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Energy Transfers and Transformations
Stress (t) & the Flux of Momentum
Molecular and Eddy Viscosity
Bulk Aerodynamic Equation for Momentum
CHAPTER 3 Energy Sources, Transfers and Transformations

Incoming Solar Vs. Outgoing Terrestrial


In the tropics the amount of incoming solar radiation exceeds the amount of terrestrial radiation that is lost to space (Figure 42). This surplus of energy in the low latitudes results in heating of the atmosphere and the evaporation of water that subsequently will be transported toward the poles. In the polar and sub-polar region, the loss of terrestrial radiation to space exceeds the incoming energy from the Sun, and an energy deficit results. The transition from deficit to surplus regions is found in the mid-latitudes (Figure 43). Much of this transport is accomplished by storms and migratory high pressure systems.

FIGURE 42 Incoming Solar Radiation Vs. Outgoing Terrestrial Radiation

The storms and high pressure systems, that carry sensible and latent heat from the tropics, transport the eastward momentum that is imparted to the air by the rotation of the Earth. It too is in surplus in the low latitudes where the circumference of the Earth is great. The crossing gradients in incoming and outgoing radiation result in strong temperature contrasts in this region which, in turn, result in a concentration of solenoids and a tendency for cyclonic and anticyclonic rotary motions to be produced. In turn these weather systems accomplish the transport from low to high latitudes.

The transport of energy by sensible and latent heat in ocean currents is shown in Figure 44.

FIGURE 43 Energy Surplus, Energy Deficit and Energy Transport

FIGURE 44

Latitude Exchanges of Energy

Energy Transfers and Transformations

We have established that the Earth-atmosphere system is fueled from below and is that it is dependent upon longwave radiation (net) and sensible and latent heat transfers from the surface. We shall now examine in more detail the radiation balance at the surface.

Stress (t) & the Flux of Momentum

Consider two horizontal plates in a viscous medium like the atmosphere (Figure 45). The lower plate is fixed in position. The upper plate is moving relative to the lower plate at a constant speed (u). The upper plate drags the air below it. This drag is propagated downward and causes a stress in the direction u. Were the lower plate not fixed it would tend to be dragged in the same direction.

FIGURE 45 Two Parallel Plates in a Viscous Medium

The moving and fixed plate experiment shown in Figure 45 permits the following inferences. A force F must be applied to keep the moving plate at the constant speed (u). If the force is not applied, the drag by the viscous fluid on the moving plate will cause it to slow dow, and eventually it will stop. The force that must be applied is proportional to the area of the plate (A) and the speed of the plate (u) and is inversely proportional to the distance between the two plates.

(EQ 74)

and

(EQ 75)

where m is the coefficient of dynamic viscosity of the fluid. Under steady state conditions with a finite height of the moving plate dz, the following equality holds.

(EQ 76)

This force when normalized to the area over which it applies is called the stress (t).

(EQ 77)

The stress may be quantified using both equations Eq. 76 and Eq. 77 as

(EQ 78)

Molecular and Eddy Viscosity

Molecular viscosity m is a function only of material type, in our studies the atmospheric gas, and the temperature of the substance.

Eddy viscosity differs from mmolecular viscosity. It is not defined by molecular motions within a gas but is characterized by the flow of the gas (fluid). The eddy viscosity exhange coefficient is symbolized by the letter K. The symbol used for eddy viscosity's exchange coefficient is K. For the atmosphere the eddy viscosity (K) is 4 orders of magnitude larger than the molecular viscosity (m). Thus, the transport and mixing within the atmosphere is not accomplished usuallyby molecular processes but rather by bulk motions of the air due to winds. The stress arising from these bulk motions is

(EQ 79)

We can also show that this stress is

(EQ 80)

where r is the density of the air and u and w are the wind direction components in the x (east-west) and z (north-south) directions. The prime marks (`) indicate that the departures from the mean values of u and v are used and that the product of these departures are averaged. This average of the product of the u and v primes is referred to as the eddy correlation term. Since ru is the u-directed momentum for a unit volume, ru'w' can be considered to be the up or down direction transport of this momentum that arises from departures from the steady wind (eddies in the wind stream). These terms are defined in Eq. 81 and Eq. 82:

(EQ 81)

(EQ 82)

These statements indicate that the observed components of the wind (u and w) can be expressed as their respective means plus a departure from their mean.

Since winds usually increase in speed with elevation above the surface it is useful to consider winds at two such levels. We will refer to the lower wind occurring at level z and the upper wind occurring at level z+i (where i is the elevation difference between the two levels). In this model

(EQ 83)

If we specify the difference between the wind at the two levels as u" then

(EQ 84)

(EQ 85)

and we can write Eq. 84 and Eq. 85 as the low order and thus large magnitude terms of a Taylor series

(EQ 86)

and

(EQ 87)

Now we equate Eq. 85 and Eq. 87

(EQ 88)

and by simplification we find that

(EQ 89)

If we do the same for vertical (w) motions we get

(EQ 90)

Now if we assume incompressibility of the air

(EQ 91)

we can write the equation for the stress as

(EQ 92)

Using Eq. 79 and Eq. 92 we can write a new equation for Km

(EQ 93)

(EQ 94)

(EQ 95)

(EQ 96)

(EQ 97)

(EQ 98)

(EQ 99)

(EQ 100)

(EQ 101)

By rearrangement of Eq. 101 we find.

(EQ 102)

Eq. 102 is now integrated and evaluated from elevation o to elevation z.

(EQ 103)

(EQ 104)

zo is the roughness length or roughness parameter and examples of zofor some typical surfaces are

At zothe wind speed must be zero (Figure 46).zois always above the zero or reference local elevation. An equation for the wind speed at a specified elevation (a) given the friction velocity (u*) and von Karmen's constant is

(EQ 105)

(EQ 106)

Solving for the friction velocity (u*) yields

(EQ 107)

FIGURE 46

Diagram of Wind Speed (u) Variation with Height (z)

(EQ 108)

(EQ 109)

(EQ 110)

Bulk Aerodynamic Equation for Momentum


Given
Eq. 97 from above

(EQ 111)

and using

(EQ 112)

we find that

(EQ 113)

and

(EQ 114)

(EQ 115)

(EQ 116)

but

(EQ 117)

and

(EQ 118)

and let

(EQ 119)

therefore

(EQ 120)

The drag coefficient CD typically has a value near 1.5 x 10-3 for a height of 1 cm over relativel smooth surfaces (a lawn)

(EQ 121)

(EQ 122)

(EQ 123)

Similarity Theory

(EQ 124)

(EQ 125)

(EQ 126)

(EQ 127)

Bulk Aerodynamic Equation for Sensible Heat and Latent Heat Transfer

(EQ 128)

(EQ 129)

Previously we used the following expression for the stress (t)

(EQ 130)

and

(EQ 131)

therefore

(EQ 132)


Climate Dynamics - 05 FEB 96
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