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residual circulation

of the water in a basin (e.g., the Mediterranean) or region to be replaced. For systems at steady state, residence time can be easily calculated for a reservoir S because the inflow (i) and outflow (q) rates are identical: TR = Sq = Si .

residual circulation The time-averaged circulation in a water body, after averaging over a long period (greater than the wind wave period and tidal period).

resolution A measure of the minimum angular separation between two sources at which they can unambiguously be distinguished as separate.

resonance In planetary dynamics, an orbital condition in which one object is periodically subjected to the gravitational perturbations caused by another object. The two bodies are usually in orbit around a third, more massive object and their orbital periods are some whole number ratios of each other. For example, Io and Europa are in a resonance as they orbit around Jupiter: Io orbits twice for every orbit of Europa. Europa and Ganymede are also in a resonance, with Europa orbiting twice for every orbit of Ganymede. In the case of Io and Europa, the resonances cause tidal heating to be a major internal heat source for both objects. Resonances can also create gaps, such as the Kirkwood Gaps within the asteroid belt (when asteroids are in resonance with Jupiter) and several of the gaps within Saturn’s rings (caused by resonances with some of Saturn’s moons).

resonance scattering A magnetohydrodynamical phenomenon: the scattering of particles by waves can be described as a random walk process if the individual interactions lead to small changes in pitch-angle only. Thus, a reversal of the particle’s direction of propagation requires a large number of such small-angle scatters. If, however, the particle motion is in resonance with the wave, the scattering is more efficient because the small-angle scatterings all work together in one direction instead of mostly cancelling each other. Thus, pitch-angle scattering will mainly occur at the magnetic field fluctuations with wavelength in resonance with

the particle motion parallel to the field:

k = ωc = ωc v µv

with k being the wave number of the waves leading to the particle scattering, v the particle speed parallel to the magnetic field, and µ the cosine of the particle’s pitch-angle.

Resonance scattering.

From the resonance condition we can see that for a given particle speed particles resonate with different waves, depending on their pitch-angle. Since the amount of scattering a particle experiences basically depends on the power density f (k ) of the waves at the resonance frequency, scattering is different for particles with different pitch-angles although their energy might be the same. Therefore, the pitch-angle diffusion coefficient κ(µ) depends on pitch-angle though µ. See slab model.

resonant absorption In solar physics, in a closed magnetic loop, resonant frequencies appear at multiples of vA/2L, where vA = B/

4πmpn is the Alfvén speed and L is the coronal length of the loop. The resonances occur because of reflections off the transition region parts of the loop. Resonant absorption occurs when the frequency of a loop oscillation matches the local Alfvén frequency. This creates a resonant layer in which there is a continuous accumulation of energy and, consequently, may result in the heating of the coronal plasma. See Alfvén speed.

resonant damping and instability In a collisionless plasma, wave damping or growth associated with the interaction between a wave and particles moving with a velocity such that the wave frequency is (approximately) Dopplershifted to zero, or to a multiple of the particle’s Larmor frequency. Thus, in a magnetized

© 2001 by CRC Press LLC

Reynolds decomposition

plasma, for a wave of angular frequency ω and wave vector k, the resonant particles have a component of velocity v along the magnetic field satisfying

ω − k v ±n ≈0 ,

where n is an integer and is the particle Larmor frequency. The n = 0 resonance is called the Landau resonance, and the other resonances (especially n = 1) are called cyclotron resonances. These resonances are of importance in wave dissipation and the pitch-angle scattering of charged particles. See cyclotron damping and instability, Landau damping and instability.

resonant layer In solar physics, a narrow, typically cylindrical, shell <250 km across in which the dissipation of Alfvén waves is thought to occur via the process of resonant absorption.

rest mass The mass of an object as measured by an observer comoving with the object, so that the observer and object are at rest with respect to one another. In relativistic physics there is a mass increase for moving objects; the term rest mass is used to denote the intrinsic, unchanging, mass of the object.

retrograde motion In observation of planets from Earth, westward motion of a planet against the background stars. Because of their motion in orbit, planets move in an average easterly direction against the background stars. However, when Earth is closest to another planet, the motion of the Earth gives a parallax which can lead to temporary retrograde (westward) motion.

retrograde orbit An orbit of a satellite of a planet, in which the orbital angular momentum lies in the hemisphere opposite that of the planet’s angular momentum.

retrograde rotation Rotation of a celestial body such that the angular velocity associated with its rotation has a negative projection on the angular velocity associated with its orbital motion; equivalently, an inclination to the orbital plane greater than 90◦. Venus and Uranus have retrograde rotation (though Uranus’ inclination of 97◦55 means that its pole is essentially lying in its orbit, with a slight retrograde net rotation).

return current Particle acceleration in solar flares is thought to result in a beam of electrons formed in the corona which propagates through the corona to dissipate in the chromosphere. The charge separation leads to the development of an equal and opposite return current which is set up in the ambient plasma in order to replenish the acceleration region.

return stroke In lightning flashes, the main current flash in a lightning stroke, in which current flows from the ground up the channel opened by the stepped leader. See stepped leader, dart leader.

reverse fault A fault where the rocks above the fault line move up relative to the rocks below the fault line. A reverse fault has the direction of motion opposite that of a normal fault. A reverse fault is a type of dip-slip fault, where displacement occurs up or down the dip (i.e., the angle that the stratum makes with the horizontal) of the fault plane. A reverse fault where the dip is so small that the overlying rock is pushed almost horizontally is called a thrust fault.

reversible process In thermodynamics, a process that occurs so slowly that the system is very close to equilibrium throughout the process. In such a case no entropy is produced (neither the entropy of the system nor that of its surroundings is changed), and the system can be returned to its original state by infinitesimally changing the external conditions.

revetment A man-made structure constructed to protect soil from erosion. Frequently made of natural stone or concrete.

Reynolds decomposition Separation of a state variable into its mean and the deviation from the mean:

x = x + x .

The over-bar denotes the average or mean over a timescale τ and the prime denotes the fluctuations or deviations from the mean. Ideally, the time scale τ should be chosen such that τ = 2π/ωg where ωg is the spectral gap frequency of the energy spectra φ(ω).

© 2001 by CRC Press LLC

Reynolds number (Re)

Reynolds number (Re) A dimensionless quantity used in fluid mechanics, defined by Re = ρvl/η, where ρ is density, v is velocity, l is length, and η is viscosity. In each case “typical” values are used. The Reynolds number gives a measure of the relative importance of acceleration to viscosity in a given situation.

Flows that have the same Re are said to be dynamically similar. Above a certain critical value, Rec, a transition from laminar flow to turbulent flow occurs. Its value serves as a criterion, low values being associated with high stability, for the stability of laminar flow. Depending on the geometry of flow, once Re becomes larger than 105, turbulence is likely to occur. In open channel flow, the hydraulic radius RH is used for the diameter of the passageway and turbulent flow occurs when R > 1000. For subsurface flow, the mean grain diameter is substituted for the diameter of the passageway, with turbulent flow occurring when R > 10. In oceanographic applications, it is often useful to consider the buoyancy Reynolds number

b ≡ (LN /K)4/3 = Iρ/ηN2

where LN is the buoyancy scale, K is the Kolmogorov scale, I is the dissipation rate of turbulent kinetic energy, and N2 is the buoyancy frequency. In this formulation, the Reynolds number describes the turbulent state of the flow based on the stratification and intensity of the turbulence.

Reynolds stress The stress, that occurs as a result of the turbulent exchange between neighboring fluid masses, defined by ρ < uiuj > to form a 9-term tensor. If the fluid is turbulent, the Reynolds stress is much larger than the viscous stress.

Rhea Moon of Saturn, also designated SV. It was discovered by Cassini in 1672. Its orbit has an eccentricity of 0.001, an inclination of 0.35◦, a semimajor axis of 5.27 ×105 km, and a precession of 10.16◦ yr−1. Its radius is 765 km, its mass, 2.49 × 1021 kg, and its density 1.33 g cm−3. It has a geometric albedo of 0.7, and orbits Saturn once every 4.518 Earth days.

rheological constitutive equation (creep law, flow law) The relation,

ε˙ = F (T , P , c , X , σ ) ,

relating creep rate (ε, in the steady-state stage of creep), temperature (T ), pressure (P ), chemical environment (c), microstructural parameter (X), and deviatoric stress (σ ). A rheological constitutive equation can be determined through a series experiments performed under well-controlled conditions with a general form as

ε˙ = ε˙0fOm21fami

2σ n exp

 

 

−

 

 

RT

 

+H0

(σ ) + P +V

 

where ε0 is a constant that often depends on the orientation of stress axis with respect to the crys-

tal lattice, f m1 and fam2 reflect the contribution

O2 i

from chemical environments (oxygen fugacity and component activity), +H0 and +V are the activation enthalpy and volume, respectively, of the controlling process, and P is the hydrostatic pressure. The term (+H0(σ ) + P +V ) defines the activation energy of the controlling process.

rheology The study of the flow of liquids and deformation of solids. Rheology addresses such phenomena as creep, stress relaxation, anelasticity, nonlinear stress deformation, and viscosity. Rheology can be a function of time scales. For instance, in geophysics, on short time scales, say less than about 1000 years, the Earth’s mantle and crust are elastic, the outer core is a fluid and the inner core is solid. On longer time scales, the Earth’s mantle beneath the lithosphere behaves as a fluid, resulting in postglacial rebound and mantle convection.

Ricci rotation coefficients In geometry, the components of a connection 1-form

γmnp = enaepb beam

where ema are the coordinate components of a complete collection of smooth orthogonal vectors em. Here epb beam is the covariant derivative of the vector em along the vector ep. (Bold indices m, n, p label the vector; a, b, c label components.) Without some other condition on

© 2001 by CRC Press LLC

gαwT

Richter magnitude scale

γmnp, this equation is definitional; the γmnp can be specified as arbitrary smooth functions. See metricity of covariant derivative, torsion.

Ricci tensor In Riemannian and pseudoRiemannian geometry, the symmetric two index tensor Rab representing the trace of the Riemann curvature tensor: Rac = Rabcb. Some works (among others, L.P. Eisenhart’s Riemannian Geometry) define the Ricci tensor by contraction in the first index, thus reversing the sign of the Ricci tensor. See Riemann tensor.

Richard’s equation Combining Darcy’s law with the continuity equation provides an expression of mass conservation and water flow in the unsaturated zone, along with a history of the tension distribution in a vertical soil column:

∂t

= ∂z

Kθ

∂z

+ 1

dθ

 

∂

 

 

∂ψ

 

where θ is the volumetric soil moisture, t is the time interval, z is depth in the unsaturated zone, ψ is the tension head, and K(θ) is the unsaturated hydraulic conductivity. For steady flow, the time derivative of the volumetric moisture content θ is zero and a single integration of the right-hand side of Richard’s equation yields qz = −K(θ)[dθ/dz + 1]. For saturated conditions dθ/dz is also zero, K(θ) becomes a constant K, the tension head becomes positive, and Richard’s equation reduces to the Laplace equation for one-dimensional flow.

Richardson, Lewis F. (1881–1953) English physicist and meteorologist. He made the first numerical weather prediction by using finite differences for solving the differential equations, but at that time (1922) the computations could not be performed quickly enough to be of practical use.

Richardson number Dimensionless parameter that compares the relative importance of mechanical turbulence (quantified by velocity shear) and convective turbulence (quantified by the buoyancy frequency) and gives the stability criterion for the spontaneous growth of smallscale waves in a stably stratified flow with ver-

tical shear of horizontal velocity, Ri, defined by

N2

Ri =

(∂V/∂z)2

where N is Brunt–Väisälä (buoyancy) frequency, V is horizontal velocity, and z is upward coordinate. The Richardson number is a measure of the ratio of the work done against gravity by the vertical motions in the waves to the kinetic energy available in the shear flow. In general, the smaller the value of Ri the less stable the flow with respect to shear instability. The most commonly accepted value for the onset of shear instability is Ri = 0.25.

For a continuously stratified fluid, the bulk Richardson number can also be defined by

Ri ≡ N2l2

U2

where N is the buoyancy frequency, l is the characteristic length scale, and U is the characteristic velocity scale of the flow.

The ratio of the buoyant destruction of turbulent kinetic energy, gαwT , to the shear production of turbulent kinetic energy, −uw (dU/dz), is called the flux Richardson number, defined by

Rif = −

uw (dU/dz)

where wT is the heat flux, uw is the Reynolds stress, g is the constant of gravity, and α = ρ−1(∂ρ/∂T ) is the thermal expansivity of sea water. For Rif > 1, buoyancy dampens turbulence faster than it can be produced by shear production. However, laboratory observations and theoretical considerations suggest that the critical value is less than unity and that a Rif 0.25 turbulence ceases to be self-supporting.

Richter magnitude scale An earthquake magnitude scale originally devised by American seismologist C.F. Richter in 1935 for quantifying energy release in local California earthquakes. It was originally defined to be the base 10 logarithm of the maximum amplitude traced on a seismogram generated by a then standard seismograph at 100 km epicentral distance. It was later substantially generalized. Since the early 1980s, the most widely used scale is the moment magnitude scale.

© 2001 by CRC Press LLC

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