## University of Louisville

# Introduction to Numerical Modeling of the Atmosphere

## Institution

University of Louisville

## Faculty Advisor/ Mentor

Benjamin MacCall

## Abstract

Using the fundamental principles of physics, a set of governing equations result that describes the dynamics of a fluid. Unfortunately these equations are nonlinear and cannot yet be solved. Instead, the equations are approximated using various numerical schemes and converted into a series of calculations to be done on a computer. There are many types of numerical schemes used; all have advantages and disadvantages that must be weighed against the physical behavior being simulated and the resources available for the computation. This work is a survey of well established finite-difference methods applied to two sets of linear equations - the advection equation and the linear shallow water equations. Finite differencing was used to approximate the partial derivatives in the equations with finite differences between discrete points in space and time. The resulting algebraic equations only approximate the original partial differential equations that lead to unwanted behavior, such as computational instability, damping, dispersion, and unphysical solutions. For the linear advection equation (LAE), forward-in-time-and-space and the Euler schemes were completely unstable. The Backward scheme was stable but experienced significant damping and dispersion. The leapfrog scheme showed little damping, but exhibited a computational mode. Lax-Wend off had no computational mode but experienced significant damping. The linear shallow water equation (LSWE), the leapfrog scheme exhibited minor damping but showed computational mode. The Lax-Wend off scheme does not have computational mode, but there is significant damping of the gravity waves. When rotation is included, both simulations exhibited the process of geotropic adjustment.

Introduction to Numerical Modeling of the Atmosphere

Using the fundamental principles of physics, a set of governing equations result that describes the dynamics of a fluid. Unfortunately these equations are nonlinear and cannot yet be solved. Instead, the equations are approximated using various numerical schemes and converted into a series of calculations to be done on a computer. There are many types of numerical schemes used; all have advantages and disadvantages that must be weighed against the physical behavior being simulated and the resources available for the computation. This work is a survey of well established finite-difference methods applied to two sets of linear equations - the advection equation and the linear shallow water equations. Finite differencing was used to approximate the partial derivatives in the equations with finite differences between discrete points in space and time. The resulting algebraic equations only approximate the original partial differential equations that lead to unwanted behavior, such as computational instability, damping, dispersion, and unphysical solutions. For the linear advection equation (LAE), forward-in-time-and-space and the Euler schemes were completely unstable. The Backward scheme was stable but experienced significant damping and dispersion. The leapfrog scheme showed little damping, but exhibited a computational mode. Lax-Wend off had no computational mode but experienced significant damping. The linear shallow water equation (LSWE), the leapfrog scheme exhibited minor damping but showed computational mode. The Lax-Wend off scheme does not have computational mode, but there is significant damping of the gravity waves. When rotation is included, both simulations exhibited the process of geotropic adjustment.