No abstract

Stuart (1960) has developed a theory of the stability of plane Poiseuille flow to periodic disturbances of finite amplitude which, in the neighbourhood of the neutral curve, leads to an equation of the Landau (1944) type for the amplitude A of the disturbance: \[ d|A|^2/dt = k_1|A|^2 - k_2|A|^4. \] If k2 is positive in the supercritical region (R > RC) where k1 is positive, then, according to Stuart, there is a possibility of the existence of periodic solutions of finite amplitude which asymptotically approach a constant value of (k1/k2)½. We have evaluated the coefficient k2 and found that there indeed exists a zone in the (α, R)-plane where it is positive. This is the zone inside the dashed curve shown in figure 1, with the region of instability predicted by the linear theory included inside the ‘neutral curve’. Stuart's theory and Eckhaus's generalization thereof could apply in the overlapping zone just above the lower branch of the neutral curve.

The Bullard—Gellman formalism is applied to investigate the existence of convergent solutions for steady kinematic dynamos. It is found that the solutions for the Bullard—Gellman dynamo, as well as for Lilley’s modification of it, do not converge. Convergent solutions have been found for a class of spherical convective cells which would be stationary in a perfect fluid in the absence of rotation and of the magnetic field. By calibrating the theoretical magnetic dipole so as to fit the observed value at the Earth’s surface, one can find a dynamo in the above class which also matches the observed equatorial magnetic dipoles. There is a dynamo which has a rate of total ohmic dissipation of only 1.8 x 1016 erg s-1 for an assumed electrical conductivity of 3 x 10~6 e.m.u.'f This is one thousandth the rate of tidal dissipation, and one hundred thousandth the rate of heat outflow from the surface of the Earth. The required velocities are of the order of 10~3 cm s_1, and the average magnetic energy density is 4 erg cm-3. The internal structure of the magnetic field in this model shows a dynamo mechanism situated in the outer part of the liquid core and is thus insensitive to possible rigidity of the material in the * inner core.

Boiling AFB, 20332 .-16. ISTSBUTION 57A TEMEN r0t ths Repor) Approved for public release; distribution ,nlimit ed. 17 :ISTRIBuTION STATEMENT (o tie abstract entered Sl atock 20. it different Irom Report) 18. SUPPLEMENTARY NOTES t9. KEY NORCS ,Continue on reverse s@ e '1 necessary and ,denrtv by block number) Lg, synthetic seismograms, nuclear explosion detection, earthquake simulation 20. ABS RACT Confln.e on r.ers. side it. nece85erv end idoneifv by Odock n-mbe, This report summarizes three research efforts performed during the past fiscal year. The firstthese efforts is a study of the theoret,.al behavior of the region-I seismic phase Lg in various tectonic provinces. Synthetic seismograms are used to determine the sensitivity of Lg to source and medium properties. The primary issues addressed concern the relationship of regional Lg characteristics to the crustal attenuation properties, the comparison of Lg in many crustal structures and the source depth dependence of Lg..... * , DO , 2 ,. 1473 EZTON 00' NOV S5 S OBSOLE E Unclassified SEC-jRITY LASSIV A10ATION :V '-iS ; ,AGE I Oare ter";

It is shown that there exist undamped solutions for perturbations of finite amplitude of plane Poiseuille flow, which are periodic in the direction of the axis of the channel. The shift in the ‘neutral curve’ as a function of the amplitude λ* of the disturbance is shown in figure 2. The solution is obtained by a perturbation method in which the eigenfunctions and the eigenvalue c are expanded in power series of the amplitude λ, as shown in (14), (15), (16) and (17). Near the neutral curve for a finite amplitude disturbance, the curvature of the mean flow shows a tendency to become negative (figure 5).