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

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 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

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.

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

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

(EQ 78)

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

We can also show that this stress is

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:

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

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

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

Using Eq. 79 and Eq. 92 we can write a new equation for K*m*

(EQ 93)

(EQ 94)

(EQ 95)

(EQ 96)

By rearrangement of Eq. 101 we find.

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

(EQ 103)

(EQ 104)

z*o* is the roughness length or roughness parameter and examples of z*o*for some typical surfaces are

- z
*o*< 0.5 cm for a smooth sea or new snow surface - z
*o*= 4 to 5 cm for a wheat field or tall grass area - z
*o*= 4 m for a forest

(EQ 105)

(EQ 106)

Solving for the friction velocity (u***) yields

(EQ 107)

FIGURE 46

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

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 C*D* 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 [Next] [Previous] [Up] [Top]

Generated with WebMaker