308 The Geophysical Structure of the Sea 



so that (IX. 12) becomes 



N = Da- D„ 



i.e. the difference in the dynamic depth of an isobaric surface at two oceanographic 

 stations gives the number of solenoids in the cross-section between the two stations 

 from the surface of the sea to the depth of the isobaric surface. 



If the sets of surfaces of two properties of sea water L^ and Lo coincide, there must be 

 a functional relationship F(Li, L,) = between them ; this represents only a purely 

 geometrical connection between the two scalar quantities Li and Lg and reveals nothing 

 of the physical relationship which probably exists between them. Examples of such 

 quantities are p, a, p and also the potential or the dynamic depth D. All these sets 

 of surfaces coincide only when there is internal equilibrium in the water mass (see 

 p. 302). The field of the second scalar quantity Lg was denoted by Bjerknes and co- 

 workers (1933) in the case where L^ = p == const, (isobaric surfaces) barotropic, 

 that means adjusted to the pressure field ; in the case where Li= t = const, (isothermal 

 surfaces) thermotropic, where it is adjusted to the temperature field and in the case 

 where L^ = S = const, (isohaline surfaces) halotropic, where it is adjusted to the 

 salinity. There will then be a set of relationships 



F(p, L,) = 0, F{t, L,) = 0, F(S, L,) = 0. 



In general, these relationships are denoted "conditions of homotropy". If no such 

 functional relationships exist, then the field of the scalar quantity Lg is barocHnic, 

 thermoclinic or haloclinic, i.e. it is "inclined" relative to the pressure field, the tempera- 

 ture field, or the salinity field; only in these cases do solenoids exist. 



If jc is a definite point in a field and x + dx is a neighbouring point, and if the geo- 

 metrical changes on transition from one point to the other are AN^^ Ap, At, AS then 

 the quantities 



J^~~A^ ~ " dFfm, ' ^^~ At " dFjdN^ ' ^^~ AS ~ dFjdN^ 



are termed homotropic coefficients of N2, and specifically each as the barotropic, 

 thermotropic and halotropic coefficients. These coefficients are entirely geometric in 

 character, since they depend on the differences of the factors at two diff'erent spatial 

 points. The behaviour of an individual small particle — e.g. on changes in pressure — 

 is on the other hand a purely physical property of the field; for example, the density is 

 given by the piezotropic coefficient of density 



dp 1 da 



'^"^dp^^^dp' 



the changes in p and a on displacement of a small particle depending on the change in 

 pressure dp. The diff'erence between the two coefficients is clearly shown by the follow- 

 ing example: Fp^ = (ApjAp) = indicates homogeneity of the mass field, while 

 yp = indicates the incompressibility of the medium. Bjerknes termed the special 

 case Tp" = yp "autobarotropy", i.e., after exchange of any two small particles the mass 

 field remains unaltered. 



