## Sunday, January 30, 2011

### When the ocean talks to the sea

Scientific Flâneurie at the beach

Part of the fun when I'm outside is about walking in a complex and theoretical labyrinth. I like to think fast and deep, and I love to ask questions and what-ifs.

This particular phenomenon kept me really busy thinking: In a warm an nice summer day, why do we have a refreshing breeze coming from the sea and reverses at night? I immediately thought that this had to do something with heat in the water and in the land, and how it escaped to the air.

Reading about it later, I learned that the absorption of heat is remarkably faster across the land that it is across the sea... Then the air above the ground is heated rapidly and becomes lighter, so it rises! (If its colder, it's "heavier" then it sinks. If it's hot, then it's "lighter" so it goes up like a balloon (this is also why we should have our air conditioners up on our walls and not down, because cold air needs to sink and circulate).

So far so good. now, in this train of thought the question is about air movement... What happens when the hot packet of air rises above the soil? Something needs to replace this hole! well... the air that is over the sea is, as you know, colder... so it spreads like butter towards the beach to fill the hole the rising air left.

This is awesome for the inquisitive mind, but are we done yet? Nope. Turns out that the air that went up is now cold and heavy, so it's not surprising if it needs to go down again... the thing is that it's not going to do it right above the place it came from, because this air aloft also spreads like butter, but way up high towards the sea! Above the sea it goes down. As you can see now, we do have a cycle: air heats up and rises over the land while cool air crawls over the beach (this is why it's so refreshing and people like it).

So when I eventually go to the beach with company, this is when the nerd comment comes from: "Did you know that we're experiencing the thermally-induced air displacement due to a natural thermal machine?" Some people prefer not to think too much about it and some won't even consider it (and this is completely fine!). On the other hand, for the people who prefer depth of knowledge instead of breadth of knowledge it is absurd to avoid thinking a little further because of some irrational fear of becoming crazy or being looked at as a nerd. "There's an occasion for everything!" some reply to me, but hey: This is the occasion because I'm witnessing it, I'm living it, and I'm enjoying thinking about it! "You should rest your mind a bit"

Very well, here's my rest:

After analyzing the cycle, I remembered that the idea of air moving, rising and sinking is in other words a force caused by a gradient in pressure. I'm naming x the axis perpendicular to the shoreline, then the force that experiences the air block that forms the sea breeze is:

$-\alpha_{s}\frac{\partial P}{\partial x}$

Here I multiplied the specific volume of air with the change of pressure between the land and the sea. We said this is a cycle and that air circulates in a closed path, so I'll use the definition of circulation from Kelvin-Bjerknes theorem from Vector Calculus:

$C=\oint \vec{V}\cdot d{l}= \oint udx+wdz$

Line integrals are very elegant indeed! Here I named u the horizontal flow (the breeze that we feel) and w is the air that sinks or rises. Now let's compute how the horizontal and vertical winds moves in time. What are the obstacles for horizontal flow relevant for this case? friction of course, so let's include it via Newton's second law:

$\frac{\partial u}{\partial t}=-\alpha _{s}\frac{\partial P}{\partial x}-ku$

The vertical is the same thing, but gravity is important here:

$\frac{\partial w}{\partial t}=-\alpha _{s}\frac{\partial P}{\partial z}-kw-g$

The problem now is that I have partial derivatives and not the expressions for each velocity component. It's easier to calculate the derivative of circulation and then plug the last two equations in:

$\frac{dC}{d t}=\oint \left( -\alpha _{s}\frac{\partial P}{\partial x}-ku \right)dx+ \left(-\alpha _{s}\frac{\partial P}{\partial z}-kw-g\right)dz$

Then plug in the equation of ideal gas and Eureka! We have an expression for sea breeze acceleration over time:

$A_{F}\textrm{cos}(\omega t)=\frac{R\Delta T}{L}\textrm{ln}\left(\frac{p_{0}}{p_{1}}\right)-kV$

Who knew we'd have a natural log in the beach? This is my share of what I call the excitement of discovery!
This is very nice! I can say that the beach can also be a wonderful laboratory of all the sciences!