Discussing the concept of wave
energy of stationary gravity
waves in the atmosphere
Tor Håkon Sivertsen
Bioforsk Plant Health and Plant Protection
Hogskoleveien 7, N-1432 Aas (Norway);
e-mail:tor.sivertsen@bioforsk.no
The influence of the topography on the movements of
the parcels of air of the atmosphere is an important
theme of dynamic meteorology. In a vertically
stratified atmosphere gravity waves or gravity
inertial waves is excited by mountains on different
spatial and temporal scales.
I am going to look closer at the concept of wave
energy of stationary gravity waves in a vertically
stratified atmosphere excited by mountain ridge with
a breadth of about 10 km.
We model this landscape and the physical situation in
a Cartesian coordinate system with the origin under
the top of the ridge, the x-axis normal to the ridge,
the y-axis parallel to the ridge. The height above the
x,y-plane is modeled by the z-axis.
We are using a macro-physical description of the
atmosphere. At a moment of time t and at a
coordinate point (x,y,z) we have got a parcel of air
described by the pressure p(x,y,z,t), the temperature
T(x,y,z,t), the density of the air ρ(x,y,z,t) and the
wind velocity v(x,y,z,t) .
Further more the parcels of air are considered not to
exchange heat with the surroundings by conduction
or radiation. Also the chemical content of each
parcel is considered not to change in time.
We then may consider the thermodynamic processes
of each parcel of air as adiabatic, and the individual
density of each parcel is merely a function of its
pressure.
We then consider a basic flow in a stratified atmosphere and the
perturbation of this basic flow by the mountain ridge, ending up
with a mathematical description of the basic flow:
The horizontal wind velocity U(z)
The pressure p0(z)
The density of the air ρ0(z)
Then we get the mathematical description of the perturbation of
the basic flow (only in the x,z-dimension):
The horizontal wind velocity U(z)+u(x,z)
The vertical wind velocity w(x,z)
The pressure p0(z)+p(x,z)
The density of the air ρ0(z)+ ρ(x,z)
The unit of the wave energy is : Joule m-3
The wave energy E is made up by a kinetic part and
an available potential and internal part according
to the definition. ν0 is the Vaisala-Brunt frequency,
and
γ
characterizes the compressibility
.
The system of linear equations for this stationary situation
is manipulated and we arrive at the following equation
of w ( the vertical wind velocity):
w zz+ w xx+l 2(z)w=0
l 2(z) is called the Scorer-parameter, and with good
approximation
we get:F
l 2(z) = ν0 2 U -2 - U zz U-1
By Fourier-transforming this equation we arrive at the
following formula:
F (w)z z -( l2- k2)F (w) =0
The solution of this equation for a constant Scorer-parameter
is:
F(w) =Ae iλz +Be - iλz
The vertical flux of wave energy is given in this
manner:
 pwdx=½ ρ0( 0)Uλ k– 1(IAI2-IBI2)
The coefficient of reflection is defined in this
manner:
r= IBI2/ IAI2
In a layered atmosphere, each layer with a constant Scorerparameter; the analytical solutions in each layer may be
linked by the knowledge of the boundary conditions between
each layer.
Eliassen and Palm invoke an analogy withelectromagnetism
when discussing reflection and transmission of energy. In
order
to maintain this analogy they use something called ‘pure
mathematical boundary conditions’:
w1= w2
and
w1z= w2z
But then they in fact sometimes violates their own main result
that
vertical the flux of wave energy is conserved through the
atmosphere
 pwdx=½ ρ0( 0)Uλ k– 1 (IAI2 - IBI2)=const
When using the ordinary kinematic and dynamic boundary
conditions you sometimes get other results than according to
the
electromagnetic analogy.
It is a very common to use concepts in one field of
science originally derived and defined in another
field of scientific knowledge. But sometimes it is
appropriate to invoke the basic system of testing,
defining parameters and measuring the parameter
values in actual situations to see the scope of the
concepts derived in this way.