On the coalescence of twisted flux tubes

Abstract

We study the problem of the coalescence of twisted flux tubes by assuming that the azimuthal field lines reconnect at a current sheet during the coalescence process and everywhere else the magnetic field is frozen in the fluid. We derive relations connecting the topology of the coalesced flux tube with the topologies of the initial flux tubes, and then obtain a structure equation for calculating the field configuration of the coalesced flux tube from the given topology. Some solutions for the two extreme cases of low-β plasma and high-β plasma are discussed. The coalesced flux tube has less twist than the initial flux tube. Magnetic helicity is found to be exactly conserved during the coalescence, but the assumptions in the model put a constraint on the energy dissipation so that we do not get a relaxation to the minimum-energy Taylor state in the low-β case. It is pointed out that the structure equation connecting the topology and the equilibrium configuration is quite general and can be of use in many two-dimensional flux tube problems.     

Magnetic equilibria resulting from a helically symmetric kink instability

Abstract

This paper explores magnetic equilibria which could result from the kink instability in a cylindrical magnetic flux tube. We examine a variety of cylindrical magnetic equilibria which are susceptible to the kink, and simulate its evolution in a frictional fluid. We assume that the evolution takes place under conditions of helical symmetry, so the problem becomes effectively two-dimensional. The initial cylindrical equilibrium field is specified in terms of its twist function k(r) = B _θ/(rB_z ) and for a variety of k(r) functions we calculate linear growth rates for the kink instability, assuming that it develops under helical symmetry with pitch τ. We find that the growth rate is sensitive to the value of τ.

We simulate nonlinear evolution of the kink using a Lagrangian frictional code which constrains the field to have helical symmetry of a given pitch τ. Ideal MHD is assumed and the plasma pressure is taken to be small in order to mimic conditions in the solar corona. In some cases the flux tube evolves to a new smooth helically symmetric equilibrium which involves a relatively small change in the maximum electric current. In other cases there is evidence of current-sheet formation.     

The rapid dissipation of magnetic fields in highly conducting fluids

Abstract

This paper treats the dynamical conditions that obtain when long straight parallel twisted flux tubes in a highly conducting fluid are packed together in a broad array. It is shown that there is generally no hydrostatic equilibrium. In place of equilibrium there is a dynamical nonequilibrium, leading to neutral point reconnection and progressive coalescence of neighboring tubes (with the same sense of twisting), forming tubes of larger diameter and reduced twist. The magnetic energy in the twisting of each tube declines toward zero, dissipated into small-scale motions of the fluid and thence into heat.

The physical implications are numerous. For instance, it has been suggested that the subsurface magnetic field of the sun is composed of close-packed twisted flux tubes. Any such structures are short lived, at best.

The footpoints of the filamentary magnetic fields above bipolar magnetic regions on the sun are continually shuffled and rotated by the convection, so that the fields are composed of twisted rubes. The twisting and mutual wrapping is converted directly into fluid motion and heat by the dynamical nonequilibrium, so that the work done by the convection of the footpoints goes directly into heating the corona above. This theoretical result is the final step, then, in understanding the assertion by Rosner, Tucker, and Valana, and others, that the observed structure of the visible corona implies that it is heated principally by direct dissipation of the supporting magnetic field. It is the dynamical nonequilibrium that causes the dissipation, in spite of the high electrical conductivity. It would appear that any bipolar magnetic field extending upward from a dense convective layer into a tenuous atmosphere automatically produces heating, and a corona of some sort, in the sun or any other convective star.     

A vortex-tube model of eddies in the inertial range

Abstract

A vortex-tube geometry of the cascade of energy to small-scale eddies, in the inertial range of fully-developed turbulence, is proposed. The model is a special case of the beta model of Frisch, Sulem and Nelkin (1978). We require that the cascade conserve the principal invariants of inviscid, incompressible flow, namely volume, topological knottedness, circulation, and, at discrete times marking the termination of steps in the cascade, energy. The process terminates in a finite time, as in any beta model, leaving behind a self-similar network of “inactive” tubes. We associate a self-similar scaling dimension D with the structure, equal to the Hausdorff dimension of the set of “active” tubes at the termination of the cascade. Because circulation Λ plays a key role in the analysis of the cascade, we refer to these vortex-tube geometries as “gamma models”. The viewpoint throughout is entirely deterministic.

We describe two examples of gamma models. In the ring geometry, an eddy is a vortex ring, and the cascade produces “rings upon rings”, so we allow cutting and fusing of tubes while conserving total helicity. In the preferred helical model, no cutting is needed, and the cascade produces an infinite progression of braided “coils upon coils”. We suggest that latter geometry as a candidate for the topology of a singularity of the inviscid limit of a Navier-Stokes flow, when modeled by discrete vortex tubes.

A crucial ingredient of a gamma model, not explicitly present in a beta model, is the possibility of “splitting” a vortex tube into sub-tubes carrying smaller circulation. We suggest a dynamical basis for this process, as an instability of tubes whose cores violate the Rayleigh criterion.

The parameters describing a gamma model are not uniquely determined by our study, but there is a “simplest” helical gamma model, involving minimal splitting and distortion of tubes. The dimension D of the structure is 13/5, with a scale factor Λ = 2^?5/4. This value of D agrees with that suggested by Hentschel and Procaccia (1982), by analogy with established results for certain branched polymers.     

The propagation of torsion along flux tubes subject to dynamical nonequilibrium

Abstract

The dynamical nonequilibrium of close-packed flux tubes is driven by the torsion in the individual tubes so that, wherever tubes with the same sense of twisting come into contact, there is reconnection of their azimuthal field components. The reconnection consumes the local torsion, causing the propagation of torsional Alfven waves into the region from elsewhere along the tubes.

The formal problem of the propagation of the torsion along twisted flux tubes is presented and some of the basic physical properties worked out in the limit of small torsion.

It is pointed out that in tubes with finite twisting the propagation of torsional Alfven waves can be a more complicated phenomenon.

Application to the sun suggests that the propagation of torsion from below the visible surface up into the corona is an important energy supply to the corona for a period of perhaps 10–20 hours after the emergence of the flux tubes through the surface of the sun, bringing up torsion from depths of 10⁴km or more. Torsion is continually supplied by the manipulation and shuffling of the field by the convection, of course.     

Spatial variability of the surface energy balance of Lake Kasumigaura and implications for flux measurements

ABSTRACT

The spatial variability of the lake surface energy balance and its causes are not well-understood. Energy balance maps (90 m resolution) of Lake Kasumigaura (172 km²), Japan, obtained by interpolating station data and bulk equations, allowed an investigation of these issues. Due to lake-scale variations in meteorological variables and small-scale fluctuations of surface temperature, T_s, surface heat fluxes differed horizontally at two distinct scales, while radiative fluxes were more uniform. As the key variable to surface flux T_s was only homogeneous for directions with a longer fetch or under calm wind conditions. Using these findings, the suitability of two flux station locations, one at the centre of the lake and another within a cove, was considered. Although both locations satisfied the fetch requirements, T_s was not always found to be homogeneous in the cove, making this location less suitable for flux measurements, an issue that, to date, has been overlooked.     

Electron temperatures during rapid subauroral ion drift events

Examples of data from DE-2 satellite instruments are presented. These illustrate the behaviour of plasma parameters in the F-region and adjacent topside ionosphere during rapid sub-auroral ion drift (SAID) events. In particular, a variety of behaviours of the electron temperature (T_e) is demonstrated, both within and equatorward of the SAID region. The Sheffield University plasmasphere-ionosphere model (SUPIM) is used to perform calculations in which a model SAID is applied to a plasma flux tube. The model results indicate that strongly elevated ion temperature (a recognised signature of SAID events) is on occasion sufficient to raise T_e to observed values by ion-electron heat transfer. On other occasions, an additional heat source is required. It is suggested that such a source for the electron gas may be due to interaction between the ring current and the plasmasphere at high altitudes. The magnitude of the downward heat flux is consistent with that necessary to produce sub-auroral red arcs. The resulting strongly heated electron gas causes vibrational excitation of molecular nitrogen in the thermosphere.     

Dependence of the power consumed by the magnetosphere on the solar wind parameters

The results of the previous studies, where the expressions were obtained for the electric current, which is generated at the bow shock front and is closed through the magnetosphere, and for the magneto-pause potential as a function of such solar wind parameters as the plasma density and velocity and the IMF intensity, are used. The power (W) consumed by the magnetosphere is equal to the Poynting vector flux S through the magnetopause. According to the special case of the Poynting theorem applied by Heikkila to the-magnetosphere, the energy flux can be expressed in terms of the electric potential (the integration is carried out over the entire surface of the magnetosphere). As a result, the required dependence, which is quadratic with respect to the IMF B _z component, has been obtained for W. It is discussed why the magnetosphere is energy-isolated at the northward IMF B _z component despite this.     

Treatment of anchor pixels in the METRIC model for improved estimation of sensible and latent heat fluxes

Abstract

Reliable estimation of sensible heat flux (H) is important in energy balance models for quantifying evapotranspiration (ET). This study was conducted to evaluate the value of adding the Priestley-Taylor (PT) equation to the METRIC (Mapping Evapotranspiration at high Resolution with Internalized Calibration) model. METRIC was used to estimate energy fluxes for 10 Landsat images from the 2005, 2006 and 2007 crop growing seasons in south-central Nebraska, USA, where each image owing to recent rainfall exhibited high residual moisture content even at the hot pixel. The METRIC model performed satisfactorily for net radiation (R_n ) and soil heat flux (G) estimation with a root mean square error (RMSE) of 52 and 24 W m^-2, respectively. A RMSE of 122 W m^-2 for H indicated the limitation of the METRIC model in estimating H for high residual moisture content of the hot pixel (Alfalfa reference ET fraction, ET _r F > 0.15). The modified METRIC model (wet METRIC or wMETRIC) incorporating the PT equation was applied to calculate H at the anchor pixels (hot and cold) for high residual moisture content of the hot pixel. The α coefficient of the PT equation was locally calibrated using hourly meteorological data from an automatic weather station and R_n and G data from a Bowen ratio flux tower. The mean α coefficient value was 1.14. The wMETRIC model reduced the RMSE of H from 122 to 64 W m^-2 and that of latent heat flux, LE, from 163 to 106 W m^-2. The RMSE of daily ET decreased from 1.7 to 1.1 mm d^-1 with wMETRIC. The results indicate that treatment of anchor pixels for high residual moisture content with the PT approach gives improved estimation of H, LE and daily ET. It is recommended that the wMETRIC model be used for estimating ET if the hot pixel has high residual moisture (i.e. reference ET fraction > 0.15).

Citation Singh, R. K. & Irmak, A. (2011) Treatment of anchor pixels in the METRIC model for improved estimation of sensible and latent heat fluxes. Hydrol. Sci. J. 56(5), 895–906.     

On the mechanism of the post-midnight winter NmF2 enhancements: dependence on solar activity

The mechanism of the N_mF₂ peak formation at different levels of solar activity is analyzed using Millstone Hill IS radar observations. The h_mF₂ nighttime increase due to thermospheric winds and the downward plasmaspheric fluxes are the key processes responsible for the N_mF₂ peak formation. The electron temperature follows with the opposite sign the electron density variations in this process. This mechanism provides a consistency with the Millstone Hill observations on the set of main parameters. The observed decrease of the nighttime N_mF₂ peak amplitude with solar activity is due to faster increasing of the recombination efficiency compared to the plasmaspheric flux increase. The E × B plasma drifts are shown to be inefficient for the N_mF₂ nighttime peak formation at high solar activity.     

Penetrative convection in the upper ocean due to surface cooling

Abstract

A new non-linear model of mixing and convection based on a modelling of two buoyant interacting fluids is applied to penetrative convection in the upper ocean due to surface cooling. In view of simple algebra, the model is one-dimensional. Dissipation is included, but no mean shear is present. A non-similar analytical solution is found in the case of a well-mixed layer bounded below by a sharp thermocline treated as a boundary layer. This solution is valid if the Richardson number, R _i , defined as the ratio of the total mixed-layer buoyancy to a characteristic rms vertical velocity, is much greater than unity. The model predicts a deepening rate proportional to R _i ^?3/4. The thermocline remains of constant thickness, and the ratio thermocline thickness to mixed-layer depth decreases as R _i ^?3/4 as the mixed layer deepens. If the surface flux is constant, the mixed-layer depth increases with time as t ^½. The vertical structure throughout the mixed layer and thermocline is given by the analytical solution, and vertical profiles of mean temperature and vertical fluxes are plotted. Computed profiles and available laboratory data agree remarkably well. Moreover, the accuracy of the simple analytical results presented here is comparable to that of sophisticated turbulence numerical models.     

Steady flows at the top of the core from geomagnetic field models: The steady motions theorem

Abstract

It is demonstrated that the steady tangential velocity v_s at the closed surface δK of a perfect fluid conductor bounded by a rigid, impenetrable exterior can be uniquely determined from knowledge of the normal component of the time varying magnetic flux density B _n, on δK. In the context of a simple earth model consisting of an electrically insulating mantle surrounding a perfectly conducting core, the assumption of steady flow provides enough extra information to eliminate the toroidal ambiguity in B _nv and to allow derivation of a unique, global flow at the top of the core from a model of the geomagnetic field.     

Mixing efficiency in stably-stratified decaying turbulence

Abstract

The behavior of the flux Richardson number R _f, as a function of the overall Richardson number Ri ₀, was investigated for a stably stratified, grid-generated, turbulent flow evolving in a closed-loop water channel. The turbulent dissipation rate ε, the buoyancy or vertical mass flux p wbar; and the rms density fluctuation ρ′ were obtained from simultaneous single-point measurements of the horizontal and vertical velocity components and density fluctuations. From these, R _f and Ri ₀ were calculated at each point in the spatially evolving flow. The resulting curves of R _f vs. Ri ₀ exhibit the full range of behavior found in the very different case studied by Linden (1980). The length scale arguments of Gibson (1980) and Stillinger et al. (1983b) provide an underlying mechanism which successfully accounts for the shape of the R _f vs. Ri ₀ curve.     

Axisymmetric equilibria of a gravitating plasma with incompressible flows

Abstract

It is found that the ideal magnetohydrodynamic equilibrium of an axisymmetric gravitating magnetically confined plasma with incompressible flows is governed by a second-order elliptic differential equation for the poloidal magnetic flux function containing five flux functions coupled with a Poisson equation for the gravitation potential, and an algebraic relation for the pressure. This set of equations is amenable to analytic solutions. As an application, the magnetic-dipole static axisymmetric equilibria with vanishing poloidal plasma currents derived recently by Krasheninnikov et al. (1999) are extended to plasmas with finite poloidal currents, subject to gravitating forces from a massive body (a star or black hole) and inertial forces due to incompressible sheared flows. Explicit solutions are obtained in two regimes: (a) in the low-energy regime β₀ ≈ γ₀ ≈ δ₀ ≈ ε₀ ? 1, where β₀, γ₀, δ₀, and ε₀ are related to the thermal, poloidal-current, flow and gravitating energies normalized to the poloidal-magnetic-field energy, respectively, and (b) in the high-energy regime β₀ ≈ γ₀ ≈ δ₀ ≈ ε₀ ? 1. It turns out that in the high-energy regime all four forces, pressure-gradient, toroidal-magnetic-field, inertial, and gravitating contribute equally to the formation of magnetic surfaces very extended and localized about the symmetry plane such that the resulting equilibria resemble the accretion disks in astrophysics.     

The structure of separated flow past a circular cylinder in a rotating frame

Abstract

This paper examines the detailed E ^1/4-layer structure of separated flow past a circular cylinder in a low-Rossby-number rotating fluid as the Ekman number E tends to zero. This structure is based on an initial proposal by Page (1987) but with some modifications in response to further evidence, outlined both in this paper and elsewhere, on the behaviour of E ^1/4-layer flows in this context. Numerical calculations for flow in an E ^1/4 shear layer along the separated free streamline are described and the mass flux from this layer is then used to calculate the higher-order flow within the separation bubble. The flow structure is found to have two forms, depending on the value of the O(1) parameter λ, and these are compared with results from published “Navier-Stokes” type calculations for the flow at small but finite values of E.     

Study on the processing method of nighttime CO2 eddy covariance flux data in ChinaFLUX

At present, using Eddy Covariance (EC) method to estimate the “true value” of carbon sequestration in terrestrial ecosystem arrests more attention. However, one issue is how to solve the uncertainty of observations (especially the nighttime CO₂ flux data) appearing in post-processing CO₂ flux data. The ratio of effective and reliable nighttime EC CO₂ flux data to all nighttime data is relatively low (commonly, less than 50%) for all the long-term and continuous observation stations in the world. Thus, the processing method of nighttime CO₂ flux data and its effect analysis on estimating CO₂ flux annual sums are very important. In this paper, the authors analyze and discuss the reasons for underestimating nighttime CO₂ flux using EC method, and introduce the general theory and method for processing nighttime CO₂ flux data. By analyzing the relationship between nighttime CO₂ flux and air fraction velocity u*, we present an alternate method, Average Values Test (AVT), to determine the thresholds of fraction velocity (u*_c) for screening the effective nighttime CO₂ flux data. Meanwhile, taking the data observed in Yucheng and Changbai Mountains stations for an example, we analyze and discuss the effects of different methods or parameters on nighttime CO₂ flux estimations. Finally, based on the data of part ChinaFLUX stations and related literatures, empirical models of nighttime respiration at different sites in ChinaFLUX are summarized.

     

Broken ergodicity in magnetohydrodynamic turbulence

Turbulent magnetofluids appear in various geophysical and astrophysical contexts, in phenomena associated with planets, stars, galaxies and the universe itself. In many cases, large-scale magnetic fields are observed, though a better knowledge of magnetofluid turbulence is needed to more fully understand the dynamo processes that produce them. One approach is to develop the statistical mechanics of ideal (i.e. non-dissipative), incompressible, homogeneous magnetohydrodynamic (MHD) turbulence, known as “absolute equilibrium ensemble” theory, as far as possible by studying model systems with the goal of finding those aspects that survive the introduction of viscosity and resistivity. Here, we review the progress that has been made in this direction. We examine both three-dimensional (3-D) and two-dimensional (2-D) model systems based on discrete Fourier representations. The basic equations are those of incompressible MHD and may include the effects of rotation and/or a mean magnetic field B _o. Statistical predictions are that Fourier coefficients of the velocity and magnetic field are zero-mean random variables. However, this is not the case, in general, for we observe non-ergodic behavior in very long time computer simulations of ideal turbulence: low wavenumber Fourier modes that have relatively large means and small standard deviations, i.e. coherent structure. In particular, ergodicity appears strongly broken when B _o?=?0 and weakly broken when B _o?≠?0. Broken ergodicity in MHD turbulence is explained by an eigenanalysis of modal covariance matrices. This produces a set of modal eigenvalues inversely proportional to the expected energy of their associated eigenvariables. A large disparity in eigenvalues within the same mode (identified by wavevector k ) can occur at low values of wavenumber k?=?| k |, especially when B _o?=?0. This disparity breaks the ergodicity of eigenvariables with smallest eigenvalues (largest energies). This leads to coherent structure in models of ideal homogeneous MHD turbulence, which can occur at lowest values of wavenumber k for 3-D cases, and at either lowest or highest k for ideal 2-D magnetofluids. These ideal results appear relevant for unforced, decaying MHD turbulence, so that broken ergodicity effects in MHD turbulence survive dissipation. In comparison, we will also examine ideal hydrodynamic (HD) turbulence, which, in the 3-D case, will be seen to differ fundamentally from ideal MHD turbulence in that coherent structure due to broken ergodicity can only occur at maximum k in numerical simulations. However, a nonzero viscosity eliminates this ideal 3-D HD structure, so that unforced, decaying 3-D HD turbulence is expected to be ergodic. In summary, broken ergodicity in MHD turbulence leads to energetic, large-scale, quasistationary magnetic fields (coherent structures) in numerical models of bounded, turbulent magnetofluids. Thus, broken ergodicity provides a large-scale dynamo mechanism within computer models of homogeneous MHD turbulence. These results may help us to better understand the origin of global magnetic fields in astrophysical and geophysical objects.     

Choosing methods for estimating dissolved and particulate riverine fluxes from monthly sampling

Abstract

In discrete water quality surveys, riverine fluxes are associated with unknown uncertainties (biases and imprecisions). Annual flux errors have been determined from the generation of discrete surveys by Monte Carlo sorting for monthly sampling, from 10 years of daily records (120 records). Eight calculation methods were tested for suspended particulate matter, dissolved solids and dissolved and total nutrients in medium to large basins (10³ to 10⁶ km²) covering a wide range of hydrological conditions and riverine biogeochemistry. The performance of each method was analysed first by type of riverine material, which appeared to be much less pertinent than the flux variability matrix. The latter combines the river flow duration in two percent of time (W_2%) and the truncated exponent (b_50sup) defining the relationship of concentration vs discharge (C–Q) at higher flows (C = aQ^b50sup). As flux variability increases (high W_2% and/or high b_50sup), averaging and rating curve methods become less efficient compared to hydrograph separation methods. Flux biases and imprecisions were plotted in the [W_2%, b_50sup] matrix for discrete monthly surveys.

Editor Z. W. Kundzewicz

Citation Raymond, S., Moatar, F., Meybeck, M., and Bustillo, V., 2013. Choosing methods for estimating dissolved and particulate riverine fluxes from monthly sampling. Hydrological Sciences Journal, 58 (6), 1326–1339.     

Impact of fractal characteristics on evaporation and infiltration in unsaturated heterogeneous soils

ABSTRACT

In this study, an approach is developed to investigate the impact of fractal characteristics of unsaturated soil between the water table and land surface on the steady-state evaporation and infiltration across a heterogeneous landscape. The soil domain is conceptualized as a collection of stream tubes of soils and the particle diameters in various stream tubes follow a fractal distribution. The saturated hydraulic conductivity of each stream tube is related to the representative particle diameter in the tube. The effective specific discharge is then integrated from the specific discharge for each stream tube and the fractal distribution. The effective evaporation and infiltration in unsaturated soils increase with the fractal dimension. The ratio of minimum over maximum diameters does not significantly affect the specific discharge in the fractal soil. The specific discharge in unsaturated fractal soils calculated by using the simple average particle diameter mostly over-predicts the actual effective specific discharge.