A. INTRODUCTION 



Forecasting ice thickness can be separated into three problems: 

 (1) determining the time required to reduce the temperature of the 

 water mass to the freezing point by theraohaline convection} (2) fore- 

 casting the temperature, wind velocity, cloudiness, humidity, depth, 

 and density of snow fall for the period for which the ice thickness 

 forecast is required; and (3) computing the thickness of the ice accre- 

 tion which will result from the predicted weather conditions. This 

 study is restricted to the last phase of this problem. 



The problem of ice thickness forecasting is* one of great complexity, 

 expressible only in terms of a complicated system of differential and 

 integral equations, the solution of which is not possible when the bound- 

 ary conditions are not simple. The first physicist to present a complete 

 mathematical theory of heat conduction was Joseph Fourier. The applica- 

 tion of Fourier's heat conduction equations to the problem of ice forma- 

 tion was first undertaken by Franz Neumann (Weber, 1910) and Stefan (1889). 

 More recently Russian and Norwegian scientists have been active in this 

 field, and it is with the description of the work of A. G. Kolesnikov 

 (1946) and Olav Devik (1931) that this study is chiefly concerned. 



After the derivation of a theoretical forecast formula the question 

 arises as to the best practical method of its application. In this con- 

 nection consideration should be given not only to the facility with which 

 the results of the formula can be obtained but also to the accuracy of 

 the evaluation. 



There are two general methods for obtaining the required ice thick- 

 ness from the formula: (l) by computation for each individual situation, 

 and (2) by taking the required hickness from graphs consisting of para- 

 metric curves of ice thickness drawn with temperature and Ice thickness 

 as abscissa and ordinate, respectively, and with meteorological factors 

 as parameters. Both of these methods will be derived and explained. To 

 quickly obtain approximate results the graphical method is recommended, 

 but for a more accurate determination in which all the parameters are 

 given individual consideration, the computational method should be employed. 



B. INCREASE OF ICE THICKNESS WITH TIME 



In approaching the problem of ice growth with time, it is desirable, 

 initially, to formulate an expression in simple terms, that is, in the 

 preliminary steps to neglect the meteorological factors of wind velocity, 

 cloud cover, and humidity and to consider the ice as formed free from the 

 blanketing influence of a snow, cover. Initially then, assume a surface 

 of still water lowered by contact with air to some temperature T Q below 

 the freezing point. There will then be formed a layer of ice whose thick- 

 ness f is a function of the time t. A solution may be reached by equating 

 the amount of heat carried up from the water below the ice sheet plus the 

 heat set free per unit of time (dt) per unit area (a3 the ice increases 

 in thickness by df ) to the total amount of heat that flows outward through 



