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In previous lessons, we were able to explain physical and chemical processes using only algebra and differential and integral calculus. Thermodynamics, moist processes, cloud physics, atmospheric composition, and atmospheric radiation and its applications can all be quantified (at this level of detail) with fairly simple mathematics. However, more math skill is required to understand and quantify the dynamics of the atmosphere.
This lesson introduces you to the math and mathematical concepts that will be required to understand and quantify atmospheric kinematics, which is the description of atmospheric motion; and atmospheric dynamics, which is an accounting of the forces causing the atmospheric motions that lead to weather. Weather is really just the wind (the motion of air in the horizontal and the vertical) and the consequences of the wind. Wind, which has both direction and speed, is best described using vectors.
The Earth is a spinning, slightly squashed sphere. The atmosphere is a tenuous thin layer on this orb, so from a human’s limited view, the Earth appears to be flat. For some applications, a simple Cartesian coordinate system, with three dimensions in the x, y, and z directions, seems like a good way to mathematically describe motion. For processes that occur on the larger scale, where the Earth’s curvature is noticeable, we must resort to using coordinates that are natural for a sphere.
The way wind direction is described sprang out of wind observations, and is now firmly implanted in the psyche of every weather enthusiast: easterly, northerly, westerly, and southerly. This wind convention, however, is quite different than that used in the equations that govern atmospheric motion, which are the basis of weather forecast models. Here we will see that a conversion between the two conventions is straightforward but requires some care.
Finally, we will see that movement of air can either be described by an observer at a fixed location (called the Eulerian framework) or by someone riding along with a moving parcel of air (called the Lagrangian framework). These two points-of-view are very different, but we will see that they are related to each other by advection, which is just the movement of air with different properties (such as temperature, pressure, and humidity) from a place upwind.
With this math and these concepts you will be ready to take on atmospheric kinematics and dynamics.
Learning Objectives
By the end of this lesson, you should be able to:
- calculate partial derivatives
- implement vector notation, the dot product, the cross product, and the del operator
- explain the different coordinate systems and how they are used
- convert between math and meteorological wind directions
- calculate temperature advection at any point on a map of isotherms (lines of constant temperature) and wind vectors
Questions?
If you have any questions, please post them to the Course Questions discussion forum. I will check that discussion forum daily to respond. While you are there, feel free to post your own responses if you, too, are able to help out a classmate.