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Surface energy balance and surface temperature

To read: Williams&Smith, p. 8-12 and 63-74; Yershov, p. 331-336

- The amount of energy delivered to the particular part of the Earth surface define the thermal state of this surface – the temperature at the ground surface
- There are many other factors that influence this temperature, but the amount of energy is the first to consider
- The direct measurements of the ground surface and permafrost temperatures are available at a very limited number of sites
- It will be good to be able to calculate those temperatures using some more easily available parameters (air temperature, characteristics of the ground surface, soil properties, …)

Surface energy balance and surface temperature

Generally, there are two ways to approach:

- From the energy balance at the ground surface
- Using a “Buffer Layer” model

More definitions: 3 major mechanisms of heat transfer:

- By radiation – via electromagnetic waves
- By conduction – via thermal oscillation of atoms and molecules
- By convection – via mass transfer

More about heat transfer by radiation:

r

1

t

a

1 = r + a + t

If a=1 – “absolute black” body

If r=1 – “absolute white” body

Most surfaces are “gray”

Albedo: α = r

More about heat transfer by radiation:

r

1

t

a

Ts>0°K

1 = r + a + t

If a=1 – “absolute black” body

If r=1 – “absolute white” body

Most surfaces are “gray”

Albedo: α = r

More about heat transfer by radiation:

r

1

Is emission

t

a

Ts>0°K

1 = r + a + t

If a=1 – “absolute black” body

If r=1 – “absolute white” body

Most surfaces are “gray”

Albedo: α = r

More about heat transfer by radiation:

r

1

Is = σ·Ts4

Is

t

a

Ts>0°K

1 = r + a + t

If a=1 – “absolute black” body

If r=1 – “absolute white” body

Most surfaces are “gray”

Albedo: α = r

Stefan-Boltzmann Law

Is = σ·Ts4

σ = 5.67·10-8 W/m2K4

For “gray” surfaces:

Is = εs·σ·Ts4

εs is long-wave emissivity

The simplest energy balance at the Earth surface:

Energy fluxes at the Earth surface:

q(1 – α) + g = Is

q – incoming solar radiation

g – geothermal heat flux

Is – emission from the Earth surface

q

qα

Is

- No atmosphere
- No water
- No vegetation

g

The simplest energy balance at the Earth surface:

From fluxes to balances:

q(1 – α) + g = Is

One day energy balance at the Earth surface:

Even this simplest energy balance can be used to estimate the temperature at the Earth surface

Q(1- α) + G = Es

- No atmosphere
- No water
- No vegetation

Q

Q·α

Es

G

The simplest energy balance at the Earth surface:

From fluxes to balances:

q(1 – α) + g = Is

One day energy balance at the Earth surface:

Even this simplest energy balance can be used to estimate the temperature at the Earth surface

Q(1- α) + G = Es

- No atmosphere
- No water
- No vegetation

Q

Q·α

Es

G

Assume very simple situation

Sc is a “solar constant”

The solar constant includes all types of solar radiation, not just the visible light. It is measured by satellite to be roughly 1.366 kilowatts per square meter (kW/m²)

The actual direct solar irradiance at the top of the atmosphere fluctuates by about 6.9% during a year (from 1.412 kW/m² in early January to 1.321 kW/m² in early July) due to the Earth's varying distance from the Sun

Day

Night

Assume: α = 0.25; εs = 0.95; Sc =1,400 W/m2; g = 0.05 W/m2

We know that σ = 5.67·10-8 W/m2K4, then:

and

Ts4 = 62.1·108 , Ts = 281 K or +8°C

If α=0.2, Ts = 12°C

The global Earth surface average temperature is 15°C

This model is too simple for the Earth, but can be used for other bodies in the Solar System with no or very thin atmosphere

Assume: α = 0.25; εs = 0.95; Sc =1,400 W/m2; g = 0.05 W/m2

We know that σ = 5.67·10-8 W/m2K4, then:

and

Ts4 = 62.1·108 , Ts = 281 K or +8°C

If α=0.2, Ts = 12°C

The global Earth surface average temperature is 15°C

This model is too simple for the Earth, but can be used for other bodies in the Solar System with no or very thin atmosphere

Sources of energy in the Earth System:

- External:
- Radiation energy from the Sun and stars ~ 6·1024 J/year
- Corpuscular, including neutrino ~ 1018 J/year
- Gravitational effects of the Moon, Sun, others ~ 1020 J/year
- Internal:
- Nuclear reactions in the Earth interior ~ 1021 J/year
- Gravitational processes ~ 4·1020 J/year
- Variations in rate of the Earth rotation ~ 1020 J/year
- Exothermal chemical reactions ~ 1019 J/year

Q(t)

Es(t)

Q(t)·α

Es(t)

G(t)

G(t)

Ts(t)

Diurnal variations in the components of the energy balanceDay

Night

Ts(t)

One-hour-sum energy balance: Q(t)·(1-α) + G(t) = Es(t),

Now Ts = Ts(t)

Q(t)

Q(t)·α

Es(t)

G(t)

Ts(t)

Seasonal variations in the components of the energy balance

“Summer”

“Winter”

Q(t)·α

Q(t)

Es(t)

Ts(t)

G(t)

G(t)

Ts(t)

One-day-sum energy balance: Q(t)·(1-α) + G(t) = Es(t),

Ground heat fluxes increase “thermal inertia” of the ground surface

Fourier low of heat conduction:

qgr(t) = - K·gradT

K is thermal conductivity

for one-dimensional T field:

Surface energy balance: atmosphere and hydrosphere are present

- Greenhouse effect
- Snow, ice and vegetation are present – albedo changing in time and space
- Redistribution of solar energy by atmosphere and ocean circulation
- Ocean as a big-capacity energy storage
- Convective heat transfer in the land-atmosphere system

Surface energy balance: atmosphere and hydrosphere are present

Modifications to the surface energy balance:

- Incoming solar radiation

q qdir + qdiff (Q Qdir + Qdiff)

- Atmosphere itself now is a source of long-wave radiation: Ia( Ea)
- The ground surface is loosing energy not only in form of long-wave radiation but also by convection:
- Sensible heat qH (H)
- Evaporation (evapotranspiration) qLE (LE)

Surface energy balance: atmosphere and hydrosphere are present

Sensible heat qH (H)

T1and T2– air temperature measured at two levels: z1and z2

ρand Ca – density and specific heat capacity of air

KHis the turbulent transfer coefficient, it depends on wind speed and on the roughness of the surface

Surface energy balance: atmosphere and hydrosphere are present

Latent heat of evaporation qLE (LE)

e1and e2are specific humidity at two levels: z1and z2

Lvis the latent heat of vaporization

Kvis the turbulent transfer coefficient for water vapour, it depends on the same parameters as KH

Also, process of evaporation depends on surface moisture conditions

Surface energy balance: atmosphere and hydrosphere are present

So, the energy balance for a real surface will be:

Surface energy balance: atmosphere and hydrosphere are present

Almost all components of the surface energy balance depends on the surface temperature Ts

Assume that we know all other input data (solar and long-wave incoming radiation, air temperature, humidity, …) and all necessary parameters in the above equations. Then we can find an unique Ts that will balance the energy balance equation for each given time interval.

Surface energy balance: atmosphere and hydrosphere are present

Almost all components of the surface energy balance depends on the surface temperature Ts

Assume that we know all other input data (solar and long-wave incoming radiation, air temperature, humidity, …) and all necessary parameters in the above equations. Then we can find an unique Ts that will balance the energy balance equation for each given time interval.

Not necessarily true

- Permafrost develops today where the net heat balance of the surface of the Earth is negative for several years
- Thermokarst depressions are usually filled with water

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