 6K after 90min, this temperature increase is only 3.7K in the other case, where less fuel has been burned. If we on the other hand do not change the fuel quantity available in Fig. 1.4, but double the neutral diffusion coefficient from 0.1[m2 s-1] to 0.2[m2 s-1], it can be shown that surface heating is less (? T=5.2K after 5400s) than in the reference solution (Fig 1.4). Instead the atmosphere above 2 -1 surface gets warmer. The inverse may be obtained in case we reduce D0 to 0.05[m s ]. The latter solution shows more heating at the surface (? T=7.3K after 5400s), but less warming of the

Section II.1. Case studies for a dry atmosphere 38

atmosphere. These results may be explained by the fact that in the first case the heat energy produced at the surface drifted upwards faster, preventing extensive heating at low levels, whereas for a smaller diffusion coefficient more heat energy remains at the lower boundary and contributes to the increasing temperature there. Thus, the solution to some degree depends on the neutral diffusion coefficient. Therefore it would be convenient to estimate the values of D0 with reference to our problem of artificial heating as good as possible, for example by means of field 2 -1 measurements. In the frame of this work we will assume D0=0.1[m s ] for further studies on heat diffusion. Evidently, the more the heat production ability of fuel, the larger the values of the surface temperature. The results shown in Fig 1.6 were obtained using propane instead of kerosene. Propane has a heat production ability of ? =5*107 [J kg-1], thus more than kerosene. Accordingly the surface temperatures are about two degrees higher than in the reference solution.

Fig. 1.6: Combustion of propane at a rate of dm/d?=2E-6[kg/(m²*s)]

320 280 240 200 160 120 Height[m] 80 40 0 269 270 271 272 273 274 275 276 277 278 279 280 281 T[K]

after 900s after 2700s after 5400s initial T profile

Up to now our calculations in this chapter were performed for an isothermally stratified atmosphere. This is a particular case of stable stratification. Fog, however, may be observed in stable as well as neutral and also unstable stratification. Therefore in the next chapter let us analyse, how an initial vertical temperature gradient that increases or decreases with height affects heat diffusion.

1.2.4. The initial stratification

As has been mentioned in chapter I.3 the method of artificial heating is most effective in calm weather conditions, which are typical for the development of radiation fog and may often be

Section II.1. Case studies for a dry atmosphere 39

observed for instance on airports. In order to see the influence of stratification on heat diffusion, we repeat the experiment that led to Fig. 1.4 for an initial vertical temperature gradient that increases with height (inversion) and for an initially adiabatic stratification. In Fig. 1.7 we let the initial temperature increase by 1K per hundred meters. The results show that compared with Fig. 1.4 the surface temperature is about 0.7K higher after 90min of heating. Here we see the effect of adiabatic heating. For every time step the surface temperature adapts adiabatically to the increased temperature of the first level above surface. This mixing process on the surface is responsible for the higher surface temperatures, which allow for slightly more heating in upper levels.

Fig. 1.7: Effect of the initial stratification (inversion)

320 280 240 200 160 120 Height[m] 80 40 0 269 270 271 272 273 274 275 276 277 278 279 280 281 282 T[K]

after 900s after 2700s after 5400s initial T profile

Directing now our attention to Fig. 1.8, where an initially neutral stratification has been imposed, we may observe the inverse effect concerning the degree of temperature increase due to less effective mixing in this case.

Fig. 1.8: Effect of the initial stratification (adiabasie)

320 280 240 200 160 120 Height[m] 80 40 0 269 270 271 272 273 274 275 276 277 278 279 280 281 282 T[K]

after 900s after 2700s after 5400s initial T proflie

2. Case studies for a moist atmosphere

In section II.1 we saw how the temperature profile in a dry atmosphere changes with time, if we artificially heat it from below. Through this analysis we got an idea, how quantity and type of fuel, a variable diffusion coefficient, various initial stratifications of the atmosphere and adiabatic mixing influence the diffusion process. In the present section we apply our model to a fog layer. Since heating causes droplet evaporation, we simulate in this way fog dissipation with time. First the fog vanishes in the lower layers and with time the fog-free zone will make its way upwards to the top of the fog or to a given height where we wish the fog to disappear. The temperature profiles will not be entirely the same as we had for a dry atmosphere, since some amount of cooling due to droplet evaporation will cause the temperatures to increase less quickly. The effect of evaporation cooling will be discussed in section 2.3. If we combust real fuel usually not only heat is released, but also some amount of water vapour. The quantity of water vapour released depends on the type of fuel. The model thus takes into consideration that analogously to heat diffusion the water vapour diffuses upward according to the value of the diffusion coefficient that itself depends on height and the temperature gradient. This water vapour release is one reason, why we cannot easily dissipate cold fog by means of real fuel, where water vapour immediately condenses and may cause the fog even to intensify. The second reason is that generally more heating is necessary to evaporate cold fog than warm fog as has been mentioned in section I.4.4. Referring to Fig. 2 in part I, we expect the effect of artificial heating to decrease noticeably when we observe a marked increase of the minimal temperature for fog dissipation. This should be true for temperatures less than about -5°C. Appropriate experiments that show this dependence on the initial fog temperature, will be shown in section 2.5.3. For our case studies in a moist (foggy) atmosphere we will not only show, how the temperature profile changes with time, but also how the vertical distribution of liquid water content (LWC), visibility and the so called total water content (LWC and water vapour content) change with time. A more detailed survey on how these profiles actually are calculated and depend on each other is given in the following section.

2.1. How temperature, total water content, LWC and visibility profiles are found

Suppose an airport covered by a fog layer of 150m height. By means of measurements we find the initial distribution of LWC and temperature in the fog layer. The relative humidity in the fog layer

Section II.2. Case studies for a moist atmosphere 41

is 100%. Using eq. A.10 in annex A these initial conditions determine the initial distribution of the saturated water vapour density. As already mentioned, we allow water vapour and LWC to diffuse, analogously to heat. Diffusion takes place, if the vertical gradient changes with height. Thus, as soon as water vapour is released by real fuel, the quantity of water vapour and also LWC (in case of condensation) is relatively large near surface and tends to diffuse upwards. For the diffusion process of water vapour and LWC we introduce for convenience a new variable “total water content” (TWC) which is the sum of the water vapour content and LWC (see eq. A.6 in annex A). Calculating the TWC profile for every time step, the LWC part may be found by subtracting the saturated water vapour density (the maximal water vapour density for the current temperature before condensation occurs) from the total water content. Finally from the LWC profile the visibility profile may be found by means of an experimental relationship between LWC and visibility (see eq. A.11 in annex A). In our studies on a dry atmosphere we took account of mixing processes that during heating tend to establish adiabatic stratification. In the presence of water droplets, however, mixing leads to the development of a pseudoadiabatic temperature gradient due to evaporation cooling. Evaporation cooling is proportional to the LWC gradient as can be seen in the second term of the lower boundary condition of the heat diffusion equation (see eq. B.3 in annex B) and in the heat diffusion equation itself (see eq. C.3 in annex C). The more the LWC gradient due to warming of the downwards mixing air and heating (fuel combustion) the more evaporation cooling occurs. TWC is well mixed, if the vapour and LWC concentration is equal throughout the atmosphere. Thus, as far as the total water content is concerned mixing processes lead to a TWC profile that is constant with height (see eq. B.5 in annex B). To sum up, the lower boundary condition of the heat diffusion equation (the value of the surface temperature) is given by the temperature gradient in the first layer, which results from heating processes on the surface (fuel combustion and mixing) in consideration of evaporation cooling or latent heat release according to the LWC gradient. The lower boundary condition of the diffusion equation for total water content is given by the TWC gradient in the first layer which results from TWC mixing and release due to fuel combustion. The lower boundary conditions (eq. B.3 and eq. B.5) give the surface values of temperature and total water content, which we need to know for the diffusion processes in the atmosphere.

2.2. Total water content diffusion

In chapter II.1 we observed how the heat generated at the surface is transported upwards changing in this way the temperature profile with time. Similar studies we now perform for the change of the TWC profile due to mixing processes and water vapour production of real fuel. In order to

Section II.2. Case studies for a moist atmosphere 42

classify real fuel in its ability to release water vapour, we assign a moisture production coefficient

? v to every fuel type. ? v of kerosene for example is 1.4[kgwater vapour/kgfuel]. The larger ? v the more is the fuel’s ability to produce water vapour during combustion. Analogously to heat diffusion, the speed at which TWC moves upwards depends on the diffusion coefficient, which is a function of the temperature gradient and height. The results of the following experiments in a foggy atmosphere are shown – as in case of a dry atmosphere – after time periods of 900s (15min), 2700s (45min) and 5400s (90min). Adiabatic (pseudoadiabatic) and vapour mixing processes are taken into account in our calculations without explicitly referring to them. In Fig. 2.1 an example for TWC diffusion is illustrated.

Fig. 2.1: Total water content diffusion (av=1.4[kg/kg], D0=0.1[m²/s], initial dT/dz=0.01[K/m])

320 280 240 200 160 120 Height[m] 80 40 0 4,0 4,5 5,0 5,5 6,0 6,5 7,0 Total water content[g/m3]

after 900s after 2700s after 5400s initial rhovv profile

We obtained this solution for an inversion-type temperature profile. In our example kerosene has been burned at a rate of dm/d?=2*10-6 [kg m-2 s-1]). Due to the positive vertical temperature gradient the initial TWC gradient also increases with height (for our assumption of an initially equal distribution of LWC in the atmosphere). If we use methane instead of kerosene, more water vapour is produced on the surface, because methane has a moisture production coefficient of 2.25[kg kg-1], thus higher than kerosene. Similarly to our results for heat transport in a dry atmosphere, doubling the neutral diffusion coefficient results in enhanced upward TWC transport. The latter implies that less TWC remains near surface. The inverse effect may be observed for a smaller diffusion coefficient, where the vapour produced diffuses upwards more slowly and concentrates near the surface. Moreover it can be shown that for an initially isothermal atmosphere, the decreased vapour mixing effect from upper levels during fuel combustion results in smaller water vapour supply near surface.

Section II.2. Case studies for a moist atmosphere 43

2.3. Ideal fuel heating and its consequences for a fog layer

In the former section we analysed, how the TWC profile changes due to vapour mixing processes and water vapour production of real fuel. Let’s now examine to what extent the presence of LWC changes our results of heat diffusion we obtained for a dry atmosphere. Heating of a fog layer will principally lead to evaporation of fog droplets. As a consequence evaporation cooling occurs. Thus, the temperature should increase not as much as in case of heat diffusion in a dry atmosphere. For the following experiment we will use ideal fuel (fuel that does not release water vapour during fuel combustion) possessing the same heat production ability as kerosene and assume an initially isothermally stratified atmosphere. This should guarantee that neither vapour mixing nor production influences the degree of evaporation cooling, i.e. for such conditions the vertically uniform TWC structure remains constant with time, because the LWC part of the total water content decreases at the same rate as the saturated water vapour part increases (owing to the temperature increase). In this way we may observe how heating actually influences the LWC content and thus visibility where vapour mixing and production plays no role. The following experiment, illustrated in Fig. 2.2.1-4 is made for the same conditions as for Fig. 1.4. Therefore through direct comparison of the temperature profiles in Fig. 1.4 and Fig. 2.2.1 conclusions about the effect of evaporation cooling can be made.

Fig 2.2.1: Heat diffusion in a fog layer using ideal fuel (dm/dt=5E- 5[kg/(m²*s)], a=4E7[J/m²])

320 280 240 200 160 120 Height[m] 80 40 0 269 270 271 272 273 274 275 276 277 278 279 280 281 T[K]

after 900s after 2700s after 5400s initial T profile

In Fig. 2.2.1 at all levels the temperature increased about 0.6K less than in the dry case after 5400s of heating. As can be seen in Fig. 2.2.2 and 2.2.3 the fog disappears with time.

Section II.2. Case studies for a moist atmosphere 44

Fig 2.2.2: Liquid water content

320 280 240 200 160 120 Height[m] 80 40 0 0,00 0,05 0,10 0,15 0,20 0,25 0,30 LWC[g/m3]

after 900s after 2700s after 5400s initial q profile

Fig 2.2.3: Visibility

320 280 240 200 160 120 Height[m] 80 40 0 0 100 200 300 400 500 600 700 800 900 1000 Visibility[m]

after 900s after 2700s after 5400s initial visibility profile

Especially in the beginning of heating good results can be achieved for lower layers. After 900s a fog free zone of about 30m height may be expected. If airplanes need a minimum fog free height of 60m above surface, our studies show that landing and departing under the present circumstances and fuel properties should be possible after approximately 2400s (40min) of heating.

2.4. Application of real fuel in a warm fog

If we use real fuel instead of ideal fuel, our results will – as has been mentioned – depend on the initial fog temperature and to a certain degree also on the moisture and heat production ability of fuel.

Section II.2. Case studies for a moist atmosphere 45

Before performing experiments on this topic, let us direct our attention to Fig. 2 in part I. These graphs tell us that for positive temperatures the usage of various real fuel (except hydrogen) is virtually equally effective as ideal fuel. The less the fog temperature, the more the minimal temperature increase necessary for fog dissipation. For real fuel this relationship depends on the ratio between the moisture and heat production coefficient. The bigger this ratio, the less is the efficacy of fog dissipation for lower temperatures. In case of hydrogen extraordinary large

-1 amounts of water vapour are released (? v=9[kg kg ]), whereas the heat produced is relatively small (? =1.4*107 [J kg-1]). Therefore hydrogen has practically no application for fog dissipation, since the initial fog temperature should be at least 290K, but fog is rarely observed for such conditions. Thus, with the exception of hydrogen, the results for fog dissipation by means of real fuel for positive initial fog temperatures should be similar to the ideal fuel solution. This may be confirmed repeating the experiment that led to Fig 2.2.1 for kerosene (? =4*107 [J kg-1] and

-1 ? v=1.4[kg kg ]) instead of ideal fuel. Since no condensation of the additional water vapour released occurs (heating obviates the additional amount of water vapour to condense), the temperature profile remains virtually the same as in Fig. 2.2.1. As for the LWC and visibility profiles we observe that in case of kerosene combustion the height of the fog free zone is 28m instead of 30m after 5400s. This small difference may be explained by the fact that due to the additional water vapour the LWC does not decrease as fast as for ideal fuel.

2.5. Approaching real atmospheric conditions

Up to now we visualized how temperature increase due to artificial heating changes the LWC and visibility profiles with time in an initially isothermally stratified fog layer where the initial LWC was chosen constant with height. Our experiments helped us to understand the physical mechanisms of fog dissipation by artificial heating. In real conditions, however, as we learned from section 2 in part I the fog temperature and LWC structure depends on the state of its evolution. For our case studies we concentrate mainly on radiation fogs. Let’s call back to mind therefore several relevant features of such fogs. If we take young radiation fog, the lowest temperatures and maximal values of LWC are often observed near the surface. Generally speaking for such fogs the temperature increases with height, whereas the LWC decreases. In radiation fogs of mature development, however, the maximum of LWC in many cases is observed in the middle of the fog layer, associated with a lifted inversion. The structure of old radiation fogs is similar to advective fogs, where LWC decreases and temperature increases with height. The fog’s upper boundary lies between the lower and upper boundary of the ground or uplifted inversion layer. According to measurements in the Ukraine from October 1960 to June 1962 (Matveev, 2000) the average height of radiation fogs is about 150m, where vertical extents

Section II.2. Case studies for a moist atmosphere 46

between 100m and 200m were observed most frequently. The following experiments we perform for young and mature radiation fogs. Old radiation fogs are not treated here, since it is probable for them to vanish in a shorter time interval by natural processes than needed for artificial dissipation.

2.5.1. Artificial dissipation of a young radiation fog

For the following experiment we assume a young radiation fog of 200m height. Its surface temperature is 270K, where the vertical temperature gradient is constant and positive (1.0K per 100m). The LWC profile shows maximal values near the surface (LWC=3.5[g m-3]) and decreases at a constant rate until 1.0[g m-3] near the top (at 180m). Within the remaining 20 meters LWC tends to zero and vanishes completely at 200m height. Above the fog layer the relative humidity decreases from 100% to 60% at a height of 600m and the temperature decreases vertically at the same rate as in the fog layer. As heating source we use kerosene that we let burn at a rate of 2*10- 6 -2 -1 2 -1 [kg m s ]. For the neutral diffusion coefficient we assume D0=0.1[m s ]. Fig. 2.3.1-4 show the temporal evolution of the temperature, TWC, LWC and visibility profiles for our young radiation fog.

Fig. 2.3.1: Heat diffusion in a young radiation fog using kerosene (fog height: 200m)

320 280 240 200 160 120

Height[m] 80 40 0 266 267 268 269 270 271 272 273 274 275 276 277 278 T[K]

after 900s after 2700s after 5400s initial T profile

Section II.2. Case studies for a moist atmosphere 47

Fig 2.3.2: Total water content diffusion

320 280 240 200 160 120 Height[m] 80 40 0 4,0 4,5 5,0 5,5 6,0 6,5 7,0 Virtual water vapour[g/m3]

after 900s after 2700s after 5400s initial rhovv profile

Fig 2.3.3: Liquid water content

320 280 240 200 160 120 Height[m] 80 40 0 0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 LWC[g/m3]

after 900s after 2700s after 5400s initial q profile

Fig 2.3.4: Visibility

320 280 240 200 160 120 Height[m] 80 40 0 0 100 200 300 400 500 600 700 800 900 1000 Visibility[m]

after 900s after 2700s after 5400s initial visibility profile

Section II.2. Case studies for a moist atmosphere 48

After 5400s the fog disappears up to a height of 105m due to our heating efforts from below. Additionally we observe that at that time the fog upper boundary dropped to a height of 180m. The reason for the fog dissipation in upper fog layers are diffusion processes that are initiated due to the interplay of the temperature and TWC (LWC and water vapour) profiles in these layers: Initially the total water content increases vertically up to a height of 180m (Fig. 2.3.2). We can see from this fact that the increase of saturated water vapour density due to the inversion type stratification is more than the decrease of LWC in the fog layer up to this height. This is not true anymore for the upper 20m, since there the LWC content rapidly decreases, whereas the temperature increase with height remains unchanged. Above the fog the relative humidity decreases such that the total water content also decreases with height. In the fog free zone TWC is identical to the water vapour density without the LWC admixture. The consequence of turbulent diffusion processes can be well seen in Fig. 2.3.2, where changes of the vertical gradient in the TWC profile are smoothed. By means of this mechanism the total water content between 125m and 280m decreases during the period of heating. In the upper fog layer this decrease is sufficient to allow full evaporation of the fog droplets and subsequent fog dissipation. Therefore the fog upper boundary dropped from 200m to 180m. The slight temperature decrease after 5400s near the top of the initial fog layer (see Fig. 2.3.1) finds its explanation in evaporation cooling. Finally it should be noted that after 5400s the visibility improved at all levels (see Fig. 2.3.4), where the zone of least success (at 140m height) lies between the dissipation processes acting in lower layers due to heat diffusion and those in upper layers due to TWC diffusion. Supposed again, planes need a vertical extent of 60m of fog free zone to land or depart, we would have to wait about one hour (see Fig 2.3.4) until all LWC is evaporated up to this height. Since the rate at which kerosene has been combusted in our example constituted 2*10-6 [kg m-2 s-1], the airport management has to supply about 720 kg of kerosene to dissipate the present young radiation fog on an area of 105 m2 up to a height of 60m.

2.5.2. Artificial dissipation of a mature radiation fog

In section I.2 we learned about the development of a mature radiation fog. In such fogs the inversion layer that has evolved from radiation cooling is lifted, where the temperature gradient under the inversion layer is near pseudoadiabatic. For the sake of comparison with the dissipation processes in a young radiation fog we assume that within the lower 100m the initial temperature profile of Fig. 2.3.1 is pseudoadiabatic (dT/dz = -0.6K/100m). Accordingly the LWC values are relatively small near the surface (1.0[g m-3]) and increase up to the lower boundary of the inversion layer. Within the inversion the initial LWC decreases similar to Fig. 2.3.3. Again we use kerosene as energy source and a neutral diffusion coefficient of 0.1[m2 s-1]. The results are shown

Section II.2. Case studies for a moist atmosphere 49

in Fig. 2.4.1-4. A comparison with Fig. 2.3.1-4 exhibits how the changed initial temperature and LWC profiles in Fig. 2.4.1 and Fig. 2.4.3 affect the process of heat and TWC diffusion and thus the improvement of visibility.

Fig. 2.4.1: Heat diffusion in a mature radiation fog using kerosene (fog height=200m)

320 280 240 200 160 120

Height[m] 80 40 0 266 267 268 269 270 271 272 273 274 275 276 277 278 T[K]

after 900s after 2700s after 5400s initial T profile

Fig 2.4.2: Total water content diffusion

320 280 240 200 160 120 Height[m] 80 40 0 3,0 3,5 4,0 4,5 5,0 5,5 6,0 Virtual water vapour[g/m3]

after 900s after 2700s after 5400s initial rhovv profile

Section II.2. Case studies for a moist atmosphere 50

Fig 2.4.3: Liquid water content

320 280 240 200 160 120 Height[m] 80 40 0 0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 LWC[g/m3]

after 900s after 2700s initial q profile

Fig 2.4.4: Visibility

320 280 240 200 160 120 Height[m] 80 40 0 0 100 200 300 400 500 600 700 800 900 1000 Visibility[m]

after 900s after 2700s initial visibility profile

Fig 2.4.4 shows that the mature radiation fog dissipates faster than a young radiation fog. Thus, it takes about 35 minutes to dissipate fog up to 60m height, which implies kerosene expenses of 420 kg for an area of 105 m2. As can be seen in Fig 2.4.3-4 the fog dissipates totally during our period of heating. The dissipation time computed by the model is 5370s (85.5min).

2.5.3. Implications of fog temperature and fuel types on fog dissipation

As we have seen in section 2.4 for fog temperatures higher than about -5°C the effect of artificial fog dissipation is similar whether we use real or ideal fuel, provided that the same amount of heat is emitted. In this section we discuss the dissipation of a mature radiation fog for different types of fuel. Again, for the sake of comparison the atmospheric initial conditions (temperature profile,

Section II.2. Case studies for a moist atmosphere 51

LWC profile) are equivalent to our experiment using kerosene in the former chapter. In Fig 2.5 to Fig. 2.8 the change of the visibility profiles with time is shown for methane, propane, coal (anthracite) and peat. In annex D the heat and water vapour production abilities for these fuel types are listed. Ts indicates the initial fog temperature near surface.

Fig 2.5: Methane (Ts=271.5K)

320 280 240 200 160 120 Height[m] 80 40 0 0 100 200 300 400 500 600 700 800 900 1000 Visibility[m]

after 900s after 5400s initial visibility profile

Fig 2.6: Propane (Ts=271.5K)

320 280 240 200 160 120 Height[m] 80 40 0 0 100 200 300 400 500 600 700 800 900 1000 Visibility[m]

after 900s after 2700s initial visibility profile

Section II.2. Case studies for a moist atmosphere 52

Fig 2.7: Coal (Ts=271.5K)

320 280 240 200 160 120 Height[m] 80 40 0 0 100 200 300 400 500 600 700 800 900 1000 Visibility[m]

after 900s after 2700s after 5400s initial visibility profile

Fig 2.8: Peat (Ts=271.5K)

320 280 240 200 160 120 Height[m] 80 40 0 0 100 200 300 400 500 600 700 800 900 1000 Visibility[m]

after 900s after 2700s after 5400s initial visibility profile

With reference to Fig. 2 in part I for our fog (initial temperature near 0°C) we do not expect the water vapour released during fuel combustion to significantly influence the heating and dissipation process. Consequently the heat production ability should determine the degree of success. This holds true if we convince ourselves that the most satisfactory dissipation results could be achieved for methane (Fig 2.5), which possesses the best heat production ability of all fuel types mentioned here. The second most effective fuel is propane (Fig 2.6), followed by kerosene (see Fig 2.4.4), coal (Fig 2.7) and peat (Fig 2.8). The heat production abilities thus reflect the results of dissipation. This is not entirely true anymore, if the initial fog temperature is 10°C less, as we can easily see in Fig 2.9 to 2.12. For these figures we did not change the initial vertical temperature and LWC gradients of our mature radiation fog, in order to see directly the effects of colder air on fog dissipation.

Section II.2. Case studies for a moist atmosphere 53

Fig 2.9: Methane (Ts=261.5K)

320 280 240 200 160 120 Height[m] 80 40 0 0 100 200 300 400 500 600 700 800 900 1000 Visibility[m]

after 900s after 2700s after 5400s initial visibility profile

Fig 2.10: Propane (Ts=261.5K)

320 280 240 200 160 120 Height[m] 80 40 0 0 100 200 300 400 500 600 700 800 900 1000 Visibility[m]

after 900s after 2700s after 5400s initial visibility profile

Fig 2.11: Coal (Ts=261.5K)

320 280 240 200 160 120 Height[m] 80 40 0 0 100 200 300 400 500 600 700 800 900 1000 Visibility[m]

after 900s after 2700s after 5400s initial visibility profile

Section II.2. Case studies for a moist atmosphere 54

Fig 2.12: Peat (Ts=261.5K)

320 280 240 200 160 120 Height[m] 80 40 0 0 100 200 300 400 500 600 700 800 900 1000 Visibility[m]

after 900s after 2700s after 5400s initial visibility profile

At first sight, as expected from Fig. 2 in part I, it is obvious from the results that our method works less effectively for lower temperatures. For all fuel types the improvement of visibility proceeds more slowly. At the same time we may observe that for a lower initial fog temperature the ratio of the moisture and heat production coefficient plays a role. Since methane releases more water vapour during combustion, our success using methane or propane turns out to be virtually the same (compare Fig. 2.9 and 2.10). Fig. 2.12 illustrates the quite unfavourable qualities of peat for cold fog dissipation due to its relatively high moisture production ability. The improvement of visibility is clearly less than we had in case of the warmer fog (Fig 2.8). Coal on the other hand is less sensitive to lower initial fog temperatures (Fig. 2.11). In Fig. 2 in part I we see that of all fuel types the properties of coal are the most close to ideal fuel. Therefore the differences in the improvement of visibility in Fig. 2.7 and Fig. 2.11 are less noticeable than in case of other fuel types.

Annex A

Model equations

The process of fog dissipation is described by two partial differential equations, the heat diffusion equation and the total water content diffusion equation which both need initial and boundary conditions to be solved. The boundary conditions are determined by the flux of heat energy and water vapour density due to external (fuel combustion) and internal (atmospheric mixing, turbulent diffusion) forcing. Some of the variables used in these equations are not directly known. Therefore four additional equations are necessary to close the system: an equation for the total water content, one for the behaviour of the diffusion coefficient, an equation determining the saturated water vapour pressure and the equation of state for the water vapour density. A seventh equation, the visibility equation, is used to relate the atmospheric water content to the visual range of the human eye in the presence of fog.

1. The heat diffusion equation

In the present model we consider heat transport by turbulent diffusion that results from artificial heating (fuel combustion) at the surface. Allowing for big values of the diffusion coefficient, turbulent diffusion hereby indirectly is meant to involve convective motion mixing processes, which also contribute to heating in the atmosphere. For this model heat transport by radiational processes are not taken into account. Strictly speaking, surface and atmospheric radiation and also radiation induced by fuel combustion (fire) to some degree affect fog dissipation. These rather complex processes could be considered in further developments of the present model, which may include for instance the Stefan Boltzmann law, Planck’s radiation law and the wave length dependent absorption properties of water vapour and liquid water. For the moment, however, we argue that in real conditions it takes at least two or three hours for a radiation fog of considerable height (100m and more) to evolve or to disperse. In contrary, artificial heating and well-developed turbulent diffusion allow for good results of fog dissipation in a time scale of one hour or less, which in our opinion justifies in a first approximation the neglect of radiation effects. We neither -5 consider molecular diffusion, since for the atmosphere molecular diffusion (Dm=2*10 ) is a factor 104 less effective than turbulent diffusion, which is of the order D=10-2 to 1. For the following the term “diffusion” will always mean “turbulent diffusion”. The governing equation for heat transport in our case thus is the heat diffusion equation (Katchurin, 1990):

Annex A. Model equations 56

? ? ? ? ? T ? ?D T ? (A.1) ?? ?z? ?z ? , where T is the temperature, D the diffusion coefficient (see chapter 1.2.4), z the height, tau the time. Equation (A.1) represents a partial differential equation of 2nd order. The two independent variables are time and height. For a constant diffusion coefficient the equation may be written in the form

? ?2 T ? D T (A.2) 2 ?? ?z .

The latter equation is used for the analytical solution solved in chapter I.4.4. In fact, temperature changes not only due to diffusion of heat, but also due to turbulent adiabatic mixing in the atmosphere (see chapter II.1) and - in case of a moist atmosphere - due to condensation and vaporisation processes (chapter II.2). As a tool to take these factors into account we may use the energy balance equation. In our model the total energy of a unit volume of air is balanced by three compounds, which are the heat energy (or internal energy), potential energy and latent heat. According to the author’s supervisor V.I. Bekrjaev the balance equation thus can be written in the form

N ? ? a cp T ? ? a g z ? Lv q, (A.3)

-3 -1 -1 where ? a [kg m ] is the air density, cp [J kg K ] the specific heat capacity of air at constant 3 -1 pressure, q [kg/m ] the liquid water content and Lv [J kg ] the heat of vaporization. The total energy per unit volume is denoted N [J m-3] (this notion is common in Russian literature). Due to the first term on the right side the total energy increases, if the temperature of the unit volume increases. The second term takes into account potential energy changes of the unit volume, if it moves upwards or downwards. Thus, the second term may serve to describe energy (temperature) changes due to turbulent adiabatic mixing. In case of moist air, the third term has to be considered, where the total energy increases, if droplets evaporate and decreases if water vapour condenses (latent heat release). Following ideas of V.I. Bekrjaev the energy balance equation (A.3) serves us to find temperature changes with time and height, if we use the total energy instead of the temperature in equation (A.1). The resulting energy diffusion equation yields

? ? ? ? ? N ? ?D N? (A.4) ?? ?z? ?z ? .

Annex A. Model equations 57

If the vertical extent of the atmospheric layer regarded (e.g. fog layer) is sufficiently low (200m – 300m), being actually often the case for radiation fogs, the air density can be taken constant with height. Using this simplification and expressing the temperature change with time we find

? ? ? ?? ? ? Lv q ?? ? ? ?? g ?z ?? ?D ? T ? ? ?? ? ?z ?z cp ? a cp T ? ? ? ?? ?? ? (A.5) Lv q ?z 1 ? ? ? a cp T ?z .

Equation (A.5) is the basic tool to describe the temperature change with time. It takes into consideration heat diffusion, turbulent adiabatic mixing as well as condensation and vaporization processes.

2. The total water content equation

The total water content (TWC) is the sum of the water vapour density and liquid water content of the fog. The corresponding equation yields:

? vv ? ? v ? q , (A.6)

-3 -3 -3 where ? vv [kg m ] is the total water content, ? v [kg m ] vapour density and q [kg m ] the liquid water content. For unsaturated air TWC coincides with the real vapour density ? v. For saturated air ? v?in eq. (A.6) is substituted by the saturation water vapour pressure ? vs. Thus, in the presence of fog ? vv is larger than ? vs for the current temperature. Strictly speaking, apart from temperature the amount of condensation nuclei plays a role for the condensation processes. For our model, however, condensation as well as evaporation occurs strictly at the time when the air temperature has reached the dew point temperature.

3. The diffusion equation for total water content

If turbulent diffusion occurs, not only heat is transported, but also water content and water vapour, i.e. total water content. The transport of TWC in the atmosphere is analogous to the transport of

Annex A. Model equations 58

heat. Thus, we can describe it in the same way, but this time it is not the temperature that changes with height and time but the total water content:

? ? ? ? ? ? vv ? ?D ? vv ? (A.7) ?? ?z? ?z ? .

4. Equation for the diffusion coefficient

The vertical profile of the diffusion coefficient in the atmosphere mainly depends on the stratification of the atmospheric layer, height and wind speed. For the present work we propose the following approximation for the diffusion coefficient: The diffusion coefficient for a neutral atmosphere is given by D0. For an unstable atmosphere the diffusion coefficient increases as instability increases, for a stable atmosphere D tends to 0. D gets larger also with height, because of the increase of the mixing length. The dependence on wind is not taken into account, we suppose calm. Adapted to the problem of artificial heating the diffusion coefficient can be qualitatively described by the equation

?d ? ? A? T ? ?a? z ?dz ? (A.8) D ? D0 ln e z0 ,

where z is the height, z0 the roughness length and ?a the adiabatic temperature gradient. Following eq. (A.8) the diffusion coefficient increases logarithmically with height and exponentially with the increase of the temperature gradient. The coefficient A can be varied to get different dependences of D on the temperature gradient. It has been proposed to choose the value of A such that for a -1 2 -1 temperature gradient of 0.5[K m ] and D0=0.1[m s ] the diffusion coefficient is about 100[m2/s]. By this way we make sure to take account of big diffusion coefficients in case of strong temperature gradients such as they occur on the surface, where fuel combustion takes place. More information concerning the choice of A can be found in chapter 1.2.1 in part II.

5. Equation for the saturated water vapour pressure

The saturation vapour pressure is a near-exponential function of temperature and may be described by the relationship

Annex A. Model equations 59

17.67 (T? 273.16) ? ? ? T? 29.66 ? (A.9) E ? E0 ?e

(Stull, 1999). Here E is the saturated vapour pressure and E0 the saturated vapour pressure at a temperature of 273.15K. The unit of E and E0 is [Pa]. T is the temperature in Kelvin.

6. Equation of state for the water vapour density

Vapour density is a function of temperature and relative humidity and can be expressed by

E f ? v ? (A.10) Rv T , where f is the relative humidity. Accordingly the saturation water vapour density can be found by -1 -1 this equation setting the relative humidity to 100%, i.e. f = 1. Rv = 461.5 [J kg K ].

7. Visibility equation

The aim of fog dissipation is the improvement of visibility. Actually visibility depends on the droplet concentration, their size and scattering characteristics. For our calculations however it will be sufficient to use an experimental equation, where visibility and water content are related by

? 0.54 L ? 46 ?q (A.11)

(Katchurin, 1990), where L is the visibility in meters and q the liquid water content in [g m-3].

Annex B

Initial and boundary conditions

The equations (A.5) and (A.7) are partial differential equations of 2nd order. In order to solve these equations we need for both of them one initial and two boundary conditions. The initial condition represents the state of the atmosphere before artificial heating sets in. The boundary conditions are given at the surface and the top of the atmospheric layer regarded.

1. Initial conditions

The initial vertical profiles of temperature, relative humidity and liquid water content (LWC) are supposed to be known by measurement. The vertical profiles of all other variables necessary for our model (? vv, ? v, ? vs, L) may be calculated from these measured meteorological parameters by means of the equations (A.6), (A.10) and (A.11).

2. Lower Boundary Conditions (LBC)

LBC for the Heat diffusion equation:

Generally the temperature of the Earth surface is known only before heating starts, i.e. from the initial condition. If we use fuel combustion as energy source, the energy flux produced is proportional to the mass of fuel burned. Thus, we get for the energy flux PN through a unit area of the Earth surface

?? ? dm P ? ?D N ? ? N( z? 0) ? ? (B.1) ?z d? ? ?z? 0 , where m [kg m-2] is the mass of fuel burned per unit area, N [J m-3] the total energy as defined in annex A (eq. A.3) per unit volume and ? [J kg-1] the heat production coefficient. The energy flux

-2 -1 is denoted PN [J m s ] (the common notion in Russian literature). Values of ? for various fuel types can be found in annex D.

Annex B. Initial and boundary conditions 61

Splitting the total energy into its compounds (internal, potential and latent heat energy) we will find after appropriate transformations for the surface layer

1 ? dm ? g Lv ? ? T ? ? ? q (B.2) ? a cp D d? ?z cp ? a cp ?z .

Transforming the differentials into finite differences we obtain for the lower boundary condition of the heat diffusion equation at the Earth surface

g Lv 1 ? dm T(z? 0) ? T(z? 1) ? ?z ? ?q(z? 1) ? q(z? 0) ? ? ?z (B.3) cp ? a cp ? a cp D d? .

As can be seen, for the calculation of the surface temperature in equation (B.3) the second, third and forth term on the right side add to the current temperature of the first level (first term). As can be seen in equation (B.3) the earth surface temperature T(z=0) is determined by adiabatic mixing processes within the first layer (first and second term on the right side), by cooling owing to evaporation or latent heat release due to condensation (third term) and by the amount of heat added during fuel combustion (forth term).

LBC for the TWC diffusion equation:

The lower boundary condition for the TWC diffusion equation is found in an analogous way as -2 -1 for the heat diffusion equation. The TWC flux P(? vv) [kg m s ] by fuel combustion is determined by the fuel’s water vapour production ability ? v [kgwater vapour/kgfuel]) and can be written as

?? ? dm P ? ?D ? ? ? ? vv ( z? 0) ? vv ? v (B.4) ?z d? ? ?z? 0 ,

Thus, the lower boundary condition of the diffusion equation of virtual vapour density yields

? v dm ? vv(z? 0) ? ? vv(z? 1) ? ?z (B.5) D d? .

Values of ? v for various fuel types are listed in annex D.

Annex B. Initial and boundary conditions 62

3. Upper boundary conditions

At the upper boundary of the atmosphere no heat and water vapour fluxes are allowed. All changes of temperature and total water content concern the layer below the upper limit. The upper boundary itself is located high enough such that the atmosphere aloft is not affected by heat diffusion, turbulence, condensation and vaporisation processes. Thus, at the top we may set the temperature and TWC constant with time and the upper boundary conditions for the equations (A.5) and (A.7) yield

T(z? top) ? const (B.6)

? vv(z? top) ? const , (B.7) respectively.

Annex C

Numerical method – finite differences

In order to solve the heat and TWC diffusion equations (A.5 and A.7) we apply the numerical method of finite differences, where the partial differentials are replaced by finite differences. In Fig. C.1 the numerical scheme for heat diffusion is illustrated for four steps by height and time.

Fig. C.1: Finite difference method: Grid point scheme

T(m,j+1) = T(1,3) 2nd level 3

zu, Du, Tu, qu 2nd layer

T(m,j) = T(1,2) 1st level j (height step) 2 T(m+1,j) = T(2,2) (defined by eq. (C.3) zd, Dd, Td, qd 1st layer

T(m,j-1) = T(1,1) T(m+1,j-1) = T(2,1) surface level 1 (defined by the lower 1 2 boundary condition) 3 4 m (time step)

1. Equation for the diffusion coefficient

As can be seen in Fig. C.1 the diffusion coefficient is calculated individually for every layer as the average value between the upper and lower grid points. For the calculation of the unknown

temperature value at the next time step the averaged value of the diffusion coefficient D of the layer over and under the current grid point is used. We call D over the grid point upper diffusion coefficient Du and D under the grid point Dd. In this way the finite difference equation of the upper diffusion coefficient Du derives to

??Tm?j? 1? Tm?j? ? ? A? ? ?a? ? k ? (C.1) Du ? D0 ln?10 zu? e .

Dd yields analogously

??Tm?j? Tm?j? 1? ? ? A? ? ?a? ? k ? (C.2) Dd ? D0 ln?10 zd? e .

In the above equations j is the index of the height steps, m the index of the time steps and k the height step (the notion k has been used in the model and is equal to ? z). zu and zd are the average heights of the corresponding layer for which Du and Dd are found.

2. Heat and TWC diffusion equation

When transforming the heat diffusion equation (A.5) to finite differences we find with respect to the above definitions for the forward step of temperature

? h h g h Lv ? Du ?Tm?j?1 ? Tm?j? ? Dd ?Tm?j ? Tm?j?1? ? ?Du ? Dd? ? Du ?qm?j?1 ? qm?j? ? Dd ?qm?j ? qm?j?1? ? 2 ? ? 2 ? ?? ? k k cp k ? a cp ? Tm?1?j ? Tm?j ? ? Lv qu ? qd ? 1? ? ? ?? a cp Tu ? Td?

(C.3)

where h is the time step (h is equal to ??). Tu and qu are the average values of temperature and water content in the layer over the current grid point, Td and qd analogously under the current grid point. The transformation of the TWC diffusion equation to finite differences yields

h ? vv( m? 1?j) ? ? vv( m?j) ? ?Du ?? vv( m?j? 1) ? ? vv( m?j)? ? Dd ?? vv( m?j) ? ? vv( m?j? 1)?? 2 ? ? (C.4) k .

Annex C. Numerical method – finite differences 65

3. Lower boundary conditions of the heat and TWC diffusion equations

For the lower boundary condition of equation (B.3) we get as numerical representation

g Lv k ? dm Tj? 1 ? Tj ? k ? ?q j ? q j? 1? ? (C.5) cp ? a cp ? a cp D0 d? and for (B.5)

? v dm ? vv( j? 1) ? ? vv( j) ? (C.6) D0 d? .

With regard to the initial and boundary conditions, for every time step and at all grid points a solution of equation (C.3) and (C.4) can be found.

Annex D

Type of fuel ? [MJ/kg] ? v [kg/kg] (heat production ability) (moisture production ability) Methane 55 2.25 Propane 50 1.6 Alcohol 31 1.2 Petrol 44 1.4 Kerosene 40 1.4 Coal (anthracite) 22 0.45 Wood 10 0.5 Peat 8 0.55 Hydrogen 14 9.0 Heat and moisture production ability of various fuel (Bekrjaev, 1977)

Acknowledgements

Without the support and struggle of many colleagues and friends in Russia and Austria the completion of this diploma work would not have been possible. First of all I would like to thank my supervisor Dr. V.I. Bekrjaev who devoted many hours of his time to smaller and bigger problems of my work, whenever I wished and Prof. Dr. I. Vergeiner who agreed to supervise my thesis formally, but in fact more than that, and made it possible by this way to finish my studies at the university of Innsbruck. In this context I would like to express special thanks also to Prof. Dr. H. Rott for endless postal and electronic communication to create a financial basis for my studies in St.Petersburg and to get acknowledged several exams I passed there. Moreover special thanks to Prof. Dr. G.G. Tarakanov for his advices and support all the two years long. I am very grateful to Mag. E. Podgaisky who has been my first contact for all university-related problems I had to face as a student from abroad and to Dr. A.I. Bogush, who without any bureaucracy could manage for me a scholarship and made it therefore possible to write the present diploma these. I would like to thank colleagues from the weather services of the major Austrian airports, Dr. H. Pümpel (Innsbruck), Mag. H. Unegg (Klagenfurt), Mag. G. Mahringer (Linz), Mag. W. Hammer (Salzburg), Mag. E. Schmidt (Graz) and Mag. M. Kerschbaum (Vienna) for their professional opinion on the topic. Special thanks to Mag. H. Unegg who established the first real contact to the Hydrometeorological University and for all the fruitful discussions and advices since then. A great help for my diploma thesis has been Tatjana Pavlova from the Main Geophysical Observatory who spent hours with me to run the program in Linux. I say thank you to everybody in the department of Russian language, especially to my teachers Natalja, Galina, Vladimir and Nikolaj for they made life much more agreeable. Thanks also to Ala, Olja and Viktoria from the foreign office for their personal advices and who sometimes made the impossible possible. Very special thanks to beloved Victoria for her never ending moral support, for all the innumerable ways through the hopelessly overcrowded metro and the constant flux of positive energy. Last but not least I would like to express my sincere gratitude to my parents for their patience and wonderful support during all days of my studies. Thank you.

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