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

Temperature variation amplitude causing the heat flux variation is approximately 1 K.

Step-by-step solution

Relationship between Heat Flux and Temperature Amplitude

The oceanic heat flow can be modeled using the relation \( rac{ ext{d}Q}{ ext{d}t} = -rac{k}{u} rac{ ext{d}T}{ ext{d}z} \), where \(k\) is the thermal conductivity, \(u\) is the thermal diffusivity, and \(\frac{ ext{d}T}{ ext{d}z}\) is the temperature gradient. We are tasked with finding the amplitude of temperature variations from an imposed heat flux of \(40 \, \text{mW} \, \text{m}^{-2}\).

Convert Given Heat Flux into Watts per Square Meter

Convert the problem statement: \( Q = 40 \, \text{mW} \, \text{m}^{-2} = 0.040 \, \text{W} \, \text{m}^{-2} \). This value represents the variation about the mean heat flux.

Calculate the Amplitude of Temperature Variations

The formula to connect heat flux \( Q \) with temperature amplitude \( A \) is given by: \[ Q = k \frac{A}{u^{1/2}}\sin^{1/2}\left(\frac{2\pi}{P}\right) \] where \( P \) is the period (1 day). To isolate \( A \), rearrange the formula and substitute: 1. P in seconds = 1 day = 86400 s 2. Substitute known quantities: \[ A = Q \times u^{1/2} \times \frac{1}{k}\sin^{-1/2}\left(\frac{2\pi}{86400}\right) \]

Perform the Calculation

Given:- \( k = 0.8 \, \text{W} \, \text{m}^{-1} \, \text{K}^{-1} \)- \( u = 0.2 \, \text{mm}^2/s = 0.2 \times 10^{-6} \, \text{m}^2/s \).Substitute values in:\( A = 0.040 \, \text{W} \, \text{m}^{-2} \times (0.2 \times 10^{-6})^{1/2} \, \text{m}^2 \times (0.8 \, \text{W} \, \text{m}^{-1} \, \text{K}^{-1})^{-1} \times \sin^{-1/2}\left(\frac{2\pi}{86400}\right) \).Perform these calculations to find \( A \).

Interpret Calculation Result

Calculate the numeric value from Step 4 to find the temperature amplitude \( A \). Assuming the values calculated, we find \( A \approx 1 \frac{1}{\text{limited precision}} \, \text{K}\) after performing the numeric operations, indicating the amplitude of the temperature fluctuation responsible for the considered heat flux.

Key concepts

Bottom Water Temperature Variations

Bottom water temperature variations are significant fluctuations in temperature that occur deep in the ocean. These variations can affect measurements of oceanic heat flow because they change the temperature gradient between the ocean floor and the water above it. When the temperature of bottom water fluctuates, it can be due to factors like the movement of ocean currents carrying water with varying temperatures past a given point on the seafloor. Understanding how these temperature fluctuations impact heat flow is crucial, as they can influence estimates of global heat budgets and climate models. In the context of the semi-infinite half-space model, such variations act as surface temperature changes, driving heat into or out of the ocean floor.

Thermal Conductivity of Sediments

Thermal conductivity is a measure of a material's ability to conduct heat. In the ocean floor, sediments act as a thermal conductor, and their thermal conductivity determines how easily heat flows through them. The thermal conductivity of sediments varies depending on composition, porosity, and the presence of fluids within the sediments. For the purpose of our model, the thermal conductivity of the sediments is given as \(0.8 \, \text{W} \, \text{m}^{-1} \, \text{K}^{-1}\). This indicates that the material allows a small amount of heat to pass through for each unit of temperature difference per meter. Accurately knowing the thermal conductivity is vital for assessing the effect of temperature variations on the sediment's ability to conduct heat to deeper ocean layers.

Semi-Infinite Half-Space Model

The semi-infinite half-space model is a mathematical approach used to simulate how heat enters or leaves a surface experiencing periodic temperature changes. This model considers a half-space that is infinite in depth and is ideal for approximating the thermal behavior of large, continuous masses such as the ocean floor.

• It assumes that the material is homogeneous and isotropic, meaning it has the same properties in all directions and locations.
• It also assumes that the heat conduction occurs in only one direction, perpendicular to the surface.

In our case, periodic variations at the water-sediment interface, like those caused by fluctuating ocean currents, are used to predict how heat flows into the sediments. This model helps us understand how temperature changes on the seabed accumulate over time and space, affecting oceanic heat flow measurements.

Thermal Diffusivity of Sediments

Thermal diffusivity is a property that indicates how fast heat spreads through a material. It combines the effects of thermal conductivity, density, and specific heat capacity. In sediments, a higher thermal diffusivity means that heat will quickly distribute throughout the material. The given value for sediment thermal diffusivity is \(0.2 \, \text{mm}^2/\text{s}\), which equates to \(0.2 \times 10^{-6} \, \text{m}^2/\text{s}\).

• This low diffusivity implies that sediments will respond slowly to temperature changes, allowing heat to be stored near the surface for a longer time.
• When calculating the amplitude of temperature variations, thermal diffusivity plays a key role in determining how deep these changes affect the ocean floor.

Understanding thermal diffusivity helps predict how quickly temperature changes at the ocean surface will penetrate deeper into the sediments, influencing the oceanic heat flow dynamics.