Environmental fluid mechanics

The study of the flows of air and water, of the species carried by them, and of their interactions with geological, biological, social, and engineering systems in the vicinity of a planet's surface. The environment on the Earth is intimately tied to the fluid motion of air (atmosphere), water (oceans), and species concentrations (air quality). In fact, the very existence of the human race depends upon its abilities to cope within the Earth's environmental fluid systems.

Meteorologists, oceanologists, geologists, and engineers study environmental fluid motion. Weather and ocean-current forecasts are of major concern, and fluid motion within the environment is the main carrier of pollutants. Biologists and engineers examine the effects of pollutants on humans and the environment, and the means for environmental restoration. Air quality in cities is directly related to the airborne spread of dust particles and of exhaust gases from automobiles. The impact of pollutants on drinking-water quality is especially important in the study of ground-water flow. Likewise, flows in porous media are important in oil recovery and cleanup. Lake levels are significantly influenced by climatic change, a relationship that has become of some concern in view of the global climatic changes that may result from the greenhouse effect (whereby the Earth's average temperature increases because of increasing concentrations of carbon dioxide in the atmosphere).  See also: Air pollution; Greenhouse effect; Weather forecasting and prediction

Scales of motion

Environmental fluid mechanics deals with the study of the atmosphere, the oceans, lakes, streams, surface and subsurface water flows (hydrology), building exterior and interior airflows, and pollution transport within all these categories. Such motions occur over a wide range of scales, ranging from eddies on the order of centimeters to large recirculation zones the size of continents. This range accounts in large part for the difficulties associated with understanding fluid motion within the environment. In order to impart motion (or inertia) to the atmosphere and oceans, internal and external forces must develop. Global external forces consist of gravity, Coriolis, and centrifugal forces, and electric and magnetic fields (to a lesser extent). The internal forces of pressure and friction are created at the local level, that is, on a much smaller spatial scale; likewise, these influences have different time scales. The winds and currents arise as a result of the sum of all these external and internal forces.

Global motion is the largest scale (greater than 5000 km or 3000 mi); synoptic scale motion is the next largest (100–1000 km or 60–600 mi). Mesoscale motion occurs over regional areas (10–100 km or 6–60 mi); local motion is commonly referred to as microscale motion (less than 100 m or 300 ft). Humans live in the microscale associated with atmospheric motion, that is, the boundary layer of air that extends about 1 km (0.6 mi) above the Earth's surface. The terrain of the Earth as well as ocean surface conditions significantly affects both the microscale and mesoscale motion of the atmosphere, and hence weather conditions. The scales of motion range from eddies on the order of centimeters to huge masses extending over thousands of kilometers.  See also: Mesometeorology; Micrometeorology

El Niño, the periodic flow of warm waters along the western coast of South America, disrupts the coastal ocean and the upwelling of cold waters, producing large amounts of precipitation, along with widespread destruction of plankton, fish, and sea birds (which prey on the fish). Major El Niño events occurred in 1925, 1941, 1957–1958, 1972–1973, 1982–1983, and 1992. It has been determined that the events are caused by changes in the surface winds over the western tropical Pacific, which periodically release and drive warm waters eastward to the South American continent.  See also: El Niño

The study of air pollution falls within the category of environmental fluid mechanics, because the air within the lower atmosphere steers (or advects) and diffuses pollutants. Atmospheric winds near the Earth's surface are generally turbulent and gusty, which helps to clear polluted areas; the velocity varies with altitude, local stability (level of turbulence), and roughness of the terrain. However, when the winds are calm, stagnant conditions can occur which subsequently prevent pollutants from being cleared from a city, resulting in high levels of bad air quality and smog. Of particular importance on the mesoscale level is acid rain (whereby rainfall removes sulfates and nitrates within the atmosphere), which has resulted in serious environmental damage. Likewise, mixing of pollutants into the upper atmosphere can cause long-term changes in the ozone layer (even though the causes, such as propellants within spray cans, may have been generated within the microscale layer). The 1991 explosion of Mount Pinatubo in the Philippines resulted in the discharge of many tons of particulates into the Earth's atmosphere; these particulates in turn acted as seed nuclei for precipitation, and were the cause of much of the flooding and climatic changes over the following few years.  See also: Weather modification

The various species found in the atmosphere have a wide range of lifetimes (residence time in the atmosphere). Species with short lifetimes have small spatial scales; those with lifetimes of years have spatial scales comparable to the entire atmosphere (Fig. 1). For example, the hydroxyl radical (OH) has a very short lifetime and small scale; methane (CH4) has a lifetime of nearly 10 years and can become mixed over the entire Earth.

Fig. 1  Spatial and temporal scales of variability for atmospheric constituents. 1 m = 3.3 ft; 1 km = 0.6 mi. (After J. H. Seinfeld and S. N. Pandis, Atmospheric Chemistry and Physics, Wiley, 1998)

Governing equations

The foundations of environmental fluid mechanics lie in the same conservation principles as those for fluid mechanics, that is, the conservation of mass, momentum (velocity), energy (heat), and species concentration (for example, water, humidity, other gases, and aerosols). The differences lie principally in the formulations of the source and sink terms within the governing equations, and the scales of motion. These conservation principles form a coupled set of relations, or governing equations, which must be satisfied simultaneously. The governing equations consist of nonlinear, independent partial differential equations that describe the advection and diffusion of velocity, temperature, and species concentration, plus one scalar equation for the conservation of mass. In general, environmental fluids are approximately newtonian, and the momentum equation takes the form of the Navier-Stokes equation. An important added term, neglected in small-scale flow analysis, is the Coriolis acceleration, 2Ω ×V, where Ω is the angular velocity of the Earth and V is the flow velocity.  See also: Conservation laws (physics); Conservation of energy; Conservation of mass; Conservation of momentum; Coriolis acceleration; Differential equation; Diffusion; Fluid-flow principles; Navier-Stokes equation; Newtonian fluid

Driving mechanisms of flow

The mechanisms which drive the flow patterns in the atmosphere and oceans are vastly different. The atmosphere is thermodynamically driven, with the major source of energy coming from solar radiation. Short-wave radiation traverses the air and becomes partially absorbed by the land and oceans, which reemit the radiation at longer wavelengths. Long-wave radiation heats the atmosphere from below, creating convection currents in the atmosphere.  See also: Atmospheric general circulation

In the oceans, periodic gravitational forces of the Sun and Moon generate tides; in addition, the ocean surface is affected by wind stress that drives most of the ocean currents. Local differences between the air and sea temperatures generate heat fluxes, evaporation, and precipitation, which ultimately act as thermodynamical forces that create or modify wind-driven currents.  See also: Ocean circulation

Environmental layers

Fortunately, not every term in the Navier-Stokes equation is important in all layers of the environment. The horizontal component of the motion is usually the most significant and is subjected to maximum frictional forces at atmosphere-ocean interfaces. This frictional force causes the formation of a boundary layer in which the velocity of air at the surface of the Earth is zero (relative to the Earth), and the velocity at the surface of the ocean is a minimum equal to the surface velocity of the water. The ocean current is primarily generated by the wind; hence, the water velocity at the surface is a maximum and decreases in depth, again as a result of frictional forces. In both instances, frictional forces cause strong velocity gradients and vorticity (rotation) within the boundary layer. Figure 2 shows the velocity distribution in the atmosphere and ocean.  See also: Boundary-layer flow

Fig. 2  Velocity distribution in the atmosphere and ocean. (After S. Eskinazi, Fluid Mechanics and Thermodynamics of Our Environment, Academic Press, 1975)

The thickness of the atmospheric boundary layer varies with the wind speed, degree of turbulence, and type of surface. For atmospheric flows, the layer is on the order of 1 km (0.6 mi) thick; within the ocean, it may be 30 m (100 ft) thick. Beyond this layer, the environmental flow is typically considered to be viscous-free (without turbulent shear), or inviscid. The rougher the terrain, or the larger the surface obstructions, the thicker the atmospheric boundary layer becomes, and the more gradual the increase of velocity with height (Fig. 3). The influence of the ground on the wind profile extends from a few hundred meters to over 500 m (1640 ft), depending on the roughness of the surface. Above this height, velocity is established from upper level meteorology. The wind speed is proportional to some power of height (empirically determined from experiments) above the surface.

Fig. 3  Atmospheric boundary-layer profiles (plots of average wind speed ū versus height z) over different terrains. Wind speeds are expressed as percentages of the upper level wind (referred to as the gradient wind) above the boundary (or surface) layer. (After E. J. Plate, Aerodynamic Characteristics of Atmospheric Boundary Layers, AEC Critical Review Series, U.S. Department of Energy, 1971)

Because there are no shear stresses, the motion of the inviscid layer is governed only by the advection, pressure, and body-force terms. In atmospheric flows, the rotation of the Earth strongly influences this layer of flow, generally referred to as the geostrophic layer. Just above the surface, the mean velocities are small; the advection terms and the Coriolis force (which depends on the velocity) are negligible compared to the shear forces (viscous terms) which appear to be constant in this inner layer. However, within the outer, or Ekman, layer advection is still negligible and the viscous forces are small; this part of the boundary layer is in equilibrium with the Coriolis, pressure, and Reynolds stresses (turbulence). The table shows typical scales of length, velocity, and time for both atmospheric and oceanic motions. Oceanic motions are slower and more confined, and tend to evolve more slowly, than atmospheric motions.

Relative importance of terms

The key to being able to obtain solutions to the Navier-Stokes equation lies in determining which terms can be neglected in specific applications. For convenience, problems can be classified on the basis of the order of importance of the terms in the equations utilizing nondimensional numbers based on various ratios of values. The Rossby number (Ro) is the ratio of the advection (or inertia) forces to the Coriolis force, Ro = V/LΩ, where V is velocity, Ω is the Earth's angular velocity, and L is a specified reference length. When the Rossby number is much less than 1, the inertia forces become insignificant, implying that these types of flows are more geostrophic. The ratio of the viscous to Coriolis forces is defined by the Ekman number, Ek = μ/ρΩH2, where ρ is density, μ is viscosity, and H is a vertical reference height (or thickness). The ratio of inertia to viscous forces is referred to as the Reynolds number, Re = ρVL/μ. The Rossby number divided by the Ekman number yields the Reynolds number, that is, Re = ρVL/μ = (V/ΩL)(ΩρH2/μ) (L2/H2) = (Ro/Ek)(L/H)2. When the Rossby number is large and the Ekman number is small, the motion is geostrophic; when the Rossby number is small and the Ekman number large, an Ekman-type boundary layer develops. As the Reynolds number increases, the ratio of the flow velocity to viscosity increases (that is, the advection terms become more important than the viscous terms), with the flow eventually becoming turbulent. Since the Ekman number is generally small and the geometric ratio (L/H) is large (Ro is on the order of unity), the Reynolds number for geophysical flow is generally large and the flow turbulent.  See also: Dimensionless groups; Geostrophic wind; Reynolds number; Turbulent flow; Viscosity

Measurements

Because of the scales of motion and time associated with the environment, and the somewhat random nature of the fluid motion, it is difficult to conduct full-scale, extensive experimentation. Likewise, some quantities (such as vorticity or vertical velocity) resist direct observations. It is necessary to rely on the availability of past measurements and reports (as sparse as they may be) to establish patterns, especially for climate studies. However, some properties can be measured with confidence.

Both pressure and temperature can be measured directly in the atmosphere and ocean with conventional instruments. In the ocean, depth is typically calculated from measured pressures obtained from instruments lowered into the sea. In the atmosphere, ground precipitation, radiative heat fluxes, and moisture content can be accurately measured. Likewise, the salinity of the ocean can be determined from electrical conductivity, and the levels monitored at shore stations. Concentration samples, collected at receptor sites over long periods of time, are examined to determine specific concentration levels and particulate sizes. These data are used to determine isopleth (concentration) levels and exposures over various atmospheric and oceanic conditions. Occasionally, inert tracer gases are released into the atmosphere to determine wind directions as well as atmospheric diffusion (turbulence levels) and plume trajectories.

Vector quantities such as horizontal winds and currents are typically measured by using anemometers and current meters. Anemometers atop buildings and towers, and current meters attached to mooring lines at fixed depths, offer fine temporal readings but are too expensive to adequately cover large areas. Instruments are routinely deployed on drifting platforms in the ocean, and balloons are released in the atmosphere. (However, such measurements are mixed in time and space.) Measuring the three-dimensional velocity components simultaneously and obtaining meaningful three-dimensional heat fluxes is difficult, and essentially relegated to small-scale laboratory experiments.  See also: Anemometer

Advances utilizing satellite imagery, Doppler radar, acoustic sounding, and lidar (laser) have made it possible to obtain highly detailed data, including turbulence information, over much broader spatial distances. Doppler radar has yielded three-dimensional velocity data and rotational characteristics within thunderstorms that can be used reliably to predict the onset of tornadoes.  See also: Doppler radar; Lidar; Meteorological instrumentation; Meteorological satellites

Modeling

There are two types of modeling strategies: physical models and mathematical models. Physical models are small-scale (laboratory) mockups that can be measured under variable conditions with precise instrumentation. Such modeling techniques are effective in examining wind effects on buildings and species concentrations within city canyons (flow over buildings; Fig. 4). Generally, a large wind tunnel is needed to produce correct atmospheric parameters (such as Reynolds number) and velocity profiles. Mathematical models (algebra- and calculus-based) can be broken down further into either analytical models, in which an exact solution exists, or numerical models, whereby approximate numerical solutions are obtained using computers.  See also: Wind tunnel

Fig. 4  Flow around two models of a tall building showing how minor design modifications can make a large difference in wind velocity at the pedestrian level. (a) Flow with vortex between the two buildings. (b) Flow with vortex removed by a slight change in the shape of the tall building. (From H. Thomann, Wind effects on buildings and structures, Amer. Sci., 63:278–287, 1975)

By far the most interesting and widely used models are the numerical models. The reason for their popularity is that it is possible to model more of the actual physics of the flow, that is, solve the Navier-Stokes equation, rather than make assumptions and eliminate key components of the physics just to obtain a solution. Although the Navier-Stokes equation is nonlinear, the partial differential equation can now be solved with some measure of confidence and reliability. In many instances dealing with environmental flows, the use of supercomputers is required.  See also: Supercomputer

Numerical methods

Several broad classes of solution techniques are employed to solve the various derivatives and terms of the Navier-Stokes equation. The most common and widely used numerical methods are finite difference schemes (which are based on the use of truncated Taylor-series expansions); finite element schemes (which use an integral approach with local weighting and basis, or shape, functions); spectral methods (in which dependent variables are transformed to wave-number space by using a global basis function, such as the Fourier transform); pseudospectral methods (which use truncated spectral series to approximate derivatives); interpolation techniques (whereby polynomials are used to approximate the dependent variables in one or more spatial directions); and particle methods (which use lagrangian particles whose trajectories are calculated within a conventional eulerian grid). Such numerical models depend strongly on boundary and initial conditions; care must be exercised to correctly initialize and specify all variables at the boundaries of the computational model. All these schemes except the particle methods require knowledge of properties such as viscosity, dispersion coefficients, and thermal conductivity; particle methods require no constitutive models for particle viscosity or thermal conductivity, but do require a large number of particles for an accurate description of the flow field. The most popular modeling approaches are the finite difference, finite element, and interpolation schemes, especially for mesoscale and synoptic-scale simulations.  See also: Computational fluid dynamics; Finite element method; Interpolation; Numerical analysis; Simulation

Capabilities

The continuing rapid improvement in computational hardware has made it possible to model more complicated problems and include more physics (or mathematical terms) in the governing equations. Simulations of environmental fluid flow over microscale and mesoscale regions without simplifications of the equations of motion are now fairly common. Arrays consisting of millions of nodes can be calculated within a few hours on supercomputers, and three-dimensional graphical displays can be generated on work stations. By using satellite, radar, and conventional surface observations as input data to meteorological models, reasonably accurate local forecasts can be made for up to several days. Advances in numerical techniques as well as computer hardware will continue, making it possible to perform more detailed calculations over broader expanses with improved accuracy over longer forecast periods.

Examples

An example of the simulation of fluid flow over a building complex is shown in Fig. 5. The three-dimensional equations of motion and species transport were solved using an adaptive finite element method. The flow field at the x-y midplane of the three-dimensional model domain (Fig. 5a) shows the development of a series of eddies as the flow moves downstream of the buildings. The flow field at the x-z vertical midplane (Fig. 5b) shows the formation of a large eddy between the two sets of buildings. Lagrangian particles, introduced upstream of the building array, clearly show the dispersion of concentration over and around the buildings (Fig. 5c). Particles are being pulled into the large eddy between the two sets of buildings.

Fig. 5  Numerical simulation of flow over a building complex, carried out on a SGI-Cray Origin 2000 parallel computer. (a) Wind field in the horizontal (x-y) midplane. (b) Wind field in the vertical (x-z) midplane. (c) Lagrangian particles depicting species transport. (After D. W. Pepper et al., eds., Development and Application of Computer Techniques to Environmental Studies, WITPress, 1998)

The flow circulation around the Great Red Spot of Jupiter (Fig. 6a) is an example of the turbulent nature of fluid flow on a large scale. The velocity field in and around the Great Red Spot was obtained by tracking small cloud features over time (Fig. 6b).  See also: Fluid mechanics; Jupiter

Darrell W. Pepper

Fig. 6  Great Red Spot of Jupiter. (a) Image from Galileo spacecraft (NASA). (b) Velocity field in and around the Great Red Spot obtained by tracking small cloud formations in sequential Voyager 1 images. (From T. E. Dowling and A. P. Ingersoll, Potential vorticity and layer thickness variations in the flow around Jupiter's Great Red Spot and White Oval BC, J. Atmos. Sci., 45:1380–1396, 1988)

ENCYCLOPEDIA ARTICLE: Environmental fluid mechanics