# Gradient Method¶

The atmosphere is in contact with both the surface of terrestrial ecosystems and the surface of the oceans. The laws of physics that govern the transport of pollutants at the interface and through it to the other medium (to or from the atmosphere, the land or the ocean) are similar but with several diversifications described below.

The horizontal wind speed at the surface of the earth is governed by its anomalies which slow it down. In neutral atmospheric conditions, where the vertical turbulence in the atmosphere is governed only from the horizontal flow of the wind, this horizontal speed of the wind increases with height (or fluctuates near the earth’s surface) in a logarithmic way:

where:

z_{o} is an empirical parameter called “surface roughness length”

Κ_{*} = 0.4 is the Von Karman, constant

u_{*} = is the friction velocity as a measure of the wind speed
deceleration due to the roughness of the surface

z = Height of wind speed measurement.

d= Height of surface obstruction (e.g. mean height of forest trees, mean height of corn field).

It is a usual practice in the above cases to find u_{*} and
z_{o} for the surface that is studied to measure the horizontal
wind speed at four (4) or more height points. Also at these four points
to measure the temperature and the concentration of water vapour or the
concentration of gases of which we wanted to measure the fluxes from or
to the surface. The flux of a gas from or to the surface is governed by
the simple law of turbulent diffusion.

For the calculation of “a flux” F with the gradient method one utilizes the bulk Richardson number to establish the degree of turbulence of the “measurement field”:

where:

g is the acceleration due to gravity,

T is the temperature in K at the mean determination height

the bracketed term in the nominator is the temperature gradient

and the squared bracketed term in the denominator is the wind speed gradient

**Richardson Classification number Comment**

For atmospheric conditions classified as **stable,** the Ri is *>*
0*.*25 there exists no vertical mixing, weak winds, strong
inversion, dampened mechanical turbulence and a negligible spreading of
a pollutant or scalar

Also for the **stable** classification where 0 *<* Ri *<* 0*.*25,
the mechanical turbulence is weakened by stable stratification

For the **neutral** classification where Ri = 0, there exists only
mechanical turbulence

For the **unstable** classification where −0*.*03 *<* Ri *<* 0 there
exists only mechanical turbulence and

convection.

Also for the **unstable** conditions where Ri *<* −0*.*04 convection
is predominant, winds

are weak, vertical motion is strong, and a pollutant or scalar spreads rapidly vertically and horizontally.

For simplification reasons, one can use the two cases below, where Ri is ≥ 0 and Ri ≤ 0.

In further considering the gradient flux theory we explore the concept of the dimensionless physical parameters of heat and velocity deceleration which are defined as follows:

and

where:

p is the air density at constant pressure

C_{p} is the air thermal capacity at constant pressure

H is the vertical heat flux

However, the Φ_{H} and Φ_{M} values are determined
empirically for Ri ≤ 0 (unstable-turbulent conditions) and Ri≥0
(stable-laminar- stratified conditions) as follows:

Hence having established Ri and calculated the appropriate Φ_{H}
and Φ_{M} the **flux density F,** using the gradient method can be
calculated from the following equation:

where:

u is the wind speed and c is the concentration of the scalar

all other symbols have been defined previously

**INPUTS**

For the input of raw slow data for the gradient method, each column contains the following comma separated variables. Columns must not have a “parameter name”:

Time (Date-Time Record)

Wind speed at height 1 (m/s)

Wind speed at height 2 (m/s)

Wind speed at height 3 (m/s)

Wind speed at height 4 (m/s)

Air Temperature at height 1 (degrees C)

Air Temperature at height 2 (degrees C)

Air Temperature at height 3 (degrees C)

Air Temperature at height 4 (degrees C)

Relative humidity height 1 (in %)

Relative humidity height 2 (in %)

Relative humidity height 3 (in %)

Relative humidity height 4 (in %)

At the reference height, Barometer (mbar)

at the reference height, Wind Direction (degrees)

The following input parameters are required:

Period: Time period for averaging depends on the value given by the user and not exceeding 30 minutes.

Limit: ‘Limit’ times the standard deviation from the mean value of the ensemble of the chosen period for averaging.

Heights: Heights of sensors. In the requested parameters of atmo-flud web application Heights = [height1, height2, height3, height4].

Z: The height at which the each of the four anemometers is installed.

**OUTPUTS**

raw data of wind speed at 4 different heights

raw data of temperature at 4 different heights

raw data of RH at 4 different heights

calculated dew point and water vapour concentrations at 4 different heights

despiked data by the moving average method for all above parameters at 4 different heights

block averaged data of all above parameters at 4 different heights

bulk Richardson number

friction velocity

QH and QE

M-O Length

Footprint weight functions of 4 different heights