# Energy balance at the bottom of a water-filled hole

1. Jul 11, 2011

### allHands

1. The problem statement, all variables and given/known data

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.

2. Relevant 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

3. 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 realise 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.