Net Momentum Research Articles

Polarized electron inner-shellionization-with-excitation of zinc atoms

The ionization-with-excitation collision process, from theground 3d104s2 1S0 state to the ionic 3d94s2 2D3/2 state of the zinc atom, has beeninvestigated using incident polarized electrons with near-threshold energies. The observed Stokes parameters of theemitted 589.4 nm photon show that the residual ion is alignedand oriented. These observations also indicate directly theelectron exchange effects in the inner-shellionization-with-excitation process. From these measured Stokesparameters the normalized state multipoles, the electricquadrupole moment of the orbital angular momentum and themagnetic dipole moment of the spin angular momentum of theexcited ionic state are derived. The combination of electronexchange and fine-structure coupling enable a net negativeangular momentum transfer to the atomic system. The contributions from various partial waves and from the spin-orbitinteraction to the ionization-with-excitation process arediscussed.

Stabilized Kuramoto-Sivashinsky system.

A model consisting of a mixed Kuramoto-Sivashinsky-Korteweg-de Vries equation, linearly coupled to an extra linear dissipative equation, is proposed. The model applies to a description of surface waves on multilayered liquid films. The extra equation makes it possible to stabilize the zero solution in the model, thus opening the way to the existence of stable solitary pulses. By means of perturbation theory, treating the dissipation and the instability-generating gain in the model (but not the linear coupling between the two waves) as small perturbations, and making use of the balance equation for the net momentum, we demonstrate that the perturbations may select two steady-state solitons from their continuous family existing in the absence of the dissipation and gain. In this case, the selected pulse with the larger value of the amplitude is expected to be stable, provided that the zero solution is stable. The prediction is completely confirmed by direct simulations. If the integration domain is not very large, some pulses are stable even when the zero background is unstable. An explanation for the latter finding is proposed. Furthermore, stable bound states of two and three pulses are found numerically.

Spectator response to the participant blast

The interplay between spectator and participant matter in heavy-ion collisions is investigated in the context of a microscopic transport model. Transport simulations show that flow patterns for the participant matter are strongly influenced by the presence of the nearby spectator matter. However, the influence is mutual. During the explosion of the participant zone, the spectator matter acquires a transverse momentum $<P^x /A>$ that shows sensitivity to the nuclear incompressibility and to the momentum dependence of the nuclear mean field (MF). An observed change in the net average momentum per nucleon, $\Delta |< {\bf P}/A >|$, can be associated with the momentum dependence of the MF. For a momentum-dependent MF and a low impact parameter in a heavy system, the spectators may emerge faster than in the initial state, accelerated by the violent participant explosion. The average excitation energy and the mass of the spectators, in contrast to the momentum, show little sensitivity to the nuclear equation of state.

The spatial inhomogeneity of turbulence above a fully rough, packed bed in open channel flow

Single point turbulence statistics measured directly above and in close proximity to the wall in a fully developed, fully rough, turbulent open channel flow are reported. In order to investigate the spatial inhomogeneity of the turbulence, the measurements were obtained over a matrix of measurement points in a plane parallel to the roughness-bed surface. The measurements were obtained with a three-component laser Doppler velocimeter (3D-LDV) system. The turbulence statistics associated with the vertical velocity component, including conditioned mean vertical velocities, rms distributions, and mean vertical momentum fluxes are emphasized. For the Reynolds and Froude numbers associated with this investigation, and with the specific roughness geometry employed in this study (a packed bed of uniform-diameter spheres), it is found that the distribution of the local mean vertical velocity, , has non-zero contributions over the roughness pattern and that this contributes to a mean net vertical momentum flux into the roughness bed. However, the net vertical momentum flux due to turbulent fluctuations is positive out of the bed, consistent with smooth-wall behavior. These results are relevant to the study of sediment entrainment and suspension/deposition as well as the exchange and transport of chemical species between the channel core flow and the fluid within the roughness bed.

Representation of Eddies in Primitive Equation Models by a PV Flux

The parametric representation of buoyancy and momentum transport by baroclinic eddies in a primitive equation ‘‘b plane’’ channel is studied through a transformation of the governing equations. Adoption of the ‘‘transformed Eulerian mean’’ and the assumption that the eddies (but not the mean flow) are quasigeostrophic in nature leads to 1) the eddies being represented symbolically by one term, an eddy potential vorticity flux, rendering a representation that incorporates both eddy momentum and eddy buoyancy fluxes, and 2) the advecting velocities being those of the residual mean circulation. A closure is employed for the eddy potential vorticity flux that directs it down the mean potential vorticity gradient. Care is taken to ensure that the resulting force does not generate any net momentum in the channel but only acts to redistribute it. The approach is investigated by comparing a zonally averaged parameterized model with a three-dimensional eddy-resolving calculation of flow in a stress-driven channel. The stress at the upper surface is communicated down the water column to the bottom by eddy form drag. Moreover, lateral eddy momentum fluxes act to strengthen and sharpen the mean flow, transporting eastward momentum from the flanks to the center of the jet, up its large-scale gradient. Both vertical momentum transfer and lateral, upgradient momentum transfer by eddies, is captured in the parameterized model. Finally, advantages of the parametric approach are demonstrated in two further contexts: 1) the spindown of a baroclinic zone and 2) the maintenance of surface winds by eddy momentum flux in the atmosphere.

General primordial cosmic perturbation

We consider the most general primordial cosmological perturbation in a universe filled with photons, baryons, neutrinos, and a hypothetical cold dark matter (CDM) component within the framework of linearized perturbation theory. We present a careful discussion of the different allowed modes, distinguishing modes which are regular at early times, singular at early times, or pure gauge. As well as the familiar growing and decaying adiabatic modes and the baryonic and CDM isocurvature modes, we identify two neutrino isocurvature modes. In the first, the ratio of neutrinos to photons varies spatially but the net density perturbation vanishes. In the second the photon-baryon plasma and the neutrino fluid have a spatially varying relative bulk velocity balanced so that the net momentum density vanishes. Possible mechanisms which could generate the two neutrino isocurvature modes are discussed. If one allows the most general regular primordial perturbation, all quadratic correlators of observables such as the microwave background anisotropy and matter perturbations are completely determined by a $5\ifmmode\times\else\texttimes\fi{}5,$ real, symmetric matrix-valued function of comoving wave number. In a companion paper we examine prospects for detecting or constraining the amplitudes of the most general allowed regular perturbations using present and future CMB data.

Average statistical characteristics of long gravity waves observed with the middle and upper atmosphere radar in the mesosphere

Average statistical characteristics of internal gravity waves (IGWs) with horizontal wavelengths longer than 60 km are obtained at the altitudes 65–80 km from the observations with the middle and upper atmosphere radar at Shigaraki, Japan (35°N, 136°E) in 1986–1995. The numbers of signals selected with narrow frequency band filters attributed to IGW having periods 0.1–6 hours vary between 870 and 2300 for different seasons and altitudes. Long‐period IGWs have larger amplitudes and larger vertical wavelengths than short‐period IGWs. For waves with periods 0.1–6 hours the mean values of the horizontal and vertical wavelengths are 100–300 km and 6–10 km, respectively. IGW amplitudes and momentum fluxes have maxima in winter and summer and minima in spring and autumn. Distributions of the azimuths of IGW horizontal wave vectors and momentum fluxes are nearly isotropic with maxima in zonal directions in summer. The westward momentum flux is larger than the eastward flux in winter and is smaller than that in summer. The differences between eastward and westward wave momentum fluxes are larger for long‐period IGWs. The net zonal IGW momentum flux is mainly directed opposite to the mean wind in the mesosphere.

A case study of air‐sea interaction during swell conditions

Air‐sea interaction data from a situation with pronounced unidirectional swell have been analyzed. Measurements of turbulence at three levels (10, 18, and 26 m above mean sea level) together with directional wave buoy data from the site Östergarnsholm in the Baltic Sea were used. The situation, which lasted for ∼48 hours, appeared in the aftermath of a gale. The wind direction during the swell situation turned slowly within a 90° sector. Both during the gale phase and the swell phase the over‐water fetch was &gt;150 km. The wind speed during the swell phase was typically 4 m s−1. During the swell phase a wind maximum near or below the lowest wind speed measuring level 10 m was observed. The net momentum flux was very small, resulting in CD values ∼0.7 × 10−3. Throughout the lowest 26 m, covered by the tower measurements, turbulence intensities in all three components remained high despite the low value of the kinematic momentum flux resulting in a reduction of the correlation coefficient for the longitudinal and vertical velocity from its typical value around −0.35 to between −0.2 and 0 (and with some positive values at the higher measuring levels), appearing abruptly at wave age c0/U10 equal to 1.2. Turbulence spectra of the horizontal components were shown not to scale with height above the water surface, in contrast to vertical velocity spectra for which such a variation was observed in the low‐frequency range. In addition, spectral peaks in the horizontal wind spectra were found at a frequency as low as 10−3 Hz. From a comparison with results from a previous study it was concluded that this turbulence is of the “inactive” kind, being brought down from the upper parts of the boundary layer by pressure transport.

On Shear-Generated Gravity Waves that Reach the Mesosphere. Part II: Wave Propagation

The propagation of Kelvin–Helmholtz (KH) shear-generated gravity waves through the summer stratosphere is investigated using ray tracing, taking into account back-reflection as well as wave dissipation due to precritical-layer breaking, radiative damping, and viscous diffusion. It is found that the transmission rate of upward–eastward waves to the mesosphere is surprisingly good, provided that the horizontal scale of the waves is large enough to prevent back-reflection. A rough upper bound on the net momentum flux into the mesosphere due to a large ensemble of mutually incoherent KH-generated clear-air turbulence events is then estimated, using reasonable-looking assumptions about the statistics of such events. It is found that on this basis the wave source cannot safely be neglected in the global angular momentum budget.

Quasilinear evolution of the ion cyclotron beam-anisotropy instability in a current carrying plasma

The evolution of particle velocity moments and wave spectra during the quasilinear saturation of the proton beam-anisotropy instability in a current carrying plasma is studied. For initial plasma moments characteristic of the near-Earth plasma sheet boundary layer, it is found that the saturation of the ion beam-anisotropy instability drives a field-aligned current, which self-consistently excites a kink-like mode. The driven current allows the wave fields of current driven mode to grow to amplitudes comparable to that of the anisotropy instability and facilitates more rapid isotropization of the unstable anisotropic beam components and saturation of the beam-anisotropy instability. This work has general relevance to the study of field-aligned instabilities in plasmas where the excited waves carry a net momentum density away from the source of instability.

Measurement of theB→DℓνBranching Fractions and Form Factor

Using a sample of 3.3 million B-meson decays collected with the CLEO detector at the Cornell Electron Storage Ring, we have studied $B^- \to D^0 \ell\nu$ and $\bar{B}^0 \to D^+ \ell\nu$ decays, where $\ell$ can be either an electron or muon. We distinguish $B \to D \ell\nu$ from other B semileptonic decays by examining the net momentum and energy of the particles recoiling against the D-lepton pairs. We find the decay rate $\Gamma(B \to D\ell\nu)$ = (14.1 +- 1.0 +- 1.2) ns-1 and derive branching fractions for $B^- \to D^0 \ell\nu$ and $\bar{B^0} \to D^+ \ell\nu$ of (2.32 +- 0.17 +- 0.20)% and 2.20 +- 0.16 +- 0.19)% respectively, where the uncertainties are statistical and systematic. We also investigate the $B \to D \ell\nu$ form factor and the implication of the result for $|V_{cb}|$.

A Plexiglas Research Pump With Calibrated Magnetic Bearings/Load Cells for Radial and Axial Hydraulic Force Measurement

A research pump intended for both flow visualization studies and direct measurement of hydrodynamic radial and axial forces has been developed. The impeller and the volute casing are constructed from Plexiglas which facilitates optical access for laser velocimetry measurements of the flow field both inside the impeller and in the volute casing. The pump housing is designed for flexibility allowing for each interchange of impellers and volute configurations. The pump rotor is supported by three radial magnetic bearings and one double acting magnetic thrust bearing. The magnetic bearings have been calibrated to characterize the force versus coil current and air gap relationship for each bearing type. Linear calibration functions valid for rotor eccentricities of up to 2/3 of the nominal bearing clearances and force level of ±58 N (13 lbf) and ±267 N (60 lbf) for the radial and axial bearings, respectively, were found. A detailed uncertainty analysis of the force calibration functions was conducted such that meaningful uncertainty bounds can be applied to in situ force measurements. Hysteresis and eddy current effects were quantified for each bearing such that their effect on the in situ force measurements could be assessed. By directly measuring the bearing reaction forces it is possible to determine the radial and axial hydraulic loads acting on the pump impeller. To demonstrate the capability of the magnetic bearings as active load cells representative hydraulic force measurements for a centered 4 vane 16 degree log spiral radial flow impeller operating in a single tongue spiral volute casing were made. At shut-off a nondimensional radial thrust of 0.084 was measured. A minimum nondimensional radial thrust of about 0.007 was observed at the nominal design flow. The nondimensional radial thrust increased to about 0.019 at 120 percent of design flow. The nondimensional axial thrust had a maximum at shut-off of 0.265 and decreased steadily to approximately 0.185 at 120 percent of design flow. Two regions of increasing axial thrust, in the flow range 75 to 100 percent of design flow, were observed. The measurements are compared to radial and axial force predictions using classical force models. The direct radial force measurements are compared to a representative set of radial force measurements from the literature. In addition, the directly measured radial force at design flow is compared to a single representative radial force measurement (obtained from the literature) calculated from the combination of static pressure and net momentum flux distribution at the impeller exit.

The Generation of Alongslope Currents by Breaking Internal Waves

Abstract Internal gravity waves propagating in a fluid of constant buoyancy frequency and approaching a uniform sloping boundary from a direction that is not in a plane normal to the boundary, and dissipating energy on reflection, generate alongslope currents. The net radiation stress or momentum flux into the breaking region is proportional to the flux of energy lost from the waves. It is supposed that the stress is balanced by a frictional boundary stress so that a steady alongslope current is generated. The dependence of the strength of the current on the steepness, AIMI, of the incident wave and on its propagation direction is examined as a function of α and β, the inclination of the boundary to the horizontal and the angle between the incident wave group velocity vector and the horizontal, respectively. Alongslope currents of several centimeters per second, generally exceeding the Lagrangian drift produced by wave reflection alone, may be generated in the ocean or in lakes. Reflection of subcritical ...

The effect of varying the source spectrum of a gravity wave parameterization in a middle atmosphere general circulation model

Climate simulations of the middle atmosphere circulation with general circulation models are now starting to include parameterizations of the momentum flux deposition due to unresolved gravity wave spectra. A current uncertainty in the application of such parameterizations is the specification of the imposed gravity wave spectrum. The aim of this work is to quantify the effect of varying within a realistic range the source spectrum of a gravity wave parameterization in a general circulation model. Results from two simulations with the gravity wave spectrum launched at two different heights, the surface and the 110‐hPa pressure level, respectively, are compared. Noteworthy differences found in the simulated middle atmosphere response include the following (1) The average temperature in the southern winter upper stratosphere is about 40 K warmer in the experiment with the surface as the launching height, virtually eliminating the typical cold polar bias that affects many general circulation models. (2) Stronger easterlies in the subtropical summer mesosphere, again in the experiment with the surface as the launching height. Diagnostics of the parameterized gravity waves indicate that in the experiment with the surface as the launching height, the net zonal momentum flux transported by the gravity waves is negative just above the troposphere at middle latitudes. This negative net momentum flux facilitates the deceleration of the mesospheric winter westerlies. The meridional circulation induced by such deceleration is thereafter responsible for the substantial polar winter warming. In contrast, in summer the negative net momentum flux limits the upper mesospheric deceleration of the easterlies. In the experiment with launching height at 110‐hPa, the gravity wave net momentum flux is instead zero by construction at the launching height.

Thermodynamic and relativistic aspects of thermal inertia in fluids

We present a generalized relativistic Lagrangian which yields the effect of heat flux q and nonequilibrium stress \ensuremath{\tau} in the energy-momentum tensor of an extended reversible fluid exhibiting thermal inertia. The actual momentum of heat (thermal momentum related to the entropy flow) follows to be many orders of magnitude larger than $\mathbf{q}{/c}^{2}$ (c is the light speed) but it is consistent with Grad's kinetic theory [in Principles of the Theory of Gases, edited by S. Flugge, Handb\"uch der Physik Vol. 12 (Springer, Berlin, 1958)] and with experiments in heat conduction. On the other hand, the net momentum of heat remains $\mathbf{q}{/c}^{2},$ in agreement with the standard relativistic result, this net momentum being the result of incomplete compensation of the actual thermal momentum and the momentum associated with self-diffusion of particles. The classical densities of mass and entropy, \ensuremath{\rho} and ${\ensuremath{\rho}}_{s},$ cease to be natural variables of energy density E in the sense of Callen [Thermodynamics and an Introduction to Thermostatistics (Wiley, New York, 1988)] whenever inertial effects prevail. This fact necessitates the use of what may be called the thermal potential ${T}^{\ensuremath{-}},$ a new quantity replacing the classical temperature T. Changes in thermodynamic formalism are related to the replacement of T by ${T}^{\ensuremath{-}}.$ The admission of a freely varied four-flux of entropy in an extended Hamilton principle implies all nonequilibrium corrections (q and \ensuremath{\tau}) to the energy-momentum tensor, making it possible to investigate the effect of nonequilibrium phenomena on the properties of associated gravitational fields.

Quantum Statistics of Cooper Pairs; Superconducting Temperature

We show that the eigenvalues of the number operator [Formula: see text] in the pair state (k1, k2), where c's and c†'s are fermion operators, are limited to 0 or 1, while the eigenvalues of the pair-number operator with the net momentum q≡ k1+k2 fixed, [Formula: see text], are 0, 1, 2,…. Both fermionic and bosonic natures of the Cooper pairs (pairons) must be used in the superconductivity theory. The Bose–Einstein condensation (BEC) of the pairons moving with a linear energy–momentum relation occurs at [Formula: see text] (3D), [Formula: see text] (2D), where n0 is the pairon density and v F the Fermi velocity. The connection between this [Formula: see text] and the BCS superconducting temperature is discussed.

Radar observations of gravity waves over Jicamarca, Peru, during the CADRE campaign

We present results obtained with the mesosphere‐stratosphere‐troposphere (MST) radar at Jicamarca, Peru, from three 10‐day experiments in January 1993, March 1994, and August 1994. Horizontal and vertical velocities were measured over ranges spanning the lower part of the stratosphere and most of the mesosphere. In the stratosphere, the fluctuating part of the wind field was found to be dominated by inertia‐gravity waves. Sinusoids of different period were fit to the velocity time series using a least squares procedure. The dominant periods of the inertia‐gravity wave motions were found to be 1.5 days for the January 1993 experiment and 2.1 days for the August 1994 experiment. For the January 1993 experiment, the amplitudes and phases of the inertia‐gravity wave oscillations were extracted for the vertical as well as the horizontal components. The vertical amplitude of the 1.5‐day period wave was small (&lt;0.1 ms−1), but measurable with the Jicamarca radar. The intrinsic periods of the inertia‐gravity waves were inferred from the fits using two different methods. The first method predicted the intrinsic period using the orbital ellipses traced out in time by fits to the horizontal winds. The second method used information taken from the vertical as well as the horizontal fits. The values of intrinsic period made using the second method were found to have much less scatter than the values inferred solely from the orbital ellipses. The momentum flux estimates in both the stratosphere and mesosphere were found to depend significantly on the exact methodology used. A technique was adopted whereby estimates were formed only when radial velocities were measured simultaneously on both beams of a coplanar beam pair. Most of the total wave stress was usually contributed by waves at periods ≥4 hours in the stratosphere and ≥1 hour in the mesosphere. In the stratosphere, localized layers of enhanced momentum flux were sometimes observed. The obvious anticorrelation between the shear of the mean wind and the momentum flux in these layers suggests that they were caused by in situ generation of waves by the Kelvin‐Helmholtz instability, rather than gravity waves propagating from lower levels. At short periods, momentum flux spectra in the stratosphere and mesosphere showed numerous positive and negative peaks. A correlation analysis revealed that the high‐frequency peaks were associated with discrete wave packets. The short‐scale waves associated with these packets were fairly isotropic in their direction of propagation, and due to cancellation they contributed little net momentum.

Compressible magnetohydrodynamic Kelvin–Helmholtz instability with vortex pairing in the two-dimensional transverse configuration

For a two-dimensional (2-D) transverse configuration, where the plasma motion occurs in a 2-D plane transverse to the magnetic field, the nonlinear evolution of the magnetohydrodynamic (MHD) Kelvin–Helmholtz (K–H) instability is investigated by means of a 2-D MHD simulation for a convective fast magnetosonic Mach number 0.35, which is defined for the total jump of the flow velocity. The compressibility and the nonzero baroclinic vector are shown to violate the conservation of the enstrophy for the 2-D MHD transverse configuration and for the 2-D fluid motion. After the nonlinear saturation of the linearly fastest growing vortices, the vortices continue to coalesce until no more vortex pairing is allowed, owing to a finite length of the simulation system. The plasma inside the vortex is rarefied strongly by the fast magnetosonic rarefaction and each vortex is associated with an eddy current, which is inertia current in nature. The plasma flow velocity is enhanced at the periphery of the vortex and the net momentum transport and shear relaxation by the instability occur as long as the vortex pairing continues. Anomalous viscosity by the K–H instability increases with the vortex pairing and its increase is due to the growth of subharmonic modes.

Solar force modeling of block IIR Global Positioning System satellites

Scientific applications of the Global Positioning System require that the space vehicles be located with an accuracy of a few centimeters. The most important uncertainties in position estimation are the result of direct and indirect solar forces. Perhaps as early as late 1996, Block IIR space vehicles will begin to replace the existing Blocks II and IIA. We give formulas for the solar force to be expected on Block IIR and evaluate their probable accuracy based on our previous experience. These a priori formulas include indirect solar forces, including the reradiation of sunlight in the form of heat from the space vehicle's body and solar panels, but do not include radiation-induced outgassing, especially from the multilayered insulation that wraps the space vehicle body. We discuss the best way to combine these a priori models with real-time tracking data to optimize ephemeris accuracy. MONG the many unique features of the Global Positioning System (GPS), one fact is of primary importance for its use in geodesy, studies of crustal dynamics, and support of such space missions as TOPEX/Poseidon: it is the first system of navigational satellites in which errors in knowledge of the Earth's gravitational field have negligible effect, and orbit accuracy is limited almost entirely by errors in modeling the force of solar radiation on the space vehicles (SVs). These SVs are high altitude (semimajor axes = 26,560 km = 4.16 Earth radii) and relatively insensitive to the higher-order gravitational harmonics, which are well known in any case via satellites such as LAGEOS. But GPS SVs have a large cross- sectional area (about 13.6 m2 for Block II/IIA) and are accelerated about 1 x 10~7 m/s by direct solar radiation. Table 1 shows our calculation (using the Aerospace TRACE program) of perturbations on a typical GPS orbit over 12 h. After the Earth oblateness J2, the sun, and the moon, solar radiation is the most important. An a priori model calculates the solar force to be expected on the vehicle as a function of its orientation and of its cross-sectional area and optical properties. Direct solar pressure can be pictured as the net momentum imparted to the S V by photons striking and recoil- ing from its opaque surfaces. Indirect solar pressure is caused, for example, by heat absorbed and reradiated from body surfaces and by outgassing, whereby solar energy and momentum is returned to space by volatile materials on and in the SV body. Indirect forces are of two kinds: predictable, such as the effects of Earthshine and SV heat flow; and anomalous, including outgassing and the so-called 7-bias force. Because none of the indirect effects, predictable or anomalous, were included in the early computer software devel- oped for GPS, many scientific GPS analysts have bypassed the modeling problem entirely and have determined SV accelerations directly as stochastic parameters to be estimated by tracking data.1'2 To attain highest accuracy, some such real-time estimation is nec- essary. Nevertheless, good a priori models are indispensable for three reasons: 1) many applications require not just precise fitting of orbits to tracking data already taken, but orbit prediction; 2) by us- ing a standard force model, workers more readily can intercompare ephemerides, which are calculated by many different agencies from data taken all over the world using different techniques; and 3) to filter real data accurately and reliably, one should know the statistics of the parameters measured—how fast they usually change, and how fast they can change—and this requires knowledge of the physics of the problem, and therefore, good a priori modeling. Our intention

An inverse-free technique for attitude control of spacecraft using CMGs

This paper presents a technique for CMG-based attitude control of spacecraft that does not require calculating the pseudo-inverse of the CMG Jacobian matrix. In order to obtain a steering law, the transpose of the Jacobian matrix is used rather than its pseudo-inverse. The transpose-based method alleviates some of the problems inherent in the existing steering laws that use the pseudo-inverse, such as high gimbal rates near singularities. The proposed method provides viable rates for the CMGs even when the gimbals pass through a singular configuration. The validity of the steering law is demonstrated using Liapunov's stability theorem, and it is shown that the method provides superior results when compared with conventional steering laws that use the pseudo-inverse of the CMG Jacobian Matrix. In addition, the redundant feature of the 4-gimbal CMG system is exploited to force the gimbals to attain certain specified angles while the spacecraft's attitude is being controlled. This is done by adding appropriate null motion to the CMGs. The specified angles for the gimbals can be arbitrary as long as their net momentum is consistent with the requirements of the spacecraft.