1. The linear sea-breeze circulation and effect of Earth’s rotation
ATM 419/563
Spring 2017
Fovell
Reference: Rotunno (1983, J. Atmos. Sci.)

2. Rough course outline
Sea-breeze circulation & 2D simulation of effect of latitude on sea-breeze
Downslope windstorms & 2D flow over topography
Planetary boundary layer schemes & 1D column model simulations
Microphysics schemes & 2D squall line simulations
Cumulus parameterizations & real-data simulations of precipitating convection
Advanced model initialization, nesting
Addressing nonlinear instability and other troubleshooting
Verifying model simulations against surface and upper air observations
Stochastic perturbations and nudging
Final project

3. Rotunno (1983) Quasi-2D analytic linear model
Heating function specified over land
Becomes cooling function after sunset
Equinox conditions
Cross-shore flow u, along-shore v
Two crucial frequencies
Heating  = 2/day (period 24h)
Coriolis f = 2sin (inertial period 45˚N)
One special latitude… where f =  (30˚N)
The two Omegas are the same…

4. Heating function
Rotunno’s analytic model lacks boundary layer mixing and
diffusion so horizontal, vertical spreading built into function

5. Streamfunction 

6. Circulation C Integrate CCW as shown Take w ~ 0; utop ~ 0
Integrate from ±infinity,
from sfc to top of atmosphere

7. Rotunno’s analytic solution

8. Rotunno’s analytic solution
Generic form
Here, B = 0, A
Here, B = 0, A > 0, so
(f2 – w2) controls behavior
Behavior =
amplitude and timing of circulation max and
magnitude and timing of wind at coastline

9. Rotunno’s analytic solution
If f > w (poleward of 30˚) equation is elliptic
• sea-breeze circulation spatially confined
• circulation in phase with heating
• circulation, onshore flow strongest at noon
• circulation amplitude decreases poleward
If f < w (equatorward of 30˚) equation is hyperbolic
• sea-breeze circulation is extensive
• circulation, heating out of phase
• f = 0 onshore flow strongest at sunset
• f = 0 circulation strongest at midnight
Elliptic, just like del^2pi = F.
F < omega circ strongest at midnight but onshore flow strongest at SUNSET
Elliptic: magnetic field around bar magnet. Hyperbolic: rock-caused wave in infinite ocean; in theory, would propagate forever.
What’s changing the behavior is f rel to omega. “What is it about the Coriolis force that can confine the circulation?”

10. Rotunno’s analytic solution
If f = w (30˚N) equation is singular
• some friction or diffusion is needed
• circulation max at sunset
• onshore flow strongest at noon
I believe for f=omega, the equation produces a discontinuity that blows up in finite time. Friction/diffusion/damping keeps the solution from blowing up.
Rotunno says circ ampl increases as f approaches omega.
In reality, Rotunno’s first term is actually (N^2-omega^2), but he neglected omega relative to N.
F < omega circ strongest at midnight but onshore flow strongest at SUNSET

11. Summary Latitude Onshore max (coastline) Circulation max f > 30˚
Noon
f = 30˚
Sunset
f = 0˚
Midnight

12. f > w (poleward of 30˚) at noon
streamfunction u w
Note onshore flow
strongest at coastline
(x = 0);
this is day’s max
Stronger onshore than offshore flow, as saw in Part I.
coast

13. f < w (equatorward of 30˚) y at three times
sunrise
Note circulation max at noon
noon (reverse sign for midnight)
“windshield wiper” at noon
Coastline onshore max SUNSET
Circ max NOON
sunset
Note coastline onshore flow
max at sunset

14. Max |C| noon & midnight
Paradox?
Why is onshore max wind at sunset but circulation max at midnight/noon?
While wind speed at coast strongest at sunset/sunrise, the wind integrated along surface larger at midnight/noon
Know coast wind strongest at sunrise/sunset because vert grad of streamfunction largest in mag then.
Midnight is onshore, so MAX circ.

15. Effect of “linear friction”
Time of circulation maximum
midnight
sunset
noon
As friction increases, the tropical seabreeze circ max becomes much earlier, the 30N circ becomes slightly earlier, the poleward circ becomes later
friction coefficient
As friction increases, tropical circulation max becomes earlier,
poleward circulation max becomes later

16. Simulation with a linearized numerical model
(not WRF)

17. Solution strategy Model starts at sunrise (6 am) with no flow
Integrate for 1 day (too short – why?)
Model is linearized
Small amount of diffusion applied (acts somewhat like friction)
Rotunno’s diurnal heating function used
Heating max at noon, zero at sunset
Cooling at night, absolute max at midnight

18. Heating function vs. time
Model starts at sunrise - 12 hour day
Especially at night:is this realistic heating/cooling function?
Is this realistic?

19. 30˚N linear case (5 m2/s diffusion)
400 km x 3 km
subdomain depicted
3 km
sea
land
Shaded: vertical velocity; contoured: cross-shore velocity

20. 30˚N linear case Circulation max @ sunset (as expected) Note non-zero
24h…
should run
several days
to spin-up
x2000
See something wrong here? Flow at sunrise on second day ≠ initial flow. Need to run LONGER to see if and when it becomes stable w/r/t time.

21. 30˚N linear case Onshore flow max ~ 4pm Note non-calm wind @ 24h…
sunset
Onshore flow
max ~ 4pm
(sunset expected)
Note non-calm
24h…
should run
several days
to spin-up
(m/s)
A little earlier than expected from model, perhaps owing to some finite amt of friction

22. Two important points Numerical models require “spin-up” time
Numerical diffusion is nearly always unavoidable and can influence the results
Here, numerical diffusion is intentionally applied, via diffusion terms added to the equations like
In models like WRF, the scheme used to solve the equations (Runge-Kutta 3rd order) has implicit diffusion

23. Variation with latitude

24. Cross-shore flow and vertical motion at noon (on 1st day)
They look quite similar

25. Cross-shore flow and vertical motion at sunset (on 1st day)
Set t 144
Big diffs now. 60N circ pooping out, winds weak across sfc… elliptic… confined. 00N winds strong, extensive… hyperbolic… spreads out unbounded.
Keep in mind diurnal forcing is SAME at all three latitudes

26. Cross-shore flow and vertical motion at midnight (on 1st day)
Night… sinking over land at all lat… but at 00N still onshore!

27. Cross-shore near-surface wind at coastline (linear model, one day)
Equator - no offshore flow
30N strongest offshore flow
Aperiodicity in 24h indicates need run model longer for it to settle

28. Circulation vs. time • Circ magnitude decreases w/ latitude (expected)
• 30N circ max at
sunset (expected)
• Poleward circ
max later than
expected (noon)
• Equator circ max
earlier than
expected (midnight)
• Consistent w/
existence of some
friction?
midnight Eq
x2000
30N
60N
90N
sunset

29. Recall: linear friction effect
Time of circulation maximum
midnight
sunset
noon
As friction increases, the tropical seabreeze circ max becomes much earlier, the 30N circ becomes slightly earlier, the poleward circ becomes later
friction coefficient
As friction increases, tropical circulation max shifts earlier,
poleward circulation max becomes later

30. t (h after sunrise) x (across-coast) Hovmoller diagrams
> 30N behavior is ELLIPTIC, closed, confined
< 30N behavior is HYPERBOLIC, spreads w/o end
AT 30N, Rotunno’s equation cannot be solved, but we see a transitional structure. Confined but spreading. Influence of the parabolic diffusion term?
>30˚ behavior is elliptic (closed,
confined)
<30˚ behavior is hyperbolic (spreads
without end)
=30˚ behavior is transitional
=0˚ note no land breeze forms

31. Something is missing in the linearized model. What is it?
Next: simulate the 2D sea-breeze using a nonlinear model (WRF)
Sea-breeze does not propagate inland!