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hydraulic-fracturing method

hydraulic-fracturing method A mensuration of crustal stress. The only method applicable to the measurement of crustal stress deeper than a depth of 100 m. Confining the upper and the lower sides of a portion of a drilled borehole with expansive stoppers, the wall of the portion of the borehole is broken, applying water pressure to the portion. The direction of resultant cracks becomes parallel to axis of maximum horizontal compressive stress. The amount of maximum and minimum horizontal compressive stresses can also be obtained from the water pressure when the cracks open and close.

hydraulic gradient (dh/dl) A dimensionless number, the driving force in flow through porous media, equal to the change in hydraulic head (dh) with a change in distance in a given direction (dl).

hydraulic head (h) The hydraulic head has units of length and is the total mechanical energy per unit weight of fluid, calculated as the sum of the pressure head (p/ρg) and the elevation head (z), or h = p/ρg + z, where p is the fluid pressure, ρg is the fluid weight or weight density (γ ), and z is the elevation head. Assumptions are that the fluid is both homogeneous and incompressible, and that kinetic energy is negligible. In unsaturated flow, the pressure head is negative and the hydraulic head is then equal to

the tension head ψ plus the elevation head, or h = ψ + z.

hydraulic jump A rapid change of the flow in a river or other open channel where the velocity drops and depth increases over a short dis-

tance. Involves a change in the Froude Number,

√

Fr = V / gD, where V is the velocity, g is acceleration of gravity, and D is hydraulic depth, from > 1 (supercritical flow) to < 1 (subcritical flow).

hydraulic radius The cross-sectional area of a flow divided by the wetted perimeter for the flow.

hydraulic routing A technique for studying the propagation of a flood through a section of river. Involves the solution of the continuity and momentum equations for a moving fluid. Com-

pare to hydrologic routing, which is a simplified method for investigation of similar problems.

hydraulic transmissivity A parameter used to describe the ability of a confined aquifer to transmit water along it, assuming flow across the aquifer is negligible. For aquifer thickness h(x, y), the transmissivity is defined as

h(x,y)

T (x, y) = K(x, y, z) dz

0

where K is the hydraulic conductivity, and z is the coordinate in the direction across the aquifer. The principal hydraulic conductivity in the z direction is zero, and therefore the transmissivity is a tensor defined in the aquifer (x, y) plane.

hydrodynamic Relating to the flow or movement of water or other fluid.

hydrodynamic instability A mean flow field is said to be hydrodynamically unstable if a small disturbance introduced into the mean flow grows spontaneously, drawing energy from the mean flow. There are two kinds of fluid instabilities. One is parcel instability such as the convective instability, inertial instability, and more generally, symmetric instability. The other kind is called wave instability, which is associated with wave propagation.

hydrogen Colorless, explosive, flammible gas; H. The most common element in the universe. Naturally occurring atomic mass 1.00794. Constituent of water. Melting point 14.01 K, boiling point 20.28 K. Two natural isotopes: 1H, 99.9885%, 2H (deuterium), 0.0115%. One known radioactive isotope, 3H (tritium), which undergoes β-decay to 3He with a half life of 12.32 years.

hydrogen burning The fusion of four hydrogen atoms (protons) to make one helium atom (alpha particle), via either the proton-proton chain or the CNO cycle. Because four hydrogen atoms are more massive than one helium atom by about 0.8%, hydrogen burning releases more energy than any other nuclear reaction in stars, 6 −7 ×1018 erg/gram (depending on lifetimes), as main sequence stars and red giants. Hydrogen burning was proposed as the main source of

© 2001 by CRC Press LLC

hydromagnetic turbulence

energy for stars as early as 1925 by Sir Arthur Eddington and Hans Bethe and others. The interaction between two protons or a proton and a heavier nucleus is a quantum mechanical process in which barrier penetration must occur. As a result, hydrogen burning occurs at a temperature of 10 to 20 million K in stars of low to high mass (0.085 to 100 solar masses). Lower mass configurations do not reach this temperature range, derive little or no energy from hydrogen burning, and are called brown dwarfs. See brown dwarf, CNO cycle, main sequence star, proton-proton chain, red giant.

hydrograph A plot showing discharge vs. time. Commonly used to illustrate flood events on rivers or creeks.

hydrographic survey A term often used synonymously with bathymetric survey. A survey of the geometry of the seafloor. Typically involves one system for establishment of horizontal position and another for vertical position or water depth. Traditionally performed from a boat, but amphibious vehicles, sleds, and helicopters have also been used.

hydrologic equation A water balance equation often used in catchment hydrology based on the law of mass conservation that states that basin outputs are related to basin inputs plus or minus changes in storage, and written in simplified form as Q = P − ET ± BS, where Q is surface water runoff at the basin outlet, P is precipitation input to the basin, ET are evapotranspiration losses of water from the basin, and BS is the change in storage of water in the basin.

hydromagnetic Pertaining to the macroscopic behavior of a magnetized electrically conducting fluid or plasma. Hydromagnetic phenomena are generally associated with large length scales compared with the Larmor radii and long time scales compared with the Larmor periods of the particles that comprise the fluid. Theoretical descriptions of such phenomena may be based on magnetohydrodynamics or kinetic theory. See magnetohydrodynamics, Vlasov–Maxwell equations.

hydromagnetics See magnetohydrodynamics.

hydromagnetic shock wave An abrupt transition between two regions in a magnetized plasma, analogous to the acoustic shock wave in air. Hydromagnetic shock waves propagate relative to the plasma, and may be regarded as the short-wavelength limit of large-amplitude magnetoacoustic waves. There are fast and slow shock waves that correspond, respectively, to the fast and slow magnetoacoustic wave modes; a large-amplitude wave of either mode may steepen to form a shock.

In a reference frame in which a shock is at rest, the flow speed of the plasma upstream of the shock is greater than the (fast or slow) magnetoacoustic wave speed, and downstream this inequality is reversed. As in the case of shocks in an ordinary gas, plasma flowing through a hydromagnetic shock undergoes heating and associated entropy production. In ideal magnetohydrodynamics a shock is infinitely thin; real hydromagnetic shocks have a structured transition region of finite thickness, comparable to the ion Larmor radius in many situations.

Hydromagnetic shock waves are routinely observed in situ in collisionless plasmas in space. The high-speed solar wind passes through a bow shock when it encounters a planetary or cometary obstacle. From time to time material ejected from the solar corona drives transient shock waves into the heliosphere; if such a shock encounters a planetary magnetic field, it may produce magnetic storms and aurorae in the planetary magnetosphere and upper atmosphere. Shock waves can also be produced internally in the solar wind as fast streams catch up with slower streams; these shocks are frequently observed from about 3 to about 10 astronomical units from the sun. See hydromagnetic wave.

hydromagnetic turbulence The turbulent, irregular fluctuations that can arise in a magnetized fluid or plasma. Just as turbulence arises at high Reynolds number in an ordinary unmagnetized fluid, hydromagnetic turbulence is associated with high magnetic Reynolds number (the ordinary Reynolds number with kinematic viscosity replaced by electrical resistivity). Mag-

© 2001 by CRC Press LLC

hydromagnetic wave

netized cosmic plasmas are likely sites of hydromagnetic turbulence, due to their large size and high electrical conductivity. Such turbulence, in turn, may be of importance in the physics of astrophysical systems, playing a role in energy or angular momentum transport, or even support against gravitational collapse.

Large-amplitude, apparently random fluctuations of magnetic field and other physical quantities are commonly observed in situ in space plasmas, especially the solar wind, which is often regarded as our best laboratory for the study of hydromagnetic turbulence. There is also strong indirect observational evidence for the existence of hydromagnetic turbulence in distant astrophysical systems.

The theory of hydromagnetic turbulence is not yet well understood, despite extensive theoretical and computational efforts.

Observationally, it is well established that there are extensive regions within the solar wind stream structure that appear turbulent in the sense that their fluctuations seem random and exhibit turbulence-like spectra; but often these same fluctuations also exhibit the properties expected for Alfvén waves. These latter properties include constant density and magnetic field strength and an alignment between velocity and magnetic field corresponding to maximum cross helicity. In the limit of perfect Alfvénic behavior, it can be shown that no turbulent energy cascade can occur. Thus there is a debate, still unsettled, about whether the interplanetary Alfvénic fluctuations are generated dynamically as part of the solar wind flow process, or are rather the remanent fossil of turbulent processes that occur near the sun and are passively convected out through the heliosphere. See Alfvén wave, Alfvénic fluctuation, cross helicity, helicity, hydromagnetic wave, magnetic helicity.

hydromagnetic wave Any long-wavelength, low-frequency wave that can propagate in a magnetized plasma; the analogous phenomenon in an ordinary gas is the sound wave. The standard constraints on the wavelength λ and frequency ω of hydromagnetic waves are that λ >> the typical Larmor radius and ω <<<<

the Larmor frequency for particles of every charge species. The theory of hydromagnetic waves can be developed in the framework of ei-

ther magnetohydrodynamics or kinetic theory; the former approach has the advantage of greater mathematical tractability, but cannot adequately describe some phenomena involving dissipation or instability that may occur in a collisionless plasma.

Either approach to the theory of hydromagnetic waves predicts several kinds of propagating waves. These may be either compressive (magnetoacoustic waves) or noncompressive (Alfvén waves). Magnetoacoustic waves can steepen to form shock waves, and in a hot collisionless plasma are subject to strong Landau damping. Alfvén waves do not steepen or Landau-damp. The discontinuity that corresponds to the short-wavelength limit of an Alfvén wave is called the rotational discontinuity.

In addition to the propagating modes, there is a one-dimensional static (nonpropagating) structure, the tangential pressure balance. In this structure, all gradients are normal to the local magnetic field, and the total transverse pressure P + B2/8π must be constant throughout the structure. The short-wavelength limit of the tangential pressure balance is called the tangential discontinuity.

Another nonpropagating structure that formally comes from magnetohydrodynamics is the “entropy wave”, which is a simple balance of gas pressure alone. The entropy wave is a useful concept in gas dynamics, but strictly speaking it cannot exist in a collisionless plasma because of rapid particle mixing across the structure. However, the entropy wave, and especially its shortwavelength limit, the contact discontinuity, is a helpful concept in space physics and astrophysics in situations involving strong shocks, in which the magnetic field may be of secondary importance and gas dynamics are a fairly good approximation.

Most kinds of hydromagnetic waves and analogous nonpropagating structures, as well as their associated “discontinuities”, have been identified in solar wind plasma and magnetic observations. See Alfvén wave, hydromagnetic shock wave, hydromagnetic turbulence, magne- toacoustic wave.

hydrosphere The water content of the Earth. The water on the Earth resides primarily in the

© 2001 by CRC Press LLC

hysteresis

oceans, and also in other surface waters in underground waters, and in glacial deposits. The term hydrosphere should also include water involved in the hydrologic cycle, i.e., water vapor in the air and droplets in clouds.

hydrostatic Related to still water. For example, hydrostatic pressure is the pressure resulting from a stationary column of water.

hydrostatic approximation An approximation in which the pressure is assumed to be equal to the weight of the unit cross-section column of fluid or air overlying the point.

hydrostatic equation The variation of pressure (p) with depth (d) in a fluid at rest is one of the fundamental relationships in fluid mechanics: dp = −ρg dz. The change in pressure (dp) of the fluid is equal to the unit fluid weight (ρg) over some vertical distance (dz), which we can integrate from the bottom of a fluid column (z = −d) to the surface (z = 0):

ps 0 dp = −ρg dz

p−d

where ps is the pressure at the surface. Integrating, ps −p = −ρg[0−(−d)], or p−ps = ρgd, which can be used to calculate the absolute pressure at any point in a static fluid. Commonly we take the fluid pressure at the surface to be zero gage pressure (ps = 0), or the fluid pressure relative to atmospheric pressure. The absolute pressure is then the sum of gage pressure and atmospheric pressure. If p is the gage pressure and the fluid is static with a constant density (ρ), then pressure increases linearly with depth: p = ρgd.

hydrostatic equilibrium The condition in a star or other object where the inward force of gravity is precisely balanced by the outward force due to the gradient of pressure, that is

dP = −GM(r)ρ , dr r

where P and ρ are the local values of pressure and density at radius r, M(r) is the mass interior to this point, and G is Newton’s constant of gravity. In the absence of hydrostatic equilibrium, a

star will expand or contract on the free-fall time scale (about 1 hour for the sun). A slight imbalance can lead to stellar pulsation (see, e.g., Cepheid variables). A large imbalance leads to either catastrophic collapse or violent explosion, or both (see supernova, type II). In general relativity in spherical systems the equation is modified to require larger |dP/dr| to maintain equilibrium; as a result, such systems are less stable when analyzed in full general relativity then as predicted from Newtonian theory.

hydrostatic pressure The hydrostatic pressure ph is the pressure in water, say the oceans, as a function of depth h,

ph = ρwgh

where ρw is the density of water and g is the acceleration of gravity.

Hygiea Tenth asteroid to be discovered, in 1849. Diameter 430 km. Orbit: semimajor axis 3.1384 AU, eccentricity 0.1195, inclination to the ecliptic 3◦.847, period 5.56 years.

hygrometer A device to measure relative humidity.

Hyperion Moon of Saturn, also designated SVII. Discovered by Bond and Lassell in 1848, it is one of the largest non-spherical bodies in the solar system. Its orbit has an eccentricity of 0.104, an inclination of 0.43◦, and a semimajor axis of 1.48 × 106 km. Its size is 205 × 130 × 110 km, its mass, 1.77×1019 kg, and its density 1.47 g cm−3. It has a geometric albedo of 0.3, and orbits Saturn once every 21.28 Earth days.

hypersonic Pertaining to speeds or flows with Mach number exceeding 5. In hypersonic motion of a body through a fluid, the shock wave starts a finite distance from the body. See Mach number.

hysteresis The property whereby a dependent variable can have different values according to whether the independent variable is increasing or decreasing. In hydrologic systems, a loop-like curve develops that relates pairs of hydraulic properties of an unsaturated porous medium because volumetric moisture content

© 2001 by CRC Press LLC

hysteresis

(θ), tension head (ψ), and hydraulic conductivity (K) covary along different curves depending on whether the soil is wetting or drying.

© 2001 by CRC Press LLC

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