Lecture 20: Taylor Proudman Theorem October 23, 2003 1 The Taylor-Proudman Theorem We have seen (last time) that if the flow is sufficiently slow and steady (small Rossby number) and frictionless (F small), then our governing momentum equation reduces to: 1 p + φ +2Ω u =0 (1) ρ Let us write it out in component form, setting Ω = Ωbz. The vertical component of Eq.(1) yields hydrostatic balance: p + ρg =0 The horizontal component yields geostrophic balance, which, in component form is: µ (u g,v g )= 1 p 2Ωρ y, 1 p (2) 2Ωρ x Now, lets suppose that ρ = const and differentiate Eq.(2) wrt z. Weget: µ ug, v Ã! g = 1 p 2Ωρ y, 1 p =(0, 0) 2Ωρ x because ρ = const. Thus, both u g and v g don t vary in the vertical! In other words, the vertical shear of the geostrophic wind vanishes if ρ is constant. We can arrive at the same conclusion in a more elegant way as follows. 1
For an incompressible fluid of constant density, (or, strictly speaking, a barotropic fluid, that is, one in which ρ = ρ(p)) then taking the curl of Eq.(1) leads to a result known as the Taylor Proudman theorem (T-P): 1 or ( Ω. ) u =0:T-P theorem (3) u =0. (4) if Ω = Ωbz. 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 Ω. The flow is 2 dimensional, as sketched in Fig.3. Thus, vertical columns of fluid remain vertical they cannot be tilted over or stretched. The columns are called Taylor Columns after G.I. Taylor who first demonstrated them experimentally. They can readily be observed in the laboratory, as we shall now see. 1.1 GFD Lab VI: Taylor columns Suppose a homogeneous rotating fluid moves in a layer of variable depth, as sketched in Fig.2. This can easily be arranged in the laboratory by placing an obstacle a bump in the bottom of a tank of water rotating on a turntable and observing the flow of water past the obstacle see Fig.2. The T-P theorem demands that vertical columns of fluid move along contours of constant fluid depth because they cannot be stretched. Near the boundary, the flow must of course go around the bump. But Eq.(4) then 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.3 and described in the legend by inducing flow past a submerged object. We see the flow being diverted around the obstacles in a vertically coherent way (as shown in picture Fig.4) as if the obstacles extended all the 1 We make use of the following vector identity: (a b) =a ( b) b ( a)+(b ) a (a ) a, set a Ω and b u, note that u = 0 and (scalar) =0. 2
Figure 1: The Taylor-Proudman theorem states that slow, steady, frictionless flow of a barotropic, incompressible fluid is 2-dimensional and does not vary in the direction of the rotation vector Ω. way through the water, thus creating stagnant Taylor columns above the obstacles. Rigidity, imparted to the fluid by rotation, is at the heart of the glorious dye patterns seen in our GFD Lab I experiment. On the right of Fig.5, the rotating fluid, brought in to motion by stirring, is constrained to move in two-dimensions. Rich dye patterns emerge in the horizontal plane but not in the vertical: flow at one horizontal level moves in lockstep with the flow at another level. In contrast a stirred non-rotating fluid mixes in three dimensions and has an entirely different character see lhs of Fig.5 2 Obervations of wind shear in the atmosphere Let s remind ourselves of the distribution of zonal winds in the atmosphere, shown in Fig.6: Clearly, µ ug, v g 6= (0, 0)!! This is because of the ever increasing tilt of the pressure surfaces with height see Fig.7 which implies, through the geostrophic relation, a 3
Figure 2: The T-P theorem demands that vertical columns of fluid move along contours of constant fluid depth because they cannot be stretched. 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. geostrophic flow which increases with height. Let us again consider an incompressible fluid (we ll return to the compressible atmosphere next time) but now allow ρ to vary thus: ρ = ρ ref + σ and σ << 1 ρ ref where ρ ref is a constant reference density, and σ called the density anomaly is the variation of the density about this reference. 2 Now take / of Eq.(??) (replacingρ by ρ ref where it appears in the denominator) we obtain, making use of the hydrostatic relation or, in vector notation u g = g σ fρ ref y ; v g = g σ fρ ref x (5) u g = g bz σ. (6) fρ ref So if ρ varies in the horizontal then the geostrophic current will vary in the vertical. To express things in terms of temperature, and hence derive a 2 Typically the density of the water in the rotating tank experiments varies about its reference value by only a few %. Thus σ ρ is indeed very small. o 4
Figure 3: We place a cylindrical tank of water on a table turning at 5 rpm. An obstacle is placed on the base of the tank which is a small fraction of the fluid depth and the water left until it comes in to solid body rotation. We now make a very small reduction in Ω (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.4 reveal that the flow moves around the obstacle as if the obstacle extended through the whole depth of the fluid. Figure 4: Dots on the surface moving around, but not over, a submerged obstacle, experimental confirmation of the schematic drawn in Fig.2. 5
Figure 5: Dye distributions from GFD Lab 1: on the left we see a pattern from dyes (colored red and green) stirred in to a non-rotating fluid; on the right we see dye patterns obtained in a rotating fluid. connection between the current and the thermal field (called thermal wind) we can use our simplified equation of state for water: ρ = ρ ref (1 αt) =ρ ref + σ (and so σ = αt ) which assumes that the density of water depends on ρ ref temperature T in a linear fashion. Then Eq.(6) can be written: u g = αg bz T (7) f This is a simple form of the THERMAL WIND relation connecting the vertical shear of the geostrophic current to horizontal temperature gradients. It tells us nothing more than the hydrostatic and geostrophic balances but it expresses these balances in a different way. So, if there are horizontal temperature gradients the geostrophic flow will vary in the vertical, just as we observe in the atmosphere. Next time we ll do a lab experiment to illustrate the thermal wind equation next. 6
Figure 6: Meridional cross-section of zonal wind (ms 1 ) under annual mean conditions (top), DJF (December, January, February ) (middle) and JJA (June, July, August) (bottom) conditions. 7
Figure 7: Zonal-mean geopotential height (m) for annual mean conditions. Values are departures from a horizontally uniform reference profile. 8