1. Gravity waves derivation ATM 562 - Fall, 2015

2. Introduction Gravity waves describe how environment responds to disturbances, such as by oscillating parcels Goal: derive “dispersion relation” that relates frequency (period) of response to wavelength and stability Simplifications: make atmosphere 2D, calm, dry adiabatic, flat, non-rotating, constant density

3. Starting equations continuity equation

4. Expand total derivatives

5. Perturbation method Calm, hydrostatic, constant density environment (“basic state”)

6. Start w/ potential temperature used log tricks and: for basic state

7. Apply perturbation method Useful approximation for x small: Base state cancels... makes constants disappear... speed of sound

8. Vertical equation Do perturbation analysis, neglect products of perturbations Rearrange, replace density with potential temperature

9. Our first pendulum equation We now know an oscillating parcel will disturb its environment and p’ plays a crucial, non-negligible role

10. Full set of linearized equations

11. Obtaining the frequency equation #1 differentiate horizontal and vertical equations of motion subtract top equation from bottom

12. Obtaining the frequency equation #2 …where we differentiated w/r/t x. Since continuity implies then Since the potential temperature equation was then (differentiated twice w/r/t x, rearranged)

13. Final steps... differentiate w/r/t t again and plug in expressions from last slide Pendulum equation.... note only w left

14. Solving the pendulum equation We expect to find waves -- so we go looking for them! Waves are characterized by period P, horizontal wavelength L x and vertical wavelength L z Relate period and wavelength to frequency and wavenumber

15. A wave-like solution where... This is a combination of cosine and sine waves owing to Euler’s relations

16. Differentiating #1

17. now differentiate again

18. Differentiating #2 now differentiate twice with respect to time Do same w/r/t z, and the pieces assemble into a very simple equation

19. The dispersion equation A stable environment, disturbed by an oscillating parcel, possesses waves with frequency (period) depending the stability (N) and horizontal & vertical wavelengths (k, m) that can be determined by how the environment is perturbed

20. Wave phase speed Example: Wave horizontal wavelength 20 km vertical wavelength 10 km and stability N = 0.01/s

21. Wave phase speed Note as you make the environment more stable waves move faster. Note that TWO oppositely propagating waves are produced.

22. Add a mean wind… intrinsic frequency flow-relative phase speed ground-relative phase speed

23. Wave phase tilt Wave tilt with height depends on intrinsic forcing frequency and stability (e.g., smaller ω … smaller cos α …larger tilt)

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