the wine cellar problem
DESCRIPTION
The Wine Cellar Problem. Geophysics’ most important contribution to the human race. The Situation. x. ?. z. Questions. What is the temperature anomaly as function of time, depth and the Fourier transform of q s (t)? - PowerPoint PPT PresentationTRANSCRIPT
The Wine Cellar ProblemGeophysics’ most important contribution to
the human race.
The Situation
x
z ?
Questions1. What is the temperature anomaly as function of
time, depth and the Fourier transform of qs(t)?
2. What constants determine the attenuation depth of the temperature anomaly?
3. What is the attenuation depth of the periodic temperature variations due to the1. Diurnal cycle?2. Annual cycle?3. Glacial cycle?
Assumptions
1. The ground is a semi-infinite homogenous half-space…so we use the 1-D, time dependent heat conducting equation
2. Constant thermal properties (, k)3. As z —> infinity the temperature T(z,t) —> To,
where To is the average surface temperature…which means we’re ignoring heat flux from the mantle,
and we have no internal heat sources…which essentially means the ground in question is an
isolated body
Deriving the Temperature Anomaly
• If qs(t) is a periodic forcing function we can assume it is of the form: . So the differential equation at the surface becomes:
• Because the heat flux is periodic and the PDE is linear we can guess the solution has the form:
Deriving the Temperature Anomaly
• Substituting T(z,t) into the diffusion equation we get:
• Which reduces to a 2nd order linear ODE:
• Which has the well known general solution:
Still Deriving…• Because we’re interested in the exponential decay
with increasing depth, we let a = 0, then select the second term and plug f(z) back into T(z,t) to get:
Still Deriving…
• And after separating out the oscillatory part:
But what about A?• Apply the boundary condition at the surface:
• If we sub T(z,t) into this bad boy we get:
And so our super final answer is…
Finally, compare T(z,t) with q(s)
There is a difference of /4 between the oscillatory parts of these two functions:
k Tz
Foe it See the
extra /4?
…meaning that the temperature anomaly at any given depth will lag behind the surface fluctuation by 1/8 of the period of the fluctuation.
Attenuation Depth• The depth at which the temperature has
negligible fluctuation w.r.t. the surface temp. In other words: where do we put our cellar??
where
• Equate this to the temperature function
Attenuation Depth…
• …and solve for z!• zo is only dependent
on and • So re-write the temperature function…
So now what?
• We want to know how the attenuation depth will vary with time and soil conditions
• …so we chose three time scales to examine
= 2f
– Diurnal: = 7.27x10-5 rad/sec– Annual: = 1.99x10-7 rad/sec– Glacial: = 1.99x10-12 rad/sec
…and we chose three soil conditions to consider: Clay Soil, Sandy Soil, & Rock
Clay Soil Sandy Soil Rock
k(W/m2/k)
0.25 0.30 2.90
x10-6 m2/s)
0.18 0.24 1.43
Diurnal Cycle
• Tiny attenuation depths!
Clay Soil Sandy Soil Rock
zo
(meters)0.07 0.08 0.20
Diurnal Cycle
0 5 10 15 20 250
2
4
6
8
10
12
Sur
face
Tem
pera
ture
Flu
ctua
tion
(kel
vin)
Time (hour)
Forcing Temperature at Surface for Diurnal Fluctuations
Glacial Cycle
• Huge attenuation depths!
Clay Soil Sandy Soil Rock
zo
(meters)425 491 1199
Annual Cycle
• Practical attenuation depths!
Clay Soil Sandy Soil Rock
zo
(meters)1.35 1.55 3.79
Annual Cycle
• We selected a wine-bearing region with substantial temperature fluctuations: Canandaguia, New York
NEW YORK CITY?! Get a rope.• Annual T = 18 kelvin• We’re assuming that the average
surface temperature, To, is the optimum temperature for storing wine: 55ºF.
Canandaguia: Clay Soil
-4 -3 -2 -1 0 1 2 3 4-12
-10
-8
-6
-4
-2
0
Temperature Fluctuation, dT (kelvin)
Dep
th (M
eter
s)Canandaguia Temperature as a Function of Depth and Time: Clay Soil
Canandaguia: Sandy Soil
-4 -3 -2 -1 0 1 2 3 4-12
-10
-8
-6
-4
-2
0Canandaguia Temperature as a Function of Depth and Time: Sandy Soil
Temperature Fluctuation, dT (kelvin)
Dep
th (M
eter
s)
Canandaguia: Rock
-4 -3 -2 -1 0 1 2 3 4-40
-35
-30
-25
-20
-15
-10
-5
0Canandaguia Temperature as a Function of Depth and Time: Rock
Temperature Fluctuation, dT (kelvin)
Dep
th (M
eter
s)
Cheers!