Lagrangian representation

In order to be consistent with other chapters, R(Ct) is defined as a positive number if the chemical is produced in the river and T(Ct) is positive if the net flux is directed from the river into the atmosphere or sediment. Note that (F(Ct) is a flux per unit volume its relation to the usual flux per area as defined, for instance, in Chapter 20, is given below (Eq. 24-15). Again we suppress the compound subscript i wherever the context is clear. The subscript Lagrange refers to what fluid dynamicists call the Lagrangian representation of the flow in which the observer travels with a selected water volume (the river slice ) and watches the concentration changes in the volume while moving downstream. Later the notion of an isolated water volume will be modified when mixing due to diffusion and dispersion across the boundaries of the volume is taken into account. 

The transformation from the Lagrangian representation to the Hamiltonian formaUsm is a special case of a Legendre transformation. A Legendre transformation, quite generally, is the following Suppose we have a function g x,y) with x and y as its two independent variables. The total differential of this function is given by... 

In order to prove equation (6), it is more convenient to work from equation (3) than from the limiting relation given in equation (5) and also to introduce the Lagrangian representation [10]. For any continuum X, let the three parameters af identify the individual point particles of continuum K for... 

A three dimensional extension of the Leibnitz rule for differentiating an integral is relevant for the derivation of the governing transport equations L In the material (Lagrangian) representation of continuum mechanics a representative particle of the continuum occupies a point in the initial configuration of the continuum at time t = 0 and has the position vector = (Ci, 2, Cs)-In this -space the coordinates are called the material coordinates. In the Eulerian representation the particle position vector in r-space is defined by r = (ri,r2,r3). The coordinates ri,r2,r3 which gives the current position of the particle are called the spatial coordinates. Let be any scalar, vector... 

There is a connection between the Lagrangian representation based on advected particles and the Eulerian representation using concentration fields. As in the case of pure advection the solution of the advection-diffusion equation can be given in terms of trajectories of fluid elements. Equation (2.6) can be generalized for the diffusive case using the Feynman-Kac formula (see e.g. Durrett (1996)) as... 

Figure 6.1 Average plankton population density as a function of the Damkohler number for logistic growth with non-uniform carrying capacity of the form K(x,y) = Kq + (5sin(27rx) sin(27ry) and chaotic mixing in the time-periodic sine-flow of Eq. (2.66). The continuous line represents results from the solution of the full partial differential equation with diffusion (Pe 104) and stars ( ) show the time-averaged plankton populations calculated from the non-diffusive Lagrangian representation.

This simplification allows us to use a convenient Lagrangian representation of the problem (6.2). Without the diffusion term the solution can be expressed by the method of characteristics through a set of ordinary differential equations... 

Neufeld et al. (1999) have shown that the roughness exponent a of the decaying chemical field is a function of the decay rate and of the Lyapunov exponent of the advection. In the large Peclet number limit we can neglect diffusion and set D = 0 so that the concentration field can be described by the Lagrangian representation (6.13) that follows the chemical dynamics within the fluid parcels advected on chaotic trajectories in the flow... 

Though these properties have not been very rigorously proved, intuitive arguments show that they are satisfactory. Moreover, Balian and Toulouse12 showed that in the continuous case the properties of a chain can be studied precisely by starting from a Lagrangian representation of the problem (see Chapter 11) and by using a transfer matrix method. In particular, these authors verified that the critical exponents are v = 1 and y = 1 [see also the article by Thouless (1975)]. 3... 

Equations 12.14 and 12.16 are solved for the surface function S and electrostatic potential 0, respectively. These coupled "Laplace-Beltrami and Poisson-Boltzmann" equations are the governing equation for the DG-based solvation model in the Eulerian representation. The Lagrangian representation of the DG-based solvation model has also been derived [72]. Both the Eulerian and Lagrangian solvation models have been shown [71, 72] to be essentially equivalent and provide very good predictions of solvation energies for a diverse range of compounds. 

The dynamics of passive particle (or quantities, e.g., tenqierature or concentration) advection in the inconqiressible, laminar flow may be described by the Lagrangian representation of the fluid element (for low Reynolds nuitibers < 1) with a time-dependent (unsteady) or time-independent (steady) velocity, U, given by 63, 64)... 

The same symbols are used for a physical properties and its Lagrangian representation. 

The first approach starts from the stationary flame equations in Eulerian coordinates. In their second-order form these equations are obtained by omitting the time derivative from the equations of type (4.19) (cf. Lagrangian representations where the distance derivative on the left-hand side is omitted). The further development, as illustrated by Wilde (1972), is essentially the same... 

Another possibility to represent the quantum mechanical Lagrangian density is using the logarithm of the amplitude X = Ina, a = e. In that particular representation, the Lagrangean density takes the following symmetrical fomi... 

V, ip, x, and t) in the PDF transport equation makes it intractable to solve using standard discretization methods. Instead, Lagrangian PDF methods (Pope 1994a) can be used to express the problem in terms of stochastic differential equations for so-called notional particles. In Chapter 7, we will discuss grid-based Eulerian PDF codes which also use notional particles. However, in the Eulerian context, a notional particle serves only as a discrete representation of the Eulerian PDF and not as a model for a Lagrangian fluid particle. The Lagrangian Monte-Carlo simulation methods discussed in Chapter 7 are based on Lagrangian PDF methods. 

The intra-cell processes are common to all PDF codes, and are treated the same in both Eulerian and Lagrangian PDF codes.8 On the other hand, inter-cell processes are treated differently in Eulerian PDF codes due to the discrete representation of space in terms of x . In PDF codes, fractional time stepping is employed to account for each process separately. Methods for treating chemical reactions and mixing are described in Section 6.9. Thus we will focus here on the treatment of inter-cell processes in Eulerian PDF codes. 

Recently in [6] we constructed effective Lagrangians of the Veneziano-Yankielowicz (VY) type for two non-supersymmetric but strongly interacting theories with a Dirac fermion either in the two index symmetric or two index antisymmetric representation of the gauge group. These theories are planar equivalent, at N —> oo to SYM [7], In this limit the non-supersymmetric effective Lagrangians coincide with the bosonic part of the VY Lagrangian. 

A new class of effective Lagrangians have been constructed to show how the information about the center group symmetry is efficiently transferred to the actual physical states of the theory [12-15] and will be reviewed in detail elsewhere. Via these Lagrangians we were also able to have a deeper understanding of the relation between chiral restoration and deconfinement [15] for quarks in the fundamental and in the adjoint representation of the gauge group. 

Riding along with a fluid packet is a Lagrangian notion. However, in the limit of dt - 0, the distance traveled dx vanishes. In this limit, (i.e., at a point in time and space) the Eulerian viewpoint is achieved. The relationship between the Lagrangian and Eulerian representations is established in terms of Eq. 2.52, recognizing the equivalence of the displacement rate in the flow direction and the flow velocity. In the Eulerian framework the... 

Equation (2) is expressed in the Eulerian frame of reference, in which the volume element under consideration is fixed in space, and material is allowed to flow in and out of the element. An equivalent representation of very different appearance is the Lagrangian frame of reference, in which the volume element under consideration moves with the fluid and encapsulates a fixed mass of material so that no flow of mass in or out is permitted. In this frame of reference, Eq. (2) becomes... 

A. Lagrangian Framework. An ideal subgrid model should be constructed on a Lagrangian hydrodynamics framework moving with the macroscopic flow. This requirement reduces purely numerical diffusion to zero so that realistic turbulence and molecular mixing phenomena will not be masked by non-physical numerical smoothing. This requirement also removes the possibility of masking purely local fluctuations by truncation errors from the numerical representation of macroscopic convective derivatives. 

The antiferromagnetic antiferromagnetic spin fluctuations which result in d-pairing in cuprate superconductor are described by the Lagrangian in lattice representation [5-7] ... 

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