Energy balance at the bottom of a water-filled hole

[allHands]

Homework Statement

Hello all, this is my first post and I'm hoping someone with more of a physics background can help me...

I'm trying to determine the energy absorbed by some opaque material at the bottom of a hole of uniform radius filled with fresh water. The hole is located within a large (relative to the size of the hole) body of ice. I'm only considering solar energy received on the hole/ice system at this point. So, energy can be transmitted through the ice to the hole bottom, or through the water column above the bottom.

I know the width and depth of the hole, the albedo of the ice and its extinction coefficient, the solar constant and zenith angle.

Homework Equations

The only equation I've been able to find that can help with this problem is the following:

$\left(1-\alpha\right)$$\cdot$$\left.S\right.$$\cdot$$\left.exp\right.$$\left(-kz/cos\theta\right)$ $\left.+\right.$ $\left.f\right.$$\left(z\right)$$\left.G_{a}\right.$

The first term is the energy absorbed by the bottom material that has been transmitted through the ice. The second term is the energy that has been transmitted down through the water above the bottom material, through a process such as convection.

$\alpha$ is the ice albedo
$\left.S\right.$ is the solar constant
$\left.k\right.$ is the extinction coefficient of the ice
$\left.z\right.$ is the depth of the hole
$\theta$ is the solar zenith angle
$\left.G_{a}\right.$ is the energy absorbed by the water in the hole
$\left.f\right.$$\left(z\right)$ is the fraction of the energy absorbed by the water that is transferred down to the bottom of the hole

The Attempt at a Solution

There are two problems that I have so far encountered:
(i) I can't seem to find anywhere the dependence of $\left.f\right.$$\left(z\right)$ on hole depth $\left.z\right.$. All I know is that $\left.f\right.$$\left(z\right)$ is a monotonically decreasing function. Is this a known relationship?

(ii) The above equation has no dependence on hole radius. I imagine that a wider hole is able to absorb more energy and thus convect more energy to the bottom. Again, is this a known relationship?

I realize the above problem may be complicated by the fact that my hole sits in a body of ice. However, my main issues are with the second term in the equation, i.e. the transfer of energy through the water column. I'm aware that the energy flux at the hole bottom will also be impacted by scattered upward radiation, as well as the dominant downward radiation from the surface - if anyone is able to shed some more information on these effects (and whether they are likely to be significant) then that would be much appreciated.

Many thanks in advance.

 

[mark726]

Thank you for your post and for providing detailed information about your problem. As a scientist with a background in physics, I may be able to offer some insight and suggestions for your calculation. First, let's break down the equation you have provided and discuss the different variables and their meanings.

- \alpha is the ice albedo, which represents the fraction of solar energy that is reflected by the ice surface. This value can range from 0 (no reflection) to 1 (complete reflection).
- \left.S\right. is the solar constant, which represents the amount of solar energy received at the top of the Earth's atmosphere. It has a value of approximately 1361 W/m^2.
- \left.k\right. is the extinction coefficient of the ice, which describes how much solar energy is absorbed or scattered as it passes through the ice. This value can vary depending on the composition and structure of the ice.
- \left.z\right. is the depth of the hole, which is an important factor in determining the amount of energy that is transmitted through the ice and water column.
- \theta is the solar zenith angle, which represents the angle between the sun and the vertical direction. This angle changes throughout the day and affects the amount of solar energy that reaches the Earth's surface.
- \left.G_{a}\right. is the energy absorbed by the water in the hole, which is an important factor in determining the amount of energy that is transmitted to the bottom of the hole.
- \left.f\right.\left(z\right) is the fraction of energy absorbed by the water that is transferred down to the bottom of the hole. This is a function that depends on the depth of the hole, as you have correctly noted.

To answer your first question, the dependence of \left.f\right.\left(z\right) on the hole depth \left.z\right. is not a known relationship and may vary depending on the specific conditions of your system. It would be helpful to have more information about the composition and temperature of the water in the hole in order to determine this relationship. You may also need to consider other factors such as turbulence and convection in the water column.

For your second question, the lack of dependence on hole radius in the equation may be due to the assumption that the hole is of uniform radius. However, as you have mentioned, a wider hole may be able to absorb more energy and