1. The linear sea-breeze circulation and effect of Earth’s rotation (and Experiment #1)
ATM 419/563
Spring 2019
Fovell
Reference: Rotunno (1983, J. Atmos. Sci.)

2. Rotunno (1983) Quasi-2D analytic linear model
Written in terms of streamfunction y
Specified heating/cooling function over land
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…

3. Streamfunction 

4. Heating function Q Heating/cooling over land, none over sea.
Rotunno’s analytic model lacks boundary layer mixing and
diffusion so horizontal, vertical spreading “built into” function

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

6. Rotunno’s analytic solution
There’s no TIME derivative, but the solution will evolve in time if only because Q varies in time

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

8. 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 spatially unbounded
• 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?”

9. 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

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

11. f > w (poleward of 30˚) at noon
sea land
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

12. f < w (equatorward of 30˚) y at three times
sunrise
(land breeze)
Note circulation max at noon
noon (reverse sign for midnight)
“windshield wiper” at noon – the land breeze is breaking up, first at the coast
Coastline onshore max SUNSET
Circ max NOON
sunset
(sea breeze)
Note coastline onshore flow
max at sunset

13. 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.

14. 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

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

16. Solution strategy Model starts at sunrise (6 am) with no flow
“cold start” = unrealistic
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

17. 30˚N linear case (5 m2/s diffusion)
400 km x 3 km
subdomain depicted
3 km
sea
land
Expect onshore max SUNSET and circ max SUNSET from analytic
Shaded: vertical velocity; contoured: cross-shore velocity

18. 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.
6AM NOON PM MIDNIGHT AM

19. 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
6AM NOON PM MIDNIGHT AM

20. 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

21. Variation with latitude in the linearized model
Got to here 2019

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

23. 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

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

25. 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

26. 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

27. 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

28. t (h after sunrise) 60˚ 30˚ x (across-coast) Hovmoller diagrams 0˚
midnight
t (h after sunrise)
sunset
noon
60˚
30˚
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
unbounded)
=30˚ behavior is transitional
=0˚ note no land breeze forms 0˚

29. 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!

30. Running WRF on Snow: 2D sea-breeze test case and Experiment #1

31. 2D WRF idealized simulation
WRF-ARW (Advanced Research WRF) v (customized)
Domain 202 x 3 x 35 points (x, y, z)
Horizontal resolution 2000 m
Vertical levels are as specified in namelist.input
Simulation starts 9/21/06Z, runs 24 h
Latitude (specified in namelist.input) is 30˚N
Longitude is 0˚
06Z is 6 AM local (about sunrise)
Left ~1/3 of domain is sea, rest is land
Differences: sounding moisture removed, surface moisture fluxes forced to be zero, sea-surface and soil temperatures changed, diffusion altered, etc..
(We’re using a customized version of the 2D sea-breeze test case, so you need to link to my code)
Uses different sounding (no moisture) than examples sb_latXX examined earlier in this PPT
So the 30˚ case for this experiment will not be identical to the sb_lat30 we just examined (and sb_lat60, etc.)

32. Setting up the environment
$ lab
$ mkdir SB2D
$ cd SB2D
$ cp $LAB/SB2D/SETUP.tar .
$ tar –xvf SETUP.tar
$ make_all_links.csh
[This creates links to files WRF needs]
[Links lead to a customized version of WRF, specifically
for 2D idealized sea-breeze simulations]
This extracts 7 files:
make_all_links.csh
namelist.input
input_sounding
control_file
hov.gs, sb_plot4.gs
sb_plotuw.gs

33. Checking on new links $ ls -al wrf.exe $ ls -al ideal.exe
lrwxrwxrwx 1 atm419 atm419lab 47 Jan 18 12:39 wrf.exe ->
/network/rit/lab/atm419lab/SOFTWARE/WRFV371_ATM419_SB2/main/wrf.exe*
$ ls -al ideal.exe
lrwxrwxrwx 1 atm419 atm419lab 49 Jan 18 12:44 ideal.exe ->
/network/rit/lab/atm419lab/SOFTWARE/WRFV371_ATM419_SB2/main/ideal.exe
Many of our experiments will utilize make_all_links and control_file files
but these will often be customized for specific tasks and not transferrable among experiments...

34. Using srun
$ hostname [by default run on headnode] headnode.rit.albany.edu
$ srun -p snow hostname [runs on Snow cluster]
snow-30.rit.albany.edu
$ srun -p snow-nwp hostname [runs on Snow NWP queue]
snow-16.rit.albany.edu
My .cshrc has an allias “salloc1” for this.
Some information:

35. namelist.input contains
&sb
sb_lat = 30.
sb_u = 0. /
sb_lat sets the latitude (in degrees) and sb_u the
mean/initial cross-shore wind speed for the simulation.
Leave unchanged for now.
Namelist is set up to run [run_hours] for 24 h, outputting once
per hour [history_interval = 60]. Leave unchanged for now.
Also these will never be changed for this experiment:
e_we = 202 … number of grid points in horizontal
e_vert = 35 … number of levels in vertical

36. input_sounding contains
sfc pres (mb) sfc theta (K) sfc mixing ratio (g/kg)
[more lines]
height (m) theta (K) mix ratio
(g/kg)
You will not need to modify this file.

37. Initialize and run model
$ srun –p snow ideal.exe [initialize model]
wrf: SUCCESS COMPLETE IDEAL INIT
$ ls -al wrfinput_d01 [initialization file; see it’s there]
$ srun –p snow time wrf.exe [runs model. Takes about 6 min]
d _06:00:00 wrf: SUCCESS COMPLETE WRF
$ ncdump –h wrfout_d01_ _06:00:00 | more
[WRF creates NetCDF files.]
[this looks inside the model output file; see next slide]
[USE TAB COMPLETION!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!]
If you execute ideal.exe WITHOUT srun, you are running on headnode

38. We specified e_we = 202 e_vert = 35
netcdf wrfout_d01_ _06\:00\:00 {
dimensions:
Time = UNLIMITED ; // (25 currently)
DateStrLen = 19 ;
west_east = 201 ;
south_north = 2 ;
bottom_top = 34 ;
bottom_top_stag = 35 ;
soil_layers_stag = 5 ;
west_east_stag = 202 ;
south_north_stag = 3 ;
variables:
char Times(Time, DateStrLen) ;
float XLAT(Time, south_north, west_east) ;
XLAT:FieldType = 104 ;
XLAT:MemoryOrder = "XY " ;
XLAT:description = "LATITUDE, SOUTH IS NEGATIVE" ;
XLAT:units = "degree_north" ;
XLAT:stagger = "" ;
XLAT:coordinates = "XLONG XLAT" ;
float XLONG(Time, south_north, west_east) ;
XLONG:FieldType = 104 ;
XLONG:MemoryOrder = "XY " ;
[lots and lots and lots more…]
We specified
e_we = 202
e_vert = 35

39. WRF uses a horizontally and vertically staggered griddx
WRF uses a horizontally
and vertically staggered
grid
< vertical grid stagger
u = east-west velocity
w = vertical velocity
s = scalar field dz
One extra W point in
vertical, one extra U point
in horizontal directions
u(i,k), w(i,k) and s(i,k)
do not represent same physical
location!

40. Making GrADS output Warning: idealized GrADS output does not
$ w2g control_file calm_lat30
Gracefull STOP
$ gr2
ga-> open calm_lat30
ga-> hov.gs
Yes, “graceful” is misspelled.
Warning: idealized GrADS output does not
preserve time… it thinks it starts 1 Jan 00Z
w2g aliases to wrf_to_grads
gr2 aliases to grads -l

41. This plots using a WRF diagnostic quantity: the wind at
10 m above surface level. More details later...

42. Experiment #1
(In two parts)

43. Part 1: 48h dry simulations
Each student will be assigned a different latitude (see class handout; also posted on class webpage)
Run ideal.exe after you modify the namelist (see following slide).
Naming format: latXX.X, where XX.X = latitude in degrees with leading zero if needed and decimal (e.g., lat30.0, lat02.5, etc..)
Copy your ctl, dat files to /network/rit/lab/atm419lab/EXP01/
…and me when you are done
Student assigned 30N still has to rerun, as am asking for 48 h simulations

44. namelist.input contains
&time_control
run_days = 0,
run_hours = 48,
run_minutes = 0,
run_seconds = 0,
[more…]
&sb
sb_lat = 30.
sb_u = 0. /
Check these lines. Change sb_lat to your assigned latitude.
Run ideal.exe and wrf.exe, then w2g

45. Computing circulation
ga-> reset
ga-> set z 1
ga-> set x 1
ga-> set t 1 49
ga-> d sum(mag(u10,v10),x=1,x=201)*2000
This is using a WRF diagnostic quantity: the wind at
10 m above ground level.

46. Experiment tasks Parts 1 and 2
Part 1 (for each latitude value assigned):
Run your case for 48 h and copy your GrADS files into the EXP01 directory.
For x = 200, plot a time series of soil temperature for your simulation at the five depths. Label the plot (i.e., put on axes and title). [See intro PPT.]
Compute circulation (see previous slide) for 5 selected cases in the EXP01 directory, and superimpose time series on a single plot. Label the plot with which latitudes you selected.
Part 2 (for each latitude value assigned):
For your assigned latitude(s), explore different initial U wind strengths (sb_u in namelist.input) to find the value that “traps” the sea breeze front at the coastline (see next slide). YOU MUST RERUN ideal.exe. Run model for 24 h only.
Plot a Hovmoller diagram of 10-m wind (U10) over that 24 h period. Save it as a PNG file. You can use hov.gs. it to me, along with the U wind value that traps the front. Do not copy your GrADS files to the EXP01 directory.
Summary: Undergrads are ing me three plots and one number. Graduate students are ing me six plots and 2 numbers.
Second part of Part 1 cannot be completed until everyone has run their simulations and put them in the EXP01 directory

47. namelist.input for Part 2
&time_control
run_days = 0,
run_hours = 24,
run_minutes = 0,
run_seconds = 0,
[more…]
&sb
sb_lat = 30.
sb_u = 0. /
To “trap” the sea-breeze front at the coastline, experiment
with different values of sb_u. Positive values are onshore,
so you want negative ones. Start with values like -6 m/s
and adjust as needed

48. Notes If you make a change to namelist.input, run ideal.exe again
Rerunning ideal.exe isn’t necessary after every change, but this is the safe strategy
This code is designed to use the slab land surface scheme (sf_surface_physics = 1). Surface moisture fluxes in that scheme are forced to be zero. Other schemes will add vapor to the atmosphere.
Handling of the soil model is not optimal. The soil temperatures should be permitted spin-up time.