Nonlinearity and Lorenz Equation — Lesson 2

This lesson covers the concept of nonlinearity in the context of instability and its role in saturating growth. The instructor explains this through the Lorenz equations, which are used to study non-linear saturation. The lesson also delves into the derivation of the Lorenz equation, the concept of Fourier, and the interaction of u dot grad theta. The instructor further explains the concept of triad interaction and the Craya-Herring method. The lesson concludes with a discussion on the Lorenz equation's applicability for large Prandtl numbers and its limitations for small Prandtl numbers.

Video Highlights

00:49 - Derivation of the Lorenz equation and its normalization.
05:44 - Derivation of the Lorenz equations for a triad.
19:18 - Derivation of the equation for theta 02 and theta 11
32:15 - Discussion of the fixed point solutions for the Lorenz equations
45:25 - Discussion on the limitations of the Lorenz equation for small Prandtl numbers.

Key Takeaways

- Nonlinearity plays a crucial role in saturating the growth in the context of instability.
- The Lorenz equations are used to study non-linear saturation.
- The concept of Fourier and the interaction of u dot grad theta are essential in understanding nonlinearity.
- Triad interaction is a significant concept in the study of nonlinearity.
- The Lorenz equation is applicable for large Prandtl numbers but has limitations for small Prandtl numbers.