Laplacian field

More than 20 years ago, Matsushita et al. observed macroscopic patterns of electrodeposit at a liquid/air interface [46,47]. Since the morphology of the deposit was quite similar to those generated by a computer model known as diffusion-limited aggregation (D LA) [48], this finding has attracted a lot of attention from the point of view of morphogenesis in Laplacian fields. Normally, thin cells with quasi 2D geometries are used in experiments, instead of the use of liquid/air or liquid/liquid interfaces, in order to reduce the effect of convection. 

In other words, in a solenoidal Beltrami field the vector lines are situated in the surfaces c = constant. This theorem was originally derived by Ballabh [4] for a Beltrami flow proper of an incompressible medium. For the sake of completeness, we mention that the combination of the three conditions (1), (2), and (3) only leads to a Laplacian field, that is better defined by a vector field that is both solenoidal (divergence-less) and lamellar (curlless). 

As in all cases, we are going to consider that within the boundary layer, we have a Laplacian field ... 

The substitutions of ffiper y) into the Laplacian field (15.7) produce a linearized homogeneous differential equation of constant coefficients ... 

A full description is beyond the scope of this review, but it is noted that the topological method identifies other chemical features in the electron density. The union of all bond paths gives a bond path network that is normally in a 1 1 correspondence with the chemical bond network drawn by chemists. The bond paths for bonds in strained rings are curved, reflecting their bent nature. In Figure 6, we show the gradient paths in the molecular plane of cyclopropane. The C—C bond paths are distinctly bent outward. The value of the Laplacian at the bond critical point discriminates between ionic and covalent bonding." Maps of the Laplacian field reveal atomic shell structure, lone pairs, and sites of electrophilic and nucleophilic attack. The ellipticity of a bond measures the buildup of density in one direction perpendicular to the... 

At this point, as was previously anticipated, Eq. (5.404) can be seen in two ways. Within the first case, abbreviated as Ml Markovian one," Eq. (5.404) may represent the Laplacian field of Eq. (5.386) that provides the unfolded function... 

Actually, this is not really diffusion-XimiiQd, but rather Laplacian growth, since the macroscopic equation describing the process, apart from fluctuations, is not a diffusion equation but a Laplacian equation. There are some crucial differences, which will become clearer below. In some sense DLA is diffusion-limited aggregation in the limit of zero concentration of the concentration field at infinity. 

By definition, the Laplacian of U represents the divergence of the attraction field, and, correspondingly, its value characterizes the density of masses at same point. Now the following question arises. What does the Laplacian tells us about the behavior of the potential To answer this question we first consider the simplest case, when U depends on one argument, x, Fig. 1.7a. Then, we can represent the derivatives as ... 

Of course, both statements can be proved from the theorem of uniqueness for the attraction field. In addition, it is appropriate to comment a linear function reaches its maximum at terminal points of the interval. The same behavior is observed in the case of harmonic functions, which cannot have their extreme inside the volume. Otherwise, the average value of the function at some point will not be equal to its value at this point, and, correspondingly, the Laplacian would differ from zero. At the same time, saddle points may exist. 

The Laplacian is constructed from second partial derivatives, so it is essentially a measure of the curvature of the function in three dimensions (Chapter 6). The Laplacian of any scalar field shows where the field is locally concentrated or depleted. The Laplacian has a negative value wherever the scalar field is locally concentrated and a positive value where it is locally depleted. The Laplacian of the electron density, p, shows where the electron density is locally concentrated or depleted. To understand this, we first look carefully at a onedimensional function and its first and second derivatives. 

The first term on the right-hand side is the expected value of the scalar Laplacian conditioned on the scalars having values r//, 33 An example of the time evolution of the conditional scalar Laplacian, corresponding to the scalar PDF in Fig. 1.11, is plotted in Fig. 1.12 for an initially non-premixed inert-scalar field. The closure of the conditional scalar Laplacian is discussed in Chapter 6. For the time being, it suffices to note the similarity between (1.36) and the IEM model, (1.16). Indeed, the IEM model is a closure for the conditional scalar Laplacian, i.e.,... 

Note that A, and , will, in general, depend on multi-point information from the random fields U and 0. For example, they will depend on the velocity/scalar gradients and the velocity/scalar Laplacians. Since these quantities are not contained in the one-point formulation for U(x, t) and 0(x, f), we will lump them all into an unknown random vector Z(x, f).16 Denoting the one-point joint PDF of U, 0, and Z by /u,,z(V, ip, z x, t), we can express it in terms of an unknown conditional joint PDF and the known joint velocity, composition PDF ... 

In sections 1.1.2, 1.19, and 3.1, we have seen applications of the Laplacian operator to the vectorial field. 

A further vector identity defines the Laplacian of a vector field as... 

In this form one sees an analogy in the vorticity equation to the other transport equations— a substantial-derivative description of advective transport, a Laplacian describing the diffusive transport, and possibly a source term. It is interesting to observe that the vorticity equation does not involve the pressure. Since pressure always exerts a normal force that acts through the center of mass of a fluid packet (control volume), it cannot alter the rotation rate of the fluid. That is, pressure variations cannot cause a change in the vorticity of a flow field. 

However, if the electric resistivity is taken into account, there appears in general a new, very fundamental circumstance the normal component of the Laplacian of an arbitrary vector also depends on the tangential components of the vector. The magnetic viscosity creates feedback, due to which the tangential components are able to modify (in particular, to amplify) the normal component. Therefore, in principle, exponential, coordinated amplification of all fields also becomes possible, i.e., a dynamo But this is a peculiar kind of dynamo that depends on the magnetic viscosity. 

A more complicated situation emerges in motion along nonintersecting surfaces with variable curvatures. If the distance between these surfaces remains finite everywhere, then the field lines do not expand infinitely in the directions normal to the surfaces. In the absence of dissipation this means that there is no unbounded growth of the normal field component. However, introduction of the finite conductivity yields an equation for the normal component which is not decoupled it contains the contribution of the Laplacian of the remaining components. At the same time, it is possible for all other components to increase exponentially with an increment which depends on the conductivity and vanishes for infinite conductivity. The authors called this mechanism of field amplification a slow dynamo, in contrast to the fast dynamo feasible in the three-dimensional case, i.e., the mechanism related only to infinite expansion of the field lines as, for example, in motion with magnetic field loop doubling. In a fast dynamo the characteristic time of the field increase must be of the same order as the characteristic period of the motion s fundamental scale. 

Hence, the early stage of the concentration profile is proportional to the Laplacian of T(r,t) rather than the temperature field itself. From the stationary solution of (25) it can be seen that V2T(r,f) is proportional to the laser intensity, neglecting temperature dependences of the coefficients and convection in a first approach. 

The occurrence of demixing morphologies characteristic for the metastable regime between the binodal and the spinodal can be understood from Fig. 17. The red dot marks the initial position of the sample with c = 0.3. Upon laser heating the temperature within the laser focus rises by AT and the distance to the binodal first increases. A stationary temperature distribution is rapidly reached and the Laplacian of the temperature field T(r,t) is obtained from the stationary solution of the heat equation (5) with the power absorbed from the laser as source term ... 

As was stressed before, what is observed is the shear field. In practice it is likely that the measured shear field is not purely of cosmological origin due to contamination or systematics effects. The geometrical properties of the cosmological shear field can however be used to derive means for testing the amount of systematics in the data. The idea is the following. Any 2D spin 2 vector field can be decomposed into a scalar and a pseudo scalar parts. They are defined through their Laplacian,... 

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