14.1 Introduction

In the atmosphere, the boundary layer may be defined as the thin layer extending from the earth's surface upward in which the airflow strongly experiences the effect of the earth's surface friction. Since the days of Osborne Reynolds (1842-1912) who experimented with the motion of viscous fluids in pipes, it has been known that when the velocity of a viscous fluid exceeds a certain critical limit, the initial laminar flow breaks down into irregular turbulent eddies, resulting in a rapid mixing of the fluid elements. The transition from laminar to turbulent flow appears to occur when the ratio of the inertial to frictional accelerations, which is called the Reynolds number, exceeds a certain critical limit.

Atmospheric motion, especially in the boundary layer, is almost always turbulent, indicating that conditions here are not quite the same as in a fluid pipe. Yet, when the atmosphere is thermally stable, several aspects of turbulent flow in which the horizontal momentum is transported vertically by the movement of small-scale eddies may be studied by applying the laws of large-scale motion. Prandtl's mixing-length hypothesis postulates that the frequent fluctuations of velocity at a point in a turbulent atmosphere near the ground are caused by the random movement of eddies, which are nothing but air in bulk of different momentum, from one level to another. In the process, the eddies carry the momentum of their original level and deliver it to the destination level and thereby produce a fluctuation of momentum at the new level. In this respect, the mixing-length may be regarded as somewhat analogous to the mean free path in molecular motion. This hypothesis enables us to parameterize the fluctuations in terms of the mean motion and work out the additional stresses, known as Reynolds stresses, which are brought about by the moving eddies. It enables us to derive an expression for the co-efficient of eddy viscosity on the model of that of molecular viscosity. Near the ground where the characteristic scale of the eddy at any level is assumed to be proportional to its distance from the ground, a logarithmic profile is found to closely represent the variation of the wind with height. This is called the surface layer in which the shearing stress varies little with height. Above this lies a layer of about a kilometer or so in height in which there is a three-way balance between the pressure gradient force, the Coriolis force and the eddy shearing stress. This is the well-known Ekman layer through which the wind speed and direction vary spirally so as to become nearly geostrophic at the top of the layer. Above the Ekman layer is the free atmosphere where the influence of the earth's surface friction is assumed to be minimal.

However, as it is known from observations, turbulence in the earth's boundary layer is greatly influenced by the thermal stability of the lower atmosphere. According to L.F. Richardson (1920), the flow can remain turbulent only so long as the rate of supply of energy by the Reynolds stresses is at least as great as the work required to be done to maintain the turbulence against gravity. Thus, it is the ratio of the gravitational stability arising from the vertical gradient of potential temperature and Reynolds's stresses, which determines the growth or decay of turbulence in the atmosphere. The importance of this ratio, which is called the Richardson number, in studies of atmospheric turbulence, can hardly be over-emphasized. Subsequent studies have shown that Richardson's work marked an important advance in our understanding of the phenomenon of atmospheric turbulence in the boundary layer. In this chapter, while we trace the early developments of the study of frictional effects in the boundary layer, we also review some of the later studies involving the effects of both friction and thermal stability.

The chapter also gives a brief introduction to the boundary layer of the ocean, created largely by the effect of windstress at the ocean surface, which is known as the Ekman layer of the ocean. A brief treatment of this problem is given, following Gill (1982).

14.2 The Equations of Turbulent Motion in the Atmosphere

One familiar with meteorological observations is aware of the presence of noise in the observations in the form of fluctuations of different frequency and amplitude. An example of this noise in the record of a sensitive anemometer exposed at a height of about 1 m or so above the ground at a fixed location on a hot summer afternoon is shown schematically in Fig. 14.1

We may interpret Fig. 14.1 as follows: If V (t) is the value of the observed windspeed at time t, the values at successive times can be averaged over a reasonably short time interval, say T, so that the observed value at any instant may be expressed

fluctuations of the windspeed, V (t), with time t at a height of about 1 m above ground on a hot summer afternoon with T is the sampling period

Fig. 14.1 An example of

> t as the sum of the averaged value V and a small deviation therefrom, V'(t). That is, V (t) = V+V'(t). Here we have used the underlining to denote the time-average and the prime to denote the deviation therefrom. The interval T is so chosen as to be long enough to average out the eddy fluctuations but short enough to reveal the trends in the large-scale flow. Therefore, if u, v, w are the components of the instantaneous velocity vector along the rectangular co-ordinate axes x, y, and z respectively, they may be expressed in terms of the time-mean and the deviations as follows:

In 11.2, we derived an expression for the force of friction per unit mass using the coefficient of kinematic viscosity, v. With the inclusion of this force in the equations of motion in large-scale viscous flow, the horizontal momentum equations may be written as du/dt + u du/dx + v du/dy + w du/dz = -(1/p)dp/dx + fv + v d2u/dz2

dv/dt + u dv/dx + v dv/dy + w dv/dz = -(1/p) dp/dy - fu + v d2v/dz2

where the last terms on the right-hand side of both the equations express the effect of molecular viscosity with v denoting the co-efficient of kinematic viscosity. With the aid of the continuity equation (11.8.2) which may be written in the form dp/dt + d(pu)/dx + d(pv)/dy + d(pw)/dz = 0 (14.2.4)

we put the momentum equations in the flux form. We do this for the u-component by multiplying (14.2.2) by p and (14.2.4) by u and adding the two to obtain the following:

d(pu)/dt + d (puu)/dx + d (puv)/dy + d (puw)/dz = -dp/dx + pfv + |d2u/dz2 (14.2.5)

where | denotes the co-efficient of molecular viscosity (= pv).

The flux form of the equation for the v-component is obtained similarly by multiplying (14.2.3) by p and (14.2.4) by v and adding the two to give d(pv)/dt + d(puv)/dx + d(pvv)/dy + d(pvw)/dz = -dp/dy - pfu + |d2v/dz2

We now partition the flow between the mean motion and turbulent fields by substituting for u, v, and w from (14.2.1) and average (14.2.5) and (14.2.6) in time by neglecting the small fluctuations in pressure and density. While averaging, we neglect the average of the product of a mean value and a fluctuation, so that the average of terms like uv' and vw' disappear, since u = v = w' = 0, and the term like puv becomes

After averaging, the Eqs. (14.2.5) and (14.2.6), with the aid of the averaged continuity equation, become respectively du/dt +u du/dx + v du/dy + w du/dz = —(1 /p) dp/dx + fv

+v d2u/dz2 — (1/p){d(p uV)/dx + d(p u'v'/dy + d(p u'w'/dz)}(14.2.7)

dv/ dt + u dv/dx + v dv/ dy + w dv/dz = —(1/p) dp/dy — fu +v d2v/dz2 — (1/p) {d(p uV)/dx + d(p v'v')/dy + d(p v'w')/dz)} (14.2.8)

The Eqs. (14.2.7) and (14.2.8) are revealing, in that they bring out not only the shearing stresses due to molecular motion (the third terms on the right-hand side of the equations) but also the shearing stresses generated by the eddies (the last terms within the second bracket on the right-hand side of the equations). The latter, i.e., the shearing stresses due to eddies are called the Reynolds stresses in honour of Osborne Reynolds. In a fully turbulent atmosphere, the components of the Reynolds stresses are of about the same order of magnitude along the co-ordinate axes, but in practice the vertical variation is much larger than the horizontal variations. So, if we retain the vertical components of the stresses only, (14.2.7) and (14.2.8) may be written in the simpler and more concise form du/dt = — (1/p)dp/dx + fv +(1/p) d(xZx + T^/dz (14.2.9)

dv/dt = —(1/p) dp/dy — fu +(1/p) d(Tzy + i'Zy)/dz (14.2.10)

where we have expressed the horizontal accelerations following mean motion by making the following substitutions:

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