Surface Energy Budget
Radiation
Sensible heat flux
Latent heat flux
Ground heat flux
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.
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.
u
t
z
 is called the dynamic viscosity and it is a property of
the fluid.
Generalizing in three dimensions
Only for people
interested in
the mathematical
approach
 ui u j 

tij  



x

x
i 
 j
i , j  x, y,z.
t xy is the x - componentof the shearing stress induced by
gradient of velocity in the y - direction.
tij is called viscous stress tensor
t 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  are
L=length
T=time
M=mass
MLT 1 L2T 1  LT 1 L1
MLT 1 L 2T 1
1 1


ML
T
1 1
LT L
So the units of  are kg m-1 s-1
The ratio v=/r is called kinematic viscosity (units m2 s-1).
 (kg m-1 s-1)
n (m2 s-1)
10-3
1. 10-6
Air
1.8 10-5
1.5 10-5
Alcohol
1.8 10-3
2.2 10-6
Glycerine
0.85 10-1
6.8 10-4
Mercury
1.56 10-3
1.2 10-7
Water
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.
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”
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.
He found that when
UD
 1900~ 2000
n
U  speed of the flow
D  diameterof the pipe
n  kinematicviscosity
The flow becomes turbulent
More in general, the ratio
Re 
UL
n
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
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.
For high Reynolds numbers the flow is turbulent
Turbulent flows are highly irregular, threedimensional, 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.
Physical meaning of the Reynolds number
Newton’s second Law
 

Du F

F  ma 

Dt m
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
Newton’s 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.
For example, for u (x component) of the wind vector (but
similar reasoning can be done for the others components)
Du
1 P 1  t xx t xy t xz 

 



Dt
r x r  x
y
z 
Du
Dt
Term of Inertia. If the others terms are zero,
the air parcel keeps moving at the same speed
1 P

r x
Pressure forces (we’re not interested at this
moment)
1  t xx t xy t xz 




r  x
y
z 
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
If U is a characteristics velocity of the flow, and L a
characteristic length, a characteristic time can be deduced
from U=L/T
Du U U 2
 
Dt T
L
Inertial
term
1  t xx t xy t xz 
U



  n 2
r  x
y
z 
L
Viscous term
The ratio between the inertial and the viscous term
give the relative importance of one respect to the other
U2
L  UL  Re
U
n
n 2
L
This is the
Reynolds number
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.
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).
Properties of turbulence
Irregularity or randomness
Highly sensitive to small
perturbations (changes in initial and
boundary conditions). Unpredictable.
Only statistical description can be
used.
Three dimensionality and rotationality
The velocity field in any turbulent flow is
three-dimensional, and highly rotational.
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
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
Mean and fluctuating variables
Wind velocity
It is useful to
split a variable
in “mean” and
“fluctuating”
part
time
Only averaged values of turbulent flows are predictable.
Instantaneous values are random.
For example for the three wind components
(u_> x, or West-East direction, v-> y or SouthNorth direction, w _> z or bottom-up direction)
u  u  u
v  v  v
w  w  w
Reynolds decomposition
where
u, v, w are the istantaneaus values
u , v , w are the mean part
u, v, w are the fluctuating part
In the analysis of observations the most common mean or
average used is the time average. For a generic variable f
T
1
f   f ( t )dt
T0
For micrometeorological observations, the averaging
time T ranges between 103 and 104.
In modeling, more used are spatial averages
1
f 
V
z  z y  y x  x
   f ( x , y , x )dx ,dy ,dz
z
y
x
with V  zyx
Average value over a grid cell
z
y
x
In some cases a spatial and time average is also
performed.
In micro- and meso- scale simulations, x and y range
between 102 and 103. z near the surface is between 10 and
102
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.
1 N
f   fi
N i 1
N  number of realizations
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.
So, by definition
f f  f
It is assumed that the average operator is such that the
average of the fluctuations is zero.
f0 f  f
For the product of two variables
fg   f  f  g  g 
fg  fg  f g  f g 
fg  f g
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
s
In the time t, the amount of
the quantity c passing through
ut
the surface s is
cuts
So the flux density ( t  1s , s  1m 2 ) is
F  uc
Splitting in mean and turbulent part and averaging
F  uc  u  uc  c  u c  uc
The term
uc
Is the the covariance, and represent the turbulent flux
density
For the momentum
tij turb  rui uj
These are called Reynolds stress, and they are
much larger in magnitude than the correspondent
viscous stress
 u
u j 
visc
i

tij
 

 x j xi 


The ratio is proportional to the Reynolds number
t ij turb
t ij visc
 Re
Turbulent mixing is much more efficient than viscous
mixing
Variances are the simplest measure of the
fluctuations
2u  u2 ,v2  v2 ,2u  w2
 are also called standard deviations. The ratio
of standard deviations over mean wind speed
are called turbulence intensities.
u
v
w
iu  , iv  , iw 
u
v
w
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.
By combining the variances, it is possible to
estimate the kinetic energy of the turbulent
part, or Turbulent Kinetic Energy (TKE)
per unit mass.

1 2
TKE  u  v2  w2
2

The kinetic energy of the flow is the sum of the
Mean Kinetic Energy (MKE) and the Turbulent
Kinetic Energy (TKE).
KE=MKE+TKE