# EARTH 135 Lecture Notes - Lecture 13: Center For Retrospective Digitization, Thermal Expansion, Multiple IntegralExam

by OC1768108

Department

Earth ScienceCourse Code

EARTH 135Professor

TanimotoStudy Guide

FinalThis

**preview**shows page 1. to view the full**4 pages of the document.**Change in ocean depth for cooling halfspace model

Test of the Cooling Halfspace Model (Section 4.23 in TS)

In this lecture, we are going to see a test of the cooling oceanic lithosphere

model. This model can explain an increasing ocean depth with age of the

ocean ﬂoor.

Let us compare the pressure under the ridge axis and the pressure under

the ocean ﬂoor at age t. We write the depth to the ridge axis as drand ocean

depth at age tas dr+d(t). We take the depth coordinate zdownward positive

and its origin (z= 0) as shown in the ﬁgure.

Pressure in the deep interior, say at z=D, should be equal whether it is

under the age zero or under the age t. Then we can write

ρwgdr+ρmg(d+D) = ρwg(dr+d) + ZD

0

ρgdz (1)

where ρwis density of sea water and ρmis density of deep mantle. Density ρ

in the integrand is a function of depth zbecause as the plate cools with age,

its density increases due to thermal contraction.

Density ρat temperature Tcan be written

ρ=ρm+ρmα(Tm−T) (2)

where αis the thermal expansion coeﬃcient. We can write Tas

T(z, t) = Ts+ (Tm−Ts)erf(z

2√κt) (3)

Then Tm−Tin (2) can be written

Tm−T= (Tm−Ts)(1 −erf(z

2√κt)) (4)

1

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