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University of California, Santa Barbara
Jeff Dozier, Mike Colee,
ESM 236: The Mountain Snowpack
Ned Bair, Karl Rittger
Final Exam, March 28, 2008
 1 
The Final Examination counts 100% of your course grade. Answer all three questions,
which are equally weighted. The exam is openbook and opennotes, but you should be
able to answer the questions without them. You do not have to repeat the question in your
answer; just use the number designations. Put your name on each page.
1.
Temperature and vapor pressure gradients in the upper part of the snowpack
The diffusion equation for heat transfer in a conducting medium is
2
2
, where
T
K
T
tC
z
temperature
time
thermal conductivity
density
specific heat
depth
T
t
K
C
z
This has an analytic solution when the medium is uniform and the temperature at the
surface is a periodic function, i.e.
0
(0, )
cos(
)
T
t
T
T
t
, where
2
P
and
P
is
the period (e.g., 86,400 seconds in a day). Under these restrictions, the equation for
temperature at any time and depth is
/
0
( , )
cos
zd
z
T z t
T
T e
t
d
2
where the damping parameter
and
is the thermal diffusivity
P
d
K
C
Using a mean temperature
o
10 C
T
and daily temperature variability
o
0
8C
T
(half amplitude), along with
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 Spring '08
 DOZIER

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