We perform direct numerical simulations of quasi-static magnetohydrodynamic turbulence, and compute various energy transfers including the ring-to-ring and conical energy transfers, and the energy fluxes of the perpendicular and parallel components of the velocity field. We show that the rings with higher polar angles transfer energy to ones with lower polar angles. For large interaction parameters, the dominant energy transfer takes place near the equator (polar angle θ ≈ π 2 ). The energy transfers are local both in wavenumbers and angles. The energy flux of the perpendicular component is predominantly from higher to lower wavenumbers (inverse cascade of energy), while that of the parallel component is from lower to higher wavenumbers (forward cascade of energy). Our results are consistent with earlier results, which indicate quasi two-dimensionalization of quasi-static magnetohydrodynamic (MHD) flows at high interaction parameters.

The field enhancement factor at the emitter tip and its variation in a close neighbourhood determines the emitter current in a Fowler-Nordheim like formulation. For an axially symmetric emitter with a smooth tip, it is shown that the variation can be accounted by a cosθ˜ factor in appropriately defined normalized co-ordinates. This is shown analytically for a hemiellipsoidal emitter and confirmed numerically for other emitter shapes with locally quadratic tips.

The Child–Langmuir law relates the voltage applied across a planar diode to the saturation value JCL of current density that can be transmitted through it in case the injection velocity of electrons is zero. The Child–Langmuir current density JCL is, at the same time: (i) the maximum current density that can be transmitted through a planar diode, (ii) the current density below which the flow is steady and unidirectional in the long time limit, and (iii) the average transmitted current density for any value of injected current density above JCL. Existing generalizations of Child–Langmuir law to nonzero velocities of injection are based on the characteristics (i) and (ii) of JCL. This paper generalizes the law to nonzero velocities of injection based on the characteristic (iii) by deriving an analytical expression for the saturation value of current density. The analytical expression for the saturation current density is found to be well supported by numerical computations. A reason behind preferring the saturation property of the Child–Langmuir current density as the basis for its generalization is the importance of that property in numerical simulations of high current diode devices.

Modeling high aspect ratio field emitter arrays is a computational challenge due to the enormity of the resources involved. The line charge model (LCM) provides an alternate semi-analytical tool that has been used to model both infinite as well as finite sized arrays. It is shown that the linearly varying charge density used in the LCM generically mimics ellipsoidal emitters rather than a Cylindrical-Post-with-an-Ellipsoidal-Tip (CPET) that is typical of nanowires. Furthermore, generalizing the charge density beyond the linear regime allows for modeling shapes that are closer to a CPET. Emitters with a fixed base radius and a fixed apex radius are studied with a view to understanding the effect of nonlinearity on the tip enhancement factor and the emitter current in each case. Furthermore, an infinite square array of the CPET emitters is studied using the nonlinear line charge model, each having a height h=1500 μm and a base radius b=1.5 μm. It is found that for moderate external field strengths (0.3−0.4 V/μm), the array current density falls sharply for lattice spacings smaller than 43h. Beyond this value, the maximal array current density can be observed over a range of lattice spacings and falls gradually thereafter.