Analytical Linear Solution Research Articles

Effect of solid dust particles on the propagation of shock wave in planar and non-planar gasdynamics

The aim of the present study is to analyze the propagation of shock wave along the characteristic path in planar and non-planar unsteady compressible ideal gas flow in presence of small solid dust particles. The analytical solution of the governing quasilinear hyperbolic system is computed in the characteristic plane and it is found that this analytical linear solution in this plane can exhibit non-linear phenomenon in the physical plane. The effect of the dust particles on the evolutionary behavior of the propagating shock wave in ideal gas flow is discussed. The transport equations leading to the evolution of shock wave is determined which introduces the conditions of shock formation. The growth and decay of compressive waves and expansive waves, respectively, in planar and non-planar ideal gas dynamics influenced by the presence of small solid dust particles, is discussed.

Inversion of fluvial channels for paleorock uplift rates in Taiwan

AbstractThe transient response of erosion to changes in rock uplift rate leads to the preservation of rock uplift history in the long profiles of rivers. However, extracting this information is nontrivial as changes in channel steepness are the result of both spatial and temporal changes in rock uplift rate, as well as other factors such as climate and rock type. We exploit an analytical linear solution for river channel profile evolution in response to erosion and tectonic uplift to investigate the rock uplift history of Taiwan. The analytical approach allows us to solve the linear inverse problem, efficiently extracting rock uplift as a function of space and time, from digital elevation data. We assess the potential of fluvial topography to resolve rock uplift rates using three approaches: (1) a synthetic resolution test, (2) analysis of the forward model to demonstrate where in space and time the fluvial topography constrains rock uplift rate, and (3) interpretation of the model resolution matrix. Furthermore, the potential to analyze large data sets reduces the influence of stochastic processes such as landslides, small‐scale river network reorganization, and also local lithological variability. In Taiwan, our analysis suggests that current rock uplift rates exceed erosion rates across much of the island and that there has been an increase in rock uplift rates since 0.5 Ma across the Central Range.

Element-free galerkin method in eddy-current problems with ferromagnetic media

This paper focuses on the application of the element-free Galerkin method to the solution of eddy-current problems in ferromagnetic media. The method is first validated by comparison with an analytical linear solution, discussing the role of the numerical parameters that affect the solution accuracy. Then, the application is extended to nonlinear and hysteretic media, adopting the fixed point iterative scheme and the classical Preisach model. The proposed method is also tailored for nonhomogenous structures. The merits and drawbacks with respect to a finite-element approach are discussed for all the considered problems.

Eddy current problems in nonlinear media by the element-free Galerkin method

The element-free Galerkin method is applied to the solution of eddy current problems in nonlinear and hysteretic media. The method is first validated by comparison with an analytical linear solution and successively applied to different nonlinear problems discussing merits and drawbacks with respect to a Finite Element approach.

Nonlinear Saturation of Topographically Forced Rossby Waves in a Barotropic Model

A quasigeostrophic barotropic model is used to examine the nonlinear saturation of forced Rossby waves and the role of wave‐wave interactions in limiting the wave growth. A simple mechanism, based on wave interference, is used to produce strong transient eddy growth and an analytical linear solution for the flow evolution is used as a starting point. Given the rigid upper bound on wave growth, set by the potential enstrophy conservation principle, the linear solution is bound to break down at high forcing amplitudes. An analytical quasi-linear solution, which guarantees potential enstrophy conservation, is formulated and its domain of validity is examined with a numerical nonlinear model. The nonlinear flow evolution is shown to bear strong similarity to the analytical quasi-linear solution and wave‐mean flow interactions are found to be always sufficient to limit wave growth. The saturation of the forced disturbances is shown to come through the deceleration of the mean flow and the modification of the topographic vorticity forcing. Overall, wave‐wave interactions prove not to be important in the saturation process in the examples considered. While the authors consider the implications of this result for the observationally more relevant case of vertically propagating Rossby waves, explicit calculations are clearly called for.

Steady-State Drawdowns in Coupled Aquifiers

The analytical linear solution that predicts the steady-state drawdowns for a coupled two-aquifer system in which pumping from an underlying artesian aquifer is balanced by a reduction in evapotranspiration from an overlying water-table aquifer was developed. An example illustrating how the drawdown equations can be used to determine whether significant drawdowns will occur in the water-table aquifer is presented. These drawdown equations can be used for preliminary analysis and planning studies and to test the convergence of digital model solutions when these more sophisticated techniques are warranted. A major limitation to using these equations is that the predicted water-table drawdowns cannot be so great that evapotranspiration would cease altogether and, thus, could not be reduced any further to balance the pumping.