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The horizontal component of Eq. 7-13 yields geostrophic balance—Eq. 7-2 or Eq. 7-4 in component form—where now z is imagined to point in the direction of Q (see Fig. 7.8). The vertical component of Eq. 7-13 yields hydrostatic balance, Eq. 3-3. Taking the curl (Vx ) of Eq. 7-13, we find that if the fluid is barotropic [i.e., one in which p = p(p)], then6:

or (since Q -V is the gradient operator in the direction of Q, i.e., z)

du lz

6Using vector identities 2. and 6. of Appendix A.2.2, setting a —> Q and b —> u, remembering that V - u = 0 and VxV (scalar)= 0.

FIGURE 7.8. The Taylor-Proudman theorem, Eq. 7-14, states that slow, steady, frictionless flow of a barotropic, incompressible fluid is two-dimensional and does not vary in the direction of the rotation vector Q.

Equation 7-14 is known as the Taylor-Proudman theorem (or T-P for short). T-P says that under the stated conditions— slow, steady, frictionless flow of a barotropic fluid—the velocity u, both horizontal and vertical components, cannot vary in the direction of the rotation vector Q. In other words the flow is two-dimensional, as sketched in Fig. 7.8. Thus vertical columns of fluid remain vertical—they cannot be tilted over or stretched. We say that the fluid is made "stiff" in the direction of Q. The columns are called Taylor Columns after G.I. Taylor, who first demonstrated them experi-mentally.7

Rigidity, imparted to the fluid by rotation, results in the beautiful dye patterns seen in experiment GFD Lab 0. On the right of Fig. 7.7, the rotating fluid, brought into gentle motion by stirring, is constrained to move in two dimensions. Rich dye patterns emerge in planes perpendicular to Q but with strong vertical coherence between the levels; flow at one horizontal level moves in lockstep with the flow at another level. In contrast, a stirred nonrotating fluid mixes in three dimensions and has an entirely different character with no vertical coherence; see the left frame of Fig. 7.7.

Taylor columns can readily be observed in the laboratory in a more controlled setting, as we now go on to describe.

7.2.1. GFD Lab VII: Taylor columns

Suppose a homogeneous rotating fluid moves in a layer of variable depth, as sketched in Fig. 7.9. This can easily be arranged in the laboratory by placing an obstacle (such as a bump made of a hockey puck) in the bottom of a tank of water rotating on a turntable and observing the flow of water past the obstacle, as depicted in Fig. 7.9. The T-P theorem demands that vertical columns of fluid move along contours of constant fluid depth, because they cannot be stretched.

At levels below the top of the obstacle, the flow must of course go around it. But Eq. 7-15 says that the flow must be the same at all z; so, at all heights, the flow must be deflected as if the bump on the boundary extended all the way through the fluid! We can demonstrate this behavior in the laboratory, using the apparatus sketched in Fig. 7.10 and described in the legend, by inducing flow past a submerged object.

We see the flow (marked by paper dots floating on the free surface) being diverted around the obstacle in a vertically

Geoffrey Ingram Taylor (1886—1975). British scientist who made fundamental and long-lasting contributions to a wide range of scientific problems, especially theoretical and experimental investigationsof fluid dynamics. Theresult,Eq. 7-14, wasfirstdemonstratedbyJosephProudman in 1915 but is now called the Taylor-Proudman theorem. Taylor's name got attached because he demonstrated the theorem experimentally (see GFD Lab VII). In a paper published in 1921, he reported slowly dragging a cylinder through a rotating flow. The solid object all but immobilized an entire column of fluid parallel to the rotation axis.

FIGURE 7.9. The T-P theorem demands that vertical columns of fluid move along contours of constant fluid depth because, from Eq. 7-14, they cannot be stretched in the direction of Q. Thus fluid columns act as if they were rigid columns and move along contours of constant fluid depth. Horizontal flow is thus deflected as if the obstable extended through the whole depth of the fluid.

FIGURE 7.10. We place a cylindrical tank of water on a table turning at about 5 rpm. An obstacle such as a hockey puck is placed on the base of the tank whose height is a small fraction of the fluid depth; the water is left until it comes into solid body rotation. We now make a very small reduction in Q (by 0.1 rpm or less). Until a new equilibrium is established (the "spin-down" process takes several minutes, depending on rotation rate and water depth), horizontal flow will be induced relative to the obstacle. Dots on the surface, used to visualize the flow (see Fig. 7.11), reveal that the flow moves around the obstacle as if the obstacle extended through the whole depth of the fluid.

FIGURE 7.10. We place a cylindrical tank of water on a table turning at about 5 rpm. An obstacle such as a hockey puck is placed on the base of the tank whose height is a small fraction of the fluid depth; the water is left until it comes into solid body rotation. We now make a very small reduction in Q (by 0.1 rpm or less). Until a new equilibrium is established (the "spin-down" process takes several minutes, depending on rotation rate and water depth), horizontal flow will be induced relative to the obstacle. Dots on the surface, used to visualize the flow (see Fig. 7.11), reveal that the flow moves around the obstacle as if the obstacle extended through the whole depth of the fluid.

coherent way (as shown in the photograph of Fig. 7.11) as if the obstacle extended all the way through the water, thus creating stagnant Taylor columns above the obstacle.