ATMO 551a
Homework 4
1. Sensible heat flux
Determine the surface sensible heat flux by calculating the heating of the planetary
boundary layer (PBL)
Below are three figures. The first two show a recent radiosonde profile at 5 am and 5 pm in
Tucson. The third figure shows the two figures superimposed on top of one another to
show the day versus night difference in the PBL thermal structure.
a. Based on the difference between these two profiles, how deep is the convective
boundary layer (measured relative to the surface)? In pressure and in altitude?
b. Based on the difference between the radiosonde temperature profiles in the
afternoon and morning, and assuming all of this difference is due to sensible heat
flux from the surface into the atmosphere, what is the average sensible heat flux
from sunrise to the late afternoon?
PBLtop


PBLtop
 C p Tafternoon  Tmorning dz
FSH 

surf
t afternoon  t morning
SHOW ALL WORK


PBLtop
  Tafternoon  Tmorning dz
 Cp
surf
t afternoon  t morning
1

Cp
g
 T
afternoon

 Tmorning dP
surf
t afternoon  t morning
ATMO 551a
Homework 4
For simplicity, assume the temperature difference increases linearly with pressure
So dT = a dP
PBLtop
FSH  
Cp
g
 T
day

Psurf
 Tnight dP
surf
t day  t night

Cp
 aPdP
PPBLtop
g t day  t night


2
2
C p a Psurf  PPBLtop
g

2 t day  t night


because Tday-Tnit at the top of the PBL is 0 and it increases approximately linearly down to
the surface. The answer is around 200 W/m2.
compare this to the solar flux
Now
The solar flux at 7:23 AM is about 50 W/m2. The air temperature increases
Below is the temperature and dew point over the same day.
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2
ATMO 551a
Homework 4
2. Nocturnal Boundary Layer Thickness
Looking at the morning radiosonde profile and the surface temperature versus time,
the surface temperature increased very quickly around 7 AM. Based on the rapid change in
the surface temperature shortly after sunrise centered approximately on 7:23 AM in the
figure, what is the depth of the nocturnal boundary layer? To keep things simple, assume
the thermal inversion the atmospheric temperature at sunrise increases linearly with
altitude up to the depth of the nocturnal boundary layer.
PBLtop
 T
end
FSH  C p

 Tbegin dz
surf
t end  t begin
Assuming the nocturnal BL temperature increases linearly with altitude, then
Tend  Tbegin Z NBL
FSH  C p 

t end  t begin 2




Assume the solar flux is converted to a sensible heat flux. FSH = Fsolar.
t end  t begin
F
Z NBL  2 solar

C p  Tend  Tbegin




The increase is from 46°F to 57°F for a change of 11 F or about 6 C over a period of about
30 minutes or 2000 seconds. The rate of surface temperature increase is therefore about
3/1000 = 0.003 C/sec.

The height of the nocturnal boundary layer is therefore about 30 m.
Given the wind, use the aerodynamic form of the surface flux to determine the approximate
drag coefficient.
FSH   a c P w' '  K H  a c P
w' '  K H

   a c p C H U  (z)   (0)
z


 z U C H
z
z
(8)
(9)

Using a representative value of the drag coefficient from the table below, determine (z) –
(0).

SHOW ALL WORK
3
ATMO 551a
Homework 4
Foreman and Emeis (2010), Revisiting the Definition of the Drag Coefficient in the Marine Atmospheric Boundary
Layer, p. 2325, DOI: 10.1175/2010JPO4420.1
What is the eddy diffusivity?
Assuming the velocity of the eddies is equal to the horizontal wind speed (which is
probably an over estimate), what is the scale size of the eddies?
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4
ATMO 551a
Homework 4
The diffusive molecular form:
Z
u Lv t
F  C Tsurf  Tair dbl  C Tsurf  Tair
 C Tsurf  Tair
p
p
p
t dbl
L u
From this we can estimate L

2
C p Tsurf  Tair 
L  

 uv t 
F











uv t 
L

Why is the dew point more variable in the daytime than at the night?

What does this tell us about the eddies?
Calculate the evaporative flux as this air passes over a lake
What is the radiative flux out of the surface
What is the sensible heat flux into the surface over night?
Energy balance
What is the IR flux out of the surface at sunrise?
What is the IR flux out of the surface in the afternoon.
Assuming the downward IR comes approximately from an altitude of 3 km (global average
is ~2 km but Tucson is quite dry so the downward IR comes from higher altitudes), what is
the approximate downward IR flux in the morning and in the afternoon?
In = F solar + Fir down
Out = Fir up + FSH.
1. Diffusion scaling:
The time to cook a hard-boiled egg is ~12 minutes. Based on your understanding of
diffusion (see eq. (16) of the diffusion lecture), approximately how long should it take for a
watermelon to cool down to the refrigerator temperature?
2. Increase in the surface evaporative flux with global warming
Suppose the Earth’s surface were to warm by 2oC while the relative humidity of the air
and the winds were to remain the same. Using the aerodynamic formula for latent heat
flux, determine the ratio of the new surface latent heat flux to the present surface latent
heat flux.
Compare the increases in the upward surface radiative flux and surface latent heat flux
in terms of W/m2. Which increase is larger?
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5
ATMO 551a
Homework 4
Why is this important to how much surface temperatures will increase as the
downward IR flux from the atmosphere into the surface increases as GHG
concentrations increase?
3. Near surface sensible heat flux
Consider a citrus tree grove where the trees are about 5 m in height and the wind at 10
m is blowing at 2 m/sec with a surface temperature of 0oC and pressure of 1000 mb.
a. Calculate an approximate eddy diffusivity
Assume the temperature at 10 m is 2oC warmer than the surface temperature
b. Calculate the vertical sensible heat flux
c. Is it up or down?
d. How large is this in comparison with the IR radiative flux from the surface? Is
it sufficient to keep the trees from freezing overnight?
Diffusive boundary layer flux
Use the diffusive flux
Calculate the flux across a puddle assuming a 2 m/s wind
Do the same calculation using the three forms of the evaporative flux equations
Diffusive flux right as the boundary layer forms versus at the far end
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6