Approximations quasi-static

When the Pecet number, the measure of the relative importance of advection to diffusion, is small, which is the case for high viscosity magmas, the temporal derivation of concentration and the advection term in Eq. (13.31) may be ignored, and quasi-static approximation may be developed. In this case, Eq. (13.31) reduces to 

Conservation of water mass in the spherical shell of melt that surrounds the bubble at r = 5 (the second boundary condition) requires. s.  

When S R (thick melt shell), and ignoring the surface tension a, then Eq. (13.40) reduces to 

We also need to find the variation of in the bubble, dp ldt can be obtained by rewriting Eq. (13.23) as  

---

Pressure induced broadening and narrowing of a whole spectrum are described by the quasi-static approximation and the perturbation theory, correspondingly. Comparing inequalities (6.13) and (2.53) one can see that the border between the stages is determined by the criterion coij 1,... 

For porous catalyst pellets with practical loadings, this quantity is typically much larger than the pellet void fraction e, indicating that the dynamic behavior of supported catalysts il dominated by the relaxation of surface phenomena (e.g., 35, 36). This implies that a quasi-static approximation for Equation (1) (i.e., e = 0) can often be safely invoked in the transient modeling of porous catalyst pellets. The calculations showed that the quasi-static approximation is indeed valid in our case the model predicted virtually the same step responses, even when the value of tp was reduced by a factor of 10. 

The main purpose of this paper is to consider a two-dimensional non-stationary (2D-I-T) problem of a nonlinear waveguide excitation by a non-stationary light beam and to study spatiotemporal phenomena arising upon propagation of the beam in a step-index waveguide, first, in the quasi-static approximation and, second, with account of MD and SS effects. 

The theoretical approach is based on the solution to the mixed type linear/nonlinear generalized Schrodinger equation for spatiotemporal envelope of electrical field with account of transverse spatial derivatives and the transverse profile of refractive index. In the quasi-static approximation, this equation is reduced to the linear/nonlinear Schrodinger equation for spatiotemporal pulse envelope with temporal coordinate given as a parameter. Then the excitation problem can be formulated for a set of stationary light beams with initial amplitude distribution corresponding to temporal envelope of the initial pulse. 

All the above results can be directly used in the problem of optical pulse propagation through the junctions provided that the quasi-static approximation is feasible. As the transmittance of a waveguide junction depends on power of a stationary component of the pulse, variation of an input pulse envelope behind the junction should be observed. 

Quasi-static approximation propagation through the junctions... 

Thus, in the quasi-static approximation, the length of the unsteady-state regime in the core of the nonlinear waveguide is finite, as in the case of the stationary light beam propagation (see Fig. 10, 11). 

Behind the junction, power of the field propagating within the core increases due to the self-focusing effect, while the pulse duration at the waveguide axis decreases. In the quasi-static approximation, this effect does not depend on the initial pulse duration. Total losses vary with power at the pulse peak similar to the case of stationary wave propagation in the structure A, i.e. they increase with the power (Fig.21, compare with Fig.l 1). 

Figure 28. Longitudinal variation of power of the light beam calculated over the computational window in nonlinear waveguide of the stmcture A, To(0)=20fs (dashed line), To(0)= lOfs (dotted line), solid line corresponds to the quasi-static approximation.

Consider now how the solution obtained in the quasi-static approximation changes if the second-order group velocity dispersion is taken into account. It is seen from Fig. 29 that in a nonlinear waveguide of the structure A with the MD and the SS effects is first observed that spatial and temporal parameters of the field vary similarly to the case of the quasi-static approximation. Then, at a given power, spatiotemporal distribution varies depending on the value and sign of the dispersion coefficient 2- The pulse duration decreases in the case of anomalous GVD (k2 < 0) and increases in the case of normal GVD ( 2 > 0). 

Figure 30. Longitudinal dependence of power of the pulse (2.17) propagating within the waveguide core for some values of 2 (A 2=0 001r0 001,0 004-0.004), in fs /pm. = 3 (solid lines), 4 (dashed lines), tq = 16fs, a = 3.0pm qsa - quasi-static approximation.

The kinetic function f(T,a) was assumed to be a first-order equation. For a quasi-static approximation, we can write the following equation for the rate of crystallization ... 

Metal nanostructures (such as particles and apertures) can permit local resonances in the optical properties. These local resonances are referred to as localized surface plasmons (LSPs). The simplest version of the LSP resonance comes for a spherical nanoparticle, where the electromagnetic phase-retardation can be neglected in the quasi-static approximation, so that the electric field inside the particle is uniform and given by the usual electrostatic solution [3] ... 

When a metallic particle is small compared with the illuminating wavelength, the variation of the electric field can be ignored, and the scattering behaviors can be described under the quasi-static approximation. Consider a smaller metallic sphere with radius a and a A placed in a uniform static electric field E — EqZ. The field potentials inside and outside the sphere, < ). and, are written by... 

Following the method of Stroud aird Hui [81], Stroud and Wood [82], and later Ma et al. [83], have derived from Maxwell s equations the general expression of the effective in the quasi-static approximation. For this, they have considered that the magnimde of the nonlinear coefficients remains sufficiently small to neglect the nonlinearity in the electric field evaluation, x // then writes... 

The electric field inside the sample is usually evaluated from the quasi-static approximation... 

---