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3 Basics of Geophysical Fluid Dynamics
Fig. 3.14 Changes of location and velocity of a parcel on a turntable
3.12.4 The Centrifugal Force in a Rotating Fluid
Consider a circular tank fille with flui on a rotating turntable. Letting the tank
rotate at a constant rate for a long time, all flui will eventually rotate at the same
rate as the tank. In this steady-state situation, the flui surface attains a noneven
shape, as sketched in Fig. 3.15. The fina shape of the flui surface is determined by
a balance between the centrifugal force and a centripetal force, that, in our rotating
fluid is provided by a horizontal pressure-gradient force provided by a slanting flui
surface. This balance of forces reads:
− g
∂η
∂r
= −Ω
2
r
(3.35)
where r is the radial distance from the centre of the tank.
The analytical solution of the latter equation is:
η(r ) =
1
2
Ω
2
g
r
2
− η o ,
(3.36)
Fig. 3.15 Sketch of the steady-state force balance between centrifugal force (CF) and pressuregradient force (PGF) in a rotating f uid. The dashed line shows the equilibrium surface level for
the nonrotating case