[MUSIC] In previous sessions, we were watching the deep circulation of the Mediterranean. But at no point do we speak of forces involved in the movement. We talked about water becoming denser, was sank and then placed according to a gradient of densities, Always putting the dense water below the one that was lighter, but there are a couple of fundamental concepts that must be seen, without necessarily entering into complexities of physics. The two concepts are, the first is the pressure gradient force, is a horizontal force which is due to a difference of pressures between one place and another. And then the Coriolis Force that you probably heard of it. These concepts have to be introduced and you have to see the consequences they have the importance of pressure gradient force is that the pressure depends on the density of the water and the height of the column. That is, the Earth is in a field of gravity, the sea is in a field of gravity. That is to say, at any point there is the force of gravity that gives the weight of water. The distribution of these weights, from the weights of the columns of water, is the which would or would give the distribution of pressures within the fluid. Therefore we speak of a force that acts on every body, and this pressure or force may vary due to changes in densities or due to changes in water heights, and here we are going to see the idea of if we change the height of the sea, how these forces are produced or if we change the density. And we see that the two things can be very related. This, therefore, is very important because in a distribution, or the distribution of sea temperatures and salinities, will give rise to a distribution of densities and this will lead to a topography. That is to say, to a distribution of the height of the sea which will be related to circulation. Let's see first, from a hypothetical framework, and that is as if it were a pool, or if it were an aquarium. Let's consider the water distributed in some columns. At this time perhaps it is not very important, but then it will be useful to see how this idea of distributing it in equal columns allows you to quickly introduce concepts. In this situation, imagine that we have that the whole density of water is uniform everywhere, and anywhere we can calculate the pressure. The pressure will be the height multiplied by the density by gravity, and this gives a pressure. And it gives rise to a distribution of increasing pressures, from here down. What happens if the densities of these columns are not the same? Imagine that we have this column is less dense, and this is more dense, and between this and this, for there is a gradient of densities. For now, remove the columns from the center. If we remove the columns from the center, we see a situation that is very similar to a situation that perhaps many have seen, and is that of communicating vessels. The two end columns, which were the lightest and most dense, we connect them by a tube, or by a communication. And this tube let the waters fill and balance. Here and here there are different pressures. Why? Because this density is greater than this, while the heights We have fixed them to the same height by construction. What will happen? There is more pressure here than here, and this water will fill this tube, until this pressure and this other are equal. Then we will realize that since this water is less dense than this, the height of part A and the height of part B. That is, this height of part A, and the height of part B, will be different. If here instead of having the two columns, we have all the columns. Obviously we find that if this is a horizontal, and we measure the pressures along this horizontal, the pressures will depend on the density and height. If we have left the heights equal, we will have the pressure on this side, will be less than the pressure on this side. And it will turn out that we will have a pressure gradient, that is, from more pressure to less pressure. And therefore, we will have a force that will go in this direction. A force that will go in this direction, and this force is called the pressure gradient force. What is the pressure gradient? It is simply the rate of change of pressure with distance. We make the difference between the pressure of two places, and we divide by the distance between these two places. The pressure gradient has units of force per unit volume. If we then divide by the density of this zone, of the water that is in this zone, will give us a force per unit mass. That in fact a force per unit mass is an acceleration. Therefore if we have water that has a horizontal acceleration, will tend to move in this direction from more pressure to less pressure. This is the first idea but the result is that we will have the isobars inclined with respect to the horizontal. Notice that the horizontal lines are incarnate the isobars are the black lines. And logically, because this water is less dense than this, we will have that to find the same pressure, we will need less height here than here. Therefore, all isobars will be inclined. and everywhere we will have a pressure gradient. I mean, here's more pressure than here. Therefore, water in principle will tend to go in this direction. Logically what happens is that this situation that we have painted, is a hypothetical situation. That is not stable. So much so that it is immediate to think that if this were so, the dense water would come down, and the less dense water would stay up. And in the end we would reach a distribution of densities where stratified water would remain with lower densities above and below. For this to happen, it would have to be that the columns could not be mixed, that the columns had no friction, could be deformed perfectly, and so on. But this is easy to imagine. Let's see another situation now. We start from the same idea, but now what we do is warm up, that is, we will warm this column, more temperature, is less and is less, and is the one that has less temperature. The columns without mixing or exchanging heat. What will happen? That this column will have warmed up more, will have less density. In fact it will expand. And therefore, without changing the weight of the column, it will be longer. In this way, each column, because we maintain the water it has, will simply tend to acquire the height corresponding to its density. The higher density, the less height, the lower density, the more expanded. And what is the result of this? A distribution of below we maintain the weight of the columns. We have the same pressure down here, because each column weighs the same as it weighed. However, the surface is now tilted. And the surface, in fact, is an isobar, because if we have the same pressure atmospheric everywhere, this would be an isobar, inclined with respect to the bottom isobar that is perfectly horizontal. The result is that the distribution of isobars will change their slope. That is, the top will have a steeper slope, the bottom one will be horizontal. When the isobar is horizontal, there is no pressure gradient rather, the pressure gradient is zero, because there is the same pressure here, as here, here, here. Therefore, the difference of pressures between any place of the same line or horizontal surface is zero. However in this part of here, the pressure gradient is important, and along a horizontal, we have a difference of pressures. This is a situation we call baroclinic. Notice that we have a distribution of isobars, that some are inclined with respect to the others. And the angle that the isobars form with the horizontal varies. This is important to keep in mind because many times, in calculations that one can do with the data that are obtained from distribution of densities. From a boat for example, we can not know exactly whether the sea is inclined or not. Because we are on the sea, and these inclinations are very small. But if we can calculate what is the relative inclination between isobars and other isobars. That is, the isobars if they were all parallel, we could not detect that they are inclined. Some respect to the others evidently not because they are not inclined one with respect to the others. But this inclination that all would have, can not be detected by direct access to density or distribution of densities. But if we can calculate the different inclination of one with respect to the others. This we will see later, that is important to do calculations. If we had a situation where for example all the water had the same density, and the sea was inclined, there would also be a pressure gradient, and would be the same for any depth. This situation would be called a barotropic situation. What slopes can we find in the sea? As a difference of 10cm in 10km, if we calculate what is the force that this represents, we calculate the pressure gradient, divided by the density. Alpha would be the inverse of density. We multiply gravity by height, divided by distance. This would be the way to calculate it, then we'll see exactly why. And this gives us 0.0001m / s ^ 2. That is, one hundred thousandth of gravity. This is a ridiculous na acceleration, very very small, but if we find that the sea is inclined for a long time, this acceleration is, which means that there is a horizontal force. So the question is, if it stays that way, what forces cause the water to not move or the water to remain inclined? When the trend would be to stay horizontal. There must be other forces, and one of the important forces that we are going to see, is the Coriolis force. The force of Coriolis, is a secondary force that appears when we have water in motion. Therefore, if we see that the sea is inclined, it is likely that there is a current whose associated Coriolis force compensates for this inclination. We will see this in the next section.