1. On the Linear Theory of the Land and Sea Breeze MPO 663 project, Falko Judt based on Rotunno, 1983 Sun and Orlanski, 1981

2. The Land Sea Breeze…. …. is much more than just and has been studied for quite some time (first quantitative study 1889)

3. Motivation Derive analytical, linear model for land-sea breeze (okay, it’ll get quite mathy) Incorporate rotational effects which are important and fundamentally determine behavior of flow Horizontal scale (How far does it push inland?)

4. Driving force is diurnal cycle of heating and cooling of land relative to sea Frequency ω ( 2π/day) 2 fundamental regimes –f > ω: “classic” flow pattern –f < ω: wave solutions, somewhat strange –f = ω (30˚ latitude) ?? → Singularity!! resonance problem

5. Atmosphere idealized as rotating, stratified fluid Characterized by parameters f (Coriolis parameter) and N (Brunt-Väisälä frequency) N,f = const. Cartesian 2-D model sealand x z

6. Equations of motion: shallow, anelastic approx., no friction BC: w(x,0,t) = 0 b = g

7. First it had been hypothesized that extent of sea breeze solely based on temp. difference But: there is a definite internal radius of deformation that determines horizontal scale

8. Let’s assume heating function Q(x,z,t) known. Eqs. (1) – (5) can be collapsed into single equation featuring a stream function

9. Forcing with period ω = 2π/day = 7.292 x 10 -5 s -1 gives us solutions of the form plugging these wave solutions into stream function equation yields

10. Now simplify N ≈ 10 -2 s -1, so N >> ω: We get: Forcing is gradient of heating!

11. Case 1: f > ω To get an easier handle on the problem, non- dimensionalize it. New coordinates: We get: Height (z)Distance (x)Time (t)

12. Equationwith point source heating can be solved, solution in physical space is: Ψ is constant on ellipses with the ratio of major to minor axis given by For increasing static stability N → flatter ellipse of this equation for ellipse Horizontal scale Vertical scale diurnal cycle

13. It can be shown that the intensity of the flow is inversely proportional to N → Explanation for weaker land breeze at night due to increased stability also shows the dilemma for f → ω

14. Now let’s make use of some more realistic heating Heating now H, not Q horizontal shapevertical decay

15. leads to the internal scale of motion. x 0 (scale of land-sea contrast) and z 0 (vertical extent) are specified externally take f = 10 -4 s -1, x 0 = 1000 m, z 0 = 500 m and λ H = 73 km just dependent on f, assuming N const.

16. How does the flow look like? http://www.atmos.ucla.edu/~fovell/H98/animations/seabreeze_rotunno_nlin.MOVhttp://www.atmos.ucla.edu/~fovell/H98/animations/seabreeze_rotunno_nlin.MOV/ at τ = π/2 (~noon) v (along coast) b p ψ u w

17. Through Bjerknes’ Circulation theorem following results can be obtained: 1.Circulation independent of x 0 (scale of land-sea contrast) 2.C independent of N (v ~ N -1, λ H ~ N) 3.C ~ (f 2 – ω 2 ) -1 -- Problematic for f → ω

18. Case 2: f < ω Redifine xi and beta as follows: Equation to solve becomes

19. sunrise noon sunset Flow concentrated along “rays” of internal-inertial waves “Perverse” result: Land breeze during daytime, almost 180˚ out of phase w/ heating

20. Distance from either side of coast influence can be felt

21. Example Yucatan Peninsula (22˚N): ω = 7.292 x 10 -5 s -1 f = 2 ω sin(22˚) = 5.463 s -1 N = 10 -2 s -1 h = 500 m 104 km

23. Role of friction According to Circulation Theorem, circulation wave leads temperature wave by 90˚ (max of circulation for max heating, not at sunset (max temperature)) Observations: Max circulation around mid-afternoon Friction leads to more realistic phase lags (for both Case 1 and Case 2); also takes care of singularity (f = ω)

24. Enhanced friction (α) bring phase lags at different latitudes into line phase lag ~ 40˚ → observations phase lag for f = ω phase lag for f = 0 phase lag for f = 10 -4 s -1 phase lags circulation - heating phase lag heating - temp

25. Summary Two fundamentally different solutions for f > ω and f < ω: Elliptic flow pattern vs. internal-inertia waves Internal radius of deformation which determines inland penetration (dependent on N and f) Friction necessary to explain “natural” behavior of flow in terms of phase lag (flow-heating) and singular latitude Observations seem to verify wave solution (Sun and Orlanski, 1981)

27. Questions? Comments? Complaints?

28. Inertial Oscillation at 30 N Wind Coriolis SundownMidnight SunriseNoon blue slides: John Nielsen-Gammon, TAMU

29. Tropical Sea Breeze Forces PGF Wind Coriolis SundownMidnight SunriseNoon

30. Tropical Sea Breeze Interpretation Inertial oscillation is too slow PGF and CF must be in phase to reinforce each other Wind oscillates at diurnal frequency

31. Midlatitude Sea Breeze Forces PGF Wind Coriolis SundownMidnight Sunrise Noon

32. Midlatitude Sea Breeze Interpretation Inertial oscillation is too fast PGF must be out of phase with CF to slow down inertial oscillation Wind oscillates at diurnal frequency

33. Alternative Midlatitude Sea Breeze Interpretation In midlatitudes, air tries to attain geostrophic balance Pressure gradient would be associated with alongshore geostrophic flow Onshore sea breeze is ageostrophic wind trying to produce alongshore geostrophic flow As if air is entering and exiting an alongshore jet streak

34. Another Alternative Midlatitude Sea Breeze Interpretation (thanks to Chris Davis) Sea breeze forcing is diabatic frontogenesis Frontogenesis produces a direct circulation Warm air rises, low-level air flows from cold to warm Intensity of circulation is proportional to the rate of change of the temperature gradient It really is governed by the Sawyer-Eliassen equation!

35. Magic Latitudes At any latitude, L = NH/ (f 2 –  2 ) 1/2 (f 2 –  2 ) 1/2 is normally of order 7x10 -5 For typical H and N, L = 150 km At 30+/- 1 degrees, (f 2 –  2 ) 1/2 is of order 2x10 -5 For typical H and N, L = 500 km

36. At 30N or 30S Diurnal heating cycle resonates with inertial oscillations Amplitude of response blows up Horizontal scale blows up Linear theory blows up