HR: 09:15h
AN: AE11B-04    [Abstracts]
TI: Derivation of the Motion of Ball Lightning in the Maser-Soliton Theory
AU: * Handel, P H
EM: handel@umsl.edu
AF: Physics Dept., U. of MO, BH, St. Louis, MO 63121 United States
AU: Carlson, G A
EM:
AF: Physics Dept., U. of MO, BH, St. Louis, MO 63121 United States
AB: The motion of ball lightning (BL) in open air, in buildings, or in airplanes is explained here on the basis of the phase differences between the harmonic wave components that make up the soliton. Consider the quasi-stationary state of ball lightning in open air, on flat terrain. In general, the occupation of the energy levels that are subject to possible population inversion due to a sudden, large, but short electric field pulse caused, e.g., by lightning, will not be perfectly uniform. As seen from the location of the BL, the population inversion may be slightly stronger in a km3 on one side (&35;1) of the BL, than in a similar volume on the other side (&35;2) of the BL. The gradient g of the population inversion points towards &35;1. In particular, the waves originating in region &35;1 with k parallel to g, will be superposed on those coming from region &35;2, forming a standing wave E = Eoexpi[k.r-(W+s)t] + Eoexpi[-k.r-(W-s)t]. Here W is the average frequency in the maser. This frequency is lower due to damping. For population inversion it is higher. If the BL is in the middle of the maser region, we denoted by s the small deviation of the frequency W+s of the waves in the &35;1 region from its average, W. Then -s is the deviation from w in the &35;2 region. We can define the component parallel to k of a vector V, such that s = k.V; or Vk = s/k. Then, the elementary standing wave becomes E =Eoexpi[k.(r- Vt)-Wt] +Eoexpi[-k.(r- Vt)-Wt] =2Eo{cos[k.(r- Vt)]}expiWt. This is a standing wave that is moving with a velocity Vk =s/k. The component of V perpendicular to k is left arbitrary, depending on boundary conditions. Choosing the origin of the coordinates at the location of the BL, for s=0 we can express the total wave, including the BL, in terms of a cosine-Fourier integral packet of standing waves, Etot =IntegralE(k)cosk.r d3x, centered on the origin. With s nonzero, Etot = IntegralE(k)cos[k.(r -Vt)]d3x, and the BL will move with velocity v opposite to the gradient of the population of levels. In general, therefore, we conclude that BL tends to move away from the region of larger population inversion, or towards the region with larger absorption coefficient. This formulation can be done in terms of the dielectric constant er and quantum mechanical matrix elements. The permittivity is e=e0[1 + C(W)]=e0[1 + C`(W) + iC"(W)]. The susceptibility introduced by a population inversion Nn-Nm per unit volume on a molecular transition of frequency Wmn and damping G is C(W) = -3i[qqFnm(Nn-Nm)]/{Me0Wmn[Grad +2i(W-Wmn)]}, where Grad =1/Trad must be replaced by the total linewidth DWmn to include all forms of broadening. The oscillator strength Fmn= 4pMWmnRmn2/3h =Trad /Tmn is related to the molecular quantum mechanical matrix element Rnm the real molecular damping rate Tmn for electron charge q, mass M. The phase velocity is v=c/n=c/(er)1/2. Therefore, for C≪1 it can be written as v=Re[c/(1+C'/2+iC"/2)]=c[(1+C'/2)]/[(1+C'/2)2+C"C"/4]=[c/(1+ C'/2)]{1 - C"C"/4(1+C'/2)2; s=-cC"C"/4(1+C'/2)3 - >-cC"2/4(1+C'/2)3>, Vk = sk/kk. The small difference present in W=kv is given by the term C". The average < > is over the whole maser volume of many km3. Thus, we obtain Vk = s/k, and C'/2=3[Fnmqq(Nn-Nm)(W-Wmn)]/{Me0Wmn[(DWmn)2+4(W-Wmn)2]} C"=3[Fnmqq(Nn-Nm)Grad]/{Me0Wmn[(DWmn)2+4(W-Wmn)2]}. This is proportional to (Nn-Nm). The linear absorption/amplification coefficient is A =WC"/2c. For balanced maser conditions, the losses caused by the end reflection coefficient R must equal the maser gain: RRexp(-2AL)=1, or AL=logR. Therefore, we get C"=(2c/WL)logR. Substituting into Vk, Vk=-c[kC"2/4k(1+C'/2)3]=-[kc3(logR)/kWWLL(1+C'/2)3]=-(lamda/2pL)2[ck(log r)2/k(1+C'/2)3]. With lamda= 0.5 m, L =1 km, and (1+C'/2)3=2, this yields Vk= 1m/s, in agreement with observations.
DE: 3304 Atmospheric electricity
DE: 3324 Lightning
DE: 3384 Waves and tides
SC: Atmospheric and Space Electricity [AE]
MN: 2005 Joint Assembly