2 geotop-summer-school2011

GEOtop: some of the dynamics

Segantini - Mezzogiono sulle Alpi

Riccardo Rigon, Stefano Endrizzi, Matteo Dall’Amico, Stephan Gruber

Wednesday, June 29, 2011

Yes, still the snow

...
What will be of the snow of the garden,
what will be of free will and of destiny
and of those who their way in the snow have lost suddenly
....

Andrea Zanzotto (La beltà, 1968)


Energy and Snow Budgets

Objectives

•Talking about the mass an energy equations of snow
•And especially the snowpack evolution

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The control volume

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Mass, Energy and Entropy of Snow

There are various layers

Snow

Unsaturated soil

Water table

For the moment we take care of the snow layers
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A snow model
As input it has precipitation and meteorological data (temperature,
relative humidity, pressure and windspeed at the ground)

These are parametrized boundary conditions

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A snow model

It also parameterizes atmospheric radiation and its components, and
turbulence.

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A snow model:
the real dynamics

Is in the transfer of fluxes (the internal layers) 8

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Snow Budgets

The snow model

The evolution of the water equivalent of snow is obtained from the mass balance
equation: solid and liquid precipitation less the water flows due to melting and
sublimation is equal to the variation in the water equivalent of the snow

mass of snow

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Snow Budgets

The snow model


mass of ice
mass of snow

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Snow Budgets

The snow model


mass of ice
mass of snow mass of liquid water

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Snow Budgets

The snow model


mass of air
mass of ice
mass of snow mass of liquid water

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Snow Budgets

The snow model


flux of liquid water
phase transition

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Snow Budgets

The snow model


phase transition
variation of mass per unit time

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Snow Budgets

The snow model


phase transition

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Snow Budgets

The snow model


phase transition

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Snow Budgets

Mass Balance

As in any budget, a surface layer must be implemented to set up boundary
conditions, and an internal layer to account for water transfer inside snow

Snow surface layer

Snow internal layers

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Snow Budgets

Mass Balance of the surface layer


Neve surface layer

mass conservation of snow
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Snow Budgets

The snow model


Snow surface layer

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Snow Budgets

The snow model


Snow surface layer

total precipitation
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Snow Budgets

The snow model


evaporation/sublimation
Snow surface layer

total precipitation
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Snow Budgets

The snow model


evaporation/sublimation
Snow surface layer

percolation

total precipitation
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Snow Budgets

The snow model

is considered negligible. Then

Or, after dividing by the liquid water density and reference volume:

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Darcian flow

The snow model

where kw and μw are the intrinsic permeability of the snow to liquid water
(m2) and the dynamic viscosity of liquid water (kg m−1 s−1)

As normally, in a snowpack, capillary forces are two or three orders of
magnitude less than those of gravity, the capillary pressure gradient can be
neglected

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Darcian flow

The snow model

Colbeck (1972) related kl and ks to the effective water saturation (S)
by means of this expression (Brooks and Corey, 1964):

where S is defined as

So:

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Darcian flow

The snow model

The intrinsic permeability of snow at saturation is a function of many physical
properties of a snow cover, including its density and grain size, and the
distribution, continuity, size, shapes and number of its pores (Male and Gray,
1981). Shimizu (1970) proposed the following relationship:

where d is the grain diameter (m), which is normally in the range of
0.04-0.2 mm for new snow, 0.2-0.6 mm for fine-grained older snow and
2.0-3.0 mm for older wet snow (Jordan, 1991) ).

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Energy budgets

The energy balance of snow at the surface

dU∗
= Rn lw + Rn sw − H − λ s E v + G + Pe
dt

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Energy budgets


dU∗
dt

Radiation budget

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Energy budgets


dU∗
dt

Radiation budget

∆R(i, t) = (1 − α∗ ) R↓sw (i, t) + R↓lw (i, Ta (t)) − R↑lw (i, Ts (t))

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Energy budgets

The energy balance of snow

dU∗
dt

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Energy budgets


dU∗
dt

Variation in energy

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Energy budgets


dU∗
dt

Variation in energy
Radiation balance

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Energy budgets


dU∗
dt

Variation in energy
Radiation balance Energy transfers due to
turbulent fluxes

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Energy budgets


dU∗
dt

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Energy budgets


dU∗
dt

Variation in energy

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Energy budgets


dU∗
dt

Variation in energy
Radiation balance

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Energy budgets


dU∗
dt

Variation in energy
turbulent fluxes

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Energy budgets


dU∗
dt

Variation in energy
turbulent fluxes

Conduction of heat
towards the ground

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Energy budgets


dU∗
dt

Variation in energy
turbulent fluxes

Conduction of heat
towards the ground

Energy brought from
precipitation
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Energy budgets


H∗
The flow of sensible heat depends on the surface temperature, it being
proportional to the temperature gradient between the surface and the height at
which the sensor is measuring the air temperature.

The coefficient of proportionality is greater when there is more turbulence.
Therefore, the coefficient is reduced in the presence of thermal stratification and
increased in conditions of de-stratification.

It is calculated by applying the similarity theory of Monin-Obukhov, which,
however, is only strictly valid in flat terrains and quasi-stationary atmospheric
conditions.
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Energy budgets


λET

Similarly, the latent heat flux depends on the specific humidity at the interface
between snow and atmosphere (by assuming saturated conditions the specific
humidity is a function solely of the surface temperature) in that it is
proportional to the humidity gradient between the surface and the height at
which the sensor is measuring the air humidity.

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Energy budgets

The Snow Energy Budget in the internal
layers

variation of the energy of snow

energy fluxes
at the boundary
phase transition

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Energy budgets

The Snow Energy Budget

energy of snow

energy fluxes
at the boundary

phase transition

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Energy budgets


energy of snow

energy fluxes
at the boundary

phase transition

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Energy budgets


heating/cooling by conduction

heating/cooling by advection
(mainly of liquid water)

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Energy budgets

The Snow Internal Energy

variation on the
energy of snow

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Energy budgets


variation on the
energy of snow

A part depends on temperature

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Energy budgets


variation on the A part depends on the
energy of snow substance

A part depends on temperature

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Energy budgets

You can believe me that the energy has
the previous form.

Or try to get it by yourself from the
basic definitions ;-)

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Energy budgets

You can believe me that the energy has
the previous form.

Or try to get it by yourself from the
basic definitions ;-)

Then you get in troubles!

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Energy budgets revisited

How does it relates with

?

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In fact, the formula takes a different
route

through the definition of entalphy

which is an equivalent of the energy (for details, Dall’Amico, 2010),
and

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And the thing complicates a little more

if you take the time variation of it:

just because of the Gibbs-Duhem identity

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Finally

One discovers that hentalphy can be approximated as a function of
temperature (and pressure actually) as:

where the derivative of hentalphy is used as quite often that has the name
of thermal capacity (at constant pressure)

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We are not there but let’s stop this story for a moment

and look at the other terms of the energy budget. The heating/
cooling by conduction

heating or
cooling or
the “heat
flux”

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temperature
gradient

heating or
cooling or
the “heat
flux”

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temperature
gradient

This is Osanger’s
theory that brings to
heating or Fourier’s law!
cooling or
the “heat
flux”

thermal
conductivity

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Metamorphisms

Snowpack dynamics at mid and higher
latitudes
Melting
Accumulation Maturation
Melting

Snow water equivalent
Runoff

Temperature

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Metamorphisms

Accumulation period

Translated in terms of energy balance. For T < 0 ºC, at the top layer

Variation of the
internal energy
of the snow

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Metamorphisms

Accumulation period

Translated in terms of energy balance. For T < 0 ºC, at the top layer

in the other layers

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Metamorphisms

Melting of the snowpack

The accumulation phase is followed by the snow melting phase.

At the beginning of the snow melting phase the snowpack is generally
made up of layers of varying density. The melting process is obviously
linked to the radiative input.
However, given the elevated albedo of snow, the direct importance of
radiation can be of limited importance.

While melting, the density of the snowpack increases and the vertical
variation tends to disappear. During the melting process the density can
fluctuate on an hourly and daily basis.

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Metamorphisms


Schematically, three phases of the melting period are distinguished:

•heating
•maturation
•flow generation

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Metamorphisms

Snowpack dynamics at mid and higher
latitudes
Melting
Accumulation Maturation
Melting

Snow water equivalent
Runoff

Temperature

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Metamorphisms

the maturation phase (T = 0 ºC)

The maturation phase of the melting process occurs when the snowpack is
an isotherm at T = 0 ºC.

From this point on, any further increase in energy produces meltwater,
which is initially trapped in the pores by surface tension.

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Metamorphisms


The snowpack does not advance linearly through these three phases: rather it
follows the daily temperature trends, and typically the melting takes place at
the surface layers in contact with the warm air.

The water then percolates downwards and recondenses, releasing latent heat,
and so contributes to raising the temperature of the snowpack.

During the night the melting snow can refreeze and so the process can carry on
for various days in a row.

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Metamorphisms


Which of the two phases exists is solely a function of pressure and
temperature, and it depends on the chemical potential of water and ice.

The phase that is present is (with very high probability) the phase with
the lower chemical potential: this is a consequence of the first and second
laws of thermodynamics.

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Metamorphisms

The equivalence of chemical potentials

µi (T, p) = µw (T, p)
identifies, in the (T,p) plane, the separation curve between phases (solid and
liquid) which is given by a Clausius-Clapeyron relationship

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Metamorphisms


Furthermore, there remains the equilibrium case, which is not well defined by
thermodynamics, when:

•T = 0 ºC

at this temperature (with p ~ 105 Pa), according to the scholastic view, phase
change occurs. This means that at this temperature both phases can co-
exist in arbitrary proportions.

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Metamorphisms


Let us suppose, however, that the temperature of the system with which the
snow is in contact is slightly greater than zero. In these circumstances the
snow is:
• slightly heated
• transformed to water

the thermal energy supplied by the system is, during this process, stored
as internal potential energy of the water and the temperature of the
remaining snow stays:

•T = 0 ºC

until all of the snow has melted. Only after this can the temperature rise.
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Metamorphisms

Melting of the snowpack energywise

Let’s assume that the pressure is constant. Then:

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Metamorphisms

And

But T=0 at the phase transition. Then

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Metamorphisms

Which can be understood if

and therefore

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Metamorphisms

Furthermore

the difference if the entalphies of water and ice are definend to be the entalphy
of fusion of ice:

Usually the specific entalphy of ice is taken as a reference and to be null. So

And therefore:

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Metamorphisms

To sum up

Where we are now able to express the flux of advected energy in terms of the
entalphy (i.e. the internal energy at constant pressure) of water expendable in
the process.

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Metamorphisms

Equations to Solve

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Phase transitions complexities

Do we forgot something ?

Capillary forces in the snow cause however a fraction of liquid water
to be retained in the snowpack and to be prevented from draining
away. Colbeck (1972) defined the irreducible water saturation (sr)
as the minimum liquid level (expressed as a fraction of porosity) to
which a snow cover can be drained at the atmospheric pressure. In a
literary review, Kattelmann (1986) showed that the irreducible water
content is highly variable, ranging from 0 to 0.4, which corresponds
for the relative saturation to ranging from 0.014 and 0.069 for a
snow of density 250 kg m−3.

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Capillary water ?
It should be noted that once liquid water is present, in the form of
capillary water, it refreezes with difficulty because of freezing point
depression, which is due to the capillary forces (surface tension) that
alter the energy balance values that lead to an estimate of the chemical
potential.

free water

capillary water

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Solutes
A similar effect is observed when, for any reason, there are solutes present
in the water.

free water

water with solute

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Freezing point depression

It can be calculated by generalising the Clausius-Clapeyron equation.
Freezing point
Specific volume Specific volume
depression
and pressure and pressure
of the ice of the water

Freezing point

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Numerics

Numerics

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Numerics

Top Boundary Conditions Energy

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Numerics

Bottom Boundary Conditions Energy

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Numerics

Top and Bottom Boundary Conditions
Mass

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Parameters

Energy balance parameters

- the air temperature above which all precipitations are liquid (2 °C)

- the air temperature below which all precipitations are snow (0 °C)

- the radiative emissivity of snow, which is close to 1 (0.98)

- the water content that the snow can retain by capillary action, expressed as a
fraction of the porosity (0.05)

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Parameters


- the saturated hydraulic conductivity of snow (~ 5.55 kg/(m2*s))

- the surface thermal conductivity of snow (~ 5.55*10^-5 m/s)

- the depth of albedo extinction (50 mm water equivalent): the albedo is
calculated with an algorithm which is a function of the age of the snow.
However, when the snow cover is less than this value it is assumed that the
snow cover is not continuous, but rather distributed in zones. In these cases
the albedo that is used is calculated as the average of the albedo calculated on
the basis of the age of the snow and the albedo of the bare soil, which must be
considered as another parameter.

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Parameters


- the “roughness length” for temperature (0.05 m): the vertical temperature
profile in the atmosphere, in turbulent conditions, is logarithmic; therefore it is
necessary to define an altitude, said “roughness length”, so that the logarithmic
profile can be considered valid for altitudes greater than this length. The
roughness length is a function of the surface roughness. It can be demonstrated
that if this parameter diminishes then there is an increase in the proportionality
coefficient between the sensible and latent heat fluxes and their respective
gradients.

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Parameters


- the roughness length for the windspeed (0.5 m*): that described for the
temperature is also valid for windspeed. The two roughness lengths are
correlated: normally the windspeed roughness length is between 7 and 10 times
greater than the temperature roughness length

* the roughness length is effectively very high with respect to real physical conditions. In reality it should
be of the order of 0.0001 m, but the parameter serves to take account of the existence of a small but
important sub-layer of atmosphere where the dynamics are laminar.
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Parameters


- soil density (1600 kg/m3)

- the thickness of thermally active soil (0.4 m), that is, the thickness of
soil that, in the absence of snow, is subject to an appreciable daily
thermal excursion

- the heat capacity of the soil (890 J/(kg * K)), considered constant, but
in reality it is highly variable in function of soil properties

- the albedo of the soil not covered by snow (0.2), variable in function
of land use

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Parameters

Run it!

Parameter Initial Boundary
estimation conditions Conditions

Run the code!

Print the result
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Parameters

Thank you for your attention.

G.Ulrici - 2000 ?

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2 geotop-summer-school2011

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