Comparison of ball analogy and plasma instability

Instabilities occur in ordinary hydrodynamics and mechanics. For practical applications it is naturally of great interest to know whether certain instability grows rapidly or perhaps so slowly that it is harmless. Early experiments showed that the plasma had a means by which it could escape from the magnetic field fast than the binary collisions rates would suggest, and indeed faster than the rate at which the compression process was increasing its temperature. This demonstrate that the magnetic field plasma configuration describe is inherently unstable, with a very short time constant. Stability theory thus is a central feature of plasma research. We studied the cases of ball analogy with the comparison of plasma instability. Consider the cases of a ball in a hole, on a level surface, and on a hill. All three are equilibrium cases. The stability depends upon the sign of the force experience by the ball as it moves in the X direction. We use a Taylor series expansion for the force F around the initial position Xo. The exp (-γt) solution damps out, but the exp (γt) solution grows rapidly, and represent instability, with γ called the growth rate. And the sufficient and necessary condition of stability is that any one of (δ²v/δXi²) > 0. Similar principal hold in plasma. Instead of a ball with mass m, we have a fluid with mass density. Each fluid element may have a displacement vector ξ representing departure from its equilibrium position. (author)