Assignment 3: EOSC 352

Assignment 3: EOSC 352
due Friday, November 5th, 2010
1. So far, we have looked at heat conduction problems without advection. However,
a more general form of the heat equation is
∂T
ρc p
∂t + ρc pu ∂T
∂x − k ∂2 T
= 0. (1)
∂x2 where u is the velocity at which material travels in the x-direction. A setting in
which u is relevant is near the surface of a glacier or ice sheet: snowfall accumulates
at the surface at a rate u. Deformation of ice deeper down in the glacier
or ice sheet then moves ice out from under the surface, keeping the elevation of
the surface fixed even though snow passes through it over time (figure 1). Suppose
we can treat u as constant, and that we have again an imposed temperature
oscillation at the surface x = 0 of the form
T (0, t) = T 0 cos(ωt).
and that heat flux dies off far below the surface,
−k ∂T
∂x
(a) Assume again that we can write
→ 0 as x → ∞. (2)
T = Re[T 0 exp(iωt + λx)].
Substitute this into the heat equation and obtain a quadratic equation for λ.
Note that for u = 0 (i.e., when there is only conduction and no advection),
this equation should reduce to
iω − k
ρc p
λ 2 = 0,
with solution
√ √ ρcp ω
λ = −
2k
− i ρcp ω
2k ,
1

corresponding to
T (x, t) = T 0 exp
( √ ) ( √ )
ρcp ω
−
2k x ρcp ω
cos ωt −
2k x . (3)
(As discussed in class, the other root does not allow the boundary condition
(2) to be satisfied).
(b) In the remainder of the assignment, you should assume that u > 0. What
would the solution for λ be if there was no heat conduction, i.e., if k =
0? Give the corresponding solution for T (x, t) in terms of real quantities
only (your answer should not contain the symbol ‘Re’). How would you
characterize this solution? How does it differ from the solution when there
is no advection and only conduction, given by (3).
(c) In general, thermal conductivity is not equal to zero, and the full quadratic
equation for λ has to be solved. The general solution for a quadratic equation
of the form aλ 2 + bλ + c is
Write λ in the form
λ = −b ± √ b 2 − 4ac
.
2a
λ = α + iβ ± γ √ 1 + iδ, (4)
giving α, β, γ and δ as real quantities that depend on ρ, c p , u, k and ω.
(d) It is not particularly easy to extract the real and imaginary parts of λ from
the solution in equation (4). However, we can approximate the square root
in (4) by using the following result: if x is small, then
√
1 + x ≈ 1 + x/2 − x 2 /8
(This is in fact the first three terms of the Taylor series for √ 1 + x.) Consider
the case of very small δ in (4). Give an approximate solution for λ in terms
of ρ, c p , u, k and ω, bearing in mind that the boundary conditon (2) has to
be satisfied. Also give the corresponding solution for T (x, t) in terms of real
quantities only (your anser should not contain the symbol ‘Re’). Compare
this qualitatively (in two sentences) with the solution for the case where
there is only conduction, equation (3), and with your solution from part (b)
for the case where there is only advection.
(e) Next, consider the case of very large δ. Use the approximation √ √ √ 1 + iδ =
iδ 1 + (iδ)
−1
to find an approximate solution for λ in this case. You will
need to use the fact that 1/i = −i. Give the corresponding solution for
T (x, t) in terms of real quantities only (your anser should not contain the
symbol ‘Re’). Again, compare this qualitatively (in two sentences) with the
solution for the case where there is only conduction, equation (3), and with
your solution from part (b) for the case where there is only advection.
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(f) For ice, we have the following parameter values: ρ = 910 kg m −3 , c p =
2.1 × 10 3 J kg −1 K −1 , k = 2.2 W m −1 K −1 . A typical rate of snowfall for
an ice sheet is u = 1 m per year. There are different tempearture cycles
relevant to a polar ice sheet. Consider two cases: (i) an annual cycle and (ii)
a Milankovitch cycle of 40,000 years associated with changes in the Earth’s
axis of rotation. Is the approximation of a small or a large δ relevant to (i)
the annual cycle and (ii) the Milankovitch cycle? Let T 0 = 10 K. Plot an
approximate solution for T (x, 0) (i.e. the temperature profile at t = 0) as a
function of x for (i) an annual cycle and (ii) a Milankovitch cycle.
(g) It is possible to identify whether advection is significant compared with conduction
without solving the temperature equation. This is done by scaling
the variables in the equation. Define dimensionless variables through
T = [T ]T ∗ , x = [x]x ∗ , t = [t]t ∗ ,
and choose your scales [T ], [x] and [t] such that the temperature equation
can be written in the form
∂T ∗
∂t ∗
∗
+ P e∂T ∂x − ∂2 T ∗
∗ ∂x = 0 ∗2 for x∗ > 0 (5)
T ∗ (0, t ∗ ) = cos(t ∗ ) at x ∗ = 0
− ∂T ∗
∂x → 0 ∗ as x∗ → 0 (6)
The parameter P e here is known as the Péclet number. Give your scales [T ],
[x] and [t] and the Péclet number P e in terms of the physical parameters ρ,
c p , u, k and ω. When do you expect advection to play a significant role?
Compare P e with the quantity δ defined in part (d). Give the Péclet number
for (i) an annual cycle and (ii) a Milankovitch cycle.
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