The complex Lorenz equations

We have undertaken a study of the complex Lorenz equations

$\text{x}\text{?}\text{= ?σx + σy}$

.

$\text{y}\text{?}\text{= (r ? z)x ? ay}$

.

$\text{z}\text{?}\text{= ?bz +}\text{1}\text{2}\text{(x}{}^1\text{y + xy}{}^1\text{)}$

. where x and y are complex and z is real. The complex parameters r and a are defined by r = r₁ + ir₂; a = 1 ? ie and σ and b are real. Behaviour remarkably different from the real Lorenz model occurs. Only the origin is a fixed point except for the special case e + r₂ = 0. We have been able to determine analytically two critical values of r₁, namely r1_c and r1_c . The origin is a stable fixed point for 0 < r₁ < r_1c, but for r₁ >r_1c, a Hopf bifurcation to a limit cycle occurs. We have an exact analytic solution for this limit cycle which is always stable if σ <b + 1. If σ > + 1 then this limit is only stable in the region r_1c < r₁ < r_lc. When r₁ >r_lc, a transition to a finite amplitude oscillation about the limit cycle occurs. The nature of this bifurcation is studied in detail by using a multiple time scale analysis to derive the Stuart-Landau amplitude equation from the original equations in a frame rotating with the limit cycle frequency. This latter bifurcation is either a sub- or super-critical Hopf-like bifurcation to a doubly periodic motion, the direction of bifurcation depending on the parameter values. The nature of the bifurcation is complicated by the existence of a zero eigenvalue.     

Arsen

Ohne Zusammenfassung     

The real and complex Lorenz equations in rotating fluids and lasers

The Lorenz equations are derived systematically from amplitude equations of weakly nonlinear dispersively unstable physical systems near criticality when weak dissipation is added. This derivation is only valid if the undamped neutral curve is not destabilised by the addition of weak dissipation. The addition of extra weak dispersive effects make some of the coefficients complex and yields a complex set of Lorenz equations. Both sets of equations are derived in examples in laser optics and baroclinic instability.     

The 3D Incompressible Euler Equations with a Passive Scalar: A Road to Blow-Up?

The three-dimensional incompressible Euler equations with a passive scalar θ are considered in a smooth domain $\varOmega\subset \mathbb{R}^{3}$ with no-normal-flow boundary conditions $\boldsymbol{u}\cdot\hat{\boldsymbol{n}}|_{\partial\varOmega} = 0$ . It is shown that smooth solutions blow up in a finite time if a null (zero) point develops in the vector B=?q×?θ, provided B has no null points initially: $\boldsymbol{\omega} = \operatorname{curl}\boldsymbol {u}$ is the vorticity and q=ω??θ is a potential vorticity. The presence of the passive scalar concentration θ is an essential component of this criterion in detecting the formation of a singularity. The problem is discussed in the light of a kinematic result by Graham and Henyey (Phys. Fluids 12:744–746, 2000) on the non-existence of Clebsch potentials in the neighbourhood of null points.     

Hot-electron-pumped K-shell X-ray laser

Amplified spontaneous inner-shell emission produced via an ultrafast burst of high-energy electrons from a femtosecond laser-produced plasma is proposed as a novel electron-pumped x-ray laser. In this scheme, a population inversion of the upper laser level is created via impact ionization of atomic inner shells by electron bombardment. Based on the requirement of a positive gain coefficient for amplifying spontaneous K _α line emission, a simple pumping threshold is found for the incident electron flux, and feasibility of the scheme is assessed for a range of low-Z elements. Published in English in the original Russian journal. Edited by Steve Torstveit. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 10, 771–776 (25 May 1998)     

Water-Mediated Epoxy/Surface Adhesion: Understanding the Interphase Region

Epoxy resins coatings are commonly found in corrosion protection coatings but the presence of water can affect their adhesion to the substrate, often weakening the adhesion of the coating to the solid, reducing its efficiency. Nevertheless, small amounts of water can enhance the epoxy/substrate interactions. In this work, the interphase region of an epoxy precursor and metal oxide substrates is investigated using molecular simulations and it is found that water accumulates between the epoxy layer and the solid substrate. At high water concentrations (9 wt %) the interaction between the epoxy precursor and the solid surface is weakened regardless of the nature of the solid, but at low water concentrations the nature of the solid surface becomes important. For hematite, the presence of water decreases the strength of adhesion but for goethite the presence of a small amount of water (3 wt %) enhances the adhesion to the surface resulting in a densification at the interface.     

Observation of the Hadronic transitions chi(b1,2)(2P)-->omegaUpsilon(1S)

The CLEO Collaboration has made the first observations of hadronic transitions among bottomonium (bbmacr;) states other than the dipion transitions among Upsilon(nS) states. In our study of Upsilon(3S) decays, we find a significant signal for Upsilon(3S)-->gammaomegaUpsilon(1S) that is consistent with radiative decays Upsilon(3S)-->gammachi(b1,2)(2P), followed by chi(b1,2)(2P)-->omegaUpsilon(1S). The branching ratios we obtain are B[chi(b1)(2P)-->omegaUpsilon(1S)]=(1.63(+0.35+0.16)(-0.31-0.15))% and B[chi(b2)(2P)-->omegaUpsilon(1S)]=(1.10(+0.32+0.11)(-0.28-0.10))%, in which the first error is statistical and the second is systematic.