[MUSIC] Now, to make progress with our equations, it is helpful to decompose them into appropriate coordinates. Cartesian coordinates have their location of interest with x pointing east, y pointing north and z pointing up in the local vertical. In this coordinates, the rotation vector omega projects onto a component on the z-axis and a component on the y-axis. That will be contributions of the Coriolis force coming from the z component and from the y component. The latter involves the vertical velocity, which is tiny. Outcomes in the equation for the vertical component of velocity, and there it is dwarfed by other terms like the gravity or the pressure gradient. So, the only contribution we retain is the one involving the vertical component of the Coriolis force. And we introduce here the Coriolis parameter f, which is two times omega times sine fie. Where fie is the latitude. As a typical value from a latitude, let's take 10- 4 radiance per second. So now, let's consider the equations of motion projected on to the z-axis first and see which terms are important. Here you see the equation for the z components. For each term, I estimate the order of magnitude noting Capital U, the order of magnitude for the horizontal velocity. For the vertical velocity the order of magnitude is capital W and because of the aspect ratio, this is U times H over L. When I scale the Lagrange derivative, I assume that the time scale involved is L over U. So, both terms of the Lagrangian derivative of the same order of magnitude. Now, we need to estimate what are the typical values. Vertical and horizontal scales have been mentioned. For the wind, a typical value of 10 meters per second represents the strong wind of the surface. A mild wind aloft. With these values, we find that the first term scales like 10 to the minus 6 while the other 2 terms scale like 10, so that's a no-brainer. To an excellent approximation, the predominant balance is between pressure gradient and gravity. This is hydrostatic balance. For motions and the scale of the planet, the pressure simply verifies hydrostatic balance. Okay, now for the horizontal. We just need to be a bit more careful about the pressure gradient. The variations that are involved here are the variations in the horizontal. Think of pressure maps for weather forecasts. At sea level, extremes go down to about 970, hectopascal or a bit less and up to 1030 hectopascal, or bit more for the higher pressures. Variations of pressure in the horizontal of 10, or few 10s of hectopascal. Let's just take 10. So, now we find that the first term is about 10 to the -4, whereas the other two are about 10 to the -3. So, it is not as much of a no-brainer, but there's no doubt the dominant balance is between the Coriolis force and the pressure gradient. To a very good approximation, the dominant balance in the atmosphere at large scales is between the pressure gradient force and the Coriolis force. This is called geostrophic balance. In other terms, the wind is such that the Coriolis force associated to this wind opposes the pressure gradient. Another way of quantifying this is to estimate the Rossby number. This is a non-dimensional number comparing the amplitude of the acceleration to that of the Coriolis force. It's U divided by f L. With the previous values, this is 0.1, this is small. For flows with small Rossby number, the effects of rotation play a dominant role and the velocities are close to geostrophic balance. Here's a useful rule of thumb. In the Northern Hemisphere, the geostrophic wind, which is a good approximation of the real wind, is along. Leaving high pressures to its right. It is more intense when, Are closer together. For instance, on the surface pressure map, you can see a low pressure system over the North Atlantic Ocean, between Iceland and Ireland. Strong winds rotate around it and clockwise. The thick lines with the symbols are called in warm fronts. Conversely, see how a very large high pressure system, that's an anticyclone, sets over southern Europe, yielding very weak winds. Good. So, let's summarize what we've seen. We've seen that in the vertical, to a very good approximation, the pressure is in hydrostatic balance. And in the horizontal, wind is what approximated by geostrophic wind. We've also seen at the beginning that the motions in the atmosphere were driven by the differential heating between the tropics which are warm and the poles which are cold. So, we'll start the next session trying to see how these different things combine and what we can expect for the global circulation of the atmosphere.
