A numerical model of the ionosphere,including the E-region above EISCAT

It has been previously demonstrated that a two-ion (O⁺ and H⁺) 8-moment time-dependent fluid model was able to reproduce correctly the ionospheric structure in the altitude range probed by the EISCAT-VHF radar. In the present study, the model is extended down to the E-region where molecular ion chemistry (NO⁺ and O⁺₂, essentially) prevails over transport; EISCAT-UHF observations confirmed previous theoretical predictions that during events of intense E×B induced convection drifts, molecular ions (mainly NO⁺) predominate over O⁺ ions up to altitudes of 300 km. In addition to this extension of the model down to the E-region, the ionization and heating resulting from both solar insolation and particle precipitation is now taken into account in a consistent manner through a complete kinetic transport code. The effects of E×B induced convection drifts on the E- and F-region are presented: the balance between O⁺ and NO⁺ ions is drastically affected; the electric field acts to deplete the O⁺ ion concentration. The [NO⁺]/[O⁺] transition altitude varies from 190 km to 320 km as the perpendicular electric field increases from 0 to 100 mV m⁻¹. An interesting additional by-product of the model is that it also predicts the presence of a noticeable fraction of N⁺ ions in the topside ionosphere in good agreement with Retarding Ion Mass Spectrometer measurements onboard Dynamic Explorer.     

Morning polar substorms and variations in the atmospheric electric field

The effects of morning magnetospheric substorms in the variations in near-Earth atmospheric electricity according to the observations of the electric field vertical component (E _z ), at Hornsund polar observatory (Spitsbergen). The E _z, data, obtained under the conditions of fair weather (i.e., in the absence of a strong wind, precipitation, and fog), are analyzed. An analysis of the observations indicated that the development of a magnetospheric substorm in the Earth’s morning sector is as a rule accompanied by positive deviations in E _z, independently of the Hornsund location: in the polar cap or at its boundary. In all considered events, Hornsund was located near the center of the morning convection vortex. In the evening sector, when Hornsund fell in the region of evening convection vortex, the development of a geomagnetic substorm was accompanied by negative deviations in E _z., It has been concluded that the variations in the atmospheric electric field E _z), at polar latitudes, observed during the development of magnetospheric substorms, result from the penetration of electric fields of polar ionospheric convection (which are intensified during a substorm) to the Earth’s surface.     

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.     

A nonlinear dynamo driven by rapidly rotating convection

In Kim et al. (Kim, E., Hughes, D.W. and Soward, A.M., “An investigation into high conductivity dynamo action driven by rotating convection”, Geophys. Astrophys. Fluid Dynam. 91, 303–332 ().) we investigated kinematic dynamo action driven by rapidly rotating convection in a cylindrical annulus. Here we extend this work to consider self-consistent nonlinear dynamo action in which the back-reaction of the Lorentz force on the flow is taken into account. In particular, we investigate, as a function of magnetic Prandtl number, the evolution of an initially weak magnetic field in two different types of convective flow – one chaotic and the other integrable. On saturation, the latter shows a systematic dependence on the magnetic Prandtl number whereas the former appears not to. In addition, we show how, in keeping with the findings of Cattaneo et al. (Cattaneo, F., Hughes, D.W. and Kim, E., “Suppression of chaos in a simplified nonlinear dynamo model”, Phys. Rev. Lett. 76, 2057–2060 ().), saturation of the growth of the magnetic field is brought about, for the originally chaotic flow, by a strong suppression of chaos.     

Convective dynamos with intermediate and strong fields

Abstract

The weak-field Benard-type dynamo treated by Soward is considered here at higher levels of the induced magnetic field. Two sources of instability are found to occur in the intermediate field regime M ～ T ^1/12, where M and T are the Hartmann and Taylor numbers. On the time scale of magnetic diffusion, solutions may blow up in finite time owing to destabilization of the convection by the magnetic field. On a faster time scale a dynamic instability related to MAC-wave instability can also occur. It is therefore concluded that the asymptotic structure of this dynamo is unstable to virtual increases in the magnetic field energy.

In an attempt to model stabilization of the dynamo in a strong-field regime we consider two approximations. In the first, a truncated expansion in three-dimensional plane waves is studied numerically. A second approach utilizes an ad hoc set of ordinary differential equations which contains many of the features of convection dynamos at all field energies. Both of these models exhibit temporal intermittency of the dynamo effect.     

Two-dimensional density currents in a confined basin

We present new experimental results on the mechanisms through which steady two-dimensional density currents lead to the formation of a stratification in a closed basin. A motivation for this work is to test the underlying assumptions in a diffusive “filling box” model that describes the oceanic thermohaline circulation (Hughes, G.O. and Griffiths, R.W., A simple convective model of the global overturning circulation, including effects of entrainment into sinking regions, Ocean Modeling, 2005, submitted.). In particular, they hypothesized that a non-uniform upwelling velocity is due to weak along-slope entrainment in density currents associated with a large horizontal entrainment ratio of E _eq ?～?0.1. We experimentally measure the relationship between the along-slope entrainment ratio, E, of a density current to the horizontal entrainment ratio, E _eq, of an equivalent vertical plume. The along-slope entrainment ratios show the same quantitative decrease with slope as observed by Ellison and Turner (, 6, 423–448.), whereas the horizontal entrainment ratio E _eq appears to asymptote to a value of E eq?=?0.08 at low slopes. Using the measured values of E _eq we show that two-dimensional density currents drive circulations that are in good agreement with the two-dimensional filling box model of Baines and Turner (Baines, W.D. and Turner, J.S., Turbulent buoyant convection from a source in a confined region, J. Fluid. Mech., 1969, 37, 51–80.). We find that the vertical velocities of density fronts collapse onto their theoretical prediction that U =-2^?2/3 B ^1/3 E _eq ^2/3 (H/R) ζ, where U is the velocity, H the depth, B the buoyancy flux, R the basin width, E _eq the horizontal entrainment ratio and?ζ?= z/H the dimensionless depth. The density profiles are well fitted with?Δ?= 2^?1/3 B ^2/3 E _eq ^?2/3 H _-1 [ln(ζ )?+?τ ], where?τ?is the dimensionless time. Finally, we provide a simple example of a diffusive filling box model, where we show how the density stratification of the deep Caribbean waters (below 1850?m depth) can be described by a balance between a steady two-dimensional entraining density current and vertical diffusion in a triangular basin.     

Properties of electric turbulence in the polar cap ionosphere

Small-scale (scales of ∼0.5–256 km) electric fields in the polar cap ionosphere are studied on the basis of measurements of the Dynamics Explorer 2 (DE-2) low-altitude satellite with a polar orbit. Nineteen DE-2 passes through the high-latitude ionosphere from the morning side to the evening side are considered when the IMF z component was southward. A rather extensive polar cap, which could be identified using the ɛ-t spectrograms of precipitating particles with auroral energies, was formed during the analyzed events. It is shown that the logarithmic diagrams (LDs), constructed using the discrete wavelet transform of electric fields in the polar cap, are power law (μ ∼ s ^α). Here, μ is the variance of the detail coefficients of the signal discrete wavelet transform, s is the wavelet scale, and index α characterizes the LD slope. The probability density functions P(δE, s) of the electric field fluctuations δE observed on different scales s are non-Gaussian and have intensified wings. When the probability density functions are renormalized, that is constructed of δE/s ^γ, where γ is the scaling exponent, they lie near a single curve, which indicates that the studied fields are statistically self-similar. In spite of the fact that the amplitude of electric fluctuations in the polar cap is much smaller than in the auroral zone, the quantitative characteristics of field scaling in the two regions are similar. Two possible causes of the observed turbulent structure of the electric field in the polar cap are considered: (1) the structure is transferred from the solar wind, which is known to have turbulent properties, and (2) the structure is generated by convection velocity shears in the region of open magnetic field lines. The detected dependence of the characteristic distribution of turbulent electric fields over the polar cap region on IMF B _y and the correlation of the rms amplitudes of δE fluctuations with IMF B _z and the solar wind transfer function (B _y ² + B _z ²)^1/2sin(θ/2), where θ is the angle between the geomagnetic field and IMF reconnecting on the dayside magnetopause when IMF B _z < 0, together with the absence of dependence on the IMF variability are arguments for the second mechanism.     

The role of vibrationally excited nitrogen in the ionosphere

Theoretical and experimental aspects of the production, transformation, diffusion and loss of N₂ in the upper atmosphere are considered. The N₂-CO₂ near-resonant system in theD andE regions is taken into account. We describe our understanding of the methods necessary to find the vibrational populations of N₂ and CO₂ (asymmetric mode of CO₂). The calculations of the vibrational temperatures in theD, E, andF regions for the mid-latitude ionosphere and an aurora are presented. The connection between the excited species and the 4.26-m radiation intensities is considered. The models for the rate coefficient of the reaction of O⁺ with N₂ and the electron density decrease resulting from N₂ in the F region are discussed.     

Penetrative convection in rotating fluids: A model for the base of stellar convection zones

Abstract

To model penetrative convection at the base of a stellar convection zone we consider two plane parallel, co-rotating Boussinesq layers coupled at their fluid interface. The system is such that the upper layer is unstable to convection while the lower is stable. Following the method of Kondo and Unno (1982, 1983) we calculate critical Rayleigh numbers R_c for a wide class of parameters. Here, R_c is typically much less than in the case of a single layer, although the scaling R_c～T^2/3 as T → ∞ still holds, where T is the usual Taylor number. With parameters relevant to the Sun the helicity profile is discontinuous at the interface, and dominated by a large peak in a thin boundary layer beneath the convecting region. In reality the distribution is continuous, but the sharp transition associated with a rapid decline in the effective viscosity in the overshoot region is approximated by a discontinuity here. This source of helicity and its relation to an alpha effect in a mean-field dynamo is especially relevant since it is a generally held view that the overshoot region is the location of magnetic field generation in the Sun.     

Relationships between field-aligned currents and convection observed by EISCAT and implications concerning the mechanism that produces region-2 currents: Statistical study

Fluid theories explain the origin of region-2 field-aligned currents as the closure of the ring current, driven itself by the azimuthal pressure gradients generated in the magnetospheric ring plasma by the sunward convection. Although the structure of pressure gradients appears experimentally complex, observations confirm that a close connection exists between the region-2 field-aligned currents and the ring current. The fluid linear theory of the adiabatic transport by convection of the ring plasma gives a first estimate of this process, and leads ultimately to phase quadrature (in terms of magnetic local time) between the region-2 field-aligned currents and the convection potential. When significant non-adiabatic processes are taken into account, such as precipitations at auroral latitudes, the theoretical phase difference rotates toward opposition. We determine experimentally the phase relationship between the region-2 field-aligned currents and the convection potential from recent statistics, depending on the magnetic activity index K_p, and performed from the EISCAT data base. For geometrical reasons of sufficient probing of region 2, it is only computed in the case of a moderate magnetic activity corresponding to 2\leqK_p<4. Region-2 field-aligned currents are found to be in phase opposition with the convection electrostatic potential at auroral latitudes. This confirms the importance of non adiabatic processes, especially ion losses, in the generation of region-2 field-aligned currents, as theoretically suggested.     

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.     

Magnetic Fluctuations for Small Magnetic Prandtl Numbers

Numerous studies of magnetic fluctuations with a zero mean-field for small magnetic Prandtl numbers (Pr _m 1) show that magnetic fluctuations cannot be generated by turbulent fluid flow with the Kolmogorov energy spectrum. In addition, the generation of magnetic fluctuations with a zero mean-field for Pr _m 1 were not observed in numerical simulations. However, in astrophysical plasmas the magnetic Prandtl numbers are small and magnetic fluctuations are observed. Thus a mechanism of generation of magnetic fluctuations for Pr _m 1 still remains poorly understood. On the other hand, in astrophysical applications (e.g., solar and stellar convection zones, galaxies, accretion disks) the turbulent velocity field cannot be considered as a divergence-free. The generation of magnetic fluctuations by turbulent flow of conducting fluid with a zero mean magnetic field for Pr _m 1 is studied by means of linear and nonlinear analysis. The turbulent fluid velocity field is assumed to be homogeneous and isotropic with a power law energy spectrum ( k ^–p ) and with a very short scale-dependent correlation time. It is found that magnetic fluctuations can be generated when the exponent p > 3/2. It is shown also that the growth rates of the higher moments of the magnetic field are larger than those of the lower moments, i.e., the spatial distribution of the magnetic field is intermittent. In addition, the effect of compressibility (i.e., u 0) of the low-Mach-number turbulent fluid flow u is studied. It is demonstrated that the threshold for the generation of magnetic fluctuations by turbulent fluid flow with u 0 is higher than that for incompressible fluid. This implies that the compressibility impairs the generation of magnetic fluctuations. Nonlinear effects result in saturation of growth of the magnetic fluctuations. Asymptotic properties of the steady state solution for the second moment of the magnetic field in the case of the Hall nonlinearity for the low-Mach-number compressible flow are studied.     

Erosion of the inner magnetosphere during geomagnetic storms

Using the empirical magnetic field model dependent on the Dst index and solar wind dynamic pressure, we calculated the behaviour of the contour B = B_s in the equatorial plane of the magnetosphere where B_s is the magnetic field in the subsolar point at the magnetopause. The inner domain of the magnetosphere outlined by this contour contains the bulk of geomag-netically trapped particles. During quiet time the boundary of the inner magnetosphere passes at the distance ∼10R_E at noon and at ∼7R_E at midnight. During very intense storms this distance can be reduced to 4–5 R_E for all MLT. The calculation results agree well with the satellite measurements of the magneto-pause location during storms. The ionospheric projection of the B = B_s contour calculated with the Euler potential technique is close to the equatorward edge of the auroral oval.     

A three-dimensional,iterative mapping procedure for the implementation of an ionosphere-magnetosphere anisotropic Ohm’s law boundary condition in global magnetohydrodynamic simulations

The mathematical formulation of an iterative procedure for the numerical implementation of an ionosphere-magnetosphere (IM) anisotropic Ohm’s law boundary condition is presented. The procedure may be used in global magnetohydrodynamic (MHD) simulations of the magnetosphere. The basic form of the boundary condition is well known, but a well-defined, simple, explicit method for implementing it in an MHD code has not been presented previously. The boundary condition relates the ionospheric electric field to the magnetic field-aligned current density driven through the ionosphere by the magnetospheric convection electric field, which is orthogonal to the magnetic field B, and maps down into the ionosphere along equipotential magnetic field lines. The source of this electric field is the flow of the solar wind orthogonal to B. The electric field and current density in the ionosphere are connected through an anisotropic conductivity tensor which involves the Hall, Pedersen, and parallel conductivities. Only the height-integrated Hall and Pedersen conductivities (conductances) appear in the final form of the boundary condition, and are assumed to be known functions of position on the spherical surface R=R₁ representing the boundary between the ionosphere and magnetosphere. The implementation presented consists of an iterative mapping of the electrostatic potential , the gradient of which gives the electric field, and the field-aligned current density between the IM boundary at R=R₁ and the inner boundary of an MHD code which is taken to be at R₂>R₁. Given the field-aligned current density on R=R₂, as computed by the MHD simulation, it is mapped down to R=R₁ where it is used to compute by solving the equation that is the IM Ohm’s law boundary condition. Then is mapped out to R=R₂, where it is used to update the electric field and the component of velocity perpendicular to B. The updated electric field and perpendicular velocity serve as new boundary conditions for the MHD simulation which is then used to compute a new field-aligned current density. This process is iterated at each time step. The required Hall and Pedersen conductances may be determined by any method of choice, and may be specified anew at each time step. In this sense the coupling between the ionosphere and magnetosphere may be taken into account in a self-consistent manner.     

Evolution of non-linear α 2-dynamos and taylor's constraint

A key non-linear mechanism in a strong-field geodynamo is that a finite amplitude magnetic field drives a flow through the Lorentz force in the momentum equation and this flow feeds back on the field-generation process in the magnetic induction equation, equilibrating the field. We make use of a simpler non-linear?α?²-dynamo to investigate this mechanism in a rapidly rotating fluid spherical shell. Neglecting inertia, we use a pseudo-spectral time-stepping procedure to solve the induction equation and the momentum equation with no-slip velocity boundary conditions for a finitely conducting inner core and an insulating mantle. We present calculations for Ekman numbers (E) in the range 2.5× 10^?3 to 5.0× 10^?5, for?α?=α ₀cos?θ?sin?π?(r?r_i ) (which vanishes on both inner and outer boundaries). Solutions are steady except at lower E and higher values of?α?₀. Then they are periodic with a reversing field and a characteristic rapid increase then equally rapid decrease in magnetic energy. We have investigated the mechanism for this and shown the influence of Taylor's constraint. We comment on the application of our findings to numerical hydrodynamic dynamos.     

Distribution of magnetic energy in αΩ-dynamos,I: The method

Abstract

In this paper a method for solving the equation for the mean magnetic energy <BB> of a solar type dynamo with an axisymmetric convection zone geometry is developed and the main features of the method are described. This method is referred to as the finite magnetic energy method since it is based on the idea that the real magnetic field B of the dynamo remains finite only if <BB> remains finite. Ensemble averaging is used, which implies that fields of all spatial scales are included, small-scale as well as large-scale fields. The method yields an energy balance for the mean energy density ε ≡ B ²/8π of the dynamo, from which the relative energy production rates by the different dynamo processes can be inferred. An estimate for the r.m.s. field strength at the surface and at the base of the convection zone can be found by comparing the magnetic energy density and the outgoing flux at the surface with the observed values. We neglect resistive effects and present arguments indicating that this is a fair assumption for the solar convection zone. The model considerations and examples presented indicate that (1) the energy loss at the solar surface is almost instantaneous; (2) the convection in the convection zone takes place in the form of giant cells; (3) the r.m.s. field strength at the base of the solar convection zone is no more than a few hundred gauss; (4) the turbulent diffusion coefficient within the bulk of the convection zone is about 10¹⁴cm²s^?1, which is an order of magnitude larger than usually adopted in solar mean field models.     

Magnetic field generation by convective flows in a plane layer: the dependence on the Prandtl numbers

Investigation of magnetic field generation by convective flows is carried out for three values of kinematic Prandtl number: P = 0.3, 1 and 6.8. We consider Rayleigh–Bénard convection in Boussinesq approximation assuming stress-free boundary conditions on horizontal boundaries and periodicity with the same period in the x and y directions. Convective attractors are modelled for increasing Rayleigh numbers for each value of the kinematic Prandtl number. Linear and non-linear dynamo action of these attractors is studied for magnetic Prandtl numbers P _m ≤ 100. Flows, which can act as magnetic dynamos, have been found for all the three considered values of P, if the Rayleigh number R is large enough. The minimal R, for which of magnetic field generation occurs, increases with P. The minimum (over R) of critical P_m for magnetic field generation in the kinematic regime is admitted for P = 0.3. Thus, our study indicates that smaller values of P are beneficial for magnetic field generation.     

The effect of stratification and compressibility on anelastic convection in a rotating plane layer

We study the effect of stratification and compressibility on the threshold of convection and the heat transfer by developed convection in the nonlinear regime in the presence of strong background rotation. We consider fluids both with constant thermal conductivity and constant thermal diffusivity. The fluid is confined between two horizontal planes with both boundaries being impermeable and stress-free. An asymptotic analysis is performed in the limits of weak compressibility of the medium and rapid rotation (τ^?1/12???|θ|???1, where τ is the Taylor number and θ is the dimensionless temperature jump across the fluid layer). We find that the properties of compressible convection differ significantly in the two cases considered. Analytically, the correction to the characteristic Rayleigh number resulting from small compressibility of the medium is positive in the case of constant thermal conductivity of the fluid and negative for constant thermal diffusivity. These results are compared with numerical solutions for arbitrary stratification. Furthermore, by generalizing the nonlinear theory of Julien and Knobloch [Fully nonlinear three-dimensional convection in a rapidly rotating layer. Phys. Fluids 1999, 11, 1469–1483] to include the effects of compressibility, we study the Nusselt number in both cases. In the weakly nonlinear regime we report an increase of efficiency of the heat transfer with the compressibility for fluids with constant thermal diffusivity, whereas if the conductivity is constant, the heat transfer by a compressible medium is more efficient than in the Boussinesq case only if the specific heat ratio γ is larger than two.     

Thermal convection in a twisted horizontal magnetic field

The onset of convection in a layer of an electrically conducting fluid heated from below is considered in the case when the layer is permeated by a horizontal magnetic field of strength B ₀ the orientation of which varies sinusoidally with height. The critical value of the Rayleigh number for the onset of convection is derived as a function of the Chandrasekhar number Q. With increasing Q the height of the convection rolls decreases, while their horizontal wavelength slowly increases. Potential applications to the penumbral filaments of sunspots are briefly discussed.