On the Linear Theory of the
Land and Sea Breeze
MPO 663 project, Falko Judt
based on Rotunno, 1983
Sun and Orlanski, 1981
The Land Sea Breeze….
…. is much more than just
and has been studied for quite some time (first
quantitative study 1889)
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?)
• 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
• Atmosphere idealized as rotating, stratified
fluid
• Characterized by parameters f (Coriolis
parameter) and N (Brunt-Väisälä
frequency)
• N,f = const.
z
• Cartesian 2-D model
x
sea
land
• Equations of motion: shallow, anelastic
approx., no friction
b=g

0
• BC:
w(x,0,t) = 0
• 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
• Let’s assume heating function Q(x,z,t) known.
• Eqs. (1) – (5) can be collapsed into single
equation featuring a stream function

gives us solutions of the form
Forcing with period ω = 2π/day = 7.292 x 10-5 s-1
plugging these wave solutions into stream function equation yields
• Now simplify
N ≈ 10-2 s-1, so
N >> ω:
We get:
Forcing is gradient of heating!
• Case 1: f > ω
• To get an easier handle on the problem, nondimensionalize it.
• New coordinates:
Height (z)
• We get:
Distance (x)
Time (t)
• Equation
with point source heating
can be solved, solution in physical space is:
of this
equation for
ellipse
diurnal cycle
• Ψ is constant on ellipses with the ratio of major to minor
axis given by
Horizontal scale
Vertical scale
• For increasing static stability N → flatter ellipse
• 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
• Now let’s make use of some more realistic
heating
Heating now H, not Q
horizontal shape
vertical decay
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.
How does the flow look like?
at τ = π/2 (~noon)
ψ
v (along
coast)
u
b
w
p
http://www.atmos.ucla.edu/~fovell/H98/animations/seabreeze_rotunno_nlin.MOV/
•
Through Bjerknes’ Circulation theorem
 
C   u.dl
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
• Case 2: f < ω
• Redifine xi and beta as follows:
• Equation to solve becomes
sunrise
Flow concentrated along
“rays” of
internal-inertial waves
noon
“Perverse” result:
Land breeze during
daytime, almost 180˚ out of
phase w/ heating
sunset
Distance from either side of coast influence can be felt
• 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
• 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)
phase lags
circulation heating
phase lag for f = 0
phase lag
heating - temp
phase lag for f = 10-4 s-1
phase lag for f = ω
• Enhanced friction (α) bring phase lags at
different latitudes into line
• phase lag ~ 40˚ → observations
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)
Questions?
Comments?
Complaints?
Inertial Oscillation at 30 N
Wind
Coriolis
Sundown
Midnight
blue slides: John Nielsen-Gammon, TAMU
Sunrise
Noon
Tropical Sea Breeze Forces
PGF
Wind
Coriolis
Sundown
Midnight
Sunrise
Noon
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
Midlatitude Sea Breeze Forces
PGF
Wind
Coriolis
Sundown
Midnight
Sunrise
Noon
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
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
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!
Magic Latitudes
• At any latitude, L = NH/ (f2 – w2)1/2
• (f2 – w2)1/2 is normally of order 7x10-5
• For typical H and N, L = 150 km
• At 30+/- 1 degrees, (f2 – w2)1/2 is of order
2x10-5
• For typical H and N, L = 500 km
At 30N or 30S
• Diurnal heating cycle resonates with inertial
oscillations
• Amplitude of response blows up
• Horizontal scale blows up
• Linear theory blows up