Ddt ddt u dudx v dvdy w ddz

Tzx(= |jd u/dz) and Tzy(= |jd v/dz) for the components of the shearing stresses due to molecular viscosity;

and tzx(= —p u' w') and xzy(= —p v' w') for the components of the shearing stresses due to eddy viscosity.

In a fully turbulent flow, the shearing stresses due to eddy viscosity are much larger than those due to molecular viscosity. So, by neglecting the small effect of the molecular viscosity, the Eqs. (14.2.9) and (14.2.10) can be further simplified to the final form du/dt = —(1/p) d p/dx + fv +(1/p) dl'zx/dz (14.2.11)

For unaccelerated balanced motion, Eqs. (14.2.11) and (14.2.12) may be looked upon as the horizontal components of the momentum equation in vector form

where we have put V' for the ageostrophic mean wind (Vg - V) and used the relations, f k x Vg = ( 1 /p)VH p; and t' = tzx i + xZy j, where i and j are unit vectors.

14.3 The Mixing-Length Hypothesis - Exchange Co-efficients

Prandtl hypothesized that turbulent fluctuations can be parameterized in terms of the mean field variables if we postulate a characteristic mixing length l' for the eddy motion somewhat like the mean free path in molecular motion. According to this hypothesis a parcel of air at level z which is displaced vertically through a height interval l' carries the mean momentum of the original level to the new level where it mixes with the air at the new level to cause a fluctuation in the mean momentum of the new level. This means that there is no mixing, whatsoever, in between the original level and the new level. The extent of the fluctuation will depend upon the magnitude of the mixing length and the vertical gradient of the mean velocity. Thus, for the u-component, u' = —l' du/dz

Since du/dz is usually positive in the boundary layer, l' is positive for downward displacement (w' < 0) and negative for upward displacement (w' > 0). We may, therefore, write

Schmidt called Ax, which has been put for pw' l', the Austausch or exchange coefficient for the x-momentum.

Observations of velocity fluctuations along the three co-ordinate axes suggest that they are of about the same order of magnitude, and, to a reasonable approximation, of about the same magnitude. So we may write, w' = l' du/dz, since w' and l' are of opposite sign by the above-mentioned convention of sign. Thus,

where lx is the mean mixing-length and Kx(= Ax/p), which is equal to lx(du/dz), is called the co-efficient of eddy viscosity for the vertical transport of x-momentum. Following an analogous procedure, we can obtain an expression for the vertical eddy transport of the y-momentum, xzy. By comparing (14.3.2) with (11.2.1), we conclude that the exchange co-efficient A has the same significance in turbulent flow as the co-efficient of kinematic viscosity ^ in molecular motion.

14.4 The Vertical Structure of the FrictionaHy-ControHed Boundary Layer

Depending upon the rate at which the shearing stress varies with height, the frictional layer has been divided into two sub-layers, as shown in Fig. 14.2

They are: (i) the Prandtl or surface layer and (ii) the Ekman or transition layer. Above the frictionally-controlled layer lies the free atmosphere in which the eddy stresses are regarded as negligible.

14.4.1 The Surface Layer

This is the bottom layer of the atmosphere in intimate contact with the earth's surface, hence dominated by friction. According to observations in a neutrally stable atmosphere, the distribution of wind with height within the first 20 m or so of the surface suggests that the shearing stress remains more or less constant within the layer. This means that the shearing stress at a height z within this layer is about the same as that at the surface, t0. Hence, we may write (14.3.2) in the form

where we have written u for the time-averaged wind u.

Since density varies only slightly within the surface layer, it may be treated as independent of height. According to Prandtl, T0/p has the dimension of velocity-squared and is written as u*2, where u* is called the friction velocity. Also, the

(c) FREE ATMOSPHERE

EKMAN or TRANSITION LAYER (100-1000 m)

(a) SURFACE LAYER

Fig. 14.2 The vertical structure of the boundary layer, showing the frictionally-affected (a) Surface layer, (b) Ekman or transition layer, and (c) the free atmosphere dimension of K is velocity times a length. Now, the question arises as to what velocity and what length should be used for K in the case. The logical choice is the friction velocity u* for the velocity and a length proportional to height z above the surface. This means that we put K = Kzu*, where k, the constant of proportionality, is von Karman constant (k = 0.4). Making the substitutions in (14.4.1) and simplifying, we obtain du/dz = u*/kz (14.4.2)

Integrating (14.4.2) with respect to z, we get the logarithmic velocity profile, u =(u*/K) ln (z/zo) (14.4.3)

where z0, which is called the roughness length, is the constant of integration chosen so that u = 0 at z = z0. The roughness length varies widely with the physical characteristics of the surface and the average height of the obstacles to airflow. Values of z0 found over some natural surfaces are: 0.5 cm over smooth lawn and snow surfaces; 3.2 cm over low grass; 3.9 cm over high grass; 4.5 cm over a wheat field. When z0 is very large as over dense vegetation, a modified logarithmic wind profile as given below has been found to represent the observations somewhat better.

where d is called the datum-level displacement.

Experience shows that inspite of several assumptions involved in the derivation, (14.4.3) gives a fairly satisfactory representation of the vertical wind profile in the earth's surface layer. Also, in this layer, since the wind direction remains more or less constant with height, it follows that the shearing stress vector is parallel to the wind vector. Hence with arbitrary orientation of the co-ordinate axes, we put, v/u = (dv/dz)/(du/dz). This relation between the wind shear and the wind may be used as the lower boundary condition for the Ekman layer.

It should be noted, however, that the logarithmic wind profile (14.4.3) holds only in a neutrally stable surface layer. The profile changes when the atmosphere becomes thermally stable or unstable. After carefully examining the vertical wind and temperature values between 0.5 m and 13 m above ground, under different stability conditions, Deacon (1949) proposed a modified wind profile du/dz = (u*/Kz0)(z/z0)-P (14.4.5)

where P is a decreasing function of the Richardson criterion Ri, defined by him for the purpose, by the expression, Ri = (95 - 90.2)/uf, where the subscripts indicate the levels (m) at which the potential temperature 9 and the wind velocity u were measured.

Since the stability of the layer is a function of Ri which depends upon the vertical variation of 9, it follows that P is less than, equal to, or greater than 1 in stable, neutral, or unstable atmosphere respectively. If (14.4.5) is integrated from z0 (where u = 0) to z, we obtain u = u*[(z/zo)1-p - 1]/{K (1 - P)} (14.4.6)

Deacon showed that when P is very nearly equal to 1, the observed wind profile approaches the logarthmic wind profile (14.4.3).

The effect of thermal stability on the wind profile can be seen readily if one examines the diurnal variation of the low-level winds at a place under different thermal stability conditions. In the early morning hours when there is an inversion of temperature with height near the surface and the atmosphere is thermally stable, the wind is light or variable at the surface but strong at some height above the ground. In the afternoon, with rapid warming of the surface, the inversion is replaced by a lapse rate of temperature and the atmosphere becomes thermally unstable. In such condition, turbulent mixing occurs which transfers momentum downward resulting in a decrease of the vertical wind shear except, perhaps, in a very shallow layer very close to the ground. The characteristic diurnal variations brought about in the vertical wind profile of the lower boundary layer by the effects of friction and thermal stability are discussed further under 'nocturnal jet' later in this section.

14.4.2 The Ekman or Transition Layer

Above the surface layer, the structure of the boundary layer is determined by the vector relation (14.2.13) which may be written in the form

where we have assumed that K is invariant with height in this layer and made use of the relation (14.4.1).

If u,v are the horizontal components of the mean velocity vector V, (14.4.7) yields the following equations:

Equations (14.4.8) and (14.4.9) can be solved to determine the departure of the observed wind from geostrophic balance in the Ekman layer. We do this in two steps, following a treatment given by Holton (1979). In the first step we ignore the presence of the surface layer and assume that the Ekman layer starts from the ground (z = 0), instead of from the top of the surface layer. The boundary conditions on u, v, then require that the velocity components disappear at the ground and assume geostrophic values at great distances from the ground. That is, the boundary conditions are:

u = 0, v = 0, atz = 0; andu = ug, v = vg, as z ^ ^ (14.4.10)

To solve (14.4.8) and (14.4.9), we multiply (14.4.9) by i = y^ - 1) and then add it to (14.4.8). The result is the complex equation

Kd2(u + vi)/dz2 - i f(u + iv) = -i f(ug + vg) (14.4.11)

We can arrive at a simple solution of (14.4.11) if we assume that the geostrophic wind does not vary with height and that we choose the x-axis along the geostrophic wind so that vg = 0. Equation (14.4.11) may then be written as d2 (u + iv - ug )/dz2 - (1 + i)2m2 (u + iv - ug )=0 (14.4.12)

The general solution of (14.4.12) may be written u + vi = ug + Aexp [(1 + i) mz]+Bexp[(-(1 + i) mz] (14.4.13)

Applying the boundary conditions (14.4.10), we note that in the northern hemisphere (f > 0), A = 0, and B = -ug.

So, using the Euler formula, exp (iy) = cos y + isiny, and equating the real and imaginary parts, we may write the permissible solution of (14.4.13) as u = ug [1 - exp (-mz) cosmz] (14.4.14)

This is the well-known Ekman spiral solution, so called in honour of the famous Swedish oceanographer, V.W. Ekman, who in 1902 first developed the theory for the boundary layer of the ocean (Ekman, 1905). The structure of the spiral is best shown by a hodograph presented in Fig. 14.3 in which the components of the wind are plotted as a function of height. Thus the points on the curve represent the values of u and v for different values of mz as one moves away from the origin. The wind becomes parallel to the geostrophic wind at a height Zg = n/m, though the magnitude of the wind at this height was slightly greater than the geostrophic wind. Zg is then the depth of the Ekman layer.

Observations show that the wind attains its geostrophic value at a height of about 1 km above ground. Using this value for Zg at a place where f = 10-4s-1, we get a value of about 5m2 s-1 for K. In Sect. 14.3, we put K = l^du/dz, where l7 is the mixing-length for an eddy.

Fig. 14.3 Hodograph of the Ekman spiral solution with points marked on the graph showing the values of mz, which is a non-dimensional measure of height z

Fig. 14.3 Hodograph of the Ekman spiral solution with points marked on the graph showing the values of mz, which is a non-dimensional measure of height z

So, if we take a value of 5 ms-1 km-1 for the wind shear du/dz , we get a value of about 30 m for l'. This value of the mixing length in the transition layer of depth about 1 km must be regarded as quite reasonable, if the mixing-length concept is to be useful.

Figure 14.4 shows a three-way balance of the frictional force F with the pressure gradient force PG and the Coriolis force CO in the Ekman layer, where V is the observed wind, Vg is the geostrophic wind, V' is the vector difference (V-Vg) (called the ageostrophic wind), f is Coriolis parameter, and k is a unit vector pointing upward from the plane of V and Vg,

However, the ideal Ekman layer solution discussed above is seldom realized in practice because of the observed large increase of the eddy exchange co-efficient with height near the surface of the earth. For this reason, the given solution is applicable to the region only above the surface layer. A more satisfactory solution can, therefore, be obtained by combining the above solution with the logarithmic wind profile solution for the surface layer. We again treat the eddy viscosity co-efficient K as constant but apply it to the region above the surface layer. Thus, instead of u + vi = 0, at z = 0, we let at the boundary between the two layers, u + v i = C0exp (i a) (14.4.16)

where C0 is the magnitude of the wind at the top of the surface layer and a is the angle between the wind and the isobars in the surface layer.

Observations show that in the surface layer the wind shear is more or less parallel to the direction of the observed wind. Applying this condition now to the wind in the Ekman layer, we write u + vi = C d (u + vi)/dz (14.4.17)

where C is a constant.

For convenience, we let z = 0 at the bottom of the Ekman layer. Then using (14.4.16) and the condition that u ^ ug as z ^ the solution of (14.4.11) may be written u + vi = [C0exp (ia) - ug] exp {- (1 + i)mz} + ug (14.4.18)

Using (14.4.18) in the boundary condition (14.4.17) and equating the real and imaginary parts, we obtain

C0cos a = mC[C0(sin a - cos a)+ug] C0 sin a = mC[-C0(sin a + cos a)+ug]

Elimination of C from these equations gives

Substituting this value of C0 in (14.4.18) and equating the real and imaginary parts, we obtain u = ug[1 - V2sinaexp(-mz)cos(mz - a + n/4) (14.4.20)

One can readily see that for a = n/4, the modified expressions (14.4.20-14.4.21) for the spiralling wind reduce to the classical version (14.4.14-14.4.15).

Experience shows that although the modified version is a slight improvement on the classical one, neither of them represents the real conditions in the atmosphere in a satisfactory manner due to effects of several other factors on the boundary layer besides friction. Transience and baroclinic effects are prominent amongst these other factors. Convective currents generated by horizontal temperature gradient in an unstable baroclinic layer create additional turbulence which modifies the turbulence due to friction. But, even in steady barotropic conditions, the ideal Ekman pattern is seldom realized.

It is further observed that even in a neutrally buoyant atmosphere, the frictional inflow into a lower pressure area generates a secondary circulation the horizontal and vertical scales of which are comparable with the depth of the boundary layer. Thus, it is not possible to parameterize this circulation in terms of the mixing-length theory. However, the circulation transfers momentum vertically and thereby reduces the angle between the boundary-layer wind and the geostrophic wind. The secondary circulation set up by friction in the boundary layer is of great importance in meteorology.

14.5 The Secondary Circulation - The Spin-Down Effect

The vertical motion in the secondary circulation referred to at the end of the previous section can be computed from the mass flux into the low-pressure area by the v-component of the boundary-layer wind given by (14.4.15). Thus,

M = /p vdz = /p ugexp (-nz/zg)(sinnz/zg)dz (14.5.1)

If it is assumed that the density remains invariant in the boundary layer, we may write the mass continuity equation in the form d(pw)/dz = —d(pu)/ dx - d(pv)/dy (14.5.2)

Integrating (14.5.2) from 0 to zg, we obtain for the vertical velocity at the top of the boundary layer the expression zg

Here, we have assumed that w = 0 at z = 0. Substituting for u and v from (14.4.14) and (14.4.15) and again assuming that ug is invariant with x, we can re-write the expression for the vertical mass flux at the top of the boundary layer where vg = 0, in the form zg

(pw)zg = -(d/dy) J pug exp (-nz/zg) sin (nz/zg) dz (14.5.3)

Comparing (14.5.3) with (14.5.1), we note that the the vertical mass flux at the top of the boundary layer is equal to the horizontal convergence of mass in the boundary layer. Since -dug/dy = Zg, where Zg is the geostrophic vorticity, it follows from (14.5.3) that the vertical velocity at the top of the boundary layer is given by wzg = ZgZ(K/2f) (14.5.4)

The relation (14.5.4) which states that the vertical motion at the top of the boundary layer is directly proportional to the geostrophic vorticity of the layer is important, since it tells us how friction communicates mass from the layer to the free atmosphere directly through a secondary vertical circulation rather than through the slow process of molecular diffusion. For example, in a typical synoptic-scale circulation vortex in midlatitudes in which Çg ~ 1.0 x 10-55s-\ K ~ 5 m2s-1 and f ~ 1.0 x 10-4 s-\ the vertical velocity given by (14.5.4) works out to be about 1.58mm s-

Figure 14.5 shows schematically the direction of the frictionally-generated secondary circulation in a cyclonic vortex in a barotropic atmosphere.

An important effect of the secondary circulation in the atmosphere is what is known as the spin-down effect. This means that in the absence of any other disturbing factor, the vorticity of the azimuthal circulation in the outflow region will continually decrease with time. This can be readily shown by taking the vorticity equation for a barotropic atmosphere and integrating it over the height interval from the top of the boundary layer zg to the tropopause level H. Since the atmosphere is

Fig. 14.5 A schematic of the secondary circulation generated by frictional convergence in the boundary layer of a cyclonic vortex in a barotropic atmosphere

treated as barotropic, the solenoidal term is zero and we write the vorticity equation (13.6.2) in the form d(Z + f)/dt = — (Z + f)(du/dx + dv/dy) = fdw/dz (14.5.5)

where we have used the continuity equation and neglected Z compared to f on the right-hand side. Neglecting the latitudinal variation of f and integrating (14.5.5) from zg to H, we obtain

If we now assume that the vertical velocity vanishes at the tropopause and that the vorticity in the layer (H — zg) remains at its geostrophic value at the level zg, then substituting the value of w(zg) from (14.5.4), we obtain the differential equation dZg/dt = —fw(zg) = — v/(fK/2H2 )Zg (14.5.7)

where we have neglected zg in comparison with H. Integration of (14.5.7) with respect to time yields

where Zg (0) is geostrophic vorticity at time 0.

It, therefore, follows from (14.5.8) that in a northern hemisphere barotropic vortex the geostrophic vorticity will decay with time to e—1 of its original value in time t = H(2/fK)—1/2. This e-folding time is actually what is called the spin-down time. For a value ofH = 10km, K = 10m2s—1, and f = 1.0 x 10—4s—1, the spin-down time is about 4 days. This time-frame should be compared with the time that would be required for spin-down to the same extent by eddy diffusion through the same height. In this latter case, using the relation, t = H2/2K, and using a value of 5 m2s—1 for K, we obtain t « 100 days.

We can understand the spin-down effect in the atmosphere in another way. Here, as the fluid elements flow outward above the boundary layer, the outflow experiences a Coriolis force to the right in the clockwise direction, which opposes the anticlockwise azimuthal velocity and thereby slows it down. Holton (1979) cites the example of spin-down in a tea-cup where a similar secondary circulation is set up by the effect of friction at the bottom of the cup. Here, as part of the vertical circulation, the outflow at the top transports fluid from low-momentum to high-momentum region and thereby, according to the principle of conservation of angular momentum, spins down the azimuthal circulation.

In nature, examples of secondary circulation set up by flow over rough terrain are so numerous that the phenomenon may be said to be almost ubiquitous. There is indication of its occurrence over deserts where long, parallel rows of sand-dunes are piled up by high winds (Hanna, 1969). According to Woodcock (1942), the flight pattern of soaring gulls indicates the presence of line updrafts under certain conditions. Cases of glider pilots experiencing long lifting bands at heights up to 2-3 km have been discussed by Kuettner (1959). Kuettner (1967) has also reported many observations of parallel cloud lines which he attributed to helical motion in the boundary layer. Elongated cloud streets and roll clouds frequently seen in satellite cloud pictures are believed to be indications of occurrence of secondary circulation in the earth's boundary layer on a large scale.

However, it is important to recognize that in the creation of most of these phenomena, friction and buoyancy are both involved. In fact, over several areas, strong surface heating and vertical lapse rates of temperature cause low-level convection and horizontal temperature gradients cause thermal winds which may augment or counteract the effect of friction. Joseph and Raman (1966) reported the occurrence of a low-level westerly wind maximum of 20-30 m s-1 at a height of about 1.5 km a.s.l. over the southern Indian peninsula during July. Above the low-level maximum, the westerly wind continuously decreased in speed with height and reversed direction to attain an easterly wind maximum of about 40-50 m s-1 at an altitude of about 14-15 km a.s.l. The low-level westerly jetstream could be explained only as the combined effect of the meridional temperature gradient and the boundary-layer friction. Under the effect of friction alone, the westerly surface wind would increase in speed to become geostrophic at the top of the boundary layer, but the effect of a horizontal temperature gradient with warm air to the north and cold air to the south produces an easterly thermal wind which counteracts and arrests the further increase in the speed of the westerly wind with height. The result is a low-level westerly jetstream of the kind observed, besides the easterly jet stream at high levels which appears to be wholly thermally generated and controlled (Saha, 1968).

14.5.1 The Nocturnal Jet

An interesting feature of the wind in the boundary layer is that at many inland stations, a jet-like strong wind appears during night at a fairly low level in the atmosphere. It is called the nocturnal jet. The basic reason for its occurrence is given by Blackadar (1957).

During the heat of the day, the boundary layer is deep and momentum is well-mixed by convective turbulence up to a height of about 1 km or more above the ground. At night, the ground cools by radiation and a stable layer develops within about 200 m of the ground. The frictional effects are then confined near the ground and a wind jet appears at the top of this stable layer where momentum is now concentrated. However, the region above (200-1000 m), which in day-time was a part of the turbulent boundary layer, is released from frictional influence, i.e., the stress term in that region suddenly drops to zero after sundown. In response, the wind begins an inertial oscillation that proceeds until mixing resumes next day. The daily cycle brings about a cyclonic rotation of the wind relative to the geostrophic wind during daytime and an anticyclonic rotation during the night. These opposite turnings of the wind between day and night have actually been observed in field experiments as well as reproduced in numerical modeling (Thorpe and Guymer, 1977).

14.5.2 Turbulent Diffusion and Dispersion in the Atmosphere

The atmosphere being turbulent, any gaseous or particulate matter released or ejected into it , such as gases, vapours, smoke, dust particles, etc., is dispersed by the eddies that may be present at the time. However, the speed and extent of dispersion in terms of height and distance traveled depends upon not only the state of turbulence but also the thermal stability of the atmosphere.

Normally, there is a large diurnal variation in the level of turbulent diffusion in the atmosphere Under thermally stable conditions and light winds as during a clear night when there is little convection and turbulence in the air, vertical mixing is restricted to the surface layers only and the released material may spread out horizontally and slowly settle down to the earth as it travels downstream. In an unstable atmosphere with strong winds as during a hot summer afternoon, there is rapid mixing of the ejected material with the upper layers of the atmosphere by large-scale convection and turbulent eddies. However, as turbulence and convection die down during the night, some of the material in the upper layer may come back to the lower layer and may even settle down to the earth. Gases, vapours and light particulate matter, such as fine dust and smoke, etc., however, may rise to higher levels and travel longer distances unless and until they are washed down by precipitation.

Atmospheric diffusion of the kind which brings about pollution in the atmosphere with harmful effects on humans, animals and plants is of great concern. In this category, one may include effluents from chemical and industrial chimneys, exhaust gases from road vehicles, aeroplanes, etc. which burn fossil fuels, radioactive leakage from atomic power plants, bomb explosions, volcanic eruptions, to name only a few. Once ejected into the atmosphere, the concentration of the pollutants in the air needs to be known in order to assess risks of adverse effects on life and property on earth. The risk is definitely greater when the pollutants are released in the atmosphere at the ground or lower levels than from chimneys or towers at sufficiently high levels where winds are usually stronger to disperse them quickly.

14.6 The Boundary Layer of the Ocean - Ekman Drift and Mass Transport

When a wind blows over a calm ocean, it forces the water at the surface to move along with it thereby producing an ocean current. Owing to viscosity of ocean water, the influence of the surface stress in dragging water is transmitted downward to a depth varying from 10 m to 100 m. In this thin layer, the ocean current decreases in strength and its direction veers with depth producing what is known as the Ekman drift in the shape of a spiral, as shown in Fig. 14.6.

In the ocean, the windstress at the surface accelerates the current in the boundary layer. Thus, along with the pressure gradient force, the forces that act in the oceanic boundary layer are practically the same as in the atmospheric boundary layer. We may, therefore, write the linearized horizontal momentum equations for a fluid on a uniformly- rotating earth in the simplified form du/dt - fv = -(1/p)dp//dx +(1/p)dX/dz (14.6.1)

where (X,Y) are the components of the stress vector, (u, v) are the components of the current velocity vector, p' is the perturbation pressure at the surface and the other terms have their usual meanings.

In the above equations, we have considered the vertical variation of the stress only and not the horizontal variation, because of the extraordinarily large scale of the horizontal variation compared with the scale of the vertical variation. Also, we have treated the density as constant within the shallow boundary layer, since water is much more incompressible than air.

From (14.6.1) and (14.6.2), it may be seen that the two forces that tend to accelerate the current are the pressure gradient force and the force due to the variation of the Ekman stress. It is convenient to consider these two effects separately. The part (up, vp) of the current velocity driven by the pressure gradient force is determined by the equations

Fig. 14.6 Schematic representation of a wind-driven ocean current in deep water, showing the decreases in velocity and change of direction at intervals of depth (the Ekman spiral).

Fig. 14.6 Schematic representation of a wind-driven ocean current in deep water, showing the decreases in velocity and change of direction at intervals of depth (the Ekman spiral).

dup/dt - fvp = -(1/p)dp'/dx; dvp/dt + fup = -(1/p)dp'/dy (14.6.3)

and assume the geostrophic value in a steady-state flow (dup/dt = dvp/dt = 0). The other part of the current velocity which is driven by the stress vector and which we denote by (uE, vE) satisfies the equations duE/dt - fvE = -(1/p)dX/dz; dvE/dt + fuE = -(1/p)dY/dz (14.6.4)

Thus, the total velocity that appears in (14.6.1) and (14.6.2) may be considered as the sum u = up + ue;v = vp + ve

Now, the windstress (X, Y) is zero outside the boundary layer. So, the integration of (14.6.4) with respect to z across the layer yields the relation (when boundary below)

p(dUE/dt - fVE) = -Xs; p(dVE/dt + fUE) = -Ys (14.6.5)

where (Xs, Ys) is the value of the stress vector at the surface, and (UE, VE) is the Ekman volume transport, relative to the pressure-driven flow, as given by the relations, UE = / uE dz, and VE = / vE dz.

Since the density is regarded as constant, the Ekman mass transports are given by (pUE, pVE). The sign of the stress term depends on whether the boundary surface is below or above the layer. The Eq. (14.6.5) applies to the case of the atmospheric boundary layer or to the ocean's bottom boundary layer in which the boundary is below the layer. In the case of the wind-driven ocean current, the boundary surface is above the layer. Hence the signs of the stress term are reversed in the oceanic case and the integral of (14.6.4) across the layer gives (when boundary above)

In the steady state, (14.6.6) gives

In steady-state flow, the Ekman transport is directed at right angles relative to the surface stress in the northern hemisphere. In the atmosphere, the transport direction is to the left of the surface stress, whereas in the ocean it is to the right, as shown schematically in Fig. 14.7, which shows the magnitude and direction of the Ekman transport above and below the ocean surface.

It follows from (14.6.5) and (14.6.7) that the sum of the Ekman mass transports above and below the ocean surface is zero.

Fig. 14.7 A schematic showing the directions and magnitudes of the Ekman mass transports in relation to the surface windstress in the atmospheric and oceanic boundary layers in the northern hemisphere. The directions are reversed in the southern hemisphere

The same argument cannot, however, be applied to the Ekman volume transports because of large differences in density between the atmosphere and the ocean. The direction of the Ekman transports in the cases of both atmosphere and ocean reverses in the southern hemisphere.

14.7 Ekman Pumping and Coastal Upwelling in the Ocean

The Ekman mass transport between neighbouring areas in the boundary layer often causes convergence or divergence of mass and thereby vertical motion which can be computed with the aid of the continuity equation, as was done for the atmospheric boundary layer in Sect. 14.5.

In the oceanic case, we again neglect the density variations and integrate the continuity equation in the form, du/dx + dv/dy + dw/dz = 0, with respect to z using the boundary condition w = 0 at the ocean surface (boundary above) and obtain d(pUE )/dx + d(pVE )/dy - pwE = 0 (14.7.1)

Substituting from (14.6.7) and neglecting latitudinal variation of the Coriolis parameter f, we obtain

Fig. 14.8 A vertical section through the center of a cyclone over the ocean in the northern hemisphere illustrating the formation of Ekman pumping in the atmosphere above and the ocean below the interface. Note the directions of Ekman transports (indicated by double-shaft arrows) in the atmospheric and the oceanic boundary layers and the raising of the thermocline in the ocean below the cyclone center where Ekman pumping is upward

Fig. 14.8 A vertical section through the center of a cyclone over the ocean in the northern hemisphere illustrating the formation of Ekman pumping in the atmosphere above and the ocean below the interface. Note the directions of Ekman transports (indicated by double-shaft arrows) in the atmospheric and the oceanic boundary layers and the raising of the thermocline in the ocean below the cyclone center where Ekman pumping is upward pwE = d(Ys/f)/dx - d(Xs/f)/dy = (1/f)k • V x (Ts) (14.7.2)

where Ts is the surface stress vector with components (Xs, Ys).

Thus, the vertical velocity wE, which is called the Ekman pumping velocity, is, according to (14.7.2), (pf)-1 times the vertical component of the curl of the surface wind stress vector. Note that it has the same sign in the ocean as in the atmosphere (14.5.4).

When a wind blows over a coastal region so as to have a strong horizontal component parallel to the coastline, the Ekman transport in the oceanic boundary layer is at right angles to the direction of the windstress and can be either inward towards the coast, or outward away from it, depending upon the direction of the windstress and the hemisphere in which the coast is located. If the Ekman transport is outward and there is divergence of water from the top, mass continuity requires cold water from deep layers to upwell to replace the water being removed. The upwelled water is rich in nutrients for plant and marine life. That is why the worlds's important fisheries are found in coastal regions where there is intense upwelling for most of the year. Famous upwelling regions are the coasts of California, northwestern Africa and Somalia in the northern hemisphere and Chile-Peru, Southwestern Africa and Southwestern Australia in the southern hemisphere. According to an estimate (Gill, 1982), coastal upwelling is 30-100 times stronger than open ocean upwelling.

When a cyclone moves over an ocean surface, the low pressure in its central area causes convergence of air and strong upward motion in the atmospheric boundary layer above the interface, while there is upwelling and divergence of water below the center of the cyclonic circulation, which results in cooling of the ocean surface. This is illustrated by a schematic in Fig. 14.8.

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