In this thesis, we have looked at the application of asymptotic methods in studying atmospheric phenomenon. Asymptotic techniques are very useful to obtain simplified systems which approximate a very complex underlying system. In this document we primarily use the simplified models to model the Hadley circulation. The Hadley circulation is an atmospheric circulation with rising air at the tropics and descending air in the subtropics(∼ 30 degree latitude). The dynamics of the Hadley cell have profound implications on the global atmosphere. It transports angular momentum, heat and moisture from the equator to the midlatitudes. The rising branch of the Hadley cell experiences increased rainfall and thunderstorms while the descending branch is marked by an increased aridity. So, understanding and accurately modeling the features of the Hadley cell is of great importance.
The thesis has been divided into four chapters. The first three chapter deal with applying the method of matched asymptotics in order to get an approximate solution valid for the entirety of the troposphere. Depending on the latitude, the system can be divided into three different layers. The innermost layer is the tropical region with dynamics on the mesoscales(500 km). The middle layer lies in the subtropics and modulates on the synoptic scales(1500 km). This is followed by a planetary scale(5000 km) outer layer in the midlatitudes. The aim of the study is to derive thesesystems using the formalism of matched asymptotics and obtain matching conditions between the solutions.
The full system of primitive equations and the non-dimensionalisation valid for large scale atmospheric flows have been described in chapter 1. Using these equations, the system valid for the tropics has been derived and its solutions have been described. The latitudinal extent of the tropical layer has been derived using scaling arguments for the tropical solutions. In chapter 2, we have looked at the shallow water system with non-dimensionalisation similar to those used in the first chapter. Shallow water formulation introduces a major simplification into the system by reducing the number of spatial dimensions by one. The tropical, subtropical and planetary layer models have been derived and their respective matching condition has been described . In chapter 3, we go back to the 3D system and derive the equations valid in the subtropics. The solution of the subtropical system yields an equation known as the Sawyer-Eliassen equation which is a second order partial differential equation. In the 3D system, the Sawyer-Eliassen equation is in 2 dimensions while inthe shallow water system it is a 1 dimensional ODE. This makes the 3D system much more difficult to solve since the PDE can be hyperbolic, parabolic or elliptic while no such complications arise in the shallow water system. A numerical solution scheme has been described for the subtropical system which solves the system when the Sawyer-Eliassen equation remains elliptic. The matching condition with the tropical boundary layer arises as the potential temperature restratification at the equator.
Chapter 4 deals with the instabilities arising in the subtropical jet. Informed by the pre-existing models of baroclinic instability, a damping model has been prescribed which incorporates the effect of baroclinic instability in the weak temperature gradient tropical mode. In the second part of the chapter, we have used the subtropical system obtained in chapter 3 to study these instabilities instead of the quasi-geostrophic system which has traditionally been used due to its simplicity. The effect of the momentum and temperature fluxes generated due to the instabilities has been studied. Since the fluctuations in linear analysis are much weaker than the mean flow, there is no interaction between the fluctuation and the mean flow. Using the method of multiple scale asymptotics, new equations have been derived to incorporate the fluxes generated due to the instabilities into the system describing the mean flow.
