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No abstract

It is found in collisionless Electron Temperature Gradient (ETG) turbulence simulations that, while zonal flows are weak at early times, the zonal flows continue to grow algebraically (proportional to time). These fine-scale zonal flows have a radial wave number such that k r ρ i > 1 and k r ρ e < 1. Eventually, the zonal flows grow to a level that suppresses the turbulence due to ExB shearing. The final electron energy flux is found to be relatively low. These conclusions are based on particle convergence studies with adiabatic ion electrostatic flux-tube gyrokinetic δ f particle simulations run for long times. The Rosenbluth-Hinton random walk mechanism is given as an explanation for the long time build up of the zonal flow in ETG turbulence and it is shown that the generation is (k ⊥ ρ e) 2 smaller than for isomorphic Ion Temperature Gradient (ITG) problem. This mechanism for zonal flow generation here is different than the modulational instability mechanism for ITG turbulence. These results are important because previous results indicated zonal flows were unimportant for ETG turbulence. Weak collisional damping of the zonal flow is also shown to be a n important effect.

A new method is presented for calculating the Stokes multipliers for a class of linear second-order ordinary differential equations. The Stokes multipliers allow the asymptotic solutions of these equations to be continued across the Stokes lines on which they are dominant. The differential equations, of the class considered here, have an irregular singular point at infinity and a singular point at the origin, which may be either regular or irregular. The Stokes multipliers, as functions of the coefficients in the differential equation, are obtained in the form of convergent infinite series, whose terms must be obtained from the solution of recursion relations, which are derived. In the case of Whittaker’s equation (when the origin is a regular singular point), the known results are obtained analytically. When the origin is an irregular singular point, numerical evaluation of the series is necessary, but the method seems to be quite efficient for use with digital computers. In the special case of an equation, with two irregular singular points, which can be transformed to Mathieu’s equation, the numerical results for the Stokes multiplier show good agreement with available known results for the characteristic exponents of Mathieu’s equation.