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, to understand and quantify the dynamics of the atmosphere requires more math skill.

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 motion of air in the horizontal and the vertical and the consequences of that motion. The motion is caused by wind and wind has both direction and speed, which are best described by 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 fixed observers on the ground (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 relative humidity) from some place upwind of the place where you are.

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

### Lesson Roadmap

Please see Canvas for a list of required assignments, due dates, and submission instructions.

### Questions?

If you have any questions, please post them to the Course Questions discussion forum in Canvas. 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.