GeoExchange Site Visit and Local Geology Field Trip#

IN THE SPACE BELOW, WRITE OUT IN FULL AND THEN SIGN THE HONOR PLEDGE:

“I pledge my honor that I have not violated the honor code during this examination.”

PRINT NAME:

If a fellow student has contributed significantly to this work, please acknowledge them here:

Peer(s):

Contribution:

By uploading this assignment through Canvas, I sign off on the document below electronically.


Part I: Geotherm: Temperature within the Earth#

Conduction is the process by which heat is transferred from the hotter end to the colder end of an object. The ability of the object of dimension \(z\) to conduct heat is known as its thermal conductivity, and is denoted \(k\). Heat spontaneously flows along a temperature gradient \(\nabla\)T = (T2-T1)/\(z\), a physical quantity that describes in which direction and at what rate the temperature changes the most rapidly around a particular location. Fourier’s Law of heat conduction is expresses the flux or the rate of heat flow (\(Q\) in \(mW/m^2\)) as:

\[ Q = k \cdot A \frac{T_2-T_1}{z}, \]

where the symbols and their typical values are provided in the table below.

drawing

The expressions below is a model calculating an approximate conductive geothermal gradient for the lithosphere. Note that these have been derived by taking a cylinder rock of length \(z\) and area \(A\) and considering Fourier’s Law. Also provided is a code snippet to plot the geotherm interactively. Your task is to experiment with and comment on this model by changing parameters (e.g. mantle heat flow, thermal conductivity). We explore this in the questions below. Note that the thermal conductivities are derived from samples at room temperature (refer Pollack et al., Journal of Geophysical Research, 1993). In the real Earth, some of these properties can vary both with depth (e.g. \(k\)) and location on the surface (e.f. T\(_s\)).

\[\begin{split} \mathrm{T}(z) = \begin{cases} \frac{Qz}{K}+\frac{A_\mathrm{o}z(b-z/2)}{K} +T_s & \text{if } z < b \\ % & is your "\tab"-like command (it's a tab alignment character) \frac{Qz}{K}+\frac{A_\mathrm{o}b^2}{2K} +T_s & \text{b $\leq$ z $\leq$ L, where L = 100 km } \end{cases} \end{split}\]

Parameter

Symbol

Typical Value

Units

Temperature at the surface

T\(_s\)

15

\(^{\circ}\)C

Heat flow in continents

Q

65

mW/m\(^2\)

Heat flow in oceans

Q

101

mW/m\(^2\)

Thermal conductivity of Granite

k

3.1

W/m/deg

Thermal conductiity of Basalt

k

1.5

W/m/deg

Heat production

A\(_\mathrm{o}\) = \(\rho\)H\(_s\)

2.0

\(\mu\)W/m\(^3\)

Characteristic depth of A\(_\mathrm{o}\)

b

10

km

Depth

z

Variable

km

# import packages
import pandas as pd
import numpy as np
import plotly.express as px
import plotly.graph_objects as go

# Define constants
Ts = 15
Q = 65
Ao = 2
b = 10
K = 3.1
L = 100 #km

# create dataframe
z = np.arange(0,L+1)
df = pd.DataFrame({'Depth (km)': z})
  
#adding column with constant value
# Note that in the Earth, this might vary with location
df['Q'] = pd.Series([Q for x in range(len(df.index))])
df['Ao'] = pd.Series([Ao for x in range(len(df.index))])
df['K'] = pd.Series([K for x in range(len(df.index))])
    
# view dataframe
#print("Initial dataframe")
#display(df)

df.loc[df['Depth (km)'] < b, 'T (Centigrade)'] = df['Q']*df['Depth (km)']/df['K'] + df['Ao']*df['Depth (km)']*(b-df['Depth (km)']/2)/df['K'] + Ts
df.loc[(df['Depth (km)'] >= b) & (df['Depth (km)'] <= L), 'T (Centigrade)'] = df['Q']*df['Depth (km)']/df['K'] + df['Ao']*b**2/(2*df['K']) + Ts

# flipped the depth 
fig1 = px.line(df, x="T (Centigrade)", y="Depth (km)")
fig = go.Figure(data=fig1.data, layout = fig1.layout)
fig.update_layout(
    yaxis = dict(autorange="reversed"),
    title_text='Conductive geotherm in continents',
    width=400, height=500
)
fig.show()