Applications to the Earths atmosphere

Since the atmosphere is essentially a thin layer of fluid on a sphere, a convenient set of axes at any point on the Earth's surface has x directed toward the east, y to the north, and z vertically upward (i.e., along the effective gravity ge, which combines the effects of the gravitational force and centrifugal force). If i, j, and k are unit vectors directed along these rotating axes, the relative velocity of the mean flow may be expressed as u = ui + vj + wk. (2.68)

Letting ge = - gk, one can rewrite the components of the momentum equation in the form:

- R tan y + R =

P

dp d x

P

(2.69)

+ — tan y + — =

P

dp dy

- 2Qu sin y +— Fy, P

(2.70)

Dw u2 + v2 Dt R =

P

dp ~3z

- g + 2Qu cos y +— Fz, P

(2.71)

where R is the radius of the Earth, ^ is its angular velocity of rotation, and y is the geographical latitude. By virtue of Eq. (2.65), the turbulent viscous force is given by

If one further assumes that the fluid is incompressible, Eq. (2.6) becomes du dv dw ,

dx dy dz which closes the system of equations.

In this section we shall be concerned with midlatitude synoptic scale motions, that is, systems of typically 103 km in the horizontal dimension and 10 km in vertical extent. For this scale, the vertical velocity (typically less than 1 cm s-1) is much smaller than the horizontal velocity (typically 103 cm s-1). Hence, to afirst approximation, terms involving w can be neglected in Eqs. (2.69)-(2.71). Similarly, because the curvature terms are also much smaller than the other terms, they too can be neglected. The resulting approximate horizontal momentum equations are

is the Coriolisparameter. To this order of approximation, Eq. (2.71) becomes ip = -pg, (2.77)

d z which is the hydrostatic approximation.

The Coriolis parameter f is the local component of the planetary vorticity normal to the Earth's surface. If the north-south particle motions are extensive enough in latitude, the values of this parameter also change. For small changes about a mean latitude y0 where f = f0, one can write df f = fo + ir y + ••• = fo + Py + ■■■- (2.78)

Atmidlatitudes, y = 45° (say), one has fo = 10-4 s-1 and P = 1.619 x 10-13 cm-1 s-1. The tangent-plane approximation with f constant is called an f plane; if we assume a linear relation between f and y, it is known as the P -plane approximation.

2.5.1 The geostrophic approximation

For synoptic motions far from the Earth's surface, turbulent friction and con-vective accelerations can be neglected altogether in Eqs. (2.74) and (2.75). Accordingly, if the response of the atmosphere to gravity leads ultimately to a steady state, that state will be given by the time-independent solution u = ug and v = vg, say, where

This is known as the geostrophic balance and describes the familiar situation in which the flow is along contours of constant pressure. * If we define the geopotential

Jo which is the work required to raise a unit mass from the Earth's surface to height z, this approximate solution becomes

where the subscript "p" refers to differentiation at constant pressure. As we shall see in Section 8.5, such a motion is also relevant to the theory of contact binaries.

Now, if / is regarded as a constant, it is a simple matter to differentiate Eq. (2.81) with respect to pressure and to make use of the fact that dQ/dp = —1/p to obtain d Ug 1 (dp\ , dVg 1 / 3p\

Thence, in combining Eqs. (2.77) and (2.82), one finds that d Ug g {dp\ j dvg g f Bp\

For the atmosphere, Eq. (2.14) implies that one can rewrite these relations in the forms:

This is the thermal wind equation, which relates the increase of the horizontal geostrophic velocity with height to the horizontal temperature gradient within a surface of constant pressure. In other words, if the surfaces of constant pressure and constant temperature do not coincide, the geostrophic wind generally has vertical shear. On the contrary, if these two families of surfaces are coincident, its velocity must be independent of height. This result implies that the Taylor-Proudman theorem is a direct consequence of the geostrophic approximation (see Eq. [2.48]).

2.5.2 Ekman layer at a rigid plane boundary

In the lowest kilometer of the atmosphere, the geostrophic solution (2.79) does not apply because the vertical viscous force generally is comparable in magnitude to the

* An empirical law that describes the approximate agreement between the geostrophic wind (Eq. [2.79]) and the actual wind was originally derived in 1857 by the Dutch meteorologist Christoph Buys Ballot (1817-1890). This rule of thumb states that in the northern hemisphere a person standing with his back to the wind has the higher pressure to his right and the lower pressure to his left; in the southern hemisphere, the lower pressure is to the right of the observer and the higher pressure to the left. Buys Ballot also noticed that the wind blows in general perpendicular to the pressure gradient and that the wind speed increases with increasing pressure gradient (see Eq. [2.79]). As we shall see in Section 2.5.2, however, in both hemispheres the wind near the ground does not flow exactly parallel to the isobars but has a component toward lower pressure because of surface friction. Buys Ballot's law, which is also known as Ferrel's law or the baric wind law, is not applicable in the equatorial regions.

pressure-gradient and Coriolis forces. In this boundary layer, the acceleration terms are still small compared to the remaining terms in Eqs. (2.74) and (2.75). For a situation in which there is a shear in the vertical direction only, we can thus write

Let us further assume that the fluid is homogeneous. Then, taking the first-order derivatives of Eq. (2.77) with respect to x and y, one readily sees that the horizontal pressure gradient does not depend on height. Hence, by making use of Eq. (2.79), one obtains d 2u dZ2

and d^ dZ2

where KV = AV/p is regarded as a constant. At ground level, in close analogy with molecular viscosity, we shall assume that eddy viscosity inhibits the tangential fluid motion. Hence, we let u = v = 0 at z = 0. (2.89)

(see Eqs. [2.17] and [2.18]). Since the flow must also match the geostrophic solution at high levels, it is also required that u ^ ug and v ^ vg as z (2.90)

The appropriate boundary-layer solution for the horizontal velocity is u = ug - e(-z/A)[ug cos(z/A) + vg sin(z/A)] (2.91)

and v = vg - e(-z/A)[vg cos(z/A) - ug sin(z/A)], (2.92)

where

This steady solution was originally obtained by Ekman (1905). Figure 2.1 illustrates the wind velocity vector as a function of the nondimensional height z/A. Owing to the combined effects of the Coriolis force and turbulent friction, the tip of the velocity vector traces a spiral as z/A decreases to zero. As the solid boundary is approached, this vector is at 45° to the left of the geostrophic velocity. As z/A = n, the wind is parallel to the geostrophic flow but slightly greater than geostrophic in magnitude. The level z = n A may be considered as the top of the viscous boundary layer. Measurements indicate that

Fig. 2.1. The velocity vector within the Ekman layer, at various heights above a solid boundary. The values of the nondimensional height z/A are, respectively, n, 5n/6, 4n/6, 3n/6, 2n/6, and n/6 (see Eq. [2.93]). The large arrow indicates the direction of the applied pressure gradient.

the wind approaches its geostrophic value at about one kilometer above the ground. Letting f = 10-4 s-1 and v = 10-1 cm2 s-1, one finds that KV = 5 x 104 cm2 s-1 and KV/v ^ 5 x 105 (see Eq. [2.66]). Note that this ideal solution is rarely observed because the coefficient KV must vary rapidly with height near the ground. On qualitative grounds, however, it gives an adequate picture of the frictional coupling between the geostrophic flow in the free atmosphere and the Earth's surface.

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