On the Linear Theory of the Land and Sea Breeze
description
Transcript of On the Linear Theory of the Land and Sea Breeze
![Page 1: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/1.jpg)
On the Linear Theory of the Land and Sea Breeze
MPO 663 project, Falko Judt
based on Rotunno, 1983
Sun and Orlanski, 1981
![Page 2: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/2.jpg)
The Land Sea Breeze….
…. is much more than just
and has been studied for quite some time (first quantitative study 1889)
![Page 3: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/3.jpg)
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?)
![Page 4: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/4.jpg)
• 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
![Page 5: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/5.jpg)
• Atmosphere idealized as rotating, stratified fluid
• Characterized by parameters f (Coriolis parameter) and N (Brunt-Väisälä frequency)
• N,f = const.
• Cartesian 2-D model
sea land
x
z
![Page 6: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/6.jpg)
• Equations of motion: shallow, anelastic approx., no friction
• BC: w(x,0,t) = 0
b = g0
![Page 7: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/7.jpg)
• 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
![Page 8: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/8.jpg)
• Let’s assume heating function Q(x,z,t) known.
• Eqs. (1) – (5) can be collapsed into single equation featuring a stream function
![Page 9: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/9.jpg)
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
![Page 10: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/10.jpg)
• Now simplify
N ≈ 10-2 s-1, so N >> ω:
We get:
Forcing is gradient of heating!
![Page 11: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/11.jpg)
• 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)
![Page 12: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/12.jpg)
• Equation with 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
![Page 13: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/13.jpg)
• 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
![Page 14: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/14.jpg)
• Now let’s make use of some more realistic heating
Heating now H, not Q
horizontal shape vertical decay
![Page 15: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/15.jpg)
leads to the internal scale of motion.
• x0 (scale of land-sea contrast) and z0
(vertical extent) are specified externally
• take f = 10-4 s-1, x0 = 1000 m, z0 = 500 m and
λH = 73 km just dependent
on f, assuming N const.
![Page 16: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/16.jpg)
How does the flow look like?
http://www.atmos.ucla.edu/~fovell/H98/animations/seabreeze_rotunno_nlin.MOV/
at τ = π/2 (~noon)
v (along
coast)
b
p
ψ
u
w
![Page 17: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/17.jpg)
• Through Bjerknes’ Circulation theorem
following results can be obtained:
1. Circulation independent of x0 (scale of land-sea contrast)
2. C independent of N (v ~ N-1, λH ~ N)
3. C ~ (f2 – ω2)-1 -- Problematic for f → ω
lduC
.
![Page 18: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/18.jpg)
• Case 2: f < ω
• Redifine xi and beta as follows:
• Equation to solve becomes
![Page 19: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/19.jpg)
sunrise
noon
sunset
Flow concentrated along “rays” of internal-inertial waves
“Perverse” result:Land breeze during daytime, almost 180˚ out ofphase w/ heating
![Page 20: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/20.jpg)
Distance from either side of coast influence can be felt
![Page 21: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/21.jpg)
• 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
![Page 22: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/22.jpg)
![Page 23: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/23.jpg)
• 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)
![Page 24: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/24.jpg)
• 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 lagscirculation - heating
phase lagheating - temp
![Page 25: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/25.jpg)
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)
![Page 26: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/26.jpg)
![Page 27: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/27.jpg)
Questions?
Comments?
Complaints?
![Page 28: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/28.jpg)
Inertial Oscillation at 30 N
Wind
Coriolis
Sundown Midnight Sunrise Noon
blue slides: John Nielsen-Gammon, TAMU
![Page 29: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/29.jpg)
Tropical Sea Breeze Forces
PGF
Wind
Coriolis
Sundown Midnight Sunrise Noon
![Page 30: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/30.jpg)
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
![Page 31: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/31.jpg)
Midlatitude Sea Breeze Forces
PGF
Wind
Coriolis
Sundown Midnight Sunrise Noon
![Page 32: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/32.jpg)
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
![Page 33: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/33.jpg)
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
![Page 34: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/34.jpg)
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!
![Page 35: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/35.jpg)
Magic Latitudes
• At any latitude, L = NH/ (f2 – 2)1/2
• (f2 – 2)1/2 is normally of order 7x10-5
• For typical H and N, L = 150 km
• At 30+/- 1 degrees, (f2 – 2)1/2 is of order 2x10-5
• For typical H and N, L = 500 km
![Page 36: On the Linear Theory of the Land and Sea Breeze](https://reader033.fdocuments.net/reader033/viewer/2022051115/56814889550346895db59eb0/html5/thumbnails/36.jpg)
At 30N or 30S
• Diurnal heating cycle resonates with inertial oscillations
• Amplitude of response blows up
• Horizontal scale blows up
• Linear theory blows up