Photo  credit:  Sinead  Farrell

Goals •  Examine how heat is redistributed within glacier ice

as its boundary is heated or cooled

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

Goals •  Examine how heat is redistributed within glacier ice

as its boundary is heated or cooled

T (z, t) = T0 +ΔTe−z ω

2κ sin ωt − z ω2κ

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Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

Goals •  Examine how heat is redistributed within glacier ice

as its boundary is heated or cooled

•  Plot the variation of temperature with depth

T (z, t) = T0 +ΔTe−z ω

2κ sin ωt − z ω2κ

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Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

Assumptions •  Ice velocity is negligible •  Advection is negligible •  Ice is homogeneous •  Ice is isotropic •  Horizontal ice surface is heated uniformly

o  (heat flow occurs only in the vertical direction)

•  Ice can be treated as infinite half-space o  Temperature far from surface equals T0 for all time t o  (ice is really thick and influence of lower boundary on near-surface temperatures is not

important)

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

Assumptions •  Ice velocity is negligible •  Advection is negligible •  Ice is homogeneous •  Ice is isotropic •  Horizontal ice surface is heated uniformly

o  (heat flow occurs only in the vertical direction)

•  Ice can be treated as infinite half-space o  Temperature far from surface equals T0 for all time t o  (ice is really thick and influence of lower boundary on near-surface temperatures is not

important)

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

Assumptions •  Ice velocity is negligible •  Advection is negligible •  Ice is homogeneous •  Ice is isotropic •  Horizontal ice surface is heated uniformly

o  (heat flow occurs only in the vertical direction)

•  Ice can be treated as infinite half-space o  Temperature far from surface equals T0 for all time t o  (ice is really thick and influence of lower boundary on near-surface temperatures is not

important)

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

Assumptions •  Ice velocity is negligible •  Advection is negligible •  Ice is homogeneous •  Ice is isotropic •  Horizontal ice surface is heated uniformly

o  (heat flow occurs only in the vertical direction)

•  Ice can be treated as infinite half-space o  Temperature far from surface equals T0 for all time t o  (ice is really thick and influence of lower boundary on near-surface temperatures is not

important)

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

Assumptions •  Ice velocity is negligible •  Advection is negligible •  Ice is homogeneous •  Ice is isotropic •  Horizontal ice surface is heated uniformly

o  (heat flow occurs only in the vertical direction)

•  Ice can be treated as infinite half-space o  Temperature far from surface equals T0 for all time t o  (ice is really thick and influence of lower boundary on near-surface temperatures is not

important)

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

Assumptions •  Ice velocity is negligible •  Advection is negligible •  Ice is homogeneous •  Ice is isotropic •  Horizontal ice surface is heated uniformly

o  (heat flow occurs only in the vertical direction)

•  Ice can be treated as infinite half-space o  Temperature far from surface equals T0 for all time t o  (ice is really thick and influence of lower boundary on near-surface temperatures is not

important)

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

•  Top ~15m of a glacier subject to: o  Diurnal variations (solar) o  Annual variations (solar &

atmospheric) o  Longer variations (climate

changes)

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

T (z, t) = T0 +Y1(z)cos(ωt)+Y2 (z)sin(ωt)

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

T (z, t) = T0 +Y1(z)cos(ωt)+Y2 (z)sin(ωt)

∂T∂t

=κ∂2T∂z2

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

T (z, t) = T0 +Y1(z)cos(ωt)+Y2 (z)sin(ωt)

∂T∂t

=κ∂2T∂z2

Y1 = −κωd 2Y2dy2

Y2 =κωd 2Y1dy2

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

T (z, t) = T0 +Y1(z)cos(ωt)+Y2 (z)sin(ωt)

∂T∂t

=κ∂2T∂z2

Y1 = −κωd 2Y2dy2

Y2 =κωd 2Y1dy2

d 4Y2dy4

+ω 2

κ 2 Y2 = 0

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

Y2 = c1ezα + c2e

−zα

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

Y2 = c1ezα + c2e

−zα

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

Temperature must be finite when z→∞

Y2 = c1ezα + c2e

−zα

α 4 +ω 2

κ 2 Y2 = 0

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

Y2 = c1ezα + c2e

−zα

α 4 +ω 2

κ 2 Y2 = 0 α = −1± i2

"

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&'ωκ

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

α 4 +ω 2

κ 2 Y2 = 0 α = −1± i2

"

#$

%

&'ωκ

Y2 = b1e−(1+i) ω

2κz+ b2e

−(1−i) ω2κ

z

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

Y2 = c1ezα + c2e

−zα

α 4 +ω 2

κ 2 Y2 = 0 α = −1± i2

"

#$

%

&'ωκ

Y2 = b1e−(1+i) ω

2κz+ b2e

−(1−i) ω2κ

z

Y2 = e−

ω2κ

zb1e

i ω2κ

z+ b2e

−i ω2κ

z"

#$$

%

&''

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

Y2 = c1ezα + c2e

−zα

α 4 +ω 2

κ 2 Y2 = 0 α = −1± i2

"

#$

%

&'ωκ

Y2 = b1e−(1+i) ω

2κz+ b2e

−(1−i) ω2κ

z

Y2 = e−

ω2κ

zb1e

i ω2κ

z+ b2e

−i ω2κ

z"

#$$

%

&''

Y2 = e−

ω2κ

za1 cos

ω2κ

z+ a2 sinω2κ

z"

#$

%

&'

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

Y2 = c1ezα + c2e

−zα

α 4 +ω 2

κ 2 Y2 = 0 α = −1± i2

"

#$

%

&'ωκ

Y2 = b1e−(1+i) ω

2κz+ b2e

−(1−i) ω2κ

z

Y2 = e−

ω2κ

zb1e

i ω2κ

z+ b2e

−i ω2κ

z"

#$$

%

&''

Y2 = e−

ω2κ

za1 cos

ω2κ

z+ a2 sinω2κ

z"

#$

%

&'

Y1 = e−

ω2κ

za3 cos

ω2κ

z+ a4 sinω2κ

z"

#$

%

&'

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

…repeat for Y1

Y2 = c1ezα + c2e

−zα

Y2 = e−

ω2κ

za1 cos

ω2κ

z+ a2 sinω2κ

z"

#$

%

&'Y1 = e

−ω2κ

za3 cos

ω2κ

z+ a4 sinω2κ

z"

#$

%

&'

T (z, t) = T0 +Y1(z)cos(ωt)+Y2 (z)sin(ωt)

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

Y2 = e−

ω2κ

za1 cos

ω2κ

z+ a2 sinω2κ

z"

#$

%

&'Y1 = e

−ω2κ

za3 cos

ω2κ

z+ a4 sinω2κ

z"

#$

%

&'

T (z, t) = T0 +Y1(z)cos(ωt)+Y2 (z)sin(ωt)

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

Y1 = −κωd 2Y2dy2

Y2 =κωd 2Y1dy2

a2 = a3a1 = −a4

Y2 = e−

ω2κ

za1 cos

ω2κ

z+ a2 sinω2κ

z"

#$

%

&'Y1 = e

−ω2κ

za3 cos

ω2κ

z+ a4 sinω2κ

z"

#$

%

&'

T (z, t) = T0 +Y1(z)cos(ωt)+Y2 (z)sin(ωt)

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

Y1 = −κωd 2Y2dy2

Y2 =κωd 2Y1dy2

a2 = a3a1 = −a4

Y2 = e−

ω2κ

za1 cos

ω2κ

z+ a2 sinω2κ

z"

#$

%

&'Y1 = e

−ω2κ

za3 cos

ω2κ

z+ a4 sinω2κ

z"

#$

%

&'

T (z, t) = T0 +Y1(z)cos(ωt)+Y2 (z)sin(ωt)

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

Y1 = −κωd 2Y2dy2

Y2 =κωd 2Y1dy2

Y2 = e−

ω2κ

za1 cos

ω2κ

z+ a2 sinω2κ

z"

#$

%

&'Y1 = e

−ω2κ

za2 cos

ω2κ

z− a1 sinω2κ

z"

#$

%

&'

Y2 = e−

ω2κ

za1 cos

ω2κ

z+ a2 sinω2κ

z"

#$

%

&'Y1 = e

−ω2κ

za3 cos

ω2κ

z+ a4 sinω2κ

z"

#$

%

&'

T (z, t) = T0 +Y1(z)cos(ωt)+Y2 (z)sin(ωt)

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

Y2 = e−

ω2κ

za1 cos

ω2κ

z+ a2 sinω2κ

z"

#$

%

&'Y1 = e

−ω2κ

za3 cos

ω2κ

z+ a4 sinω2κ

z"

#$

%

&'

T (z, t) = T0 +Y1(z)cos(ωt)+Y2 (z)sin(ωt)

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

T (0, t) = T0 +ΔT sin(ωt)

T (∞, t) = T0

Y2 = e−

ω2κ

za1 cos

ω2κ

z+ a2 sinω2κ

z"

#$

%

&'Y1 = e

−ω2κ

za3 cos

ω2κ

z+ a4 sinω2κ

z"

#$

%

&'

T (z, t) = T0 +Y1(z)cos(ωt)+Y2 (z)sin(ωt)

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

T (0, t) = T0 +ΔT sin(ωt)

a1 = ΔTa2 = 0

T (∞, t) = T0

Y2 = e−

ω2κ

za1 cos

ω2κ

z+ a2 sinω2κ

z"

#$

%

&'Y1 = e

−ω2κ

za3 cos

ω2κ

z+ a4 sinω2κ

z"

#$

%

&'

T (z, t) = T0 +Y1(z)cos(ωt)+Y2 (z)sin(ωt)

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

Y2 = e−

ω2κ

zΔT cos ω

2κz+ a2 sin

ω2κ

z#

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&

'(Y1 = e

−ω2κ

za2 cos

ω2κ

z−ΔT sin ω2κ

z#

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&

'(

T (0, t) = T0 +ΔT sin(ωt)

a1 = ΔTa2 = 0

T (∞, t) = T0

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

Y2 = e−

ω2κ

zΔT cos ω

2κz

#

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&

'(Y1 = e

−ω2κ

z−ΔT sin ω

2κz

#

$%

&

'(

T (z, t) = T0 +Y1(z)cos(ωt)+Y2 (z)sin(ωt)

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

Y2 = e−

ω2κ

zΔT cos ω

2κz

#

$%

&

'(Y1 = e

−ω2κ

z−ΔT sin ω

2κz

#

$%

&

'(

T (z, t) = T0 +Y1(z)cos(ωt)+Y2 (z)sin(ωt)

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

T (z, t) = T0 +ΔTe−z ω

2κ sin(ωt)cosz ω2κ

− cos(ωt)sin z ω2κ

#

$%

&

'(

Y2 = e−

ω2κ

zΔT cos ω

2κz

#

$%

&

'(Y1 = e

−ω2κ

z−ΔT sin ω

2κz

#

$%

&

'(

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

sin ωt − z ω2κ

"

#$

%

&'

T (z, t) = T0 +Y1(z)cos(ωt)+Y2 (z)sin(ωt)

T (z, t) = T0 +ΔTe−z ω

2κ sin(ωt)cosz ω2κ

− cos(ωt)sin z ω2κ

#

$%

&

'(

Y2 = e−

ω2κ

zΔT cos ω

2κz

#

$%

&

'(Y1 = e

−ω2κ

z−ΔT sin ω

2κz

#

$%

&

'(

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014

T (z, t) = T0 +ΔTe−z ω

2κ sin ωt − z ω2κ

#

$%

&

'(

T (z, t) = T0 +Y1(z)cos(ωt)+Y2 (z)sin(ωt)

Y2 = e−

ω2κ

zΔT cos ω

2κz

#

$%

&

'(Y1 = e

−ω2κ

z−ΔT sin ω

2κz

#

$%

&

'(

T (z, t) = T0 +ΔTe−z ω

2κ sin(ωt)cosz ω2κ

− cos(ωt)sin z ω2κ

#

$%

&

'(

Julia Ruth SIO 234 with Dr. David Sandwell Homework #3 Presentation Fall 2014