## An Introduction to Hydrodynamics and Water Waves by Bernard LeMehaute

By Bernard LeMehaute

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The vector sum of (2wU s sin ¢) and (2wU E sin ¢) is (2wU sin ¢) along a horizontal perpendicular to the direction ofU and to the left ofU. Finally, the geostrophic inertial force is composed of three terms: 1. A horizontal component 2pwU sin * perpendicular to and to the left of U. In the southern hemisphere, ¢ is negative, and the horizontal component 2pwU sin is directed to the right of U. 2. A horizontal component 2pw W cos directed toward the west (due to the negative sign appearing in the previous expression), and generally negligible as the motions are generally so nearly horizontal that W /U is very small. *

In the sou~hern hemisphere, the same reasoning applies, and the deflection is always found to the left of the direction of motion (Fig. 4-5). 2 Let us now consider a fluid particle which rotates with the earth. Its velocity about the earth's axis creates a centrifugal force. Since the fluid particle moves at the same speed as the earth, this force just balances the component of gravity force which is perpendicular to the earth's axis. If now it moves eastward with respect to the earth along a parallel in the northern hemisphere, the particle experiences an additional centrifugal force.

Hence, the viscous stresses are dependent on the existence of deformation and not rotationality. 1 Two-Dimensional Motion Consider a fluid element ABCD at time t (Fig. 2-16). The velocity components u and v are functions of x and y such that du = (oufox) dx + (oufoy) dy and dv = (oufox) dx + (ovfoy) dy. At time t the space coordinates of A are x, y and of D are x + dx, y + dy. The coordinates of A and D at time t + dt are given in Equation 2-1. D'{x y + dx + (u + du) dt + dy + (v + dv) dt Rotation or x { D' y + dx + u dt + (~~ dx + ~; dy) dt + dy + v dt + (~~ dx + ~>Y) dt Adding and subtracting ~ovfox) dy dt to the x coordinate and ~oufoy) dx dt to the y coordinate leads to the form for the coordinates of D' expressed in Equation 2-2.