The Wine Cellar Problem

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The Wine Cellar Problem

• Geophysics most important contribution to the
  human race.

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The Situation
x
?
z
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Questions

• What is the temperature anomaly as function of
  time, depth and the Fourier transform of qs(t)?
• What constants determine the attenuation depth of
  the temperature anomaly?
• What is the attenuation depth of the periodic
  temperature variations due to the
• Diurnal cycle?
• Annual cycle?
• Glacial cycle?

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Assumptions

• The ground is a semi-infinite homogenous
  half-spaceso we use the 1-D, time dependent heat
  conducting equation
• Constant thermal properties (k, k)
• As z gt infinity the temperature T(z,t) gt To,
  where To is the average surface temperature
• which means were ignoring heat flux from the
  mantle, and we have no internal heat sources
• which essentially means the ground in question
  is an isolated body

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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

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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

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Still Deriving

• Because were 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

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Still Deriving

• And after separating out the oscillatory part

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But what about A?

• Apply the boundary condition at the surface
• If we sub T(z,t) into this bad boy we get

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And so our super final answer is
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Finally, compare T(z,t) with q(s)

• There is a difference of ?/4 between the
  oscillatory parts of these two functions

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.
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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

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Attenuation Depth

• and solve for z!
• zo is only dependent on k and w

• So re-write the temperature function

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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
• w 2pf
• Diurnal w 7.27x10-5 rad/sec
• Annual w 1.99x10-7 rad/sec
• Glacial w 1.99x10-12 rad/sec

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• 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
k (x10-6 m2/s) 0.18 0.24 1.43
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Diurnal Cycle

• Tiny attenuation depths!

Clay Soil Sandy Soil Rock
zo (meters) 0.07 0.08 0.20
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Diurnal Cycle
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Glacial Cycle

• Huge attenuation depths!

Clay Soil Sandy Soil Rock
zo (meters) 425 491 1199
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Annual Cycle

• Practical attenuation depths!

Clay Soil Sandy Soil Rock
zo (meters) 1.35 1.55 3.79
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Annual Cycle

• We selected a wine-bearing region with
  substantial temperature fluctuations
  Canandaguia, New York
• NEW YORK CITY?! Get a rope.
• Annual DT 18 kelvin
• Were assuming that the average surface
  temperature, To, is the optimum temperature for
  storing wine 55ºF.

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Canandaguia Clay Soil
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Canandaguia Sandy Soil
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Canandaguia Rock
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Cheers!