## The Thermal Wind Equation

We saw in Section 5.2 that isobaric surfaces slope down from equator to pole. Moreover, these slopes increase with height, as can be seen, for example, in Fig. 5.13 and the schematic diagram, Fig. 5.14. Thus according to the geostrophic relation, Eq. 7-8, the geostrophic flow will increase with height, as indeed is observed in Fig. 5.20. According to T-P, however, dug/dz = 0. What's going on?

The Taylor-Proudman theorem pertains to a slow, steady, frictionless, barotropic fluid, in which p = p(p). But in the atmosphere and ocean, density does vary on pressure surfaces, and so T-P does not strictly apply and must be modified to allow for density variations.

Let us again consider the water in our rotating tank, but now suppose that the density of the water varies thus:

Pref where prff is a constant reference density, and a, called the density anomaly,

is the variation of the density about this reference.8

Now take d/dz of Eq. 7-4 (replacing p by pref where it appears in the denominator) we obtain, making use of the hydrostatic relation Eq. 3-3:

du notation (see Appendix

So if p varies in the horizontal then the geostrophic current will vary in the vertical. To express things in terms of temperature, and hence derive a connection (called the thermal wind equation) between the current and the thermal field, we can use our simplified equation of state for water, Eq. 4-4, which assumes that the density of water depends on temperature T in a linear fashion. Then Eq. 7-17 can be written:

where a is the thermal expansion coefficient. 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, Eq. 3-3 and Eq. 7-4, but it expresses these balances in a different way.

We see that there is an exactly analogous relationship between dug/dz and T as between ug and p (compare Eqs. 7-18 and 7-3). So if we have horizontal gradients of temperature then the geostrophic flow will vary with height. The westerly winds increase with height because, through the thermal wind relation, they are associated with the poleward decrease in temperature. We now go on to study the thermal wind in the laboratory, in which we represent the cold pole by placing an ice bucket in the center of a rotating tank of water.

7.3.1. GFD Lab VIII: The thermal wind relation

It is straightforward to obtain a steady, axially-symmetric circulation driven by

8Typically the density of the water in the rotating tank experiments, and indeed in the ocean too (see Section 9.1.3), varies by only a few % about its reference value. Thus a/pref is indeed very small.

FIGURE 7.12. We place a cylindrical tank on a turntable containing a can at its center, fill the tank with water to a depth of 10 cm or so, and rotate the turntable very slowly (1 rpm or less). After solid-body rotation is achieved, we fill the can with ice/water. The can of ice in the middle induces a radial temperature gradient. A thermal wind shear develops in balance with it, which can be visualized with dye, as sketched on the right. The experiment is left for 5 minutes or so for the circulation to develop. The radial temperature gradient is monitored with thermometers and the currents measured by tracking paper dots floating on the surface.

FIGURE 7.12. We place a cylindrical tank on a turntable containing a can at its center, fill the tank with water to a depth of 10 cm or so, and rotate the turntable very slowly (1 rpm or less). After solid-body rotation is achieved, we fill the can with ice/water. The can of ice in the middle induces a radial temperature gradient. A thermal wind shear develops in balance with it, which can be visualized with dye, as sketched on the right. The experiment is left for 5 minutes or so for the circulation to develop. The radial temperature gradient is monitored with thermometers and the currents measured by tracking paper dots floating on the surface.

radial temperature gradients in our laboratory tank, which provides an ideal opportunity to study the thermal wind relation.

The apparatus is sketched in Fig. 7.12 and can be seen in Fig. 7.13. The cylindrical tank, at the center of which is an ice bucket, is rotated very slowly anticlockwise. The cold sides of the can cool the water adjacent to it and induce a radial temperature gradient. Paper dots sprinkled over the surface move in the same sense as, but more swiftly than, the rotating table—we have generated westerly (to the east) currents! We inject some dye. The dye streaks do not remain vertical but tilt over in an azimuthal direction, carried along by currents that increase in strength with height and are directed in the same sense as the rotating table (see the photograph in Fig. 7.13 and the schematic in Fig. 7.14. The Taylor columns have been tilted over by the westerly currents.

For our incompressible fluid in cylindrical geometry (see Appendix A.2.3), the azimuthal component of the thermal wind relation, Eq. 7-18, is:

where vg is the azimuthal current (cf. Fig. 6.8) and f has been replaced by 2Q. Since T increases moving outward from the cold center (dT/dr > 0) then, for positive Q, dvg/dz> 0. Since vg is constrained by friction to be weak at the bottom of the tank, we therefore expect to see vg > 0at the top, with the strongest flow at the radius of maximum density gradient. Dye streaks visible in Fig. 7.13 clearly show the thermal wind shear, especially near the cold can, where the density gradient is strong.

We note the temperature difference, AT, between the inner and outer walls a distance L apart (of order 1°C per 10 cm), and the speed of the paper dots at the surface relative to the tank (typically 1 cm s-1). The

FIGURE 7.14. A schematic showing the physical content of the thermal wind equation written in the form Eq. 7-20: spin associated with the rotation vector 2Q [1] is tilted over by the vertical shear (du/dz) [2]. Circulation in the transverse plane develops [3] creating horizontal density gradients from the stable vertical gradients. Gravity acting on the sloping density surfaces balances the overturning torque associated with the tilted Taylor columns [4].

FIGURE 7.14. A schematic showing the physical content of the thermal wind equation written in the form Eq. 7-20: spin associated with the rotation vector 2Q [1] is tilted over by the vertical shear (du/dz) [2]. Circulation in the transverse plane develops [3] creating horizontal density gradients from the stable vertical gradients. Gravity acting on the sloping density surfaces balances the overturning torque associated with the tilted Taylor columns [4].

tank is turning at 1 rpm, and the depth of water is H ~ 10 cm. From the above thermal wind equation we estimate that ag AT -i ug ~--— H ~ 1cms 1, g 2Q L

This experiment is discussed further in Section 8.2.1 as a simple analogue of the tropical Hadley circulation of the atmosphere.

7.3.2. The thermal wind equation and the Taylor-Proudman theorem

The connection between the T-P theorem and the thermal wind equation can be better understood by noting that Eqs. 7-17 and 7-18 are simplified forms of a more general statement of the thermal wind equation, which we now derive.

Taking Vx (Eq. 7-13), but now relaxing the assumption of a barotropic fluid, we obtain [noting that the term of the left of Eq. 7-13 transforms as in the derivation of Eq. 7-14, and that Vx (1/pVp) = 1/p2Vp xVp = 1/p2Vp x Vp]:

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