1. Richard Rotunno National Center for Atmospheric Research, USA Dynamic Mesoscale Mountain Meteorology Lecture 3: Mountain Waves

2. Topics Lecture 1 : Introduction, Concepts, Equations Lecture 2: Thermally Driven Circulations Lecture 3: Mountain Waves Lecture 4: Mountain Lee Vortices Lecture 5: Orographic Precipitation

3. Jane English Mt. Shasta Mountain Waves

4. Bob Houze Wave Clouds at Three Levels

5. Dane Gerneke Multiple Wavelength Wave Clouds

6. Rotor Cloud in lee of the Sierra Nevada R. Symons

7. Air Parcel Behavior in a Stable Atmosphere Potential Temperature z E A B

8. Mountain Waves under Stable Conditions

9. Back to Inviscid, Adiabatic Governing Equations Conservation of momentum energy mass

10. momentum Flow Past an Impermeable 2D Obstacle

11. momentum Vorticity Equation Lecture 2  2D version  Baroclinicity

12. 2D Eqns in Terms of Vorticity and Streamfunction vorticity energy Lecture 2 

13. vorticity energy Simplify

14. vorticity energy 2D Linear Theory of Internal Gravity Waves combine vorticity and energy eqns  Lighthill (1978, Chap. 4)

15. plane-wave solution relation of and to b  wavenumber vector phase velocity

16. Example phase motion

17. Internal Gravity Waves: Group Velocity Form wave energy eqn  For plane-wave solution wavelength average gives 

18. Internal Gravity Waves: Group Velocity phase velocity wavenumber vector In this example Lighthill (1978, Chaps. 3-4)

19. momentum Linear Theory of Mountain Waves

20. vorticity energy Linear Theory of Mountain Waves combine 

21. momentum Equation + Boundary Conditions Energy radiation or decay as

22. Example: Sinusoidal Hill Look for sol’n of the form  Gov’n eqn  b.c. 

23. Case 1  Sol’n  use radiation condition to indicate which sign to choose in the exponents

24. dispersion relation including U for steady sol’n must choose minus sign in dispersion relation and so

25. Doppler-shifted=Natural Freq

26. Case 2  Sol’n  use decay condition to indicate which sign to choose in the exponents

27. Doppler-shifted>Natural Freq No Waves

28. Steady-State Linear Theory: Isolated Hill Fourier Transform

29. Radiation Condition for Inverse Fourier Transform Sol’n Decay Condition for F.T. of B. C.  Fourier Transform Eqn 

31. Summary - Internal gravity waves are peculiar (upward energy but downward phase propagation) -These basic ideas from IGW theory are necessary for the intepretation of mountian waves - Literature is vast: more general situations such as U=U(z), N=N(z) have been studied to understand flow response in particular atmospheric conditions