On the frontal condition for finite amplitude α2Ω-dynamo wave trains in stellar shells

Abstract

In a previous paper, Bassom et al. (Proc. R. Soc. Lond. A, 455, 1443–1481, 1999) (BKS) investigated finite amplitude αΩ-dynamo wave trains in a thin turbulent, differentially rotating convective stellar shell; nonlinearity arose from α-quenching. There asymptotic solutions were developed based upon the small aspect ratio ε of the shell. Specifically, as a consequence of a prescribed latitudinally dependent α-effect and zonal shear flow, the wave trains have smooth amplitude modulation but are terminated abruptly across a front at some high latitude θ_F. Generally, the linear WKB-solution ahead of the front is characterised by the vanishing of the complex group velocity at a nearby point θ_f; this is essentially the Dee–Langer criterion, which determines both the wave frequency and front location.

Recently, Griffiths et al. (Geophys. Astrophys. Fluid Dynam. 94, 85–133, 2001) (GBSK) obtained solutions to the α²Ω-extension of the model by application of the Dee—Langer criterion. Its justification depends on the linear solution in a narrow layer ahead of the front on the short O(θ_f—θ_F) length scale; here conventional WKB-theory, used to describe the solution elsewhere, is inadequate because of mode coalescence. This becomes a highly sensitive issue, when considering the transition from the linear solution, which occurs when the dynamo number D takes its critical value D _c corresponding to the onset of kinematic dynamo action, to the fully nonlinear solutions, for which the Dee—Langer criterion pertains.

In this paper we investigate the nature of the narrow layer for α²Ω-dynamos in the limit of relatively small but finite α-effect Reynolds numbers R _α, explicitly ε^½ ? R ² _α ? 1. Though there is a multiplicity of solutions, our results show that the space occupied by the corresponding wave train is generally maximised by a solution with θ_f—θ_F small; such solutions are preferred as evinced by numerical simulations. This feature justifies the application by GBSK of the Dee—Langer criterion for all D down to the minimum D _min that the condition admits. Significantly, the frontal solutions are subcritical in the sense that |D _min| ≤ |D _c|; equality occurs as the α-effect Reynolds number tends to zero. We demonstrate that the critical linear solution is not connected by any parameter track to the preferred nonlinear solution associated with D _min. By implication, a complicated bifurcation sequence is required to make the connection between the linear and nonlinear states. This feature is in stark contrast to the corresponding results for αΩ-dynamos obtained by BKS valid in the limit R _{2 α} ? ε^½, which, though exhibiting a weak subcriticality, showed that the connection follows a clearly identifiable nonbifurcating track.     

Tangential discontinuities and the optical analogy for stationary fields. v. formal integration of the force-free field equations

Abstract

This paper demonstrates the appearance of tangential discontinuities in deformed force-free fields by direct integration of the field equation ? x B = αB. To keep the mathematics tractable the initial field is chosen to be a layer of linear force-free field B_x = + B ₀cosqz, B_y = — B ₀sinqz, B_z = 0, anchored at the distant cylindrical surface ? = (x ² + y ²)^1/2 = R and deformed by application of a local pressure maximum of scale l centered on the origin x = y = 0. In the limit of large R/l the deformed field remains linear, with α = q[1 + O(l ²/R ²)]. The field equations can be integrated over ? = R showing a discontinuity extending along the lines of force crossing the pessure maximum. On the other hand, examination of the continuous solutions to the field equations shows that specification of the normal component on the enclosing boundary ? = R completely determines the connectivity throughout the region, in a form unlike the straight across connections of the initial field. The field can escape this restriction only by developing internal discontinuities.

Casting the field equation in a form that the connectivity can be specified explicitly, reduces the field equation to the eikonal equation, describing the optical analogy, treated in papers II and III of this series. This demonstrates the ubiquitous nature of the tangential discontinuity in a force-free field subject to any local deformation.     

Dependence of geomagnetic activity during magnetic storms on the solar wind parameters for different types of streams

The dependence of the maximal values of the |Dst| and AE geomagnetic indices observed during magnetic storms on the value of the interplanetary electric field (E _y ) was studied based on the catalog of the large-scale solar wind types created using the OMNI database for 1976–2000 [Yermolaev et al., 2009]. An analysis was performed for eight categories of magnetic storms caused by different types of solar wind streams: corotating interaction regions (CIR, 86 storms); magnetic clouds (MC, 43); Sheath before MCs (Sh_MC, 8); Ejecta (95); Sheath (Sh_E, 56); all ICME events (MC + Ejecta, 138); all compression regions Sheaths before MCs and Ejecta (Sh_MC + Sh_E, 64); and an indeterminate type of storm (IND, 75). It was shown that the |Dst| index value increases with increasing electric field E _y for all eight types of streams. When electric fields are strong (E _y > 11 mV m⁻¹), the |Dst| index value becomes saturated within magnetic clouds MCs and possibly within all ICMEs (MC + Ejecta). The AE index value during magnetic storms is independent of the electric field value E _y for almost all streams except magnetic clouds MCs and possibly the compressed (Sheath) region before them (Sh_MC). The AE index linearly increases within MC at small values of the electric field (E _y < 11 mV m⁻¹) and decrease when these fields are strong (E _y > 11 mV m⁻¹). Since the dynamic pressure (Pd) and IMF fluctuations (σB) correlate with the E _y value in all solar wind types, both geomagnetic indices (|Dst| and AE) do not show an additional dependence on Pd and IMF δB. The nonlinear relationship between the intensities of the |Dst| and AE indices and the electric field E _y component, observed within MCs and possibly all ICMEs during strong electric fields E _y , agrees with modeling the magnetospheric-ionospheric current system of zone 1 under the conditions of the polar cap potential saturation.     

An explicit solution for static unbounded helical dynamos

Abstract

The Lortz dynamo with helical symmetry is re-examined. It is shown that by imposing appropriate boundary conditions the set of possible solutions can be broken down into various classes characterized by the behavior of the mean magnetic field. It is found that, as the cylindrical radius, s, tends to zero, <B_Φ> ～ 0(s^j), <B_z> ～ const + 0(s^j?i), where j>5. It is proved that the azimuthal wavenumber associated with the j=5 class is necessarily equal to 2. The existence of at least one cylindrical surface inside which the dynamo is self-sustained is demonstrated. A new simple explicit solution is obtained. The topology the magnetic field is studied and three-dimensional pictures of the magnetic field lines are exhibited. Finally, a criterion for reversal of the magnetic field as a function of radius is ohtained and is applied to our solution.     

Non-axisymmetric α2Ω-dynamo waves in thin stellar shells

Linear α²Ω-dynamo waves are investigated in a thin turbulent, differentially rotating convective stellar shell. A simplified one-dimensional model is considered and an asymptotic solution constructed based on the small aspect ratio of the shell. In a previous paper Griffiths et al. (Griffiths, G.L., Bassom, A.P., Soward, A.M. and Kuzanyan, K.M., Nonlinear α²Ω-dynamo waves in stellar shells, Geophys. Astrophys. Fluid Dynam., 2001, 94, 85–133) considered the modulation of dynamo waves, linked to a latitudinal-dependent local α-effect and radial gradient of the zonal shear flow. These effects are measured at latitude θ by the magnetic Reynolds numbers R _α f(θ) and R _Ω g(θ). The modulated Parker wave, which propagates towards the equator, is localised at some mid-latitude θ_p under a Gaussian envelope. In this article, we include the influence of a latitudinal-dependent zonal flow possessing angular velocity Ω_*(θ) and consider the possibility of non-axisymmetric dynamo waves with azimuthal wave number m. We find that the critical dynamo number D _c?=?R _α R _Ω is minimised by axisymmetric modes in the αΩ-limit (Rα→0). On the other hand, when Rα?≠?0 there may exist a band of wave numbers 0?m?m ^? for which the non-axisymmetric modes have a smaller D _c than in the axisymmetric case. Here m ^? is regarded as a continuous function of R _α with the property m?→0 as R _α→0 and the band is only non-empty when m??>1, which happens for sufficiently large R _α. The preference for non-axisymmetric modes is possible because the wind-up of the non-axisymmetric structures can be compensated by phase mixing inherent to the α²Ω-dynamo. For parameter values resembling solar conditions, the Parker wave of maximum dynamo activity at latitude θ_p not only propagates equatorwards but also westwards relative to the local angular velocity Ω_*(θ _p ). Since the critical dynamo number D _c?=?R _α R _Ω is O (1) for small R _α, the condition m ^??>?1 for non-axisymmetric mode preference imposes an upper limit on the size of |dΩ_*/dθ|.     

Resistive instabilities in rapidly rotating fluids: Linear theory of the convective modes

Abstract

This paper analyzes the linear stability of a rapidly-rotating, stratified sheet pinch in a gravitational field, g, perpendicular to the sheet. The sheet pinch is a layer (O ? z ? d) of inviscid, Boussinesq fluid of electrical conductivity σ, magnetic permeability μ, and almost uniform density ρ _o; z is height. The prevailing magnetic field. B _o(z), is horizontal at each z level, but varies in direction with z. The angular velocity, Ω, is vertical and large (Ω ? V_A/d, where V_A = B₀√(μρ₀) is the Alfvén velocity). The Elsasser number, Λ = σB² ₀/2Ωρ₀, measures σ. A (modified) Rayleigh number, R = gβd²/ρ₀V² _A, measures the buoyancy force, where β is the imposed density gradient, antiparallel to g. A Prandtl number, P_K = μσK, measures the diffusivity, k, of density differences.     

Turbulent transport of magnetic fields. V. Distribution of magnetic energy in a simple α2-dynamo

Abstract

In this paper we analyse the stationary mean energy density tensor T_ij = B_iB_j for the x ²-sphere. This model is one of the simplest possible turbulent dynamos, originally due to Krause and Steenbeck (1967): a conducting sphere of radius R with homogeneous, isotropic and stationary turbulent convection, no differential rotation and negligible resistivity. The stationary solution of the (linear) equation for T_ij is found analytically. Only T_rr , T _θθ and T _φφ are unequal to zero, and we present their dependence on the radial distance r.

The stationary solution depends on two coefficients describing the turbulent state: the diffusion coefficient β≈?u²?τ_c/3 and the vorticity coefficient γ ≈ ?|?×u|²?τ_c/3 where u(r, t) is the turbulent velocity and _c its correlation time. But the solution is independent of the dynamo coefficient α≈??u·?×u?τ_c/3 although α does occur in the equation for T_ij . This result confirms earlier conclusions that helicity is not required for magnetic field generation. In the stationary state, magnetic energy is generated by the vorticity and transported to the boundary, where it escapes at the same rate. The solution presented contains one free parameter that is connected with the distribution of B over spatial scales at the boundary, about which T_ij gives no information. We regard this investigation as a first step towards the analysis of more complicated, solar-type dynamos.     

Large-scale structure of the magnetic field in the outer heliosphere

Summary Magnetic field structures at great distances from the Sun have been analyzed qualitatively for a simple vacuum reconnection model of the interplanetary and interstellar magnetic field. In dependence on the mutual orientation of the main solar dipole _s and the local interstellar fieldB ₀ , either an open or closed configuration of the large-scale field is formed. For(_s B ₀ )>0, the field lines are represented by a system of magnetic lines open towards interstellar space. In the case of(_s B ₀ )<0 there exist two zero-points and a separating surface below the heliopause separating the open lines of the interstellar field from the closed lines of the interplanetary field. The magnetic field configuration is characterized by a certain asymmetry, which is considered for(_s B ₀ )=0.     

An antidynamo theorem for partly symmetric flows

Abstract

It is shown that magnetic fields generated by flows v _r,(r,t)e_r+v_T where v_T is an arbitrary toroidal component (e_r˙v_T≡V≡v_T≡0), cannot be maintained indefinitely against ohmic dissipation. The poloidal field variable max |r ² B _r| is shown to decay strictly monotonically with an undetermined decay rate. A bound on the growth of the toroidal field norm ∥T∥₁ is established solely dependent on the rate of conversion of poloidal to toroidal field, so that when the poloidal field is negligible then ∥T∥₁ decays strictly monotonically. The main application of these results is to models of stellar evolution based on axisymmetric differential rotation and spherically symmetric contraction. This symmetric velocity theorem overlaps with two already known theorems, namely the toroidal velocity theorem where v _r≡0 and the radial velocity theorem where v_T≡0. The new theorem does not entirely include the already established ones, principal differences being in the rates of decay and the field variables for which the decay is proven (see Table 1).     

Cusp Modeling and Observations at Low Altitude

Cusp properties have been investigated with an open-field line particle precipitation model and Defense Meteorological Satellite Program (DMSP) satellite observations. Particular emphasis is placed on the effects of IMF B_y, since previous studies focus mostly on IMF B_z. The model-data comparisons for various IMF configurations show that the model captures the large-scale features of the particle precipitation very well, not only in the cusp region, but also in other open-field line regions such as the mantle, polar rain, and open-field line low-altitude boundary layer (LLBL). When the IMF is strongly duskward/dawnward and weakly southward, the model predicts the occurrence of double cusp near noon: one cusp at lower latitude and one at higher latitude. The lower latitude cusp ions originate from the low-latitude magnetosheath whereas the higher latitude ions originate from the high-latitude magnetosheath. The lower latitude cusp is located in the region of weak azimuthal E × B drift, resulting in a dispersionless cusp. The higher latitude cusp is located in the region of strong azimuthal and poleward E × B drift. Because of a significant poleward drift, the higher latitude cusp dispersion has some resemblance to that of the typical southward IMF cusp. Occasionally, the two parts of the double cusp have such narrow latitudinal separation that they give the appearance of just one cusp with extended latitudinal width. From the 40 DMSP passes selected during periods of large (positive or negative) IMF B_y and small negative IMF B_z, 30 (75%) of the passes exhibit double cusps or cusps with extended latitudinal width. The double cusp result is consistent with the following statistical results: (1) the cusp’s latitudinal width increases with |IMF B_y| and (2) the cusp’s equatorward boundary moves to lower latitude with increasing |IMF B_y|.     

The electric field produced in the mantle by the dynamo in the core

A rigorous singular perturbation theory is developed to estimate the electric field E produced in the mantle M by the core dynamo when the electrical conductivity σ in M depends only on radius r, and when |r?_rln σ| ? 1 in most of M. It is assumed that σ has only one local minimum in M, either (a) at the Earth's surface ?V, or (b) at a radius b inside the mantle, or (c) at the core-mantle boundary ?K. In all three cases, the region where σ is no more than e times its minimum value constitutes a thin critical layer; in case (a), the radial electric field E_r ≈ 0 there, while in cases (b) and (c), E_r is very large there. Outside the critical layer, E_r ≈ 0 in all three cases. In no case is the tangential electric field E_S small, nearly toroidal, or nearly calculable from the magnetic vector potential A as ??_tA_S. The defects in Muth's (1979) argument which led him to contrary conclusions are identified. Benton (1979) cited Muth's work to argue that the core-fluid velocity u just below ?K can be estimated from measurements on ?V of the magnetic field B and its time derivative $\text{?}{}_{\mathrm{t}}\text{B}$. A simple model for westward drift is discussed which shows that Benton's conclusion is also wrong.In case (a), it is shown that knowledge of σ in M is unnecessary for estimating E_S on ?K with a relative error |r?_r 1nσ|^?1from measurements of E_S on ?V and knowledge of ?_tB in M (calculable from ?_tB on ?V if σ is small). Then, in case (a), u just below ?K can be estimated as $\text{?}\text{r}\text{×}\text{E}{}_{\mathrm{S}}\text{/B}{}_{\mathrm{r}}$. The method is impractical unless the contribution to E_S on ?V from ocean currents can be removed.The perturbation theory appropriate when σ in M is small is considered briefly; smallness of σ and of |r?_r ln σ|^?1 a independent questions. It is found that as σ → 0, B approaches the vacuum field in M but E does not; the explanation lies in the hydromagnetic approximation, which is certainly valid in M but fails as σ → 0. It is also found that the singular perturbation theory for |r?_r ln σ|^?1 is a useful tool in the perturbation calculations for σ when both σ and |r?_r ln σ|^?1 are small.     

The α-effect in 2D fast dynamos

We look at the large-scale dynamo properties of spatially periodic, time dependent, helical 2D flows of the form u(x, t)?=?(? _y ?ψ?(x, y, t), ?? _x ?ψ?(x, y, t), ?ψ (x, y, t). These flows act as kinematic fast dynamos and are able to generate a mean magnetic field uniform and constant in the xy-plane but whose direction varies periodically along z with wavenumber k. Using Mean Field Electrodynamics, the generation mechanism can be understood in terms of a k-dependent α-effect, which depends on the magnetic Reynolds number, R _m . We calculate this effect for different motions and investigate how its limit as k?→?0 depends on R _m and on the properties of the flows such as their spatial structure or correlation time. This work generalises earlier studies based on 2D steady flows to motions with time dependence.     

Maximally-efficient-generation approach in the dynamo theory

Abstract

We propose a method of derivation of global asymptotic solutions of the hydromagnetic dynamo problem at large magnetic Reynolds number. The procedure reduces to matching the local asymptotic forms for the magnetic field generated near individual extrema of generation strength. The basis of the proposed method, named here the Maximally-Efficient-Generation Approach (MEGA), is the assertion that properties of global asymptotic solutions of the kinematic dynamo are determined by the distribution of the generation strength near its leading extrema and by the number and distribution of the extrema.

The general method is illustrated by the global asymptotic solution of the α²-dynamo problem in a slab. The nature of oscillatory solutions revealed earlier in numerical simulations and the reasons for the dominance of even magnetic modes in slab geometry are clarified.

Applicability of the asymptotic solutions at moderate values of the asymptotic parameter is also discussed. We confirm this applicability using comparisons with complementary asymptotic expansions and numerical simulations. In particular, this justifies application of the MEGA solutions to estimation of the generation threshold.     

α2-Dynamos and taylor's constraint

Abstract

An idealised α²ω-dynamo is considered in which the α-effect is prescribed. The additional ω-effect results from a geostrophic motion whose magnitude is determined indirectly by the Lorentz forces and Ekman suction at the boundary. As the strength of the α-effect is increased, a critical value α? _c is reached at which dynamo activity sets in; α? _c is determined by the solution of the kinematic α²-dynamo problem. In the neighbourhood of the critical value of α? the magnetic field is weak of order E ^1/4(μηρω)^½ due to the control of Ekman suction; E(?1) is the Ekman number. At certain values of α?, viscosity independent solutions are found satisfying Taylor's constraint. They are identified by the bifurcation of a nonlinear eigenvalue problem. Dimensional arguments indicate that following this second bifurcation the magnetic field is strong of order (μηρω)^½. The nature of the transition between the kinematic linear theory and the Taylor state is investigated for various distributions of the α-effect. The character of the transition is found to be strongly model dependent.     

Magnetic instability in a rapidly rotating cylindrical annulus with a finitely conducting inner core

Abstract

We consider the stability of a toroidal magnetic field B = B(s*) (where (s*,φ,z*) are cylindrical polar coordinates) in a cylindrical annulus of conducting fluid with inner and outer radii s_i and s _o rotating rapidly about its axis. The outer boundary is taken to be either insulating or perfectly conducting, and the effect of a finite magnetic diffusivity in the inner core is investigated. The ratio of magnetic diffusivity in the inner core to that of the outer core is denoted by ηη→0 corresponding to a perfectly conducting inner core and η→∞ to an insulating one. Comparisons with the results of Fearn (1983b, 1988) were made and a good match found in the limits η→0 and η→∞ with his perfectly conducting and insulating results, respectively. In addition a new mode of instability was found in the eta;→0 regime. Features of this new mode are low frequency (both the frequency and growth rate →0 as η→0) and penetration deep into the inner core. Typically it is unstable at lower magnetic field strengths than the previously known instabilities.     

Hydromagnetic waves in a differentially rotating Annulus I. A test of local stability analysis

Abstract

A cylindrical annulus containing a conducting fluid and rapidly rotating about its axis is a useful model for the Earth's core. With a shear flow U ₀(s)∮, magnetic field B ₀(s)∮, and temperature distribution T _o(s) (where (s, ∮, z) are cylindrical polar coordinates), many important properties of the core can be modelled while a certain degree of mathematical simplicity is maintained. In the limit of rapid rotation and at geophysically interesting field strengths, the effects of viscous diffusion and fluid inertia are neglected. In this paper, the linear stability of the above basic state to instabilities driven by gradients of B ₀ and U ₀ is investigated. The global numerical results show both instabilities predicted by a local analysis due to Acheson (1972, 1973, 1984) as well as a new resistive magnetic instability. For the non-diffusive field gradient instability we looked at both monotonic fields [for which the local stability parameter Δ, defined in (1.4), is a constant] and non-monotonic fields (for which Δ is a function of s). For both cases we found excellent qualitative agreement between the numerical and local results but found the local criterion (1.6) for instability to be slightly too stringent. For the non-monotonic fields, instability is confined approximately to the region which is locally unstable. We also investigated the diffusive buoyancy catalysed instability for monotonic fields and found good quantitative agreement between the numerical results and the local condition (1.9). The new resistive instability was found for fields vanishing (or small) at the outer boundary and it is concentrated in the region of that boundary. The resistive boundary layer plays an important part in this instability so it is not of a type which could be predicted using a local stability analysis (which takes no account of the presence of boundaries).     

Chaotic behaviour of interplanetary magnetic field under various geomagnetic conditions

In the present study, the deterministic chaotic behaviour of interplanetary magnetic field (IMF) under various geomagnetic conditions of low and high solar active periods was analyzed, using the time series of IMF |B| and B_z, by employing chaotic quantifiers like, Lyapunov exponent, Tsallis entropy, correlation dimension, and non-linear prediction error. We have investigated whether the chaotic behaviour of interplanetary magnetic field would modify, when it produces major geomagnetic storms, and how it depends on the phase of solar activity. The yearly average values of Lyapunov exponent for the time series of IMF |B| and B_z, show solar flux dependence, whereas those values of entropy, correlation dimension and non-linear prediction error had no significant solar flux dependence. The yearly average values of entropy for quiet periods are higher compared to those values for major storm periods belonging to low/high solar active conditions, for both the time series |B| and B_z.     

Non-local effects in the mean-field disc dynamo I. An asymptotic expansion

Abstract

We reconsider thin-disc global asymptotics for kinematic, axisymmetric mean-field dynamos with vacuum boundary conditions. Non-local terms arising from a small but finite radial field component at the disc surface are consistently taken into account for quadrupole modes. As in earlier approaches, the solution splits into a local part describing the field distribution along the vertical direction and a radial part describing the radial (global) variation of the eigenfunction. However, the radial part of the eigenfunction is now governed by an integro-differential equation whose kernel has a weak (logarithmic) singularity. The integral term arises from non-local interactions of magnetic fields at different radii through vacuum outside the disc. The non-local interaction can have a stronger effect on the solution than the local radial diffusion in a thin disc, however the effect of the integral term is still qualitatively similar to magnetic diffusion.     

THE PROBLEM OF LONG-TERM STORAGE IN RESERVOIRS

Abstract

The geometry of a meandering stream depends strongly on the relative stream size (Q ^2/5 / g ^1/5)/D, on the valley slope, Sv, and on the charge, Q _s/Q, where Q and Q _s are the fluid and sediment discharges respectively, g is acceleration due to gravity and D is the mean sediment size. The geometry depends less strongly on the relative settling size of sediment, D/(v ^2/3 / g ^1/3), where v is the kinematic viscosity. For constant values of Q, S _v and D, the effect of increase of charge reduces the meander length, M _L, and the mean channel surface width, B, whereas meander width, M _B, bend radius, R _M, and mean channel depth, H, increase, For a constant value of (Q ^2/5 / g^1/5)/D the values of M _L, M _B, R _M and B increase with the increase of valley slope but the value of H tends to decrease.     

Diagnostics of the magnetosphere based on the outer belt relativistic electrons and penetration of solar protons: A review

The present-day state of the studies of the outer radiation belt relativistic electrons and the boundary of the solar proton penetration into the magnetosphere during magnetic storms is briefly reviewed. The main attention is paid to the results from studying the interrelation between these structural formations and other magnetospheric plasma structures. It has been indicated that the relationship between the position of the maximum of belt of relativistic electrons injected during magnetic storms (L _max) and the magnetic storm amplitude (|Dst|_max = 2.75 × 10⁴/L _max⁴) can be used to predict the extreme latitudinal position of such magnetospheric plasma formations as a trapped radiation region boundary, the nighttime equatorial boundary of the auroral oval, and westward electrojet center during a storm. Using the examples of still rare studies of the solar proton boundary dynamics in the magnetosphere based on the simultaneous measurements on several polar satellites, it has been demonstrated that a change in the geomagnetic field topology during magnetic storms can be diagnosed.