1. Tor Håkon Sivertsen Bioforsk Plant Health and Plant Protection Hogskoleveien 7, N ‑ 1432 Aas (Norway); e-mail:tor.sivertsen@bioforsk.no Discussing the concept of wave energy of stationary gravity waves in the atmosphere

2. 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.

4. 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).

5. 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.

6. 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 p 0 (z) The density of the air ρ 0 (z)

7. 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 p 0 (z)+p(x,z) The density of the air ρ 0 (z)+ ρ(x,z)

9. 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. ν 0 is the Vaisala-Brunt frequency, and γ characterizes the compressibility

11. .

13. 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) = ν 0 2 U -2 - U zz U -1

14. By Fourier-transforming this equation we arrive at the following formula: F (w) z z -( l 2 - k 2 )F (w) =0 The solution of this equation for a constant Scorer-parameter is: F(w) =Ae iλz +Be - iλz

15. The vertical flux of wave energy is given in this manner:  pwdx= ½ ρ 0 ( 0)Uλ k – 1 (IAI 2 -IBI 2 ) The coefficient of reflection is defined in this manner: r= IBI 2 / IAI 2

16. In a layered atmosphere, each layer with a constant Scorer- parameter; 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’: w 1 = w 2 and w 1z = w 2z

17. But then they in fact sometimes violates their own main result that vertical the flux of wave energy is conserved through the atmosphere ò pwdx= ½ ρ 0 ( 0)Uλ k – 1 (IAI 2 - IBI 2 )=const When using the ordinary kinematic and dynamic boundary conditions you sometimes get other results than according to the electromagnetic analogy.

18. 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.