Surface Energy Budget
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Surface Energy Budget. Q E. Q H. Q *. Q G. Q * =Net radiation Q E =Latent Heat Flux Q H =Sensible Heat Flux Q G =Ground Heat Flux. Radiative fluxes are positive if directed toward the surface (gain of energy for the surface)

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Surface Energy Budget

QE

QH

Q*

QG

Q*=Net radiation

QE=Latent Heat Flux

QH=Sensible Heat Flux

QG =Ground Heat Flux

Radiative fluxes are positive if directed toward the surface (gain of energy for the surface)

Non-Radiative fluxes are positive if directed away from the surface (loss of energy for the surface)


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Soil heat flux is (usually) positive during the day (directed away from the surface, it is a loss), and negative during night (directed towards the surface, it is a gain).


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Q (directed away from the surface, it is a loss), and negative during night (directed towards the surface, it is a gain).G

Heat conduction in soil

Heat conduction: process of heat transfer from molecule to molecule by vibration

The flux density of heat conducted in the soil (QG) isproportional to the vertical gradient of soil temperature (TG). k is the thermal conductivity (units W m-1 oK-1 )


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Some definitions (directed away from the surface, it is a loss), and negative during night (directed towards the surface, it is a gain).

Heat= total amount of energy of molecules

Temperature = proportional to the average kinetic energy of the molecules

Specific heat (c) =quantity of heat required to raise temperature of unit mass by one degree Celsius.

Heat capacity (C) = quantity of heat required to raise temperature of unit volume by one degree Celsius.


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The axis z is direct downward, so (directed away from the surface, it is a loss), and negative during night (directed towards the surface, it is a gain).dT/dz is positive when it increases with depth, and negative when it decreases with depth

T

z

T

z


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Thermal conductivity is the ability of the matter to conduce heat. Mineral matter is good conductor, water intermediate, air very poor. Mineral soils have larger thermal conductivity than organic soils

.

Mineral soil

Organic soil

Wet soils have larger thermal conductivity than dry soils (addition of water gives non linear increase in k)

Wet soil

Dry soil


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From the budget equation for a layer of soil heat. Mineral matter is good conductor, water intermediate, air very poor. Mineral soils have larger thermal conductivity than organic soils

Ds

z

z+Dz

Assuming that energy fluxes are only function of depth (z), and that positive fluxes are downward.


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then heat. Mineral matter is good conductor, water intermediate, air very poor. Mineral soils have larger thermal conductivity than organic soils

m is the mass of soil in the layer

In differential form

If r (soil mass density, units kg m-3) and c (soil specific heat, units J kg-1 oK) are not function of z.

Fourier’s equation of heat conduction

Thermal diffusivity (units m2 s-1)


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Thermal diffusivity is the ability to diffuse temperature waves.

Heat capacity is the product of the soil mass density and the soil specific heat. CS=cr


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Thermal properties of soils waves.

C=rc where r is soil density.

Change in heat content per unit volume is equal to DQ=CDT

Soil heat capacity (CS)

CS =Cm qm + Co qo + Cw qw + Ca qa

Where qis volume fraction occupied by mineral (m), organic (o), water (w), and air (a), and Cm,o,w,a is the heat capacity if the mineral, organic, water, and air constituents.


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The daily and annual periodic forcing by the sun creates a surface temperature wave which propagates into the soil

  • To understand the physics of the phenomena, let study a simple case:

  • Thermal diffusivity constant

  • Sinusoidal surface temperature


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Using this equation as Boundary Condition for the Fourier heat conduction equation, it can be shown that the solution is:

Phase leg of the wave

Amplitude at depth z

The wave amplitude decays exponentially with depth. It is e-1(=0.37) of the surface value at

zD is called the damping depth

The damping depth increases with the thermal diffusivity


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z=0 heat conduction equation, it can be shown that the solution is:

z=zDp/8

z=zDp/4

z=zDp/2

Dowward

Upward


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The time lag for the wave maximum and minimum to reach lower depths is given by:

Where t1 and t2 are the times at which the wave maximum and minimum reaches depths z1 and z2

The maximum of ground heat flux is reached 1/8 of the cycle before the maximum temperature (3hrs in the daily cycle, and 1.5 months in the annual cycle)

The ratio between the maximum ground heat flux at the surface, and the amplitude of the temperature wave at the surface is:


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Thermal admittance is the ability of soil surface to accept or release heat following a change in soil heat flux. Units are J m-2 oC-1 s-1/2

All other things being equal, soil with higher thermal admittance will have smaller surface temperature wave.

High thermal admittance (wet areas, clay, bare rocks)- relatively small temperature range

Low thermal admittance (dry, sandy organic soils), large daily temperature range


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Temperature or release heat following a change in soil heat flux. Units are J m

time=0

time=P/4

time=P/2

time=P3/4

At z=3zD the wave amplitude is 5% of the surface value

Ground Heat Fluxes

Temperature tendency

Heating

Cooling


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