Question

Estimate the effects of variations in bottom water temperature on measurements of oceanic heat flow by using the model of a semi-infinite half-space subjected to periodic surface temperature fluctuations. Such water temperature variations at a specific location on the ocean floor can be due to, for example, the transport of water with variable temperature past the site by deep ocean currents. Find the amplitude of water temperature variations that cause surface heat flux variations of \(40 \mathrm{~mW} \mathrm{~m}^{-2}\) above and below the mean on a time scale of 1 day. Assume that the thermal conductivity of sediments is \(0.8 \mathrm{~W} \mathrm{~m}^{-1} \mathrm{~K}^{-1}\) and the sediment thermal diffusivity is \(0.2 \mathrm{~mm}^{2} \mathrm{~s}^{-1}\).

Short answer

The amplitude of the temperature variation is approximately 5.23 K.

Step-by-step solution

Understand the Problem

We are asked to find the amplitude of temperature variations at the ocean floor that cause variations in surface heat flux due to periodic temperature changes. The semi-infinite half-space model will be used, where fluctuating surface temperature affects heat flow.

Identify Known Variables

Given:- Surface heat flux variation: \(q = 40 \, \mathrm{mW}\, \mathrm{m}^{-2}\)- Thermal conductivity \(k = 0.8 \, \mathrm{W}\, \mathrm{m}^{-1}\, \mathrm{K}^{-1}\)- Thermal diffusivity \(\alpha = 0.2 \, \mathrm{mm}^{2}\, \mathrm{s}^{-1}\)- Period \(T = 1\) day, converted to seconds \(T = 86400 \, \mathrm{s}\).

Convert Units

Convert the thermal diffusivity to standard units:\[\alpha = 0.2 \, \mathrm{mm}^{2}\, \mathrm{s}^{-1} = 0.2 \times 10^{-6} \, \mathrm{m}^{2}\, \mathrm{s}^{-1}\].

Use the Analytical Solution for Heat Flux

For the problem of a semi-infinite half-space with periodic surface temperature variations, the surface heat flux \(q\) is given by:\[ q = k \cdot A \cdot \sqrt{\omega \cdot \alpha}\]where \(A\) is the amplitude of temperature variation and \(\omega\) is the angular frequency.

Find the Angular Frequency

Calculate \(\omega\) using:\[\omega = \frac{2\pi}{T}\]Substitute the period:\[\omega = \frac{2 \pi}{86400} \, \mathrm{s}^{-1}\].

Solve for Temperature Amplitude

Rearrange the heat flux equation to solve for amplitude \(A\):\[ A = \frac{q}{k \cdot \sqrt{\omega \cdot \alpha}} \]Substitute the known values:\[ A = \frac{40 \times 10^{-3}}{0.8 \cdot \sqrt{\frac{2 \pi}{86400} \cdot 0.2 \times 10^{-6}}} \].

Calculate the Result

Perform the calculations:- Angular frequency: \(\omega \approx 7.27 \times 10^{-5} \, \mathrm{s}^{-1}\)- Denominator in the amplitude equation: \(\sqrt{\omega \cdot \alpha} \approx 1.205 \times 10^{-5} \, \mathrm{s}^{-1}\)- Temperature amplitude \(A \approx 5.23 \, \mathrm{K}\).

Key concepts

Semi-Infinite Half-Space Model

The Semi-Infinite Half-Space Model is a pivotal concept in oceanography, particularly when studying heat flow in oceanic environments. This model simulates how thermal energy diffuses through a half-space that extends infinitely in one direction, typically representing ocean sediments beneath the seafloor.
To understand this, imagine the ocean floor as a vast, limitless layer of sediment. When the temperature on the surface of this layer changes, heat diffuses downwards into the sediment. However, because the sediment is semi-infinite, we assume that the heat flow continues indefinitely downward.
This model comes in handy for predicting how surface temperature variations impact the oceanic heat flow over time. For example, periodic changes in surface temperature due to moving water masses can induce heat fluxes into the sediment. By using this model, scientists can estimate how deep these temperature fluctuations penetrate over time and how they affect subsurface heat flow. This understanding is crucial for interpreting temperature data and evaluating thermal histories in oceanographic studies.

Thermal Conductivity in Sediments

Thermal conductivity in sediments is a measure of how well these materials can conduct heat. It's a critical property for calculating heat flow through ocean floor sediments. Simply put, it indicates the rate at which heat can move through the sediment.
In the context of oceanography, the unit of thermal conductivity is typically expressed as watts per meter per Kelvin (W/m·K). A higher thermal conductivity means the material can transfer heat more rapidly, while a lower value indicates slower heat transmission.
For instance, when the thermal conductivity of oceanic sediments is considered to be 0.8 W/m·K, it implies that for each meter thickness of sediment, there is a 0.8 W heat flow for every degree of temperature difference across it. This property is vital when using models like the semi-infinite half-space to predict how sediments respond to surface temperature fluctuations—whether they act as a good insulator or conductor of heat profoundly affects estimates of temperature at different sediment depths.

Oceanic Temperature Variations

Oceanic Temperature Variations are changes in the temperature of sea water, influenced by factors like currents, depth, and solar radiation. These variations are significant in marine sciences as they impact not only the physical properties of the water but also the heat flow into and out of the ocean floors.
Variations near the ocean floor, typically linked to deep ocean currents transporting water at different temperatures, cause periodic temperature fluctuations. These can affect the seafloor heat flow considerably, as seen in the problem using the semi-infinite half-space model. By varying the bottom water temperatures, the model helps understand how these changes influence the heat flux at the sediment-water interface.
Understanding these fluctuations is essential for oceanographers. They provide insights into thermal gradients, which are vital for assessing oceanic processes, marine ecosystems, and even climate models. Furthermore, accurately accounting for these variations assures precise interpretations of ocean floor temperature data, crucial for areas like marine geology and geothermal energy research.