Newtonian Jerky Dynamics and Inertial Instability

Newtonian jerky dynamics is applied to inertial instability analysis to study the nonlinear features of atmospheric motion under the action of variable forces. Theoretical analysis of the Newtonian jerky function is used to clarify the criteria for inertial instability, including the influences of the meridional distributions of absolute vorticity (ξg) and planetary vorticity (the β effect). The results indicate that the meridional structure of absolute vorticity plays a fundamental role in the dynamic features of inertial motion. Including only the β effect (with the assumption of constant ξg) does not change the instability criteria or the dynamic features of the flow, but combining the β effect with meridional variations of ξg introduces nonlinearities that significantly influence the instability criteria.Numerical analysis is used to derive time series of position, velocity, and acceleration under different sets of parameters, as well as their trajectories in phase space. The time evolution of kinematic variables indicates that a regular wave-like change in acceleration corresponds to steady wave-like variations in position and velocity, while a rapid growth in acceleration (caused by a rapid intensification in the force acting on the parcel) corresponds to track shifts and abrupt changes in direction. Stable limiting cases under the f-and β-plane approximations yield periodic wave-like solutions, while unstable limiting cases yield exponential growth in all variables. Perturbing the value of absolute vorticity at the initial position (ξ0) results in significant changes in the stability and dynamic features of the motion. Enhancement of the nonlinear term may cause chaotic behavior to emerge, suggesting a limit to the predictability of inertial motion.