Mark J. McGuinness

THE COMPLEX LORENZ EQUATIONS

We have undertaken a study of the complex Lorenz equations x = −σx + σy . y = (r − z)x − ay . z = −bz + 12(x∗y + xy∗) . 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 r1c and r1c . The origin is a stable fixed point for 0 r1c, a Hopf bifurcation to a limit cycle occurs. We have an exact analytic solution for this limit cycle which is always stable if σ + 1 then this limit is only stable in the region r1c rlc, 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.

The heat source of Ruapehu crater lake; deductions from the energy and mass balances

Abstract Regular observations of temperature, outflow rates and water chemistry of Crater Lake, Mt. Ruapehu, New Zealand have been made for the last 25 years. These data have been used to derive a model of the dynamics of the lake, and determine the input of energy, mass, and chloride from the volcano to the Crater Lake. The recent, relatively quiescent state of the volcano, when virtually no heat has been input to the lake, has also enabled an assessment to be made of the surface heat loss characteristics, which play an important role in the model of the lake. The modelling suggests that since about 1982 the ratio of the volcanic heat to mass added to the base of the lake is about 6 MJ/kg, which is not compatible with heating of the lake by magmatic steam alone. Thus, only about 50% of the heating has been by magmatic steam. It is suggested that heat could be transferred from a magmatic source to the region below the lake by a heat-pipe mechanism, commonly associated with geothermal systems. The simultaneous upward movement of vapour phase, and downward movement of liquid phase from condensed vapour allows efficient heat transfer without overall mass transfer. The permeability necessary to supply the required heat is of the order of 10 darcy, and is consistent with a rubble filled vent. For at least the last five years, there has been a characteristic pattern in the Crater Lake temperature record, with alternate heating and cooling phases. The heating phase generally lasts for one or two months, while the cooling phase lasts for six months to a year. A possible explanation for this cyclic behaviour is the presence of a layer of liquid sulphur under Crater Lake, acting as a partial barrier between the heat-pipe and the lake. The unusual variations of the viscosity of liquid sulphur with temperature will mean that at temperatures greater than 160°C, the layer of sulphur becomes highly viscous and would block any upwards steam flow and hence stop the heat input to Crater Lake, so producing a cooling phase. This blockage would last until the heating from below raised the temperature of the sulphur beyond the high-viscosity region, so gases could again pass through the sulphur.

The real and complex Lorenz equations in rotating fluids and lasers

Abstract 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 real and complex Lorenz equations and their relevance to physical systems

Abstract We summarize some recently obtained results on real and complex Lorenz equations and discuss their possible significance in relation to real fluid dynamical processes.

Arnold tongues in human cardiorespiratory systems.

Arnold tongues are phase-locking regions in parameter space, originally studied in circle-map models of cardiac arrhythmias. They show where a periodic system responds by synchronizing to an external stimulus. Clinical studies of resting or anesthetized patients exhibit synchronization between heart-beats and respiration. Here we show that these results are successfully modeled by a circle-map, neatly combining the phenomena of respiratory sinus arrhythmia (RSA, where inspiration modulates heart-rate) and cardioventilatory coupling (CVC, where the heart is a pacemaker for respiration). Examination of the Arnold tongues reveals that while RSA can cause synchronization, the strongest mechanism for synchronization is CVC, so that the heart is acting as a pacemaker for respiration.

A theoretical model of the explosive fragmentation of vesicular magma

Recent experimental work has shown that, when a vertical column of rock under large pressure is suddenly depressurized, the column can ‘explode’ in a structured and repeatable way. The observations show that a sequence of horizontal fractures forms from the top down, and the resulting blocks are lifted off and ejected. The blocks can suffer secondary internal fractures. This experiment provides a framework for understanding the way in which catastrophic explosion can occur, and is motivated by the corresponding phenomenon of magmatic explosion during Vulcanian eruptions. We build a theoretical model to describe these results, and show that it is capable of describing both the primary sequence of fracturing and the secondary intrablock fracturing. The model allows us to suggest a practical criterion for when such explosions occur: firstly, the initial confining pressure must exceed the yield stress of the rock, and, secondly, the diffusion of the gas by porous flow must be sufficiently slow that a large excess pore pressure is built up. This will be the case if the rock permeability is small enough.

Geothermal heat pipe stability: Solution selection by upstreaming and boundary conditions

In a geothermal reservoir, the heat pipe mechanism can transfer heat very efficiently, with vapor rising and liquid falling in comparable quantities, driven by gravity. For a given heat and mass flux that is not too large, there are two possible steady solutions with vapor-liquid counterflow, one liquid-dominated, and one vapor-dominated. Numerical solution of the equations for two-phase vertical counterflow displays intriguing stability behaviour. If pressure and saturation are fixed at depth, and heat and mass flux specified at the top, the vapor-dominated solution is almost always obtained. That is, for a variety of boundary values, the solution settles to the vapor-dominated steady-state, and only for very special values is it possible to obtain the liquid-dominated case. Similarly the liquid-dominated solution is almost always obtained if the boundary conditions are reversed, with pressure and saturation fixed at the top and heat and mass flux specified at depth.This behaviour is here explained in two complementary ways. It is shown to be a consequence of upstream differencing of the flow terms in the numerical method. It is also shown to be expected behaviour for wavelike saturation solutions. Hence the observed behaviour is not only a direct consequence of the numerical method used, but is fundamental to geothermal heat pipes.

Amplitude equations at the critical points of unstable dispersive physical systems

The amplitude equations that govern the motion of wavetrains near the critical point of unstable dispersive, weakly nonlinear physical systems are considered on slow time and space scales Tm ═ εmt; Xm ═ εmx (m ═ 1, 2,...). Such systems arise when the dispersion relation for the harmonic wavetrain is purely real and complex conjugate roots appear when a control parameter (μ) is varied. At the critical point, when the critical wavevector kc is non-zero, a general result for this general class of unstable systems is that the typical amplitude equations are either of the form ( ∂/∂T1 + c1∂/∂X1) (∂/∂T1 + c2∂/∂X1) A ═ ±αA ─ βAB, ( ∂/∂T1 + c2∂/∂X1) B ═ (∂/∂T 1 + c1∂/∂X1) |A|2, or of the form ( ∂/∂T1 + c1∂/∂X1) (∂/∂T1 + c2∂/∂X1) A ═ ±αA - βA |A|2. The equations with the AB-nonlinearity govern for example the two-layer model for baroclinic instability and self-induced transparency (s. i. t.) in ultra-short optical pulse propagation in laser physics. The second equation occurs for the two-layer Kelvin-Helmholtz instability and a problem in the buckling of elastic shells. This second type of equation has been considered in detail by Weissman. The AB-equations are particularly important in that they are integrable by the inverse scattering transform and have a variety of multi-soliton solutions. They are also reducible to the sine-Gordon equation ϕξƬ ═ ± sin ϕ when A is real. We prove some general results for this type of instability and discuss briefly their applications to various other examples such as the two-stream instability. Examples in which dissipation is the dominant mechanism of the instability are also briefly considered. In contrast to the dispersive type which operates on the T1-time scale, this type operates on the T2-scale.

Heat transport in McMurdo Sound first‐year fast ice

We have monitored the temperature field within first-year sea ice in McMurdo Sound over two winter seasons, with sufficient resolution to determine the thermal conductivity from the thermal waves propagating down through the ice. Data reduction has been accomplished by direct reference to energy conservation, relating the rate of change of the internal energy density to the divergence of the heat current density. Use of this procedure, rather than the wave attenuation predicted by the thermal diffusion equation, avoids difficulties arising from a strongly temperature dependent thermal diffusivity. The thermal conductivity is an input parameter for ice growth and climate models, and the values commonly used in the models are predicted to depend on temperature, salinity, and the volume fraction of air. The present measurements were performed at depths in the ice where the air volume is small and the salinity is nearly constant, and they permit the determination of the absolute magnitude of the thermal conductivity and its temperature dependence. The weak temperature dependence is similar to that predicted by the models in the literature, but the magnitude is smaller by ∼10% than the predicted value most commonly used in climate and sea ice models. In the first season we find an additional scatter in the results at driving temperature gradients larger than ∼10–15 °C/m. We suggest that the scatter arises from a nonlinear contribution to the heat current, possibly associated with the onset of convective motion in brine inclusions. Episodic convective events are also observed. We have further determined the growth rate of the ice and compared it with the rate explained by the heat flux from the ice-water interface. The data show a sudden rise of growth rate, without a rise in heat flux through the ice, which coincides in time and depth with the appearance of platelet ice. Finally, we discuss the observation of radiative solar heating at depth in the ice and demonstrate that the absorption exceeds that in the ice alone; dust or algae must contribute to the absorption.

Mean action time for diffusive processes

For a number of diffusive processes involving heat and mass transfer, a convenient and easy way to solve for penetration time or depth is to consider an averaged quantity called mean action time. This approach was originally developed by Alex McNabb, in collaboration with other researchers. It is possible to solve for mean action time without actually solving the full diffusion problem, which may be nonlinear, and may have internal moving boundaries. Mean action time satisfies a linear Poisson equation, and only works for finite problems. We review some nice properties of mean action time, and discuss some recent novel applications.