- Hydrostatic Equilibrium
- The Dry Adiabatic Lapse Rate
- The Buoyant Force on a Parcel of Air
- The Stability of Dry Air
- The Stability of Moist Air
- Airflow Near the Ground
- Determination of Roughness Length from Wind Profiles
- Examples of Wind Profiles
- Turbulent Boundary Layer Theory
- Determination of Friction Velocity from Wind Profiles

The rate of change of pressure P with height z is:

dP/dz = -rho.g,

where g = 9.8 m/s^{2}, *rho* = P/RT_{v}, and T_{v} is the virtual temperature. This gives the **hydrostatic equation**:

dP/P = (-g/RT_{v})dz.

Integrating from the surface (z = 0, P = P_{0}) we get the pressure at height z:

P(z) = P_{0}exp(-gz/R<T_{v}>),

where <T_{v}> is the mean virtual temperature in the layer.

For dry air in an adiabatic process

C_{P}dT = RTdP/P.

Since dP/P is given by the hydrostatic equation we get for the **dry adiabatic lapse rate** *GAMMA*:

dT/dz = -g/C_{P}= -GAMMA= 0.975×10^{-2}K/m,

or about 1°C per 100m height.

For *unsaturated* moist air the adiabatic lapse rate is almost the same as the dry adiabatic lapse rate.

A volume V of dry air with density *rho* has weight *rho*.Vg. The upward force due to the displacement of surrounding air with density *rho*' is *rho*'Vg. Therefore the **net buoyant force F per unit mass** is:

F = (rho' -rho)Vg/rho.V = (T - T')g/T',

where T and T' are the temperatures of the parcel of air and the surrounding air respectively. For moist air we should use virtual temperatures:

F = (T_{v}- T_{v}')g/T_{v}'.

Let the actual lapse rate of the surrounding air be

gamma= -dT'/dz.

If *gamma* < *GAMMA*, then in a vertical displacement of the air we have

-dT'/dz < dT/dz, or dT < dT'.

Suppose initially T = T'. After an upward displacement, T < T' and the buoyant force is downwards. After a downward displacement, T > T' and the buoyant force is upwards. These results show that the air is **stable**.

If *gamma* = *GAMMA*, then the stability of the air is **neutral**.

If *gamma* > *GAMMA*, then the air is **unstable**.

Let *theta*' be the **potential temperature** of the surrounding air. Then the stability criteria can be written:

If d(theta')/dz > 0, then the air is stable.

If d(theta')/dz = 0, then the air is neutral.

If d(theta')/dz < 0, then the air is unstable.

Let *GAMMA*_{s} be the **pseudoadiabatic lapse rate** of vertically displaced saturated moist air. It is always true that *GAMMA*_{s} < *GAMMA*. The different classes of stability with actual lapse rates *gamma* are as follows (See Fig. 9):

gamma<GAMMA_{s}: Always stable.

gamma=GAMMA_{s}: Stable when unsaturated, neutral when saturated.

GAMMA_{s}<gamma<GAMMA: Stable when unsaturated, unstable when saturated.

gamma=GAMMA: Neutral when unsaturated, unstable when saturated.

gamma>GAMMA: Always unstable.

Fig. 9. Diagram of lapse rates and classes of stability.

It can be shown that when a thick layer of air is lifted, for example by flow over a mountain, the following processes occur:

(a) The lapse rate in dry air approaches the dry adiabatic lapse rate - stable air becomes less stable, and unstable air becomes less unstable.

(b) The lapse rate in moist air can be made stable, neutral, or unstable, depending on **the lapse rate of the wet bulb potential temperature** *theta*_{w}:

If d(theta_{w})/dz > 0, then the air isconvectively stable(moisture condenses at the top of the layer first).

If d(theta_{w})/dz = 0, then the air isconvectively neutral.

If d(theta_{w})/dz < 0, then the air isconvectively unstable(moisture condenses at the bottom of the layer first).

If the wind is strong, turbulent mixing of the air near the ground makes the surface layer neutrally stable. Then the wind speed u has a vertical profile given by:

du/dz = A/z,

where A is independent of the height z, but depends on u at a standard height and on the nature of the surface (See Fig. 10). Integration gives u(z) = A.ln z + B, where B is a constant chosen to make u = 0 at height z_{0}. The height z_{0} is called the **roughness length** of the surface. Now B can be writtten B = A.ln(z_{0}), and the wind profile is written

u(z) = A.ln(z/z_{0}).

Fig. 10. Wind speed profile at the surface.

Table 8 gives examples of roughness lengths for different surfaces. The value of z_{0} is approximately one tenth of the size of the objects causing the roughness.

Surface | z_{0} |
---|---|

Rough grass | 0.01 m |

Field crops | 0.1 m |

Forest, towns | 1.0 m |

Since A is the same at different heights z_{1} and z_{2}, we have

A = u(z_{1})/ln(z_{1}/z_{0}) = u(z_{2})/ln(z_{2}/z_{0}).

Solving for z_{0} we get

z_{0}= exp[(u(z_{2}).ln z_{1}- u(z_{1}).ln z_{2})/(u(z_{2}) - u(z_{1}))] .

Given: z_{1} = 1m, z_{2} = 2m, u(z_{1}) = 4.0m/s, u(z_{2}) = 4.8m/s. Then z_{0} = 0.031m. This is a typical value for rough grass.

Examples of wind profiles for roughness length 0.1 m and different wind speeds at height 10m are shown in Table 9 and Fig. 11.

Roughness length z_{0}=0.1m |
|||
---|---|---|---|

Height z | u_{10}=2.5m/s | u_{10}=5.0m/s | u_{10}=10m/s |

1m | 1.25 | 2.5 | 5.0 |

3m | 1.85 | 3.7 | 7.4 |

10m | 2.5 | 5.0 | 10.0 |

30m | 3.10 | 6.2 | 12.4 |

100m | 3.75 | 7.5 | 15.0 |

Fig. 11. Wind speed profiles for the same roughness length z_{0}=0.1m and different speeds at height 10m.

Examples of wind profiles for different roughness lengths and the same wind speed at height 10m are shown in Table 10 and Fig. 12.

Wind speed 5m/s at height 10m | |||
---|---|---|---|

Height z | z_{0}=0.01m | z_{0}=0.1m | z_{0}=1.0m |

1m | 3.3 | 2.5 | 0 |

3m | 4.1 | 3.7 | 2.4 |

10m | 5.0 | 5.0 | 5.0 |

30m | 5.8 | 6.2 | 7.4 |

100m | 6.7 | 7.5 | 10.0 |

Fig. 12. Wind speed profiles for different roughness lengths and the same wind speed 5m/s at height 10m.

The wind profile equation can be derived using turbulent boundary layer theory. The **mixing length** L in turbulent flow is the distance a lump of air moves until it mixes with the surrounding air.

Suppose a lump of air at height z+L with horizontal velocity u(z+L) falls to height z and mixes with the surrounding air with horizontal velocity u(z). The momentum lost per unit volume is

rho(u(z+L) - u(z)) =rho.u' =rho.L.du/dz,

where u' is the turbulent part of the velocity.

Observations show that L is proportional to z:

L = kz,

where k is a universal constant (von Karman's constant)

k = 0.40.

Now the wind profile equation becomes

u' = kz.du/dz.

Suppose w' is the downward speed due to turbulence. Then the momentum lost per unit horizontal area per unit time creates a horizontal **shearing stress** *tau* (a horizontal force per unit horizontal area) given by:

tau=rho.u'w'.

Since *rho*.u' is the momentum density of the turbulence, and w' is the downward speed of the momentum density, the shearing stress *tau* is the *downward flux of turbulent momentum*.

The **friction velocity** u_{*} is defined by

tau=rho.u_{*}^{2}.

When the turbulence is the same in both the horizontal direction and the vertical direction we assume

u_{*}= u' = w'.

Now the wind profile equation can be written

du = (u_{*}/k)dz/z.

Comparison of this with the wind profile equation given earlier shows that u_{*}/k = A, and we can write the wind profile equation as:

u(z) = (u_{*}/k)ln(z/z_{0}).

To calculate u_{*}^{2} from u(z_{1}) and u(z_{2}) we eliminate z_{0} from the equations:

u(z_{1}) = (u_{*}/k)ln(z_{1}/z_{0}) and u(z_{2}) = (u_{*}/k)ln(z_{2}/z_{0}).

This gives:

u_{*}= k[u(z_{2}) - u(z_{1})]/ln(z_{2}/z_{1}),

where k = 0.40.

Given: z_{1} = 1m, z_{2} = 2m, u(z_{1}) = 4.0m/s, u(z_{2}) = 4.8m/s. Then u_{*} = 0.462 m/s.

*By R. H. B. Exell, revised 2004. King Mongkut's University of Technology Thonburi.*

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