Global Wave Forecasts Using Spacecraft Data 



and should be fairly optimum for the time and space scales and quality of data 

 expected. The very high frequencies are always treated as fully developed and 

 can therefore be handled quite simply. 



The geographical variations are represented by remembering the spectrum 

 at discrete grid points. As can be seen from Figs. 3, 4, and 5 the variation of 

 significant wave height at adjacent grid points 120 nautical miles apart on a 

 square grid is quite large. Thus a tighter grid spacing has been chosen. If the 

 number of grid points is doubled on the square grid, each grid point represents 

 about 6400 square nautical miles. This is near the limit of the useful precision 

 from the presently available wind data. A similar, though larger, grid-point 

 system is used in most numerical dynamical weather prediction work. 



The time scale chosen must be compatible with the space scale such that 

 the fastest wave components do not skip grid points during a particular time 

 step. Because of the general smoothness of both the wind and wave fields for 

 the longer faster components, the time constraint can be relaxed slightly without 

 serious error. The time interval was therefore chosen as 2 hours to be an in- 

 tegral divisor of the normal 6 -hourly meteorological charts. 



Since the fields of wave spectra adequately represent an integration of all 

 that has happened previously, the initial value problem given by a single initial 

 wave field and the wind field for the next time interval are adequate to compute 

 the change for the next time step. Thus, only the latest information is used for 

 successive time steps. 



Since wave energy propagates along great circle routes, it would be desir- 

 able for the lines connecting the grid points to be great circles. Then the energy 

 could travel directly from one grid point to another grid point. It would also be 

 desirable for the spacing of these points to be constant for the entire ocean and 

 for as many directions as possible. 



To have a conformal, equal-area, equidistant representation of the earth 

 with great circles shown as simple curves (say, circular arcs or straight lines) 

 we would be forced to work on a sphere. However, a usable network of nearly 

 equally- spaced grid points which are defined by great circles cannot be developed 



On plane projections, both geometrical and conventional (i.e., mathematical), 

 the straight-line great circle property is the rarest one of the desired proper- 

 ties. It occurs only in the gnomonic projection, a geometric projection of no 

 more than a hemisphere onto a tangent plane by means of diverging rays origi- 

 nating at the center of the sphere. On this projection, great circles are repre- 

 sented by straight lines. The primary disadvantage is that of great areal dis- 

 tortion. Many other projections exist which preserve either angles and small 

 shapes, or areas, or distances; but on all of them the problems of accounting for 

 the three properties not preserved is very complicated. On the other hand, the 

 gnomonic projection possesses radial symmetry and the various distortions can 

 be expressed as relatively simple functions of the distance between the point of 

 tangency of the projection and the point in question. The equations, describing 

 this projection and its superimposed grid, can be found in Baer and Adamo (25) 

 or developed relatively easily if one is familiar with spherical geometry. 



513 



