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or, in component form in the local Cartesian geometry of Fig. 6.19,

The geostrophic balance of forces described by Eq. 7-2 is illustrated in Fig. 7.1. The pressure gradient force is of course directed away from the high pressure system on the left and towards the low pressure system on the right. The balancing Coriolis forces must be as shown directed in the opposite sense and consequently the geostrophically balanced flow must be normal to the pressure gradient that is along the contours of constant pressure as Eq. 7-3 makes explicit. For the northern hemisphere cases f> 0) illustrated in Fig. 7.1, the sense of the flow is clockwise around a high pressure system, and anticlockwise around a low. (The sense is opposite in the southern hemisphere.) The rule is summarized in Buys-Ballot's (the 19th-century Dutch meteorologist) law:

If you stand with your back to the wind in the northern hemisphere, low pressure is on your left

("left" ^ "right" in the southern hemisphere).

We see from Eq. 7-3 that the geostrophic flow depends on the magnitude of the pressure gradient, and not just its direction.

hemisphere case, f > 0 .) The effect of Coriolis deflecting flow ''to the right'' (see Fig. 6.10) is balanced by the horizontal component of the pressure gradient force, -1/pVp, directed from high to low pressure.

Consider Fig. 7.2, on which the curved lines show two isobars of constant pressure, p and p + Sp, separated by the variable distance Ss. From Eq. 7-3, ii 1 i i 1 Sp

Since Sp is constant along the flow, | ug | a (Ss)-1; the flow is strongest where the isobars are closest together. The geostrophic flow does not cross the pressure contours, and so the latter act like the banks of a river, causing the flow to speed up where the river is narrow and to slow down where it is wide. These characteristics explain in large part why the meteorologist is traditionally preoccupied with pressure maps: the pressure field determines the winds.

Note that the vertical component of the geostrophic flow, as defined by Eq. 7-3, is zero. This cannot be deduced directly from Eq. 7-2, which involves the horizontal components of the flow. However, consider for a moment an incompressible fluid (in the laboratory or the ocean) for which we can neglect variations in p. Further, while f varies on the sphere, it is almost constant over scales of, for example, 1000 km or less.5 Then Eq. 7-4 gives

FIGURE 7.2. Schematic of two pressure contours (isobars) on a horizontal surface. The geostrophic flow, defined by Eq. 7-3, is directed along the isobars; its magnitude increases as the isobars become closer together.

dx dy

Thus the geostrophic flow is horizontally nondivergent. Comparison with the continuity Eq. 6-11 then tells us that dwg/dz = 0; if wg = 0 on, for example, a flat bottom boundary, then it follows that wg = 0 everywhere, and so the geostrophic flow is indeed horizontal.

In a compressible fluid, such as the atmosphere, density variations complicate matters. We therefore now consider the equations of geostrophic balance in pressure coordinates, in which case such complications do not arise.

7.1.1. The geostrophic wind in pressure coordinates

To apply the geostrophic equations to atmospheric observations and particularly to upper air analyses (see below), we need to express them in terms of height gradients on a pressure surface, rather than, as in Eq. 7-4, of pressure gradients at constant height.

Consider Fig. 7.3. The figure depicts a surface of constant height z0, and one of constant pressure p0, which intersect at A, where of course pressure is pA = p0 and z 5 constant 5 zq z 5 constant 5 zq

FIGURE 7.3. Schematic used in converting from pressure gradients on height surfaces to height gradients on pressure surfaces.

^Variations of f do matter, however, for motions of planetary scale, as will be seen, for example, in Section 10.2.1.

height is za = z0. At constant height, the gradient of pressure in the x-direction is dp\ = pc - po dx ) Sx '

where Sx is the (small) distance between C and A and subscript z means ''keep z constant.'' Now, the gradient of height along the constant pressure surface is dz\ = Zb - Zo axjp ~

Sx where subscript p means ''keep p constant.'' Since zC = z0, and pB = po, we can use the hydrostatic balance equation Eq. 3-3 to write pc - p0 = pc - Pb = _ dp = zb - z0 = zb - zc ~ dz = gP'

Therefore from Eq. 7-6, and invoking a similar result in the y-direction, it follows that

In pressure coordinates Eq. 7-3 thus becomes:

where Zp is the upward unit vector in pressure coordinates, and Vp denotes the gradient operator in pressure coordinates. In component form it is,

The wonderful simplification of Eq. 7-8 relative to Eq. 7-4 is that P does not explicitly appear and therefore, in evaluation from observations, we need not be concerned about its variation. Just like p contours on surfaces of constant z, z contours on surfaces of constant p are streamlines of the geostrophic flow. The geostrophic wind is nondivergent in pressure coordinates if f is taken as constant:

Equation 7-9 enables us to define a streamfunction:

which, as can be verified by substitution, satisfies Eq. 7-9 for any yg = yg(x,y,p, t). Comparing Eq. 7-10 with Eq. 7-8 we see that:

Thus height contours are streamlines of the geostrophic flow on pressure surfaces: the geostrophic flow streams along z contours, as can be seen in Fig. 7.4.

What does Eq. 7-8 imply about the magnitude of the wind? In Fig. 5.13 we saw that the 500-mbar pressure surface slopes down by a height Az = 800 m over a meridional distance L = 5000 km; then geostrophic balance implies a wind of strength u = g/f Az/L = ff x 5^ « 15ms-1.

Thus Coriolis forces acting on a zonal wind of speed ~15ms_1, are of sufficient magnitude to balance the poleward pressure gradient force associated with the pole-equator temperature gradient. This is just what is observed; see the strength of the midlevel flow shown in Fig. 5.20. Geostrophic balance thus "connects" Figs. 5.13 and 5.20.

Let us now look at some synoptic charts, such as those shown in Figs. 5.22 and 7.4, to see geostrophic balance in action.

7.1.2. Highs and lows; synoptic charts

Fig. 7.4 shows the height of the 500-mbar surface (contoured every 60 m) plotted with the observed wind vector (one full quiver represents a wind speed of 10 ms-1) at an instant in time: 12 GMT on June 21, 2003, to be exact, the same time as the hemispheric map shown in Fig. 5.22. Note how

Gmt Plot Vector Map

FIGURE 7.4. The 500-mbar wind and geopotential height field at 12 GMT on June 21, 2003. [Latitude and longitude (in degrees) are labelled by the numbers along the left and bottom edges of the plot.] The wind blows away from the quiver: one full quiver denotes a speed of 10 ms-1, one half-quiver a speed of 5 ms-1. The geopotential height is contoured every 60 m. Centers of high and low pressure are marked H and L. The position marked A is used as a check on geostrophic balance. The thick black line marks the position of the meridional section shown in Fig. 7.21 at 80° W extending from 20° N to 70° N. This section is also marked on Figs. 7.5, 7.20, and 7.25.

FIGURE 7.4. The 500-mbar wind and geopotential height field at 12 GMT on June 21, 2003. [Latitude and longitude (in degrees) are labelled by the numbers along the left and bottom edges of the plot.] The wind blows away from the quiver: one full quiver denotes a speed of 10 ms-1, one half-quiver a speed of 5 ms-1. The geopotential height is contoured every 60 m. Centers of high and low pressure are marked H and L. The position marked A is used as a check on geostrophic balance. The thick black line marks the position of the meridional section shown in Fig. 7.21 at 80° W extending from 20° N to 70° N. This section is also marked on Figs. 7.5, 7.20, and 7.25.

the wind blows along the height contours and is stronger the closer the contours are together. At this level, away from frictional effects at the ground, the wind is close to geostrophic.

Consider for example the point marked by the left ''foot'' of the ''A'' shown in Fig. 7.4, at 43° N, 133° W. The wind is blowing along the height contours to the SSE at a speed of 25 ms-1. We estimate that the 500 mbar height surface slopes down at a rate of 60 m in 250 km here (noting that

1° of latitude is equivalent to a distance of 111 km and that the contour interval is 60 m). The geostrophic relation, Eq. 7-7, then implies a wind of speed g/f Az/L = 981 x „ = 24 ms-1, close to that

observed. Indeed the wind at upper levels in the atmosphere is very close to geostrophic balance.

In Fig. 7.5 we plot the Ro (calculated as |u-Vu|/fu|) for the synoptic pattern shown in Fig. 7.4. It is about 0.1 over most of the region and so the flow is

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Fig. 7.4. The contour interval is 0.1. Note that Ro ~ 0.1 over most of the region but can approach 1 in strong cyclones, such as the low centered over 80° W, 40° N.

to a good approximation in geostrophic balance. However, R0 can approach unity in intense low pressure systems where the flow is strong and the flow curvature large, such as in the low centered over 80° W, 40° N. Here the Coriolis and advection terms are comparable to one another, and there is a three-way balance between Coriolis, iner-tial, and pressure gradient forces. Such a balance is known as gradient wind balance (see Section 7.1.3).

7.1.3. Balanced flow in the radial-inflow experiment

At this point it is useful to return to the radial inflow experiment, GFD Lab III, described in Section 6.6.1, and compute the Rossby number assuming that axial angular momentum of fluid parcels is conserved as they spiral into the drain hole (see Fig. 6.6). The Rossby number implied by Eq. 6-23 is given by:

where r1 is the outer radius of the tank. It is plotted as a function of r/r1 in Fig. 7.6 (right).

The observed Ro, based on tracking particles floating on the surface of the fluid (see Fig. 6.6) together with the theoretical prediction, Eq. 7-12, are plotted in Fig. 7.6. We see broad agreement, but the observations depart from the theoretical curve at small r and high Ro, perhaps because of the difficulty of tracking the particles in the high speed core of the vortex (note the blurring of the particles at small radius evident in Fig. 6.6).

According to Eq. 7-12 and Fig. 7.6, R0 = 0 at r = r1, R0 = 1 at a radius r1/V3 = 0.58r1, and rapidly increases as r decreases further. Thus the azimuthal flow is geostrophically balanced in the outer regions (small R0) with the radial pressure gradient force balancing the Coriolis force in Eq. 6-21. In the inner regions (high R0) the v^/r term in Eq. 6-21 balances the radial pressure gradient; this is known as cycl0straphic balance. In the middle region (where R0 ~ 1) all three terms in Eq. 6-21 play a role; this is known as gradient wind balance, of which geostrophic and cyclostrophic balance are limiting cases. As mentioned earlier, gradient wind balance can be seen in the synoptic chart shown in Fig. 7.4, in the low pressure regions where R0 ~ 1 (Fig. 7.5).

FIGURE 7.6. Left: The R0 number plotted as a function of nondimensional radius (r/r1) computed by tracking particles in three radial inflow experiments (each at a different rotation rate, quoted here in revolutions per minute [rpm]). Right: Theoretical prediction based on Eq. 7-12.

FIGURE 7.7. Dye distributions from GFD Lab 0: on the left we see a pattern from dyes (colored red and green) stirred into a nonrotating fluid in which the turbulence is three-dimensional; on the right we see dye patterns obtained in a rotating fluid in which the turbulence occurs in planes perpendicular to the rotation axis and is thus two-dimensional.

FIGURE 7.7. Dye distributions from GFD Lab 0: on the left we see a pattern from dyes (colored red and green) stirred into a nonrotating fluid in which the turbulence is three-dimensional; on the right we see dye patterns obtained in a rotating fluid in which the turbulence occurs in planes perpendicular to the rotation axis and is thus two-dimensional.

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