Question

Earthquakes below the Yellowstone caldera originate at very shallow depths, about 4 kilometers on average. Below this depth, the rocks are at least \(400^{\circ} \mathrm{C},\) too hot and weak to store elastic energy. Based on this data, answer the following questions: a. What is the minimum geothermal gradient in the first 4 kilometers beneath the Yellowstone caldera, assuming an average surface temperature of \(0^{\circ} \mathrm{C}\left(32^{\circ} \mathrm{F}\right) ?\) b. At about what depth is the groundwater below the Yellowstone caldera hot enough to "boil" and therefore capable of generating a geyser?

Short answer

a. 100°C/km; b. About 1 km.

Step-by-step solution

Understanding Geothermal Gradient

The geothermal gradient is the rate at which the Earth's temperature increases with depth. Given that at a depth of 4 kilometers the temperature reaches 400°C and the surface temperature is 0°C, we need to find the gradient in °C per kilometer.

Calculating Geothermal Gradient

To find the geothermal gradient, subtract the surface temperature from the temperature at depth and then divide by the depth in kilometers. The formula is: \(\text{Gradient} = \frac{\text{Temperature at depth} - \text{Surface temperature}}{\text{Depth}}\).

Applying Values to Calculate Gradient

The temperature at 4 kilometers depth is given as 400°C and the surface temperature is 0°C. Substituting these into the formula gives: \[\text{Gradient} = \frac{400 - 0}{4} = 100\,^{\circ}\mathrm{C/km} \].

Analyzing Geyser Formation

Water boils at 100°C at surface pressure, but boiling point increases with depth due to higher pressure. For simplicity, assume water needs to reach approximately 100°C to "boil" in this context.

Determining Depth for Boiling Water

Using the calculated geothermal gradient, determine the depth at which the temperature reaches approximately 100°C by using the formula \(\text{Depth} = \frac{\text{Boiling Temperature} - \text{Surface Temperature}}{\text{Geothermal Gradient}}\). Substituting values gives: \[\text{Depth} = \frac{100 - 0}{100} = 1\, \text{km} \].

Key concepts

Understanding Geothermal Gradient

The geothermal gradient is a crucial concept in geology. It describes the rate of temperature change within Earth's crust as you move deeper underground. This gradient is often expressed in degrees Celsius or Fahrenheit per kilometer or mile depth. In the context of the Yellowstone caldera, where the surface temperature is approximately 0°C, and at a depth of 4 kilometers, the temperature rises to 400°C. To determine the geothermal gradient, we consider the temperature increase over the depth.
This increase helps us understand not just temperature conditions underground but also influences geological activity. The geothermal gradient provides insights into subterranean heat flow and helps us predict geological phenomena like volcanic activity.

Exploring Earthquake Origin

Earthquakes at Yellowstone originate from very shallow depths, typically around 4 kilometers. Why so shallow? Primarily because at deeper layers, the rocks are incredibly hot, reaching around 400°C. At such high temperatures, rocks cannot effectively store elastic energy.
Elastic energy stores in rocks when they deform and attempt to snap back to their original shape. However, at extreme temperatures, rocks become too malleable and weak, thus inhibiting energy storage. This means deeper sections cannot generate the strain needed to trigger earthquakes. Most seismic activities, therefore, initiate in shallow, cooler rock layers.

Mechanics of Geyser Formation

Geysers are a fascinating natural phenomenon, and Yellowstone is famous for having several of them. But how do they form? Below the surface, water seeps into the earth and gets heated by the incredibly high geothermal gradient. As water temperature rises, it approaches boiling conditions. Depth and pressure influence this boiling point significantly, so even though water boils at 100°C at surface levels, it can withstand higher temperatures below the surface.

• As water boils underground, it turns into steam and expands, creating pressure.
• This pressure builds until it finds a way to escape, usually through cracks or fissures in the earth.
• Once the pressure is released, it results in the spectacular eruptions we recognize as geysers.

The specific geothermal conditions at Yellowstone help facilitate these conditions, supporting geyser activity.

Elastic Energy Storage in Rocks

Elastic energy storage in rocks is a foundational aspect of understanding earthquakes. Rocks in the Earth's crust can store energy by deforming elastically. Think of them as springs that absorb energy when stressed. However, not all conditions allow for this storage to happen efficiently.
At the Yellowstone caldera, it's noted that beyond 4 kilometers in depth, the temperature becomes too high—around 400°C. Rocks at such high temperatures lose their ability to store elastic energy. Instead, they become more ductile and can flow rather than snap back like a stretched rubber band when stress is released.
This characteristic is crucial because it limits the potential for earthquakes below certain depths, focusing seismic activity to cooler, more elastic zones of the crust.