= L3/T2, the dimension of the first term on the left-hand side of Eqn. (2.19) is the term on the right-hand side of Eqn. (2.19) is [kA] = L4/T, the dimension of the second and third term on the left-hand side is also L4/T. Therefore, it is clear that the dimension of the first term on the left-hand side is wrong. In order to have a dimensionally correct equation, there should be an additional factor in this term, which has a dimension of TL. A careful re-examination of the derivation of this equation reveals that the first term in Eqn. (2.19) should have an additional factor of j in the denominator:
A more interesting example of dimensional analysis is to introduce the nondimensional variable as the following. Equation (2.20) can be rewritten as
where dI, dw, and dk all have a dimension of length. This equation can be solved by using one of these length scales and introducing a nondimensional depth. The details of such an application are discussed in Section 5.5.1.
The atomic bomb explosion Dimensional analysis can provide very useful information, especially when the system has a simple geometry. One of the classic examples used to demonstrate the power of dimensional analysis is the case of the strong shock waves associated with a point source explosion.
There are three key physical quantities; the amount of energy released, [E] = ML2/T2, the density of air, [p] = M /L3, and the time [T] = T. This system has a very simple geometry. Since there is no intrinsic length scale imposed, the distance of the shock wave propagation must depend on a power combination of the three basic quantities, i.e., energy, density, and time d = Eapj TY. Since d has a dimension of L, the only possible combination is a = 1/5, j = -1/5, and y = 2/5; thus we have the equation for the propagation of the shock waves d = (E/p)1/512/5 (2.22)
The most beautiful example of the power of dimensional analysis was demonstrated by G.I. Taylor. After the first test of the atomic bomb, he was invited to watch a movie recorded during the experiment. Although all the information was classified and thus not available, using the scaling analysis outlined above, G.I. Taylor was able to infer the amount of energy released from the atomic explosion by substituting the time evolution of the shock front he saw in the movie. The reader who is interested in the details of dimensional analysis and the exciting story of G.I. Taylor is referred to the classic book on dimensional analysis by Sedov (1959).
2.2.5 Important nondimensional numbers in dynamical oceanography v
where v is the viscosity, ^ is the angular velocity of the Earth, and L is the horizontal length scale. The Ekman number is the ratio of frictional force and the Coriolis force. This nondimensional number is introduced in the study of the frictional boundary layer in a rotating frame. Within this boundary layer the frictional force is balanced by the Coriolis force, but outside this boundary layer the frictional force is negligible in comparison with the Coriolis force.
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