# Heat Flow and Diffusion
GEOL 301/303 F23/W24

## Introduction
This topic follows the material on seismic tomography because we use the seismic tomography image of a mantle plume beneath the west Antarctic [Seroussi et al., 2017](http://u.pc.cd/ihQ) to think about delivery of thermal energy to base of the lithosphere.

However to keep it simple, we will think of the heat energy as being delivered to the base of the Antarctic crust. Below is the tomography figure from Seroussi, 2017:

The A-A' cross section, with the red spot on figure panel (b), is where observations of metling processes have been made.

The big-picture idea behind this discussion is that earth's processes at any one time are a superposition of many different time scales. When we interpret a snapshot of data taken at any one time it is good to keep this in mind.

In this case, the observed melting can have many influences: warmer currents? If so is that part of a natural fluctuation? Is it part of a trend caused by human influences? Both? Underlying all of this is the time scale on which the mantle plume has deliveder heat energy to the base of the crust and by conductive heat transfer tries to bring the temperature gradient into equilibrium.

## Simulations of diffusion
There is a "heat equation" (also called the "diffusion equation") that governs how the anomaly is relaxed over time, i.e. equilibrates.

We looked at two Matlab codes in class to help us gain intution: [Diffusion_1D](https://filedn.com/lhT56ADuefxuE99NAXo9VKH/websites/geol301303comboF23W24/heat-flow-diffusion/Diffusion_1D.m) and [Diffusion_2D](https://filedn.com/lhT56ADuefxuE99NAXo9VKH/websites/geol301303comboF23W24/heat-flow-diffusion/Diffusion_2D.m). These codes are from the [Matlab File Exchange](https://www.mathworks.com/matlabcentral/fileexchange/38088-diffusion-in-1d-and-2d).

You can run these to gain intuition about how a temerature anomolay "diffuses". you can adjust the parameter "nt" to make them run longer.

A key observation is that initially the "diffusion" is fast, but as the anomaly is reduced, the speed is reduced.

## The Heat/Diffusion Equation
We will wait to actually write this down. Here the point we want to make is that it is a *differential equation*. In fact a *partial differential equation*.

We should start with some simpler examples of differential equations to get comfortable. We will later return to the form of the heat equation and try to get a physical understanding of why it has the form that it does.

## Examples of (ordinary) Differential Equations
Differential equations are equations that contain the derivatve of a function. The goal is to solve the equation for the function itself. If you are a math nerd, then this is an amusing game ...

The key thing to think about in solving differential equations, conceptually, is:
- derivatives tell us how something is changing (i.e. the famous "derivatives are rates of change" phrase).
- To solve a differential equation and know the solution at some future time, you are (speaking broadly) adding up all the changes between *now* and *then*. In calculus, integrals add things up!
- Thus you hear the phrase "intergrate" the equation. In a simple case this can be a literal statement; in more complex cases it's not as obvious how to do this, but the principle is the same.

### Example 1: Finding the position of a car
Suppse you are driving a car at a speed of $$ 70 \frac{miles}{hr} $$. This is your derivative (it is constant) or rate of change.

Your differential equation is then $$ \frac{dx}{dt}= 70 $$

To get your position at any future time "t", what do you do? You cumulatively add up all the positional changes over the time interval of interest.

Allowing for some sloppy algebra:

$$ \int \,dx = \int 70\,dt $$
This gives us $$ x(t) = 70 \cdot t + C $$

So the total change in position at any future time "t" is $70 \cdot t$ *relative to your starting position*. This is what the "constant of integration" expresses.

To get a definitive position you of course need to know your startting position. This is done by specifying an *initial condition*. For example, you may start 15 miles north of San Luis Obispo: $x(0)=15$. This allows you to find the constant $C$ (which will be 15 in this case) and this get your absolute position.

### Example 2: Isotope Age Dating

This is a superficial discussion of radioactive decay - it's complex (at least to me). Here is a good reference in terms of geochronology: Costal et al., 2020 [Clocks in Magmatic Rocks](https://www.annualreviews.org/content/journals/10.1146/annurev-earth-080320-060708).

The term "carbon dating" is probably something you have heard of; here C^14^ → N^14^. This is used in many places, even dating earthquakes on fault scarps.

#### Radioactive Decay
The probability of a atom (from a collection) to decay in a time period can be called its decay constant (although it's a probablility). We can consider other isotope systems in geology, with U → Pb being a common one in geochronology (the U, Pb are in the mineral zircon, commonly).

Each isotoped has a "decay rate" constant - although it is really a *probability* that any one atom from a collection parent atomes decays to its daughter atom in a given time period.

For carbon it is $\lambda = 1.2 \cdot 10^{-4} \frac{1}{yr}$.

To model this process mathematically:

Suppose there are $P(t)$ atoms of the "parent" atom (e.g. C, U). Then at time $t+\delta t$ there are $P+\delta P$ parent atoms remaining.

What is $\delta P$ quantitatively? In other words how ==many parent atoms have decayed in this time interval==?

This is expressed at the "number of atoms" *times* probability of a decay *times* the time interval, i.e. $$ \delta P = -\lambda P \delta t $$

#### The differential equation
The differential equation can be written as $$ \frac{dP}{dt} = - \lambda P(t) $$

What function has derivative proportional to itself? The exponential function: $$ \frac{d (e^{2\cdot t})}{dt} = 2 e^{2t} $$

This mean that the solution of our differential equation is:

$$ P(t) = P_{0} \cdot e^{-\lambda \cdot t} $$

where $P_{0}$ is the *initial amount of the parent isotope* in the sample (for example, when the magma solidified and the clock starts on the Parent → Daughter decay process).

Using this solution, and the decay constant for C^14^ you can determine that the *half-life* of C^14^ is ~ 5,730 years: Replace the left side of the equation with $\frac{1}{2} P_{0}$ (the $P_{0}$ cancels) and solve for $t$.

#### Fault scarp dating
An application to *paleoseismology*.


Now read the following example:

#### Isotopic Age Dating Example - Granitic Pluton
text


Using the decay rate constant above you can calculate that the half-life of Uranium is ~ 4/5 billion years (the age of the earth!)

Heres a case study - date the granitic pluton intrusion:

Suppose your mass spectrometer results are as follows:


Then apply:


Filling in the numbers gives the age:


## Heat Flow/Flux out of earth's surface:
Here is a nice figure from [Davies and Davies, 2010](https://se.copernicus.org/articles/1/5/2010/)


You can see high heat flow in places you expect it (e.g. mid-oceanic ridges), but you may not have expected a relatively high heat flow in Antarctica, for example.

Let's examine heat flow via **conduction** through a section of earth material (e.g. the crust):


What effects the heat transfer?
* Density of the material, $\rho$
* [Thermal conductivity](https://en.wikipedia.org/wiki/Thermal_conductivity_and_resistivity), $\kappa$
* [Specific heat](https://en.wikipedia.org/wiki/Specific_heat_capacity) of the material, $C_{P}$

You can look up the units and further descriptions at the linked sites.

==These three parameters combine to make the parameter of *thermal diffusivity*==, $D=\frac{\kappa}{C_{P} \cdot \rho}$, which is a part of the heat equation we will see later.

But we can apply it in a highly simplified example ...

## Time scale of heat transfer in the crust
How long will it take the thermal energy anomaly of the mantle plume impinging on the base of the Antarctic crust (we are ignoring the upper mantle part of the lithosphere) to equilibrate with the top of the crust at the earth surface boundary?

A so-called *characteristic time scale* can be approximated from the (yet to be derived) heat equation:

$$ \Delta t = \frac{L^{2}}{D}$$

where $L$ = the length (thickness of the crust) and $D$ = thermal diffusivity.

### Simplified example

The western Antarctic (the side we are interested in) crust is ~ 20 km thick.

What is the crust made up of?

Let's be naive and use the most common crustal mineral *plagioclase feldspar* as "representing" the crust.

Can we find the *thermal diffusivity*, $D$, of plagioclase?

Sure {see: [Heat Transfer in Plagioclase Feldspars](https://pubs.geoscienceworld.org/msa/ammin/article-abstract/97/7/1145/45650/Heat-transfer-in-plagioclase-feldspars
)}


Looking over th abstract, we might use $D = 0.8 \frac{mm^{2}}{s}$

Substituting these two values into $\Delta t = \frac{L^{2}}{D}$, and making sure we do unit conversions correctly, and then convert seconds to years, we get $\Delta t \approx 16$ million years.

## Take Home Point
Our take home point of the earth working on many different time scales is true. It can hard to even know what time scales are affecting something you are researching.

## Another example of a deep earth process influencing surficial geology

**Dynamic Topography**

I will just leave you with a few links to articles - but the idea is that sedimentation patterns can respond to slight changes in the "tilt" of the surface ... which can be influenced by deeper mantle processes *as a superposition on top of "shallower" processes*, analogous to the point (we tried) to make above with many time scales superimposing.

- [Drainage and Sedimentary Responses to Dynamic Topography](https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2019GL084400)
- [Eroding dynamic topography](https://agupubs.onlinelibrary.wiley.com/doi/full/10.1002/grl.50310)
- [Observation and Simulation of Solid Sedimentary Flux: Examples From Northwest Africa](https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2019GC008262)
- [Dynamic topography and vertical motion of the U.S. Rocky
Mountain region prior to and during the Laramide orogeny](https://par.nsf.gov/servlets/purl/10325502)