Surface Energy Budget

1
Surface Energy Budget
The exchanges of heat, moisture and momentum
between the air and the ground surface depend on
the state of the atmosphere close to the surface.
Since the air is a fluid, some basic notions of
turbulence are needed to understand atmospheric
motions close to the surface.
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Viscous fluids
The viscosity causes an irreversible transfer of
momentum from the points where the velocity is
large to those where it is small. It is an
internal resistance of fluid to deformation
In 1D
The momentum flux density t (also called shearing
stress) due to viscosity is proportional to the
gradient of the velocities.
m is called the dynamic viscosity and it is a
property of the fluid.
3
Generalizing in three dimensions
Only for people interested in the mathematical
approach
4

• is a momentum flux density, so its dimensions
  should be those of momentum (M L T-1), per unit
  time (T-1) , per unit surface (L-2).
• As a consequence, the dimensions of m are

Llength Ttime Mmass
So the units of m are kg m-1 s-1 The ratio vm/r
is called kinematic viscosity (units m2 s-1).
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Fluid particle of a viscous fluid adhere to solid
surface. If the surface is at rest, the fluid
motion right at the surface must vanish (No slip
boundary condition).
Viscosity dissipates kinetic energy of the fluid
motion. Kinetic energy is converted to heat.
To maintain the motion, the energy has to be
continuously supplied externally, or converted
from potential energy, which exist in the form of
pressure and density gradients in the flow.
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Osborne Reynolds (English, 1842-1912) The
internal motion of water assumes one or other of
two broadly distinguishable forms either the
elements of the fluid follow one another along
lines of motion which lead in the most direct
manner to their destination, or they eddy about
in sinuous paths the most indirect possible
Laminar
Turbulent
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In which conditions the flow is laminar and in
which turbulent ?
Reynolds made an experiment the 22nd of February
1880 (at 2 pm).
He varied the speed of the flow, the density of
the water (by varying the water temperature), and
the diameter of the pipe. He used colorants in
the water to detect the transition from Laminar
to Turbulent.
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He found that when
The flow becomes turbulent
More in general, the ratio
with L typical length scale of the flow, is
called Reynolds number.
The critical value of the transition from laminar
to turbulent flow changes with the type of flow
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Reynolds number is very important in fluid
mechanics
The flow behaves in two different ways for low
and high Reynolds numbers.
For Low Reynolds numbers the flow is Laminar
A laminar flow is characterized by smooth,
orderly and slow motions. Streamlines are
parallel and adjacent layers (laminae) of fluid
slide past each other with little mixing and
transfer (only at molecular scale) of properties
across the layers. A small perturbation does not
increase with time. The flow is regular and
predictable.
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For high Reynolds numbers the flow is turbulent
Turbulent flows are highly irregular,
three-dimensional, rotational, and very diffusive
and dissipative. A small perturbation increases
with time.
They cannot be predicted exactly as function of
time and space. Only statistical averaged
variables can be predicted.
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Physical meaning of the Reynolds number
Newtons second Law
For a fluid parcel
It links the acceleration of a fluid parcel,
with the forces acting on the parcel.
Since momentum is conserved, you can think at the
Newtons second law as a budget equation. If the
sum (in vector sense) of all the forces acting on
a parcel is different than zero, there is a
change in momentum.
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For example, for u (x component) of the wind
vector (but similar reasoning can be done for the
others components)
Term of Inertia. If the others terms are zero,
the air parcel keeps moving at the same speed
Pressure forces (were not interested at this
moment)
Viscous forces. This can be seen also as budget
(input-output) of momentum fluxes induced by
viscosity
We focus only on the inertia and viscous terms
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If U is a characteristics velocity of the flow,
and L a characteristic length, a characteristic
time can be deduced from UL/T
Inertial term
Viscous term
The ratio between the inertial and the viscous
term give the relative importance of one respect
to the other
This is the Reynolds number
14
A high Reynolds number means that the inertial
terms in the equation of motion are far greater
than the viscous terms. However, viscosity cannot
be neglected, because of the no-slip boundary
condition at the interface.
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The value of the critical Reynolds number (for
the transition from laminar to turbulent flows)
varies a lot from one case to the other (between
103 and 105). However in the atmosphere typical
values are between 106 and 109.
Atmosphere is turbulent
Moreover, in the generation or damping of
turbulence in the atmosphere a very important
role is played by the buoyancy effects (we will
see later on).
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Properties of turbulence
Irregularity or randomness
Highly sensitive to small perturbations (changes
in initial and boundary conditions).
Unpredictable. Only statistical description can
be used.
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Three dimensionality and rotationality
The velocity field in any turbulent flow is
three-dimensional, and highly rotational.
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Diffusivity or ability to mix properties
Very efficient in diffuse momentum, heat and
mass. Macroscale diffusivity of turbulence is
usually many order of magnitude larger than the
molecular diffusivity.
This is one of the most important properties
concerning the applications. It is largely
responsible of the dispersion of pollutants
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Multiplicity of scales of motion and
dissipativeness
Turbulent flows are characterized by a wide range
of scales. The energy is transferred from the
mean flow, to the larger eddies. There is a
continuous transfer of energy from the largest
to the smallest eddies. Viscous dissipation of
energy occurs in the smallest eddies. In order to
maintain the turbulent motion energy must be
supplied continuously.
Energy
dissipated
20
Mean and fluctuating variables
It is useful to split a variable in mean and
fluctuating part
Wind velocity
time
Only averaged values of turbulent flows are
predictable. Instantaneous values are random.
21
For example for the three wind components (u_gt x,
or West-East direction, v-gt y or South-North
direction, w _gt z or bottom-up direction)
Reynolds decomposition
In the analysis of observations the most common
mean or average used is the time average. For a
generic variable f
For micrometeorological observations, the
averaging time T ranges between 103 and 104.
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In modeling, more used are spatial averages
Average value over a grid cell
Dz
Dy
Dx
In some cases a spatial and time average is also
performed.
In micro- and meso- scale simulations, Dx and Dy
range between 102 and 103. Dz near the surface is
between 10 and 102
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In laboratory and some modeling and theoretical
studies the ensemble average is used. This is an
arithmetical average over a very large
(approaching infinite) number of realization of
a variable, obtained by repeating the experiment
over and over again under the same general
conditions.
Nearly impossible to do in real atmospheric
conditions
Theoretically the three averages are equal only
for homogeneous and stationary flows. These
conditions are very difficult to satisfy in
micrometeorological applications. However, very
often it is expected that an approximate
correspondence between the results of the three
averages exists.
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So, by definition
It is assumed that the average operator is such
that the average of the fluctuations is zero.
For the product of two variables
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Co-variances or turbulent fluxes
Let consider a quantity c (per unit volume, ex
Mass per unit volume density). If it is only
transported, its flux density is a function of
the wind speed
Splitting in mean and turbulent part and averaging
The term
Is the the covariance, and represent the
turbulent flux density
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For the momentum
These are called Reynolds stress, and they are
much larger in magnitude than the correspondent
viscous stress
The ratio is proportional to the Reynolds number
Turbulent mixing is much more efficient than
viscous mixing
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Variances are the simplest measure of the
fluctuations
s are also called standard deviations. The ratio
of standard deviations over mean wind speed are
called turbulence intensities.
In the atmosphere, turbulence intensities are
less than 10 in the nocturnal boundary layer,
10-15 in a near-neutral surface layer, and
greater than 15 in a unstable and convective
boundary layers.
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By combining the variances, it is possible to
estimate the kinetic energy of the turbulent
part, or Turbulent Kinetic Energy (TKE) per unit
mass.
The kinetic energy of the flow is the sum of the
Mean Kinetic Energy (MKE) and the Turbulent
Kinetic Energy (TKE).
KEMKETKE