SECT. 5] TIDES 791 



currents in an idealized channel with dimensions of the River Elbe. A general 

 representation of methods appropriate for computing the tides in rivers is 

 given by Hensen (1958). 



These results in the one-dimensional case suggested that an attempt should 

 be made on the two-dimensional problem. A first start was made with North 

 Sea tides using a grid with large mesh-size and a small number of grid points to 

 avoid wearisome computations. Later on, the Swedish electronic computer 

 became available for tide and storm-surge computations with small grid-size 

 in the North Sea (Hansen, 1956; Fischer, 1959). These first attempts were not 

 completely satisfactory although approximate solutions were obtained. 

 Recently this work has been continued with electronic computers of large 

 capacity and again tides and wind-caused motion in the North Sea are being 

 considered. The North Sea is limited by a boundary in the north running from 

 the Firth of Forth to a point north of Bergen on the Norwegian coast. In the 

 south the boundary is situated in the Straits of Dover. On these boundaries the 

 principal M2 constituent is prescribed in the form 



i = ii cos (yt + ^2 sin at. 



These values are derived from observations. On all other boundaries (i.e. 

 along the coastline) the values are calculated and the normal component of 

 velocity is zero. Skagerrak and Cattegat are included in the area under in- 

 vestigation, and it is assumed that for all intents and purposes the entrance to 

 the Baltic is closed. The hydrodynamical meaning of this assumption is that 

 there are no currents through these straits and, as on coastal boundaries, 

 the normal component of velocity is zero. In brief, the variations of sea-level 

 and currents in the whole North Sea are uniquely determined by these para- 

 meters ; no observations, other than the ^-values on the northern boundary 

 and in the Straits of Dover, are needed. In the difference equation the Coriolis 

 force and the variation of depth are taken into account. The friction term is 

 determined as follows : 



The coefficient r has a constant dimensionless value r = 3 x 10~3. This number 

 seems to be applicable to problems in estuaries and in open seas as well as in 

 the ocean. The mesh-size was 1= 18.5 km and the time step t= 147 sec. Con- 

 vective terms have been dropped out of the equation. Investigations in shallow 

 water have shown that these terms are only important in areas with large 

 variations of depth ; this is not the case in the open North Sea. From friction 

 and divergence terms there arise non-linear influences effective especially in 

 regions of small depth. In Fig. 20 the tide curves for spring and neap are drawn 

 up according to observations in Heligoland, and the computed tide curve in the 

 grid-point next to this island is reduced to a uniform scale. 



In order to obtain information as to the accuracy of computed tides and tidal 

 currents, these have been compared with observed data. In Fig. 21, for each 



