2. THE EQUATIONS OF MOTION OF SEA-WATER 



Carl Eckart 



1. Introduction 



Prior to the twentieth century, hydrodynamics concerned itself almost 

 exclusively with the motion of idealized fluids. The most common is the in- 

 compressible, homogeneous fluid. The autobarotropic fluid (also called the 

 polytropic fluid by some), whose density is a function only of its pressure, also 

 received attention, notably in the theory of acoustics. More recently it has been 

 studied in connection with steUar structure. These two classic lines of develop- 

 ment were brought to a definitive stage by Rayleigh (1894) and Lamb (1932), 



In the twentieth century, it became clear that the hydrodynamics of real 

 fluids cannot be developed independently of thermodynamics. The first to 

 attempt a fusion of these two disciphnes was G. Jaumann (1911 ; 1918). In a 

 series of elaborate studies, he and E. Lohr (1916; 1924; 1924a) made very 

 important contributions to the field. Two other independent attempts were 

 made by V. Bjerknes and his collaborators (1933) and by C. Eckart (1940). 

 Since 1947, an extensive literature on the subject has appeared, consisting 

 largely of applications to acoustics and physical chemistry. The most recent 

 review of this work, by Meixner and Reik (1959), contains a complete biblio- 

 graphy. 



A review of earlier work in geophysics was published in 1956 (Eckart and 

 Ferris, 1956); this also contained a much simplified account of the results of 

 the recent research in thermodynamics. The air and sea-water were treated as 

 chemically pure substances, and the irreversible phenomena of viscosity and 

 heat conduction were not treated. 



There are other more extensive and systematic accounts of the hydro- 

 dynamics of chemically pure fluids, but these are usually written from the 

 aerodynamical rather than the geophysical point of view. One such account is 

 that of Oswatitsch (1956). These treatments devote special attention to the 

 theory of shock waves and similar discontinuities. The present account will not 

 consider such discontinuities. Since the velocities of sea-water under natural 

 conditions are sub -sonic, shock waves in the common sense of the term are not 

 to be expected. Other forms of discontinuous flow, especially those appro- 

 priately described as highly oblique, very weak shocks, may be of importance, 

 especially in turbulent motion. 



In the following pages, sea-water will be treated as a solution of a single 

 compound ("salinity") in water, and the equations of motion will include 

 viscosity, heat conduction and diffusion. The equations are a special case of 

 those derived by Jaumann (1911 ; 1916) and Eckart (1940) and are the same 

 (except for notation) as those given in Section 11 y of the article by Meixner 

 and Reik (1959). The derivation of the equations will only be outlined here; 

 more complete derivations will be found in the references just cited. The present 

 objectives will be to introduce certain approximations and to transform the 

 resulting equations into forms that exhibit their essential mathematical structure. 



[MS received December, 1959] 31 



