Chapter XI 



The Ocean at Rest (Statics of the Ocean) 



1. The Basic Static Equation and the Conditions for Static Equilibrium 



If a water mass in the sea is at rest relative to the Earth, the only external force 

 acting on it will be the conservative force of gravity. In the stationary state its effect 

 is balanced exactly by the resistance of the masses underneath. The elastic force of the 

 substratum is thus opposed by the weight of the water masses and any vertical dis- 

 placement is extinguished, when both effects are equal (i.e. when the weight of the 

 water masses above any surface is equal to the pressure exerted upwards by the water 

 masses underneath this surface). The condition for internal equiUbrium thus requires 

 that no resultant of the gravity and the pressure force should act in the direction of 

 the gravitational level surfaces. A horizontal cross-section through a water column 

 enclosed between two vertical walls will carry a greater weight of water the deeper it is 

 placed. At a depth z it shall be p^^. At a small distance dz below this there will be a 

 pressure 



dp 

 A = A + 7- ^-. 



The increase in pressure p.-^^ — p^ will be identical with the weight of the water masses 

 per unit area between the two surfaces : 



P2— Pi= pg dz. 

 From these two equations the "basic static equation" is obtained 



1 dp 



Since the negative derivative of the potential <P with respect to z is equal to the gravi- 

 tational acceleration, equation (XI. 1) can also be written in the form 



d0 = - adp. (XI.2) 



It contains the simplest statement about the three-dimensional fields of potential, 

 mass and pressure in hydrostatic equilibrium. The gradient of the potential is per- 

 pendicular to the level surfaces and the pressure gradient is vertical to the iso- 

 baric surfaces. Since they have opposite directions the equi-potential surfaces and the 

 isobaric surfaces must coincide if there is hydrostatic equilibrium. The equation 

 (XI. 2) states further that at any point the ratio of the thickness of a thin potential 

 sheet d0 to the thickness of a thin isobaric sheet dp will be constant and taken with a 

 negative sign must be numerically identical with the mean specific volume in this layer. 

 From this it follows that in the case of static equilibrium the isosteric surfaces must 



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