Single Fourier Mode Research Articles

Resistive tearing in line-tied magnetic fields: Slab geometry

The resistive tearing-mode instability of a current carrying plasma sheet is investigated including the stabilising photospheric line-tying boundary conditions. This end condition prohibits a single Fourier mode and so requires a series expansion in harmonics of the fundamental sheet excitation. Equilibria in which there exist field lines that do not connect to the photosphere are unstable provided the ratio of the sheet length to characteristic transverse scale is smaller than a critical value that depends on the equilibrium profile. Line-tying has a strong stabilising effect on the fundamental periodic mode. That tearing mode harmonic which develops close to the instability threshold, leads to a configuration with one X point and one 0 point. Its linear growth rate follows the usual constant-ψ scaling with resistivity γ ∼ S-3/5, where S is the magnetic Reynolds number.

Nonlinear instability in long time calculations of a partial difference equation

The stability of long time ( t → ∞) calculations of nonlinear partial difference equations is examined. The model chosen is a discretisection in space and time of the Korteweg de Vries equation. The standard linearized von Neumann stability analysis, although necessary, is not sufficient, since it ignores the quadratic nature of the nonlinearity in the problem. Nonlinear stability is analysed by perturbing a constant solution with a period 3 Fourier mode. This leads to a nonlinear system of ordinary differential equations, the stability of which is examined by phase plane analysis. The results obtained contrast strongly with those obtained from the linear analysis. A numerical illustration shows how a single Fourier mode initial condition through side band growth, eventually causes blow-up in a long time calculation.

Instability and flow over topography

Abstract The stability of quasi-geostrophic β-plane flow over topography is examined. The approach is to first calculate the stationary, asymmetric response to a uniform zonal current flowing over topography, and then calculate the stability properties of the total, zonally asymmetric, field. Under many circumstances this flow is unstable, barotropically and or baroclinically particularly if the asymmetric flow is of large amplitude. This is demonstrated first using a long-wave approximation, which examines the stability with respect to perturbations of large meridional scale. So-called form-drag instability then ensues. This may be thought of as a special, nonlocal, form of isosceles triad interaction involving the zonal flow interacting with the topography and another “free” mode of topographic scale. For topography consisting of a single Fourier mode, instability then arises only if the zonal current is eastward and exceeds that required for resonance. However, in general other triads exist in which the asymmetric flow, if its amplitude is large enough, is always unstable, for any value of the zonal current. In particular, flow with the zonal current slightly below the resonant value can be unstable. This implies that resonantly amplified stationary waves, sometimes cited as possible mechanisms for blocks, will decay rapidly through their interaction with other modes, unless further nonlinear equilibration occurs. Certain integral constraints prove useful in ascertaining necessary conditions for instability, both for topographic instability and the zonally symmetric (but continuously stratified) baroclinic instability problem.

Fundamental Mode Decay Constants for Small Two- and Three-Dimensional Pulsed Neutron Moderator Assemblies

A method has been developed to calculate the fundamental mode decay constants in two- and three-dimensional pulsed neutron moderator assemblies using the separable form of the scattering kernel in the transport equation. The method uses the Fourier transform of the integral transport equation and is an extension of the method developed by Sahni to treat monoenergetic criticality problems for two- and three-dimensional geometries. The new kernel of the integral transform equation is factored into components depending on only one of the dimensions of the assembly. This property is further exploited by use of a single Fourier mode approximation in one or more dimensions while the kernels in the remaining dimensions are retained in their respective forms. In our numerical work, three simple forms of the scattering cross section are used for calculating the matrix elements of the relevant equations accurately. Numerical results are presented for the asymptotic decay constant in a one-dimensional slab, a one-dimensional cylinder, two-dimensional infinite rectangular prisms, and three finite cylinders of different height-to-diameter ratios. The relation between the asymptotic decay constant and the geometrical buckling in the transport and diffusion approximations are also calculated for interpreting the results in terms of extrapolation lengths.

Nonlinear Saturation of the Trapped-Ion Mode

A nonlinear model of the collisional trapped-ion mode is presented, in which the energy in long-wavelength instabilities is transferred to short-wavelength modes which are then damped by ion-bounce resonances. Near marginal stability, the saturation of a single unstable Fourier mode is computed. Far from marginal stability, steady-state nonlinear solitary waves containing many Fourier modes are found. Particle transport is estimated in both cases.