454 Water Bodies and Stationary Current Conditions at Boundary Surfaces 



Hadamard classification (1903)t it is thus a discontinuity surface of zero order for the 

 density and of the first order for the pressure. The horizontal movements in each 

 water body must thus be parallel to the boundary surface since otherwise the surface 

 could not remain at rest. 



There are kinematic and dynamic boundary conditions that must be satisfied at 

 the discontinuity surface (see p. 324). The kinematic condition (equation X.29) re- 

 quires that 



(ill — iio) cos (nx) + (vi — V2) cos (ny) + (m'i — u'a) cos (nz) = 0, (XIV. 1) 



where /; is the direction of the normal to the boundary surface ; u^, v^, w^ are the velocity 

 components of the lighter and U2, v^, Wo are those for the heavier water type. The 

 dynamic condition (equation X.29) requires that the pressure should be the same on 

 both sides of the boundary surface (pressure equal counter pressure) 



P^-P2== 0. (XIV.2) 



If w, V, vv are the total acceleration components and X, Y, Z are the components of the 

 forces, the equation of motion for the lighter water body 1 can then be written in the 

 form: 



dPi = Pi [(A^i - it,) ^x + ( n - i\) dy + (Zi - vi-i)] dz. (XIV.3) 



An analogous equation will apply for the heavier water body 2. The equations 



dpi = and dp2 — 



will then give the equations for the isobaric surfaces according to the motion in each 

 water body while the dynamic condition (XIV.2) will give the equation of the boundary 

 surface 



[(Pi ^1 - P2 ^2) — (Pi wi — p2 W2)] dx + [(pi Ti — P2 Y^ — (pi Vi - P2V2)] dy + 



[(Pi Zi - P2 Z2) - (pi vvi - P2 vva)] dz = 0. (XIV.5) 



In the most general form these are the equations for the slope of the isobaric surfaces 

 in each of the water bodies and for the inclination of the boundary surface. 



If the water bodies, each in itself, are both homogeneous (pi and pn = const.), the 

 motion is non-accelerated {ii — v — w = Q) and is directed straight along the y-axis 

 (ui = 112 — and \\\ = H'2 = 0), then there will be a static equilibrium in each water 

 body and 



^i=/''i, -^1 = ^ and X2=fv2, Zg = g. 



Further, if the slope of the isobaric surfaces in the (.vz)-plane is denoted by 

 dz\dx == tan ^ and that of the boundary surface by dz\dy = tan y, then the above 

 equations will give 



f f 



tan ^^= --^vy; tan /Sg = - - V2, (XIV. 6) 



t According to the classification of such surfaces introduced by Hadamard (1903), a discontinuity 

 surface at which the velocity and the density (temperature and salinity) change abruptly by a finite 

 amount from one to the other side, is termed a discontinuity surface of zero order. It is defined to be of 

 Ihe first order when the characteristic properties of the water bodies at the surface change continuously 

 but their derivatives normal to the surface are subject to abrupt changes. 



