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.

• Cartesian 2-D model

sea land

x

z

• Equations of motion: shallow, anelastic approx., no friction

• BC: w(x,0,t) = 0

b = g0

• 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

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

• 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, non-dimensionalize it.

• New coordinates:

• We get:

Height (z) Distance (x) Time (t)

• 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

• 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?

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

• 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

.

• Case 2: f < ω

• Redifine xi and beta as follows:

• Equation to solve becomes

sunrise

noon

sunset

Flow concentrated along “rays” of internal-inertial waves

“Perverse” result:Land breeze during daytime, almost 180˚ out ofphase w/ heating

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)

• 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

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 Sunrise Noon

blue slides: John Nielsen-Gammon, TAMU

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 – 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

At 30N or 30S

• Diurnal heating cycle resonates with inertial oscillations

• Amplitude of response blows up

• Horizontal scale blows up

• Linear theory blows up