## P P

which is a more general statement of the ''thermal wind'' relation. In the case of constant P, or more precisely in a barotropic fluid where p = p(p) and so Vp is parallel to Vp, Eq. 7-19 reduces to 7-14. But now we are dealing with a baroclinic fluid in which density depends on temperature (see Eq. 4-4) and so p surfaces and p surfaces are no longer coincident. Thus the term on the right of Eq. 7-19, known as the baroclinic term, does not vanish. It can be simplified by noting that to a very good approximation, the fluid is in hydrostatic balance: 1/pVp + gz = 0, allowing it to be written:

When written in component form, Eq. 7-20 becomes Eq. 7-16 if 2Q —► fz and p —►

The physical interpretation of the right hand side of Eqs. 7-16 and 7-20 can now be better appreciated. It is the action of gravity on horizontal density gradients trying to return surfaces of constant density to the horizontal, the natural tendency of a fluid under gravity to find its own level. But on the large scale this tendency is counterbalanced by the rigidity of the Taylor columns, represented by the term on the left of Eq. 7-20. How this works is sketched in Fig. 7.14. The spin associated with the rotation vector 2Q [1] is tilted over by the vertical shear (du/dz) of the current as time progresses [2]. Circulation in the transverse plane develops [3] and converts vertical stratification in to horizontal density gradients. If the environment is stably stratified, then the action of gravity acting on the sloping density surfaces is in the correct sense to balance the overturning torque associated with the tilted Taylor columns [4]. This is the torque balance at the heart of the thermal wind relation.

We can now appreciate how it is that gravity fails to return inclined temperature surfaces, such as those shown in Fig. 5.7, to the horizontal. It is prevented from doing so by the Earth's rotation.

7.3.3. GFD Lab IX: cylinder "collapse" under gravity and rotation

A vivid illustration of the role that rotation plays in counteracting the action of gravity on sloping density surfaces can be carried out by creating a density front in a rotating fluid in the laboratory, as shown in Fig. 7.15 and described in the legend. An initially vertical column of dense salty water is allowed to slump under gravity but is ''held up'' by rotation, forming a cone whose sides have a distinct slope. The photographs in Fig. 7.16 show the development of a cone. The cone acquires a definite sense of rotation, swirling in the same sense of rotation as the table. We measure typical speeds through the use of paper dots, measure the density of the dyed water and the slope of the side of the cone (the front), and interpret them in terms of the following theory.

### Theory following Margules

A simple and instructive model of a front can be constructed as follows. Suppose that the density is p1 on one side of the front and changes discontinuously to p2 on the other, with p1 > p2 as sketched in Fig. 7.17. Let y be a horizontal axis and y be the angle that the surface of discontinuity makes with the horizontal. Since the pressure must be the same on both sides of the front then the pressure change computed along paths [1]

FIGURE 7.15. We place a large tank on our rotating table, fill it with water to a depth of 10cm or so, and place in the center of it a hollow metal cylinder of radius r1 = 6 cm, which protrudes slightly above the surface. The table is set into rapid rotation at 10 rpm and allowed to settle down for 10 minutes or so. While the table is rotating, the water within the cylinder is carefully and slowly replaced by dyed, salty (and hence dense) water delivered from a large syringe. When the hollow cylinder is full of colored saline water, it is rapidly removed to cause the least disturbance possible—practice is necessary! The subsequent evolution of the dense column is charted in Fig. 7.16. The final state is sketched on the right: the cylinder has collapsed into a cone whose surface is displaced a distance Sr relative to that of the original upright cylinder.

FIGURE 7.15. We place a large tank on our rotating table, fill it with water to a depth of 10cm or so, and place in the center of it a hollow metal cylinder of radius r1 = 6 cm, which protrudes slightly above the surface. The table is set into rapid rotation at 10 rpm and allowed to settle down for 10 minutes or so. While the table is rotating, the water within the cylinder is carefully and slowly replaced by dyed, salty (and hence dense) water delivered from a large syringe. When the hollow cylinder is full of colored saline water, it is rapidly removed to cause the least disturbance possible—practice is necessary! The subsequent evolution of the dense column is charted in Fig. 7.16. The final state is sketched on the right: the cylinder has collapsed into a cone whose surface is displaced a distance Sr relative to that of the original upright cylinder.

FIGURE 7.16. Left: Series of pictures charting the creation of a dome of salty (and hence dense) dyed fluid collapsing under gravity and rotation. The fluid depth is 10 cm. The white arrows indicate the sense of rotation of the dome. At the top of the figure we show a view through the side of the tank facilitated by a sloping mirror. Right: A schematic diagram of the dome showing its sense of circulation.

FIGURE 7.16. Left: Series of pictures charting the creation of a dome of salty (and hence dense) dyed fluid collapsing under gravity and rotation. The fluid depth is 10 cm. The white arrows indicate the sense of rotation of the dome. At the top of the figure we show a view through the side of the tank facilitated by a sloping mirror. Right: A schematic diagram of the dome showing its sense of circulation.

and [2] in Fig. 7.17 must be the same, since they begin and end at common points in the fluid:

for small 5y, Sz. Hence, using hydrostatic balance to express dp/dz in terms of pg on both sides of the front, we find:

dy g p1 - p2 Using the geostrophic approximation to the current, Eq. 7-4, to relate the pressure

FIGURE 7.17. Geometry of the front separating fluid of differing densities used in the derivation of the Margules relation, Eq. 7-21.

gradient terms to flow speeds, we arrive at the following form of the thermal wind

FIGURE 7.18. The dome of cold air over the north pole shown in the instantaneous slice across the pole on the left (shaded green) is associated with strong upper-level winds, marked ® and © and contoured in red. On the right we show a schematic diagram of the column of salty water studied in GFD Lab IX (cf. Figs. 7.15 and 7.16). The column is prevented from slumping all the way to the bottom by the rotation of the tank. Differences in Coriolis forces acting on the spinning column provide a "torque" which balances that of gravity acting on salty fluid trying to pull it down.

FIGURE 7.18. The dome of cold air over the north pole shown in the instantaneous slice across the pole on the left (shaded green) is associated with strong upper-level winds, marked ® and © and contoured in red. On the right we show a schematic diagram of the column of salty water studied in GFD Lab IX (cf. Figs. 7.15 and 7.16). The column is prevented from slumping all the way to the bottom by the rotation of the tank. Differences in Coriolis forces acting on the spinning column provide a "torque" which balances that of gravity acting on salty fluid trying to pull it down.

equation (which should be compared to Eq. 7-16):

where u is the component of the current parallel to the front, and g = gAp/p1 is the ''reduced gravity,'' with Ap = p1 - p2 (cf. parcel ''buoyancy,'' defined in Eq. 4-3 of Section 4.2). Equation 7-21 is known as Margules relation, a formula derived in 1903 by the Austrian meteorologist Max Margules to explain the slope of boundaries in atmospheric fronts. Here it relates the swirl speed of the cone to g', y, and Q.

In the experiment we typically observe a slump angle y of perhaps 30° for Q ~ 1 s-1 (corresponding to a rotation rate of 10 rpm) and a g ~ 0.2 m s-2 (corresponding to a Ap/p of 2%). Equation 7-21 then predicts a swirl speed of about 6cms-1, broadly in accord with what is observed in the high-speed core of the swirling cone.

Finally, to make the connection of the experiment with the atmosphere more explicit, we show in Fig. 7.18 the dome of cold air over the north pole and the strong upper-level wind associated with it. The horizontal temperature gradients and vertical wind shear are in thermal wind balance on the planetary scale.

### 7.3.4. Mutual adjustment of velocity and pressure

The cylinder collapse experiment encourages us to wonder about the adjustment between the velocity field and the pressure field. Initially (left frame of Fig. 7.15) the cylinder is not in geostrophic balance. The ''end state,'' sketched in the right frame and being approached in Fig. 7.16 (bottom) and Fig. 7.18, is in ''balance'' and well described by the Margules formula, Eq. 7-21. How far does the cylinder have to slump sideways before the velocity field and the pressure field come in to this balanced state? This problem was of great interest to Rossby and is known as the Rossby adjustment problem. The detailed answer is in general rather complicated, but we can arrive at a qualitative estimate rather directly, as follows.

Let us suppose that as the column slumps it conserves angular momentum so that Qr2 + ur = constant, where r is the distance from the center of the cone and u is the velocity at its edge. If ri is the initial radius of a stationary ring of salty fluid (on the left of Fig. 7.15) then it will have an azimuthal speed given by:

2QSr,

if it changes its radius by an amount Sr (assumed small) as marked on the right of Fig. 7.15. In the upper part of the water column, Sr < 0 and the ring will acquire a cyclonic spin; below, Sr > 0 and the ring will spin anticyclonically.9 This slumping will proceed until the resulting vertical shear is enough to satisfy Eq. 7-21. Assuming that tan y ~ H/ \8r\, where H is the depth of the water column, combining Eqs. 7-22 and 7-21 we see that this will occur at a value of 8r - Lp = vgH/2Q. By noting that10 g'H x N2H2, Lp can be expressed in terms of the buoyancy frequency N of a continuously stratified fluid thus:

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