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This chapter focuses on the scalar advection equation and develops a Galerkin discretization using the techniques described in Chapter 4. It includes an extended presentation of the discontinuous Galerkin formulation for advection equations. Eigenspectra of the advection operators in both two and three dimensions are reviewed, which are relevant for explicit time stepping. Two forms of a semi-Lagrangian method for advection (strong and auxiliary forms) that could potentially prove very effective in enhancing the speed and accuracy of spectral/hp element methods in advection-dominated problems are discussed. A new section on stabilization techniques is introduced that discusses filters, spectral vanishing viscosity, and upwind collocation.

This chapter focuses on internal gravity waves in a stable thermal stratification. When the amplitude of the fluid velocity is small relative to the amplitude of the phase velocity, a linear analysis, which neglects advection, provides insight to the relation between the wavelength and frequency of internal gravity waves. Furthermore, when thermal and viscous diffusion play relatively minor roles the system can be further simplified by neglecting diffusion. The chapter first describes the linear dispersion relation before discussing the computer code modifications and simulations. In particular, it explains what modifications would be needed to convert one's thermal convection code to a code that simulates internal gravity waves, including the nonlinear and diffusive terms. Finally, it considers the computer analysis of wave energy.

The Earth’s atmosphere is far from equilibrium; it is constantly in motion from the combined effects of gravity and planetary rotation, is constantly absorbing and emitting radiation, and hosts ongoing chemical reactions which are ultimately fuelled by solar photons. It has fluxes of material and energy across its boundaries with the planetary surface, both terrestrial and marine, and also emits a continual outward flux of infrared photons to space. The gaseous atmosphere is manifestly a kinetic system, meaning that its evolution must be described by time dependent differential equations. The equations doing this under the continuum fluid approximation are the Navier–Stokes equations, which are not analytically solvable and which support many non-linear instabilities. We have also seen that the generation of turbulence is a fundamentally difficult yet central feature of air motion, originating on the molecular scale. Non-equilibrium statistical mechanics may offer insight into which steady states a system far from equilibrium as a result of fluxes and anisotropies may migrate, without the need for detailed solution of the explicit path between the states. However, it does not seem possible to demonstrate mathematically that such steady states exist for the atmosphere. A physical view of the planet’s past and probable future suggests that the past and future evolution of the sun and its outgoing fluxes of energy may mean that the air-water-earth system may never have been or will ever be in a rigorously defined steady state. Also, to the human population, the detailed, time-dependent evolution is what matters in many respects. Nevertheless, non-equilibrium statistical mechanics is a discipline which should be applicable in principle to yield information about approximate steady states. These steady states may as a practical matter be definable from the observational record, for example the ice ages and the intervening periods evident in the geological record, or between states with two differing global average abundances of a radiatively active gas such as carbon dioxide. There has been great progress recently in non-equilibrium statistical mechanics, stemming from recent work on the concept of the maximization of entropy production.

Climate is one of the most important determinants of biotic structure and function in the alpine. High winds and low temperatures are defining elements of this ecosystem, requiring adaptations of the alpine biota. Interaction between topography and snowcover strongly influences spatial heterogeneity of microclimate, which in turn influences and is influenced by the distribution of vegetation. For nearly 50 years investigators have used Niwot Ridge to examine and document the climate and its interaction with the biota of the alpine tundra. This chapter reviews some of the many findings of these ongoing bioclimatic investigations. Climate studies started on Niwot Ridge in October 1952 when Professor John W. Marr and his students set up a transect of climate stations across the Front Range between Boulder and the Continental Divide (Marr 1961). There were originally 16 stations in groups of four representing different slope exposures in what he defined as the Lower and Upper Montane Forest, the Subalpine Forest, and the Alpine Tundra ecosystems of the Front Range. After 1 year, the network was reduced to four stations, called Al, Bl, Cl, and Dl, which all had ridge-top locations and ranged from lower montane (Al) to high alpine (Dl). From time to time, these stations were supplemented by other stations that supported particular studies. This was especially true during the International Biological Programme years in the early 1970s when focus on work on the Saddle research site of the Ridge began. Following the establishment of Niwot Ridge and Green Lakes Valley as a Long-Term Ecological Research (LTER) site in 1980, even more intensive climatological work has been conducted. The construction of the Tundra Laboratory in August 1990 facilitated intensive winter climatological studies. Geographical locations and elevational data on most of the stations has been provided by Greenland (1989) and is also found in the LTER electronic database (http://culter.colorado.edu/). The climate of Niwot Ridge is characterized by large seasonal and annual variability with very windy and cold winters, wet springs, mild summers, and cool, dry autumns.

This chapter, which discusses the basics of fluid dynamics, including the Lagrangan and Eulerian description, advection, and the substantive derivative, introduces the Navier-Stokes equation and explains the distinction between inertial and non-inertial flows. It discusses stress and strain tensors and the basic concepts of rheology, including linear and non-linear visco-elastic descriptions. The chapter presents ideas of flow driven by external stress, and of the stochastic fluctuations that arise when flow is directed through a porous solid matrix, including dispersive flow, and its link with Brownian dynamics. It introduces the Péclet number and describes the various types of dispersive flow, both in model geometries and in a random porous medium. The chapter concludes with a discussion of non-local dispersion.

The diffusion equation is derived and solved for simple geometries. Fourier series are described, and superposition is used to combine simple solutions into more complicated ones. Advection is combined with diffusion, and compartment models defining diffusion between contacting systems (e.g. a pill, the gut, the bloodstream and tissues) are described.

The finite-volume method is introduced as a consequence of a general (mass, tracer, energy) conservation problem. The wave equation is derived from this fundamental conservation principle. The concept of numerical fluxes is presented in the context of the Riemann problem, descriptive of the transport of a discontinuity in an advective (hyperbolic) system. The concept of fluxes is used to derive reflection and transmission coefficients at a material interface. The solutions of the scalar and elastic wave equations are presented for both homogeneous and heterogeneous materials.

The discontinuous Galerkin method is introduced as a special type of finite-element method in which the solution fields are allowed to be discontinuous at the element boundaries. This requires the use of the same fluxes as introduced in the chapter on the finite-volume method. The solution field is interpolated using Lagrange polynomials. The discontinuous Galerkin principle leads to an elemental system of equations. Communication between elements is possible through the fluxes. The method is presented for scalar and elastic wave equations for both homogeneous and heterogeneous media. The method can be considered a mixture of the spectral-element and the finite-volume methods.

The first part of this chapter presents a description of the GATE rain rate data (Polyak and North, 1995), its two-dimensional spectral and correlation characteristics, and multivariate models. Such descriptions have made it possible to show the concentration of significant power along the frequency axis in the spatial-temporal spectra; to detect a diurnal cycle (a range of variation of which is about 3.4 to 5.4 mm/hr); to study the anisotropy (as the result of the distinction between the north-south and east-west transport of rain) of spatial rain rate fields; to evaluate the scales of the distinction between second-moment estimates associated with ground and satellite samples; to determine the appropriate spatial and temporal scales of the simple linear stochastic models fitted to averaged rain rate fields; and to evaluate the mean advection velocity of the rain rate fluctuations. The second part of this chapter (adapted from Polyak et al., 1994) is mainly devoted to the diffusion of rainfall (from PRE-STORM experiment) by associating the multivariate autoregressive model parameters and the diffusion equation coefficients. This analysis led to the use of rain data to estimate rain advection velocity as well as other coefficients of the diffusion equation of the corresponding field. The results obtained can be used in the ground truth problem for TRMM (Tropical Rainfall Measuring Mission) satellite observations, for comparison with corresponding estimates of other sources of data (TOGA-COARE, or simulated by physical, models), for generating multiple rain samples of any size, and in some other areas of rain data analysis and modeling. For many years, the GATE data base has served as the richest and most accurate source of rain observations. Dozens of articles presenting the results of the GATE rain rate data analysis and modeling have been published, and more continue to be released. Recently, a new, valuable set of rain data was produced as a result of the TOGA-COARE experiment. In a few years, it will be possible to obtain satellite (TRMM) rain information, and a rain statistical description will be needed in the analysis of the observations obtained on an irregular spatial and temporal grid.

Advection–diffusion models provide a continuous description of tracers in the ocean. The advection–diffusion equation is obtained by combining the conservation equation for an infinitely small box with the equations describing advective and diffusive fluxes. The advection flux is proportional to the current speed and to the tracer concentration. Molecular diffusion is due to the random motion of molecules and the resulting net flux is given by the first Fick law. It is significant at the microscopic scale only like at the sea/air interface. Eddy diffusion is due to the more or less random motions of a turbulent fluid. By analogy with molecular diffusion, net flux is also calculated with the first Fick law. Integration of the advection-diffusion equation yields the spatial and temporal evolution of the tracer in the ocean. This is applied to radium transport in coastal waters and to the deliberate release of SF₆ in the thermocline.

This chapter deals with solute transport, which is a very important part of the study of soil physics, describing the movement of nutrients, salts and contaminants. Its application is particularly important for knowing the rate of loss of various solutes carried in water moving out of the soil, for nutrient budgets and for increasing understanding of nutrient cycling. It can provide knowledge of the amount and concentration of fertilizer nutrients below the root zone and can help in the design of management schemes. Movement of pesticides and other toxic compounds such as heavy metals, viruses and radioactive materials can also be modelled. This chapter describes the advection–dispersion equation, which is the fundamental equation describing the movement of solutes in soil. It then presents a simplified analytical solution and a numerical solution.

The velocity of a fluid at each point of space-time is a vector field (or flow). It is best to think of it in terms of the effect of fluid flow on some scalar field. A vector field is thus a first order partial differential operator, called the material derivative in fluid mechanics. The path of a speck of dust carried along (advected) by the fluid is the integral curve of the velocity field. Even simple vector fields can have quite complicated integral curves: a manifestation of chaos. Of special interest are incompressible (with zero divergence) and irrotational (with zero curl) flows. A fixed point of a vector field is a point at which it vanishes. The derivative of a vector field at a fixed point is a matrix (the Jacobi matrix) whose spectrum is independent of the choice of coordinates.

Both Cartesian tensor and vector notation will be used in this text. The notation xi means the i-component of the vector x = (x1 x2, x3). When used in the argument of a function [e.g., f(xi)], xi, represents the whole vector, so that f(xi) stands for f(xl,x2,x3). Repeated indices indicate summations over all coordinate directions, (uiui = u2i = u21 + u22 + u23). Two special and frequently used tensors are the unit tensor ∂ij and the alternating tensor ε_ijk. The unit tensor has components equal to unity for i = j and zero for i ≠ j. The alternating tensor has components equal to +1 when the indices are in cyclical sequence 1, 2, 3 or 2, 3, 1 or 3, 1,2; equal to -1 when the indices are not cyclical; and equal to zero when two indices are the same. The vorticity vector is defined by the relation The symbol = is used throughout to represent a definition or identity. Conditions near the sea surface are usually very anisotropic. It is often desirable to distinguish between the horizontal and vertical directions. We shall do so by using an x, y, z coordinate system with the origin at mean sea level and the z -axis pointing upward. Unless otherwise specified, the x and y directions will be toward east and north. The vertical velocity will be denoted by W; the horizontal velocity by the vector U with components U and V. Unity vectors in the x, y, z directions are denoted by i, j, k. The usual vector operation symbols will be used only to represent operations within the horizontal plane. For example, In a fluid one has to distinguish between local changes and changes that are experienced by an individual fluid element as it moves about. The former can be recorded by a fixed sensor and is represented by the partial time differential. The individual change could only be recorded by a sensor that would float with the element. It is denoted by the total time differential In a treatise that covers such a variety of topics, some use of the same symbols for different properties is inevitable.

Particles are composed of solids and/or liquids, thus the bulk optical properties of these media must be known before propagation modeling within a medium of suspended particles (called aerosols when in air) can begin. We return to our discussion of propagation in the atmosphere and oceans of the earth that began in Chapters 7 and 9, and we now include attenuation by small particles. Particles vary in size, shape, concentration, and composition. Size and concentration distributions are described in the following two sections. The composition of the most common particles is presented in the last section. Unfortunately, a representation of shape variation does not exist. As mentioned in Chapter 4 (Section 4.4.2 on Mie scattering), a collection of real aerosols will have a range of different radii. This is called a polydisperse medium. Various models are used to represent particle size distributions. One commonly used model for particle number density as a function of radius is the modified gamma distribution function, as given by . . . ρp(r) = Arα exp(−brγ), (10.1) . . . where A, b, α, and γ are empirically determined parameters. This function represents the number of particles per unit volume and unit radius as a function of radius r. The total particle number density is obtained by integrating ρp(r ) over all r.

Since evaporation represents some 60% of precipitation over land surfaces, it is crucial for hydrologic purposes to know with some degree of certainty the magnitude of the water vapor flux into the atmosphere. Actual evaporation (E) from drying land surfaces is often formulated, in hydrology, as a fraction of some measure of potential evaporation (Ep), which can be written as a bulk transfer relationship: . . . Ep =CE up(qs* -q) (10.1) . . . where CE is the bulk mass transfer coefficient for water vapor, u is the mean wind speed at reference height z above the surface, r is the density of the air, q is the mean specific humidity at z, and q*s is the saturation specific humidity at the temperature of the surface (Ts) (Brutsaert, 1982, 1986).

So far our treatment of fluid motions has not emphasized the behavior of fluids residing within porous geological materials. Let us now turn to this topic and, in doing so, make use of our insight regarding purely fluid flows. The general topic of fluid behavior within porous geological materials is an extensive one, forming the heart of such fields as groundwater hydrology, soils physics, and petroleum-reservoir dynamics. In addition, this topic is an essential ingredient in studies concerning the physical and chemical evolution of sedimentary basins, and the dynamics of accretionary prisms at convergent plate margins. In view of the breadth of these topics, the objective of this chapter is to introduce essential ingredients of fluid flow and transport within porous materials that are common to these topics. Our first task is to examine the physical basis of Darcy’s law, and to generalize this law to a form that can be used with an arbitrary orientation of the working coordinate system relative to the intrinsic coordinates of a geological unit that are associated with its anisotropic properties. We will likewise examine the basis of transport of solutes and heat in porous materials. We will then develop the equations of motion for the general case of saturated flow in a deformable medium. In this regard, several of the Example Problems highlight interactions between flow and strain of geological materials during loading, because this interaction bears on many geological processes. Examples include consolidation of sediments during loading, and responses of aquifers to loading by oceanic and Earth tides, and seismic stresses. We will concentrate on the description of diffuse flows within the interstitial pores of granular materials, as opposed to flows within materials containing dual, or multiple, pore systems such as karstic media, or media containing both interstitial and fracture porosities. We will consider unsaturated, as well as saturated, conditions. For simplicity, the subscript h is omitted from the notation of quantities such as specific discharge q and hydraulic conductivity K.